^NCIENT INDIAN ASTRONOMY PLANETARY POSITIONS AND ECLIPSES
pQrvScSiyamatebhyo yadyat srestham laghu sphutam bljam tattadtbSvikalamaham rahasyamabhyudyato vaktum "I shall state i n full the best of the secret lore of astronomy extracted from the different schools of the ancient teachers so as to make It easy and clear." — PancasiddhSnUl^. 1.2.7
ANCIENT INDIAN ASTRONOMY PLANETARY POSITIONS A N D ECLIPSES
S. Balachandra Rao,
M.SC..
Ph.o.
Principal and Professor of Mathematics National College, Basavanagudi, Bangalore
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© 2000 S. Balachandra Rao (b. 1944—) I S B N 81-7646-162-8
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ACKNOWLEDGEMENTS I acknowledge my sincere gratitude to the prestigious Indian National Science Academy (INSA), New Delhi, and in particular to Dr. A . K . Bag, for sponsoring my research project on the subject. In fact, this project forms the genesis of the present book. I express my special thanks to Prof. K . D . Abhayankar (formerly of Osmania University), Dr. K . H . Krishnamurthy (Bangalore) and Dr. B.V. Subbarayappa (Bangalore) for their continuous encouragement. I record my appreciation to my research assistant Smt. Padmaja Venugopal for her contribution in completing the manuscript. For the sake of continuity of presentation of the subject I have used material from my earlier two titles : (1) Indian Astronomy—An Introduction, Universities Press, Hyderabad, 2000 and (2) Indian Mathematics and Astronomy—Some Landmarks, 2"^ Ed., 2"'' Pr., 2000, Jnana Deep Publications, 2388, Rajajinagar II Stage, Bangalore-10. I am highly indebted to all the authors and publishers of the titles listed it\ the Bibliography as also to the learned reviewers of my earlier works on the subject. Valuable suggestions are indeed welcome from discerning readers. S. Balachandra R a o
PREFACE
The present book, Ancient Indian Astronomy—Planetary Positions and Eclipses, is mainly addressed to students who are keenly interested in learning and becoming proficient in the concepts, techniques and computational procedures of Indian astronomy. These form an integral part of our Indian culture and are developed by the great savants of Indian astronomy over the past more than two millennia. A comparative study of the popular Indian traditional texts like Khanda Khadyaka of Brahmagupta (7th cent. A . D . ) , SHrya Siddhanta (revised form, c. 10th-11 th cent. A.D.) and Graha Laghavam of Ganesa Daivajfia (16th cent.) is presented. The procedures and algorithms described succinctly in these traditional texts for (i) the mean and true positions of the sun and the moon, the tdrdgrahas ("star-planets") viz.. Mars, Mercury, Jupiter, Venus and Saturn and (ii) computations of lunar and solar eclipses are elaborated with actual examples. A unique and pioneering feature of the present book is providing ready-to-use computer programs for the above-cited procedures. The 'source codes' (listings) of these computer programs are presented after Chapter 14. The advantage of these programs is that students and researchers in the field of Indian astronomy can readily use them for computations of planetary positions and eclipses according to the popular Indian astronomical texts used in our work and for a comparative study. It is humbly claimed that a sincere attempt to contribute significantly to the field of Indian astronomy is made here by (i) providing suggested improved procedures for computing lunar and solar eclipses, and (ii) suggesting bijas (corrections) for planetary positions to yield better results comparable to modem ones. The effect of the phenomenon of precession of equinoxes and the resulting ayandmsa, relevant to Indian astronomy, is presented in Appendix-1. The computation of lagna, the orient ecliptic point (ascendant), according to the traditional Indian method is included in Appendix-2. A detailed Bibliography of the original Sanskrit works and also of the secondary sources.in English is provided after the appendices. Fairly exhaustive Glossaries of technical words (both from Sanskrit to English and vice versa) and Index, for ready reference, form the last part of the book.
S. Balachandra R a o
DIACRITICAL MARKS FOR ROMAN TRANSLITERATION OF DEVANAGART
Short Vowels
a
i
u
r
Long Vowels
I
a
7
u
Visarga
e
ai
0
au h
Consonants
k
kb
t
th
fif
h
d
1 n
dh
ph
bh
b
n
1
th
t
CompoundConsonants
1
P
ch
c
m
n
dh
d
ks
tr
Anus\«ra y
r
/
V
s
s
s
h
Of II
m
jn
LIST OF FIGURES
Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig Fig. Fig. Fig. Fig. Fig. Fig. Fig.
2.1: 2.2: 2.3: 2.4: 3.1: 3.2: 3.3: 3.4: 7.1: 9.1: 9.2: 9.3: 9.4: 9.5: 9.6: 9.7: 10.1: 10.2:
Celestial sphere Equator and poles Altitude of pole star and latitude of a place Ecliptic and equinoxes Celestial longitude and latitude Right ascension and declination Azimuth and altitude Hour Angle & Declination Epicyclic Theory Nodes of the moon Earth's shadow cone and the lunar eclipse Angular diameter of the shadow cone Ecliptic limits Half durations of lunar eclipse Parallax of a body Angular diameter of the shadow cone Solar eclipse Angle MES at the beginning and end of solar eclipse
15 16 16 17 19 20 20 21 55 74 75 75 76 78 84 85 99 100
Fig. Fig. Fig. Fig.
12.1: 12.2: 12.3: A-1.1:
ifg/ira epicycle Retrograde motion of Kuja Stationary Points Precession of equinoxes
127 140 141 266
CONTENTS
Acknowledgements
vii
Preface Diacritical
1.
2.
3.
4.
v
Marks for Roman Transliteration of Devnagari
ix
List of Figures
x
INTRODUCTION—HISTORICAL SURVEY
1
1.1
Astronomy in tiie Vedas
1
1.2
Vedatiga Jyotisa
3
1.3
Siddhantas
S
1.4
Aiyabhata I (476 A . D )
5
1.5
Post-Aryabhatan astronomers
9
1.6
Contents of Siddhantas
11
1.7
Continuity in astronomical tradition
12
1.8
A i m and scope of the present work
13
ZODIAC ANDCONSTELLATIONS
15
2.1
Introduction
15
2.2
Equator and Poles {Visuvadvrtta and Dhruva)
15
2.3
Latitude of a place and altitude of Pole Star
16
2.4
Ecliptic and the Equinoxes
16
2.5
Zodiac
17
CO-ORDINATE SYSTEMS
19
3.1
Introduction
19
3.2
Celestial longitude and latitude (Ecliptic system)
19
3.3
Right ascension and declination (Equatorial system)
20
3.4
Azimuth and altitude (Horizontal system)
20
3.5
Hour angle and declination (Meridian system)
21
YUGA SYSTEM AND ERAS
22
4.1
Mahayuga, Manvantara and Kalpa
22
4.2
K a l i Era
23
xii
Ancient Indian Astronomy 4.3 4.4
5.
Introduction Working method to find Ahargana since the Kali epoch Ahargana according to Khanda Khadyaka {KK) Ahargatta according to Graha Laghavam (GL) Ahargana from the Christian date; finding the weekday Tables 5.1 to 5.3 for finding Ahargana
55 25 27 29 35 38-41 42
6.1 6.2
42 42 43 44 45 45
Introduction Mean positions of the sun and the moon Table 6.1: Revolutions of the sun, the moon, etc., in a Kalpa Table 6.2: Daily mean motions of the sun, the moon etc. Table 6.3: Mean positions of planets at the Kali epoch Mean positions of the sun and the moon 6.3.1 According to Suryasiddhdnta
(SS)
45
6.3.2 According to Kharida khadyaka (KK)
49
6.3.3 According to Graha laghavam (GL)
51
Table 6.4: Mean sidereal longitudes for 21-3-1997
54
T R U E POSITIONS O F T H E S U N A N D T H E M O O N
55
7.1 7.2
55 55 57 57 58 61 62 63 64
7.3 7.4 7.5 7.6 7.7
8.
25
MOTIONS OF T H E SUN AND T H E MOON
6.3
7.
23 24
AHARGANA 5.1 5.2 5.3 5.4 5.5
6.
yikrama Era Salivahana Saka Era
Introduction Epicyclic theory and Mandaphala Table 7.1: Peripheries of Epicycles of Apsis Mandaphala according to SS for the sun and the moon Table 7.2: Sines according to SUrya Siddhanta Bhujantara correction Further corrections for the moon True longitudes of the sun and the moon according to KK True longitudes of the sun and the moon according to GL
TRUE DAILY MOTIONS OF T H E SUN AND T H E M O O N
67
8.1 8.2 8.3 8.4 8.5
67 69 70 71 72
According to 55 True daily motions of the sun and the moon according to KK True daily motions of the sun and the moon according to GL Instant of conjunction of the sun and the moon Instant of opposition of the sun and the moon
xiii
Contents 9.
LUNAR ECLIPSE
73
9.1
73
Introduction
9.2
Indian astronomers on eclipses
73
9.3
Cause of lunar eclipse
74
9.4
Angular diameter of the shadow cone
75
9.5
Ecliptic limits for the lunar eclipse
76
9.6
Half durations of eclipse and of maximum obscuration
78
9.7
Lunar eclipse according to SS
79
9.8
Lunar eclipse according to KK
84
9.9
Lunar eclipse according to G L
91
10. S O L A R E C L I P S E
99
10.1
Cause for solar eclipse
10.2
Angular distance between the sun and the moon at the
99
beginning and end of solar eclipse
100
10.3
Computations of solar eclipse according to SS
101
10.4
Computations of solar eclipse according to G L
10.5
Computations of solar eclipse according
t
106 o
11. M E A N P O S I T I O N S O F T H E S T A R - P L A N E T S
1
1
3 119
( K U J A , B U D H A , G U R U , S U K R A A N D SANI) 11.1 11.2
Introduction
119
Table 11.1: Revolutions of planets in a Mahdyuga {SS)
119
D^iflrtMra correction for the planets
120
11.3
Mean positions of planets according to/lIT
122
11.4
Mean positions of planets according to G L
123
Table 11.2: Dhruvakas and Ksepakas
123
12. T R U E P O S I T I O N S O F T H E S T A R . P L A N E T S 12.1
125
Manda correction for the tdragrahas
125
Table 12.1: Peripheries of manda epicycles {SS)
125
Table 12.2: Revolutions of mandoccas in a Kalpa and their positions at the beginning of Kaliyuga
126
12.2
Sfghra correction for the taragrahas
127
12.3 12.4 12.5 12.6 12.7
Table 12.3: Peripheries of sighra epicycles (55) Working rule to determine the sighra correction Application of manda and ijg/ira corrections to faragra/ias True daily motion of the tdrdgrahas Retrograde motion of the tdrdgrahas Rationale for the stationary point Table 12.4: Stationary points for planets
129 130 133 137 139 141 142
xiv
Ancient Indian Astronomy 12.8 Bhujantara correction for the tdrdgrahas 12.9 . True positions of the tdrdgrahas according to KK
143 143
12.10 True positions of the tdrdgrahas according to GL
149
Table 12.5: Manddrikas of tdrdgrahas
149
Table 12.6: Sighrdrikas of tdrdgrahas
149
12.11 A comparison of true planets according to different texts
158
Table 12.7: Eight planets'combination
158
13. S U G G E S T E D I M P R O V E D P R O C E D U R E S F O R E C L I P S E S
160
13.1 13.2
Computation of lunar eclipse Computation of solar eclipse (for the world in general)
160 163
13.3
Solar eclipse for a particular place
166
14. S U G G E S T E D BIJAS ( C O R R E C T I O N S ) F O R P L A N E T S ' P O S I T I O N S 14.1
Introduction
170
14.2
Bijas for civil days and revolutions, mandoccas, epicycles etc., of planets
171
14.2.1 C i v i l days in a Mahdyuga Table 14.1: C i v i l days in a Mahdyuga 14.2.2 Revolutions of bodies in a Mahdyuga Table 14.2: Revolutions of bodies in a Mahdyuga 14.2.3 Peripheries of manda epicycles Table 14.3: Peripheries of manda epicycles Table 14.4: Earth's eccentricity and coefft. of sun's manda equation Tables 14.5 to 14.13: Eccentricities and peripheries of manda epicycles of planets 14.2.4 Mandoccas of planets Tables 14.14 and 14.15: Mandoccas of planets
'
170
172 172 173 173 173 174 175 176-180 181 181-184
14.3
14.2.5 Peripheries of Sighra epicycles of planets Table 14.16: Peripheries of Sighra epicycles Tables 14.17 to 14.24: Mean heliocentric distances and 5/^hra peripheries of eight planets from Budha to Pluto Moon's equations
14.4
The case of Budha and Sukra
191
14.5
Mean positions of bodies at the Kali epoch Table 14.25: Mean positions of bodies at the Kali epoch Table 14.26: Mandoccas of planets at the Kali epoch
193 193 194
14.6
Revolutions of bodies in a XaZ/jfl Table 14.27: Revolutions of bodies in a Kalpa Conclusion
194 194 195
14.7
184 184
185-189 189
xv
Contents COMPUTER PROGRAMS
196
APPENDICES - 1 PRECESSION OF EQUINOXES
266
- 2 L A G N A (ASCENDANT) BIBLIOGRAPHY A . Sanskrit Works B . Secondary Sources in Englisii G L O S S A R Y O F T E C H N I C A L T E R M S IN INDIAN A S T R O N O M Y I English to Sanskrit II Sanskrit to English INDEX
1
269 272 272 274 276 276 276 285
1
INTRODUCTION—HISTORICAL SURVEY Yathd sikha mayurdndm ndgdndm manayo yathd I tadvad veddriga sdstrdndm jyotisam (ganitam) murdhani sthitam II "Like tiie crests on the heads of peacocks, like the gems on the hoods of the cobras, stronomy (Mathematics) is at the top of the Veddiiga sastras—the auxiliary branches he Vedic knowledge". (Veddriga Jyotisa, R - V j , 35; Y - V J , 4) Astronomy i n the Vedas The above verse shows the supreme importance given to astronomy mathematics) among the branches of knowlege ever since the Vedic times.
(and
Even like many other branches of knowledge, the beginnings of the science of astronomy in India have to be traced back to the Vedas. In the Vedic literature, Jyotisa is one of the six auxiliaries (sadarigas) of die corpus of Vedic knowledge. The six veddrigas are : (1) Siksd (phonetics) (2) Vydkarana (grammar) (3) Chandas (metrics) (4) Nirukta (etyomology) (5) Jyotisa (astronomy) and <6) Kalpa (rituals). It is important to note that although in modern common parlance the word Jyotisa is used to mean predictive astrology, in the ancient literature Jyotisa meant all aspects of astronomy. O f course, mathematics was regarded as a part of Jyotisa. The Veddriga Jyotisa is the earliest Indian astronomical text available. Even during the time of the early mandalas of the Rgveda the astronomical knowledge necessary for the day-to-day life of the people was acquired. The Vedic people were conversant with tlie knowledge required for their religious activities, like the time (and periodicity) of the full and the new moons, the last disappearance of the moon and its
2
Ancient Indian Astronomy
first appearance etc. This type of information was necessary for the monthly rites like darsapiirnamdsa
sand seasonal rites like cdturmdsya.
The naksatra system consisting of 27 naksatras (or 28 including Abhijit) was evolved long back and was used to indicate days. It is pointed out that Agrahdyana, an old name for the Mrgasira
naksatra, meaning "beginning of the year" suggests that the sun used
to be in that asterism at the vernal equinox. This corresponds to the period of around 4000 B . C . The Rohini legends in the Rgveda point to a time in the late Rgveda period when the vernal equinox shifted to the RohinT asterism (from
Mrgasira).
The later sacrificial session called Gavdmayana was especially designed for the daily observation of the movements of the sun and of the disappearance of the moon. This must have given the priests and their advisors sufficient knowledge of a special kind, even like the "saros" of the Greeks, for predicting the eclipses. There is evidence, in the Rgveda that this specialized knowledge about the eclipses was possessed by the priests of the Atri family. During the Yajurveda period it was known that the solar year has 365 days and a fraction more. In the Taittiriya samhita it is mentioned that the extra 11 days over the twelve lunar months (totalling lo 354 days), complete the six rius by the performance of the ekddasa-rdtra i.e., eleven-nights sacrifice. Again, the same samhita says that 5 days more were required over the savana year of 360 days to complete the seasons adding specifically that "4 days are too short and 6 days too long". The Vedic astronomers evolved a system of five years' yuga. The names of the five years of a yuga are : 1. Samvatsara 2.
Parivatsara
3. Iddvatsara 4. Anuvatsara and 5. Idvatsara This period of a yuga (of 5 years) was used to reckon time as can be seen from the statements like, "Dirghatamas,
son of Mamata, became old even in his tenth
yugcC{, i.e. between the age of 45 and 50 years {Rgveda 1.158.6). The two intercalary months, Amhaspati (^Kl.25.8).
and Samsarpa
to complete the yuga (of 5 years) were known
Introduction-Historical
Survey
3
In the Yajurveda, a year comprising 12 solar months and 6 rtus (seasons) was recognized. The grouping of the six rtus and the twelve months, in the Vedic nomenclature, is as follows : Seasons
Months
1.
Vasanta rtu
Madhu and Mddhava
2.
Grisma rtu
Sukra and Suci
3.
Varsa rtu
Nabha and Nabhasya
4.
Sarad rtu
Isa and Urja
5.
Hemanta rtu
Saha and Sahasya
6.
Sisira rtu
Tapa and Tapasya
The sacrificial year commenced with vasanta rtu. The Vedic astronomers had also noted that the shortest day was at the winter solstice when the seasonal year Sisira began with Uttarayana and rose to a maximum at the summer solstice.
1.2 Vedariga Jyotisa The purpose of the Veddriga Jyotisa was mainly to fix suitable times for performing the different sacrifices. The text is found in two rescensions—Rgveda Jyotisa and Yajurveda Jyotisa. Though the contents of both the rescensions are the same, they differ in the number of verses contained in them. While the Rgvedic version contains only 36 verses, the Yajurvedic version contains 44 verses. This defference in the number of verses is perhaps due to the addition of explanatory verses by the adhvaryu priests with whom it was in use. In one of the verses, it is said, "I shall write on the lore of time, as enunciated by sage Lagadha." Therefore, the Veddriga Jyotisa is attributed to Lagadha. According to the text, at the time of its composition, the winter solstice was at the beginning of the constellation Sravisthd (Delphini) and the summer solstice was in the middle of the Aslesd constellation. Since Varahamihira (505 A . D . ) stated that in his own time the summer solstice, w^s at the end of three quarters of Punarvasu and the winter solstice at the end of the first quater of Uttrardsddhd, there had been a precession of the equinoxes (and solstices) by one and three-quarters of a naksatra, i.e. about 23°20'. Since the rate of precession is about a degree in 72 years, the time interval for a precession of 23°20' is about 72 x 23''20' i.e., 1,680 years prior to Varahamihira's time. This takes us back to around 1150 B . C . Generally, the accepted period of Veddriga Jyotisa is between 12th and 14th centuries B . C .
4
Ancient Indian
Astronomy
The Veddiiga Jyotisa belongs to the last part of the Vedic age. The text proper can be considered as the record of the essentials of astronomical knowledge needed for the day-to-day life of the people of those times. The Veddiiga Jyotisa is the culmination of the knowledge of astronomy developed and accumulated over thousands of years of the Vedic period upto 1400 B . C . In the Veddiiga Jyotisa, a yuga of 5 solar years consists of 67 lunar sidereal cycles, 1830 days, 1835 sidereal days, 62 synodic months, 1860 tithis, 135 solar naksatras, 1809 lunar naksatras and 1768 risings of the moon. It also mentions that there are 10 ayanas and visuvas and 30 rtus in a yuga. The practical way of measuring time is mentoned as the time taken by a specified quantity of water to flow through the orifice of a specified clepsydra (water-clock) as one nddikd i.e. 1/60 part of a day. One can find in the Veddriga Jyotisa very useful presentation of the various calendrical items prevalent during those times like (i) (ii)
the solstices increase and dercrease of the durations of days and nights in the ayanas
(iii)
the solstitial tithis
(iv)
the seasons
(v) (vi) (vii)
omission of tithis table of parvas yogas
(which developed later as one of the five limbs of a fullfledged
paricdriga) (viii) (ix) (x)
finding the parva naksatras and the parva tithis the visuvas (equinoxes) the solar and other types of years
(xi)
the revolutions of the sun and the moon (as seen from the earth)
(xii)
the times of the sun's and the moon's transit through a naksatra
(xiii)
the adhikamasa (intercalary month)
(xiv)
the measures of the longest day and the shortest night, etc.
The Veddriga Jyotisa mentions that the durations of the Tdngest and the shortest days on the two solstitial days are of ratio 3 : 2 i.e., 36 and 24 ghatikas (or nddikds) which correspond to 14 hours, 24 minutes and 9 hours, 36 minutes respectively. This means the dindrdhas i.e. the lengths of half-days come to be 7'' 12"* and 4''48'" respectively. It is calculated that around 1400 B . C . , the sun's maximum declination used to be about 23°53'. However, our ancient Indian astronomers took it as 24°. Now, the latitude 0 of a place can be found using the formula :
Introduction-Historical
Survey
5
sin (ascensional difference) = tan <)| tan 5 where 8 is the declination of the sun. The correction due to ascensional difference in this case is l''12"' i.e., in angular measure,
1* 12" x 1 5 ° = 18°. Now, using the above
formula, we get the latitude of the place, (]> = 35° approximately. Therefore, the place of composition of the Veddriga Jyotisa appears to be in some region around the northern latitude of 35°. 1.3 Siddhantas The astronomical computations described in the Veddriga Jyotisa were in pracdcal use for a very long time. Around the beginning of the Christian era, say a century on either side of it, a new class of Indian astronomical literature emerged. The texts representing this development are called siddhdntas. The word "siddhanta" has the connotation of an established theory. These siddhdnta texts contain much more material and topics than the Veddriga Jyotisa. Along with the naksatra system, the twelve signs of the zodiac viz.. Mesa. Vrsabha etc., were introduced. A precise value for the length of the solar year was adopted. Computations of the motions of the planets, the solar and lunar eclipses, ideas of parallax, determination of mean and true positions of planets and a few more topics formed the common contents of the siddhdntic texts. A very significant aspect of that period, in the history of Indian astronomy, was the remarkable development of newer mathematical methods which greatly promoted mathematical astronomy. Needless to say, the unique advantage of the famous H i n d u invention of decimal numerals—adopted wprld over now-^made computations with the huge numbers very handy and even enjoyable to the ancient Indian astronomers. According to the Indian tradition, there were principally 18 siddhdntas : SUrya, Paitdmaha, Vydsa, Vdsistha, Atri, Pardsara, Kdsyapa, Ndrada, Gdrgya, Marici, Arigira.
Lomasa
(Romakal),
Manu,
Paulisa, Cyavana, Yavana, Bhrgu and Saunaka. However,
among these only five siddhdntas were extant during the time of Varahamihira (505 A.D.) viz., Saura (or Surya), Paitdmaha (or Brahma), Vdsistha, Romaka and Paulisa. These five siddhdntas were ably collected together by Varahamihira and preserved for the posterity as his
Paricasiddhdntikd.
1.4 A r y a b h a t a I (476 A . D . ) Aryabhata I, different from his namesake of the tenth century, was bom in 476 A . D . and composed his very famous work, Aryabhatiyam, when he was 23 years old. He mentions in his monumental text that he sets forth the knowledge honoured at Kusumapura, identified with modem Patna in Bihar.
6
Ancient Indian
Astronomy
The Aryabhatiyam consists of four parts (pddas) : Gltikd, Ganita, Kdlakriyd and Gola. The first part contains 13 verses and the remaining three parts, forming the main body of the text, contain totally 108 verses. In the Gitikdpdda, we are introduced to : (i)
the large units of time viz, Kalpa, Manvantara and Yuga (different from that of the Veddrigajyotisa);
(ii) (iii)
circular units of arc—degrees and minutes ; and linear units viz., yojana, hasta, and arigula.
The numbers of revolutions of planets in a (mahd-) yuga of 43,20,000 years are given in the Gitikdpdda. Further, the positions of the planets, their apogees (or aphelia) and nodes are also given. Besides these, the diameters of the planets, the inclinations of the orbital planes of the planets with the eclipdc and the peripheries of the epicycles of the different planets are also included. The topic of great mathematical importance, in this part, is the construction of the tables of Jyd, the trigonometric function "sine". It is significant that so much of information is packed, as i f in a concentrated capsule form, in just ten verses. The second part of the Aryabhatiyam,
the Ganita pdda contains 33 stanzas
essentially dealing with mathematics. This part deals with the following important mathematical topics: geometrical figures, their properties and mensuration vyavahdra);
(Ksetra
arithmetic and geometric progressions; problems on the shadow of the
gnomon (sanku-chdyd); equations (kuttaka).
simple, quadratic, simultaneous and linear indeterminate
In fact, the most signiUcant contribution of Aryabhata, in the
history of world maUiematics, is his method of solving a first order indeterminate equation: to find solutions of ax + by = c, in integers (where a and b are given integers). The kdlakriyd pdda, the third part of the Aryabhatiyam
contains 25 verses explaining
the various units of time and the method of determination of positions of planets for a given day. Calculations concerning the adhikamasa (intercalary month), ksyatithis, angular speeds of planetary motions (in terms of revolutions), the concept of weekdays are all included in this part of the text. The Golapdda forms the fourth and the last part of the Aryabhatiyam.
It contains
50 stanzas. Important geometrical (and trigonometric) aspects of the celestial sphere are discussed in the Golapdda.
The important features of the ecliptic, the celestial
equator, the node, the shape of the earth, the cause of day and night, rising of the zodiacal signs on the eastern horizon etc., find a place in this last part of the text.
Introduction-Historical
Survey
7
In fact, much of the contents of the Golapdda discussed under a chapter called triprasna
of the Aryabhatiyam
are generally
(three problems of time, place and
direction) in the later siddhantic texts. Another very important topic included in the chapter is on the lunar and solar eclipses. The system of astronomy expounded in the Aryabhatiyam
is generally referred to
as the auddyika system since the Kali beginning is reckoned from the mean sunrise (udaya) at LMtikdy a place on the earth's equator. However, we learn from Varahamihira and Brahmagupta that Aryabhata I propounded another system of astronomy called drdha-rdtrika
in which the day is reckoned from the mean midnight (ardha-rdtri)
at
Lankd. The important parameters are different in the two systems. However, Aryabhata's text of the drdha-rdtrika system is not available now. Its parameters can be recovered from Brahmagupta's Khandakhadyaka
and some later works.
The following are some of the innovative contributions of Aryabhata I : 1.
A unique method of representing huge numbers using the alphabets for the
purposes of metrics and easy memorization. The method followed
by Aryabhata is different from the now popular methods of
Katapayddi
and Bhutasahkhyd
However,
which
also
serve
the
same purpose.
Aryabhata's method was not followed by later astronomers perhaps due to the inconvenience of pronunciation and lack of meanings of the words formed. 2.
The value of n is given as 3.1416, which is correct to the first four decimal places, for the first time in India. Aryabhata gives the value of Jt as the ratio of 62,832 to 20,000. But he cautiously points out that the value is "dsanna" i.e. approximate. The great Kerala astronomer, Nllakaritha Somayaji (1500 A . D . ) provides the explanation that n is incommensurable (or irrational). This achievement
of
Aryabhata I, as early as in the fifth century, is truly
remarkable in view of the fact that it was only thirteen centuries later, in 1761, that Lambert proved that n is irrational (i.e. cannot be expressed as ratio of two integers). Again, it was yet more than a century later, in 1882, that Lindemann established the fact that n is transcendental i.e. it cannot be the root of an algebraic equation of any degree. 3.
Sine tables : The importance of the trigonometric functions like sine (jyd) and cosine (kotijyd) in Indian astronomy can hardly be exaggerated.
8
Ancient Indian
Astronomy
Aryabhata I gives the rule for the formation of the sine-table just in one stanza! Accordingly, the sine values for the angles from 0° to 90° at intervals of 3°45' can be obtained. The values thus obtained compare well with the modem values. It is important to note that for an angle 0, the "Indian-sine" (jyd) of the angle 9 is related to the modem sine values by the relation Jyd (9) = ^ sine where /? is a predefined constant value of die radius of a circle. For example, Aryabhata, as also Surya siddhdnta, take the value R = 3438' so that Jyd (9) = 3438'sin 9 Brahmagupta takes R = 150'. Aryabhata also gives the following relations for the trigonometric ratios of "allied" angles like 9 0 ° + 9, 1 8 0 ° + 9 and 2 7 0 ° + 0 : J(i) sin (90°-h 9)
= sin 90° - versine 9 = cos 9
(ii) sin (180° + 9) = sin 90° - versine 90° - sin 9 = - sin 9 (iii) sin (270° + 9) = (sin 9 0 ° - versine 90°) - (sin 90° - versine 0) = - cos 9 where versine 9 = 1 - cos 0 4.
Earth's shape and rotation : Now it is well known that the earth is spherical (or spheroidal) in shape and that it rotates about its own axis once a day causing day and night. Aryabhata clearly maintains that : (i)
The earth is spherical—"circular in all directions" (see Golapdda, 6).
(ii)
Halves of the globes of the earth and the planets are dark due to their own shadows; the other halves facing the sun are bright. It is truly creditable that Aryabhata recognised that the earth and the other planets are not self-luminous but receive and reflect light from the sun.
(iii)
Again, Aryabhata was the first to state that the rising and setting of the sun, the moon and other luminaries are due to the relative motion caused by the rotation of the earth about its own axis once a day. He says, "Just as a man in a boat moving forward sees the stationary objects (on either side of the river) as moving backward, just so are the stationary stars seen by the people at Latika (i.e. on the equator) as moving exactly towards the west" (Golapdda, 9).
Introduction-Historical 5.
Survey
9
The period of one sidereal rotation (i.e., with reference to the fixed stars in the sky) of the earth, as given by Aryabhata works out to be 23'' 56'" 4.1''. The
corresponding modern value is 23''56"* 4.091'^. The accuracy of
Aryabhata is truly remarkable. Aryabhata I (476 A . D . ) is regarded as the major expounder of systematic and scientific astronomy in India. The unparalleled popularity of Aryabhata I and his system of astronomy is demonstrated by the fact that the remarkable development of astronomy in Kerala in 14th to 19th centuries is based exclusively on the Aryabhatan system. 1.5 Post-Aryabhatan astronomers The cryptic and aphoristic style of Aryabhata would have made it extremely difficult to understand his text but for the detailed exposition of the system by Bhaskara I (c.600 A.D.). In his commentary on the Aryabhatiyam, as also in the works Mahd—and Laghu • Bhdskariyams, Bhaskara I (to be distinguished from his more popular namesake of the 12th century) has very ably expounded Aryabhata's astronomy with examples and copious references. As mentioned earlier, Varahamihira (505 A . D . ) brought together five systems of astronomy, extant during his period, in his remarkable work, Pahcasiddhdntikd. He mentions that among the five systems, the Suryasiddhdnta is the best. Even to this day the most popular astronomical text is Suryasiddhdnta, though in its revised form. It is believed that the SHryasiddhdnta in its current verson was composed around 1000 A . D . The parameters in the two texts are totally different. W h i l e the siddhdntas proper are large texts consisting of broad theories and a large number of topics generally these texts are not handy for practical computations for day-to-day use. Further, very large numbers w i l l have to be dealt with which are very inconvenient and lead to errors. Therefore, besides these siddhdntas, two other types of texts on astronomy have been in vogue. These are called tantras and kararias. Conventionally, siddhdntas choose the beginning of the Mahdyuga (43,20,000) years of Kalpa Siddhdntas
(432 x 10^ years) as the epoch. After the Suryasiddhdnta, are
Siddhdntasiromarii
Brahmasphutasiddhdnta
of
Brahmagupta
(628
two popular A.D.)
and
of Bhaskara II (1114 A.D.). A large number of commentaries and
even super-commentaries are written particularly on the
Suryasiddhdnta.
The tantra texts have comparatively fewer topics and explanations. These works choose the more canvenient epoch viz., the beginning of the Kaliyuga (the midnight of
10
Ancient Indian
Astronomy
17/18 February 3102 B . C . or the sunrise of February 18). For example, the Aryabhatiyam and Nilakantha Somayaji's Tantrasangraha (c 1500 A . D . ) are tantra texts. However, for practical computations and making pahcdhgas the most useful handbooks are the karana texts. In these, practical algorithms are provided taking a convenient contemporary date as the epoch. The advantage of a recent epoch is that one now deals with smaller numbers for the ahargaria (the number of civil days elapsed since the epoch). Further, since corrected positions of planets for a recent date have been given with necessary bijasarnskdras (corrections), die computations based on these kararia handbooks are more accurate. The wellknown kararia texts are Brahmagupta's Kharidakhddyaka (7th cent.), Bhaskara II's Kararmkutuhalam (12th cent.) and Ganesa Daivajiia's Grahaldghavam (16th cent.). A large number of such handbooks and tables (sarariis) were composed during different periods, even as late as in the nineteenth century. Some of the famous Indian astronomers and their major astronomical works are listed below. The dates in brackets refer to the approximate dates of composition of the works : Author
Works
1.
Aryabhata I (499 A.D.)
Aryabhatiyam, Aryasiddhdnta
2.
Varahamihira (b. 505 A.D.)
Pahcasiddhdntikd, Brhatsamhitd
3.
Bhaskara I (c. 600 A.D.)
Bhasya on Aryabhatiyam. Mahdbhdskariyam, Laghubhdskariyam
4.
Brahmagupta (b. 591 A.D.)
Brahmasphutasiddhdnta. Kharidakhddyaka
5.
Vatesvara (880 A.D.)
Vatesvarasiddhdnta
6.
Mafijula (932 A.D.)
Laghumdnasam
I.
Aryabhata II (950 A.D.)
Mahdsiddhdnta
8.
Bhaskara II (b. 1114 A.D.)
Siddhdntasiromarii. Karariakutuhalam
9.
Paramesvara (c. 1400 A.D.)
Drggariitam, Suryasiddhdnta vivarariam. Bhatadipikd, etc.
10. Nilakantha Somayaji (1465 A.D.)
Tantrasarigraha. A ryabhatlyabhdsya
II. Ganesa Daivajfia (1520 A.D.) 12. Jyesthadeva (1540 A.D.)
Grahaldghavam Yuktibhdsd
13. Candrasekhara Samanta (b. 1835 A.D.) Siddhdntadarpariah 14. Sarikara Varman (19th cent.) 15. Verikatesa Ketkar (1898 A.D.)
Sadratnaindld Grahagariitam. Jyotirgariitam
Introduction-Historical
Survey
11
1.6 Contents of Siddhantas Various topics of interest in Indian astronomy are discussed in different chapters. A chapter is called adhyaya or adhikara. The following is generally the distribution of the topics into the different adhikdras in a typical siddhdntic text. 1. Madhyamddhikdra The
word madhyama
means the average or 'mean' positions of planets. Here, by
'planets' are meant the sun, the moon and the so-called tdrd—grahas
viz.. Mercury,
Venus, Mars, Jupiter and Saturn. In order to calculate the mean angular velocities, the numbers of revolutions completed in a mahdyuga
(of 43,20,000 years) or a kalpa
(432 X 10^ years) by the planets as also by the special points viz., the apogee (called mandocca)
of the moon and the moon's ascending node (popularly called Rdhu) are
given. The procedure to calculate the ahargana (the number of civil days from the epoch) of the given date is also explained in this chapter. The total number of civil days in a mahdyuga is also specified. Then, the motion of a planet from the epoch to the given date is given by Motion = (No. of revns. x Ahargaria x 3 6 0 ° ) / ( N o . of civil days in a Mahdyuga) in degrees. When the nearest integral multiple of 360° (i.e, completed number o f revolutions) is dropped from the above value, we get the mean position of the planet in degrees, etc. for the given date. 2. Spastddhikdra In this chapter the procedure to obtain the "true" position of a planet, from the mean position, is discussed. The word spasta means correct or true. For obtaining the true positions from the mean, two corrections are prescribed; (i) (ii)
manda, applicable to the sun, the moon and the five planets and sighra, applicable only to the five planets {tdrdgrahas) viz., Budha, Sukra, Kuja, Guru and Sani.
The manda correction takes into account the fact that the planets' orbits are not circular. This correction corresponds to what is called "the equation of the centre" in modem astronomy. The sighra correction corresponds to conversion of the heliocentric positions of planets to the geocentric. 3. Triprasnddhikdra This chapter deals with the "three questions" of direction {dik), place {desa) and time {kdla). Procedures for finding the latitude of a place, the times of sunrise and sunset.
12
Ancient Indian
Astronomy
variations of the points of sunrise and sunset along the eastern and westem horizon, gnomon problems and calculation of lagrux (ascendant) are discussed. 4. Candra - and Surya • Grahanadhikdra In these two chapters the computations of the lunar and the solar eclipses are discussed. The instants of the beginnings, the middle and the endings, effects of parallax, regions of visibility, possibility of the occurrence, totality etc. of the eclipses are considered. Their computational procedures are elaborated. In fact, for Indian astronomers the true testing ground for the veracity of their theory and procedures very much depended on the successful and accurate predictions of eclipses. O f course, as and when minor deviations between computations and observations were noticed, necessary changes and corrections (bija sarnskdrd) were suggested. Besides these four important topics, the siddhdntic texts contain many other topics, which vary from text to text, like the first visibility of planets, moon's cusps, mathematical topics like kuttaka (indeterminate equations), spherical trigonometry and the rationales of the formulae used, etc. 1.7 Continuity in astronomical tradition A characteristic feature of Indian astronomy is the unbroken continuity in the tradition starting from Vedic period upto the recent times. Starting from simple observations and a simple calendar, relevant to the contemporary needs during the Vedic times, there has been a gradual progress in the extent of astronomical topics considered, mathematical techniques developed, and refinement and sophistication in the computational algorithms, always aimed at greater accuracy during the siddhdntic period of evolution spread over nearly fifteen centuries. The existing popular siddhdnta texts, like the Siirya siddhdnta, with
elucidations and illustrations by a large
commentaries etc. For example, the Aryabhatiyam
number
are made clearer
of commentaries,
super-
carries highly learned and exhaustive
commentaries by Bhaskara I, Paramesvara and Nilakantha Somayaji among others. Prthudakasvamin's commentary on the Kharidakhddyaka of Brahmagupta, besides those by Bhattotpala and Amaraja, is extremely useful. Bhaskara II has written his own commetary, Vdsand bhdsya on his magnum opus, Siddhdntasiromarii.
In fact, often
the commentaries improve upon the parameters and computational techniques of the original texts to yield better results. While Manjula (or Munjala, c. 932 A . D . ) and SrTpati (c. 1000 A . D . ) introduced additional corrections for the moon, Nilakaritha Somayaji (c.1500 A . D . ) revised the model of planetary motion in his Tantrasarigraha for obtaining better positions of the inferior planets, Budha and Sukra. Inspired by the ideas of Paramesvara (c.1400 A . D . ) , Nilakantha (c.1500 A.D.) developed a heliocentric model in which all planets move round the Sun in eccentric
Introduction-Historical
Survey
13
orbits. It is a significant achievement before Copernicus came into picture. Nilakaritha's revised model was successfully adopted by all later astronomers
of Kerala, like
Jyesthadeva, Acyuta Pisarati and Citrabhanu. It is also noteworthy that the knowledge of astronomy was never restricted to any particular region, but spread throughout India. While Candrasekhara Samanta of Orissa made quite a few important innovations, like an additional correction to the month, independently, Kerala became the pocket of tremendous development during 14th to 19th centuries. O f course, congenial social milieu and patronage must have played an important role in the development of astronomy more during certain periods and in certain regions and less at other times and regions. 1.8 Aim and scope of the present work In the present book we have studied in detail the procedures and algorithms presented in the traditional Indian astronomical texts, for the computations of the mean and true positions of the sun, the moon and the five tdrdgrahas (Budha, Sukra, Kuja, Guru and Sani). The methods of computations of lunar and solar eclipses are presented. The algorithms are documented with ready-to-work computer programs for the benefit of
students
and
Kharidakhddyaka
researchers.
Three
representative
siddhdntic
texts
of Brahmagupta (7th cent.), the revised form of the
viz.,
the
Suryasiddhdnta
(assigned to 10th or 11th cent.) and Ganesa Daivajiia's Grahaldghavam (early 16th cent.) are chosen in the present work for a comparative study. For some chosen dates of eclipses, both ancient and current, computations according to the above texts are presented in Chapters 9 and 10. The procedures for determining the mean and true positions—with a detailed understanding of the manda equation—of the sun and the moon are discussed in Chapters 6 and 7. Similarly the mean positions and true positions of the tdrdgrahas planets")—with an analysis of the sighra
("star
equation—are discussed with examples in
Chapters 11 and 12. However, initially, an introductory description of the zodiac and the different co-ordinate system is provided in Chapters 2 and 3. The basic ideas of the system—peculiar to traditional Indian astronomy - and determination of the ahargaria (accumulated civil days since chosen epochs) are discussed with relevant examples in Chapters 4 and 5. In this report, in addition to the above, we have made an attempt to (i)
provide improved procedures and parameters for computations of lunar and solar eclipses (Chapter 13); and
(ii)
suggest bijasarnskdras (i.e., corrections), in Chapter 14, to the various relevant parameters for computations of planetary positions based on the known modem values and formulae. The parameters considered in this context are:
14
Ancient Indian
Astronomy
(a)
The proposed number of civil days in a mahayuga (or a kalpa);
(b)
Numbers of revolutions of the bodies in a mahdyuga (or a kalpa) which when divided by the number of civil days yield the mean daily motions (in fractions of a revolution); The peripheries of the manda epicycles of all the planets indicating how these vary with the eccentricities of their heliocentric orbits; Mandoccas (apogees) of the planets at the beginning of each century over a period of 5,200 years from 2000 A . D . backwards upto - 3200;
(c) (d) (e)
Revolutions of the mandoccas of the planets in a kalpa (432 X 10' years);
(0
The peripheries of the sighra epicycles which are dependent on the mean heliocentric distances of the planets;
(g)
The mean epochal positions of all the planets at, the Kali beginning (i.e. the mean midnight between 17th and 18th February, 3102 B.C.) giving both the tropical (sdyana) and the sidereal (nirayana) values;
(h)
The mean epochal positions of the mandoccas (apogees) of planets at the Kali beginning.
In continuation of the present work, we propose to suggest, in due course, improved parameters and algorithms for the various traditional texts, in line with their simplified forms of expressions and for their chosen individual epochs. Several other karana granthas will be taken up for detailed study. A l l the algorithms, even with our proposed bijas, will be documented with computer programs. This will greatly facilitate future research workers in the field of Indian astronomy to compute in seconds the true positions of planets and eclipses for any date in any century according to various Indian texts and to compare the values without any effort on their part.
ZODIAC AND CONSTELLATIONS 2.1
Introduction
Stars and planets, on a clear night, appear as luminous points as though placed on a hemispherical dome. This imaginary sphere of arbitrarily large radius is called celestial sphere (khagola). This sphere has no real physical existence and in fact the stars and planets are at different large distances from the observer. Since the relative angular distances of the celestial bodies are of interest in spherical astronomy, their actual linear distances are not considered. In Fig. 2.1, A and B are two celestial bodies and O is the position of the observer which is taken as the centre of the celestial sphere. The lines OA and OB joining A and B to the observer's position 0 cut the celestial sphere at the points a and b respectively. Angle aO b is the same as angle AdB, the angular distance between the celestial bodies A and B as seen from O. Thus we observe that although the two objects A and B are at different distances from the observer, the angular distances between them remains the same as though the two bodies lie on the celestial sphere with O as centre.
e
\ 0
y
^
N
Fig. 2.1: Celestial sphere
The radius of the celestial sphere is taken arbitrarily so large that the entire earth can be considered as just a point at the centre of this very huge imaginary sphere. This means that wherever may the observer be on the surface of the earth, he can always be considered as at the centre of the celestial sphere. However, it is important to note that all observers at different places on the earth do not see the same part of the celestial sphere at a given time. 2.2 Equator and Poles (Visuvadvrtta and Dhruva) The earth is rotating about its own axis pp'. The axis pp' is produced both ways to meet the celestial sphere at P and F ' which are called the celestial north and south poles (uttara and daksina dhruva). The great circle qr on the earth whose plane is perpendicular to the axis pp is the earth's equator and the points pmdp' are the terrestrial north and south poles.
16
Ancient Indian
Astronomy
Correspondingly, the great circle QR on the celestial sphere is called the celestial equator (visuvad vrtta). The points P and P' are the celestial poles. It is clear that the celestial equator QR is the intersection of the celestial sphere with the plane of the earth's equator. 2.3 Latitude of a place and altitude of Pole Star The terrestrial (or geographical) latitude <|i of a place on the surface of the earth is the angular distance of the place from the earth's equator. In other words, the latitude of a place is the angle made by the line joining the earth's centre to the place with the plane of the earths equator. In F i g 2.2, AdR is the latitude ^ of the place A.
Rg. 2.2: Equator and poles
Altitude of a celestial body is its angular distance from the horizon. To put it in the ordinary language, the altitude of a luminary is the angle through which the observer has to raise his eyes, above the horizon, to see the body. In F i g 2.3, E is the centre of the earth and O is the position of the observer on the earth. The latitude ()) of the observer O is the angle OEq made by the line Fig. 2.3: Altitude of pole star and EO with the plane of the terrestrial equator Eq. In the latitude of a place figure, r denotes the radius of the earth and R the arbitrarily large radius of the celestial sphere (R» r). The altitude of the pole star P is ndP (or arc nP) made by the line OP with the horizon sn of the place. Now, we have Latitude ^ = OEq = arc Oq = QdZ=aic = arc PQ-arc
QZ PZ
= 90° - (arc Zn - arc Pn) = 90°-(90°-arcPn) = arc Pn Altitude of P Thus, the latitude of a place is equal to the altitude of the pole star at that place. For example, an observer in Bangalore can locate the pole star at about 13° above the horizon. 2.4 Ecliptic and the Equinoxes The sun appears to move round the earth, as seen from the earth, from west to east with respect to fixed stars, continuously and comes back to the same position after a year. This motion of the sun, for an observer on the earth, is apparent and is a relative motion caused by the revolution of the earth round the sun in a year.
Zodiac and
Constellattons
17
The apparent annual path of the sun round the earth, with respect to fixed stars is a great circle 5, 52 (Fig 2.4) called the ecliptic. The points of intersection of the ecliptic 5] 52 with the celestial equator QR are called equinoxes denoted by y and n. The equinoxial point y where the sun during his annual motion along the ecliptic crosses the celestial equator from the south to the north is called the Vernal Equinox or the first point of Aries and the other equinoxial point is called Autumnal Equinox or the first point of Libra.
Fig. 2.4: Ecliptic and equinoxes
The angle between the planes of the ecliptic and the celestial equator is called obliquity of the ecliptic, denoted by e. The value of e is about 23°30'. The ecliptic is called apanumdala or krdnti vrtta. 2.5 Zodiac Consider two small circles parallel to the ecliptic lying at an angular distance of 8° on either side of die ecliptic. The positions of stars and planets are considered with reference to this circular belt, called Zodiac (bhacakra). The zodiac is divided into 12 equal parts, each part of 30° extent, called signs (rdsi). The twelve signs of the zodiac are counted starting from the vernal equinox which is called the first point of Aries. Each sign (or rdsi) is characterized by a group of stars called constellation. These are named after the objects or animals or human forms which these are supposed to resemble. The twelve groups of stars, characterizing the twelve signs, are called zodiacal constellations. The sun moves from one sign to the next in the course of a solar month. He is at the first point of Aries i.e., at the vernal equinox, around March 22, and at the first point of Libra i.e., at the autumnal equinox around September 23 each year. Table 2.1 gives the names of the twelve constellations, the Indian equivalenj names of the rdsis, the imaginary shapes of the clusters of stars and the angular extent of each sign (in degrees). Table 2.1 : Signs of the Zodiac Signs
Rdsis
Shape of die Constellation
Angular Extent
1.
Aries
Mesa
Ram
0° -
30°
2.
Taums
Vrsabha
Bull
30° -
60°
3.
Gemini
Mithuna
Twins
60° - 90°
4.
Cancer
Karkataka
Crab
90° - 120° (Contd...)
Ancient Indian Astronomy Signs
Rasis
Shape of the Constellation
Angular Extent
5.
Uo
Simha
Lion
120° - 150°
6.
Virgo
Kanyd
Virgin
150° - 180°
7.
Libra
Tula
Balance
180° - 210°
8.
Scorpio
Vrscika
Scorpion
210° - 240°
9.
Sagittarius
Dhanus
Archer
240° - 270°
10.
Capricorn
Makara
Sea goat
270° - 300°
11.
Aquarius
Kumbha
Water carrier
300° - 330°
12.
Pisces
Mina
Fish
330° - 360°
Note : The signs and Rdsis shown in the first two columns of the table are equivalent when the first point of Aries (i.e, the vernal equinox) coincides with the first point of Mesa of the Indian system. However, currently there is a difference of about 23°49' (in 1997) between the two, due to a phenomenon called the "precession of the equinoxes", which will be discussed in Appendix - 1.
3
CO-ORDINATE SYSTEMS 3.1 Introduction In order to determine and specify the location of a celestial body in the sky different systems of co-ordinates are evolved. Each of these systems is useful in a particular context to locate a celestial body. The following are the different systems of co-ordinates : 1. Celestial longitude and latitude (or ecliptic system) 2. Right ascension and declination (or equatorial system) 3. Azimuth and altitude (or horizontal system) 4. Hour angle and dedination (or meridian system) We shall discuss these systems of co-ordinates in the following sections. 3.2 Celestial longitude and latitude (Ecliptic System) Let 5 be the position of a planet on the celestial sphere. The ecliptic is represented by CL and its poles by ATandAf. The arc KS is produced to meet the ecliptic CL at M (Fig. 3.1). The angular distance SM (equal to angle SO\f) of the planet from the ecliptic is called latitude of the planet 5 and is denoted by p. The latitude is northern if S lies on the same side of the ecliptic as the north pole P and negative otherwise. O f course, i f S is on the ecliptic, the latitude p = 0.
^
K' 3^. j , ^ , ^ ^ ^ . ^ , ,^
The angular distance yM, measured eastward along the ecliptic from the first point of Aries is called the celestial longitude of the planet S and denoted by X. The celestial longitude X varies from 0 ° to 360". For example, on January 1, 1995 at 5.30 a.m. (1ST) the longitude X, of the sun was 280°05'. This means that the sun was at 10°05' in the sign of Capricorn whose range is 270° to 300° in the zodiac. Ifowever, in the fiidian system, the position of the sun is not considered as in Makara rdsi since in this system die longitude X , is 256°18', less by 23°47' due to the precession of die equinoxes 5_, and this falls in Dhanus (whidi extends ftom 240° to 270°) of the Indian zodiac.
20
Ancient Indian The longitude
Astronomy
measured from the vernal equinox y is called "tropical" (sdyana)
and the longitude A,, used in Indian astronomy is called "sidereal" (nirayana). The longitudes in the two systems are related by : The Indian system of reckoning the celestial longitude is called sidereal since the motions of planets are reckoned with reference to fixed stars. The word "sidereal" is used as a reference to fixed stars. 3.3 Right ascension and declination (Equatorial system) In this system the celestial equator QR is the great circle of reference and the first point of Aries is the origin. In F i g . 3.2 let S be a celestial body and M be the foot of the secondary through it to the celestial equator. Now, yM measured along the celestial equator is called the right ascension {R.A.) denoted by a and SM is called the declination 5 of the body 5. The right ascension is measured from 0 ° to 360°. But, usually the R.A. of a heavenly body is expressed in time units as hours, minutes and seconds (by dividing the angle in degrees by 15 to get hours etc.)
^^9-
f^'Q*^* ascension and declination
When a %ody is to the north of the celestial equator, its declination is said to be north and when the body is to the south, declination is also south. In the course of the sun's motion, in a year, his declination 5 increases from 0° (March 22) to about 23°27'N (June 22), then decreases from 23°27'N to 0° (Sept. 23). For the next six months, the sun will go "south" of the celestial equator; 8 increases from 0° (Sept 23) to 23°27'S (Dec. 22) and then it decreases from 23°27'S to 0° (March 22). The northern course (Dec. 22 to June 22) and the southern course .(June 22 to Dec. 22) of the sun are respectively referred to as the uttardyana and the daksiridyana. However the difference in the dates of observance is due to the precession of the equinoxes. 3.4 Azimuth and altitude (Horizontal system) In this system die celestial horizon is die reference circle and the north point n is the origin. In Fig. 3.3, let 5 be a celestial body and SM be the secondary to the horizon; M is the foot of this "vertical" through S. The angular distance nM from the north point n measured along the horizon is called the azimuth of the body S. The angular distance SM along the vertical through S is called the altitude of 5. The azimuth A Fig. 3.3: Azimuth and altitude
Co-ordinate Systems
21
is always measured eastward from the north point n and varies from 0° to 360°. Sometimes the azimuth is also measured westward in whiich case a specific mention is made to that effect. The altitude " a " of a body is the angular distance from the horizon measured along the vertical through the body. It varies from 0° to 90° on either side of the horizon. The horizontal system of azimuth and altitude is suitable for local short interval observations. These coordinates are affected by diurnal motion and also vary from one place of observation to another. 3.5 Hour Angle and Declination (Meridian system) Here, the celestial equator QR is the reference circle and the visible point of intersection of the meridian with the equator is the origin. Let 5 be a celestial body and M be the foot of the secondary to the celestial equator through 5. A s defined earlier (section 3.3), the angular distance SM from the celestial equator, along the secondary through S, is the declination 8 of the body S (Fig. 3.4). ^9- 3.4: Hour Angle & Declination The angular distance QPM
arc QM) is called the hour angle " / i " of the celestial
body 5. Thus, the hour angle of 5 at any instant is the angle between the meridian of the place and the declination circle through S. The hour angle h is always measured westward from the meridian. However in some context i f h is measured eastward, then it will be referred to as the eastern hour angle. The hour angle varies from 0° to 360°. While the hour angle is affected by the diurnal motion, the declination 8 is not affected. Further, h changes from one place of observation to another.
4 YUGA SYSTEM AND ERAS 4.1 Mahayuga, Manvantara and Kalpa On the macroscopic scale of time, the yuga system is evolyed in Indian astronomy. Many important elements of planets and other parameters are given in terms of the number of revolutions in the course of a long period of time called yuga. While in the Veddriga Jyotisa the word yuga was used to mean a period of 5 years, in later works the word meant a large period of time. So far as Indian astronomy is concerned, the yugas of large periods of time have been used to indicate the rates of motion of planets and other important points of astronomical significance. This technique enabled them to express these constants as integers, though very large, thus avoiding very inconvenient ifractions. One Mahdyuga of 43,20,000 years comprises four yugas (or yugapddas) viz., Krta, Tretd, dvdpara and Kali. Aryabhata took them all to be of equal duration, 10,80,000 years. But other astronomers, except Lalla and Vatesvara, have taken the four yugas having their durations in the ratio 4:3:2:1. Thus, we have: Krta Yuga
:
17,28,000 years
Tretd Yuga
•
12,96,000 years
Dvdpara Yuga
•
8,64,000 years
Kali Yuga
4.32,000 years
Total period of Mahdyuga
•
43,20,000 years
Note that a Mahdyuga is ten times a Kaliyuga in its duration. 1 Manvantara 14 Manvantaras
= 71 Mahdyuga = 30,67,20,000 years = 14 x 30672 x lO'* = 4.29408 X 10^ years.
In between two successive manvantaras
there is a sandhyd period equal to the
duration of a Krtayuga. For the fourteen manvantaras there are 15 sandhyds. A Kalpa is formed by the fourteen manvantaras along with their sandhyds so that 1 Kalpa = =
(14 X 30,67,20,000) + (15 x 17,28,000) 432 X lO''years
Thus, a Kalpa is one thousand times a mahdyuga.
Yuga System and Eras
23
A t present, we are under what is called Svetavaraha kalpa in which already six manvantaras
have elapsed and we are now in the seventh one called Vaivasvata
manvantara. In this manvantara 11 Mahayugas have elapsed and we are now in the 28th Mahdyuga. Again, in this running Mahayuga. the first three yugas viz., Krta. Tretd, and Dvdpara
are over. The fourth yuga namely, Kaliyuga
is currently running. Indian
astronomers are agreed on the point that the present Kaliyuga commenced at the midnight between the 17th and 18th February 3102 B . C . (by Julian reckoning). For the year 1997 A . D . , 5098 years have elapsed since the Kali epoch and the actual number of civil days (called Kali Ahargana) as on the midnight of January 1-2, 1997 is 18,61,895. The Indian astronomical siddhdntas assumed that at the commencement of the Kalpa, all the planets including Ketu were in conjunction (i.e., at the same celestial longitude) at the first point of Mesa and the ascending node (Rdhu) of the moon was 180° away i.e., at the first point of Tuld. A s pointed out earlier, the yuga theory has been adopted by Indian astronomers to express the mean motions of planets and other important geometrical points. These details will be discussed in a subsequent chapter.
4.2 Kali Era In Indian astonomical texts, generally, this era is adopted. A s pointed out earlier the Kaliyuga is supposed to have commenced at the midnight between 17th and 18th of February, 3102 B . C . (Julian), the day following the midnight being a Friday. Actually an astonomical reference to the Kali era is available in the famous text Aryabhatiyam composed by the renowned astronomer and mathematician, Aryabhata I (476 A.D.). He says that he was 23 years old when 3600 years have elapsed in the Kaliyuga. That year corresponds to 499 A . D . when the celebrated astronomer composed his immortal text. It has been inferred that this era was in vogue for long before Aryabhata I although earlier records to that effect are yet to be unearthed. The Kali era is more advantageous compared to later eras for the simple reason that it covers the antiquity of our Indian civilization adequately which other eras cannot since those were started later. However, for recording of contemporary events many other eras were adopted. The beginning of Kaliyuga is characterized by the end of the Mahdbhdrata war (Aryabhata refers to the end of the Dvdparayuga as "Bhdratdt purvam") and also the demise of Sri Krstia as per the purdnas. 4.3 Vilcrama E r a This era is widely used in most states of north-west India which follow the pdrnimdnta lunar calendar ( i.e. lunar month ranges from a full-moon to the succeeding full-moon). This era is used in Gujarat also, although the amdnta (new-moon to new-moon) lunar month is followed, but the lunar year starting from the Diwali new-moon. The starting epoch of the Vikrama saka is 58 B . C . , but the source of its origin is not exactly known. The popular belief is that the era was started by king
24
Ancient Indian Astronomy
Vikramdditya of Ujjayini to commemorate his victory over the sakas or scythians; bul inscriptional or other evidences for this are not available. Actually this era was associated with Malavas and hence it was known as Malava era for a very long time from the early 5 th century A . D . Earlier it was also called Krta era, but it is not known why it was called so. It is also argued that this era was started by the Gupta emperor, Chandra Gupta II who defeated the sakas. 4.4 Salivahana Saka E r a -.This era starts from 78 A . D . and is very widely used for both solar and lunar calendars. Even like the Vikrama era, the origin of the saka era is not well understood. One theory is that the Kusana emperor, Kanishka, the king who came after Asoka, is believed to have started this era from the date of his accession. It is opined by some that Kanishka used the old saka era omitting 200; this opinion presumes that there was a saka era 200 years earlier. However, the date of Kanishka is quite uncertain. In fact, no Indian text makes any reference in support of this theory.
AHARGAISJA
5.1 Introduction For the purpose of finding the mean positions of planets on any day, the total number of civil days elapsed since a chosen epoch is first determined. Then this duration of time multiplied by the mean daily motion of a planet gives the amount of motion of the planet during that period. From this motion the completed number of revolutions (multiples of 360°) is removed. The remainder when added to the mean position of the planet at the beginning of the epoch gives the mean position for the required day. The number of days elapsed since the chosen fixed epoch is called ahargana which literally means a "heap of days". The calculation of the ahargaria depends on the calendar system followed. Since in the traditional Hindu calendar both luni-solar calendar and solar calendar, to which the former is pegged on to, are followed the intercalary months {adhikamasa) play an important role in calculating the ahargaria. The process of finding the ahargana essentially consists of the following steps : (i)
Convert the solar years elapsed (since epoch) into lunar months ;
(ii)
A d d the number of adhikamasas during that period to give the actual number of lunar months elapsed till the beginning of the given year ;
(iii)
A d d the number of lunar months elapsed in the given year ;
(iv)
Convert these actually elapsed lunar months into tithis (by multiplying by 30);
(v) (vi) Note :
A d d the elapsed number of tithis in the current lunar month; and finally Convert the elapsed number of tithis into civil days. While finding the adhikamasa if an adhikamasa is due after the given lunar month of the current lunar year, then I is to be subtracted from the calculated number of adhikamasas. This is because an adhikamasa which is yet to come in the course of the current year will have been already added.
5.2 Working method to find Ahargana since the Kali epoch Before evolving a working algorithm to find Ahargaria, we shall list some useful data for the purpose according to Siirya Siddhdnta. In a Mahdyuga of 432 x lO'* years, we have
26
Ancient Indian
Astronomy
Number of sidereal revolutions of the moon : 577,53,336 Number of revolutions of the sun
:
43,20,000
Number of lunar months in Mahdyuga of 432 X 10^ The difference of the above
: 534,33,336
Number of Adhikamasas in a Mahdyuga
= Number of lunar months - (12 X Number of solar years) = 534,33,336 - (12 x 43,20,000) = 534,33,336-518,40,000 = 15,93,336
Suppose we want to find the Ahargana for the day on which ' V luni-solar years, "y" lunar months and "z" tithis have elapsed. Then, we have the number of adhikamasas in x completed luni-solar years given by ;cj = INT[(;c) (15,93,336/43,20,000)] where I N T (i.e. integer value) means only the quotient of the expression in the square brackets is considered. Now, since in the given luni-solar year y lunar months and z tithis have elapsed, we have : Number o f lunar months elapsed since the epoch = l2x + Xi+y
+ z/30
where the number of elapsed tithis z is converted into a fraction of a lunar month. The average duration of a lunar month is 29.530589 days. Therefore, the number of civil days /v' elapsed since epoch N^=\m[(\2x-\-x^+y\-z/30)
x 29.530589] -i-1
Here also, only the integer part of the expression in the square brackets is considered. Since in our calculations we have considered only mean duration of a lunar month, the result may have a maximum error of 1 day. Therefore, to get the actual ahargana N, addition to or subtraction from
of 1 may be necessary.
This is decided by the verification of the week-day. The tentative ahargana is divided by 7 and the remainder is expected to give the weekday counted from the weekday of the chosen epoch. For example, the epoch of Kaliymia is known to have been a Friday. Therefore, when N\ is divided by 7, if the remainder is 0, then that day must be a Friday, if 1 then Saturday etc. However, i f the calculated weekday is different from the actual weekday, then 1 is either added to or subtracted from so as to get the calculated and the actual weekday the same. Accordingly, the actual ahargaria N =
N^±\.
Ahargana
27
It is important to note tiiat the method described above is a simplified version of the actual procedure described variously by the siddhdntic texts. Note :
While finding the number of adhikamasas
in the aforesaid method if an adhikamasa
is due after the given lunar month in the said lunar year, then subtract 1 from
to get
the correct number of adhikamasas.
Example : Find the Kali Ahargana corresponding to Caitra krsna trayodasi of saka year 1913 (elapsed) i.e. for 12 April, 1991. Number of Kali years = 3179 + 1913 = 5092, since the beginning of saka i.e. 78 A . D . corresponds to 3179 years (elapsed) of Kali. Therefore. Adhikamasds in 5092 years
= (15,93,336/43,20,000) x 5092 = 1878.0710
Taking the integral part of the above value,
= 1878.
Now, an adhikamasa is due just after the Caitra mdsa under consideration. Although the adhikamdsa is yet to occur, it has already been included in the above value of X|. Therefore, Corrected value of x^ = 1877. Since the month under consideration is Caitra, the number of elapsed lunar months in the lunar year, y = 0. The current tithi is trayodasi of krsna paksa so that the elapsed number of tithis is 15 + 12 = 27. i.e., z = 27. Therefore, Number of lunar months completed = (5092 x 12) + 1877 + 0 + 27/30 = 62,981.9. (Tentative) number of civil days yv'
= INT [62,981.9 x 29.530589] + 1 = INT [18,59,892.603]+ 1 = 18,59,893
Now, dividing A'' by 7, the remainder is 0; counting 0 as Friday, 1 as Saturday etc., the remainder 0 corresponds to Friday. Also, from the calendar. 12 April, 1991 was a Friday. Therefore, we have Ahargaria, N= 18,59,893 since the Kali epoch. 5.3 Ahargana
according to Khanda Khddyaka (KK)
Deduct 587 from the saka year, multiply the remainder by 12, to this result add the number of lunar (synodic) months elapsed from the sukla paksa of Caitra month; multiply the sum by 30 and add to it the number of tithis elapsed; put down the result increased by 5 separately in two places. In one place divide by 14945; by the quotient diminish the result in the other place, and divide it by 976; by the quotient of intercalary months reduced to lunar days, increase the result in the original place; put down the result below
28
Ancient Indian
Astronomy
(i.e. in the other place), multiply it by 11 and add to it 497; put down the sum below (i.e. in the other place) and divide by 111573, diminish it by the quotient obtained from the sum in the first place; divide the new result by 703 and by the quotient of omitted lunar days (or ksaya tithis), diminish the result. The final result is the ahargana and begins from Sunday. The epoch of Brahmagupta's Khandakhadyaka is the mean midnight between 22nd and 23rd March 665 A . D . (587 saka) at Ujjayini. Brahmagupta's celebrated commentator Prthudakasvami works out the following example. Example : Find the KK ahargana at the end of 11 tithis and one lunar month from the sukla paksa of the Caitra in the saka year 786 (864 A.D.). Following the steps given above, we have : (i)
786-587
= 199
(ii)
(199xl2) + l
=
2389, total solar months.
(iii)
(2389 x 3 0 ) + l l
=
71681,total solar days.
iv)
71681 +5 14945 ) 71686 days
71681 +5 71686 days - 4d., 41gh., Alpa
4 days, 47 gh. 47 pa. (v)
71681 days, \2gh., npa
Now, dividing 71681 days, \2gh., 13 pa by 976, we get 73 as the quotient which is the number of adhikamasas (intercalary months). The remainder is 433 days, \2gh., 13pa. To this add Mgh. (to be explained later). Hence the true remainder relating to intercalary months
is taken
as 433 days,
29gh., \ 2>pa. Note :
(vi)
(vii)
1 day = 60 ghatikas (or nadikas) and 1 gh (or nadi= 60 palas (or vighaiis)
73 adhikamasas =73 x 3 0 = 2190 lunar days. Adding this to the earlier obtained [in (iv)] number of lunar days. 71681. we get 2190 + 71681 = 73871. This is the total number of tithis elapsed. Multiplying 73871 ((obtained in (vi)| by 11 and adding 497 we get 813078.
This number is put down in two places : 813078 813078 Dividing this Subtracting from the above by 111573, we get 7days, llgh., I4pa. (viii)
7 days. 17g/i., \4pa.. we get 813070 days, 42^^/1., 46pakis
Now, dividing 813070 days, 42g/i., 46/x(. by 703, wc get the quolicnl 1156 and the remainder as 402 days, 42 gh.. 46pa. This is increased by 14 gii. (lo
Ahargana
29 be explained later). The true remainder relating to the omitted lunar days (ksaya tithis) is thus taken as 402 days, 56 gh., 46 palas.
The quotient 1156 representing the integral number of omitted lunar days is now subtracted
from
the
total
tithis,
73871
and
hence
the
ahargaria
is
now
73871 - 1156 = 72715. To obtain the week day, dividing 72715 by 7, the remainder is 6. Therefore, the day, at the end of which the KK ahargaria is 72715 is a Friday (considering the remainder 1 as representing Sunday etc.) Note :
(1) In items (v) and (viii) above, \7gh., and \4gh. respectively were added. For explanation see Khandakhadyaka. Tr, by P.C, Sengupta. 1933. pp.6-7. (2) The epoch of Khandakhadyaka viz. March 23, 665 A . D . {saka 587) was a Sunday.
5.4 Ahargaria according to Graha Laghavam (GL) Epoch : March 19,1520 A . D . (Julian), Monday. (i)
Subtract 1442 from the saka year (elapsed) of the required date.
(ii)
Divide the remainder by 11. The quotient is called cakra (cycles) s C
(iii)
Multiply the remainder [obtanined in (ii)] by 12 and to the product add the number of lunar months elapsed counting Caitra as 1. The sum thus obtained is called Mean Lunar Months (Madhyama mdsa gaiia) =M
(iv)
Number of Adhikamdsas =
(v)
M + 2 C + 10 , , . (take the quotient)
True lunar months (Spasta mdsa gana) = Mean lunar months + No. of adhikamdsas = M+ INT
" M + 2 C + lO" "
(vi)
sTM
J
Mean ahargaria (madhyama ahargaria) = MAH = [{(True lunar months}) x 30 + (No. of tithis elapsed = [(TM x 30) + {TI) + T + ( 7 Cakras)] 6 6
(where Tl = No. of tithis elapsed in the given lunar month) (vii)
Ksaya dinas
= INT [
(Madhyama Ahargaria)]
= INT[-^(MAH)] 64
= KD
[i.e., take the quotient of 77 (MAH)] 64 (viii) Ahargaria (Savana dinas) i.e. Civil days s TAH = Mean Ahargaria - Ksaya dinas
30
Ancient Indian
(ix)
Astronomy
= MAH -KD = MAH - 77 (MAH) 04 However, since the average values of the various parameters are considered in the above computations, 1 day may have to be either added to or subtracted from T A H in (viii) to get the actual ahargaria. This is done by finding the weekday : (a)
Multiply the cakras by 5 i.e. find SC. A d d Savana Ahargaria to this
(b)
i.e. find (5C + TAH) Divide the result of (a) by 7 and find the remainder : Let R = Remainder of
(c)
\5C+TAH
If = 0, Monday; R=\: Tuesday etc. If the calculated weekday is a day next to the actual weekday, then subtract 1 from TAH and if it is one day less than the actual weekday, then add 1 to TAH. {See Note (1) and (2) appearing later}.
Example 1 : Sa.Saka 1534, Vaisakha Sukla PUrrtintd, Monday = M a y 16,1612 A . D . (Gregorian) (i)
Subtracting 1442 from 1534 : 1534 - 1442 = 92 years (from the epoch)
(ii) (iii)
Divide the remainder in (i) by 11 : the quotient, cakras = 8 = C and the remainder = 4 'Multiplying the remainder from (ii) i.e. 4 by 12 and adding the number of lunar months elapsed in the given year : ( 4 x 12)+ 1 = 49 = M
is the Madhyama mdsa garia (iv)
N o . of Adhikamdsas _ A / + 2 C + l Q _ 4 9 + 2(8) + 10 33
33
= — ; Quotient = 2 (v)
M + No. of Adhikamdsas = 49 + 2 = 51 =
(vi)
is the Spasta mdsa garia
(a) Mean Ahargaria (Madhyama Ahargaria) = (TM X 30) + (No. of tithis elapsed in the given lunar month) = (51 x 3 0 ) + 14= 1544 (b) A d d INT 7 [ q i.e. I N T (8/6) = 1 to the result of (vi)(a) 6 .-. 1544+1 = 1545 s M A / /
Ahargana
(vii)
31
Ksayadirms = I N T
= INT (viii)
(ix)
(MAH\
64 ri545^ = 2A = KD 64
Savana Ahargaria
(i.e. N o . of civil days in the running cakra)
= MAH - KD = 1 5 4 5 - 2 4 = 1521 = r A / / Week day verification : 5 C + ry4// = 5 (8) + 1521 = 1561 .•. R = Remainder of
^^1
=0
That is, the weekday comes out as Monday. Since the weekday obtained from calculation is the same as the actual weekday (known), nothing need be added to or subtracted from TAH. Therefore True Ahargana = 1521 No. of cakras = 8 F i n d i n g the corresponding C h r i s t i a n date (see section 5.5): No. of civil days since epoch = 8 (4016) + 1521 = 33,649 Kali Ahargana of GL epoch
= 16,87,850
Therefore, Kali Ahargana of = 17,21,499 the given date From : Table 5.1 1600 A . D . Table 5.2
12
:
17,16,982
:
4,383
Table 5.3 May 14 Total
134 :
17,21,499
Therefore, the given day corresponds to M a y 14, 1612 A . D . (Gregorian) Note 1 :
Sometimes wiien {Saica year - 1442) is divided by 11. to get Cakras the remainder could be 0. In that case even 2 may have to be added to or subtracted from the obtained Savana dinas to get the true Alvtrgana for the weekday. See the following example.
Example 2 : Saka 1574 Caitra Sukla Pratipat Ravivara {i.e. April 7, 1652 A . D . , Sunday} (i) (ii)
1574 - 1442 = 132, years since the epoch C = INT
132 = 12 11
Remainder of
is 0
Ancient Indian Astrono\ (iii)
(0xl2) + 0 = 0 s M
(iv)
No. of
Adhikamdsas
i.e. Adhikamdsas (V) (vi)
= INT
M + 2 C + 10 "0 + 24+ l O ' = INT 33 33
=INT
34 = 1 33
TM = M + No. of adhikamasas = 0 + 1 = 1 Mean Ahargana A f A / / = (1 X 30) + 0 + I N T [7 (12)] o = 30 + 2 = 32
(vii)
Ksaya dinas, KD = I N T
^MAH\
64
= INT
i,e. KD = 0 (viii)
Sdvana Ahargaria = MAH - /TD = 32 - 0 = 32 .•. Cakras = 12, Ahargaria = 32 (in the running Cakra)
(ix)
Weekday verification : R = Remainder of i.e. of
i.e. of
i^^'^J^^
60 + 32
^92^
or /?= 1
i.e. Tuesday. But the actual weekday is Sunday. Therefore subtracting 2 from the Sdvana Ahargaria, we get Actual Ahargaria = 32 - 2 = 30. Thus, Cakras = 12 and Ahargaria = 30. (X)
Christian date (see section 5.5) : No. of civil days since epoch = 12 (4016)+ 30
=48,222
Kali Ahargaria of GL epoch = 16,87,850 Therefore, Kali Ahargaria of the given date : 17,36,072 From Tables 5.1 to 5.3, we have 1600 A . D . (G)
17,16,982 18,993 52 97 April 7 17,36,072 Total corresponding to April 7,1652 A . D . (G)
Ahargana Note 2 ; (i)
33 Sometimes tliere could be an Adhikamasa in a particular given lunar year, If the given date is before the adhikamdsa of that lunar year, then subtract 1 from the no. of adhikamdsas obtained in the calculation.
('•)
i f the given date is after the adhikamdsa
of that lunar year, then add 1 to
the no. of adhikamdsas obtained in the calculation. This is demonstrated in the following example. Example 3 : Saka
1555 Caitra
Sukla Pratipat,
Friday. In this lunar year, Vaisakha is the
adhikamdsa [March 11,1633]. We shall find the ahargana for the given date : (i)
1 5 5 5 - 1442= 113
(ii)
Cakras, C = I N T
{U3^ = 10, Remainoer = 3 11
(iii)
(Remainder x 12) + No. of elapsed lunar months = (3 x 12) + 0 = 36 s M
(iv)
No. of adhikamdsas = INT
36 + 2 ( 1 0 ) - H 0 M + 2C+ 10 = INT 33 33
= INT
66 = 2 (Note : Remainder = 0) 33
Since the given date falls before the adhika Vaisakha masa, subtract 1 from the number obtained above. Therefore,, the actual adhikamdsas = 2 - 1 = 1. (v)
True lunar months = M + No. of adhikamdsas = 36 + 1 = 37 s TA/
(vi)
Mean Ahargaria MAH= (37 X 30) + Tithis elapsed + INT ^10^ = 1110 + 0 + INT 6 = 1110+1 = 1111.
(vii)
Ksayadirms KD = I N T
(viii)
Savana Ahargaria
MAH 64
= INT
1111 = 17 64
= MAH - KD
= n i l - 17= 1094 3 7>1// (ix)
Weekday verification : 5C + r A / f = 5 (10)+ 1094= 1144
^C} 6 \
/
elapsed
34
Ancient Indian Remainder
1144
Astronomy
= 3 i.e. Thursday; but actual weekday : Friday
Therefore, the true Ahargaria = TAH+ 1 = 1095 (X)
Christian date (see section 5.5) : No. of civil days since epoch = 10(4016)+1095 = 41,255 Kali Ahargaria of GL epoch : 16,87,850 .-. Kaii Ahargaria of the date : 17,29,105 From Tables 5.1 to 5.3, we have 1600 A . D . 33 March 11 Total
17,16,982 12,053 70 17.29.105
Corresponding to March 11,1633 A . D . (Gregorian) Example.4 : Saka 1530 (Bhadrapada is adhikamdsa) Kdrtika Sukla Pratipat, Saturday. (i) (ii) (iii)
1 5 3 0 - 1442 = 88 ,r88^ = 8, Remainder = 0 C= Cakra = mi 11 Mean lunar months A/=(0xl2) +7 =7 (since Kdrtika is the 8th month i.e. 7 months are over) .
(iv)
N o . o f Adhikamdsas = INT
7 + 16+10 Af + 2C+ 10 = INT 33 33
= 1. Remainder 0 Since Kdrtika
month (i.e. the given month) occurs after
the adhika
Bhadrapada mdsa, add 1 to the calculated no. of adhikamdsas. Therefore, no. of adhikamdsas = 1 + 1 = 2 (v)
True lunar months TM = M+ N o . of adhikamdsas =7+2=9
(vi)
Mean Ahargana MAH = (TM X 30) + (Tithis elapsed in the given month) + I N T = ( 9 x 3 0 ) + 0 + INT = 271
6
^C^ 6
Ahargana
(vii) (viii)
35
Ksaya dinas = I N T Savana
= 4 = KD
Ahargana
TAH^MAH(ix)
64
KD^ni
- 4 = 267
Weekday verification : 5C
+ r/l// = 40 + 267 = 307
^307^ Remainder of = 6 i.e. Sunday. But the given weekday is Saturday. 7 .•. True Ahargana ~ TAH - 1 = 266 Cakras = 8 and ahargana = 266 (x)
Christian date : No. of civil days since epoch = 8 (4016) + 266 = 32,394 Kali Ahargaria of GL epoch : 16,87,850 .-. Kali Ahargaria of given date : 17,20,244 From Tables 5.1 to 5.3, we have 1600 A . D . 08
: ;
17,16,982 2922
December 6 : 340 Total : 17,20,244 corresponding to December 6,1608 A . D . (Gregorian) 5.5 Ahargana from the Christian date; finding the weekday In the earlier sections we saw how to obtain the ahargaria from the lunar date with the epoch of (i) The beginning of the Kaliyuga (according to the Siirya Siddhdnta) (ii) The Kharida Khddyaka and (iii) The Graha
Laghavam.
Now, we shall discuss the method of finding the ahargaria for a given Christian date (either Julian or Gregorian). In Table 5.1 the Julian days, and the ahargaria for the epochs of the Kaliyuga, Kharida Khddyaka and the Graha Laghavam are given for the beginnings of the century years from—3200 (J) to 2200 A . D . (G). Epochs Chosen : (i)
Julian days (JD) : The reckoning of the Julian days starts from the mean noon ( G M T ) on January 1, 4713 B . C . , Monday. On that day at the mean noon ( G M T ) , JD = 0. The so-called Julian calendar was adopted by Julius Ceasar (100 B . C . to 44 B.C.)
36
Ancient Indian
Astronomy
This was revised by Pope Gregory XIII'in 1582 A . D . on the advice of his astronomer. According to the erstwhile Julian calendar system October 4, 1582 was a Thursday. Pope Gregory ordered that the next day would be October 15, 1582, Friday. However, the continuity of the week days was maintained. In Table 5.1, the beginnings of centuries from—3200 to 1500 A . D . are suffixed with "(J)" which means for those centuries the system of Julian calendar is applicable. This is so upto October 4, 1582 A . D . Then for the beginnings of centuries beyond that date the suffix " G " (Gregorian) is added. For any year before Christ (B.C.) for mathematical convenience a negative sign (-) is perfixed to one less than the numerical value of the year. For example, 46 B . C . is ^ 5 and 3102 B . C . is -3101. This convention is adopted since 1 B . C . is considered as the "0" year of the Christian era. (ii)
Epoch of the Khanda Khddyaka {KK) The epoch chosen by Brahmagupta in this Kararia text is the mean midnight between 22rid and 23rd of March 665 A . D . (J) at Ujjayini.
(iii)
Epoch of the Graha Ldghavam {GL) : Ganesa Daivajha, in his Graha Laghava has adopted the mean sunrise (at Ujjayini) of March 19, 1520 A . D . (J), Monday.
Finding the Ahargarias (a)
First, from Table 5.1 for the beginning of the Christian century (column 1) in which the given date lies—the Julian days. Kali ahargaria, KK and GL ahargarias are given in columns 2 to 6. For example, consider March 21, 1997. Now, the century beginning for 1997 is 1900. In Table 5.1, against the entry 1900 (G), we have :
(b)
Julian days
Kali ahar
KK ahar
GL ahar Ca. ahar
2415020
1826554
450989
34 2160
In Table 5.2 the total clasped days are given for the beginning of the years of a century. In our example, the year is 97. Against the entry 97 in Table 5.2, we have Days 35429
(c)
Graha Ldghavam Ca.
Ahar
8
3301
In Table 5.3, the accumulated days for the dates of the different months in a year are given. Here, for the months of January and February in a leap year, the dates given under the second column headed by the letter " B " must be used. For all other months of any year and for January and February of a non-leap year, the dates in the first column under the letter " C " must be used.
Ahargana
37
In our example, for March 21, the number of days elapsed in the year is 80 (in the column under " C " ) . Now, the total J D and the different aharganas are obtained by adding the corresponding number of days from items (a), (b) and (c) above. For example, for March 21. 1997, we have :
1900 (G) 97 March 21 Add Note :
Julian days
Kali Ahar.
KKAhar
2415020 35429 80 2450529
1826554
450989 35429 80 486498
35429 80 1862063
Graha Lag. Ch. Ahar 342160 83301 80 425541
While adding the ahargana numbers for the epoch of the Graha Laghava if the total ahargana is greater than or equal to 4016, then the number should be divided by 4016 and the quotient (an integer) must be added to the cakras. In the example considered, the ahargana number 5541, under C L , exceeds 4016 and hence dividing 5541 by 4016, the quotient is 1 and the remainder is 1525. Therefore, adding 1 to the cakra number 42 we get cakras (elapsed) as 43 and the balance ahargana as 1525.
Finding the, weekday. (a)
From the J D to find the weekday, divide J D by 7. If the remainder is 1, it is Tuesday: i f the remainder is 2 it is Wednesday etc. In the example, when J D = 2450529 is divided by 7, the remainder is 4 and hence March 21, 1997 is a Friday.
(b)
From the Kali ahargaria, to find the weekday, divide the ahargaria by 7. If the remainder is 1, then it is Saturday; i f the remainder is 2, it is Sunday etc, In the example, the Kali ahargaria is 1862063. When this number is divided by 7, the remainder is 0 and hence the weekday is Friday.
(c)
From the KK ahargaria find the remainder R by dividing the ahargana by7 If /? = 1. then the weekday is Monday; i f /? = 2. it is Tuesday etc. In our example, for March 21, 1997, the KK ahargaria is 486498. Dividing this by 7, the remainder /? = 5. Therefore the weekday is Friday.
(d)
The weekday from the GL ahargaria : Multiply the number of cakras C by 5 and add to this product the ahargaria A (after reducing the cakras) i.e find (5C + A). Dividing (5C + A) by 7, let the remainder be R. If /? = 0:
Monday; i f R=l:
Tuesday
etc.
In
the
example
C = 4 3 a n d A = 1525. Therefore, 5 C + A = 5 (43) + 1525 = 1740. Dividing 1740 by 7, /? = 4 and hence the weekday is a Friday.
considered,
38
Ancient Indian Astronomy Table 5.1: Ahargana—Kali, Khandakhadyaka, Graha Ldghavam and JD Chris. Year
Julian Days
Kali
Kharida Khdd.
Graha
Ldghava
Ahargaria
Ahargaria
Ca.
Ahargaria
-3200 (J)
552258
-36208
-1411773
-430
1%11
-3100 (J)
588783
317
-1375248
-All
3203
-3000 (J)
625308
36842
-1338723
-412
3584
-2900 (J)
661833
73367
-1302198
-403
3965
-2800 (J)
698358
109892
-1265673
-393
330
-2700 (J)
734883
146417
-1229148
-384
711
-2600 (J)
771408
182942
-1192623
-375
1092
-2500 (J)
807933
219467
-1156098
-366
1473
-2400 (J)
844458
255992
-1119573
-357
1854
-2300 (J)
880983
292517
-1083048
-348
2235
-2200 (J)
917508
329042
-1046523
-339
2616
-2100 (J)
954033
365567
-1009998
-330
2997
-2000 (J)
990558
402092
-973473
-321
3378
-1900 (J)
1027083
438617
-936948
-312
3759
-1800 (J)
1063608
475142
-900423
-302
124
-1700 (J)
1100133
511667
-863898
-293
505 886
-1600 (J)
1136658
548192
-827373
-284
-1500 (J)
1173183
584717
-790848
-275
1267
-1400 (J)
1209708
621242
-754323
-266
1648
-1300 (J)
1246233
657767
-717798
-257
2029
-1200 (J)
1282758
694292
-681273
-248
2410
-1100 (J)
1319283
730817
-644748
-239
2791
-1000 (J)
1355808
767342
-608223
-230
3172
-221
3553
-900 (J)
1392333
803867
-571698
-800 (J)
1428858
840392
-535173
-212
3934
-700 (J)
1465383
876917
-498648
-202
299
-600 (J)
1501908
913442
^62123
-193
680
-500 (J)
1538433
949967
^25598
-184
1061
^ 0 0 (J)
1574958
986492
-389073
-175
1442
-300 (J)
1611483
1023017
-352548
-166
1823
-200 (J)
1648008
1059542
-316023
-157
2204
-100 (J)
1684533
10^6067
-279498
-148
2585
0(J)
1721058
1132592
-242973
-139
2966 (Contd...)
Ahargaria
39
Chris.
Kali
Khanda Khdd.
Graha
Laghava
Ahargana
Ahargana
Ca.
Ahargana
Julian Days Year 100 (J)
1757583
1169117
-206448
-130
3347
200 (J)
1794108
1205642
-169923
-121
3728
300 (J)
1830633
1242167
-133398
-111
93
400 (J)
1867158
1278692
-96873
-102
474
500 (J)
1903683
1315217
-60348
-93
855
600 (J)
1940208
1351742
-23823
-84
1236
700 (J)
1976733
1388267
12702
-75
1617
800 (J)
2013258
1424792
49227
-66
1998
900 (J)
2049783
1461317
85752
-57
2379
1000 (J)
2086308
1497842
122277
-48
2760
1100 (J)
2122833
1534367
158802
-39
3141
1200 (J)
2159358
1570892
195327
-30
3522
1300 (J)
2195883
1607417
231852
-21
3903
1400 (J)
2232408
1643942
268377
-11
268
1500 (J)
2268933
1680467
304902
-2
649
1500 (G)
2268923
1680457
304892
-2
639
1600 (G)
2305448
1716982
341417
7
1020
1700 (G)
2341972
1753506
377941
16
1400
1800 (G)
2378496
1790030
414465
25
1780
1900 (G)
2415020
1826554
450989
34
2160
2000 (G)
2451545
1863079
487514
43
2541 2921 3301
2100 (G)
2488069
1899603
524038
52
2200 (G)
2524593
1936127
560562
61
Table 5.2: Ahargana for Year Beginnings Year
Days
0 1 2 3 4 5 6 7
0 365 730 1095 1461 1826 2191 2556
Graha Laghava Ca. Ahar. 0 0 365 0 730 0 0 1095 0 1461 1826 0 0 2191 0 2556
Year
Days
8 9 10 11 12 13 14 15
2922 3287 3652 4017 4383 4748 5113 5478
Graha Laghava Ahar. Ca. 2922 0 3287 0 0 3652 1 1 367 1 1 732 1 1097 1462 1 (Contd...)
40
Ancient Indian Astronomy Year
Days
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57
5844 6209 6574 6939 7305 7670 8035 8400 8766 9131 9496 9861 10227 10592 10957 11322 11688 12053 12418 12783 13149 13514 13879 14244 14610 14975 15340 15705 16071 16436 16801 17166 17532 17897 18262 18627 18993 19358 19723 20088 20454 20819
Graha Laghava Ca. Ahar 1 1828 1 2193 1 2558 1 2923 1 3289 1 3654 2 3 2 368 2 734 2 1099 2 1464 2 1829 2 2195 2 2560 2 2925 2 3290 3 3656 3 5 3 370 3 735 3 1101 3 1466 3 1831 3 2196 3 2562 3 2927 3 3292 3 3657 4 7 4 372 4 737 4 1102 4 1468 4 1833 4 2198 4 2563 4 2929 4 3294 4 3659 5 8 5 374 5 • 739
Year
Days
58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
21184 21549 21915 22280 22645 23010 23376 23741 24106 24471 24837 25202 25567 25932 26298 26663 27028 27393 27759 28124 28489 28854 29220 29585 29950 30315 30681 31046 31411 31776 32142 32507 32872 33237 33603 33968 34333 34698 35064 35429 35794 36159
Graha Laghava Ca. Ahar. 5 1104 5 1469 5 1835 5 2200 5 2565 5 2930 5 3296 5 3661 6 10 6 375 6 741 6 1106 6 1471 6 1836 6 2202 6 2567 6 2932 6 3297 6 3663 7 12 7 377 7 742 7 1108 7 1473 7 1838 7 2203 7 2569 7 2934 7 3299 7 3664 8 14 8 379 8 744 8 1109 8 1475 8 1840 8 2205 8 2570 8 2936 8 3301 8 3666 9 15
41
Ahargana Table 5.3: Ahargana for Days of a Year Dates
Jan.
Feb. Mar.
Apr.
May.
Jun.
Jul.
Aug.
Sep.
Oct.
Nov.
Dec.
B
C 0
1
0
31
1
2
1
32
60
91
121
152
182
213
244
274
305
335
2
3
2
33
61
92
122
153
183
214
245
275
306
336
3
4
3
34
62
93
123
154
184
215
246
276
307
337
155
185
216
247
277
308
338
156
186
217
248
278
309
339
249
279
310
340
4
5
4
35
63
94
124
5
6
5
36
64
95
125
6
7
6
37
65
96
126
157
187
218
7
8
7
38
66
97
127
158
188
219
250
280
311
341
8
9
8
39
67
98
128
159
189
220
251
281
312
342
160
190
221
252
282
313
343
161
191
222
253
283
314
344
315
345
316
346
9
10
9
40
68
99
129
10
11
10
41
69
100
130
11 12 13
12
11
42
70
101
131
162
192
223
254
284
13 14
12
43
71
102
132
163
193
224
255
285
13
44
72
103
133
164
194
225
256
286
317
347
134
165
195
226
257
287
318
348
135
166
196
227
258
288
319
349
289
320
350
14
15
14
45
73
104
15
16
15
46
74
105
16
17
16
47
75
106
136
167
197
228
259
17
18
17
48
76
107
137
168
198
229
260
290
321
351
18
19
18
49
77
108
138
169
199
230
261
291
322
352
262
292
323
353
19
20
19
50
78
109
139
170
200
231
20
21
20
51
79
110
140
171
201
232
263
293
324
354
294
325
355
295
326
356
21
22
21
52
80
111
141
172
202
233
264
22
23
22
53
81
112
142
173
203
234
265
23
24
23
54
82
113
143
174
204
235
266
296
327
357
267
297
328
358
268
298
329
359
24
25
24
55
83
114
144
175
205
236
25
26
25
56
84
115
145
176
206
237
26
27
26
57
85
116
146
177
207
238
269
299
330
360
178
208
239
270
3QP
331
361
179
209
240
271
301
332
362
272
302
333
363
27
28
27
58
86
117
147
28
29
28
59
87
118
148
30
29
-
30
31
30
31
-
31
29
-
88
119
149
180
210
241
89
120
150
181
211
242
273
303
334
364
90
-
151
-
212
243
-
304
-
365
6
MOTIONS OF THE SUN AND THE MOON
6.1 Introduction It was pointed out, while discussing the Yuga system of Indian astronomy, that the mean modon of each planet i f given in numbers of revolutions made by the planet in the course of a mahayuga of 432 x 10'* years or a kalpa of 432 x l O ' years. The advantage of choosing such a long period of time is that the motions of planets could be expressed in integral numbers of revolutions completed, avoiding inconvenient fractions. The yuga model is used to compute the mean positions of planets. These positions are then corrected to get the true positions. 6.2 Mean positions of the sun and the moon The following is the usual procedure for finding the mean position of any planet : Let
be the mean position at an epoch. If d is the daily mean motion of the planet in
degrees and A is the ahargana from the epoch, then the mean motion from the epoch till the given day is Xj = (A x d). The mean position on the given day is X = Xp + Xj = Xq + (A X ^ . In the texts where the beginning of Kali yuga is used as the epoch, Ahargana is calculated from the mean
midnight between
17th and 18th of February 3102 B . C .
Therefore, in this method, the Ahargana A is obtained till the mean midnight preceding the given day. Further, the mean midnight is as at thp Ujjayini (23° l l ' N latitude and 75° 47' E longitude) meridian passing through Lanka on the equator. Therefore, when we calculate the positions of the planets, corrections will have to be applied to account for (i)
the time interval between the midnight at Ujjayini and the midnight at the given place;
(ii)
the duration from the local midnight to the given time. In addition to these some more important corrections will also have to be applied. These will be discussed later.
Motions of the Sun and the Moon
43
Some astronomical texts consider the mean sunrise at LaAka on Feb 18, 3102 B . C . as the starting time of the Kali era. The mean motion o f the sun. the moon and other planets are given in terms o f revolutions (each o f 360° extent) completed in the course of a Kalpa of 432 x l O ' years. Table 6.1 gives the needed details for finding the positions of the sun and the moon. For other planets the details will be considered later. From Table 6.1. we notice that while the revolutions of the sun in a kalpa are the same according to the different texts, those of the moon and its apogee and node (Rahu) are different. Further, the number o f civil days in a Mahdyuga (or multiplied by 1000 for a Kalpa) are slightly different according to Aryabhata I and the Siirya Siddhdnta, for example. These differences have resulted from the corrections made periodically, and give rise to slightly different mean daily motions. Table 6.1: Revolutions of the sun, the moon etc. in a Kalpa (1 Kalpa = 432 x l o ' years = 1000 Mahdyugas) Planets and points
Ravi
Candra
Candra's Civil days in a Candra's Apogee (Asc.) Node Mahdyuga of {Mandocca) {Rdhu) 432 X 10'* yrs.
Aryabhata I 432,00.00.000 5775.33.36.000 48.82.19.000
23.21.16.000
1.57,79,17.500
Brahmagupta 432.00.00.000 5775.33.36,999 48.82,19,000 (Khattda
23.22.26,000
1.57,79,17,800
432.00.00.000 5775.33.36,000 48,82,03.000
23.22.38.000
1.57.79.17.828
48,82,08,674 Aryabhata II 432.00.00.000 5775,33.34.000
23,23,13.354
1,57,79.17.542
23.23.11,168
1.57.79.16.450
khddyaka) Siirya Siddhdnta
{Mahd Siddhdnta) Bhaskara II
432.00,00,000 5775.33.00.000 48.82.05.858
{Siddhdnta Siromarii) The daily mean motions are given in Table 6.2 according to die Siddhanta Siromarii of Bhaskara II as compared to the modem values and those of the Kharidakhddyaka of Brahmagupta and the Siirya Siddhdnta. The values are given in degrees, minutes, seconds of arc and two further subdivisions of a second viz., 1/60 and 1/36(X) of second of arc.
44
Ancient Indian
Astronomy
Table 6.2: Daily mean motions of the sun, the moon etc. Planets & points Ravi
Surya
Siddhdnta
Siddhdnta
Siromani
0 ° 5 9 08 10 09.7
Candra Kuja
Modem Astronomy
Kharida khddyaka
0 ° 5 9 08 10 21
0 ° 5 9 08.2
0° 59 08
13° 10 34 52 02
13° 10 34 53 00
13° 10 34.9
13° 10 31
0°31 26 28 10
0°31 26 28 07
0°31 26.5
0°31 26 4° 05 32
Budha's 4° 05 32 20 42
4° 05 32 18 28
4° 05 32.4
0 ° 0 4 59 08 48
0° 04 59 09 09
0° 04 59.1
0 ° 0 4 59
1° 36 07 43 37
1° 36 07 44 35
1° 36 07.7
1°36 07
Sani
0° 02 00 22 53
0°02 00 22 51
0° 02 00.5
0° 02 00
Candra's Mandocca
0° 06 40 58 42
0° 06 40 53 56
0° 06 40.92
0° 06 40
- 0 ° 0 3 10.77
- 0 ° 0 3 10
Sighrocca Guru 5ukra's Sighrocca
Candra's - 0 ° 0 3 10 44 43
Pdta (Rahu)
- 0 ° 0 3 10 48 20
It is clear that the daily motion of a celestial body is given, in revolution, per day, by ^
_
Number of revolutions in a Mahd yuga Number of civil days in a Mahd yuga
Example : According to the Siddhdntasiromarii, the number of revolutions completed by the moon is 577,53,300 in a Mahdyuga having 157,79,16,450 civil days. Therefore, moon's daily motion, '^moon = (577,53,300/157.79,16,450) X 360 (in degrees) Similarly, for the sun, the mean daily motion, d,^„
= (43,20,000/157,79,16,450) X 360 (in degrees) =
(43,20,000 X 360/157,79,16,450) degrees
Now, as explained earlier, the mean motion for any given day is obtained by multiplying the mean daily motion of the celestial body by the Ahargana of the day, that is, X\=dy.A The mean motion
thus obtained must be added to the initial mean position at the
epoch viz, Xq. According to Bhaskara II, the following are the initial mean positions (dhruvakas),
for the different celestial bodies at the Kali epoch. The values as per the
Surya Siddhdnta are also given.
Motions of the Sun and the Moon
45
Table 6.3: Mean positions of planets at the Kali epoch Planets & points
Siddhanta Siromani
Surya Siddhanta
Ravi
0°0 0
0°0 0
Candra
0°0 0
0°0 0
Kuja
359° 03 50
0°0 0
Budha Sighrocca
357° 24 29
0°0 0
Guru
359° 27 36
0°0 0
Sukra's Sighrocca
358" 42 14
0°0 0
Sani
358° 46 34
0°0 0
77° 45 36
77° 7 48
Candra's Mandocca
125° 29 46
90° 0 0
Candra's Pdta (Rahu)
153°12 58
180° 0 0
Ravi's Mandocca
6.3 Mean positions of the sun and the moon For the computation of the mean positions of the planets, we choose the beginning of the Kaliyuga, viz, the midnight of 17/18 February 3102 B . C . as the epoch. Further, it is assumed that the sun, the moon and the five planets {tdrdgrahas) were at the beginning of the zodiac i.e., 0" Mesa. The mean longitude of a planet at the mean midnight at Ujjayini on a day with Kali Ahargaria A is given by Mean longitude -Ax
Mean daily motion
= /4 X No. of revolutions/No. of civil days where the number of revolutions of the concerned body and the number of civil days correspond to a Mahdyuga of 43,20,000 years. In the following example, for the midnight preceding March 21, 1997, the mean positions of the sun, the moon and the moon's apogee and node are calculated. 6.3.1 According to Suryasiddhdnta {SS) (i)
Mean Longitude of the Sun In a Mahdyuga, the number of revolutions of the sun is 43,20,000 and the number of civil days is 157,79,17,828. Now, on March 21, 1997 the elapsed Kali Ahargana is 18,62,063. Therefore, Mean longitude of the sun = 18,62,063x43,20,000/157,79,17,828 = 5097 revolutions, 334° 14' 0 9 " Now, ignoring the completed revolutions viz. 5097, the mean longitude of the sun : 334° 14'09"
46
Ancient Indian Astronomy (ii)
Mean Longitude of the Moon In a Mahdyuga, the number of revolutions of the moon is 577,53,336 and the number of civil days is 157,79,17,828. At the midnight preceding March 21, 1997, the elapsed Kali Ahargaria is 18,62,063. We have Mean longitude of the moon = 18,62,063 x 57,53,336/157,79,17,828 = 68,153 revolutions, 117° 48' 2 5 " Ignoring the number of completed revolutions, the mean longitude of the moon : 117° 48' 25".
Note :
In the above computations, since the mean longitudes are given in terms of revolutions, the decimal part is multiplied by 360° to get it in degrees, the decimal part of the degree by 60 to get minutes and lastly the decimal part of the minute by 60 to get seconds.
(iii)
Mean Longitude of the Moon's Apogee {Mandocca) In a Mahdyuga,
the number of revolutions of the moon's mandocca is
48,82,03; and the number of civil days is 157,79,17,828. At the epoch, the mandocca was 90°. Therefore for the Kali Ahargana of 18,62,063, the mean motion of the moon's mandocca is 48,82,03 X 18,62,063/157,79,17,828 = 576 revolutions 41° 5 9 ' 4 6 " Removing the number of completed revolutions, the mean motion of the moon's mandocca since the epoch is 41° 59'46". Now adding this mean motion to the longitude of the mandocca at the epoch, we have the mean longitude of the moon's mandocca = 90° + 41° 59' 46" = 131° 59' 46" (iv)
Mean Longitude of the Moon's Node (Rahu) The number of revolutions of the ascending node of the moon, called Rahu, in a mahdyuga
is 2,32,238. Rahu was assumed to be at 180° from the
beginning of the zodiac at the epoch i.e., the beginning of the Kali yuga. Therefore, Mean longitude of Rahu = 1 8 0 ° - mean motion for the Ahargaria. Note:
Mean motion is subtractive since Rahu moves backwards along the ecliptic,
In the example, we have Mean motion of Rahu = (No. of revolutions/No. of civil days) X Ahargaria = (2,32,238/157,79,17,828) X 18,62,063 = 274 revolutions, 21° 0 3 ' 3 1 "
Motions of the Sun and the Moon
47
Removing the completed number of revolutions, the motion of Rahu is 21° 0 3 ' 3 1 " . Since the motion is backwards, this value is to be subtracted from the initial position at the epoch i.e., 180°. Therefore, Mean longitude of Rahu = 180° - (21° 03' 31") = 158°56'29" Note:
If the mean longitude comes out to be negative then 360° must be added to render it non-negadve.
The opposite node, called the descending 'node of the moon (Ketu) is exactly 180° away from Rahu. Therefore, the mean longitude of Ketu = 338° 56' 29". Desantara correction Desantara is the longitude of a place measured east or west from the standard (or prime) meridian. In traditional Indian astronomy the prime meridian is the great semicircle of the earth passing through the north and south poles, Ujjayini and Lafika where Lanka was assumed to be on the earth's equator. The sun completes a rotation (of 360°) in one day due to its diurnal motion. Therefore, the angular distance covered by the sun in 1 hour is 360°/24 i.e. 15°. This means i f the sun rises at 6 a.m. (1ST) at place A, then the sunrise at another place B, 15° to the west of the first place will be 1 hour later i.e. at 7 a.m. (1ST) provided the two places are on the same latitude (i.e. on the same small circle parallel to the equator). Similarly, at place C to the east of A, the sunrise will have taken place earlier. If C is 15° to the east of A (but on the same small circle parallel to the equator), then the sunrise at C will be 1 hour before that at A i.e. at 5 a.m. (1ST). In Indian astronomy, the mean positions of planets are calculated either for the midnight or for the sunrise at the standard meridian. Then to get the position at the corresponding time (i.e., either midnight or sunrise) at another place, on a different meridian, we have to apply a correcton due to the difference in the longitudes of the given place and the standard meridian. This correction is called desantara correction. Since the standard meridian passes through Ujjayini, in terms of the modem terrestrial longitudes (referred to Greenwich meridian), we have Desantara = [(Long, of the place—Long, of Ujjayini) / 15] in hours where the longitudes are in degrees. Since the longitude of Ujjayini is 75°E 47' (east of Greenwich), we have Desdntara = [(Long, of the place—75° 47')/15] in hours
...(1)
If the correction is required in ghatikas, since the earth rotates at the rate of 360° per 60 ghatikds (or 6 ° per gh.), we have Desdntara = [(Long, of the place—75° 47') /6] in gh.,
...{!)
48
Ancient Indian Astronomy The correction to be applied to the mean sun or moon (obtained for Ujjayim midnight)
due to Desdntara is given by Desdntara correction = (Desdntara in hours)x d/24 -
(Desdntara in ghatikds) X d/60
...(3)
where d is the mean daily motion of the sun or die moon as the case may be. The above correction is to be added to or subti-acted from the earlier obtained mean positon (for Ujjayim midnight) according as the place is to the west or east of Ujjayini. If the place is to the west of Ujjayini, the midnight occurs later than at Ujjayini during which time interval the celestial body will have moved further and hence the desdntara correction in that case is additive. Similarly, i f the place is to the east of Ujjayini, the midnight there would have occurred earlier and hence the celestial body will have moved less; therefore the correction is subtractive. Thus, from (1) (2) and (3) we have Desdntara correction = - (X - X,,,) X rf/360 where X,
= Longitude of the place
Xf, = Longitude of Ujjayini and d
= Mean daily motion of a celestial body
The negative sign indicates that the correction is subtractive when 'K>X^ i.e., the place is to
the east of Ujjayini. For a place lying to the west of Ujjayini (i.e., X < X^),
the correction automatically becomes additive. Note that for a place with a westem longitude (with reference to Greenwich) X must be taken negative. Note :
In the Indian astronomical texts, the Desdntara
is obtained from the linear
distance (in yojanas) of the place from Ujjayini. We shall now apply the Desdntara correction to the sun, the moon and moon's apogee and node for Bangalore at the local midnight between 20th and 21st March, 1997. Taking the longitudes of Bangalore and Ujjayini respectively as 77° E 35' and 75° E 47', we have ( X - X j / 3 6 0 = 1.8/360 (i)
Desdntara
correction for the sun :
Since the daily mean motion of the sun is 59'. 136078 according to the SUrya Siddhdnta, we have Desdntara correction for the sun = (1.8/360) x 59'.136078 = 17".7. Therefore, the mean longitude of the sun on 21-3-1997 at the preceding local mean midinight of Bangalore is 334° 14' 09" - 17".7 = 334° 13' 51".3
Motions of the Sun and the Moon (ii)
49
Desdntara correction for the moon : Daily mean motion of the moon : 13°. 176352. Therefore, Desdntara correction = (1.8/360) X 13°. 176352 = 3' 57" Mean longitude of the moon at the local midnight preceding 21-3-1997 at Bangalore is 117° 48' 2 5 " - 3' 57" = 117° 44' 28"
(iii)
Desdntara correction for the moon's apogee (mandocca) : Daily mean modon of the moon's apogee is 6'.6829747 Therefore, for the moon's apogee, Desdntara correction = (1.8/360) x 6'.6829747 = 2". The mean longitude of the moon's apogee at the mean midnight
preceding
21-3-1997 at Bangalore is 1 3 1 ° 5 9 ' 4 6 " - 2 " = 131° 5 9 ' 4 4 " (iv)
Desdntara correction for the moon's node : Daily mean motion of the moon's node is - 0°.052985 or - 190".746 so that Desdntara correction = - (1.8/360) x 190".746 = - 0".95373 = - i " approximately.
Thus, the mean longitude of Rahu for the local mean midnight preceding 21-3-1997 at Bangalore is 158° 56' 29" - (- 1") = 158° 56' 30" 6.3.2 According to Khanda khddyaka (KK) (i)
Mean Longitude of the Sun In the Khanda khddyaka, the mean longitude of the sun is given by : ,
(A X 800) + 438
, .
If the above value is multiplied by 360°, we get X in degrees from which the integral multiples of 360° (i.e. the completed number of revolutions) have to be removed. Example : For March 21, 1997, the KK Ahargaria is 4,86,498 i.e., A
= 4,86.498. Therefore (4,86,498x800)+438 292207 =
1331.928523 revolutions.
Removing the completed number of revolutions viz., 1331, we get X i.e., X
= 0.928523 rev. = 334° 16'06"
50
Ancient Indian
Astrononvj
(Compare this with the SS value : 334° 1 4 ' 0 9 " for the same date and time.) (ii)
Mean Longitude of the Moon In KK the mean longitude of the moon is given by : -
( A x 6 0 0 ) + 417.2 , . 16393 revolutions
~
4929 Example
With A
minutes of arc.
= 4,86,498, we get
X
= 17806.33302 revolutions -98'.70115642 = 119° 53' 1 4 " - 9 8 ' 4 2 " = 118° 14'32".
(iii)
Mean Longitude of the Moon's Apogee (Mandocca) In KK the mandocca of the moon is given by M =
A-453.75 revolutions + mm. 3232 39298
Example : With A = 4,86,498, we have M = 150.3849783 revns. + 12' 2 3 " = 138° 4 7 ' 5 5 " (omitting the revns.) (iv)
Mean Longitude of the Moon's Node (Rdhu) In KK the mean longitude of the moon's ascending node is given by n
=-
A-372 degrees of arc rev + 6795 514656
Example : With A = 486498, the mean Rahu is n
= - [71.54172185 rev. + 0°.945287726] = - [ 7 1 ' ' 1 9 5 ° 1' 12" + 0 ° 5 6 ' 4 3 " ] = - [ 7 r 195° 57'55"] = 164° 0 2 ' 0 5 "
by removing the completed revns. (-71) and adding 360°. Note:
Lalla (c. 786 A.D.) in his celebrated text Sisya-dhi-vrddhida gives a bija (conrection) for the mean longitude of the moon's node. According to Lalla's rule, the bija is at the rate of - 96' for the lapse of every 250 years since 499 A.D. (the year of composition of the Aryabhatiyam). This means that for the year Y A.D. the bija to the mean longitude of Rahu is (- 96') x ( K - 499)/250.
Thus, in the example considered, for March 21, 1997, Bija = - 96' X (1997 - 499)/250 = - 96' x 1498/250
Motions of the Sun a n d the Moon
51
= -575'.232 = - 9 ° 3 5 ' 14'I" Now, as calculated earlier. Mean Rahu
164° 02'05'
Bija (correcdon)
- 9 ° 35'14' I"
Corrected Rahu
154° 26' 51
Desantara correction according to KK (i)
According
to the Khatjda khddyaka, the daily mean motion of the sun,
d = 59'%" Here, X = 77° 35' and
= 75° 47' with reference to Greenwich. Therefore,
Desdntara correciton for the sun : = - (77°35' - 75° 47') x 59* 8 " / 3 6 0 ° = - 17".74 We have, for the preceding midnight of 21-3-1997, Mean sun
334° 16'06'
De'santara correction
:
Corrected mean sun (ii)
- 17".74 334° 15' 4g".26
For the moon, the mean daily motion d= 13° 10'31"= 13°.175278 wcidX-\=
1°.8
.-. Desdntara correction for the moon =
- 1.8 X 13.175278/360 degrees
=
-3'57".15
Mean moon
Note:
118° 14'32"
De'santara correction
- 3 ' 57". 15
Corrected mean moon
118° 10'34". 85
The desdntara corrections for the apogee and the node of the moon are negligible and hence left out.
6.3.3 According to Graha laghavam (GL) (i)
Mean Longitude of the Sun Let Cakra and the Ahargana (A), according to GL be determined for the given date. Then, the mean longitude of the sun is given by A 70
A - Cakra x 1.81972 150x60
+ 349.683 in degrees Here, for the sun, according to the GL Dhruvaka = 1 ° 4 9 ' 11" = 1°.81972 and Ksepaka = 349° 41'= 349°.683 at the mean sunrise of the epoch.
Ancient Indian
Astronomy
Example : For March 21, 1997, we have Cakra C = 43 and Ahargana A = 1525 according to GL. Therefore, the mean longitude of the sun at the mean sunrise on that day at Ujjayini is ' X = 1525 X 1 - J L 70 150x60 - 43 X 1.81972 + 349.683 degrees = 4 rev. 334° 2 8 ' 4 8 ' Removing the completed number of revolutions, the mean longitude of the sun, X = 334° 2 8 ' 4 8 " at the mean sunrise. Note :
The mean daily motion of the sun is 0° 59' 08" according to GL. Accordingly, the mean longitude of the sun at the preceding midnight of 21-3-1997 is 334° 14'01".
(ii)
Mean Longitude of the Moon In the case of the moon, we have Dhruvaka = 3 ° 4 6 ' 1 1 " and Ksepaka = 349° 06' The mean longitude of the moon is given by X
= A x 1 4 - 1 4 x A / 1 7 - A / ( 1 4 0 x 60)
- Ca/:rax 3.76972+ 349.1 in degrees. Example : For 21-3-1997, we have Cakra = 43 and A = 1525 Therefore, X
= 20280°.93804 = 56'^" 120° 56' 17" i.e. 120° 56' 17"
Note :
The moon's mean daily motion, according to GL is 790' 35 ". Therefore,
the moon's
mean longitude at the preceding midnight is 117° 38'38".
(iii)
Mean Longitude of Moon's Apogee (Mandocca) The moon's mandocca is given by M = A / 9 - A / ( 7 0 X 60) - 272.75 x Cakra + 167.55 in degrees.
Example : For Cakra = 43 and A = 1525, we have M = -11391°.61865 = - [ 3 1 ' ^ ^ 231° 37'07"] Ignoring the completed revolutions and then adding 360°, we get M = 128° 2 2 ' 5 3 " . Note :
The mean daily motion of the moon's apogee is 6' 42" so that the mean longitude of the moon's apogee at the preceding midnight is 128° 21' 13".
(iv)
Mean Longitude of Moon's Node (Rdhu) In G L , the mean longitude of the moon's ascending node (Rahu) is given by Rahu = [360 - ( A / 1 9 + A / 4 5 x 60) - (212.83 x cakra) + 27.63] degrees.
Motions of the Sun and the Moon
53
Example : WitK A = 1525 and caitra =43 for 21-3-1997 the mean longitude of Rahu is Rahu
=-9205°.027833 = -[25'*^-205°01'40"]
. Omitting the completed revolutions and adding 360°, Rahu = 154° 58' 2 0 " at the sunrise. Desantara corrections We have seen earlier that due to the difference in the terrestrial longitudes of the given place and Ujjayini there will be a corresponding change in the celestial longitude of each heavenly body. We shall obtain the Desdntara corrections for the sun and the moon according to GL. The desdntara correction is given by - (X - X^) d / 360 where X = longitude of the sun Xg = longitude of Ujjayini d (i)
= mean daily motion of die body.
According to the Graha ldghava, the daily mean motion of the sun ^ = 59' 8" Here, X = 7 7 ° 35' and X^ = 75° 47'. Therefore Desdntara correction for the sun = - ( 7 7 ° 3 5 ' - 7 5 ° 47') X 5 9 ' 8 " 360° = - 1.8° X 5 9 ' . 1 3 / 3 6 0 ° = - 17".74 We have, for March 21, 1997 at mean sunrise at Ujjayini Mean Sun Desdntara cor. Corrected mean sun
(ii)
= = =
334° 28' 48" - 17".74 3340 28' 30".26
For the moon, the mean daily motion d = 13° 10' 35" and X - Xo = - 1.8°. Therefore, Desdntara cor. for the moon = - 1 . 8 X 13°. 176398/360° = - 3 ' 5 7 " . 18 Mean moon Desdntara cor. Corrected mean moon
Note :
120° 56' 17" ~ 3' 57". 18 120° 52' 19".82
The Desdntara corrections in respect of the apogee and the node of the moon are negligibly small.
54
Ancient Indian Astronomy
A comparison of the mean longitudes In the preceding sections, we have computed the mean longitudes of the sun, the moon, the moon's ^ g e e and the node for 21-3-1997. In Table 6.4 the mean longitudes according to the Surya Siddhdnta, the Grahaldghavam and die Khandakhadyaka, are compared with the modem values. Table 6.4: Mean sidereal longitudes for 21.3-1997 (at the preceding midnight) SS
KK
GL
Modem
Ravi
334° 14'09"
334° 16'06"
334° 14'01"
334° 31'48"
Candra
117° 48'25"
118° 14'32"
117° 38'38"
117° 55' 06"
Candra's Mandocca
131° 59'46"
138° 47'55"
128° 21'13'-
126° 16' 13"
Candra's Pdta (Rahu) 158° 56'29"
154° 26'51"
154° 59'08"
155° 03'51"
Note:
The mean longitude of Rihu according to KK is given after applying Lalla's correction. The modem sidereal positions are as per the Ind. Ast. Ephemeris.
7
TRUE POSITIONS OF THE SUN AND THE MOON 7.1 Introduction In obtaining the mean positions of the sun and the moon, it was assumed that these bodies move in circular orbits round the earth with uniform angular velocities. However, by observations it was found that the motions are non-uniform. The procedure for calculating the major corrections to the mean positions, to obtain the true positions, is related to die epicyclic dieory which is explanined in the following section. 7.2 Epicyclic theory and Mandaphala The theory is that while the mean sun or the moon moves along a big circular orbit (dotted in Fig.7.1), the actual (or true) sun or moon moves along another smaller circle, called epicycle, whose centre is on the bigger circle. The bigger circle ABP with the earth E as its centre
is called the kaksavrtta.
Let A be
the
position of the mean sun at a certain time. The line AEP is called the apse line (or nicoccarekhd) and AE is the trijyd
J'
(radius) of this orbit. The
epicycle, with A as centre and a prescribed radius (smaller than A f ) is called the nicoccavrtta. Let the apse line PEA cut the epicycle at U and ^V. The
Fig. 7.1: Epicyclic Theory
two points U and N are respectively called the mandocca (apogee) and the mandanica of the sun. Note that as the sun moves along the epicycle, he is farthest from the earth when he is at II and nearest when at N. The epicyclic theory assumes that as the centre of the epicycle (i.e. mean sun) moves along the circle ABP in the direction of the signs (from west to east) with the velocity of the mean sun, the true sun himself moves along the epicycle with the same velocity but in the opposite direction (from east to west). Further, the time taken by the sun to complete one revolution along the epicycle is the same as that taken by the mean sun (i.e., centre of the epicycle), to complete a revolution along the orbit.
56
Ancient Indian Astronomy
Now, in Fig.7.1, suppose the mean sun moves from A to A\ Let A'E be joined cutting the epicycle at U' and N' which are the current positions of the apogee and the mandanica. While the mean sun is at A', suppose the true sun is at S on the epicycle so that U'A'S = [/'£A..Join ES cutting the orbit (i.e. circle ABP). Then A' is the madhya (mean sun) and S" is spasta (or sphuta) Ravi. The difference between the two positions viz, A'ES" (or arc A T ) is called the equation of centre (or mandaphala). Now, in order to obtain the true position of the sun, it is necesary to get an expression for the equation of centre which will have to be applied to the mean position. In Fig.7.1, SC and A'D are drawn perpendicular to U't^E and CW£ respectively. The arc AA' (or Ajfe4'), the angle between die mean sun and die apogee is called the mean anomaly of the sun (mandakendra). We have, in the right-angled triangle A'DE, sin A£A' = sin D£A' = A'D/A'E so that A'D = RsmAA' (where R = A'E) called mandakendrajyd. From the similar right-angled triangles SCA' and A'DE, we have SC/SA'=A'D/A'E so that SC =
A'DxSA'/A'E
Since SA' and A'E are respectively the radii of the epicycle and the orbit, these are proportional to the circumferences of the two circles; that is, SA'/A'E = circumference of epicycle / circumference of orbit SC = (circumference of epicycle/circumference of orbit) x A'D Taking the circumference of the orbit as 360°, we have 5'C= (circumference of the epicycle) x
Mandakendrajyd/360°.
Now taking SC approximately the same as A'S", we have Equation of centre (Mandaphala) = (circumference of the epicycle)
(mandakendrajyd)/360°
= (r/R) (R sin m) where R sin (m) is the "Indian sine" of the anomaly m of the sun. The maximum value of the equation of centre is r, the radius of the epicyle. By observation this can be obtained as the maximum deviation of the sun's position from the calculated mean position. Note that when the sun is at his apogee or perigee, the mean and true positions coincide since sin (m) is 0 when /n = 0° or 180°. The maximum equation of centre for the sun was observed by Bhaskara II to be 2° 11' 30" (i.e. 131'.5) which is the value of r. Therefore,
True Positions of the Sun and the Moon
57
Circumference of the epicycle of the sun = (131.5/3438) X 360° = 13°.66 This value is given by B h a s k a r a II. Note :
The same epicycle theory is applied to the moon also. In the case of the moon, Bhaskara II has given the maximum equation of centte as 302'. Most texts have taken the epicycles as of varying radii and not fixed.
Table 7.1: Peripheries of Epicycles of Apsis Bodies
Aryabhatiyam
Khanda
Saura siddhanta Surya Siddhanta
Khddyaka
(Vardhamihira)
Ravi
13° 30'
14°
14°
13° 40' to 14°
Candra
31° 30'
31°
31°
31° 40' to 32°
Kuja
63.0° to 81.0°
70°
70°
72° to 75°
Budha
22.5° to 31.5°
28°
28°
28° to 30°
Guru
31.5° to 36.5°
32°
32°
32° to 33°
Sukra
9.0° to 18.0°
14°
14°
11° to 12°
Sani
40.5° to 58.5°
60°
60"
48° to 49°
From Table 7.1 we notice that while the Khanda Khddyaka and the Saurasiddhdnta (as given by Varahamihira) take the epicycles as of constant periphery (and hence radius), Aryabhatiyam and the later Siirya Siddhdnta take them as varying between two limits. 7.3 Mandaphala according to SS for the sun and the nxoon Now, how are these perpheries of die epicycles used to detemiine the equations of centre (mandaphala)'? We will follow die procedure given by die Siirya Siddhdnta in this section. For example, in the case of the sun, the periphery varies from 13° 40' to 14°. Therefore, the
radius
r
varies
from
( 1 3 ° 4 0 ' / 3 6 0 ° ) x 3438' to (14°/360°) x 3438'.
i.e., from
130'.517 to 133'.7. But then we must know how to find the actual value of r, at the given moment,
between die given limits. For this, the Sutya Siddhdnta gives die following
rule : "The degrees of the sun's epicycle of the apsis (manda paridhi) are fourteen,...at the end of the even quadrants; and at die end of die odd quadrants, diey are twenty minutes less." There are four quadrants : the odd quadrant endings are 90° and 270° and the even quadrant endings are 180° and 360° (or 0°). Let m be the mean anomaly (mandakendra) of the sun where m = Mandocca of the sun - Mean longitude of the sun. Since at m = 90° and /n = 270°
the periphery is minimum and at m = 0 ° and
; n = 180° it is maximum (i.e. 14°), we can formulate : Corrected periphery = 14° - (20' x I sin m I)
58
Ancient Indlcm. Astrononiy
assuming that the variation is periodic sinusoidally. Correspondingly, we have the corrected radius of the epicycle of the sun's apsis : r = ( 3 4 3 8 7 3 6 0 ° ) [14° - ( 1 / 3 ) ° I sin m I ] Similarly, in the case of the moon, the periphery of the epicycle varies from 3 1 ° 4 0 ' to 32°. Hence, the corrected radius is give by r = ( 3 4 3 8 7 3 6 0 ° ) [32° - ( 1 / 3 ) ° I sin m I ] where m= Moon's mandocca - Moon's mean longitude. Having found out the corrected radius of the epicycle, the Mandaphala = r sin m so that with the corrected r, we have the following : Sun's Mandaphala = ( 3 4 3 8 7 3 6 0 ° ) [14° - ( 1 / 3 ) ° I sin m I ] sin m = [133.7 sin m - 3.183 (sin m). I sin m I ] minutes of arc Moon's Mandaphala = (3438/360) [32° - ( 1 / 3 ) ° I sin m I ] sin m = [305.6 sin m - 3.183 (sin m) I sin m I ] minutes of arc The Mandaphala is additive for m < 180° and subtractive for m > 180°. Table 7.2: Snies according to Siirya Siddhdnta (/? = 3438 and « ' = 3437.75) SI. No.
Arc (9)
Arc (6) (min)
(Hindu)jyd
R' sin (9) (True) 224.84
RsmXQ)
Difference
1.
3° 45'
225
225
2.
7° 30'
450
449
224
448.72
3.
11° 15'
675
671
222
670.67
4.
15° 00'
900
890
219
889.76
5.
18° 45'
1125
1105
215
1105.03
6.
22° 30'
1350
1315
210
1315.57
7.
26° 15'
1575
1520
205
1520.48
8.
30° 00'
1800
1719
199
1718.88
9.
33° 45'
2025
1910
191
1909.91
10.
37° 30'
2250
2093
183
2092.77
11.
41° 15'
2475
2267
174
2266.67 2430.86
12.
45° 00'
2700
2431
164
13.
48° 45'
2925
2585
154
2584.64
14.
52° 30'
3150
2728
143
2727.35
15.
56° 15'
3375
2859
131
2858.38 (Contd...)
True Positions of the Sun and the Moon SI. No.
Arc (9)
59
Arc (9) (min)
(Hindu) jya
R sin (9)
Difference
/?' sin (9) (True)
16.
60° 00'
3600
2978
119
2977.18
17.
63° 45'
3825
3084
106
3083.22
18.
67° 30'
4050
3177
93
3176.07
19.
71° 15'
4275
3256
79
3255.31
20.
75° 00'
4500
3321
65
3320.61
21.
78° 45'
4725
3372
51
3371.70
22.
82° 30'
4950
3409
37
3408.34
23.
86° 15'
5175
3431
22
3430.39
24.
90° 00'
5400
3438
7
3437.75
Note :
The circumference of a circle, in arc, is 2nR = 360° = 21,600' so that
R = 2\ .600'/27t = 3437'.7468
Aryabhata I has taken the value of A as 3438'.
Example : Find the equations of centre and hence the true longitudes of the sun and the moon at the mean midnight preceding March 21, 1997 at Bangalore. We have already computed in Chapter 6, the mean longitudes after the desdntara correction, for the midnight preceding the given date at Bangalore, and the values are : Mean longitude of the sun
Note :
;
334° \y
51"
Mean longitude of the moon
j j 7° 44' 28"
Moon's Mandocca
131° 59'44"
Sun's Mam/occa
77° 17'40"
According to the Surya Siddhdnta the sun's apogee {mandocca) completes 387 revolutions in a Kalpa of 432 x l o ' years. At that rate of motion the position of the sun's mandocca at the beginning of the Kaliyuga (i.e., February
17/18, 3102 B.C.) works out to be
77° 7' 48". Therefore, for the Kali Ahargana of 18,62,063, corresponding to March 21,
1997, the motion of the suii's mandocca is = [18,62,063/(157.79,17,828 x 10^)] x 387 x 360 x 60' = 9'.8645 = 9'52"
Therefore, for the given date Sun's mandocca = 77° 7' 4 8 " + 9'52" = 77° 17' 4 0 " (0
Sun's equation of centre {mandaphala) and true longitude Sun's mean longitude Sun's Mandocca
334° 13'51" :
77° \T 40"
60
Ancient Indian
Astronomy
Therefore Sun's anomaly {mandakendra) m = Sun's mandocca - sun's mean longitude = 77° 17 4 0 " - 3 3 4 ° 1 3 ' 5 1 " = 103° 0 3 ' 4 9 " (by adding 360°) = 103°.06361 and hence the equation of centre is additive (m < 180°). The rectified periphery = 14° - ( 1 / 3 ) ° I sin m I = 13°.67529343 Sun's equation of centre {Mandaphala) = 133'.7 sin m - 3'. 183 (sin m) I sin m I = (133'.7) (0.9741197) - (3'. 183) (0.948909) = 130'.2398-3'.020378 = 127'.219422 = 2" or 13" Therefore, at the mean local midnight at Bangalore, preceding 21-3-1997, True longitude of the sun = (Mean longitude of the sun + Equation of centre of the sun) = 334° 1 3 ' 5 1 " - 2 ° 0 7 ' 1 3 " = 336° 2 1 ' 0 4 " (ii)
Moon's equation of centre and true longitude Moon's mean longitude
:
Moon's A/a/idocca
117° 44'28" 131° 59'44"
Therefore, we have Moon's anomaly {mandakendra) m = Moon's mandocca - Moon's mean longitude = 131° 5 9 ' 4 4 " - 117° 4 4 ' 2 8 " = 14° 15' 16"= 14°.254 Hence the equation of centre is additive (m < 180°). Corrected periphery of the epicycle = 32° - ( 1 / 3 ) ° I sin w I = 31°.91793 Moon's equation of centre = 305'.6 (sin m) - 3'. 183 (sin m). I sin m I = (305'.6) (0.246228) - (3'. 183) (0.060628) = 75'.24742 - 0'. 19298 = 75'.0544188 = 1° 15'03" Therefore, at the mean local midnight at Bangalore,
True Positions of the Sun and the Moon
61
True longitude of the moon = (Mean longitude of the moon + Equation of centre of the moon) = 117° 4 4 ' 2 8 " + 1° 1 5 ' 0 3 " = 118° 5 9 ' 3 1 " 7.4 Bhujantara
correction
The true midnight of a place differs from the mean midnight by an amount of time called "equation of time". The equation of time is caused by (i) the eccentricity of the earth's orbit ; and (ii) the obliquity of the ecliptic with the celestial equator. The correction to the longitude of a planet due to the part of the equation of time caused by the eccentricity of the earth's orbit is called Bhujantara.
The other correction
caused by the obliquity of the ecliptic is called Udayantara. While all the Siddhdntie
texts consider the Bhujantara
correction, the other
correction—Udayantara—was first introduced by Sripati (about 1025 A . D . ) and later followed by Bhaskara-II and others. We shall discuss the Bhujantara correcdon which is mentioned in the Surya
Siddhanta.
The eccentricity of the earth's orbit results in the equation of the centre of the sun (mandaphala) which is converted into time at the rate of 15° per hour or 6° per ghatikd. This rate of conversion is due to the fact that the earth rotates about its axis at the rate of 360° in 24 hours (or 60 ghatikds.). The resulting amount in time unit is the equation of time caused by the eccentricity of the earth's orbit. Thus, the equation of time (due to the eccentricity) = [ (Equation of centre of the sun)/15 ] hours = [ (Equation of centre of the sun)/6 ] Now, to get the Bhujdntara
ghatikds
correction for the sun Or the moon or any other planet,
the equation of time obtained above must be multiplied by the motion of the planet per hour or per ghatikd as the case may be. That is, Bhujdntara
correction for a planet = [ Equation of time in hours ] x [ (Daily motion)/24 ] = [ (Equation o f centre of the sun)/15 ] x [ (Daily motion)/24 ] = [ (Equation of centre of the sun) ] x (Daily motion)/360
where the factors in the numerator are in degrees and the daily motion is that of the planet. If the time unit used is ghatikd, then Bhujdntara
correction
= (Eqn. of time in ghatikd) x (Daily motion)/60 ] = [ (Eqn. of centre of the sun in degrees)/6 ] x [ (Daily motion of the planet)/60 ] = [ Eqn. of centre of the sun in degrees ] x [ Daily motion of the planet/360 ]
62
Ancient Indian Astronomy
where the daily motion of the planet is in degrees and hence the Bhujdntara is also in degrees.
correction
However, i f the daily motion of the planet is in minutes of arc, then Bhujdntara
correction in degrees
= (Eqn. of centre of the sun in degrees) x (Daily motion of the planet)/2l600 Further, the Bhujdntara of centre of the sun is so.
correction is additive or subtractive according as the equation
For example, in the case of the moon, its mean daily motion is 13°. 176352 or 790'.58112. Therefore, we have Bhujdntara
correction (mean)
= (Eqn. of centre of the sun) x 790'.58112/21600' = Eqn. of centre of the sun/27.321674 Note :
Brahamagupta takes the denominator approximately as 27 in his Khandakhadyaka.
It is important to note that to obtain the actual (and not the .mean) Bhujdntara correction of a planet, we have to use the true daily motion of the planet for the given day. Example : Find the Bhujdntara given that on a certain day
corrections for the longitudes of the sun and the moon
True daily motion of the sun True daily motion of the moon
59*.65 :
Equation of centre of the sun
855'.23 + 2°7'32"= 127'.53
Therefore, we have (i)
True Bhujdntara
correction of the sun
= (Eqn. of centre of the sun) x (Daily motion of the sun)/216(X) = 127'.53 X 59'.65/21600' = 0'.3521835 = 0' 2 1 " Since the equation of centre of the sun is additive, the Bhujdntara additive. (ii)
True Bhujdntara
correction is also
correction of the moon
= (Eqn. of centre of the sun ) x (Daily motion of the moon)/21600 = 127'.53 X 855'.23/21600' = 5'.0494205 = 5' 3" Here also, the correction is additive since the sun's equation of centre is so. 7.5 Further corrections for the moon We have applied so far an important correction namely, the equation of centre, to the mean position of the moon. Besides this correction, the other two corrections applied viz, Desantara
and Bhujdntara
arc mainly to get the true position of the moon at the
true local midnight at the place of observation.
True Positions of the Sun and the Moon
63
However, to get the true apparent position of the moon at least two more important corrections will have to be applied, of course, ignoring other minor corrections due to planetary perturbations. (i)
£v£crio« = ( 1 5 / 4 ) ' m e sin ( 2 ^ - 0 ) = 7 6 ' 2 6 " s i n
(24-(|»)
where m is the ratio of mean daily motions of the sun and the moon, e is the eccentricity of the moon's orbit, ^ = ( M - S), the elongation of the moon from the sun and ^ = M-P, the mean anomaly of the moon (P being the moon's perigee). (ii)
Variation = 39' 30" sin (2^) In the above formulae, S and M are respectively the mean longitudes of the sun and the moon. The Surya Siddhanta, being an earlier text, does not mention these corrections. However, Manjula (932 A.D.), Bhaskara-II (1150 A . D . ) and later Indian astronomers have recognized the evection correcdon in addition to the equation of centre.
Besides these, the famous
Orissa
astronomer Samanta Candrasekhara Simha discovered independently a fourth correction called annual equation. According to Candrasekhara, Annual equation = (11' 27".6) sin (Sun's distance from apogee) In fact, Candrasekhara's coefficient viz., \\'21".6 is very close to the known modem value. Tycho Brahe took the coefficient wrongly as 4' 30". 7.6 True longitudes of the sun and the moon according to KK The method of Khandakhadyaka
is demonstrated
for the example of midnight
preceding 21-3-1997. 1.
Sun's true longitude
(i)
The sun's mandocca
= 80°
(fixed according to Brahmagupta) Mandakendra (A/)
= Mean sun - sun's mandocca = 334° 1 6 ' 0 6 " - 8 0 °
M (ii)
= 254° 16'06"
Mandaphala = -—• sin M A
is the equation of centre in radian measure where b=]4° M = Mandakendra (i.e. anomaly)
and R = 360° and
14° i.e. Mandaphala =
sin M
(in radian measure)
360° Therefore, Mandaphala
= - s i n A/
(in degrees, multiplying by 180°/n)
= - - s i n (254° 16'06") 71 = 2° 8 ' 4 1 "
64
Ancient Indian Astronomy True longitude of the sun = Mean sun + Mandaphala = 334°16'06" +
2° 0 8 ' 4 1 "
i.e.. True longitude of the sun = 336°24'47" (Compare this with the Surya Siddhanta 2.
value : 336° 21'04")
Moon's true longitude (i)
The moon's mandocca (apogee) = 138° 47' 55" Mandakendra
(anomaly) M
= Mean moon - Moon's apogee = 118° 1 4 ' 3 2 " - 138° 47' 55" = 339° 2 6 ' 3 7 "
(ii)
Mandaphala
(equation of centre)= — sin M
(in radian measure)
A
where a = 31°, /? = 360° and M -
Mandakendra.
-31° .'. Mandaphala =
x sin M (radian measure) -31° 180 . ,^ . = X sin M (in degrees) 360° n V s ; = - ^ sin Af in degrees 271
I.e. Mandaphala
=1°43'57"
Now, the true longitude of the moon = Mean moon + mandaphala = 119° 58' 29" 7.7 IVue longitudes of the sun and the moon according to GL Ganesa Daivajiia's algorithm for the sun's mandaphala
as in his GL is described
below : (1)
The sun's mandocca = 7 8 ° (taken as fixed) Mandakendra,
MK = Mandocca—Mean
Sun.
Now, the Bhuja of MK is determined as follows : (i)
If MK is negative, then the effective MK is obtained by adding 360°.
(ii)
If 0 <
< 90°, then Bhuja
= MK
(iii)
If 90° <
(vi)
If 180°
(v)
< 180°, then Bhuja
= 180° - MK 80°
If 270°
rue Positions of the Sun and the Moon
65
Let XX = Bhuja/9 Numerator = XX (20 - XX) and Denominator = 57 - ^ (Numerator) Then, Mandaphala (MP) is given by MP = Numerator / Denominator Note :
Ganesa Daivajna totally avoids trigonometric functions; still the results thus obtained are reasonably good though not very accurate.
Ixample : For the midnight preceding 21-3-1997, we have the mean sun: 334° 14' 1". 1.
True longitude of the sun Mandakendra, MK = Mandocca - Mean sun = 7 8 ° - 3 3 4 ° 14' 1" i.e., MK= 103° 45' 59" Since MK lies between 90° and 180°, Bhuja : 180° - M/c: = 180° - 103° 45' 59" le. Bhuja
=76°14'1"
Let XX = Bhuja/9 = 8° 28' 13" = 8.4702778 Numerator
=XX{1Q-XX) = 8.4702778 (20 - 8.4702778) = 97.658889
, , , Mandaphala
Numerator =— ; Denominator
where Denominator = 5 7 - 1 / 9 (Numerator) = 46.149 97.658889 = 2° 6'58". 19 = 46.149
i.e. Mandaphala .•. True longitude of the sun
= Mean longitude + Mandaphala = 334° 14' l " + 2 ° 6 ' 5 8 " = 336° 20' 59".
2.
True longitude of the moon For the example considered we have Mandakendra
MK = Mandocca - Mean moon = 128° 2 1 ' 1 3 " - 1 1 7 ° 3 8 ' 3 8 " = 10° 42' 35"
Let
YY=Bhuja/6
Numerator
=
(30-YY)/YY
and Denominator = 56 - (Numerator)/20 Mandaphala {MP) =
Numerator Denominator
66
Ancient Indian Astronomy
Since MK lies between 0° and 90°, Bhuja = 10° 42' 35" 6 Numerator iDenominator Therefore,
=
(30 - IT) KK
= 50.3625
=
56-^^=^=53.481667
Mandaphala {MP) = 0° 56' 30"
True longitude of the moon = Mean longitude + Mandaphala = 117° 3 8 ' 3 8 " + 0° 56' 30" = 118° 35'08".
8
T R U E DAILY MOTIONS O F T H E SUN AND T H E MOON
8.1 According to SS The mean daily motions of the sun and the moon, as given by the Surya Siddhanta are respectively 59'. 1361592 and 790'.5811287. Due to their non-uniform motion the true daily motions of the sun and of the moon go on changing from day to day. The method to find the true daily motion from the mean motion is given in the Surya Siddhanta as follows : True daily motion = n±{n-n').P°x
[sine difference at (m - a)] / (360° x 225')
where P ° is the periphery of the epicycle of the sun (or the moon), in is the mean longitude and a is that of the apogee (mandocca) of the sun (or the moon) and "sine difference" is the tabulated difference in the sine table corresponding to the mean anomaly, namely (m - a) from Table 7.2; n and n are the mean daily motions of the sun (or the moon) and the corresponding apogee. Rationale : Suppose L and L' are the true longitudes of the sun (or the moon) on two consecutive days. Then, we have L
= m±P°x
[sin (m - a)] 7360° and
L ' = (m + n ) ± ( P ° / 3 6 0 ° ) s i n {[(m + / i ) - ( a + «')]} so that True daily motion = L'-L
= n± ( P ° / 3 6 0 ° ) [sin { [ ( m - a ) + (n-n')]-
= n±[P°x(n-
n')/360°
sin (m - a)]
x 225'] x Tab. diff of sines at (m - a)
In the case of the sun, the daily motion of its mandocca (apogee) n is negligible so that True daily motion of the sun = « ± P ° X M X [tab. diff. of sines at (m - a ) ] / ( 3 6 0 ° x 225') Note :
The correction to the mean daily motion n is additive if the anomaly (of the sun or the moon) is between 90° and 270°(i.e. 2nd and 3rd quadrants) and subtractive if the anomaly is between 270° and 90° (i.e. in 1st and 4th quadrants).
68
Ancient Indian Astronomy
Example (i)
Find the true daily motions of the sun and the moon on March 22, 1991. True daily motion of the sun We shall use the following values in respect of the sun as on March 22, 1991 at the preceding midnight : Sun's mean daily motion
n = 59' 8"
Sun's mean longitude
334° 51'30"
Sun's apogee (mandocca)
77° 17' 39"
Sun's mean anomaly
102° 26' 09"
Tabular difference of sines
51'
(corresponding to 180° - 102° 26' 09" = 77° 33' 5 1 " - see Table 7.2, between serial numbers 20 and 21). Corrected periphery of the sun's epicycle : P = 13°.6745 Therefore, the correction to the sun's daily motion = P X « X Tab. sin diff./(360 x 225) = 13.6745 X 59.13 X 51/(360 X 225) = 0'.5091 = 0 ' 3 1 " Since the sun's anomaly 102° 26' 09" lies between 90° and 270° (i.e. 2nd & 3rd quadrants), the coi-rection is additive. Hence Sun's true daily motion = 5 9 ' 0 8 " + 0 ' 3 1 " = 59' 39" (ii)
True daily motion of the moon In the case of the moon the following are the values for March 22, 1991 at the preceding midnight :
Moon's mean daily motion
n = 790'35"
Mean daily motion of the moon's apogee
«' = 6'.683
Moon's mean longitude
47° 44'48"
Moon's apogee (mandocca)
247° 57' 34"
Moon's mean anomaly
200° 12' 46"
Tabular sine diff. for (200° 12' 46" - 180°)
210' (SI. Nos. 5 & 6 in Table 7.2)
Moon's corrected periphery of the epicycle
:
p = 31°.8848
Therefore, Correction to the moon's daily motion = (n - «') X /> X Tab. sin diff.7(360° x 225') = (790'.583 - 6'.683) x 31°.8848 x 2 1 0 ' / ( 3 6 0 ° x 225') = 64'.8 = 6 4 ' 4 8 " Since the moon's anomaly 200° 12'46" lies between 90° and 270°, the correction is additive. Hence
True Daily Motions of the Sun and the Moon
69
Moon's true daily motion = 7 9 0 ' 3 5 " + 6 4 ' 4 8 " = 8 5 5 ' 2 3 " = 14° 15' 2 3 " 8.2 IVue daily motions of the sun and the moon according to KK (i)
IVue daily motion of the sun
To fmd the true daily motion of the sun the method of the Khandakhadyaka
can be
explained as follows. Let n be the mean daily motion and An be the correction. True daily motion of the sun =n + An Here, n = 59'.133 (according to KK) is the mean daily motion of the sun: An - -^cos M R at of daily motion.
where b = 14°, R = 360°, M = mandakendra and
Ar
is the rate
Example : True daily motion of the sun on April 29, 1995 Ahargana. A = 485806 (according to KK) Mean longitude of the sun
=
12° 13'52"
Mandakendra
=
292° 13' 52"
Mandaphala
=
2° 3' 45"
True longitude of the sun
=
14° 17' 37"
True daily motion of the sun
=
n + An 59'.133 + An
and
An = — cos (292° 13' 52") x 59'. 133 360° = -0'.8306.
Therefore, true daily motion of the sun = 59'.133 -0'.8306 = 58'.3024 = 58' 18" (U)
IVue daily motion of the moon True daily motion of the moon is obtained by using the formula (n + An) where n is the mean daily motion and An is given by the formula An = — cos m R
AM A (Mandocca) At ~ At
where b = 3\°,R = 36O°,m = Mandakendra,AM/At
= 190'34",
mean daily motion and ^(^^^f"""""^ = 6' 40" per day according to KK. Example : True daily motion of the moon on April 29, 1995. We have Ahargaria A = 485806 Mean longitude of moon
=
0° 12' 23"
Mandakendra, m
=
298° 29' 16"
the
moon's
Ancient Indian Astronomy
70 Mandaphala
=
0°4'32"
True longitude of the moon
=
True daily motion of die moon
=
0° 16' 55" n+An 790'34" + A n
where A n =
-31°
cos m
360°
AM At
A {Mandocca) At
= - 31 [cos (298° 29'16")] (790' 34" - 6' 40")/360 = -30'44" Therefore, the moon's true daily motion : n + An = 790' 34" - 30'44" = 759' 50" i.e. 12° 3 9 ' 5 0 " 8.3 TVue daily motions of the sun and the moon according to GL The algorithm for the true daily motion of the sun according to GL is described below. (1)
Find the sun's mandakendra {MK). (i) (ii)
If 0 ° < M ^ : ^ 9 0 ° , then Koti =
90°-MK
If 9 0 ° < m a : ^ 180°, then Koti =
MK-90°
(iii)
If 180°
(vi)
If 2 7 0 ° < M K < 3 6 0 ° , then Koti =
MK-270°
Thus, the corresponding Koti is determined. In fact, Koti = 90° - Bhuja (2)
Divide Koti by 20 to get degrees etc.
(3)
Subtract item (2) from 11°
(4)
Multiply item (3) by item (2)
(5)
Divide item (4) by 13 to get in minutes etc. of arc.
(6)
If 90°
(7)
If 270°
Example : The sun's Mam/aifeem/ra (anomaly) is I'' 13° 46' 18". Find the true daily motion of the sun. Following the procedure described above, we have (1)
MK = 43° 46' 18". Since 0°
(2)
(46° 13'42")/20 = 2° 1 8 ' 4 1 "
(3)
1 1 ° - ( 2 ° 18'41") = 8° 4 1 ' 1 9 "
90°, we get Koti = 90° - (43° 46'18")
True Daily Motions of the Sun and the Moon
71
(4)
8° 4 1 ' 1 9 " X 2° 18'41"= 8.68861 X 2.3118 = 20.08275918
(5)
20.08275918/13= 1 '.544827629 = 1' 32".69
(6)
Since 0°
TVue daily motion of the moon (1)
Find the moon's Mandakendra
(2)
Koti =90°-
Bhuja {MK Bhuja).
Bhuja.
(3)
Divide item (2) by 20 to get liptis (minutes of arc).
(4)
Subtract item (3) from 11 liptis.
(5)
Multiply item (4) by item (3)
(6)
Multiply item (5) by 2.
(7)
Divide item (6) by 6 and add it to (6) to get gatiphala.
(8)
If MK > 90° and MK < 210° (i.e. II and III quadrants), add the gatiphala to the moon's mean daily motion viz., 790' 35". If 270°
Example : The moon's mandakendra is 2" 7° 43' 34". Find the true daily motion of the moon. (1)
The moon's mandakendra bhuja = 67° 43' 34".
(2)
Koti = 90° - 67° 43' 34" = 22° 16' 26".
(3)
{Kotil20) liptis =V 6".%.
(4)
l l ' - r 6 " . 8 = 9'53".2.
(5)
(9.8863056) X 1.1 = 10'52".
(6)
1 0 ' 5 2 " X 2 = 21'44".
(7)
2 1 ' 4 4 " / 6 = 3'37".
(8)
2 1 ' 44" + 3' 37" = 25' 2 1 " is gatiphala.
(9)
Since 0 < MK < 90°, (8) is to be subtracted from the mean daily motion of the moon viz., 790' 35". Therefore, the true daily motion of the moon is 790' 35" - 25' 2 1 " = 765' 14".
8.4 Instant of conjunction of the sun and the moon The instant of conjunction on a newmoon day is necessary for computing a solar eclipse. r • • Instant of conjunction ^
Sun's longitude - Moon's longitude , = -r-rt -. °—j : x 24 hours (Difference of daily motions of sun and moon)
Example : To calculate the instant of conjunction of the sun and the moon on April 29, 1995: At the midnight preceding April 29, 1995
72
Ancient Indian Astronomy True tongitude of the sun = True longitude of the moon = Difference =
14° 13'37" 0° 16' 55" 13° 56'42"
Instant of conjunction
836'42" 836' 42" (759'50"-58'18")^
=
28*'37"" 26* 4*'37"" 26'a.m. (April 30) i.e. the instant of conjunction of the true sun and the true iftoon takes place at 4** 37"" 26' in the morning of April 30, 1995. 8.5 Instant of opposition of the sun and the moon In the calculations of a lunar eclipse, we require the instant of the full-moon. That instant in hours is given by , . . r Instant of opposition = 24 x
(Sun's longitude + 1 8 0 ° ) - M o o n ' s longitude ^ , :—-^^7-—; —9, :—r (Moon s true daily motion) - (Sun s true daily motion)
Example : To calculate the instant of opposition on March 3, 1988. At the midnight preceding 3-3-1988 we have True longitude of the sun
=
3 j go 4 3 ' 5 "
True longitude of the moon
=
j 30'^ 2 4 ' 32"
True daily motion of the sun
=
60' 23"
True daily, motion of the moon
=
722' 15"
Instant of opposition (hrs.) _
(Sun's longitude + 180°) - Moon's longitude (Moon's true daily motion) - (Sun's true daily motion)
(318° 43' 5"'+ 180°) - 130° 24' 32" 722' 1 5 " - 6 0 ' 2 3 "
= iflf
x24h
=
18h4'"42*.3
Thus, the instant of opposition of the sun and the moon (i.e. full-moon) is at IS'M"' 4213. Note :
The above procedures give a fairly correct instant of newmoon or fullmoon. However, the exact instants can be worked out by applying the procedures iteratively.
9
LUNAR ECLIPSE
9.1 Introduction In ancient and medieval Indian astronomical texts great importance is given to the phenomenon and computations of eclipses. Their theories and computations in respect of positions of the heavenly bodies—especially the sun and the moon—were put to test on the occasions of eclipses (Grahana or Uparaga). As and when disagreements occurred between the observed and the computed positions, the great savants of Indian astronomy revised their parameters and, if necessary, even the computational techniques. Improving the computations of eclipses—based on observations over long periods of time—was an important target of the
Siddhdtic
astronomers. The scholarly Kerala astronomer, Nilakantha Somaydji (1444-1545), paying glowing tributes to his paramaguru, Paramesvara (1360-1455), remarks "Paramesvara ... having observed and carefully examined eclipses and conjunctions for 55 years composed his Samadrgganitam ". 9.2 Indian astronomers on eclipses The real scientific causes of the lunar and solar eclipses were well known to the ancient Indian astronomers even in the pre-Aryabhatan period. The procedures explained in the then extant Panca Siddhantas bear this out. Aryabhata (476 A.D.) explains the causes of the two types of eclipses briefly thus : Chddayati sail suryam sasinam mahati ca
bhucchdyd
"The moon covers the sun and the great shadow (of the earth) eclipses the moon." — Aryahbhatiyam 4,37. Varahamibira (c. 505 A.D.) explains at length, in his Brhat Samhitd, the real causes of the eclipses and demolishes the irrational myths entertained by the ignorant. He says: Bhucchdydm
svagrahatie bhdskaram arkagrahe pravisati
induh
"At a lunar eclipse the moon enters the shadow of the earth and at a solar eclipse the moon enters the sun's disc...". — Br. Sam. 5,8.
74
Ancient Indian Astronomy
Varahamihira gives all credit to the ancients for the knowledge of the causes of eclipses in saying : Evamuparaga Rdhurakdranam
kdranamuktamidam asminniyuktah
divyadrgbhirdcaryaHf sdstra sadbhdvah II
I — Br Sam. 5, 13.
"In this manner, the ancient seers endowed with divine insight have explained the causes of eclipses. Hence the scientific fact is that the demon Rahu is not at all the cause of eclipses". 9.3 Cause of lunar eclipse On a full moon day the sun and the moon are on opposite sides of the earth. The sun's rays fall on one side of the earth, facing the sun, and a shadow will be cast on the other side. When the moon enters the shadow of the earth, a lunar eclipse occurs. This happens when the sun and the moon are in opposition i.e. the difference between the celestial longitudes of the sun and the moon is 180°. However, a lunar eclipse does not occur on every full-moon day. This is so because the plane of the moon's orbit is inclined at about 5° with the ecliptic. If the moon's orbit were in the plane of the ecliptic, then there would have been a lunar eclipse every full-moon day. Generally on a full-moon day the moon will be either far above or far below the plane of the ecliptic and hence fails to pass through the shadow of the earth. But, on that full-moon day, when the moon does pass through the earth's shadow, a lunar eclipse occurs. In order that an eclipse of the moon may take place, the moon must come sufficiently close to the ecliptic. This means that the moon, on the full-moon / day, must be close to one of the nodes of the moon. In Fig 9.1 the orbit of the moon intersects with the ecliptic at two points N and A^'. These two points are referred to as the ascending and the descending nodes of the moon. They are called Rahu and Ketu in Indian astronomy.
N
Ecliptic Moon's t Orbit Fig. 9.1: Nodes of the moon
The lunar eclipse is said to be total when the whole of the moon passes through the shadow. The eclipse is partial when only a part of the moon enters the shadow. In Fig 9.2, S and E represent the centres of the sun and the earth respectively. Draw a pair of direct tangents AB and CD to the surfaces of the sun and the earth, meeting SE produced in V. A transverse pair of tangents AD &nd BC meeting SE in X. If these lines arc imagined to revolve round SE as axis, they will generate cones. There is thus a conical shadow BVD, with V as its vertex, into which no direct ray from the sun can enter. This conical space is called the umbra.
Lunar Eclipse
75
N Fig 9.2: Earth's shadow cone and the lunar eclipse The spaces, around the umbra, represented by VBL and VDN form what is called the penumbra from which only a part of the sun's light is excluded. It is to be noted that the passage of the moon through the penumbra does not give rise to any eclipse. It results only in diminution of the moon's brightness. When the moon is at A/, (see Fig. 9.2) it receives light from portions of the sun next to A; but rays from parts near C will not reach the moon at My Therefore, the brightness is diminished, the diminution being greater as the moon approaches the edge of the umbra. A n eclipse is considered as just commencing when the moon enters the umbra or the shadow cone. 9.4 Angular diameter of the shadow cone In Fig 9.3, the angular diameter of the cross-section of the shadow cone is represented by arc MN. Let the semi-angle MEV subtended by MN at the centre of the earth be a. We have p = the sun's horizontal parallax = EAX p
= the moon's horizontal parallax = EkB
s
= the sun's angular semi-diameter = S E A
9
= semi-vertical angle of the shadow-cone = E^B
Now, in triangle MEV, we have a + 8 = a = /7'-e
= E'kB
so that ...(1)
Ancient Indian Astronomy
76 Similarly, we have from triangle AEV, Q = s-p From (1) and (2) we get a=p'-is-p) or
...(2)
a=p+p'-s
...(3)
Since p,p' and s are known, the angular semi-diameter a of the shadow cone is determined using (3). However, it is found that as the earth's atmosphere increases, due to absorption, the calculated radius of the shadow cone increases by about 2 per cent. Therefore, for the prediction of lunar eclipses, the following expression is used : a = i5\/50) (p+p'-s)
...(4)
As an example, using the mean values the moon's hor. parallax, p' = 57' the sun's hor. parallax,
p = 8" and
the sun's semi-diameter,
s = 16'
The angular semi-diameter of the shadow cone, from (4) is given by a = (51/50) [8"+ 5 7 ' - 1 6 ' ] or
a = 41'.956 = 42'
To be more accurate, it is found that a varies from a minimum of 3 7 ' 4 9 " to a maximum of 44' 37". In fact, the maximum value of a is reached when the moon is nearest to the earth (i.e. at perigee) and the earth itself being at the same time farthest from the sun (at the apogee of the sun) i.e., when p' is maximum and s minimum value being attained when the conditions are reversed i.e. when the moon is farthest from the earth (i.e. at apogee) and the earth itself, at the same time, being closest to the sun i.e., when p is minimum and 5 maximum. 9.5 Ecliptic limits for the lunar eclipse We noted earlier that the possibility of an eclipse on a full-moon day is restricted due to the inclination of the moon's orbit to the plane of the ecliptic. In F i g . 9.4, NM represents the moon's orbit; M and C are the centres of the moon and the shadow respectively when the eclipse is about to take place. Let Ci be the position of the shadow's centre when the moon is at the node N. Let A^C, be denoted by ^ and t be the time required by the moon to go from N to M and for the shadow's centre to go from C , to C .
Fig. 9.4: Ecliptic limits
Lunar
Ecl^se
77
The geocentric longitude of C is the sun's longitude plus 180°. Hence, in finding the maximum value of ^ for which an eclipse is possible, we are actually finding the maximum angular distance of the sun from the other node. Suppose the sun's longitude increases at the rate of 6 and (j) be the angular velocity of the moon in its orbit. For simplicity, let us take 6 and (j) as constants. Then, we have NM = ^t and NC = ^ + ^t If r| is the angular distance C M and / the inclination M^C, we have from triangle M A ^ C (regarded as a plane triangle)
C M 2 = NC^ + NM'^ - INC.
NM
cos i
i.e.,
Ti2 = (4 + 6r)2 + (
or
Ti^ = ^2 _ 2^ ^
cosi - 6) + f2 (6^ + (j>2 _ 2^
cos i]
Now, r\ is minimum when t is given by
4 ((j) cos / - 6) - r [6^ +
- 26
cos/] = 0
In fact, we have Tlmin = Tlo = 4
...(1)
Let 9 = 6/^ so that (1) becomes 4 = Tio[l -2
...(2)
In (2), 9 is the ratio of the earth's orbital angular velocity to that of the moon which is the same as the ratio of the moon's sidereal period to the duration of a year. Taking the mean values, we have q -
3/40
Also, since / - 5 ° . 1 5 , we get from (2) 4
-I0.3T1O
...(3)
When the moon is about to enter the umbra shadow cone, we have = a + i' where a is the angular semi-diameter of the shadow cone and s' is the moon's angular semi-diameter. We have from (4) of the previous section a
= {5\/50)
(p+p'-s)
For a partial lunar eclipse to be possible, it is evident that
For a total lunar eclipse, 4
S10.3(a-j')
Example : Taking the following values s = the sun's semi-diameter
= 16' = 960"
. j ' = the moon's semi-diameter
=
1 5 ' 3 5 " = 935"
p = hor. parallax of the sun
=
8"
p' = hor. parallax of the moon
= 3422"
78
Ancient Indian Astronomy
we have (i) The condition for a partial eclipse of the moon to be possible : 4 < 10.3 [ (5 l / 5 0 ) ( p + p ' + = 10.3 [(51 /50) (8" + 3422" - 960") + 935"] = 9°.8863 or (ii)
4 < 9°.9. The condition for a total lunar eclipse : 4 < 10.3 [ (51/50) (p + p'-s)-s'] = 10.3 [ (51/50) (8" + 3422" - 960") - 935"] or 4 < 10.3 [(51/50) (2471")-935"] = 4°.5361 or
4
< 4°.6
These values of ^ are called the ecliptic limits for the occurrence of a lunar eclipse. However, since the quantities used in the above derivation are only mean, considering the actual variations, it is found that for a partial eclipse the maximum value of ^ is 12°.l and the minimum value of ^ is 9°.l which are respectively called the superior and inferior ecliptic limits. 9.6 Half durations of eclipse and of maximum obscuration The next important step is to determine the instants of the beginning and the end of a lunar eclipse as also of the maximum obscuration. For this, we need to fmd the durations of the first half and the second half of the total duration of the eclipse. This is explained in F i g . 9.5.
Fig. 9.5: Half durations of lunar eclipse A half duration is the time taken by the moon, relative to the sun, so that the point A in the figure moves through OA. We have OA^
= O X ? - A X f = ( 0 £ + £X,)2-AXf
Lunar Ek:lipse
79
where OE = di = semi-diameter of the shadow EXj
= d2 = semi-diameter of the moon
Pi
= AX I = latitude of the moon (Viksepa)
when its centre is at Xy Half duration =
^id,+d,)^-^}
(Moon's daily motion - Sun's daily motion)
Since the actual moment of the beginning of the eclipse, and hence the moon's latitude then, are not known the above formula is used iteratively. By a similar analysis, the half-duration of maximum obscuration (or totality as the case may be) is given by : Half duration of max. obsn. =
~
^
where DM and DS are the daily motions of the moon and the sun respectively. The thus obtained half durations of the eclipse and the maximum obscuration are (1) subtracted from the instant of the opposition to get the beginning moments; and (2) added to the instant of the opposition to obtain the ending moments. Finally, the magnitude (pramanam) of the eclipse is given by ^
f H -
Amount of obscuration (Grasa) Angular diameter of the moon
Obviously, if the magnitude is greater than or equal to 1, then the eclipse is total; otherwise, it is partial. It is also clear, from F i g . 9.5, that i f the sum of the angular semi-diameters of the moon and the shadow is less than the latitude of the moon, there w i l l be no eclipse. 9.7 Lunar eclipse according to 55 The procedure of computaion of a lunar eclipse is described in the Siirya Siddhanta in Chapter 4 (Candragrahanam) of die text. This is demonstrated in the following example. The parameters required for the computaion of a lunar eclipse are : (i) True longitudes of the sun, the moon and the moon's node (Rahu), (ii) The true daily motions of these three bodies, (iii) The latitude of the moon and (iv) The angular diameters of the earth's shadow (Bhucchdyd)
and of the moon.
80
Ancient Indian Astronomy
Example : Lunar eclipse on September 27, 1996. Since the longitude of the moon's node (Rahu) according to SS is not very accurate, we use the true longitudes of the sun, moon etc., from the Ind. Ast. Eph. as at 5-30 a.m. (1ST). However the prodecure of SS is adopted.
(i)
True longitude of the sun
=
160° 21' 0 1 "
True longitude of the moon
= 338° 44' 27"
Trur longitude of Rahu
=
True daily motion of the sun
= 58' 5 1 "
True half daily motion of the moon
= 432' 32".7
True daily motion of the moon
= 861'
Node's daily motion
=
-3'ir'
Instant of opposition
=
S'' 24*" (I.S.T.)
164° 10' 14"
To find the sun's diameter c . .A A. Sun s corrected diameter =
5 8 ' 5 1 " X 650O>' ,^o-y,. , = 6487^13 5o JO
43,20,000 X 6487>.13 _ 57,753.336 and
4 8 ^
^
= 32'20"
where 58' 58" is the sun's mean daily motion and 57,753,336 is the number of moon's revolutions in a Mahayuga and 6500 yojanas is the sun's mean diameter. Here y stands for yojana, a distance unit. (ii)
To find the moon's diameter r . . . . 861' X 480^ ^^^y^ Moon s corrected diameter = , „ = 524-*^.2 7oo 25 and = 34' 56"
X4
where 480^ is the moon's mean diameter and 788' 25" is the moon's mean daily motion. (iii)
True longitudes of sun, moon and node at the opposition Motion of the sun in 2^54"'
= 3 B 31 ^xz 3 ^
^^,^„
: . Sun's true longitude at opposition = 160° 2 1 ' 0 1 " + 7 ' 7 " = 160° 2 8 ' 0 8 " Motion of the moon in 2^ 54*" = 2 ' ' 5 4 ' " x 4 3 2 ' 3 2 " . 7 / 12'' = 104'32"= 1° 4 4 ' 3 2 " .•. True longitude of the moon at the opposition
Lunar Ek:Hpse
81 ^
3 3 8 ° 4 4 ' 2 7 " + l < ' 4 4 ' 3 21" '
= 340° 28' 5 9 " Node's longitude at opposition : = -23 .-. Node's true longitude = 164° 10' 14" - 23'1" = 164° 9 ' 5 1 " (iv)
Diameter of the earth's shadow Earth's corrected diameter
=
True daily motion of moon x 1600^ Mean daily motion of moon
788'25" Sun's corrected diameter - earth's diameter 6487>'.13-1600>' = 4887>'.13
Diameter of the earth's shadow = Earth's corrected diameter - 360^.9 = 1747>'.2-360>'.9 = 1386>'.3 .-. Earth's shadow diameter in arc = 1386.3/15 = 9 2 ' 2 5 " (v)
The moon's latitude at the middle of the eclipse and the amount of greatest obscuration : (Longitude of Moon - Longitude of Node) at the oppositiion = 340° 28' 5 9 " - 164° 09' 5 1 " = 176° 1 9 ' 0 8 " Bhuja of the above difference s 3 ° 4 1 ' .-. Jya (3° 41') = 221' Moon's latitude at the instant of opposition 2 7 0 ' X 221' 3438'
= 17'21
As determined earlier Semi-diameter of the eclipsed body (moon)
=
Semi-diameter of the eclipsing body (earth's shadow)
= 46' 12'
17' 28'
Their sum
=
63'40'
De^^wcf moon's latitude
=
17'21'
.". Obscured portion (grasa)
= 46' 19"
Since grasa is greater than the moon's diameter the lunar eclipse is total.
82
Ancient Indian Astronomy (vi)
The durations of the eclipse and of total obscuration
Diameter of the eclipsing body (shadow)
92'25"
92' 25"
Diameter of the eclipsed body (moon)
34'56"
34'56"
Their sum and difference
127'21"
57' 29"
Half sum and half difference
63'40"
28'44"
Squares of die above
4053 301
825
Deduct t!.e square of the moon's latitude Their square roots (a)
3752
301 524
61' 15"
22' 53"
Half duration of the eclipse 61'15" X 6 0 " (Daily motion of moon - Daily motion of sun) 61'15"X60" 802'9"
Note;
= 4" 34''
The superscripts n and v denote nadis and vinadis (b)
Half duration of totality 22' 53'
X
60"
802' 9"
=
1"42''
To get a more accurate value of the moon's latitude more iterations are to be carried out. Moon's motion in 4" 34" =
4" My y Rfil' = 1° 5' 3 1 " 60" Node's motion in 4" 34" =
4" 34"
X
3' 11'
= 14"
60" Moon's longitude at opposition
340° 28' 59"
340° 28' 59"
Add and subtract moon's motion in 4" 34"
(+) 1 ° 5 ' 3 1 "
(-) 1 ° 5 ' 3 1 "
341° 34' 30"
339° 23' 28"
164° 9 ' 5 1 "
164° 9 ' 5 1 "
(-) 14"
(+) 14"
164° 9' 37"
164° 10'05"
177° 24' 53"
175° 13' 23"
Node's longitude at opposition Subtract and add node's motion in 4" 34"
Long., of Moon - Long, of Node Jya values of the above Moon's latitude at the end and
158'
287'
12' 24"
22' 32"
4053
4053
153
507
at the beginning of the eclipse Square of half the sum of the diameters Deduct the squares of latitudes Their square roots
3900
3546
62' 26"
59' 32"
Lunar Eclipse
83
Corrected second half-duration
=
=4" 40"
802' 9"
60" X 5 9 ' 3 2 " = 4" 27" 802' 9" Apparent instants of beginning and end of totality
Corrected first half-duration (vii)
Motion of die moon in l ' ' 4 2 "
l"42"x861'
= 24' 23"
60" Node's motion in l " 4 2 "
l"42"x3'10"
5"
60" Moon's longitude at opposition
340° 28' 59"
340° 28' 59"
Add and subtract moon's motion
(+)0°24'23"
(-) 0° 24' 23"
in 1"42" Moon's long, at the end and
340° 53' 22"
340° 04' 36"
at the beginning of totality Node's longitude at opposition
164° 9 ' 5 1 "
164° 9 ' 5 1 "
Subtract and add node's motion in
(-)5"
(+) 5"
1"42" Node's long, at the end and at die beginning of totality Moon's distance from die node
164° 9'46"
164° 9' 56"
176° 42'
175° 53'
Jya values
197'
246'
Moon's latitude at die end and at the beginning of totality Square of half-difference of diameters of the moon and shadow Deduct the squares of latitudes
15' 28"
19' 19"
825
825
Their square roots
239
373
586 24'.21
452 21'.26
Second half-interval of totality = ^'^IL^JT o02 9 First half-interval of totality
= 1" 49" = 0^ 43"'
2 r . 2 6 X 60" _ j„ 802' 9"
_ ^f, 3g,„
Summary of the lunar eclipse
Note :
Beginning of the eclipse
6*42"*
Beginning of the totality
7*49"'
Middle of the eclipse
8* 24""
End of the totality
8* 59"*
End of the eclipse 10*06" The timingt coincide with the ones given in the Ind. Am. Eph.
84
Ancient Indian Astronomy
9.S Lunar eclipse according to KK Before proceeding witii the procedure of computation of lunar eclipse according to the Khanda Khadyaka, let us familiarize ourselves with certain basic concepts, assumptions and the formulae derived therefrom. Fig. 9.6: Parallax of a body Let O be the position of an observer on the surface of the earth whose centre is E and radius p (Fig.9.6). The centre of a celestial body—like the sun or the moon—is represented by B. Suppose the radius of the earth OE makes the angle OBE=n, called the horizontal parallax of the body B. In the right-angled triangle OEB, we have sin n = p/R. Since R is considereably large and the angle n is sufficiently small, we take R as the distance between the earth and the body B and n ~ p/R According to the Khanda Khadyaka, the moon's mean daily inotion is 7 9 0 ' 3 1 " and , , ,, _p_ Moon's daily motion 790' 31" M o o n s hor. parallax, 7t„, = - ^ = rj^ = — — — = 52 42 . It is assumed, in traditional Indian astronomy, that all planets move with the same linear velocity. Therefore, if R and /^,„ are respectively the distance of a heavenly body B and of the moon, then we have R X Daily angular motion of B =
X Daily angular motion of the moon
= Constant. It follows that the parallax of a body _ £ _ _p_ ^~ R ~ y?,„
Moon's daily inotion R ~ 15 R
(i) True angular diameter of the sun Mean angular dia. of the sun _ ... . ^ . = — f: X True daily motion of the sun Mean daily motion of sun =
32'31" , „ X True daily motion of the sun 59 o
= ^ X True daily motion of the sun. (ii) True angular diameter of the moon Mean angular diameter ^ , ., = —— f-— X True daily motion Mean daily motion
Lunar Eclipse
85 32' 790'31' =
10
X Trae daily motion of the moon
X True daily motion of the moon
(iii) Angular diameter of the shadow : Let S and E be the centres of the sun and the earth and M be the position of the moon. In Fig. 9.7, the sun's horizontal parallax = E^M and the moon's horizontal parallax = E^B. B
Moon's plane M
Fig. 9.7: Angular diameter of the shadow cone Now, in triangle BEM, we have A
EBM + EMB = MEB' = MEV + VEB' A
But VEB' = SEB, the sun's angular semi-diameter. Therefore, the angular diameter of the shadow A A A A = 2MEV = 2[EBM
+ EMB - SEB] at the plane of the moon's orbit,
i.e. the angular diameter of the shadow = 2 [Sun's hor. parallax +SDM Moon'sMDM hor. parallax - Sun's angular diameter] 11 = 2 ~15~^~[5~-40^^^ = (1 / 6 0 ) [8 SDM + 8 MDM - 33 SDM] = (1/60) [8 M D M - 2 5 SDM] where MDM = Moon's true daily motion and SDM = Sun's true daily motion. Note :
Indian astronomers have assumed that the horizontal parallax of the body is equal to the true daily motion of the body divided by 15.
Now, with these basic concepts and expressions we set out to
demonstrate
Brahmagupta's procedure, as in his Khanda Khadyaka, with an example. Example : Lunar eclipse on 10-12-1992 according to KK I
77ie longitudes at the preceding midnight : (1)
True longitude of the sun = 233° 48' 23"
(2)
True daily motion of the sun = 1° 1' 13" = 61' 13" = SDM
Ancient Indian Astronomy
At
(3)
True longitude of the moon = 51° 50' 35"
(4)
True daily motion of the moon = 13° 12' 53" = 792' 5 3 ' s MDM
(5)
True longitude of the moon's node (Rahu) = 237° 13' 58".
the instant of opposition the difference in the longitudes of the sun and
the moon should be 180°. A t the preceding midnight Moon's longitude - Sun's longitude = 178° 2' 12" Therefore, for the moon
to be in opposition with the sun, it has to travel
= 1° 57' 48" (relative to the sun). The time taken for this is given by ^"^^^"^^^
X
24'' = 3 ' ' 5 1 ' " 5 H
12° 11'40" where 12° 11' 40" = Moon's daily motion - Sun' daily motion. s MDM-SDM Therefore, the instant of opposition : 3'' 51"* 5 1 ' A . M . At the instant of opposition: (1)
True sun
=
True sun at midnight + motion for 3*51'" 51'
(2)
True Moon
(3)
Node (Rahu)
=
233° 4 8 ' 2 3 " +
' 24*
=
233° 58' 14"
=
51° 50'35" + 2° 7 ' 4 0 "
=
53° 58' 15"
=
237° 1 3 ' 5 8 " -
xS^srSl'
X 3*.8641667 24
237° 13' 5 8 " - 3 1 " 237° 13' 27" (4)
Moon - Node
176° 44'48"
.•. Moon's latitude
270' sin (176° 44' 48") 15' 19"
(5)
Moon's angular diameter =
10 — - x Moon's true daily motion 247 10 247
(6)
X
792' 53" = 32' 6
Angular diameter of the shadow cone
= (1/60) [8 X moon's true daily motion - 25 x sun's true daily motion] = 80' 13"
Lunar Eclipse
87 (7)
Sum of the semi-diameters of the shadow and the moon = (1/2) (112' 19") = 56'9".5
(8)
Portion of the moon obscured (grasa) : (Sum of the semi-diameters) - (Moon's latitude) = 56' 09".5 - 15' 19" = 40' 50".5
Now, the portion obscured > Moon's diameter. Therefore the lunar eclipse is possible and total. Note :
(i) If the portion obscured is positive and < Moon's diameter, then eclipse is partial (ii) If the portion obscured is negative, then there is no eclipse. (iii) If the portion obscured > Moon's diameter, then the eclipse is total. (9)
Difference of semi-diameters of the shadow and the moon =
[ 8 0 ' 1 3 " - 3 2 ' 16"]
= 24' 3".5 (10) Moon's true daily motion - sun's true daily motion MDM - SDM = 792' 53" - 61' 13" = 731' 40". First approximation to half-durations (1)
Half-duration of the eclipse The half-duration is given by the formula V(^,+^,)2-p2 (MDM-SDM) where dy and ^2 are the semi-diameters of the shadow and the moon, P is the moon's latitude at the opposition. Therefore, V ( 5 6 ' 9 " . 5 ) 2 - ( 1 5 ' 19")2 Half duration =
(2)
73^40^^
Half-duration of total obscuration The formula to calculate the half duration of total obscuration: V(cf,-d,)2-P^ (MDM-SDM)
, _ V(24'3".5)^-(15'19"g ""^^ "
731'40"
, ^ ""^^
, ^
where dy and d2 are the semi-diameters of the shadow and the moon and P, the moon's latitude. Second approximation to the beginning of the eclipse The first approximation to the beginning of the eclipse is 1* 46"" 20' = 106'".33 before the instant of opposition as obtained in item (1) above. We shall determine the true moon, Rahu, the moon's latitude and hence the second approximation to the first half-duration of the eclipse.
Ancient Indian Astronomy
88
1.
Moon's true motion in this time is 106"" 33 ^ 24 X 6 ( P
2.
X 792'. 883 =
58'33"
Moon's true longitude at the (first approx. to) beginning of the eclipse 53° 58' 15" - 58' 33' = 52° 59' 4 2 "
3.
Node's longitude at the (first approx. to) beginning of the eclipse : 237° 13' 27" +
106'".33 X 190".7 = 237° 13'27" + 14" 24x60^ = 237° 1 3 ' 4 1 "
4.
M o o n - N o d e = 52° 5 9 ' 4 2 " - 237° 1 3 ' 4 1 " = 175° 46' 1" (by adding 360°)
5.
Moon's celestial latitude at the (first approx. to) beginning of the eclipse = 270'sin (175° 4 6 ' 1 " ) = 1 9 ' 5 6 "
To get a higher degree of accuracy, the above values must be calculated at the improved instant of opposition. Now, the second approximation to the first half-duration is V ( 5 6 ' 9 " . 5 ) 2 - ( 1 9 ' 56")^ 731'40" = l''43'"20^ 1.
Moon's motion in 1 * 43"" 20^ = 56' 54"
2.
Node's movement in 1* 43"" 20^ = 13".7
3.
Moon's true longitude at the (second approx. to) beginning of the eclipse :
4.
Moon's longitude at the opposition - 5 6 ' 5 4 " = 53° 0 1 ' 2 1 " Node's longitude at the (second approximation) beginning of the eclipse : 237° 13' 27" + 13".7 = 237° 13' 40".7 Therefore, we have M o o n - N o d e = 53° 0 1 ' 2 1 " -
237°13'40".7
= 175°47'40".3 Moon's celestial latitude at the (second approx.) beginning 270' sin (175° 47' 40".3) = 19' 48" Third approximation to the beginning : The third approx. to the first half-duration :
Lunar Ek:lipse
89
731.66
V(75' 57".5) (36' 21".5) = l ' ' 4 3 ' " 2 6 '
A t the third approx. to the beginning of the eclipse, we have 1.
Moon's motion for l * 43"* 26' is 56' 57"
2.
Moon's longitude at the beginning : 53° 1' 18"
3.
Node's longitude at the beginning : 237° 1 3 ' 4 1 "
4.
Moon - Node = 175° 47' 37"
5.
Moon's celestial latitude at the beginning = 1 9 ' 4 8 "
Fourth approximation to the beginning If we proceed as mentioned above, the first half duration is obtained as 1*43'" 26'. Therefore the fourth approximation to the instant of the beginning of the eclipse: 3* 51"* 5 1 ' a.m. - 1* 43"* 26' = 2* 08"" 25'. Second approximation to the end of the eclipse We have obtained the half-duration of the eclipse: 1* 46"* 20' as the first approximation. This is also the first approx. to the second half duration. 1.
Moon's true longitude at the end of the eclipse (first approx.) : 53° 5 8 ' 1 5 " + 58' 33" = 54° 5 6 ' 4 8 "
2.
Moon's node at the end (first approx.) = 237° 13' 13"
3.
M o o n - n o d e = 177° 4 3 ' 3 5 "
4.
Moon's latitude at the (first approx. to) the end : 10'42".7
The second approximation to the second half duration i.e. to the end of the eclipse is given by l''48"'29^ from the instant of opposition. In the same way as mentioned above,
the moon's
longitude, the node's longitude and the
moon's
latitude
for
1* 48"* 29' after the opposition are calculated. Using the calculated data we obtain the third approximation as 1*48"" 3 1 ' for the second half duration. First approximation to the beginning of totality For the totality of the eclipse also the iterative process should be carried out to obtain a higher degree of accuracy. As obtained earlier. First half-duration of totality : 0* 36"" 31'. 1. Moon's motion in this time = 2 0 ' 6 " 2.
Node's motion in 0* 36"" 3 1 ' = 4".8
Therefore at the beginning of totality : 3.
Longitude of the Moon : 53° 38' 9"
4.
Node's longitude = 237° 13' 32"
90
Ancient Indian Astronomy 5.
Moon's celestial latitude = 270' sin (Moon - Node) =
16'54"
Continuing the process of iteration, we have the second approxiniation for the beginning of totality as O* 33"* 4 2 ' before the opposition. Third approxiniation to the beginning of totality
: 0* 33"" 5 5 '
Fourth approximation : 0* 33*" 54' Therefore, the beginning of totality : 3* 51"* 5 H - 0* 33"* 54^ = 3* H * " 57^ Similarly the exact instant of the end of totality is calculated by the method of iteration. First approximation to the end of totality : The first approx. to the second half of totality : 0* 36"* 3 1 ' Moon's motion for 0* 36'" 31'
=
20' 6"
Node's motion in this
=
4 . 8 " = 5"
Moon's true longitude
=
53° 58' 15" + 2 0 ' 6 " = 54° 18'21"
Node's longitude
=
237° 13'27"
Moon - Node
=
5 4 ° i g ' 2 1 " - 237° 13' 22" = 1 7 7 ° 4 ' 59"
Moon's celestial latitude
=
270'sin (177° 4'59") = 1 3 ' 4 4 "
time
— 5"=
237° 13'22"
Second approximation to the end of totality
Summary of the lunar eclipse 3''51"'51'A.M. (LMT)
Instant of opposition 1.
(a) Duration of the 1st half interval (b) Duration of Ilnd half interval
2.
l''48"'3l'
(a) Duration of 1st half of totality
: 0''33"'54'
(b) Duration of Ilnd half of totality
: 0* 39"" 1'
Therefore, we have Beginning of the eclipse
2*08"' 25' A . M .
Beginning of the totality
3* 17" 5 7 ' A . M .
Lunar Eclipse
91
Middle of the eclipse
S^Sl^Sl'A.M.
End of the totality
4* 30" 52' A . M .
End of the eclipse
5*40"'22'A.M.
The timings are the Local Mean Time at Ujjayini. 9.9 Lunar eclipse according to GL The importance of the procedure for computations of eclipses in Ganesa Daivajna's Graha Ldghavam (GL) lies in the fact that the use of trigonometric ratios is dispensed with. A l l the same, on account of improved values of the astronomical elements, predictions of eclipses according to GL are fairly reliable. The following are some important formulae for the parameters used for computations according to GL : 1.
Sun's angular diameter Sun's true daily motion - 55
+ 10 aiigulas
^ 2.
., , . . . Moon's true daily motion . , Moon s angular diameter = ——^ angulas
3.
Shadow's angular diameter =
[(3/11) X Moon's angular diameter + 3 x Moon's angular diameter - 8]
angulas Note :
1 angula = 3' of arc i.e. 3 kalas. Therefore, angular diameter in minutes of arc when divided by 3 gives the same in angulas. 11
4.
Latitude of the moon, Sara «
—(M-R)
where Af and/? are respectively the longitudes of the moon and Rahu. If iM-R)> Note :
90°. its Bhuja must be taken.
The approximate formula follows from a general approximation given by 72 6 Ganesa Daivajfia: 120 sin 6 •» - ^ w h e n 0 is small (Ref, : GL, Prasnadhikara. Sl.ll) Now, the latitude (Viksepa or Sara) of the moon P = 270'sin ( A / - / ? ) » 270 x ^ ^ ^ ^ ^ ( A / - / ? ) 162 i.e. P = ^ j - (Af - R) minutes of arc (i.e. kalas) Now, dividing by 3 p , | | ( M - / e ) = il(A/-/?)
angulas
Ancient Indian Astronomy
The approximations in this case are justified since under the possible circumstances of an eclipse (M - R) is indeed small. If {M-R)>m°, 5.
then its Bhuja = (M-R)
- 180°
The amount of obscured portion, Grasa = (1/2) [Chadaka dia. + Chadya dia.] - Sara In the case of a lunar eclipse, the Chadaka (eclipser) and the Chadya (eclipsed) bodies are respectively the earth's shadow and the moon.
6.
Mdnaikya Khanda = (1/2) [Chadaka + Chadya ] diameter so that we have Grasa = Mdnaikya Khanda - Sara Therefore (0
If Mdnaikya Khanda < Sara (i.e. Grdsa < 0) there will be no eclipse;
(ii)
If Grdsa > Chadya diameter i.e. if Grdsa > Moon's diameter, then the eclipse is total.
Such an eclipse is called Khagrdsa
Grahana.
E x a m p i c : Find (i)
the angular diameters of the sun, the moon and the earth's shadow cone,
(ii)
Sara
(iii)
Grdsa and
(iv)
whether a lunar eclipse is possible given that
Tnic Sun
:
g' 0° 12' 06", Tnie moon : 2 ' 0° 12' 06"
Rahu
1' 28° 23' 18", SDM = 61' U "
MDM
823' 50"
at the instant of fullmoon where SDM and MDM are respectively the true daily motions of the sun and the moon, (i) 1.
Angular diameter of the sun : SDIA = (\/5)
(SDM - 55) + \0 angulas
= (l/5)(61'll"-55)+10=11^8l4P" whert! 1 arigula = 60 pratyarigulas 2.
Angular diameter of the moon: MDIA = ^ 2 3 ^ ^ 74
3.
nang7pra
Angular diameter of the earth's shadow cone : SHDIA = [3 (MDIA)/] 1 + 3 (MDIA) - 8] angulas
Lunar Eclipse
93 = (3/11+3) M D M - 8
angulas
= (36/11) (MDM) - 8
angulas
= 36 X (11"** 7P'*)/11 - 8 = 28*"« 23l*« (ii)
Sara : Here, we have M - / ? = 60° 1 2 ' 0 6 " - 2 3 8 ° 23' 18" = 181° 4 8 ' 4 8 " since M - /? > 1 8 0 ° . Bhuja = 1° 48' 4 8 " Therefore, 5ara = 11 (1° 48' 48")/7 = 2*"« 48P"'
(iii)
Grdsa = Mdnaikya Khanda -
5ara
Here, Mdnaikya Khattda = (1/2) (SHDIA + MDIA) i.e.
Mdnaikya Khanda = (1/2) (28""« I S P " + 11*"' 7P") = (1/2) (39*"» 30P") = 19*"« 4 5 P " Gra^a = 1 9 ' ' " « 4 5 P " ' - 2 « ' « 4 8 P ' » = 16''"«57P"
(iv)
Since Mdnaikya Khanda > Sara, the eclipse is possible. Further, since Grdsa > MDIA.
(i.e. le""* 57P''' > l l * " * 7P"') the lunar eclipse is total
(i.e , Khagrdsa
grahana).
In that case, we have Khagrdsa = Grdsa - Chadya diameter = Grdsa -
MDIA
= 16*"8 57P"-11*"8 7 P " = 5»ng5opra (v)
Half-durations of the eclipse and of totality 1.
A d d sara to the mdnaikya khanda and multiply this sum by 10; multiply this product by grdsa and then take the square-root of product. Take one-sixth of the square root and subtract it from square-root. If the result is divided by the Candra-bimba (i.e. moon's diameter), we get the madhyasthiti (in ghatikds) i.e. half-duration of the eclipse.
2.
the the the the
In the case of a total lunar eclipse, a half of the difference between the moon's diameter and the diameter of the earth's shadow (called manantara khanda) must be taken. To that difference add sara, then mulitply by 10. The product must be multiplied by the khagrdsa. Take the square-root of the result and divide the same by 6 and
94
Ancient Indian Astronomy
1.
subtract it from the square-root. Divide the remainder by the diameter of the moon. This will be the marda (i.e. half duration of the totality). In symbols, this means Half duration of the eclipse : Let jc = V[(1/2) {SHDIA + MDIA) + Sara ]xlQx
Grdsa
Then, the half-duration =
2.
{x-x/6)/(MDIA)
= 6oiiz4) ^'"'^'^"^ Half-duration of totaility : Let :y = V ( l / 2 ) (SHDIA - MDIA) + Siro ] x 10 x Khagrdsa Then, the half-duration of totality =
\y-y/6]/MDIA
Example : In the example considered in this section fmd the half-durations of the eclipse and of the totality. 1. Half-duration of the eclipse (Sthiti) Mdnaikya khanda
19*"* 45*^
Sara
2*"«48P"
Add 22*"' 33*^ Multiplying the above sum by 10, wc have 10 x 22'"'8
= 225'"*8 30P"
Multiplying the above value by grdsa viz. 16'"* (225-30) X (16-57)= 3822-13
we get
Square-root : V3822-13 =61'"'«49P'* Dividing by 6: 61*"8 49P'*/6= lO*"" 18P" Subtracting lO""* ISP'* from 61*''«49P''» wc get 51'"«3lP''' Dividing the above quantity by the moon's diameter viz. ll*"«7P'*. we get (51 '"'8 31 P")/( 11
7P'*) = 4*'' 38^'«
Therefore, the half-duration of the eclipse : 4*'' 38^'«. 2.
Half-duration of totality (Marda) Shadow's diameter
28*"* 23^'*
Moon's diameter
jjanSyPr*
Manantara Khanda
=
(1/2) (28"* 23'*' -11*^ 7'**)
Umar
Eclipse
95 = (1 /2) (1T*"* 1 e n
=
8'"8 38P"
Adding Sara : 2*^8 48P''' to the above, we get 11'*"8 2 6 ? " Multiplying by 10 : 10 x 11*^8 2 6 ? " = 114'"'8 20?" Khagrdsa = 5*^8 5 0 ? " Multiplying, we get 666-56 Squre-root : 25*"8 5oP« Dividing by 6:4*"8i8Pra Subtracting : 2 l ' ^ 8 32Pra Dividing by the moon's diameter MDIA i.e. 11*"8 yP^^ we get |j^=l8h56vig Therefore, the half-duration of totality marda is 18''56^*8 Note :
1 ghatika = 60 vighatika (or palas)
First and second halves of eclipse and of totality The difference, (True sun - Rahu) called vyagu at the instant of the opposition is considered and its bhuja is determined. The product 2 x bhuja (in degrees) is put in two places as palas. (i)
If the vyagu is in an even quadrant i.e. i f 90° < vya^« < 180° or 270° < vyagu < 360° then (2 X bhuja) in palas is subtracted from and added to the madhya sthiti (i.e. half duration in gh. obtained earlier), respectively, to get the corrected sparsa and moksa half-durations.
(ii)
If the vyagu is in an o ^ J quadrant i.e. i f 0° < vyagu < 90° or 180° < vyagu < 270°, then (2 x bhuja) in palas is added to and subtracted from the madhya sthiti (i.e. the half-duration in gh. obtained earlier), respectively, to get the corrected sparsa and moksa half-durations.
Similar operations are carried out to get the first and the second half-durations of totality by considering the marda duration instead of the sthiti. Example : Now, in the example considered, we have at the instant of opposition True Sun
:
8 ' 0 ° 12'06"
Rahu
••
7 ' 2 8 ° 23'18"
Vyagu
jo 4 g ' 4 3 "
Since 0° < vyagu < 90°, bhuja = 1° 48' 48". Now, multiplying this bhuja by 2, we get 3 . 37 - 36 = 4 palas Again, since vyagu is in I quadrant (i.e. odd), 4 palas, is added to and subtracted from the madhya sthiti. Thus, we have :
96
Ancient Indian 3gP«'as
48^ 3gpalas
^ 4l»las 4gh 42Palas
_ 4Pal»s ^A^alas
Madhyasthiti (gh.) Add and subtract 4 palas
:
ThcTtfore, sparsa sthiti and moksa sthiti
Astronomy
4^''42?'^'". the first Half-duration 4gh 34Pai« ^^^^^^ half-duration
Similarly, by considering the marda (i.e. l*** 56?*''*) the half-durations of totality are (jgh55palas)^4palas (j gh jgpalas) _ 4palas i.e. 2**^ and l^**
52?*'**
for the first and the second halves of totality respectively.
Instants of beginning and ending of eclipse and of totality 1.
The parvanta,
in this case the instant of opposition, is the middle of the
eclipse. B y subtracting from and adding to the instant of the middle, we get the beginning and ending moments of the eclipse respectively. Thus, we get the sparsa kdla and the moksa kdla of the lunar eclipse. 2.
Similariy, subtracting from and adding to the instant of the middle of the eclipse the first and second halves of totality,_we get the instants of the beginning (sammilanam) and the ending (unnmilanam) of the eclipse.
Example : We have (i)
Instant of fullmoon Less sparsa sthiti
(ii)
408'' 4gh
;
.'. sparsa kdla
368"
Instant of fullmoon
408"
Add moksa sthiti
;
.-. moksa kdla
458''
(iii) Instant of fullmoon
408"
Qgpalas 4gpalas
22Palas 4gpalas Qpalas
2Sh
388"
.: Sammilana kdla (iv) Instant of fullmoon
42palas
34Palas
48"
Less sparsa marda
48Palas
408"
4gpalas 4gpalas ^2P^las
Add moksa marda
18"
:. Unmilana kdla
428"
4Qpalas
Summary of the eclipse
Middle of the eclipse
gh. 36 38 40
End of the totality End of the eclipse
42 45
Beginning of the eclipse Beginning of the totality
:
palas 06 48 48 40 22
Lunar Eclipse
97
The timings are from the local mean sunrise. Note :
Computer programs for (i) computations of the positions of the sun, the moon and Rahu and (ii) the lunar eclipse according to the Crahalaghavam are provided in PROGRAMS 7.3 and 9.3. The output for a worked example follows. The example woriced out, using PROGRAMS 7.3 and 9.3, is for May 2, 1520 A.D. (J), Wednesday. This eclipse occurred during the time of Ganesa Daivajfia [his epoch in GL is March 19, IS20 A.D. (J)]. The date of the lunar eclipse is taken from the Epigraphia Indica, Vol VI, Page 237. GRAHALAGHAVA POSITIONS O F S U N , M O O N A N D R A H U
(CHRISTIAN) D A T E T I M E (AFTER SUNRISE) N A M E OF THE PLACE L O N G I T U D E (-ye F O R WEST)
Y E A R : 1520 M O N T H : 5 D A T E : 2 HOURS : 0 MINS : 0 UJJAYINI D E G : 75 M I N ; 45
L A T I T U D E (-ve F O R SOUTH) D E G : 23 M I N : 11 WEEK DAY WEDNESDAY CHAKRAS : 0 A H A R G A N A : 44 [EPOCH : 19-3-1520 (J)] RAVISPHUTA M E A N RAVI A T UJJAYINI SUNRISE
33°
3'
0"
0°
0'
0"
33°
3'
(f'
M O T I O N F O R 0 HRS. 0 M I N .
0°
0'
0"
M E A N RAVI A T G I V E N T I M E
33°
3'
0"
MEAN ANOMALY
44°
57'
0"
+ 1° 34°
32'
23"
35'
23"
DESHANTARA CORRECTION M E A N RAVI AT L O C A L SUNRISE
MANDA PHALA T R U E RAVI
PRESS A N Y K E Y TWICE C H A N D R A SPHUTA M E A N M O O N A T UJJAYINI SUNRISE
208°
51'
34"
0°
C
0"
208°
51'
34"
0°
0'
0"
M E A N MOON AT GIVEN TIME
208°
51'
34"
MOON'S MANDOCCA
172°
25'
42"
MOON'S M A N D A KENDRA (ANOMALY)
323°
34'
8"
M A N D A P H A L A (EQN. O F C E N T R E )
-2°
58'
52"
BHUJANTARA CORRECTION
0° 205°
3'
25"
56'
7"
DESHANTARA CORRECTION M E A N M O O N AT L O C A L SUNRISE M O T I O N F O R 0 HRS. 0 M I N .
TRUE MOON
:
Ancient Indian Astronomy
98 R A H U SPHUTA M E A N R A H U A T UJJAYINI SUNRISE
25°
18'
4"
0°
0'
0"
25°
18'
4"
0'
0"
DESHANTARA CORRECTION M E A N R A H U A T L O C A L SUNRISE M O T I O N F O R 0 HRS. 0 M I N . M E A N R A H U AT GIVEN TIME
18'
25°
D O Y O U W A N T E C L I P S E / P L A N E T S C O M P U T A I O N ? (E/P) ? E L U N A R ECLIPSE ACCORDING TO GRAHA LAGHAVAM AT 6 H R S * * * T R U E . S U N : 34.58963
T R U E M O O N : 205.9354 N O D E : 25.30124
SUN'S TRUE DAILY MOTION
57' 30"
MOON'S TRUE DAILY MOTION
736' 15"
M O O N ' S DISTANCE F R O M OPPN.
8° 3 9 ' 1 5 "
T I M E O F OPPN. A F T E R M I D N I G H T (LMT)
24H-21 M - 3 7 S
T R U E S U N A T OPPN.
35° 19' 22"
T R U E M O O N A T OPPN.
215° 19' 22"
N O D E A T OPPN.
25° 15' 39"
M O O N ' S D I A M E T E R (in Angulas)
9.949405
S H A D O W ' S D I A M E T E R (in Angulas)
24.56169
ECLIPSE IS POSSIBLE (NORTHERN) S H A R A (in Angulas) : 15.81192 G R A S A (in Angulas)
1.443631 • L U N A R ECLIPSE IS PARTIAL*
M A D H Y A STHITI (in Ghatis)
1.829996
SPARSHA STHITI (in Gh.)
2.165401
M O K S H A STHITI (in Gh.) S U M M A R Y OF THE 1
1.494592
AFTER MIDNIGHT PRECEDING
2/5/1520
LOCAL
MEAN
TIME
SPARSHA (BEGINNING) T I M E
23 H - 2 9 M - 39 S
M A D H Y A (MIDDLE) O F E C L .
24 H - 21 M - 37 S
M O K S H A (ENDING) T I M E
24 H - 5 7 M - 29 S
Ok
10
SOLAR ECLIPSE
10.1 Cause for solar eclipse On a new-moon day, the sun and the moon are on the same side of the earth (see F i g . 10.1). The rays of the sun S which fall on the surface of the moon M, facing the sun, are prevented from reaching the earth. A shadow cone is caused by the moon on the side facing the earth. In fact, a solar eclipse is caused under the following conditions ; (i) The sun and the moon must be in conjunction i.e. it must be a new-moon day; and (ii)
The new-moon must be close to one of the nodes (Rahu or Ketu)
On account of the inclination of the moon's orbit with the ecliptic (at an angle of about 5°), there will not be solar eclipse on every new-moon day. Only on those new-moon days when the moon is close to the ecliptic, and hence close to one of the nodes, a solar eclipse is possible. Since the moon's radius is much smaller than that of the earth, the shadow cone formed by the tangents to the surfaces of the sun and the moon can cover only a portion of the earth. Therefore, a solar eclipse is visible only from a limited portion of the earth's surface. In Fig 10.1, the shadow cone of the moon, formed by the tangents at the surfaces of the sun and the moon, has its vertex at V. This shadow cone is called "umbra". The "penumbra" region is obtained by drawing the internal tangents to the surfaces of the sun and the moon. For the region on the surface of the earth represented by the arc HK of the umbra the sun is completely obscured by the moon; hence there is a total solar eclipse for that portion of the earth's surface. For portions of the earth's surface which lie in the penumbra
H H' K K D Fig. 10.1: Solar eclipse
100
Ancient Indian Astronomy
region, such as the point L , the moon covers only a part of the sun and hence there will be a partial solar eclipse. In fact, the total solar eclipse is possible due to the fact that the moon's angular diameter at times is greater than that of the sun (although the actual linear diameter of the moon is quite small as compared to that of the sun). However, sometimes on the occasion of a solar eclipse, the angular diameter of the moon is less than that of the sun so that the moon obscures only a central circular portion of the sun leaving a ring portion of the sun bright. Such an eclipse is called annular solar eclipse. This is illustrated in Fig. 10.1 where the centres of the sun, moon and earth are at S, M and £* respectively. For the portion of the earth's surface between H' and K', the solar eclipse is annular. 10.2 Angular distance between the sun and the moon at the beginning and end of solar eclipse In Fig. 10.2, the penumbra formed by the internal tangents between the surfaces of the sun and the moon are shown. Suppose the tangent AB is also tangential to the earth's surface at C. Then to an observer at C , the sun is just about to enter or leave the penumbra, marking the beginning or the end of the partial phase of the solar eclipse.
Penumbra
Fig. 10.2: Angle MES at the beginning and end of solar eclipse Let D denote MES; we have D
=
MEB + BES
...(])
But, under the conditions of the beginning or the end of the partial phase of the solar eclipse, MB is almost perpendicular to EB. Therefore, MEB
=
s'
the moon's angular semi-diameter, so that we have D
=
BES
+ s'
Also BES = OBE + EOB ...(2) A A But. OBE = CBE, the horizontal parallax of B, which is approximately equal to the horizontal parallax of the moon, p' i.e., OBE = p Further, we have
101
Solar Ek:lipse
EOB » EOA = AES - EAC = s - p where s is the sun's angular semi-diameter and p is the sun's horizontal parallax. Therefore, from (1) and (2), we have D
=
s
s' + p' - p
...(3)
This gives the angular distance between the sun and the moon with respect to the centre E of the earth at the beginning or the end of a partial solar eclipse. Now, using the mean values : Sun's semi-diameter,
s = 16'
Moon's semi-diameter, / Sun's hor. parallax,
= 15'
p = 8"
Moon's hor. parallax, p' = 57' we get values of the angle MES at the beginning and end of the partial phase of the solar eclipse given by D = MES = s + s'+
p'-p
= 16'+ 15' + 5 7 - 8 " = 8 7 ' 5 2 " 10.3 Computations of solar eclipse according to 5 5 The procedure of computations of the solar eclipse is somewhat lengthy and complicated, as compared to the lunar eclipse, mainly on account of the effect of the parallax. In this section we consider the procedure for the computations of solar eclipse as given in the Siirya Siddhanta (55). Since the positions of the moon and Rahu, in particular, are somewhat inaccurate if calculated as per 55, we use the true positions of the sun, the moon and Rahu, as given in the Indian Astronomical Ephemeris for the date considered. However, the procedure adopted is as per 55. Example : Solar eclipse on 24th October, 1995 at Bangalore (according to 55). I True longitudes at 5:30 A . M . (1ST) : 1.
True longitude of the sun
=
186° 18'9"
2.
True longitude of the moon
=
183° 46' 5"
3.
True longitude of Rahu
=
182° 43"
4.
Sun's true daily motion
=
59' 46"
5.
Moon's true daily motion
=
14° 1 ' 3 7 " = 841'37"
The latitude and the longitude of Bangalore are taken respectively as 1 2 ° N 5 8 ' , 7 7 ° E 3 5 ' . II Instant of conjunction : K^h ^
^
Diff. in true longitudes of the sun and the moon ^ Difference in their daily motions
^ 186° 1 8 ' 9 " - 1 8 3 ° 4 6 ' 5 " ^
^ 5 . 3 0 - + 4*40-
14° 1 ' 3 7 " - 0 ° 5 9 ' 4 6 " = 10* lO'" = 25" 25" after midnight i.e., 11" 40" after 5-30 A . M .
102
Ancient Indian Astronomy
Note :
The nadis and vinadis are denoted by n and v ; 1 day = 60 nadis (Ghatikas); 1 nadi = 60 vinadis.
I l l Longitude at tiie instant of conjunction : 1.
Motion of the sun in 11"40"
=
^"
^ 60" since 4* 40"" = 1 r 40" after s" 30"'(IST).
= 11'37"
.-. Longitude of the sun = 186° 18' 9" + 11' 37" = 186° 29' 46" 2.
Moon's motion in 11" 40" = ^ 1" 40" x 14° 1'37' ^ 60" .-. Longitude of the moon = 183° 46' 5" + 2° 43' 4 1 " = 186° 29' 46"
3.
Node's motion in 11" 40" = ' ' " 4 0 " x ( - 3 ' l l " ) 60" .-. Longitude of the node = 182° 4 3 ' - 0 ' 3 7 " = 182° 4 2 ' 2 3 "
Note :
Here, 4'' 40"' = 11" 40^" is the time interval for the conjunction from 5* 30" a.m. (1ST)
I V To i l n d the true diameters of the sun and the moon , 1.
e • ^. Sun s diameter
6500^ x 5 9 ' 4 6 " =
= 32 46 11,858.75^
o 2.
^
A-
>
480^
X
14° 1' 37"
Moon s diameter =
= 34 4 11,858.75^ In the above expressions, the moon's mean daily linear motion in yojanas is about 11,858.75. According to the siddhantas, it is assumed that the sun, the moon and the planets move at a common linear velocity. Now according to SS, the circumference of the moon's orbit is 3,24,000 yojanas. The sidereal period of the moon being 27.32167416 days, the mean linear daily motion is given by 3,24,000/27.32167416= 11,858.71693 yojanas. Similarly, in the case of the sun, the circumference of the orbit is given as 43,31,500 voya/icu. Here also, the sidereal year being 365.25875648 days, the mean linear daily motion comes out approximately as 11,858.75 yojanas. V Orient ecliptic point (lagna) etc. at the moment of true conjunction 1.
The tropical lagna is 264" 30' 45"
T 2.
^ • . • • -^ Orient sine (Udayajya) :
sin e X /? sin (lagna) ZZSQ' 3
Solar Eclipse
j.03
wiiere e = 24°, obliquity of the ecliptic (as in SS) .: Udayajya =-U2S'.375
=
-\428'.
3.
Meridian ecliptic point ( M E P ) = 179° 42'
A
\ *
4.
Meridian sine (Madhyajya) =
A-
•
J,
•
-N
^ sin € X / ? sin (M£P) . ^^^g, = 7 m arc
sin ' ^ ^ ] = 0 ° 7 ' 2 2 "
inangle.
This is the declination 8 of the MEP The meridian zenith distance = 5 - (]) where 5 is the declination of MEP.
is the latitude of the place and
Declination of MEP = 0° 7' 19" N Latitude of the place = 12° 58' N .-. Meridian zentih distance = - 12° 50' 38" /?sin ( - 1 2 ° 50'38") = - 7 4 6 ' The sine of the ecliptic zenith distance Drkksepa : „, , Orient sine x R sine of Meridian zenith distance We have =
- M 7 8 ' y — 764.' 343g/
=
317'; square of 317 = 100489
Deducting the above value from the square of R sin (meridian zenith distance), we get V ( - 764)2 - 100489 = 695' 10" That is, the sine of ecliptic zenith distance 695'
sRsinz
The sine of the ecliptic altitude (Drggati) =
- /?2 sin^ z = V(3438)2 - (695)^ = 3367'
To find the divisor (cheda) and the sun's parallax in longitude (lambana). We have the divisor (R sin 30°)^ sine of ecliptic altitude _ ( 1 7 1 9 ) 2^ _
Cheda =
-
3367
-^^^
i.e. Divisor (cheda) = 878' where R sin 30° is given by 1719. Sayana Ravi at the inst. of conjn. = sun's sidereal longitude + Ayandmsa = 186° 29' 46" + 23° 47' 54" = 210° 17' 4 0 " At the instant of conjunction : Longitude of the meridian ecliptic point = 179° 4 1 ' 5 3 "
Ancient Indian Astronomy Longitude of the sun {Sayana Ravi)
= 210° 17' 40"
Interval in longitude (i.e., the difference) = 329° 24' 13" R sine of the above is the numerator. Parallax in longitude (lambana)
= ^ ^ i n (329° 24'13") Cheda = -2"0"
Therefore, the corrected instant of conjunction = 11" 40" - 2" 0" = 9" 40" from 5-30 a m . (1ST). Time of true conjunction = 25",25" from the midnight correction
= -2"0"
Time of app. conjunction = 23" 25^ from the midnight. Now, calculating the parameters again, at the apparent conjn, we get 1.
Nirayana sun at the apparent conjunction : 186° 27' 47"
2.
Rahu = 182° 43' - 30" = 182° 42' 29"
3.
True moon : = 186° 0 1 ' 4 8 "
4.
Orient
ecliptic point
(sayana
lagna)
at
the
moment
of
23" 25" is
253° 39' 0 2 " 5.
« . . . . ^sine Orient sine (Udayajya) =
6.
Meridian ecliptic pgint (MEP) = 166° 39' 33"
n 7.
Meridian sine (Madhyajya) =
»>!
A-
•
JL
•
-X
X
R sin (253° 39') ,^^^,0-, ^ ^ = - 1376 .92
/? sin 6 xR sin (MEP) = 322 40
Latitude of the place (^)
= 12° 58' N
Declination of MEP (8)
= 5° 23' 07" N
Meridian zenith distance, 8-(t) = - 7° 3 4 ' 5 3 " i.e., 7° 3 4 ' 5 3 " (5) Now, R sin (- 7° 34' 53") 8.
= - 453' 35"
The sine of the ecliptic zenith distance (drkksepa) We have drkksepa = 4 1 5 ' 38" = /? sin z
9. 10.
The sine of the ecliptic latitude (Drggati) = 3412' 47" To find the divisor (cheda) and the sun's parallax in longitude (lambana) : Divisor (cheda) = 8 6 5 ' 5 1 " r parallax in longitude (lambana) = - 2 " 44" Corrected time of apparent conjunction = 25" 25*^ - 2" 44" = 2"2" 41" (from the midnight).
Solar Ek:Upse
105
Repeating the above process iteratively, for getting convergent values, in the third approximation, we get the following readings. At the time of coi\junction (Approxn. 3) True longitude of sun
186° 2 5 ' 4 "
True longitude of moon
185° 2 3 ' 2 4 "
Longitude of node
182° 4 2 ' 3 8 "
Sayana Ravi (Trop. sun)
210° 1 2 ' 5 8 "
Orient ecliptic point (Lagna)
249° 3 6 ' 8 " -1344'.974
Orient sine (Udaya Jya) Meridian ecliptic point (MEP)
161° 5 0 ' 3 "
Meridian sine (Madhya Jya)
435'58"
Declination of the meridian
7° 1 7 ' 6 " - 5 ° 40'54"
Meridian zenith distance Sine of eel. zen. dist. (Drkksepa)
313'14"
Sine of eel. altitude (Drggati)
3423'42"
Divisor (cheda)
863' 5"
(Parallax in longitude) Lambana
-2na.59vin.
Cor. time of apparent conjunction
llna.llvin.
= - ( 1 H-
= 8//-58 A / - 3 6 5
A t the time of apparent coi^n. (after 3 iterations) True longitude of sun
186° 2 2 ' 6 "
True longitude of moon
184° 4 1 ' 3 7 "
Longitude of node
182° 42' 4 7 "
5ayana Ravi (Trop. sun)
:
210° 1 0 ' 0 "
Parallax in latitude (Nati)
4' 2 7 "
Moon's latitude at apparent conjn.
:
Moon's apparent latitude at conjn.
:
9' 2 0 " 13'47"
Sun's angular diameter
32'46"
Moon's angular diameter
34' 4 "
Obscum. at apprnt. conjn. (Grasa)
19'38"
11 M - 2 8 5)
Eclipse is partial Magnitude of the eclipse : 0.5994744 Calculation of the half-duration of the eclipse The square of the sum of the semi-diameters
: (33' 25")^ = 1 1 1 6 - 4 0
The square of the moon's latitude
: (13' 47")^ = 189 - 58
Ancient Indian
106
Astronomy
Subtracting : 926 - 42 Here, the moon's apparent latitude at the apparent conjunction is considered. Square root of the above diff. = V 9 2 6 - 4 2 = 3 0 ' 26".5 Half-interval =
60" X 30' 26".5 Diff. in true daily motions of sun and moon 60"X30'26".5 = 2" 19" 783'51"
Beginning of the eclipse = Time of apparent conjunction - (2" 19") = 22" 27" - 2" 19" = 20" 08" s S* 03"" 12" End of the eclipse = 22" 27" + 2" 19" = 24" 46" s 9* 54"* 24* To obtain more accurate values for the beginning and the end of the eclipse, further iterations of the above procedure must be carried out. After a few iterations, the values converge reasonably to yield the following : First half duration of the eclipse
: 2na.
25 vin. = 0 / / - 5 8 M - O S
Second half duration of the eclipse
: 2na.
15 vin. = 0 / / - 5 3 M - 4 8 5
Summary of the solar eclipse on 24 / 10 / 1995 at Bangalore Beginning of the eclipse
lOna.
\vin.
Middle of the eclipse
22na.
21vin.=
= 8//-00M-35 5 8//-58M-36S
End of the eclipse
24na 4lvin.=
9H-52M-25S
10.4 Computations of solar eclipse according to GL Example : Solar eclipse on December 15, 1610 A . D . (G), Wednesday at VaranasF. The cakras = 8 and the ahargaria = 1005 I At the sunrise : The mean longitudes of the sun, the moon and Rahu according to the Graha ldghavam (GL) at the sunrise are : Mean sun
:
Mean moon Moon's mandocca
8M°10'33" :
Rahu Now, according to the pahcdhga, the end of Amavasya
8-'5° 3 9 ' 2 5 "
8'' 17° 27' 2 1 " 2Ml°4r59"
is at 12** 36''"''" after sunrise.
The true positions of the bodies at the end of the Amavasya
are as following :
True sun
8' 5° 2 5 ' 5 7 "
True moon
8'5° 20'41"
Rahu
2 - M l ° 4 1 ' 19"
Sun's true daily motion
61' 15"
Moon's true daily motion
726' 30"
Solar Eclipse
107
II At the instant of conjunction The instant of conjunction of the sun and the moon is at IS** 0 4 ^ ° ' ' " after local sunrise. At that instant, by using the true daily motions of the sun and the moon, we have True sun
8* 5° 26' 2 5 "
True moon
8^ 5" 26' 2 5 "
Rahu
2 M I ° 4 1 ' 18" = True sun - Rahu
Virahvarka
8*5° 2 6 ' 2 5 " - 2 ' 1 1 ° 41' 18" 5* 23" 4 5 ' 0 7 " i.e., 173° 45'07" III Find the Lambana (i)
(Parallax in longitude)
Find the Lagna (Orient eel. point) at the instant of the conjunction. Subtract 90° from the Lagna
to get the Tribhona
lagna
(the
Nonagesimal). (ii)
Find the
declination (Kranti) of the sayana (tropical) Tribhona
lagna. (iii)
If the latitude of the place (tj)) and die declination of die (sayana) tribhona lagna (8) are both north (or both south), then take their sum (i.e., + 8) called Natdmsas and accordingly it is north (or south). On the other hand, i f 8 and ^ are of opposite signs, then take their difference and the sign of the greater value is taken.
(iv)
Divide the Natdmsas by 22 and square the result. If the squared result is greater than 2 subtract 2 from it and take half of the result ; add
this to the squared result. A d d 12° to this
to get hara i.e., divisor (or cheda). (V)
Take the numerical difference between the true sayana Ravi (S) and sayana Tribhoria lagna (TL). Consider the Bhuja of this difference. [IS-rLlorl80°-l5-rLI
or 1 5 - T L I - 180° or 360° - 1 5 - 7 L I
according as 15 - TL! is in I, II, III, IV quadrant.] Divide the Bhuja (in degrees) by 10 and subtract the result from Bhjua 14 and then multiply the balance by I.e. find 10 14°-
Bhuja"" 10
Bhuja" 10
This is the numerator. (Vi)
Then, dividing the numerator in (v) by the divisor (hara) of (iv) we get the lambana (in ghatikds).
Ancient Indian Astronomy
108
(vii)
(a) If the Tribhonalagna < True sun, then subtract the lambana (in gh.) from the instant of conjunction (in gh.). (b) If the Tribhotialagna > True sun, then add the lambana (in gh.) to the instant of conjunction (in gh.).
Example : 15 - 12 - 1610 A . D . (i.e. Sd. sa. 1532). Instant of conjunction is 13^4"). Ayandmsa
determination according to Grahaldghavam :
Subtract 444 from the Sdlivdhana Saka year. This is Ayana "liptis". Dividing this by 60, we get Ayandmsa (in degrees) for the given year. „ '_ , . _ , 1532j-444 . 1088 ,_ono/ For Sa. sa. 1532 : Ayanamsa = — deg. = —rr- = 18° 08 60 60 (i)
Nirayana lagna at the instant of conjunction
1 1 ' 2 ° 46' 17"
Subtract
3'
Nirayana tribhona lagna (sid. Nonagesimal)
8 ' 2 ° 46' 17" i.e.
(ii) (iii) Note :
242° 4 6 ' 1 7 "
A d d Ayandmsa
18° 08'
Sayana tribhoria lagna (Trop. Nonagesimal)
260° 5 4 ' 1 7 "
Declination of Sdyana tribhona lagna
8 = - 2 3 ° 40'46"
Latitude of the place (^) : 25° 26' 4 2 " (South) In Indian astronomy, the latitude of the place is treated as "South" if it is in the northern hemisphere of the earth for determining naiamsas.
In fact, natdmsas = 8 - 0 algebraically (if
8 - 0 = 4 9 ° 0 7 ' 2 8 " "South".
i^^''=-L(490r28") = 2 ° 1 3 ' 5 8 " . 5 Square of 2° 13' 58".5 = 4 / 5 9 / 9 . 3 Since this is greater than 2, subtract 2 from it and divide by 2 to get 1/29/34.6 A d d the above result to the squared result : 1/29/34.6 + 4/59/9.3 = 6/28/43.9 A d d 12° : 1 8 ° 2 8 ' 4 3 " . 9 This is hara (i.e. divisor)
Solar Eclipse
(v)
109
Tribhona tagna
:
8' 2° 46' 17"
True sun
:
8* 5° 26' 2 5 "
True sun - Tribhona lagna=
2° 40' 8"
This is less than 9 0 ° (i.e. I quadrant) and negative. The Bhuja = 2° 40' 8' Therefore, \o_BhuJa°
^ Bhuja" 10
10
= (14° - 0 ° 16' 0".8) (0° 16' 0".8) = 3° 39' 54".7 This is the numerator. , (vi)
J L Numerator , ., _ Lambana = -—— ghatikas Hara 3° 3 9 ' 5 4 " 7 1. 18°28'43".9 (where
(vii)
l * * = 60 palas; 1 pala = 60 vipalas)
Since Tribhonalagna < true sun, Lambana is subtractive from the instant of conjunction to get the middle of the solar eclipse. Therefore, we have Instant of conjunction =
13** 04''
Lambana = - 0 * * l l ' ' . 9 Middle of the eclipse :
12** 52.1'"^'"
(a) Lambana X 13 = (-) 0** 11'' X 13 = -2'''23*"' (in liptis) (b) Spasta Ravi - Rahu : i.e.,
Vyagu
Lambana
= 5* 23° 4 5 ' 0 7 " = 173° 45' 0 7 " =
Corrected Vyagu Since m°
8' 5° 26' 2 5 " - 2 ' 11° 4 1 ' 18"
42'44"
- 2'23"
:
-2'23"
5' 23° 42' 4 4 " : 173° 42' 4 4 " is in H quadrant ( 9 0 ° < V^ag« < 180°) subtract it from
180° to get the bhuja = 6° 17' 16". Since 6 ° 17' 16" < 14°, there will be an eclipse. 11 Sara i.e. Sara Note :
~ :y
^^^^ ^y^S** bhuja x 6° 17' 16" =
5^""
1 angula = 60 pratyangulas ; 1 angula = 3 kalas Sdyana Tribhoria Lagna : 8* 20° 54' 17" (1/9) Sara
: (-)1°6'
(-ve since/amZ>a«a is so)
10
Ancient Indian Astronomy
= 8* 19° 4 8 ' 1 7 " = 259° 48' 17" Kranti
:-23°.59746057 i.e., 5 =
Latitude of the place, Natamsa,
^=
5 - (j)
- ( 2 3 ° 35'51")
25° 3 4 ' 3 5 "
:
- 4 9 ° 10'26"
Finding Nati : (i) (ii) (iii)
Natamsa I 10 : (S) 49° 10' 26" / 10 = 4 ° 55' 2".6 (S) 18° - 1 Natamsa / 10 I
= 18° - (4° 55' 2".6) = 13° 04' 57".4
Multiply items in (i) and (ii) : 4 ° 55' 2".6 x 13° 4' 57".4 = 4.91738 X 13.082611 = 64°.33217 s Numerator (Num.) Taking the above as kalas, N= 6 4 ' 2 0 " s 1 ° 4 ' 2 0 "
(iV)
(v)
6° 18' - ( i i i ) : 6 ° 18' - (1° 4' 20") = 5° 13' 4 0 " = Denominator (Den.)
Nati =^~^ Den.
=
= 12°"*.30585715 5° 13'40" = 12'««18.4'""''(5)
Note : (vi)
Note : (vii)
In considering N it is taken in kalas and the same number as in degrees in N as numerator. Nati
: (-) 12""^ 18'"'.4
Sara
:
.: Spasta Sara '
: (-) 2^"« 25'".4
5^
Since Natamsa is south, the nati is taken negative. Sun's true daily motion •Spasta Sara X ^
= 61' 15" = SDM = H 2 ^ * 25.4''™ x
Sun's dia. (Siirya bimba) = 11^"* 08'"''' ^ (vni)
, ,. Moon s diameter
i.e., Candra bimba
=
sSDIA
Moon's daily motion
=
= 9^8 49?'''' = MDIA 74
(ix)
Mdnaikyakhandardha = (1/2) (SDIA + MDIA) = (1/2) ( 1 l'"'S 08'"''' + 9""* 49/"«) = (1/2) (20^"* 57'"''') = lO'"'* 28''™
(x)
SHryagrdsa
= Mdnaikyakhatiddrdha
- Sara
\ 11
Solar Ellipse = (1/2) (SDIA + MDIA)-
(xi)
Sara
= l O ' ^ * 28^*^" - 1 ' " ' * 28.4'"'"
(1 / 2 ) ( S D M + MDIA) + Sara = \(f"« 2'^'" + 2"''« 28.4'"''' = 12'"'«56.4^''''
(xii)
Multiplying item (xi) by 1 0 : 10 x (\2"^^ 56.4'"''') = 129'"'* 24'""
(xiii)
Multiplying (xii) by Suryagrasa, we get 129""* 24'"'" x 8""* = 1035.2
(xiv)
Square-root of item (xiii) is V1035.2 = 32.17452408""*
(xv)
Dividing item (xiv) by 6, we get 5.36242068 angulas
(xvi) (xvii)
Subtracting (xv) from (xiv), we get =26.8121034
angulas
Dividing item (xvi) by the moon's diameter we get sthiti (in Gh) i.e.S././ri = ^ ^ - « ^ ^ ^ Q ^ - ^ ° " ^ 2.731283896** =
2** 43'''' 52"''
( s i ' ' 5 " " 330 Sparsa kdla and Mok^a kdla (i) (ii)
Sthiti (in C/i) X 6 = 16°.38770338 Subracting item (i) from Tribhoria lagna
we get
8' 2° 46' 17"
(-)
16° 2 3 ' 1 6 "
Sparsa kdla Tribhona lagna (iii)
:
7 M 6° 23' 0 1 "
Natdmsa = (Krdnti of Sparsa kdla Tribhoria lagna)—(Lat. of the place) = - 2 1 ° 2 4 ' 3 9 " - 25° 2 6 ' 4 2 " i.e. Natdmsa
(iv) (v)
: (-) 4 6 ° 5 1 ' 2 1 "
Natdmsa I 22 = (-) 2° 07' 47" Ravi at the newmoon
8' 5° 26' 25"
Sun's true daily motion
6\'\5" (SDM)
Sun's motion for sthiti of 2** 44''^"' = Ravi at the newmoon
2S'' 44''/"' 61'15" x ^ , =2'47" 60** : 8' 5 ° 26' 25" (-)
(vi)
2' 47"
Ravi at Sparsa kdla
:
8' 5° 23' 38"
Tribhoria lagna (of Sparsa kdla)
:
7 ' 1 6 ° 22'17"
12
Ancient Indian Astronomy Rfi\i-Tribhona
(vii) (viii) (ix)
lagna
:
(8* 5° 23' 38") - (7* 16° 22' 17")
=
19°r21"
Dividing item no. (vi) by 10 we get 1° 54' Subtracting item no. (vii) from 14° we get 14° - 1° 54' = 12° 06' Multiplying item no. (vii) and item no. (viii) we get 12° 06' X 1° 54' = 22° 59' = Numerator
(x) (xi)
From item no. (iv) we have Natamsa / 22 as 2° 0 7 ' Squaring the above result (i.e.) 2° 07' and subtracting 2 from it we have (2° 07')2 - 2 = (4° 2 8 ' ) - 2 = 2° 28'
(xii)
Considering half of item no. (xi) we get (1/2) (2° 28') = 1° 14'
(xiii)
By adding item no. (xii) to the square of 2° 07' we have
(xiv)
Adding 12 to item no. (xiii) we get
(1° 14')+ (4° 28')= 5° 42' (5° 42') + 12 = 17° 4 2 ' s Hara (i.e. denominator) (XV)
Lambana = ^ ^ ^ ^ ^ ^ = ^ ^ ^ H l ' ' ' Denommator \ 70 4 2 '
Moksa kdla (i)
il""
Lambana
Multiplying sthiti (in gh) by 6 we get the result in degrees 28h M^ip X 6 = 16° 24'
(ii)
Adding Tribhona lagna (TBL) to item no. (i) we get 16° 24' + 8* 2° 46' 17" = 8 M 9 ° 10' 17" .-. Moksa kdla TBL : 8' 19° 10' 17" Krdnti of Moksa kdla TBL : (-) 23° 42' 2 8 " Less latitude of the place : (-) 25° 2 6 ' 4 2 " .-. Natdmsa = - 4 9 ° 09' 10"
(iii)
Natdmsa/22
(iv)
Square of item no. (iii) is (2° 14')^ = 4 ° 59'
(v) (vi) (vii)
=2° 14'
Subtracting 2 from item no. (iv) we get (4° 59') - 2 = 2° 59' Half of item no. (v) is (1/2) (2° 59') » 1° 29'.5 By adding item no. (iV) with item no. (vi) we get (4°59') + (I°29') = 6 ° 2 8 '
(viii)
Add 12 to item no. (vii) i.e. (6° 28') + 12 = 18-28 s Hdra i.e. denominator
Solar Eclipse (ix)
113
Moksa kala Ravi : 8' 5° 29' 12" Moksa kdla TBL : 8' 19° 10' 17" Difference
(x) (xi) (xii)
:
13° 4 1 ' 0 5 "
Dividing item no. (ix) by 10 we get 1° 22' Subtracting item no. (x) from 14 we get 14° - 1° 22' = 12° 38' Multiplying item no. (x) and (xi) we have 12° 38' X 1° 22' = 17-15 (Numerator)
To calculate the denominator, the procedure explained in Spdrsika lambana is to be followed. Hence the denominator = 1 8 - 2 8 (xiii)
Numerator Denominator
17-15 18-28
(xiv)
Middle of the eclipse
Qeh^^lio
13gh 04lipti
By subtracting sthiti
2^ 44''P''
(Mean) Sparsa kdla
jQgh 2o"Pti
Sparsakdla
(_)
lambana
igh nlipti
9gh os'iPti
:. True Sparsakdla (XV)
= Lambana (+ve)
j3gh Q^lipti
Middle of the eclipse
44lipti
Sthiti Mean Moksakdla Moksakdla
(by adding)
15gh 48lipti + 0«" 56"P'*
lambana
168" 44lipti
: . True Moksakdla
Summary of the solar eclipse on 15 - 12 - 1610 A.D. Beginning :
9 8 " 03"?''
s 9* 37"" 1V a.m. ( L M T )
Middle
: 128" 52''P''
= 11 ^ 8'" 48* a.m. ( L M T )
End
: 168h 44''P''
= 12* 41"" 36* a.m. ( L M T )
10.5 Computations of solar eclipse according to KK We shall now apply Brahmagupta's procedure for computing solar eclipse, to the following example, as explained in his Khanda Khadyaka
{KK)
Example : January 5, 1992 at Bangalore. The KK Ahargana : 4,84,596 (from Tables 5.1 to 5.3) The true longitudes of the sun, the moon and Rahu at the preceding midnight are
114
Ancient Indian Astronomy True longitude of Sun
259°38'18"
True longitude of Moon
259° 17'44"
Longitude of Rahu
255° 15' 05"
Sun's true daily motion
61'26"
Moon's true daily motion
731'41"
Difference of the daily modons of the sun and the moon
669'48"
Moon's distance from conjunction : Sun - Moon
=
259° 3 8 ' 1 8 " 259°17'44" 0° 20' 34"
Time of conjunction (True) : 0° 20' 34" X 24* 669'48"
=
0*44'" 13*
True motion of the sun in 44"* 13* 44m ,35
1440'"
x
6 1 ' 2 6 " = 1'53"
True sun at conjunction : 259° 38' 18" + 1'53" 259° 40' 11' True Sun = True Moon at conjunction = 2 5 9 ° 40' 11' Motion of the Node in 44"" 13* =
190" X 44"* 13* 1440'"
Node at conjunction : 255° 1 5 ' 5 " - 0 ' 5" 255° 15' 00" At 0* 44"" 13* on Jan 5 at Bangalore : Sdyana Lagna : 6* 24° 7' = 204° 7' Nonagesimal = Sdyana Lagna - 90° = 2 0 4 ' ' 7 ' - 9 0 ° = 114° 7' Declination of nonagesimal = s i n - ' (sin 24° sin 114° 7') = 21°47'30"
= 0'5"
Solar EcUpse
115
(Moon-Node) at the instant of conjunction : 259° 40' 11" - 2 5 5 ° 15'0" 4° 25' 11" Moon's latitude, P = 270' sin (Moon - Node) = 270'sin (4° 25' 11") = 20'48" Moon's diameter = True daily motion of Moon x = 731'14" x ^ Sun's diameter
= 29'36"
= 61'26" x | ^ = 33'47"
Sun's true longitude = 259° 38' 18" Ayandmsa Sayana Ravi
= S
Sun's declination
23° 45'
= 283° 23' 18" = sin" ' (sin24° sin 283° 23' 18") = - 2 3 ° 18' 30"
At the instant of conjunction we have : Longitude of the orient ecliptic point
=
204° 7'
Longitude of the nonagesimal N
= = = =
114° 7'
Declination of the nonagesimal Node's latitude Moon's celestial latitude P
Sun's app. diameter
= =
20' 48" 731'41" 33' 47" 29' 36"
Moon's app. diameter Latitude of the place
255° 15' 61'26"
Sun's apparent daily motion Moon's app. daily motion
21°47'30"
=
13°
z = Declination of nonagesimal + Lat. of Moon - Lat. of place = 21° 47' 30"+ 2 0 ' 4 8 " - 13° = S-N
9° 8 ' 1 8 "
= 283° 23' 1 8 " - 1 1 4 ° r = 169° 16' 18
The equation of apparent conjunction = - 4cos z' X sin {S - N) gh
116
Ancient Indicm Astronomy = - 4 cos (9° 8' 18") X sin (169° 16' 18") = - 0 * 1 7 ' " 38* Time of apparent conjunction = 0* 44"" 13* - 0* 17"" 38* = 0* 26"" 35* 2nd Iteration Sun's motion in (- 0* 17"* 38*) - 0 * 17"" 38* X 6 1 ' 2 6 " 24* sayana Ravi = 283° 23' 18" -
0'45"
= 283° 22' 33" Sayana Lagna : 6* 19° 58'
= 199° 58'
The nonagesimal
= 109° 58'
Declination of the nonagesimal
= s i n " ' (sin 24° x sin 109° 58') = 22° 2 8 ' 3 1 "
Moon's motion in ( - 0 * 17"" 38*) ^ - 0 ^ 17"* 3 8 * x 7 3 1 ' 4 r 24*
^ _ 5.57,,
Therefore, Moon's app. longitude at conjunction : 259° 40' 1 1 " - 8 ' 5 7 " = 259° 31' 14" (Moon-Node) at apparent conjunction : 259° 31' 1 4 " - 2 5 5 ° 1 5 ' 0 " = 4 ° 16' 14" Moon's latitude, p
= 270' sin (4° 16' 14") = 20' 6"
At the apparent conjunction, we have Longitude of orient ecliptic point
=
199° 58'
Longitude of the nonagesimal
=
109° 58'
Declination of the nonagesimal
= 22° 2 8 ' 3 1 "
Moon's latitude
=
20'6"
Sun's app. daily motion
=
61'26"
Moon's app. daily motion
=
731'41"
Latitude of the place
=
13°
z Note :
= Decl. of nonagesimal + Moon's lat. - lat. of place
= 22° 2 8 ' 3 1 " + 2 0 ' 6 " - 13° = 9° 48' 37" In traditional texts the northern latitudes of places are taken as negative. Equation of apparent conjunction = - 4 cos (9° 48' 37") x sin (173° 24' 33") = - 0* 10"' 52*
Solar Eclipse
117
3rd Iteration Time of apparent conjunction
= 0*44'" 13' - 0* 10"" 52' = 0*33"'21*
At the apparent conjunction : Sun's noUon in (- 0* 10" 52')
= - <^ "^T 24*
61'26"
= -0'28" Therefore, Sayana Ravi S
= 2 8 3 ° 23' 18" - 0' 28" = 283° 22' 50"
Sayana Lagna =6'2\°
= 201° 27'
IT
Longitude of the nonagesimal
= 111° 27'
Declination of the nonagesimal
= sin" ' (sin 24° sin 111° 27') = 22° 14'41"
Moon's motion in (-0* 10"* 52') ^ - 0 ^ ^0"*52'><731'41" ^ 3 , , , 24* Moon's long, at app. conjn.
= 259° 40' 11" - 5' 31" = 259° 34' 40"
(Moon - Node) at app. conjn.
= 259° 34' 40" - 255° 15' 0" = 4 ° 19' 40"
Moon's latitude, p = 270' sin (4° 19' 40") = 20' 22" Thus, at the apparent conjunction, Vfe have Long, of orient ecliptic point
= 201° 27'
Longitude of nonagesimal
= 111° 27'
Declination of nonagesimal
= 22° 14'41"
Moons' latitude
= 20' 22"
Sun's app. daily motion
= 61'26"
Moon's app. daily motion
= 731'41"
Latitude of Bangalore
= 13°
.'. z
= Decl. of nonagesimal + Moon's lat. - lat. of place = 22° 14'41"+ 2 0 ' 2 2 " - 1 3 ° = 9° 35'3"
S - N = 283° 22' 50" - 111° 27' = 171° 55' 50" Eqn. of apparent conjn. = - 4 cos (9° 35' 3") X sin (171° 55' 50") =
-0*13"'17'
4th Iteration : Eqn. of apparent conjunction : - 0 * 12"'53'
118
Ancient Indian Astronomy 5th Iteration : Eqn. of apparent conjunction : - 0 * 12"" 51* After 5 Iterations : Sun's diameter
= 33' 47"
Moon's diameter
= 29' 36"
Sum of semi-diameters
= ^ ^ y ^ = 31' 4 r ' . 5
Portion obscured = (Sum of semi-diameters) - M o o n ' s latitude = 31'41".5-20'18" =
11'23".5
Half duration of the eclipse Consider the quadratic equation : (31' 4l".5f
= {tx 670'.25)^ + [20' 18" +1 (Change of Moon's lat./hr)]^
Now, Moon's lat. P = sin (4°.5) sin (Moon - Node) ^
= sin4°.5 cos
(M-R)
dt
dt
= sin 4°.5 [cos (4° 18' 38")] (130.16194) = 61'47" where M and R are true longitudes of Moon and Node. The quadratic equation reduces to (31' 41".5)2 = {tx 670'.25)2 + [20' 18" + f (61' 47")]^ _ - 2508.4033 ± V6292087.1 + 4 x 453052.24 x 592.277 2 X 453052.24 Considering the positive and negative signs in the above, we get First half duration = - 0* 56"" 12* Second half duration = 0* 48*" 14* Summary of the solar eclipse Time of geometric conjunction Equation of apparent conjunction Time of apparent conjunction
-12"* 51* 0*31'" 22*
.-. Time of the beginning of the eclipse: 0*31'" 2 2 * - 0 * 5 6 ' " 12* = 23* 35"'10* in the night between January 4th and 5th. Time of the end of the eclipse
: 0* 31"' 22* + 0* 48"' 14* = 1*19"'36*
Duration of the eclipse
: 0* 56'" 12* + 0* 48"' 14* = 1*44"" 26*
on the 5th of January, 1992.
11
M E A N POSITIONS O F T H E
STAR-PLANETS
( K U J A , B U D H A , G U R U , S U K R A A N D SANI)
11.1
Introduction The mean positions of the five taragrahas ("star-planets")
namely, Kuja, Budha,
Guru, Sukra and Sani are determined by the same procedure as in the case of the sun and the moon. Mars, Mercury, Jupiter, Venus and Saturn are called "star-planets" since these appear small like stars in contrast to the sun and the moon. Let
be the mean longitude of a planet at the chosen epoch, d be the daily mean
motion of the planet and A be the Ahargana, the number of days elapsed since the epoch upto the day under consideration. Then, the mean longitude of the planet is given by X = Xo +
Axd
As in the case of the sun and the moon, we choose the beginning of Kaliyuga i.e. the mean midnight between 17th and 18th, February 3102 B . C . as the epoch. The mean daily motion (in revolution) is given by : d = No. of revolutions in a kalpa I No. of civil days in a kalpa where 1 kalpa = 432 x 10^ years. The details are given in Table 11.1. Table 11.1: Revolutions of planets in a Mahayuga (Surya Siddhanta) (Number of civil days in a Mahayuga 157,79,17,828) Planet Kuja Budhsk-sighrocca Guru Sxikia-sigghrocca Sani Note :
No. of Revolutions
Mean daily modon (d)
22,96,832
0°.5240193
1,79,37,060
4°.0923181
3,64,220
0''.0830963
70,22,376
1 ".6021464
1,46,568
0°.0334393
In the case of Budha and Sukra, the positions of their Sighrocca are considered. This aspect will be discussed in the next chapter.
120
Ancient Indian Astronomy
Example : Mean longitudes of the five planets at the midnight between March 21 and 22, 1991. Now, the Kali Ahargana : 18,59,872. (1)
Mean longitude of Kuja = Ahargana x Mean daily motion =
1859872x0.5240193
= 974608°.82 = 88° 49' 12" (after removing the nearest integral multiple of 360°) (2)
Mean longitude of Budha's sighrocca = 1859872 x 4 ° . 0 9 2 3 1 8 1 = 7611187°.8 = 67° 48' (after removing the nearest integral multiple of 360°)
(3)
Mean longitude of Guru = 1859872 xO°.0830963 = 154548°.48 = 108° 2 8 ' 4 8 " (after removing the nearest integral multiple of 360°)
(4)
Mean longitude of Sukra's sighrocca = 1859872 X 1°.6021464 = 2979787°.2 = 67° 12' (after removing the nearest integral multiple of 360°)
(5)
Mean longitude of Sani = 1859872 x 0 ° . 0 3 3 4 3 9 3 = 62192°.818 = 272° 4 9 ' 5 "
Note :
(after removing the nearest integral multiple of 360°) In the above computations greater accuracy can be obtained by considering more than 8-digits display in a calculator and double precision (16 digits) on a computer
11.2 Desantara
correction for the planets
While considering the mean positions of the sun and the moon in Chapter 6 it was explained that due to the difference in the longitude of the prime meridian
(Ujjayini)
Mean Positions of the
Star-Planets
121
and the place under consideration, a correction called Desantara correction has to be applied to the mean longitudes of the sun and the moon. This correction due to Desantara has to be applied to the mean longitudes of the remaining planets also. As explained in Sec. 6.4, Desantara correction = - (X - X^) d/360 where X = (terrestrial) longitude of the place w.r.t. Greenwich Xg = longitude of Ujjayini = 75° 47' (E) d
= Mean daily motion of the planet.
The negative sign indicates that the correction is subtractive when A,>>,p i.e. the place is to the east of Ujjayini. On the other hand, for a place to the west of Ujjayini (i.e. X < X^), the correction becomes additive. In modem calculations terrestrial longitudes X and Xq are taken with respect to Greenwich. Example : Apply the Desantara correction to the five planets for the midnight preceding March 22, 1991 at Bangalore. Now, we have Longitude of Ujjayini,
X^ = 75°47'
E
Longitude of Bangalore,
X = 77° 35' E
so that ( X - \ o ) / 3 6 0 = 1.8/360 If this factor is multiplied by the mean daily motion d of a planet, we obtain the Desantara correction for the planet and it is subtractive since X > X^. We shall calculate this correction and apply it to each of the five planets as shown below using d as given in Table 11.1. (1)
Desantara correction for Kuja = - (1.8/360) X 0°.5240193 = - 0° 0' 09" Therefore, the mean longitude of Kuja at the midnight at Bangalore : 88° 49' 1 2 " - 0 9 " = 88° 4 9 ' 0 3 "
(2)
Desantara correction for Budha's sighrocca = - (1.8/360) X 4°.0923181 = - 0 ° 01' 14" Therefore the mean longitude of Budha's sighrocca Bangalore
at the midnight at
= 67° 48' - 1' 14" = 67° 46' 46" (3)
Desantara correction for Guru = - ( 1 . 8 / 3 6 0 ) x 0 ° . 0 8 3 0 9 6 3 = - 0 ° 0 ' 1".5 Therefore, the mean longitude of Guru at the midnight at Bangalore = 108° 28' 48" - r.5 = 108° 28' 46".5
(4)
Desantara correction for Sukra's sighrocca = - (1.8/360) X 1°.6921464 = - 0° 0' 29"
122
Ancient Indian Astronomy Therefore, the mean longitude of Sukia's' sighrocca at Bangalore at midnight : = 67° 1 2 ' - 2 9 " = 67° 11'31" (5)
Desantara correction for Sani = - (1.8/360) X 0°.0334393 = - 0 ° 0' 0".6 = - 1" Therefore, the mean longitude of Sani at the midnight at Bangalore = 272° 49' 5" - 1" = 272° 49' 4 "
11.3 Mean positions of planets according to Let A be the ahargana of the given date with respect to the Kharida khadyaka epoch. Then the mean positions of the five taragrahas at the midnight preceding the given date are given by the following expressions: A - 4 9 6 + 0.25
A ^ ^ ^ " ^ + 1 ^
. , ™ " - °f ^'"^
(i)
Mean Kuja
•
(ii)
Mean Guru
A - 2 1 1 3 + 0.2 A , ^ ^ 1 ^^^"^•-T6262r''^S-
(iii)
Mean Sani
(iv)
Mean Sighrocca
A — 2491 5
-
A x 100-2181 8797
A
A 7H04
.
.
of Budha (v)
(A - 37.25) X 10 A-l\2 • ^ ^^^"^--^^3-
Mean Sighrocca
.
of Sukra Example : We shall obtain according to the three texts, the mean longitudes of the taragrahas
at the midnight
between February 4th and 5th, 1962. In this section, we
consider the Kharida khadyaka.
The KK Ahargaria =473670 = A. Using the above
expressions (i) to (v) the mean longitudes for the given date are as follows : (1)
Mean Kuja at Ujjayini midnight Desantara correction
(2)
-10"
Mean Kuja at Bangalore midnight
271° 37' 24"
Mean Guru at Ujjayini midnight
304° 40' 0 2 "
Desantara correction
(3)
271° 37' 34"
-02"
Mean Kuja at Bangalore midnight
3 0 4 ° 4 0 ' 00"
Mean Sani at Ujjayini midnight
275° 27' 07"
Desantara correction Mean Sani at Bangalore midnight
-01" 275° 27' 06"
Mean Positions of the Star-Planets (4)
123
Mean Sighrocca of Budha at Ujjayini midnight
72° 46'24"
Desantara correction
(5)
- 0 1 ' 15"
Mean Sighrocca of Budha at Bangalore midnight
72° 45' 09"
Mean Sighrocca of Sukra at Ujjayini midnight
310° 20'30"
Desantara correction
-29" 310° 20'01"
Mean Sighrocca of Sukra at Bangalore midnight 11.4 Mean positions of planets according to GL
As in the case of the sun and the moon, the mean positions of the five taragrahas are determined using the cakras and ahargaria obtained with respect to the epoch of the Graha ldghavam. For this purpose we use the following table of dhruvakas and ksepakas. Table 11.2 : Dhruvakas and Ksepakas Planets Dhruvaka Ksepaka
Kuja
Budha
Guru
Sukra
Sani
(-)4*3°27'
( - ) 0 ' 2 6 ° 18'
(-1 l M 4 ° 2 '
(-)7' 15° 42'
(-) 1* 25° 32'
7* 2° 16'
7'20° 9'
9*15° 21'
10'7° 8'
8' 29° 33'
Here, ksepaka of a planet is its mean {nirayana) position at the epoch and dhruvaka is the residue of the motion of the planet at the end of a cakra. The dhruvaka \s subtractive for all the five planets as shown in T^ble 11.2. In the case of Budha and Sukra, their respective Sighra anomalies (and not Sighroccas) are given. The procedure for determining the mean longitude of each planet is explained below with an example. For the given date let C be the Cakras and A be the Ahargana with reference to the GL epoch. Let £) and AT respectively be die Dhruvaka and die Ksepaka of a planet as in Table 11.2. (i) Mean Kuja ^
deg. - ^
min. - (C x D) deg. + K deg.
(ii) Mean Guru A A — deg. - ~ min. + (C x D) deg. + K deg. (iii) Mean Sani ^
deg. +
min. + (C x D) deg. + K deg.
(iv) Mean Sighra kendra of Budha 3/4 A 3A deg. + — deg. - — min. + (C x D) deg + K deg. ZO
JO
Ancient Indian Astrononry
124 (v) Mean Sighra kendra of Sukra y Note :
deg. +
deg. + iCxD)
deg. + K deg.
In expressions (i) to (v) above Dhruvaka D is negative. The Ksepakas are for the mean sunrise at Ujjayini on 19-3-1520 (J), the epoch.
Example : The mean positions of the taragrahas at the midnight preceding February 5, 1962. Here, Cakras = 40 and Ahargana =744, at the sunrise on the previous day, 4-2-1962. (1)
(2)
Mean Kuja at f/jjaymr sunrise
: 275° 4 0 ' 5 0 "
Desantara correction
-0°0'
Motion of Kuja for 18 hours
+ 0 ° 2 3 ' 35"
Mean Kuja for the given time at Bangalore
276° 04'
Mean Guru at Ujjayini
302° 05' 22"
sunrise
Desantara correction
-0°
Motion of Guru for 18 hours
(3)
(4)
(5)
15"
0' 0 2 "
0°03'
45"
Mean Guru for the given time at Bangalore
302° 09' 05"
Mean Sani at Ujjayini sunrise
282° 13' 50"
Desantara correction
- 0 ° 0' 0 1 "
Motion for 18 hours
+ 0°01'
Mean Sani for the given time at Bangalore
282° 15' 29"
30"
162°56'
17"
Desantara correction
-0°
57"
Motion for 18 hours
+ 2° 19' 48"
Budha Sighra kendra at the given time at Bangalore
165° 15' 08"
Mean Sighra kendra of Budha at
Ujjayini
sunrise
0'
Mean Sighra kendra of Sukra at Ujjayini sunrise
7°32'
Desantara correction
- 0 ° 0' 11"
Motion for 18
hours
Sukra Sighra kendra at the given time at Bangalore Note :
10"
+ 0°27'
54"
45"
8° 0' 28"
In GL the ;nean Sighra kendras of Budha and Sukra are directly determined instead of using the Sighroccas.
12
T R U E POSITIONS O F T H E S T A R - P L A N E T S
12.1. Manda correction for the taragrahas The mean. positions of the planets viz., Kuja, Budha's sighrocca, Guru, Sukra's sighrocca and Sani are obtained, as explained in the previous chapter, assuming uniform circular motion for them. Since by observations it was found that the motion of each pknet was non-uniform, suitable corrections were devised. These are called inanda and sighra corrections. The manda correction for the five planets is similar to that for the sun and the moon discussed in Chapter 7. The points 5 and 5 ' in Fig. 7.1 now represent respectively the true and the mean planet. The manda phala in the case of a planet is given by Mandaphala = (r/R) (R sin in) where R sin m is the Indian sine of the planet's anomaly m and r is the radius of the epicycle of the "apsis" as distinguished from the epicycle of "conjunction" which will be discussed shortly. The radius r of the manda epicycle is variable (see Table 12.1) as given in the Siirya Siddhanta. Table 12.1: Peripheries of manda epicycles (SS) Planet
Periphery of manda epicycle At the end of odd quadrants
At the end of even quadrants
Kuja Budha Guru Sukra
72° 28° 32° 11°
75° 30° 33° 12°
Sani
48°
49°
The corrected periphery p for any given manda anomaly m is : /'=Pe-(/'e-/'o) where Pg and Pg are respectively the peripheries of a planet at the end of even quadrants (i.e. at m = 180° and m = 360° or 0°) and at the end of odd quadrants (at in = 90° and m = 270°); m is the manda anomaly of the planet given by Manda anomaly = Mandocca of the planet - Mean planet For example, in the case of Kuja, the corrected periphery of the inanda epicycle is given by
126
Ancient Indian Astronomy /j = 7 5 ° - ( 7 5 ° - 7 2 ° > l s i n m l
= 75° - 3° I sin m I The mandoccas of the five planets are given in Table 12.2 in terms of the revolutions completed in the course of a kalpa. The mandoccas at the beginning of the Kaliyuga are also provided in Table 122 according to the Siirya Siddhanta. Table 12.2: Revolutions of mandoccas in ii Kalpa and their positions at the beginning of Kaliyuga Planet
No. of revns. in a Kalpa
Mandocca at the beginning of Kaliyuga
Kuja
204
4' 9° 57' 36"
Budha
368
7' 10° 19' 12"
Guru
900
5'21°0'0"
Sukra
535
2M9°39'0"
39
Sani
7' 26° 36' 36"
The method of calculating the mandocca of a planet on a given day, with Ahargana A, using Table 12.2 is as follows : Mandocca = {Mandocca at the beginning of Kali) + (No. of revns. in a Kalpa) x 360° x A/{No.
of civil days in a Kalpa).
The number of civil days in a Kalpa = 1577917828000. Examples : (i)
Find the mandocca of Guru as on March 22, 1991. We have, for the given day, A = 1859872. According to Table 12.2, the number of revolutions of Guru's mandocca in a Kalpa is 900 and its position at the beginning of the Kaliyuga is 5' 21°. Therefore, Guru's mandocca = 5' 21 ° + (900 X 360° x 1859872/1577917828000)
(ii)
= 5*21° + 0° 23'= 5 ' 2 1 ° 23' Find the manda correction for Jupiter (Guru) as on March 22, 1991 at the preceding midnight. We have Guru's manda anomaly, m = Guru's mandocca - mean longitude of Guru = (5* 21 ° 23') - ( 3 M 8° 29') = 2^ 2° 54' = 62° 54' = 62°.9 Guru's corrected periphery of the manda epicycle, /' = P . - ( / ' « - P „ ) l s i n / « l = 33°-(33°-32°)lsin(62°.9)l
True Positions of the Star-Planets
127 = 3 3 ° - l ° l s i n (62°.9) I = 32°.109787 = 3 2 ° 6 ' 3 5 "
Therefore, we have Guru's mandaphala = 3438' x (/7°/360°) x sin m = 3438' X (32°. 109787/360°) X sin (62°.9) = 272'.98239 = 4 ° 3 2 ' 59" Since Guru's anomaly m is less than 180°, the mandaphala additive. Note ;
is
In the Surya siddhanta, the anomaly m of a planet is defined by m = Mandocca - mean planet and the corresponding mandaphala is additive for »i< 180° But, generally, in other siddhanta texts, m is defined as mean planet minus its mandocca. Accordingly, the resulting mandaphala is subractive and addidve respectively for m < 180° and m > 180°. However, the resulting effect is the same, from both the methods, in correcting the mean position of a planet by the manda equation.
12.2 Sighra correction for the taragrahas The sighra correction corresponds to the "elongation" in the case of Budha and Sukra from the sun and the annual parallax in the case of Kuja, Guru and Sani. The manda correction is applied to the mean longitude of a planet to get the "true-mean" or manda corrected (mandasphuta graha) position of the planet. Now, the concept of the sighra correcdon is explained with the help of Fig. 12.1. Let the circle CDFG,
with the earth at
the centre E, represent the kaksdvrtta (or deferent circle) of a planet. Just like the manda epicycle, a sighra epicycle is prescribed with a specified variable radius for each planet. Let C be the centre of the sighra epicycle of the planet. While C moves along the deferent circle, the planet moves along its epicycle. The epicycle in this case is called sighra-nicocca-vrtta. Let CEF cut the epicycle at U and N which are respectively sighrocca and sighranica {sighra apogee andperigee) of the sighra epicycle. The centre C of the epicycle moves along the deferent circle with the velocity of the corrected planet {mandasphuta graha). Let the planet move
M J
/
N L
Fig. 12.1: Sighra epicycle
Ancient IndUm Astronomy
128
from U' to M akmg the qncycle so that arc U'M is equal to arc C ' C. Join EM cutting the deferent at M ' . Then C is the mandasphuta graha and A / ' is the tnie planet (sphuta graha). Therefore, the correction to be made to the longitude of the "true mean" planet (i.e. manrfa-corrected planet) is the zicCM'. is called sighraphala. C L,C'
The correction, arc C A f ' in angular measure,
Now, in order to obtain an expression for the arc C A/',,draw
P and MQ perpendiculars respectively to C £ , EM and U 'E.
The angle C 'EC which is the angle between the sighrocca and the mandasphutagraha is called the sighra kendra or the anomaly of conjunction. From Fig. 12.1, we have C'L = R sin (sighra anomaly) EL = R cos (sighra anomaly) Also, arc ( / ' A f = arc C ' C and angle f / ' C ' A f = angle C ' £ C and hence the triangles MC'Q
and C'EL are similar. Therefore, MQ/MC'
=
C'L/C'E
MQ
=C'LxMC'/C'E = R sin (sighra anomaly) x Radius of epicycle / R = R sin (sighra anomaly) x Epicyclic periphery / 360° = dohphala.
Again, from the same similar right-angled triangles, we have C'Q/C'M
=
EL/EC
.: C'Q =
ELxMC'/C'E
= R cos (sighra anomaly) x Radius of epicycle / R = R cos (sighra anomaly) x Epicyclic periphery / 360° = kotiphala Now, from F i g . 12.1, we have sphutakoti EQ = EC' + C'Q = R+ kotiphala The kotiphala is positive or negative according as the sighra anomaly is in the fourth and first quadrants (i.e. between 270° and 90°) or in the second and third quadrants (i.e. from 90° to 270°). Then we have sighrakarna
EM = ^EQ^ + MQ^ = ^(C'E
+ C'Q)^
+ MQ'^
the hypotenuse of the right-angled triangle MEQ. From the similar triangles EC 'P and EMQ, we have C'P/C'E
=
MQ/EM
True Positions of the Star-Planets
.-. C'P = MQxEC'/EM
129 = (Dohphala xR) I sighrakarna
Then the sighraphala, arc C 'M',
is the arc corresponding to C'PasR
sin (sighra
anomaly). It is important to note that in the Siirya siddhanta, we have Sighra anomaly = sighrocca - mean planet In the case of the superior planets viz, Kuja, Guru and Sani, their mean sighrocca is the same as the mean longitude of the sun. In the case of Budha and Sukra, their mean longitude is taken to be that of the sun while their sighroccas are special points. In the siddhdntie texts while the revolutions of the other mean planets, in a Kalpa or a Mahayuga are given, in the case of Budha and Sukra, the revolutions of their sighroccas are given. Thus, we have, according to the Siirya siddhanta : (i)
For the superior planets viz., Kuja, Guru and Sani, Sighra anomaly = Mean sun - mean planet
(ii)
For the inferior planets viz. Budha and Sukra Sighra anomaly = Planet's sighrocca - mean sun In both the cases, we have R sin (Sighraphala) = (r/k) (R sin m)
where r is the corrected radius of the sighra epicycle of the planet, k is the sighra hypotenuse (sighrakarna) and R sin m is the Indian sine of the sighra anomaly in of the planet. It is important to note that the radius r of the sighra epicycle is a variable even as in the case of the manda epicycle. The peripheries of the sighra epicycles of the five star-planets are listed in Table 12.3. Table 12.3: Peripheries of sighra epicycles (SS) Periphery of sigh ra epicycle At the end of the odd quadrants
At the end of the even quadrants
Kuja
232°
235°
Budha
132°
133°
Guru
72°
70°
Sukra
260°
262°
Sani
40°
39°
Planet
As in the case of the manda epicycle, the radius and the periphery of the sighra epicycle are variable. The corrected periphery p for a given sighra anomaly is given by P = Pe-(Pe-Po)
Isinml
130
Ancient Indian Astronomy
where is the periphery of a planet's epicycle at the end of an even quadrant, Pg is that at the end of an odd quadrant and m is the sighra anomaly. The sighra anomaly, as pointed out earlier, is given by Sighra anomaly = sighrocca - mean planet. Example : Find the Sighra anomaly and hence the corrected periphery of the Sighra epicycle in the case of Budha and Sani for the midnight preceding March 22, 1991 at Bangalore. For the given date, time and place, the mean positions (after the desantara correction) are as given below :
(i)
Mean longitude of the sun
: 334° 51' 30"
Mean Sighrocca of Budha
: 67° 46' 46"
Mean longitude of Sani
: 272° 49' 04"
Therefore, we have Budha's sighra anomaly m = 67° 4 6 ' 4 6 " - 3 3 4 ° 5 1 ' 3 0 " = 9 2 ° 55' 16" (after adding 360°) = 92°.921 The corrected periphery of Budha's Sighra epicycle is
= 1 3 3 ° - ( 1 3 3 ° - 1 3 2 ° ) (0.0161) = 132°.98392 or 132° 5 9 ' 2 " (ii)
We have the Sighra anomaly of Sani, m = Sighrocca of Sani - Mean Sani = Mean sun - Mean Sani = 334° 5 1 ' 3 0 " - 2 7 2 ° 4 9 ' 0 4 " = 62° 02' 26" = 62°.0406 The corrected periphery of Sani's Sighra epicycle p = 39°-
(39° - 40°) I sin 62°.0406 I
= 3 9 ° + (1°) (0.88328) = 39°.88328 = 39° 53' The sighra anomaly and the corrected Sighra periphery for the remaining planets can also be found similarly. 12.3 Working rule to determine the sighra correction After finding the Sighra anomaly m and the corrected periphery of the Sighra epicycle for a planet, the Slghra correction is determined as follows (with R = 3438) :
True Positions of the Star-Planets
(i) (ii) (iii)
Dohphala - {p°/im°)
131
X R sin (m)
Kotiphala = ( p V 3 6 0 ° ) x R cos (m) Sphutakoti =R±
Kotiphala
where the positive or the negative sign is taken according as m lies between 270° and 90° (i.e. I V and I quads.) or between 90° and 270° (i.e. II and HI quads.) (iv)
Sighrakarna (or sighra hypotenuse) (Sphutakoti)'^ + {Dohphalaf
(v)
Then, we have R sin {Sighraphala) = {Dohphala I Sighrakarna) x R :. Sighraphala = s i n " ' (I Dohphala I Sighrakarna I)
(vi)
The Sighraphala is additive or subtractive according as the Sighra anomaly is less than 180° or greater than 180°' From the above working rule, we have Sphutakoti =R± (/j/360) Rcosm
= R[\ ± {p/360) cos m]
Dohphala = (p/360) R sin m Sighrakarna = V(pV3602) /?2 (sin^ m) + /?2 [1 ± (p/360) cos m]^ = y? V ( p V 3 6 0 2 ) (sin^ m) + (pV3602) (cos^ m) ± 2 (p/360) (cos m) + 1 = /? V(pV3602) ± 2 (p/360) (cos m) + 1 Therefore, we have Sighraphala = sin" ' [(p/360) (sin m ) / V ( p V 3 6 0 2 ) ± 2 (p/360) (cos m) + 1 ] = sin"
[r sin
± 2r cos m + 1 ]
where r = (p/360) is the corrected radius of the epicycle. In the above formula, of the alternative signs (±), the positive sign is to be taken if m is greater than 270° but less than 90° (i.e. 270° < / M < 3 6 0 ° or 0° < m < 90°) and the negative sign if 90°
/?i = 92°.921
Ancient Indian Astronomy Sani's corrected periphery, p = 39°.88328 Sani's sighra anomaly,
m = 62°.0406
Sighra correction for Budha : (0
Dohphala
= (132.98392/360) x 3 4 3 8 ' x sin (92°.921) = 1268'.3464
(ii)
(iii)
Kotiphala
= (132.98392/360) x 3 4 3 8 ' x cos (92°.921)
Sphutakoti
= -64'.717731 = 3438' - 64'.717731 = 3373'.2823
(iv)
Sighrakarna
= V(3373'.2823)2 + (1268'.3464)2 = 3603'.8502
(v)
R sin (Sighraphala) = 1268'.3464 x 3438'/3603'.8502 = 1209'.9767
Therefore Sighraphala
= sin" ' [1209.9767/3438] = 20°.606145
= 20° 36' 2 2 " Since the Sighra anomaly m = 92°.921 is less than 180° the STghra correction is additive. i.e. the if^hra correction = + 20° 36' 22" Sighra correction for Sani : (i)
Dohphala
= (39.88328/360) x 3438' x sin (62°.0406)
(ii)
Kotiphala
= (39.88328/360) x 3438' x cos (62°.0406)
(iii)
Sphutakoti
= 178'.57648 = 3438' + 178'.57648 = 3616'.5765
(iv)
Sighrakarna
= V(336'.4284)2 + (3616'.5765)2
= 336'.4284
= 3632'. 1907 (v)
R sin (Sighraphala) = 336'.4284 X 3438'/3632'.1907 = 318'.44166
Therefore, Sighraphala
= sin"' [318.44166/3438] = 5°.3145878 = 5° 18' 5 3 "
Since the Sighra anomaly, m = 62°.0406 is less than 180° the correction is addidve.
True Positions of the Star-Planets
133
:. Sighra correction = + 5° 18' 53" 12.4 Application of manda and sighra corrections to taragrahas In the case of the five taragrahas viz, Budha, Sukra, Kuja, Guru and Sani, the manda and the sighra corrections are applied successively one after the other, according to the prescribed rule, to get the true positions of the planet. In fact, though the prescribed rules slightly differ from text to text, essentially the application is an iterative process for getting a convergent value as the true position. Surya siddhanta gives the following procedure for applying the manda and the sighra corrections successively : 1st Operation : To the mean planet add half of the sighra correction. Let MP be the mean longitude of the planet (after desantara correction) and SE^ be the sighra correction calculated for MP. Then, the position of the planet after the first opwration is given by P, = M P + ( l / 2 ) 5 £ , 2nd Operation : To the position thus obtained from the 1st operation, add half of the corresponding manda correction. For the first corrected position, P j , suppose the corresponding manda correction is MEy Then, the position of the planet after this second correction is given by P2 = P , +(1/2)M£:, 3rd Operation : From the position thus corrected, find the manda correction and apply it entirely to the original mean position of the planet. Thus, the manda correction ME2 is determined corresponding to the twice corrected position of the planet, namely P2 and then it is applied to the original mean position MP of the planet. That is, the position after the 3rd operation is given by P2 = MP + ME2
where the manda correction ME2 is calculated taking P2 as the mean planet. Finally, the fourth operation is effected to get the true position of the planet. 4th Operation : From the position of tKei planet obtained after the 3rd operation, find the sighra correction and apply the whole of it to the same. This means that the sighra correction
obtained from the position P 3 of the planet
is applied entirely to P3. The position P4 after this fourth correction is, therefore, given by P4 = P3 + 5£2
134 Note :
Ancient Indian Astronomy The above four operations may be repeated in the same order, treating P4 as the new mean position of the planet. A repeated application of this cycle of four operations refine the position of the plan^, iteratively by successive approximadons.
Example: Find the true position of Sani by applying the four operation t\ die manda and the sighra corrections, for the midnight preceding March 22, 1991 at Bangalore. We have the following details : Mean longitude of Sani (MP) (after Desantara correction) : 272° 49' 4 " Mean longitude of the sun : 334° 51' 30" Corrected periphery of Sani's sighra epicycle : 39°.88328 Sighra anomaly of Sani : 62°.0406 1st Operation : We found in section 12.3 that for the mean position MP of Sani, Sighra equation = + 5 ° 18' 53" = 5 £ , Therefore, after the 1st correction, Sani's position P\ is P2 = A / P + ( 1 / 2 ) S £ , = 272° 4 9 ' 0 4 " + 2° 3 9 ' 2 7 " = 275° 2 8 ' 3 1 " 2nd Operation : The position of Sani after the 2nd operation, P2 is given by P2 =
Py+il/2)MEy
where M E j is the manda correction corresponding to the first corrected position P j of Sani. J, ^Vee lhave for the given day, the Kali Ahargaria A = 1859872. Using Table 12.2,
1^
l i ' s mandocca = {T 26° 36' 36") + (39 x 360° x 1859872/1577917828000)
ij
=7* 26° 3 7 ' 3 6 "
Therefore, Sani's manda anomaly, m = Sani's mandocca - mean longitude of Sani = 7* 26° 3 7 ' 3 6 " - 9 ' 2 ° 4 9 ' 0 4 " = 10* 23° 48' 32" = 323°.8089 ^ Rani's corrected periphery of the manda epicycle, p = 49°-
(49° - 48°) I sin 323°.80891
= 48°.40952
^
True Positions of the Star-Planets
135
Therefore, Sani's mandaphala = 3438' x (48°.40952/360*') x sin (323°.8089) = -272'.98549. i.e.
M£, =-4°32'59"
It is negative since m > 180°. Therefore the position of Sani after this second operation is given by P2 = P,+(1/2)M£', = 275° 28' 31" - (1/2) (4° 32' 59") = 275° 2 8 ' 3 1 " - 2 ° 16' 30" = 273° 12'01" 3rd Operation : The position after the 3rd operation is given by P2=MP
+ ME2
where MP = 272° 49' 04", the mean longitude of Sani and ME2 is the manda correction obtained for the last corrected position P 2 . Now, Sani's new manda anomaly, m = Sani's mandocca - P2 = 236° 37' 3 6 " - 2 7 3 ° 12'01" = 323° 25' 35" = 323°.4263389 Sani's corrected periphery of the manda epicycle p = 49° - 1 ° XI sin 323°.426389 I = 48°.404145 Therefore, Sani's mandaphala (new) ME2 = (3438' X 48°.404145/360°) x sin (323°.426389) = -275'.4397 = - 4 ° 3 5 ' 26" The position of Sani after this third operation is therefore given by P^ = MP + ME2 = 272° 4 9 ' 0 4 " - 4 ° 35' 26" = 268° 13'38" 4th Operation : Finally, the last correction 5^2. the second sighra correction, is determined for the last corrected position i.e., P 3 and applied to the same. Thus, the fourth corrected position P4 is given by
136
Ancient Indian Astronomy
The new sighra anomaly of Sani m = Sighrocca of Sani - P 3 = 334° 51' 3 0 " - 2 6 8 ° 13' 38" = 66° 3 7 ' 5 2 " = 66°.63111 The corrected periphery of Sani's sighra epicycle p is given by p = 3 9 ° + l ° x l s i n 66°63111 I = 39°.9179701 or 39° 5 5 ' 0 5 " Next, we shall calculate the Dohphala, Kotiphala, Sphutakoti, and Sphutakarna to obtain the resulting sighraphala. (i)
Dohphala
= (p°/360°) X /? X sin (m) = (39.9179701/360) x 3438' x sin (66°.63111) = 349'.94546
(ii)
Kotiphala
= (p°/360°)x/?xcos(m) = (39.9179701/360) x 3438' x cos (66°.63111) = 151'.20939
(iii)
Sphutakoti
= /? + Kotiphala = 3589'.2094
t _
(iv)
Sighrakarna
= ^{Sphutakoti)^ + {Dohphala)^ = V(3589.2094)2 + (349.94546)^ = V l 2 8 8 2 4 2 4 + 122461.82 = 3606'.2288
(v)
Sighraphala
= sin" ' (I Dohphala/Sighrakarria
I)
= sin" ' (1 349.94546/3606.3388 I) = 5°.5686983 = 5°34'07"s5£2 Since the sighra anomaly, m = 66°.63111 < 180°, the sighra correction is additive. Therefore, the finally corrected true longitude of Sani is P4 = P3 + 5£2
True Posttions of the Star-Planets
137
= 268° 13' 38"+ 5° 3 4 ' 0 7 " = 273° 4 7 ' 4 5 " 12.5 TVue daily motion of the
taragrahas
The mean daily motions of the five stat-planets viz. Kuja, Budha's sighrocca, Guru, Sukra's sighrocca and Sani are obtained assuming their uniform motion. However, due to the manda and sighra corrections the motions are non-uniform. Therefore, to get the true daily motion of a star-planet, we have to apply the relevant corrections. According to the Siirya siddhanta, the following procedure is prescribed for obtaining the true daily motion : (1)
To the mean daily motion n of the planet, apply the correction due to manda which is similar to the one applied in the case of the sun and the moon (see Chapter 8). For this purpose, from the third operation in the process of getting the true position, the planet's longitude P2 and hence die equated manda anomaly may be used. From this we get die planet's manda corrected daily motion n^.
(2)
From the planet's sighrocca mean daily motion ni subtract the planet's manda corrected daily motion n, (obtained from the previous step). This gives the planet's equated daily synodical motion (/ij - / i j ) . Note that in the case of Kuja, Guru and Sani, their sighrocca is the mean longitude of the sun.
(3)
Let K be the sighrakarna (hypotenuse) of the planet used in the last operation for finding the true position.
The excess of the sighrakarria over the radius of the deferent circle is given by Excess = {sighrakarna - radius) = {sighrakarria - 3438') where sighrakarria is in minutes of arc. i.e.. Excess = (A"-/?)'. Then, the correction due to sighra to n for getting the true daily motion of the planet is given by Sighra cor. = (Excess / sighrakarna) x (Equated syn. motion). = {K-R)
{n2-nO/K
The sighra correction thus obtained is additive or subtractive according as the sighra anomaly of the planet is less than or greater than 180°. The sighra correction thus obtained is applied to the planet's manda equated daily motion obtained earlier; that is. True daily motion = /ij + [{K- R) (AJ2 -
ni)/K].
138
Ancient Indian Astronomy
Example : Find the true daily motion of Sani at midnight preceding March 22, 1991. The mean daily motion of Sani : 0°.0334393 i.e. n = 0°2' 0" Mean longitude of Sani (after Desantara cor.) : 272° 49' 04" Mean longitude of sighrocca
: 334° 51' 30"
(i.e. the mean sun) Mandocca of Sani
: 236° 37" 36"
In the third operation for getting the true position of Sani, we have obtained : Sani's equated longitude (P2) : 273° 1 2 ' 0 1 " Therefore, Sani's equated manda anomaly, m = Mandocca - P2 = 236° 3 7 ' 3 6 " - 2 7 3 ° 1 2 ' 0 1 " = 323° 25' 35" after adding 360° Tabulated difference of sines : 183' (for 360° - 323° 25' 3 5 " = 36° 34' 2 5 " see Table 7.2 between si. nos. 9 and 10). The corrected periphery of Sani's manda epicycle is given by
= 4 9 ° - l ° i s i n (323° 25' 35") I = 4 8 ° . 404145 Therefore, we have the correction to Sani's motion (due to manda) = nxp°x
Tab. sin diff./(360° x 225') in min. of arc
= 2' x 4 8 ° . 4 0 4 1 4 5 x 183'/(360° x 225') = 0'.218715 = 13" Since Sani's manda anomaly, m = 323° 25' 35" lies in the 4th quadrant, the manda correction to the daily motion is subtractive i.e. the correction to the daily motion due to manda is - 1 3 " . Now, mean daily motion of Sani, n = 0° 2' 0" the inanda correction
=-13"
Hence, Sani's inanda corrected daily motion ; j , = 0° r 4 7 " Now, the mean daily motion of Sighrocca
True Positions of the Star-Planets
139
(i.e. daily modon of the sun)
«2 = 59' 0&"
Deduct Sani's manda corrected motion, r47"fromn2 Therefore Sani's equated daily synodical motion (n2-n]) : 5 7 ' 2 1 " The variable hypotenuse (sighrakarna) used in the last process for finding the true place of Sani, A = 3606'. 2288. Its excess over the deferent radius (= 3438') is K-R
= 3606'.228« - 3438' = 168'.2288
Therefore, the equation of motion due to sighra is given by sighra correction = (Excess/sighrakarna) =
X (Equated syn. motion)
(K-R)(n2-nO/K
= (168'.2288/3606'.2288) x 57'.35 = 2'.6753493 = 2 ' 4 0 " Since the variable hypotenuse is greater than the radius, the correction is additive i.e. sighra correction = + 2' 40". Hence, Sani's manda equated daily motion : r 4 7 " (i.e. after manda correction) sighra correction
: + 2' 4 0 "
Hence, Sani's true daily motion : 1'47"+ 2 ' 4 0 ' = Note :
4'27"
IfrtI is the mean daily motion of the planet after the manda correction, n2 is the mean daily motion of the planet's siglira, K is the hypotenuse (sighrakanxa) in the last operation of finding the planet's true position and R = 3438' is the constant radius of the deferent circle, then True daily motion = [n, + (Wj - n,) (A"- R)/K\ which, on simplification, can also be written as : True daily motion = [n^ - (Wj -
R/K\
In the case of Budha and Sukra, the sun in their mean position and their sighroccas are separately obtained along with the mean positions of other planets. Therefore, in the above formula,
the sighrocca daily motion and O] is the manda corrected daily
motion of the concerned planet (Budha or Sukra) which is the same as that of the sun. 12.6 Retrograde motion of the taragrahas The star-planets move from west to east, relative to fixed stars, as seen from the earth due to their natural motion. However, during certain periods each of these planets appears to move backwards i.e. from east to west. Its celestial longitude keeps on
140
Ancient Indian Astronomy
decreasing instead of increasing, day by day for some time. This apparent backward motion is called vakra gati (retrograde motion). The phenomenon of retrograde motion is caused by the difference in the velocities of the earth and the planet i.e., the relative velocity. This phenomenon is demonstrated in F i g . 12.2.
/ Fig. 12.2: Retrograde motion of Kuja In F i g . 12.2, the motion of Mars (Kuja) relative to the earth is shown in the heliocentric model. The earth's linear speed is about 18.5 miles per second while that of Mars is 3.5 miles less i.e. about 15 miles per second. A s the earth overtakes Mars, the latter appears to move backwards as seen from the earth. The direct motion of Mars eastward is shown at positions 1,2 and 3, retrograde at 4 and 5 westward and again direct motion eastward at 6 and 7. The rule for determining the retrograde motion of a planet is given in the Siirya siddhanta as follows : The retrograde motion (vakra gati) of the different star-planets commences when the sighrakendra (i.e. sighra anomaly) in the fourth process of determining true positions, is as follows : Kuja
164°
Budha
144°
Guru
130°
Sukra
163°
Sani
115°
That is, the retrograde motion of Kuja, for example, commences when Sighrocca (i.e. Sun) - Kuja = 164°. The point when the motion of a planet changes from direct to retrograde is called a stationary point. The planet remains retrograde for some days and then again its motion changes from retrograde to direct. This point of change is the second stationary point.
True Positions of the Star-Planets
141
At both the stationary points the planet has no apparent motion (i.e. the relative velocity is zero). 12.7 Rationale for the stationary point Let A/ be the mean planet, P he the true planet on the epicycle of radius p, E be the earth and S the sun (Fig. 12.3). If n is the mean daily motion of the sun, t be the number of days since the sun S was at the first point of Mesa, PMK = 6 and PEM = E, then the celestial longitude L of the planet is given by L = nt-Q + E where nt is the longitude of the sun. Therefore
Fig; 12.3: Stationary Points
dL/dt = n-dQ/dt-^dE/dt ...(1) Let PM=p and EM = r where the radii pandr are constants. In Fig. 12.2 we have MA =p cos 0 and PA=p sin 8. Therefore, £ 4 = £ A f + AM = r + p c o s 0 and hence tan £ = PA/EA = p sin Q/{r+p cos 9) so that £ = tan"' [p sin e/(r + p cos 9)] Differentiating this expression with respect to t we get dE/dt = ide/dt) [p^ + rp cos %V[i^ -^p^ + lrp cos 9] Substituting (2) in (1), we get dL/dt = n + {dQ/dt) [(p- + rp cos 8)/(/^+p^ + 2r p cos 9)] ...(3) If n is the mean daily motion of Uie sun and n' that of the planet and a is a suitable constant, then
e = (n-n') (r + a) so that dQ/dt = {n-n') Substituting (4) in (3), we get
...(4)
dL/dt = [np2 + n'r^ + rp{n-^ n') cos 9]/[r2 -\-p^ + 2rp cos 9] At the stationary point where the retrograde motion begins, we have dL/dt = 0. Therefore, np^ + n'r^ + rp (/i + «') cos 9 = 0 so that Example
:
In
the
case
cos 9 = - (np2 + n'p-yrp (n + nO of Kuja, considering the mean
values,
n = 0° .98560265, n' = O°.5240193,p = 2 3 3 ° . 5 , r = 3 6 O ° . . Here, p a n d r
we
have
are taken as
Ancient Indian Asxronomy
142
peripheries of the planet's sighra epicycle and of the mean orbit which are proportional to their radii. Substituting these values, we get Q _ - [0.98560265 (233.5)^ + 0.5240193 (360)^] [(233.5) (360) (1.5096219)] = - [53737.271 +67912.901]/[126898.82] = - 121650.17/126898.82 = - 0.9586391 Therefore, G= 180°-cos"'(0.9586391) = 163°.4636 Siirya siddhanta has taken this value as 164° The other stationary point is given by 360° - 9, noting that cos 0 = cos (360° - 9). In the above example, since 0 = 164° according to the Siirya siddhanta, the second stationary point is 3 6 0 ° - 1 6 4 ° = 196°. This means that Kuja will be retrograde during the period when its sighrakendra (or sighra anomaly) lies between 164° and 196°. Similarly, the corresponding limits for other planets can be calculated. Remark : According to modem astronomy, the stationary value of angle 0 is given by COs0 = [fl»/2/,l/23/[^_^l/2fcl/2 + tj
where a is the mean distance of the planet from the sun, b is the mean distance of the earth from the sun. Now, taking b as one astronomical unit, we get cose=-[a'/V[a-fl'^^+
n
where a is the mean distance of the planet from the sun in astronomical units. Note : 1 astronomical unit = Earth's mean distance from the sun. Table 12.4 gives the stationary values of 9 according to different Indian texts as compared to the modem values for the five taragrahas. Table 12.4: Stationary points for planets Planet
Mean distance a in ast. unit
9
0
Modern
Siirya siddhanta
0 0 Bhaskara II Brahmagupta & Lalla
Kuja
1.52369
163°.215
164°
163°
164°
Budha
0.3871
144°.427
144°
145°
146°
Guru
5.20256
125°.565
130°
125°
125°
Sukra
0.7233
167°.005
163°
165°
165°
Sani
9.55475
114°.466
115°
113°
116°
Note :
Brahmagupta has given the stationary value of 6 for Guru as 125 in his Brahmasphuta siddhanta (BS) and as 130° in his KItandakhadyaka (KK).
True Positions of the Star-Planets
143
The stationary points 0 given in Table 12.4 are those at which the respective planets change their motion from direct to retrograde i.e. the beginning of the retrograde motion (vakrdrambha).
The
other
stationary
points
where
the
retrograde
motion
ends
(vakratydga) are given by (360° - 0). 12.8. Bhujdntara correction for the tdrdgrahas We noted earlier that the true midnight at a place differs from the mean midnight by an amount of time called "equation of time." The correction to the celestial longitude of a planet due to the part of the equation of time caused by the eccentricity of the earth's orbit is called Bhujantara correction. A s pointed out earlier, in the case of the five star-planets also, Bhujdntara correction = (Eqn. of centre of the sun) (Daily motion of the planet) / 360° where the equation of centre is in degrees. If the daily motion of the planet is in minutes of arc, then Bhujantara correction (in minutes) = (Eqn. of centre in min.) (Planet's daily motion in min.) / 21600' Example : Find the Bhujdntara correction for Sani at the midnight preceding March 22, 1991 at Bangalore. We have for the given date and time. Sun's equation of centre = + 2 ° 7' 32' = 127'.53 Sani's true daily motion = 4' 27" = 4 ' . 45 Therefore, Bhujdntara correction
= (127'. 53) ( 4 ' . 45)/21600' = 0'.0262735 = r ' . 5 7 6
In fact, the Bhujdntara correction is negligible for most of the planets. However, in the case of the moon the Bhujdntara correction is quite pronounced. 12.9 True positions of the tdrdgrahas according to KK After obtaining the ahargana for the given date according to the Khanda khadyaka (KK) the mean positions of Ravi and the tdrdgrahas are computed as explained in the earlier chapters. The method of obtaining the true positions of the tdrdgrahas is demonstrated below with an example. Example : The true position of Kuja for the midnight preceding February 5, 1962. The KK ahargaria : 473670 Mean Ravi : 290° 57'
144
Ancient Indian Astronomy
For the superior planets viz. Kuja, Guru and Sani, the mean Ravi is considered as their sighrocca. Mean Kuja at Bangalore midnight : 271° 3 6 ' 3 3 " Sighra Kendra = Sighrocca - Mean Kuja = 290° 5 7 ' - 2 7 1 ° 3 7 ' = 1 9 ° 20' Since the Sighra Kendra < 90°, its bhuja = 19° 20'. Then, according to KK, Sighra equation =
_ tan" ' 0.212121 x t a n
Bhuja
VI
Here, Sighra equation = 7° 36' 14" s - S £ , Therefore, (1/2) 5 £ , = 3° 48' 7" Let Py=MP + (1/2) SEi = 271° 36' 33" + 3° 48' 07" = 275° 24' 40" Longitude of the Mandocca of Kuja = 110° (fixed) Mean anomaly = P j - Mandocca = (275° 24' 40") - 110° = 165° 24' 4 0 " Now, considering the Sun's manda equation for the above anomaly, we have Sun's manda equation = ~ - sin (165° 24' 40") = -0°33'40" .-. Mandaphala = 5 x sun's manda equation = - 2° 48' 20" s A/£, Let P2 = (\/2)MEi
+P,
= 275° 24' 40" + (- 1° 24' 10") = 274° 0' 30" Mandocca of Kuja : 110° (fixed) Mean manda anomaly = P 2 ~ Mandocca = (274° 0' 30") - 110° = 164° 0' 30" Sun's manda equation = ^ .-. Mandaphala
sin (164° 0' 30") = - 0° 36' 49"
= 5 X sun's manda equation
= -3°4'9"sA/£. Now, let
P-^ = MP + ME2 = 271° 36' 33" - 3° 4' 9" = 268° 32' 24"
Considering the mean sun as the sighrocca of Kuja, we have Sighra anomaly = Mean sun - P 3 = 290° 57' - 268° 32' 24" = 22° 24' 36" .-. Sighra equation =
^ - t a n - ' 0.212121 x t a n
= 8° 47' 56" = 5£-,
Bhuja^
True Positions of the Star-Planets
145
Let ^4 = ^3 + SE2 = 268° 32' 24" + 8° 47' 5 6 " = 277° 20' 2 0 " In the similar manner true Guru, true Sukra, true Budha and true Sani are also calculated. IVue longitude of Guru We have the mean longitude of Guru : 304° 39' 13" at Bangalore midnight Sighra anomaly = Sighrocca - Mean Guru = Mean sun - Mean Guru = 290° 57' - 304° 39' 13" = 346° 17' 4 7 " Bhuja = 360° - 346° 17' 4 7 " = 13° 42' 13" Sighra equation
tan"'
Bhuja^
0.6666 X tan
it \
P
= -2°16'20"s5£i Let/», = M P + ( l / 2 ) 5 £ , = 304° 39' 13" - 1° 08' 10" = 303° 31' 0 3 " Longitude of Mandocca of Guru = 160° (fixed) Mean anomaly of Guru =P\-
Mandocca
= 303° 3 1 ' 0 3 " - 1 6 0 ° = 143° 3 1 ' 0 3 " Sun's manda equation = — sin (143° 31' 3") = - 1° 19' 29" 71
.-. Guru's Mandaphala^ Therefore,
Sun's equation x y
= - 3° 1' 4 1 " s M £ ,
(1 / 2 ) M E i = - 1 ° 30' 50"
Let P 2 = ' ' i + (1 / 2 ) A/£, = 303° 31' 0 3 " - 1 ° 30' 50" = 302° 0' 13" Mean manda anomaly ^Pj-
Mandocca = 302° 0' 13" - 160° = 142° 0' 13"
Sun's manda equation = — sin (142° 0' 13") = - 1° 22' 18" 7t
16 .-. Guru's Mandaphala = Sun's equation x y = - 3" 8' 7" s MEj Let P^ = MP-i- ME2 = 304° 39' 13" - 3° 08' 07" = 301 ° 31' 06" Guru's Sighra anomaly = Mean sun - P 3 = 290° 57' - 301° 31' 6' = 349° 25' 54" Bhuja = 10° 3 4 ' 6 " Bhuja T 'r , Bhuja -1 0.6666 X tan Sighra equation = — y - - tan
Ancient Indian Astronomy
146
Then, the true longitude of Guru is given by ^4 = ^3 +
= 301° 31' 06" - 1° 45' 22" = 299° 45' 44"
True longitude of Sani We have the mean longitude of Sani : 275° 27' 08" s MP (at Bangalore midnight) Sighra anomaly of Sani = Mean sun - mean Sani = 290° 57' - 275° 27' 08" = 15° 29' 52" Sani's Sighra equation
= ^ ^ y ^ _ tan
' 0.8 X tan
Bhuja
= l°32'10"s5£t Therefore,
(1 /2) SE^ = 0° 46' 5"
L e t P , =MP + (1/2) SEi = 275° 27' 8" + 0° 46' 5" = 276° 13' 13" Longitude of Mandocca of Sani = 240° (fixed) Sani's manda anomaly
=P\-
Mandocca
= 276° 13' 13" - 240° = 36° 13' 13" Sun's manda equation
= ^
sin (36° 13' 13")
= -l°18'59" .-. Sani's mandaphala
60 = Sun's equation x — = - 5 ° 38'30" s M E ,
Therefore, (1 /2) M E ,
= - 2° 49' 15"
Now, P2 = Pi + (1/2) M £ , = 276° 13' 13" - 2° 49' 15" = 273° 23' 58' Sani's manda anomaly
= P2
-Marulocca
= 273° 23' 58" - 240° = 33° 23' 58" Sun's manda equation
= - 1° 13' 35"
.•. Sani's manda equation = Sun's equation x = - 5 ° 15' 2 3 " s M £ 2 Now,
P^=MP
+ ME2 = 275° 27' 8" - 5° 15' 23" = 270° 11' 45"
Sani's Sighra anomaly = Mean sun - ^3 = 290° 57' - 270° 11' 45" = 20° 45' 15"
True Positions of the Star-Planets
147
Sani's Sighra equation = ^^^J^ _ tan
' 0 . 8 X tan
Bhuja
= 2° 2' 34" = 5^2 The true longitude of Sani is given by ^4 = ^3 + SE2 = 270° 11' 4 5 " + 2° 2' 34" = 272° 14' 19" True longitude of Budha We have the mean Sighrocca of Budha at Bangalore midnight Sighrocca of Budha
= 72° 33' 5 1 "
Mean sun
= 2 9 0 ° 57'
Therefore, Budha's Sighra anomaly = 141° 36' 5 1 " Budha's Sighra equation
=
_ tan
' 0.4634146 x t a n
'Bhuja'
= 17°43'10"HS£i
Therefore, ( 1 / 2 ) 5 £ : , = 8 ° 5 1 ' 3 5 " Let
Py=MP-\-
(1/2) 5 E , = 290° 57' + 8° 51' 3 5 " = 299° 48' 3 5 "
Longitude of Mandocca of Budha = 2 2 0 ° (fixed) Budha's manda anomaly
=P\-
Mandocca
= 290° 48' 35" - 220° = 79° 48' 35"
Sun's manda equation
— sin (79° 4 8 ' 35") = - 2 ° 11'34"
.'. Budha's
Mandaphala
= Sun's equation x 2 = - 4 ° 23' 9" = MEi
Now, P2 = P , + (1/2) M £ , = 299° 48' 35" - 2° 11' 34" = 297° 37' 01' Again, we have Budha's manda anomaly
= ^2 ~ Mandocca = 297° 37' 1" - 220° = 77° 37' 1"
Sun's manda equation
— sin (77° 37' 1") n = - 2 ° 10' 34".5
.-. Budha's Mandaphala
= Sun's equation x 2 = - 4° 21' 9" s M f j
Let P^ = MP + ME2 = 290° 57' - 4° 21' 09" = 286° 35' 5 1 "
148
Ancient Indian Astronomy
where MP is the mean sun. In the case of the inferior planets viz., Budha and Sukra, the mean sun itself is taken as the mean planet. Budha's Sighra anomaly = P3 - Sighrocca = 286° 35' 51" - 72° 33' 51 " = 214° 02' 0" Bhuja = 360° - (214° 2' 0") = 145° 58' 0" Sighra equation = ^ ^ y ^ _ tan ' 0.4634146 xtan
Bhuja
= 16° 25' 29" s Now, ^4 = ^3 + SE2 = 286° 35' 51" + 16° 25' 29" = 303° 1' 20" is the true longitude of Budha. True longitude of S u k r a We have the mean longitude of the Sighrocca of Sukra = 310° 17' 46' at Bangalore midnight. Sighrocca of Sukra = 3 1 0 ° 17'46" Mean sun
= 290° 57'
Therefore, Sukra's Sighra anomaly = 19° 20'46" Bhuja -1 {Bhuja Sukra's Sighra equation ion = — y - - tan 0.1612903 x t a n 2 = 8° 5' S4' = SEy Therefore, (1/2)
= 4° 2' 57"
Let P , =MP + {\/2)SEy = 290° 57' + 4° 2' 57" = 294° 59' 57" Longitude of Mandocca of Sukra = 80° (fixed) Mean manda anomaly =P\-
Mandocca
= 294° 59' 57" - 80° = 214° 59' 57" Sun's manda equation = ^
sin (214° 59' 57")
= 1° 16'41" In the case of Sukra his mandaphala is given by the same expression as the sun's manda equation Sukra's mandaphala
= Sun's equation = 1° 16' 4 1 " = ME^
Therefore (I / 2 ) ME^
= 0° 38' 20"
Let P2 = P\ + (1/2) A/£j = 294° 59' 57" + 0° 38' 20" = 295° 38' 17"
True Positions oj the Star-Planets
149
Sukra's manda anomaly = ^2 - Mandocca = 295° 38' 17" - 80° = 215° 38' 17" Sun's manda equation
= — sin (215° 38' 17") 7t = 1°17'54"
Sukra's mandaphala = Sun's equation = 1° 17' 54" = ME2 Let P^ = MP-¥ ME2 = 290° 57' + 1° 17' 54" = 292° 14' 54" where MP is the mean sun which is taken as the mean longitude of Sukra. Sighra anomaly = Sighrocca - P3 = 310° 17' 46" - 292° 14' 54" =18°2'52" Sukra's Sighra equation = ^ ^ y ^ _ tan ' 0.1612903 xtan
Bhuja)
Now, the true longitude of Sukra is given by P4 = P3 + SE2 = 292° 14' 54" + 7° 33' 24" = 299° 48' 18" 12.10. True positions of the tdrdgrahas according to GL Based on cakras and the ahargana, as per the Grahaldghavam {GL) the mean positions of the tdrdgrahas are first determined. Then the true positions can be computed as explained below with an example. Table 12.5: Manddiikas of tdrdgrahas 0
1
2
3
4
5
29 12 14 6
57
• 124
21 27 II
85 28 39 13
109
Sukra
0 0 0 0
33 48 14
35 55 15
Sani
0
19
40
60
77
89
Kuja Budha Guru
6
I
130 1 36 i 57 15 93
Table 12.6: Sighraiikas of taragrahas 0
1
2
3
4
5
6
7
8
9
10
Kuja
0
117
Sukra
85 246
279 178 98 302
325 199 106 354
365 212 108 402
393 212 102 440
368
0 0
228 150
400
Budha Guru
58 41
195 89 461
155 66 443
Sam
48
54
57
53
45
0
25 63
81 47 126
174 117 68 186
0
15
28
39
f
11
12
209 89
0
36 326 18
0 0 0 0
150
Ancient Indian Astronomy
Example : The true positions for the taragrahas on April 3, 1996 at the sunrise. According to die Grahldghavam, for the given date, cakras = 43 and die ahargaria = 1173 (i) True longitude of Kuja The mean longitude of Kuja at the sunrise on the given day : 1 1 ' 3 ° 5 3 ' 2 5 " = A/P. The mean longitude of the sun : 11' 17° 32' 53". Sighra anomaly = Sighrocca - Mean Kuja. In the case of the superior planets viz., Kuja, Guru and Sani, their Sighrocca is the mean sun. Therefore for Kuja, Sighra anomaly = 1 F 17° 32' 53" - 11* 3° 53' 25" = 0 M 3 ° 39' 28" < 180°. Dividing the Sighra anomaly by 15, the quotient is 0 (called ahka). In Table 12.6 for Sighrahkas, in the case of Kuja the difference between the entries in the 1st and 2nd columns (corresponding to Sighra anomaly 0° and 15°) is 58. Therefore, for Sighra anomaly of 13° 39' 28", the corresponding entry would be :
X 58 = 52° 48' 36"
Sighrphala = (1/10) (52° 48' 36") = 5° 16' 5 1 " = 5 £ , Half of Sighraphala = (1/2) (5° 16' 51") = 2° 38' 25" =
SE/2
Since the Sighra anomaly < 180° the Sighraphala is additive (otherwise it would be subtractive). Now, Mean Kuja + (1/2) Sighraphala =MP + SE^/l = 11* 3° 53' 25"+ 2° 38' 25" = 11* 6° 3 1 ' 5 0 " s P , the first Sighra corrected Kuja. Mandocca of Kuja = 4* = 120° (fixed). Manda anomaly of
P | = Mandocca - Pj = 4 * 0 ° 0 ' 0 " - 11*6°31'50" = 4* 23° 28' 10"< 180°
Bhuja = 6* - 4* 23° 28' 10" = 1 * 6° 31' 50" = 36° 31' 50" Ahka
- Quotient of ^ . , 3 6 ° 31'50" = Quotient of j-^ =2
The remainder in the above division = 6° 31' 50".
True Positions of the Star-Planets
151
In Table 12.5 for mandahkas the entries in columns against 2 and j for Kuja are 57 and 85. Their difference is 85 - 5 7 = 28. Therefore for the remainder
6° 3 1 ' 5 0 " the
proportionate value is 6° 31' 'SO" 5 - ~ ^ x 2 8 = 1 2 ° Adding
this
to
the
preceding
ir25" entry
in
the
table
we
gel
5 7 ° + 12° 11'25" = 69° 11'25". Dividing this by 10, we get Mandaphala = 69° 11' 25"/10 = 6° 55' 9" s M £ , Since the manda anomaly (i.e. 143° 28' 10") is less than 180°, the mandaphala is additive. Manda spasta Kuja = Mean Kuja + Mandaphala = MP + ME^ = 11* 3° 53' 25"+ 6° 55' 9" = 1 1 ' 1 0 ° 4 8 ' 3 4 " = ^2 The I" Sighra anomaly of Kuja, obtained earlier, is 0* 13° 39' 28". Subtracting from this the mandaphala, obtained above viz., MEy we get 2"'' Sighra anomaly =0* 13° 39' 28" - 6° 55' 9" = 0*6° 44' 19" Dividing the above by 15, the quotient is 0 and the remainder is 6° 44' 19". From Table 12.6 for Sighrahkas, the entries in columns under 0 and 1 are respectively 0 and 58 and their difference is 58. Therefore, the corresponding proportionate value for the remainder 6° 44' 19" is fi0 4 4 ' I Q " 15 X 58 = 26° 3 ' 2 1 " Dividing the above result by 10, we get the second corrected Sighraphala =
26° 3 ' 2 1 " — = 2° 36' 20" s SE2
The true longitude of Kuja is given by Manda spasta Kuja + Second cor. Sighraphala s
+ 5^2 = 11' 10° 48' 34" + 2° 36' 20" = 11* 13° 2 4 ' 5 4 "
(According to Ind Ast. Eph., True Kuja = 11*13° 28' 3") (ii) True longitude of Guru The mean longitude of Guru at the sunrise on the given date : 8* 18° 50' \5" = MP Guru's
Sighra anomaly = Sighrocca - Mean Guru
152
Ancient Indian
Astronomy
= Mean Sun - M P = 1 P 17° 32' 53" - 8* 18° 50' 15" = 2* 28° 42' 38" = 88° 42' 38" < 180° Dividing the above Sighra anomaly by 15 i.e. (88° 42'53")/15, the quotient is 5 (called ahka) and the remainder is 13° 4 2 ' 3 8 " . In Table 12.6 for Sighrahkas, for Guru, the entries under columns 5 and 6 are respectively 98 and 106 and their difference, 1 0 6 - 9 8 = 8. The proportionate value for the remainder 13° 42' 38" is given by 13° 42' 38" "^^Y^ X 8 = 7° 18'44". Adding this to the preceding entry (headed by column 5) viz. 98, we get 9 8 ° + 7° 18'44" = 1 0 5 ° 18'44" Dividing this value by 10, we get SEi H Sighraphala = 10° 31' 5 3 " or (1/2) Sighraphala = 5° 15' 57" and it is additive since the Sighra anomaly < 180°. Therefore P ' Sighra corrected Guru = Mean Guru + (1/2) Sighraphala = 8* 18° 50' 15" + 5° 15' 57" = 8' 24° 6' 12" = Pj Mandocca of Guru = 180° (fixed) Manda anomaly of Guru s Mandocca - P( = 6' 0 ° 0' 0 " - 8' 24° 6' 12" = 9* 5° 53' 48" = 275° 5 3 ' 4 8 " = MA Since 270° < M A < 360°, its Bhuja = 360° - 275° 53' 4 8 " = 84° 06' 12". Dividing the Bhuja by 15, the quotient, ahka =5 and remainder = 9° 6' 12". In Table 12.5 for manddnkas, for Guru, the entries under columns 5 and 6 respectively are 55 and 57; their difference is 57 - 55 = 2. The proportionate value for the remainder 9° 6' 12" is given by ^ - ^ ^ x 2 = 1° 12' 5 0 " Adding this value of the preceding entry (headed by column 5) viz., 55, we get 5 5 ° + 1 ° 12'50" = 56° 1 2 ' 5 0 " Dividing this value by 10, we get Mandaphala = - 5 ° 37' 17" s M E , Since the manda anomaly MA is greater than 180°, the mandaphala ME^ is subtractive. Therefore, Manda spasta Guru = MP + ME^ = 8' 18° 50' 15"+ ( - 5 ° 37' 17") = 8' 13° 1 2 ' 5 8 " s P ,
True Positions of the Star-Planets
153
Sighrakendra (obtained earlier) = 2" 28° 42' 38" Now, Sighrakendra -
Mandaphala = 2' 28° 42' 38" + 5° 37' 17" = 3* 4 ° 19' 55'
i.e., 2"'* Sighra kendra
= 94° 19' 55'
Dividing the above value by 15 the quotient ahka is 6 and the remainder 4° 19'55". FroiTi Table 12.6 for Sighrahkas, the entries under columns 6 and 7 are respectively 106 and 108 and the difference, 1 0 8 - 106 = 2. The proportionate value is given by 4° 19' X
2 = 0° 34' 39"
Adding this value to the preceding entry (under column 6) we get 1 0 6 ° + 0° 3 4 ' 3 9 " = 106° 3 4 ' 3 9 " Dividing this value by 10 we get 2"'' Sighraphala = 106° 34' 39"/10 = 10°39' 28" = 5^2 Since the Sighrakendra < 180°, the Sighraphala is additive. Now the true longitude of Guru is given by True Guru = Manda spasta Guru + 2"^ Sighraphala = P2 + SEj = 8' 13° 12'58"+ 10° 3 9 ' 2 8 " = 8* 23° 52' 26" [According to bid. Ast. Eph., true Guru is 8' 22° 19' 22"]. (iii) True longitude of Sani Mean Sani at the sunrise on the given date at Bangalore
: 11' 9° 28' 31" = MP
Mean Sun
: 11* 17° 3 2 ' 5 3 "
Sighra anomaly of Sani = Mean Sun - MP = 11* 17° 3 2 ' 5 3 " - 11* 9° 2 8 ' 3 1 " = 8° 4' 22" < 180° Dividing the Sighra anomaly by 15, the quotient is 0 (aiika) and the remainder is 8° 4' 22". The entries in Table 12.6 for Sighrankas in the case of Sani, under columns headed by 0 and 1 are respectively 0 and 15 and their difference is 1 5 - 0 = 15. The proportionate value for the remainder 8° 4' 22" is 00
A'29"
x l 5 = 8 ° 4 ' 22". Dividing this value by 10, we get
Ancient Indian Astronomy
154 P' Sighraphala of Sani
= ^° ^^22" ^
4^' 26" = S £ ,
Therefore. (1/2) Sighraphala
= (1/2) 5 E , = 0° 24' 13" (additive) =MP + {\/2) 5 £ ,
Sfg/ira corrected Sani
= 11* 9° 28' 3 1 " + 0° 24' 13" = 11* 9° 52' 44" = P , Mandocca of Sani
= 8* = 240° (fixed)
Manda anomaly =
Mandocca-P^ = 8* - 11 * 9° 52' 44" = 8* 20° 7' 16" s 260° 7' 16"
Therefore, (Manda) Bhuja = 260° 7' 16" - 180° = 80° 7' 16" Since Manda anomaly > 180°, the mandaphala is subtractive. Dividing the Bhuja by 15, the quotient is 5 (anka) and the remainder is 5° 17' 16". The entries in Table 12.5, for manddiikas, in the case of Sani, the entries in columns under 5 and 6 are respectively 89 and 93 and their difference is 93 - 89 = 4. The corresponding proportionate value for the remainder 5° 17' 16" is ro
,7'
^
' 1 ^ ' ^ x 4 = l ° 2 4 ' 36".
1/:"
Adding this proportionate value to 89 (under col. 5) we get 90° 24' 36". Dividing this value by 10, we get 9° 2 ' 2 8 " . Therefore, Mandaphala
of Sani = 9° 2' 28" = yW£,
Manda spasta Sani
=MP—ME^ = 1 1 ' 9 ° 2 8 ' 3 1 " - 9 ° 2' 28" = 11*0° 26' 03" s P ,
Now, the earlier obtained Sighra anomaly is 8° 4' 22". Subtracting the mandaphala (which is negative), we gel 2"'^ Sighra anomaly = 8° 4' 22" - (- 9° 2' 28") = 8° 4' 22" + 9° 2' 28" = 17° 6' 50" < 180° Dividing the above value by 15, we gel the quotient (arika) as 1 and the remainder 2° 6' 50". In Table 12.6 for Sighrahkas in the case of Sani the entries under columns headed by 1 and 2 are respectively 15 and 28 and their difference is 2 8 - 1 5 = 1 3 . The proportionate value for the remainder 2° 6' 50" is given by
True Positions of the Star-Planets
155
2° 6' 50"
X 13 = 1 ° 4 9 ' 5 5 "
Adding this value to 15 (under column 1), we get 16° 4 9 ' 5 5 " . Dividing this value by 10, we get 2"'' Sighraphala = °
= 1° 41' = S E j
Now, the true longitude of Sani is given by True Sani =
+
= 11' 0° 26' 3" + 1° 4 1 '
= 11* 2° 07' 03" (According to Ind. Ast. Eph., true Sani is 11* 05° 39' 56"). (iv) True longitude of Budha In the case of the inferior planets, Budha and Sukra, the mean Sighra anomaly, rather than the Sighrocca is first determined according to the Grahaldghavam. For the date, under consideration, we get Budha's Sighra kendra = 1* 15° 21' 0 8 " = 45° 21' 08" < 180° Dividing the Sighrakendra by 15, the quotient is 3 {ahka) and the remainder is 21'08". The entries in Table 12.6 for Sighrahkas, in the case of Budha, under columns headed by 3 and 4 are respectively 117 and 150 and their difference is 1 5 0 - 117 = 33. 21' 08" The proportionate value for the remainder is — j - ^ — x 33 = 0° 46' 30". Adding this value to 117° (under column 3), we get 117° 46' 30". Dividing this value by 10, we get Sighraphala of Budha Therefore, Sighraphaldrdha
= 11 ° 46' 39" s S £ , = (1 / 2 ) S £ , = 5° 53' 20"
Since the Sighra anomaly < 180°, the Sighraphaldrdha is additive. Therefore Sighra corrected Budha = M P + (1/2) S £ , = 11*17° 3 2 ' 5 3 " + 5° 53' 20" = 11*23° 26' 13" = where MP is the mean longitude of Budha, taken same as that of the mean sun. Budha's Mandocca = 7* = 210° Budha's manda anomaly (for Pj) = Mandocca - P , = 7* - 11* 23° 26' 13" = 7* 6° 33' 47" = 216° 33' 47" > 180°
156
Ancient Indian Astronomy Therefore, Bhuja = 2 1 6 ° 33' 47" - 180° = 36° 33' 47" Since the manda anomaly > 180°, the mandaphala is subti active. Dividing
the Bhuja
by
15, the quotient
is 2 (arika) and the remainder
is
6° 33' 47". The entries in Table 12.5, for mcuiddhkas, in the case of Budha, under columns 2 and 3 are respectively 21 and 28. Their difference is 2 8 - 2 1 =7. The proportionate value for the remainder 6° 33' 47" is given by 6° 33' 4 7 " ^ ^ j * ^ ^ ^ x 7 = 3° 3'46". Adding this value to 21° we get 24° 3' 46". Dividing this value by 10, we have Mandaphala = 2° 24' 2 3 " (subtractive) = M £ , Manda Spasta Budha =MP-¥ ME^ = 11M7°32' 5 3 " - 2 ° 24'23" = 11* 15° 08' 30" = Now, 2"'' Sighra anomaly = Sighra kendra (obtained earlier) -
Mandaphala
= 4 5 ° 21' 08" - (- 2° 24' 23") = 47° 45' 3 1 " Dividing the above value by 15, we get quotient {arika) as 3 and the remainder 2° 45' 31". In the entries from Table 12.6 for sighrahkas in the case of Budha, are 117 and 150 under columns 3 and 4 respectively. Their difference is 1 5 0 - 117 = 33. The proportionate value for the remainder 2° 4 5 ' 3 1 " is given by 9O AC' Ol// ^ ^g-^' X 33 = 6° 04' 08" Adding this value to 117° (under column 3), we get 123° 04' 08". Dividing this value by 10, we get 2"'' Sighraphala = 12° 18' 25" = 5^2 Therefore, we have True Budha
= P 2 + SE2 = 11*15° 08' 3 0 " + 1 2 ° 18' 25" = 11* 27° 26' 55"
(According to Ind Ast. Eph., True Budha is 11* 25° 45'04"). (v) True longitude of S u k r a For the given date, we have Sukra's Sighra kendra =4* 19° 57' 3 1 " = 139° 57' 3 1 " < 180°.
True Positions of the Star-Planets
157
Dividing the Sighra kendra by 15, the quotient is 9 (ahka) and the remainder is 4° 57'31". The entries from Table 12.6 for Sighrahkas in the case of Sukra, under columns 9 and 10 are respectively 461 and 443 and their difference is 4 4 3 - 4 6 1 = - 1 8 . Therefore, the proportionate value for the remainder 4° 5 7 ' 3 1 " is r 4° 5 7 ' 3 1 " 15
'^) = ~ 5 ° 5 7 ' 0 1 "
Adding this value to 461 (under column 9), we get 461° - 5° 5 7 ' 0 1 " = 455° 2 ' 5 9 " Dividing this value by 10, we get Sighraphala = 45° 30' 18" = SE^ Therefore, Sighraphaldrdha
= 1/2 SEy = 22° 4 5 ' 0 9 " .
Since the Slghra anomaly < 180°, the Sighraphaldrdha
is additive. Therefore
P' Sighra corrected Sukra = M P + (1 / 2 ) S £ , = 11* 17° 3 2 ' 5 3 " + 22° 4 5 ' 0 9 " '
= a ' 1 0 ° 18'02" = P ,
'(
where M P is the mean Sukra, taken same as the mean sun. Mandocca of Sukra = 3* = 90° Manda Kendra
= Mandocca - P , = 3* - 0* 10° 18' 02" = 2* 19° 41' 58"
Therefore,
Bhuja = 79° 41' 58".
Dividing the above value by 15, the quotient is 5 (arika) and the remainder 4° 41'58". The entries from Table 12.5 for manddrikas, in the case of Sukra, under columns 5 and 6 are both 15. Therefore, for the remainder 4° 4 1 ' 5 8 " the proportionate value is 0. Hence I I
Mandaphala = 15°/10 = 1 ° 30' = ME^ Since manda kendra < 180°, M £ | is additive. Manda spasta Sukra = M P + M £ | = 11* 17° 3 2 ' 5 3 " + 1° 30'= 11* 1 9 ° 0 2 ' 5 3 " s P 2 A
' —
' —
2™ Sighra anomaly = Sighra kendra (obtained earlier) - M £ j = 139° 5 7 ' 3 1 " - 1 ° 3 0 '
158
Ancient Indian Astronomy
= 1 3 8 ° 2 r 3 1 " < 180°. Dividing the above value by 15, the quotient is 9 and the remainder is 3° 2 7 ' 3 1 " . The entries from Table 12.6 for Sighrdrikas, in the case of Sukra, under columns 9 and 10 are respectively 461 and 443. Their difference, 443 - 4 6 1 = - 18. The proportionate value for the remainder 3° 2 7 ' 3 1 " is given by 3° 27' 3 1 " x ( - 18) = - 4 ° 0 9 ' 0 1 " Adding this to the previous value 461 (under column 9), we get 461 ° - 4 ° 0 9 ' 0 1 " = 456° 5 0 ' 5 9 " Dividing this value by 10, we get 2"'' Sighraphala = 45° 4 1 ' 06" = SEjSince the Sighra anomaly < 180°, SE2 is additive. Therefore, we have True Sukra = P j + ^£'2 = 11* 19° 0 2 ' 5 3 " + 45° 4 1 ' 0 6 " = 1* 04° 43' 59" (According to Ind. Ast. Eph., true Sukra is 1* 5° 34' 23"). 12.11. A comparison of true planets according to different texts In this section the true longitude of the sun, the moon, Ketu and the obtained according to the Siirya siddhanta, the Khandkhddyaka and the
tdrdgrahas
Grahaldghavam
are compared among themselves and with the ones according to the modem computations. As an example, the famous Asta-grahakUta (conjunction of eight planets) of February 5, 1962 in Makara rdsi is considered. In Table 12.7, the true longitudes of the eight heavenly bodies in conjunction at the midnight between 4th and 5th of February 1962 are provided for comparison. Table 12.7: Eight planets' combination SS
KK
GL
Modem
Ravi
292° 08'
292° 06'
292° 06'
292° 09'
Candra
286° 52'
288° 31'
288° 57'
288° 53' 294° 37'
Planets
Ketu
298° 38'
294° 20'
294° 50'
R. Budha
291° 49'
302° 57'
293° 15'
293° 52'
Sukra Guru
297° 30'
299° 50'
294° 46'
294° 10'
299° 09'
299° 46'
296° 20'
295° 14'
Sani Kuja
274° 09'
272° 14'
278° 0'
280° 27'
277° 07'
277° 21'
279° 22'
278° 52'
True Posttions of the Star-Planets
159
We observe that the values are remarkably comparable. The true positions of the planets according to GL are extremely close to the modern values. In view of the fact that the Indian texts precede modem computations by more than 500 years (in the case of GL) and even as ancient as 7th century A . D . (KK), the traditional Indian computations are truly praiseworthy. O f course, Budha of KK is off the mark. Particularly, in respect of the Crahalaghavam, the accuracy of the result based on the text must be greatly appreciated in the light of its author totally dispensing with trigonometric ratios.
13
SUGGESTED IMPROVED PROCEDURES FOR ECLIPSES
In Chapters 9 and 10 we studied the procedures for computing lunar and solar eclipses according to the Surya siddhanta, the Khanda Khadyaka and the Graha laghavam. We observed that the results obtained according to these ancient and medieval Indian texts were reasonably good. The correctness of the timings of the eclipses accomplished by the traditional Indian astronomers, with periodically introduced bija correctons, is truly remarkable. However, in the light of better formulae and improved values of parameters, we can introduce a few bijas (corrections) in the procedures and the relevant parameters to yield timings of eclipses comparable to the ones obtained from modern computations. The presently suggested improved procedures are inspired by the works of the great savant in the field of Indian astronomy, the late Prof. T.S. Kuppanna Saslri. 13.1 Computation of lunar eclipse The suggested improved procedure for computaions of lunar eclipse is demonstrated with an example providing explanatory comments in relevant places. Example : Lunar eclipse on April 13, 1968. The instant of fullmoon : 10* 22"' a.m. (1ST) = 10" 5 5 " " from the mean sunrise. i.e. 25" 55"'" from the mean midnight. At the instant of fullmoon 1.
Moon's
true
angular
diameter
(Candra
bimba):
MDIA = 2 1939.6
+ (61.1) cos A 2 ] / 6 0 in minutes of arc where A 2 is the moon's anomaly (measured from its perigee). Here, MDIA = 29' 56" 2.
Sun's true angular diameter (Ravi bimba): SDIA = 2 [961.2 + (16.1) cos A^]/&) in minutes of arc. Here, SDIA = 32' 0 2 " In the expression for SDIA,A^
is the sun's anomaly measured from the
perigee. 3.
Diameter of the earth's shadow (Chaya bintba) :
Suggested Improved Procedures for Eclipses
SHDIA = 2[2545.4 + (228.9) cos
161 - (16.4) cos A 8 J / 6 0
in minutes of arc where A 2 = Moon's anomaly and Ag = Sun's anomaly. Here, SHDIA = 96'.92 4.
Moon's latitude, p = 25' 30" S
5.
Vyarkendu sphuta Nddi gati = 14'.0378 = VRKSN the rate of motion of the true (Moon - Sun) per nadi. This is given by {MDM-SDM)/ where MDM=
True daily motion of the moon and SDM = True daily motion
of the sun. One day = 60 nadi; 1 nadis= 60 vinadis 6.
Bimba Yogdrdham= {MDIA + SHDIA)/!
= (29'.933 + 96'.92)/2 = 63'.43 s D
7.
Bimba Viyogdrdham = {SHDIA - MDIA)/2 = (96'.92 - 29'.933)/2 = 33'.49 = £>'
8.
X = p X (1 - 1/205) = 25'.5 x 204/205 = 25'.376 5 = - 2 5 ' . 3 7 6 where P is the moon's latitude from item (4) above.
9.
m = V/?A5A^x(l + 1/205) = 14'.0378 X 206/205 = 14'. 1063 where VRKSN = Vyarkendu sphuta nddi gati, obtained in item (5) above.
10.
If \ X\
11.
Virdhucandra, VRCH = (True Moon - True Rahu). If VRCH < 0, add 360° to render it positive.
12.
Calculate 61 )t I ( 1 - 1 / 6 0 ) / m vinadis (i)
if VRCH is in an odd quadrant (i.e. I or III), then subtract the above value from the instant of fullmoon to get the instant of the middle of the eclipse.
(ii)
If VRCH is in an even quadrant (i.e. II or I V ) , then add the above value to the instant of fullmoon to get the instant of the middle of the eclipse. In the current example, we have
162
Ancient Indian Astronomy 61 X l ( l - l / 1 6 0 ) / m = 6 x 2 5 . 3 7 6 x 159/(160 x 14.1063) = 10.73 vinadis Now, VRCH= 184° 41'. Since 180° < VRCH < 270°, i.e. VRCH is in III quadrant (odd), the above value is subtractive from the instant of fullmoon. Therefore, Middle of the eclipse = 25" 55"'" - 10'"".73 = 25" 44^27 from midnight = 10'' 17"" 4 2 ' a.m. 13.
Half-duration of the eclipse HOUR =
-
X^/m
= V(63' . 4 3 ) 2 - ( 2 5 ' . 3 7 6 ) V l 4 ' . 1063 = 4" 7^.25 =1*38"'54* where D is obtained in (6) and X in (8) above. 14.
End of the eclipse = Middle of the eclipse + HOUR = 25" 44''.27 + 4" 7^25 = 29" 51 ".52 from midnight = 11* 56"'36* a.m.
15.
Beginning of the eclipse = Middle of the eclipse -
HOUR
= 2 5 " 4 4 " . 2 7 - 4 " 7 " . 2 5 = 2 r ' 3 7 " . 0 2 from midnight
16.
= 8* 38"' 50* a.m. Half-duration of totality THDUR
=
•^{D'f-X^/m
= V(33'.49)2 - (25'.376)Vl4'.1063 = 1" 32" = 0*36'" 48* 17.
Beginning of the totality = Middle of the eclipse -
THDUR
= 25" 44".27 - 1" 32" = 24" 12".27 from midnight = 9* 40'" 54* a.m. 18.
End of the totality = Middle of the eclipse + THDUR = 25" 44".27 + 1" 32" = 27" 16".27 = 10* 54"" 30* a.m.
Suggested Improved Procedures for
Eclipses
163
Summary of the lunar eclipse (i)
Beginning of the eclipse
8* 3i'" 50' a.m.
(ii)
Beginning of the totality
9* 40"" 54'a.m.
(iii)
Middle of die eclipse
10* 17"'42'a.m.
(iv)
End of the totality
10*54'"30'a.m.
(V)
End of the eclipse
11*56'" 36'a.m.
Pramanam (Magnitude of the eclipse) = iD-\X\-
\
X\/\400)/MD1A
= (63'.43 - 25'.376 - 25'.376/1400)/29'.933 = 1.272. 13.2 Computation of solar eclipse (for the world in general) In the case of solar eclipse, with an example.
suggested improved procedure is demonstrated here
Example : Solar eclipse on October 24, 1995. 1. Instant of newmoon Sun's true daily motion Moon's true daily modon Moon's daily latitude variation Node's daily motion
10* 05'" a.m. (1ST) s 10" 13"'" from 6 a.m. (1ST) 59'.65851 s SDM 858'.9944 = MDM 78'.37068 -0'.2069092
2. At the instant of newmoon (0
Ravi sphuta
186° 29'51"
{ii)
Candra sphuta
186° 29'51"
("0
Rahu sphuta
182° 39' 44"
(IV)
Moon's latitude
0° 20'46" s p (in deg.)
(v)
Sun's true diameter
32'. 16244 s 5 D M
(vi)
Moon's true diameter
33'.352321 =MDIA
(vii)
Ravi sphuta nadi gati
0'.99431 s S D A / / 6 0
i.e. the true rate of motion of the sun per nddi (1/60 of a day) (viii) Candra spasta nddi gati
•
14'.31657 s AfDM/60
(ix)
:
13'.32226H VKAA^G
Vyarkendu spasta nddi gati
i.e. the true rate of motion of (Moon - Sun) per nddi by (MDM 3. Solar eclipse for the world in general (/)
Moon's horizontal parallax is given by PAR = [3447.9 + 224.4 cos (A2)]/60 in minutes of arc
SDM)/60.
164
Ancient Indian Astronomy
where A2 = Moon's anomaly from its perigee. In the example, PAR = 60'.265 (H)
Let D = PAR + (MDIA+
SDIA)/2
where the second expression is the Ravicandra bintba yogdrdham. Here, D = 60'.265 + (33'.35231 + 32'.16244)/2
{iii)
i.e.
Z) = 93'.02237
Let
£/ = PAR + (MDIA
-SDIA)/1
where the second expression is the Bimba viyogdrdham. In the example, we have D ' = 60'.265 + (33'.35231 - 32'.16244)/2 i.e. (iv)
(v)
D ' = 60'.85994.
Let X = P X 60 X 204/205 minutes of arc where P is the latitude of the moon (in degrees) obtained in item (2)(iv) above. Here X = 20'.65857 m = VRKNG X 206/205 where VRKNG above.
is the vyarkendu nddigati obtained in item (2)(ix)
In the example, rh = 13'.38725 4.
If I X, I < D , then a solar eclipse is possible. If I X, I < D ' , then the solar eclipse is total. In the example, we have X = 20'.65857, D = 93'.02237 and D ' = 60'.85994 Since \ X \
5.
Virdhucandra = True Moon - Rahu s VRCH at the instant of newmoon. Here, VRCH = 186° 29' 5 1 " - 182° 39' 44" i.e.
Note : 6.
VRCH =3°
50'OT
If VRCHkO, then add 360° to make it positive and call the new value VRCH.' Evaluate 6 x 99 x I X I / ( 6 0 x 100 x m ) i.e. 99 I X I / (1000 n i ) where X (in minutes of arc) is obtained in (3)(/v) above. In the example, we have 99 I X 1/(1000 m ) = 9.3 vinadis
7.
Middle of the eclipse is obtained as follows : (/) (//)
If VRCH is in an odd (I or III) quadrant, subtract the above value from the instant of newmoon to get the middle of the eclipse. If VRCH is in an even (II or IV) quadrant, add the above value to the instant of newmoon to get the middle of the eclipse.
Suggested Improved Procedures for Eclipses
165
In the example, VRCH = 3° 50' 07" is in I quadrant. Therefore, Middle of the eclipse is given by MIDDLE=
10" 13"-9''.3 = 10" 04" = lO* O r . 6 a.m. (1ST)
Half interval of the eclipse : the half duration of the solar eclipse is given by HDUR =
^D'^-7}/rh
In the example under consideration. HOUR = V(93.02237)2 - (20.65857)^/13.38725 = 6.775059 9.
nddis = 2* 42"" 36'
(/) Beginning of the eclipse, Sparsakdla =
MIDDLE-HOUR
= 10" 04" - 6" 47" = 3" 17" after 6 a.m. (1ST) = 7'' 18"" 48' a.m. (1ST) (ii) End of the eclipse, Moksakdla = MIDDLE
+ HDUR
= 10" 04" + 6" 47" = 16" 51" after 6 a.m. (1ST) = 12* 44"" 24'p.m. (1ST) 10.
Half-duradon of totality of the eclipse. THDUR = V ( D ' ) 2 - X V m In the example under consideration THDUR = V(60'.85994)2 - (20'.65857)2 /13'.38725 = 4" 17"= 1*42'" 48' (i) Beginning of totality =
MIDDLE-THDUR
In the example, we have Beginning of totality = 10" 0 4 " - 4 " 17" 5" 47" after 6 a.m. (1ST) 8* 18"" 48'a.m. (1ST) (II) End of totality
MIDDLE + THDUR 14" 21" after 6 a.m. (1ST) 11''44"'24' a.m. (1ST)
166
Ancient Indian Astronomy
Summary of the solar eclipse for the world I.S.T.
Remark:
Beginning of the ecUpse
T* 18"^48'a.m.
Beginning of the totaHty
S* IS"" 48* a.m.
Middle of the echpse
10* 01"" 36*a.m.
End of the totahty
l l ' ' 44'" 24*a.m.
End of the eclipse
12* 44"" 24* p.m.
According to Ind. Ast. Eph. the beginning and the and of the solar eclipse are respectively 7* 22" and 12*43".
13.3 Solar eclipse for a particular place The circumstances of a solar eclipse for a particular place, using its terrestrial coordinates, are computed by the procedure demonstrated below. We consider the example of the solar eclipse that occurred on October 24, 1995. The beginning, middle and end of the eclipse are determined for Diamond Harbour (Latitude: 22° N 12', Longitude : 88° E 13'). Now, as computed in section 13.2, the solar eclipse on October 24, 1995 commences at 7* 18"" 48* a.m. (I.S.T.) somewhere in the world. Therefore, the commencement of the solar eclipse at Diamand Harbour cannot be earlier than the above time. 1. (i)
(ii)
Choosing a conveniently small time interval (step length STP), say SrP = 0.1 hour (i.e. 6 minutes of dme), determine the Lagna at intervals of STP starting with the time (//I hours 1ST) of commencement of the eclipse for the world in general. The Lagna is determined for the latitude of the particular place (Diamond Harbour, in the example) under consideration at the above time intervals, The correspondingly changing longitudes of the moon are also determined at the above time intervals. This can be done using the moon's true nadi gati (i.e.
MDM/60).
2.
Let Vicandra lagna = {Lagna - Moon) = VCHLG at each time interval mentioned above. If VCHLG < 0°, then add 360° and the new (positive) value be called VCHLG.
3-
Drgjya = sin e cos <[) sin S - cos € sin (]) where e = obliquity of the ecliptic
5 = ndksatrika kdla (sidereal time) at the time
4.
Sahku = V1 -
5.
Samskdra gunakam = sin {VCHLG)
{Drgjydf
= sin {Lagna - Moon) = SAMSGUN
Suggested Improved Procedures for E:clipses
6.
167
Parama navina lambanam = PAR - 0.0017 (1 - cos 2(1)) = where PAR is the moon's horizontal parallax obtained under item (3)(/) in section 13.2
7.
Laniiana
gurpkam
= PARy x Sariku =
8.
Samskarakam
= lAMBGUN
x SAMSGUN
9.
Lambanam
= LAMBGUN
x cos {VCHLG)
= LAMBGUN
LAMBGUN = SAMSKR
X cos {Lagna - Moon) = LAMB
10.
5/?/i«ra lambanam
11.
Vyarkendu = (True moon - True sun) = VYARKND in minutes of arc.
12.
M
= VYARKND
= LAMB X (1+5AA/5A/?) s
+
= (Moon - Sun) + Sphuta 13-
SPLAMB
lambanam
A'AT/ = P/4/?, X Drgjyd where
is obtained in item (6) above and Drgjyd in item (3)
14.
Viksepam = Moon's latitude (in minutes of arc)
15.
X = {NATI + viksepam) x (1 + SAMSKR) where SAMSKR is obtained in item (8) above.
16.
UCD = {MDIA/2) x ( 1 + SAMSKR) + {SDIA/2) where MDIA and SDIA are the angular diameters of the moon and the sun respectively obtained under items (2) (v) and {vi) in section 13.2
17.
LCD = ^X^ + M^ where X is obtained in (15) above and M in item (12)
18.
DIFFD =
19.
A l l the above calculations are carried out successively for the time intervals [with time step-length STP as explained in item (1) above] until DIFFD attains the maximum value.
UCD-LCD
Suppose this maximum of DIFFD
(i.e. UCD - LCD) is attained at the
{K- 1)'" time-interval. Let A^, B^ and C^ be the values oi' DIFFD
20.
at the
{K-2)"*,{K1)'" and A ' " time-intervals respectively. Let TP be the TIME at which the maximum DIFFD is attained. The computations are continued further until the sign of DIFFD changes (from negative to positive) i.e. the product of two successive values [say {K- 1)'" and A ' " ] of DIFFD
is negative. Then let
DIFF] = DIFFD {K) - DIFFD {K - ]) DIFF = I DIFFD {K) I x TIME] =
TIME-DIFF
STP/DIFF]
168
Ancient Indian Astrononvy
where DIFFD (K) and DIFFD (K(K21.
if
1)
are values of DIFFD
at A ' " and
intervals.
The computations are continued until the sign of DIFFD changes again (this time from positive to negative) i.e. the product of two successive values of DIFFD, [DIFFD iK-l)]x
[DIFFD (K)] < 0
Note that the values of K ( i.e. the order of the time interval) are different in items (20) and (21) above. Let
Note :
DIFFl
= DIFFD (K) - DIFFD
DIFF
= I DIFFD (K) I x
TIMEl
=
(K-\)
STP/DIFFl
TIME-DIFF
In (20) and (21) above, TIME = 1 + (/: - 1) x STP
22.
If TIME] = 0, then the solar eclipse is not visible at the place.
23.
Beginning of the eclipse =11] + TIME]
24.
End of the eclipse = / / I + TIMEZ
In (23) and (24) above, HI is the time (in hrs. 1ST) of the commencement of the eclipse for the worid in general (see item (1) («') above]. The beginning and end of the solar eclipse, obtained in (23) and (24) are for the particular place under consideration. Note :
25.
Instant of maximum obscuration : Let A , = fi„ -
A2 = (Co - fi„) - (Bo - Ao)
and PRMGR = TP- STP/2 - STP x A , / A 2 where A^, Bq and Cq as also TP are obtained in item (19) above. 26.
Summary of the solar eclipse at the place: (i) (ii)
Beginning of the eclipse = H] + TIME 1 If
(A, + A 2 / 2 ) 2 - 2 x B , x A 2 < 0 ,
then
the
eclipse
is
partial,
otherwise total. Here,
= (B^ + D') - £>
where D and D' are obtained under item (3) (/) and {ii) in section 13.2. If the eclipse is total, let BECT= PRMGR - I STP/2 A j I x V ( A , + A 2 / 2 ) 2 - 2 B , A j and EECT = PRMGR +1 STP/2 A^' x V ( A , + A 2 / 2 ) 2 - 2 f l , A j (iii)
Beginning of the totality
= / / I + BECT
(iv)
Instant of maximum obscuration (i.e. of parama grdsa)
(v)
= H]+ PRMGR End of the totality
(vi)
End of the eclipse = H] + TIME 2
= H] + EECT
and
Suggested Improved Procedures for Eclipses
(vii)
169
Magnitude of the eclipse, pramanam = [B" - (2 A j + A 2 ) V 8 A j J / S D / A
Example : For the solar eclipse on October 24, 1995 at Diamond Harbour (West Bengal), after going through the procedure described above, we get the following details: I.S.T. Beginning of die eclipse
T* 3 r
Beginning of die totality
8* Si"* a.m.
Middle of die eclipse
8* 54'" a.m.
End of die totality
8* 55"" a.m.
End of die eclipse
10* 19'" a.m.
a.m.
Pramanam (Magnitude) = 1.0018 Note : Remark;
The timings are rounded off correct to a minute. According to Ind. Ast. Eph. the beginning and the end of the eclipse at Diamond Harbour are respectively 7* 32" 08* a.m. (1ST) and 10* 17" 50* a.m. (1ST).
14
S U G G E S T E D BijAS ( C O R R E C T I O N S ) F O R P L A N E T S ' POSITIONS 14.1 Introduction In the earlier chapters on computations of planets' mean and true positions as well as eclipses, we observed that there were deviations, though small, in the traditionally computed values from the modern ones. The deviations are quite natural in the light of the fact that the traditional Indian astronomical texts were composed thousands of years ago. In the Indian astronomical tradition, the practice of introducing bijas (corrections) to the parameters has been in vogue for long. The Indian astronomers were aware that the values of the governing parameters, given by them, would be valid only for a century or so and that future competent astronomers should provide further improvements. For example, the celebrated Kerala astronomer, Paramesvara states : Kdldntare tu samskdrascintyatdm
ganakottainaih
"In course of time, the (necessary) corrections must be decided by the expert mathematicians." In fact, Paramesvara in his extensive work on computations of eclipses—Grahanamandana—observes in all humility that the times of contact etc. of an eclipse as given by him may at times differ slightly from observed positions {Kdlo aneiia ca siddhah kaddcidapi bhidyate svalpam). In the famous karana text, Laghumdnasam, Manjula (or Munjala) composed five slokas separately in the arya metre while the main text of 60 slokas was in anustubh metre. In the arya verses Manjula has given the planetary details for his epoch. Giving reasons for the separate treatment of the five verses, the commentator Siiryadeva Yajvan suggests : The epochal positions stated in those (five) verses will not serve for more than 100 years and after every century thereafter these will have to be replaced by new verses giving new epochal positions. Again, the famous astronomer Jyesthadeva (c. 16th century) in his Malayalam text Drkkarana
described the long series of revisions introduced over centuries in the
Aryabhatan system of astronomy. He says :
Suggested Btjas (Corrections) for Planets' Positions
171
(i)
In the Kali year "giritimga" (i.e. 3623 = 522 A.D.), his work {Aryabhatjyam) was composed He had adjusted the (planets') revolutions by reduction and addition in such a way that there was no zero correction at the beginning of Kali.
(ii)
In course of time, deviations were observed in (the results arrived at by) this computation. Then, in the Kali year mandasthala (3785 = 684 A.D.), several astronomers gathered together and devised, through observation, a system wherein (the correct mean longitudes were to be found) by multiplying the current Kali year minus "girituriga" (Kali 3623 i.e. the epoch). This system was named parahita and many followed it, assuring themselves of its accuracy.
(iii)
When a long time had elapsed, there occurred substantial deviations. Then a noble brahmana,
Parames'vara residing on the coast of the Western Ocean,
revised the Parahita system by means of astronomical observations in the Kali year ''rangasobhanu" (iv)
(4532 s 1431 A . D . )
The work Tantrasarigraha by Nilakantha (Somayaji), with revised constants, is for twelve years later.
(v)
The revolutions given therein (i.e. in the Tantrasahgraha) too, becoming imperfect (in course of time), observations were continued by the astronomers on the West Coast for 30 years, from the Kali year "jaustava" (4678 = 1577 A.D.) through the Kali year ' 'jhanasevd nu" (4709 = 1607 A.D.) and, by observation, the astronomical tradition was revised accurately.
Henceforth too, deviations between the calculated and the observed positions of planets should be carefully observed and revisions effected. Again, there is a detailed statement in the Brhat-tithi-cintdmani, by Ganesa Daivajfia (16th cent. A.D.) describing how the sdstra (science) which is tathya (accurate) at one period of time becomes slatha (inaccurate) and needs samsthapana (firm establishment) in any later period. The celebrated astronomer Nilakaiitha Somayaji, referring to a certain commentator of Manjula's "Mdnasam"
who laments, "Alas! we have precipitated into a calamity -
mahati sahkate patitdh smah") points out : "... One has to realize that the five siddhantas had been correct at a particular period. Therefore, one should search for a (new) siddhanta that does not show discord with actual observation (at the present time). Such accordance with observation has to be ascertained by (astronomical) observers during times of eclipses etc." 14.2 Bijas for civil days and revolutions, mandoccas, epicycles etc., of planets In Indian astronomy, computations of true positions of planets are based mainly on the following parameters :
172
Ancient Indian Astronomy
(i)
Number of civil days in a Mahayuga (or Kalpa)
(ii)
Mean rate of daily motion given in terms of the number of revolutions in a
(iii)
The rates of motion of the mandoccas (apogees) of the planets in terms of
Mahayuga (or Kalpa); revolutions in a Mahayuga (or Kalpa) ; (iv)
The peripheries of the manda and the sighra epicycles of planets; and
(v)
The epochal positions of bodies and special points.
In the present chapter we propose suitable bijas (corrections) to the above parameters based on modem scientific computations. The parameters related to the items mentioned above are now considered one by one. 14.2.1 Civil days in a Mahayuga In a Mahayuga, the sun completes 432 x 10"* revolutions; the period of one revolution with reference to fixed stars being defined as a sidereal solar year. Taking the modern value of the sidereal sun's daily motion as 5£)A/ = 3548". 1928098 the duration of a sidereal solar year becomes 365.2563627378105 days. However, allowing a maximum error of ± 5 in the eighth digit in the value of SDM, correspondingly, the duration of a sidereal solar year lies between 365.2563627429576 days and 365.2563627326635 days. Accordingly, the number of civil days in a Mahayuga of 432 x 10"^ years turns out to be 1577907487 days (ignoring the decimal part). However, if the longer period of a Kalpa of 432 x i o ' years is considered, we can have a more accurate figure for the number of civil days as 1577907487027. Now, the numbers of civil days in a Mahayuga according to some of important traditional texts and our proposed value are compared in Table 14.1. Table 14.1: Civil days in a Mahayuga Texts
No. of civil days
1.
Aryabhata I (Aryabhatiyam)
1,57,79,17,500
2.
Brahamagupta's Khandakhadyaka
1,57,79,17,800
3.
Suryasiddhdnta (SS)
1,57,79,17,828
4.
Aryabhata II (Mahdsiddhdnta)
1,57,79,17,542
5.
Bhaskara II (Siddhanta siromani)
1,57,79,16,450
6.
Proposed modern value
1,57,79,07,487
Our proposed bija to the SS value is - 10,341. It may be noted that Bhaskara II (1114 A . D . ) suggested a bija of - 1,378 to the SS value.
173
Suggested Bijas (Corrections) for Planets' Positions
14.2.2 Revolutions of bodies in a Mahayuga The mean motion of a heavenly body is determined from its number of revolutions in a Mahayuga of 432 x 10'* years. In our proposed values for the revolutions of the sun, the moon and other bodies, we have considered the rate of daily change in the sidereal longitude of these bodies. In Table 14.2, the numbers of revolutions of the bodies according to the traditional texts are compared with our suggested modern values. Table 14.2: Revolutions of bodies in a Mahayuga Bodies 1.
Ravi
2.
Candra
3.
Aryabhata I Brahmagupta
Surya siddhanta Bhaskara II 43,20,000
Proposed Modern
43,20,000
43,20,000
43,20,000
43,20,000
5,77,53,336
5,77,53,336
5,77,53,336
Candra's Mandocca
4,88,219
4,88,219
4,88,203
4,88,206
4,88,125
4.
Rahu
2.32,226
2,32,226
2,32,238
2,32,311
2,32,269
5.
Kuja
22,96,824
22,96,824
22,96,832
22,96,828
22,96,876
6.
Budha's
1,79,37,020
1,79,37,000
1,79,37,060
1,79,36,999
1,79,37,034
3,64,224
3,64,220
3,64,220
3,64,226
3,64,195
70,22,388
70,22,388
70,22,376
70,22,389
70,22,260
1,46,564
1,46,564
1,46,568
1,46,567
1,46,656
5,77,53,300 5,77,52,986
sighrocca 7.
Guru
8.
Sukra's sighrocca
9.
Sani
Note :
The traditional siddhantas have given the revolutions of the bodies in a Kalpa of
432 X 10^ years. However, in Table 14.2, we have reduced them for a Mahayuga of A 432 x 10 years (by dividing thefiguresby 1000) rounding off to the nearest integer. 14.2.3 Peripheries of manda epicycles We explained, in earlier chapters, the procedure of the manda equation (i.e. the equation of centre) to be applied to the mean sun and the moon as also to the mean planets. The expression for the manda equation is given by £ = -=; sin m
...(1)
A
where a is the periphery (in degrees) of the manda epicycle, R = 360° and m is the manda anomaly of the body. The corresponding modern formula for the equation of centre, considering the first two terms, is 5 2 11 4 E = sin 2m ...(2) sm m + 4^
174
Ancient Indian
Astronomy
where e is the eccentricity of the body's elHptical orbit. Generally, since e is small, ignoring the higher powers of e, the equation of centre is approximated as £: = (2e)sinm
...(3)
However, in proposing bija to the peripheries of the manda epicycles of the heavenly bodies, we now consider even the higher powers of e viz. e^, e and e'^. . o 1 3 .J Let C| = 2 « - —e-^ and ^2 = — T
11£ 24
Then (2) can be written as £ =
sin m + ^2 sin 2m
= e, sin m + 2^2 sin m cos m or
E = {ei + 2^2 cos m) sin m
Here, it is interesting to note that the co-efficient of sin m is a variable and that most of the traditional texts indeed have taken the coefficient of sin m viz. a/R as a variable. In Table 14.3, the proposed peripheries (in degree) for the different celestial bodies based on (4) above, are given using the current values of the eccentricities of the orbits of the bodies. Table 14.3: Peripheries of manda epicycles Planet
Coefft. of Equation of Centre
Periphery (Deg) Min.
Max.
Ravi
11.80781
12.31284
Candra
36.8085
42.22932
Budha
109.8002
Sukra
4.869
184.7195 4.952738
Min.
Max.
D
M
S
D
M
S
1
52
5
51
45
1
57
34
29
6
43
15
17
28
30
29
23
56
0
46
29
0
47
17
Kuja
59.30061
74.92371
9
26
16
11
55
28
Guru
32.69004
36.89168
5
12
10
5
52
17
Sani
37.41842
43.03509
5
57
19
6
50
57
Uranus
31.42751
35.29051
5
0
6
5
36
59
1
1
9
1
2
33
19
34
37
36
45
46
Neptune Pluto
6.404953 123.0071
6.550653 230.9876
The second and third columns give respectively the minimum and the maximum values of the peripheries (in degrees) of the manda epicycles. Correspondingly, the third and the fourth columns provide respectively the minimum and the maximum values of the coefficients (in deg., min and sec.) of sin m of the manda equation (1). We have included Uranus, Neptune and Pluto also in the list of planets. The eccentricity e of the earth's orbit is given by
Suggested Bijas (Corrections) Jor Planets' Positions
175
e = 0.01675104 - (0.0000418) T - (1.26 x 10" ^) 7^ where 7" is the number of Julian centuries (of 36525 days each) completed since the epoch 1900.0 A . D . Accordingly, over centuries, the eccentricity e and hence the periphery of the manda epicycle change. In Table 14.4, the eccentricity of the earth's orbit and the corresponding minimum and maximum co-efficients of sin m in the equation of the centre, are given for the beginnings of the centuries. Table 14.4: Earths' eccentricity and coefft. of sun's manda equation Year
Eccentricity
Coefft. of sun's equation of centre
e A.D.
Min.
Max.
D
M
S
D
M
S
1900
.01675104
1
52
45
1
57
34
1800
.016792714
1
53
1
1
57
52
1700
.016834136
1
53
18
1
58
10
1600
.016875306
1
53
34
1
58
28
1500
.016916224
1
53
50
1
58
45
1400
.01695689
1
54
6
1
59
3
1300
.016997304
1
54
22
1
59
20
1200
.017037466
1
54
38
1
59
37
1100
.017077376
1
54
54
1
59
55
1000
.017117034
1
55
9
2
0
12
900
.01715644
1
55
25
2
0
29
800
.017195594
1
55
40
2
0
45 2
700
.017234496
1
55
56
2
1
600
.017273146
1
56
11
2
1
19
500
.017311544
1
56
26
2
1
35
400
.01734969
1
56
41
2
1
52
In Table 14.4, the earth's eccentricity for the beginning of century years from 1900 A . D . backwards to 400 A . D . is evaluated on the basis of the expression for the same with respect to 1900.0 A . D . as the epoch. However, in the currently published ephemerides, the eccentricities of the orbits as also other elements, of the planets are computed with respect to J2000 (corresponding to 2451545.0 Julian Ephemeris Days). Based on these formulae, the eccentricities, the minimum and the maximum values of the peripheries of the manda epicycles are given for the planets and the sun in Table 14.5 to 14.13 at intervals of 300 years. By using these tables, the accurate manda equation of a planet can be computed for a date in any century between - 3400 (i.e. 3401 B . C . ) and 2000 A . D .
Ancient Indian Astronomy
176
Table 14.5: Earths eccentricity and periphery of the Sun's manda epicycle Year 2000 1700 1400
Eccentricity
Min.
1.667218E-02 .0167863 1.690042E-02
Max.
D
M
S
D
M
S 13
11
45
12
12
15
11
49
55
12
20
21
11
54
39
12
25
29
1100
1.701454E-02
11
59
22
12
30
38
800
1.712866E-02
12
4
5
12
35
46
500
1.724278E-02
12
8
48
12
40
54
12
200
.0173569
13
31
12
46
3
-100
1.747102E-02
12
18
14
12
51
12
-400
1.758514E-02
12
22
57
12
56
20
-700
1.769926E-02
12
27
39
13
1
29
- 1000
1.781338E-02
12
32
22
13
6
38
12
37
5
13
11
47
12
41
47
13
16
56
-1300 - 1600
.0179275 1.804162E-02
- 1900
1.815574E-02
12
46
29
13
22
5
-2200
1.826986E-02
12
51
12
13
27
14
-2500
1.838398E-02
-2800
.0184981
12
55
54
13
32
24
13
0
36
13
37
33
-3100
1.861222E-02
13
5
18
13
42
43
-3400
1.872634E-02
13
10
0
13
47
52
Table 14.6: Budha's eccentricity and periphery of manda epicycle Year
Eccentricity
2000 1700
.205656 .2055802
1400 1100 800
.2055043 .2054285 .2053527
D 109 109 109 109
Min. M 50 49
S 46 10 34
D 184 184
Max. M 47 42
S 43 52
184 184 184
38 33 28
0 9 18
184 184
23 18
109
47 45 44
500 200 - 100
.2052769 .2052011
109 109
42 41
.2051253
109
39
184
13
26 35 44
-400 -700
.2050495 .2049737
109 109
0 24
184 184
8 4
53 2
- 1 0 0 0 .2048979 - 1300 .2048221 - 1600 .2047463 - 1900 .2046704
109
38 36 34
48
183
109 109 109
33 31 30
12 36 0
183 183 183
59 54 49 44
11 19 28 37
59 23 47 11 35
(Conld...)
iy
Suggested Bijas (Corrections) for Planets' Positions
Year
Eccentricity
177
Min. D
M
Max. S
D
M
S
-2200
.2045946
109
28
24
183
39
46
-2500
.2045188
109
26
47
183
34
55 5
-2800
.204443
109
25
11
183
30
-3100
.2043672
109
23
35
183
25
14
-3400
.2042914
109
21
59
183
20
23
Table 14.7: Sukra's eccentricity and periphery of manda epicycle Year
Eccentricity
Min.
Max.
D
M
S
D
M
S
1
4
52
54
2000
6.72385E-03
4
48
1700
6.87199E-03
4
54
19
4
59
25
1400
7.02013 lE-03
5
0
36
5
5
55
53
5
12
26
1100
7.16827E-03
5
6
800
7.31641E-03
5
13
10
5
18
57
500
7.46455E-03
5
19
27
5
25
28
200
7.612619E-03
5
25
44
5
31
59
-100
7.76083E-03
5
32
0
5
38
31
17
5
45
2
-400
7.90897E-03
5
38
-700
8.0571 lE-03
5
44
33
5
51
34
- 1000
8.20525E-03
5
50
49
5
58
5
- 1300
8.35339E-03
5
57
5
6
4
37
-1600
8.50153E-03
6
3
21
6
11
9
- 1900
8.64967 lE-03
6
9
37
6
17
42
-2200
8.79781 lE-03
6
15
52
6
24
14
-2500
8.94595E-03
6
22
8
6
30
46
-2800
9.09409E-03
6
28
23
6
37
19
-3100
9.24223E-03
6
34
38
6
43
52
-3400
9.39037E-03
6
40
53
6
50
25
Table 14.8 : Kuja's eccentricity and periphery of manda epicycle Year
Eccentricity
Min.
Max.
D
M
S
D
M
S
2000
9.353136E-02
59
25
25
75
7
11
1700
9.317429E-02
59
13
37
74
48
14
1400
9.281723E-02
59
1
48
74
29
•18
1100
9.246017E-02
58
49
59
74
10
22
800
9.210311E-02
58
38
8
73
51
27 (Contd...)
178 Year
Ancient Indian Astronomy Eccentricity
Min. D
Max.
M
S
D
M
S
500
9.174605E-02
58
26
17
73
32
33
200
9.138899E-02
58
14
25
73
13
40
- 100
9.103193E-02
58
2
32
72
54
47
-400
9.067487E-02
57
50
38
72
35
56
-700
9.03178 lE-02
57
38
43
72
17
5
- 1000
8.996076E-02
57
26
48
71
58
15
- 1300
8.960369E-02
57
14
52
71
39
25
- 1600
8.924663E-02
57
2
55
71
20
37
- 1900
8.888957E-02
56
50
57
71
1
49
-2200
8.85325 lE-02
56
38
58
70
43
2
-2500
8.817546E-02
56
26
58
70
24
16
-2800
8.781839E-02
56
14
58
70
5
31
-3100
8.746133E-02
56
2
56
69
46
46
-3400
8.710428E-02
55
50
54
69
28
2
Table 14.9: Guru's eccentricity and periphery of manda epicycle Year
Eccentricity
2000 1700 1400 1100
4.826387E-02 4.865026E-02 4.903666E-02 4.942306E-02
32 32 33 33
800 500 200 - 100 -400 -700 - 1000 - 1300
4.980946E-02 5.019586E-02 5.058226E-02 5.096866E-02 5.135506E-02 5.174146E-02 5.212786E-02 5.251426E-02
33 33 34 34 34 34 35 35
- 1600 - 1900 -2200 -2500
5.290066E-02 5.328706E-02 5.367346E-02 5.405987E-02
35 35 36 36 .
-2800 -3100 -3400
5.444626E-02 5.483266E-02 5.521906E-02
36 36 37
D
Min. M
Max. M
S 44
D 36 37 37 37
19
23 2 39 16 52 26 0 33 33 36 5
38 38 38 39 39 39 39 40
33 48 2 16
34 2 29 55
40 40 41 41
35 54
31 45 0
20 44 7
41 42 42
51 10 29
38 53 8 22 37 51 6 21 35 50 4
50 8 27 46 4 23 42 1 20 38 57 16
13 32
S 5 47 30 14 58 44 30 18 6 56 46 37 30 23 17 12 8 5 3
Suggested Bijas (Corrections) for Planets' Positions
179
Table 14.10: Sani's eccentricity and periphery of manda epicycle Year 2000 1700 1400 1100 800 500 200 -100 -400 -700 - 1000 - 1300 - 1600 - 1900 -2200 -2500 -2800 -3100 -3400
Eccentricity 5.378298E-02 5.488584E-02 .0559887 5.709156E-02 5.819442E-02 5.929728E-02 6.040014E-02 .061503 6.260586E-02 6.370873E-02 6.481158E-02 6.591444E-02 .0670173 6.812016E-02 6.922302E-02 7.032588E-02 7J42874E-02 .0725316 7.363446E-02
D 36 36 37 38 38 39 40 40 41 42 42 43 44 44 45 46 46 47 48
Min. M 6 47 28 9 50 30 11 51 31 12 51 31 11 50 30 9 48 27 6
S 34 43 43 35 19 56 24 44 57 1 58 46 27 59 24 41 50 50 43
D 41 42 43 44 44 45 46 47 48 49 50 51 52 53 54 55 55 56 57
Max. M 18 12 6 1 55 50 44 39 34 29 24 20 15 11 7 2 58 54 51
S 39 42 52 11 37 11 52 42 38 43 55 15 43 18 1 51 49 55 8
Table 14.11: Uranus' eccentricity and periphery oi manda epicycle Year 2000 1700 1400 1100 800 500 200 - 100 -400 -700 -1000 - 1300 - 1600 - 1900 -2200 -2500 -2800 -3100 -3400
Eccentricity 4.697621E-02 4.75507 lE-02 4.81252 lE-02 4.86997 lE-02 4.92742 lE-02 4.984872E-02 5.04232 lE-02 5.09977 lE-02 5.15722 lE-02 5.2I4671E-02 5.27212 lE-02 5.32957 lE-02 5.38702 lE-02 5.44447 lE-02 5.50192 lE-02 5.55937 lE-02 5.61682 lE-02 5.674272E-02 5.73172 lE-02
D 31 32 32 32 33 33 34 34 34 35 35 35 36 36 36 37 37 37 38
Min. M 49 11 33 55 17 38 0 22 43 5 26 48 9 31 52 14 35 56 17
S 45 38 28 16 1 45 27 6 43 18 51 21 50 16 41 3 22 40 56
D 35 36 36 37 37 38 38 39 39 39 40 40 41 41 42 42 43 43 44
Max. M 47 15 43 11 39 6 34 2 30 58 26 54 22 51 19 47 15 43 12
S 54 37 23 11 1 53 47 43 41 41 44 48 55 4 14 27 42 59 19
Ancient Indian Astronomy
Table 14.12: Neptune's eccentricity and periphery of manda epicycle Year 2000 1700 1400 1100 800 500 200 -100 -400 -700 -1000 - 1300 - 1600 -1900 -2200 -2500 -2800 -3100 -3400
Eccentricity 8.61101 lE-03 8.53559E-03 8.460169E-03 8.38475E-03 8.30933E-03 8.23391 lE-03 8.15849E-03 8.08307 lE-03 8.00765E-03 7.93223E-03 7.8568 l l E - 0 3 7.78139E-03 7.70597E-03 7.63055 lE-03 7.555131E-03 7.47971 lE-03 7.40429E-03 7.32887E-03 7.25345 lE-03
D 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
Min. M 7 4 1 58 55 52 48 45 42 39 36 32 29 26 23 20 16 13 10
S 59 48 36 25 13 2 50 39 27 16 4 53 41 29 17 6 54 42 30
D .6 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 5 5
Max. M 15 12 9 6 2 59 56 52 49 46 42 39 36 32 29 26 22 19 16
S 59 40 20 0 41 21 2 42 23 4 44 25 6 46 27 8 49 30 11
Table 14.13: Pluto's eccentricity and periphery of manda epicycle Year 2000 1700 1400 1100 800 500 200 - 100 -400 -700 - 1000 -1300 - 1600 - 1900 -2200 -2500 -2800 -3100 -3400
Eccentricity .2488723 .2486784 .2484844 .2482905 .2480965 .2479026 .2477086 .2475147 .2473207 .2471268 .2469328 .2467389 .2465449 .246351 .246157 .2459631 .2457691 .2455752 .2453812
D 123 123 123 123 123 123 123 123 122 122 122 122 122 122 122 122 122 122 122
Min. M 22 19 16 13 9 6 3 0 56 53 50 47 43 40 37 34 30 27 24
S 42 29 16 2 49 35 20 6 51 37 21 6 51 35 19 2 46 29 12
D 232 232 231 231 231 231 231 230 230 230 230 229 229 229 229 229 228 228 . 228
Max. M 20 6 53 40 27 14 1 47 34 21 8 55 42 29 16 2 49 36 23
S 2 52 42 32 23 14 5 56 47 39 31 23 15 7 0 52 45 38 32
Suggested Byas (Corrections) for Planets' Positions
181
14J.4 MandoccoM of planets In the siddhdndc texts, except in the Surya siddhanta, the mandoccas (apogees) of planets have been taken as fixed. To a great extent they were not off the mark since the rates of motion of apogees are very small. The mandoccas of the various bodies as taken by the traditional texti are listed in Table 14.14. Table 14.14: Mandoceas of planeti Planeu
Aryabhatiyam
Khaitda
Ravi Budha
78° 210°
Khadyaka i(f 22(f
Sukra Kuja
90°
80°
118°
110°
130°
125° 30' 04"
Guru
180°
160°
171° 16'
167° 25' 12"
Sani
236°
240°
236° 37'
240° 51'51"
Suryasiddhanta Modem (500 A.D.) (499 A.D.) 77° 14' 220° 26'
74° 15' 14" 231° 12'09"
79° 49'
187° 16' 55"
The moon's mandocca changes rapidly and hence not included in the above table. Ravi's mandocca values according to the various texts are comparable to the one obtained using the modem formula (for sidereal apogee). In fact, the mandocca (modem) for the beginnings of 1400 A.D. and 1700 A.D. arc respectively 77° 8'46" and 78° 6'46". The famous Kerala astronomers of that period, like Madhava, Parames'vara and Nilakantha Somayaji, were significantly right in taking the mandocca of the sun around 77° to S0°. The (sidereal) mandocca of the sun for the beginning of 2000 A.D. is 79° 4' 55". Table 14.15 gives the {nirayana) mandoccas of the planets (except the moon) at the century beginnings from 2000 A.D. back to - 3100 (i.c. 3101 B.C.). For evaluating the coirect mandocca of any planet for any date during these centuries, the tables can be used. For intermediate dates proportional changes can be introduced by linear interpolation. In Table 14.15, the trans-saturnine planets viz.. Uranus, Neptune and Pluto are also included. Table 14.15: Mandoccas of planets (in Deg., Min., Sec.) •
Year 2000 1900 1800 1700 1600 1500 1400 1300 1200 1100
79 78 78 78 77 77 77 76 76 76
RAVI 4 45 26 6 47 28 8 49 30 10
BUDHA 55 31 8 46 26 5 46 27 9 52
233 233 233 233 232 232 232 232 232 232
35 26 16 7 57 47 38 28 19 9
49 15 41 7 33 58 24 49 14 40
187 187 187 187 187 187 187 187 187 187
SUKRA 42 42 41 40 39 37 36 35 33 31
KUJA 44 6 18 21 15 59 34 0 17 24
132 131 131 130 130 129 129 129 128 128
12 2 45 '22 18 40 51 58 25 13 58 28 42 31 4 55 38 6 17 11 (Contd...)
182
Year 1000 900 800 700 600 500 400 300 200 100 0 - 100 -200 -300 -400 -500 -600 -700 -800 -900 -1000 -1100 - 1200 - 1300 - 1400 - 1500 - 1600 - 1700 - 1800 -1900 -2000 -2100 -2200 -2300 -2400 -2500 -2600 -2700 -2800 -2900 -3000 -3100
Ancient Indian Astronon^y RAVI 75 75 75 74 74 74 73 73 73 72 72 72 72 71 71 71 70 70 70 69 69 69 68 68 68 67 67 67 66 66 66 65 65 65 64 64 64 63 63 63 62 62
51 32 13 53 34 15
55 36 17 58 38 19 0 41 21 2 43 23 4 45 25 6 46 27 7 48 28 9 49 30 10 51 31 11 51 32 12 52 32 12 52 32
35 18 2 46 30 14 58 43 27 10 54 37 20 3 45 26 7 46 25 3 41 17 52 25 58 29 59 27 54 19 42 4 24 42 57 11 23 33 40 45 47 47
232 231 231 231 231 231 231 230 230 230 230 230 230 229 229 229 229 229 229 228 228 228 228 228 228 228 227 227 227 227 227 227 226 226 226 226 226 226 225 225 225 225
BUDHA 0 50 40 31 21 12 2 52 43 33 24 14 4 55 45 36 26 16 7 57 48 38 28 19 9 0 50 40 31 21 11 2 52 42 33 23 14 4 54 45 35 25
5 30 55 20 44 9 33 58 22 46 10 34 58 22 46 9 33 56 20 43 6 29 52 14 37 0 22 45 7 29 51 13 35 57 18 40 1 23 44 5 26 47
SUKRA 187 29 27 187 187 24 187 22 187 19 187 16 187 13 187 10 187 7 187 4 0 187 186 56 186 52 48 186 44 186 186 40 35 186 186 31 186 26 21 186 186 16 186 11 5 186 0 186 54 185 48 185 42 185 185 36 29 185 23 185 185 16 9 185 2 185 184 55 184 48 41 184 184 33 184 25 184 17 184 9 1 184 53 183
23 11 51. 22 43 55 57 51 35 10 35 52 59 57 45 25 55 16 27 30 23 7 41 7 23 30 27 16 55 24 45 56 58 51 35 9 34 50 56 54 42 21
127 127 126 126 125 125 125 124 124 123 123 122 122 121 121 121 120 120 119 119 118 118 117 117 116 116 116 115 115 114 114 113 113 112 112 112 111 111 110 no 109 109
KUJA 44 27 17 36 50 44 23 51 56 58 30 4 3 9 36 14 . 9 19 42 22 15 26 48 29 21 31 54 34 27 36 0 38 33 40 6 42 39 44 12 45 45 47 18 49 51 51 24 53 57 56 30 59 4 2 37 5 10 9 14 43 16 19 49 24 22 31 55 38 28 45 1 54 35 3 8 13 41 24 14 36 47 49 21 3 (Contd...)
Suggested Bijas (Corrections) for Planets' Positions Year 2000 1900 1800 1700 1600 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0 -100 -200 -300 -400 -500 -600 -700 -800 -900 - 1000 -1100 - 1200 - 1300 - 1400
GURU 170 170 170 169 169 169 169 169 168 168 168 168 168 167 167 167 167 167 166
28 15 2 50 37 24 12 0 47 35 23 11 0 48
36 25 13 2 51 166 40 166 29 166 18 166 7 165 57 165 46 165 36 165 25 165 15 165 5 164 55 164 46
25 32 44 2 26 55 30 11 57 49 48 52 3 19
SANI 11 38 4 30 57 23 50 16 43 10 37 3 30 57
249 248 248 247 246 246 245 245 244 244 243 243 242 241 42 241 24 12 240 51 47 240 19 30 239 46 18 239 13 14 238 40 16 238 25 237 41 237 4 236 33 235 10 235 54 234 45 234
8
35 3 30 58 26 54 21
44
233 49 50 233 17
URANUS 56 10 27 48 13 41 14 50 29 13 0 51 45 44 46 51 1 14
31
229 229 228 228 228 228 228 228 228 228 228 228 228 228 227 227 227 227 227 227
10 5 59 54 49 43 38 33 27 22 17 11
9 46
204 204 204 204 204 204 204 204 204 204 204 203 203 203 203 203 203 203 203
38 55 12 31 49 9 29 50 11 33 56 19
53 51
1 28
19 19 19 19 19
49
56
19
25
48
39
24
33
59 34 203 8 203 43 203 17 203 51 203 24 203 58 203 31 203 4 203 37 203
48
203
52 28 16 227 23 44 227 17 16 227 12 52 227 6 31 227 1 14 226 55 1 226 50 51 226 45 39 45 226
PLUTO
17 15 14 12 10 9 7 5 4 2 0 59, 57 56 54
44
44 9
34
20 20 20 20 20 20 20 20 19 19 19
0 0 0 0 0 0 0 0 59 59 59 59 59 59 59 59 59
50
58
55
44
37 31 24 18 11 4 58
51 44
37 30 23 16 9 2
46
55
19 19
58
48
45 43 42
25 57
19 19
58
41
58
34
19
58
!9
58
39
29 2 36
58
38
11
36 35
47 24
19 19 19
27 19 12
34 32
41
19
58 57 57
5 57 •50
2
19
57
42
40
57
43
226
34
10
226
28
43
203 31 203 30 48 203 28 203 27 20 52 203 26 24 203 24
20
19 19
57
35 27
15
1
19
57
20
42
19
57
12
25
19
57
4
9
19
56
57
53
19
56
49
203 203
23
39
19
22
26
19
45
- 1500
163 59
5
- 1600 - 1700
163 50 163 41
49 226 1 13 225 55 41 225 50
35 3
30 12 53 34 15 55 34 14 52 31
1 55 50
3 232 45 24 232 13 53 231 42 29 231 10 13 230 38 6
NEPTUNE
23 0 36 12
6
164 36 164 26 164 17 164 8
230 5 229 13 229
183
51 226 23 0 226 17 12 226 12 29 226 6
57 29
41 56
33
(Contd...)
184
Ancient Indicui Astronomy
Year -1800 - 1900 -2000 -2100 -2200 -2300 -2400 -2500 -2600 -2700 -2800 -2900 -3000 -3100
GURU 163 32 163 23 163 163 162 162 162 162 162 162 162 162 161 161
15 7 58 50 42 35 27 20 12 5 58 51
28 52 24 5 53 51 56 11 33 5 45 35 33 40
SANI 228 32 13 228 0 48 227 29 27 226 58 9 226 26 56 225 55 46 225 24 40 224 53 37 224 22 39 223 51 44 223 20 52 222 50 5 222 19 21 221 48 41
URANUS
NEPTUNE
225 45 39 34 28 23 17
1 33 5 37 9 41
203 21 203 20 203 18 203 17 203 16 203 15
3 53 44 37 30
225 225 225 224 224 224 224 224
14 46 18 51 23 56 29 2
203 203 203 203 203 203 203 203
25 21 18 16 16 17 19 22
225 225 225 225 225
12 6 1 55 50 44 39 34
14 13 12 11 10 9 8 7
14
PLUTO 19 19 19 19 19 19 19 19
56 56 56
19 19 19 19 19 19
56 55
26 18 10 2 54
55 55 55
46 38 29
55 55 55 54 54 54
21 13 5 57 48 40
14.2.5 Peripheries of Sighra epicycles of planets The peripheries of the sighra epicycles of the different planets taken by the various siddhantic texts are presented in Table 14.16. In fact, it is found that'the periphery of the sighra epicycle of a planet is elated to its mean heliocentric distance a. For the inferior planets viz.. Budha and Sukra. the periphery of the sighra epicycle is given by p = (360° a) where a is in astronomical units. One astronomical unit (a.u.) is defined as the mean distance of the earth from the sun. In Table 14.16, the peripheries of the sighra epicycles of the planets according to the traditional texts are compared with the modem values (considering the mean heliocentric distances of the planets). In the case of the superior planets viz., Kuja, Guru, Sani and the trans-saturnine planets, the periphery is given hy p = (360°/a). Table 14.16: Peripheries of Sighra epicycles Planets Budha Sukra Kuja Guru Sani Note :
Mean heliocentric Aryabhatiyam Khatida Suryasiddhanta distance a Khadyaka 132 132 - 133 0.3870986 130.5 - 139.5 260 - 262 0.7233316 256.5 - 265.5 260 1.523692 5.202561 9.554747
229.5 - 239.5 67.5 - 72 36.5 - 40
The peripheries are in degrees.
234 72 40
232 - 235 70 - 72 39-40
Modern 139.3555 260.3994 236.2682 69.1967 37.6776
Suggested Bijas (Corrections) for Plcaiets' Positions
185
According to modem astronomy, the mean heliocentric distance a i.e., the semi-major axis of the elliptical orbit as well as its eccentricity e of a planet gradually and slowly change with time. The semi-minor axis b of the orbit is given by b = a V l -e^ at any given time. The actual distance (radius-vector) of a planet from the sun varies between the semi-minor axis b and the semi-major axis a. Accordingly, it is conjectured that the periphery of the sighra epicycle varies (i) from (360Va) to {360°/b) for a superior planet and (ii) from (360 b)° to (360 a)" for an inferior planet. The eccentricities, the semi-major and the semi-minor axes of the orbits of the planets, from Budha to Pluto, are considered as they vary over the centuries and the corresponding minimum and maximum values of the peripheries of the sighra epicycles are computed. The results are provided in Tables 14.17 to 14.24 at the beginnings of the century years from 2000 A.D. backwards to - 3400 (i.e. 3401 B.C.) at intervals of 300 years. TaUc 14.17: Budha's mean heliocentric distance and Sighra periphery Year
Helio. Dist. (A.U.)
Maximum
Minimum D
M
S
D
M
S
2000
.38709893
136
22
39
139
21
20
1700
.38709695
136
22
44
139
21
17
1400
.38709497
136
22
50
139
21
15
1100
.38709299
136
22
55
139
21
12
800
.38709101
136
23
1
139
21
9
500
.38708903
136
23
6
139
21
7
200
.38708705
136
23
11
139
21
4
-100
.38708507
136
23
17
139
21
2
-400
.38708309
136
23
22
139
20
59
-700
.38708111
136
23
28
139
20
57
- 1000
.38707913
136
23
33
139
20
54
- 1300
.38707715
136
23
39
139
20
52
-1600
.38707517
136
23
44
139
20
49
-1900
.38707319
136
23
50
139
20
46
-2200
.38707121
136
23
55
139
20
44
-2500
.38706923
136
24
0
139
20
41
-2800
.38706725
136
24
6
139
20
39
-3100
.38706527
136
24
11
139
20
36
-3400
.38706329
136
24
17
139
20
34
186
Ancient Indian Astronomy Table 14.18: Siikra'i mean heiiocentrk dkteiKc and Sighra pcripiicry
Year 2000 1700 1400 1100 800 500 200 - 100 -400 -700 -1000 - 1300 - 1600 - 1900 -2200 -2500 -2800 -3100 -3400
Helio. Dist. (A.U.) .72333199 .72332923 .72332647 .72332371 .72332095 .72331819 .72331543 .72331267 .72330991 .72330715 .72330439 .72330163 .72329887 .72329611 .72329335 .72329059 .72328783 .72328507 .72328231
D 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260
Minimum M 23 23 23 23 23 23 23 23 22 22 22 22 22 22 22 22 22 22 22
S 36 32 27 23 18 13 9 4 59 55 50 45 40 36 31 26 21 17 12
D 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260
Maximum M 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 22 22
S 58 54 51 47 43 40 36 33 29 26 22 18 15 11 8 4 0 57 53
Table 14.19: Kuja's mean heliocentric distance and Sighra periphery Year 2000 1700 1400 1100 800 500 200 - 100 -400 -700 - 1000 - 1300 - 1600 - 1900 -2200 -2500 -2800 -3100 -3400
Helio. Dist. (A.U.) 1.52366231 1.52387894 1.52409557 1.5243122 1.52452883 1.52474546 1.52496209 1.52517872 1.52539535 1.52561198 1.52582861 1.52604524 1.52626187 1.5264785 1.52669513 1.52691176 1.52712839 1.52734502 1.52756165
Minimum D M 236 16 236 14 236 12 10 236 236 8 6 236 4 236 236 2 0 236 58 235 56 235 54 235 235 52 235 50 235 48 46 235 44 235 • 42 235 40 235
S 22 21 20 19 18 17 17 16 15 15 14 13 13 12 12 12 11 11 10
D 237 237 237 237 237 237 237 237 236 236 236 236 236 236 236 236 236 236 236
Maximum M 18 16 13 11 8 6 3 1 58 56 53 51 48 46 43 41 38 36 33
S 37 7 37 7 38 8 39 9 40 12 43 14 46 18 49 21 54 26 59
Suggested Bijas (Corrections) for Hanets' Positions
187
Table 14.20: Guru's mean heliocentric distance and Sighra periphery Year 2000 1700 1400 1100 800 500 200 - 100 -400 -700 -1000 - 1300 -1600 - 1900 -2200 -2500 -2800 -3100 -3400
Helio. Dist. (A.U.) 5.20336301 5.2015409 5.19971879 5.19789668 5.1%07457 5.19425246 5.19243035 5.19060824 5.18878613 5.18696402 5.18514191 5.1833198 5.18149769 5.17967558 5.17785347 5.17603136 5.17420925 5.17238714 5.17056503
D 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69
Minimum M 11 12 14 15 16 18 19 21 22 24 25 27 28 30 31 33 34 36 37
S 9 36 4 31 59 26 54 21 49 17 44 12 40 8 36 4 33 1 29
D 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69
Maximum M 16 17 19 20 22 23 25 26 28 29 31 32 34 36 37 39 40 42 43
S 1 33 6 38 10 43 15 48 .21 53 26 59 33 6 39 13 46 20 54
Table 14.21: Sani's mean heliocentric distance and Sighra periphery Year 2000 1700 1400 1100 800 500 200 -100 -400 -700 - 1000 - 1300 - 1600 - 1900 -2200 -2500 -2800 -3100 -3400
Helio. Dist. (A.U.) 9.53707032 9.54611622 9.55516212 9.56420802 9.57325392 9.58229982 9.59134572 9.60039162 9.60943752 9.61848342 9.62752932 9.63657522 9.64562112 9.65466702 9.66371292 9.67275882 9.68180472 9.69085062 9.69989652
D 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37
Minimum M 44 42 40 38 36 34 32 29 27 25 23 21 19 17 15 13 10 8 6
S 50 42 33 25 17 9 1 54 47 40 33 27 21 15 9 4 59 54 49
D >7 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37
Maximum M 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12
S 10 9 9 9 9 10 11 13 15 18 20 24 27 31 35 40 45 51 57
188
Ancient Indicm Astronomy Table 14.22: Uranus* mean heliocentric distance and Sighra periphery
Year 2000 1700 1400 1100 800 500 200 - 100 -400 -700 - 1000 - 1300 - 1600 -1900 -2200 -2500 -2800 -3100 -3400
Helio. Dist. (A.U.) 19.19126393 19.18670318 19.18214243 19.17758168 19.17302093 19.16846018 19.16389943 19.15933868 19.15477793 19.15021718 19.14565643 19.14109568 19.13653493 19.13197418 19.12741343 19.12285268 19.11829193 19.11373118 19.10917043
D 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18
Minimum M 45 45 46 46 46 46 47 47 47 47 48 48 48 49 49 49 49 50 50
S 30 46 2 18 34 51 7 23 39 55 11 27 43 0 16 32 48 4 20
D 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18
Maximum M 46 47 47 47 47 48 48 4«' 49 49 49 50 50 50 50 51 51 51 52
S 45 3 21 39 57 15 33 52 10 28 46 4 23 41 59 18 56 54 13
Table 14.23: Neptune's mean heliocentric distance and Sighra periphery Year 2000 1700 1400 1100 800 500 200 - 100 -400 -700 - 1000 - 1300 - 1600 - 1900 -2200 -2500 -2800 -3100 -3400
Helio. Dist. (A.U.) 30.06896348 30.07271936 30.07647524 30.08023112 30.083987 30.08774288 30.09149876 30.09525464 30.09901052 30.1027664 30.10652228 30.11027816 30.11403404 30.11778992 30.1215458 30.12530168 30.12905756 30.13281344 30.13656932
D 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11
M mimum M 58 58 58 58 57 57 57 57 57 57 57 57 57 57 57 57 56 56 56
S 20 15 10 4 59 54 48 43 37 32 27 21 16 11 5 0 54 49 44
D 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11
Maximum M 58 58 58 58 58 57 57 57 57 57 57 57 57 57 57 57 56 56 56
S 22 17 11 6 0 55 50 44 39 33 28 23 17 12 6 1 56 50 45
Suggested Bijas (Corrections) for Planets' Positions
189
Table 14.24: Plato's mean heliocentric distance and Sighra periphery Year
Helio. Dist. (A.U.)
2000 1700 1400 1100 800 500 200 -100 -400 -700 -1000 -1300 -1600 -1900 -2200 -2500 -2800 -3100 -3400
39.48168677 39.48399413 39.48630149 39.48860885 39.49091621 39.49322357 39.49553093 39.49783829 39.50014565 39.50245301 39.50476037 39.50706773 39.50937509 39.51168245 • 39.51398981 39.51629717 39.51860453 39.52091189 39.52321925
Minimum
Maximum
D
M
S
D
M
S
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
7 7 7 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
•5 3 1 59 57 55 53 51 50 48 46 44 42 40 38 36 34 32 30
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
24 24 24 24 24 24 24 24 24 24 24 24 24 24 23 23 23 23 23
51 47 43 39 36 32 28 25 21 17 13 10 6 2 59 55 51 48 44
143 Moon's equations In computing the position of the moon, according to the siddhantic texts, there has always been a noticeable deviation. The ancient Indian astronomers suggested the well known corrections, besides the manda equation, which we call evection and variation. TTie equation of centre (manda connection) was known in India even before Aryabhata I (476 A.D.). In fact, Aryabhata himself gives the coefficient in the manda equation as 300'. 25 for the Moon. Brahmagupta in his Uttara Khanda Khadyaka gives the same as 301'. 7. However, it must be pointed out that out of the actual equation of centre, a part of it is combine'd with the second correction ("evection") and the combined equation is given in later siddhantic texts. In fact, this combined equation for the moon was first given, among the Indian astronomers, by Mai^jula (or Mufijala, 932 A.D.) in his Laghumdnasam. P C . Sengupta points out, "In form the equation is most perfect, it is far superior to Ptolemy's; it is above all praise." While the credit of discovering the moon's second equation, among the Hindu astronomers, undoubtedly goes to Mafijula, it was Bhaskara II (1114 A.D.) who introduced it into his siddhanta. The third equation for the moon's position, "variation" was introduced in Indian astronomy by Bhaskara II in 1152 A.D., four centuries before Tycho Brahe discovered it in the west.
190
Ancient Indian Astronomy
The honour of introducing the fourth equation, to the moon's position, now called "annual equation" goes to the highly dedicated astute asronomer from Orissa, Mm. Samanta Chandrashekhara Simha of the last century. He called it "Digarnsa" samskdra and incorporated it in his remarkable text, Siddhanta darparia. The constant coefficient in Chandrashekhara's equation is \\'21".6. It is important to note that Tycho Brahe had given the coefficient as 4'30". The modem value is 11'10". Thus, Chandrashekhara Samanta's value is far closer to the modem value. This accuracy of his value is truly remarkable in the light of the fact that the Samanta was trained exclusively in the orthodox Sanskrit tradition and totally ignorant of the English education or the western development of astronomy. The modem expressions for the three above-said equations of the moon are as follows: 1. Evection = 4586" sin (2D - g) where D = M-S, the mean elongation of the moon (from the sun), M and S being the mean longitudes of the moon and the sun respectively and g is the mean anomaly of the moon (from its perigee). In the context of Indian astronomy, the mean anomaly (manda kendra) is measured from the apogee (mandocca). If the perigee and the apogee of the moon are denoted respectively by P and A, then we have Mean anomaly, g = M-P = M-(A + l80°) = (M-A)-180° so that the evection equation becomes Evection = 4586" sin [2D-{M-(A = -4586" sin
+ \ 80°)) ]
[2D-(M-A)]
However, as defined in the SHrya siddhanta, Manda anomaly = Mandocca - Mean longitude = A- M in which case Evection = - 4586" sin [2D + MA] where MA=A-M, the manda anomaly of the moon. In terms of the mean longitudes of the sun (5) and the moon (M) and the mandocca A, we have Evection = - 4586" sin (M -2S + A)
...(1)
2. Variation = 2370" sin (2D) ...(2) where D = M-S, the moon's elongation from the sun. Chandrashekhara Simha in his Siddhanta Darparia has taken this equation as [R sin 2(M- S)]/90 or 38' 12" sin 2D i.e. 2292" sin (2D) where R = 3438'. 3. Annual equation = - 668" sin (g') where g' is the sun's mean anomaly i.e. g' = S-P'[P' : Sun's perigee] Here also, considering the sun's anomaly measured from his apogee (mandocca) A', as is the case in the siddhdntas, we have Annual equation
= - 668" sin [S -(A'+\
80°)]
|
't
Suggested Bijas (Corrections) for Planets' Positions = 668" sin (S-A') i.e. Annual equation = - 668" sin (A' - S)
191
...(3)
where (A' - S) is the sun's manda kendra (as defined in the Surya siddhanta). Remark : In fact, the moon's annual equation happens to be a fraction of the sun's mandaphala (equation of centre). According to the modern values of the concerned coefficients, we have Sun's equation of centre Annual equation of the moon
= 6910" sin (g') = - 668" sin (g') "*" 668 ~" 1 The ratio of the latter to the former = - r r r r = T T T T 6910 10.34 Chandrashekhara Samanta has approximated this ratio to - 1/10 and taken Moon's annua! equation = Sun's manda equation. As a respectful tribute to Mm. Samanta Candras'ekhara Simha, we shall continue to use the names Turigdntara, Paksika and Digamsa, given by him, respectively for eveclion, variation and the annual equation. Thus, from (1), (2) and (3), we have the three equations of the moon, besides the usual mandaphala, given by (i)
Turigdntara Samskdra (Evection) = - 4586" sin (M - 2S + A)
(ii)
Paksika Samskdra (Variation)
(iii)
= 2370" sin 2 (M - 5)
Digamsa Samskdra (Annual equation) = - 668' sin {MK) =
(sun's
equation) where MK is the sun's manda kendra (anomaly of apsis) defined by MK = A' -S {A' : Sun's mandocca) Note :
In the Tuhgantara equation, given by Candras'ekhara Simha, a part of the (modern) equation of centre is combined with the evection equation. In the earlier siddhantic texts also, the mdnda equation included only a major part of the (modem) equation. However, in the proposed equations we are suggesting in this work, the three equations (i) to (iii) above correspond to the three equations adopted in modem astronomy.
14.4 The case of Budha and Sukra The traditional Indian astronomical texts have always treated Budha and Sukra differently from the remaining taragrahas viz., Kuja, Guru and Sani in the context of determining their true positions. (i)
While the mean positions of the superior planets are taken as they are, in the case of Budha and Sukra (the inferior planets) two special points called Budha sighrocca and Sukra sighrocca are considered.
192
Ancient Indian Astronomy
The position of the mean Ravi is itself taken asthe positions of both mean Budha and mean Sukra. Again, while working out the sighra equation, the argument of the relevant sine function is taken, for example in the case of Budha, as {B - ^) as per the SUrya siddhanta convention, where B and R are respectively the Budha sighrocca and the mean Ravi. In the case of the superior planets, the order of the terms in the argument is reversed. For example, for Guru, the argument in the sine term of the sighra equation is (R - G) where G is the mean position of Guru. For the superior planets the mean Sun (Ravi) is considered as their sighrocca. Nilakantha Somayaji (1444 - 1550 A.D.) points out in his Tantra sahgraha that it is incorrect to have a differential treatment to the inferior and superior planets and that the sun is the common centre for the sighra equation to all the planets. This is truly a remarkable breakthrough in the history of mathematical astronomy in general and in Indian astronomy in particular (see Ramasubramanian et al.. Current Science, May 1994). In fact, Nilakantha's innovation, prompted by his paramaguru Parames'vara (1380 - 1460 A.D.), is highly suggestive of a heliocentric model of planetary motion, much before Copernicus. Following the innovation introduced by the celebrated Kerala astronomer Nilakantha, we propose the consideration of the sighra anomaly in the same way for all the planets, namely Sighra Kendra = Sighrocca - Mean planet (in the SUrya siddhanta style) where the sighrocca is the mean sun for all the planets. (ii)
In the case of Budha (Mercury), the major reason why its calculated true position, according to siddhdntas, goes generally off the mark is that the manda periphery taken for Budha sighrocca by the siddhdntas is far below its actual modem value.
The manda periphery, as prescribed in the SUrya siddhanta, varies from 28° to 30° for Budha. But, actually, as pointed out in Table 14.3, the periphery of Budha's manda epicycle should vary from about 109°.8 to about 184°.7. The coefficient of the manda equation depends on the eccentricity e of the planet's orbit. In fact, the coefficient, to a first approximation is given by 2e. In the case of Budha, e = 0.205656 currently and hence the co-efficient of the manda equation turns out to be about 23°.566. In our proposed improved values for the parameters (vide Table 14.3 and 14.6) we have considered even higher powers of e upto and the first two terms in the expansion for the equation of centre. Accordingly, for the beginning of 2000 A . D . , the variation of the periphery of the manda epicycle is from 109° 50'46" to 184° 47'43" (dependihg on the varying manda anomaly). Correspondingly, the coefficient of the manda equation varies from about 17°.48 to 29°.41. In fact, our ancient Indian astronomers had not taken note of the fact that among the planets known to them, Budha's orbit has the largest eccentricity. This lapse explains why Budha's true position always remained an enigma to them
Suggested Bijas (Corrections) for Planets' Positions
193
despite all the bija samskdras introduced by them. Of course, now after the discovery of the three trans-saturnine planets, we know that Pluto's orbit has the largest eccentricity (0.2488723) while the immediately next is Mercury's orbit (0.205656). 14.5 Mean positions of bodies at the Kali epoch The mean positions of the sun, the moon (with its mandocca, anomaly from the apogee and the ascending node, Rahu), and all the planets, upto Pluto, at the Kali epoch (mean midnight between 17/18 February 3102 B.C. at Ujjayini) are computed according to modern formulae. The sdyana (tropical) as well as the nirayana (sidereal) longitudes are listed in Table 14.25. The mean ayanamsa adopted for the Kali epoch is - 4 6 ° 34' 52" based on the recommendations of the Calendar Reform Committee-of the Govt, of India. Table 14.25: Mean positions of bodies at the Kali epoch Date (Julian)
:
18/2/-3101
Time (1ST)
:
0 Hours 27 Minutes (local midnight)
Place
•
Ujjayini
Longitude
75°46'E
Latitude
23°11'N
Julian Days
: 588466
Week day : FRIDAY
•Mean longitudes of Planets* Ayanamsa : - 46° 34' 52' Body Candra Apogee Rahu •Ravi Apogee Sukra Budha Kuja Guru Sani Uranus Neptune Pluto
Nirayana
D 297 66 142
Sayana M 41 31 12
S 10 36 36
D 344 113 188
M 16 6 47
S 2 28 28
301 15 333
37 59 52
54 32 43
348 62 20
12 34 27
46 24 35
267 289 26 349
27 2 41 49
27 33 57 59
314 335 73 36
2 37 16 24
19 25 49 51
344 250 317
50 1 21
57 11 11
31 296 3
25 36 56
49 3 3
194
Ancient Indian Astronomy Table 14.26: Mandoccas of planets at the KiUi epoch. *Mean Mandoccas (ApogM9> of Pbnets* Ayanamsa : — 46° 34' 52"
Mandocca of Candra Ravi Kuja Budha Guru Sukra Sani Uranus Neptune Pluto Note :
D 66 15 109 178 191 293 325 349 70 333
Tropical M 31 59 2 5 54 54 46 2 1 0
S 36 32 33 2 28 36 24 35 11 18
D 113 62 155 224 238 340 12 35 116 19
Sidereal M 6 34 37 39 29 29 • 21 37 36 35
S 28 24 25 54 20 28 16 27 3 10
In Tables 14.25 and 14.26, (i) Apogee is the mandocca and the anomaly,(Anom.) is the manda kendra measured from the apogee; (ii) in the case of Budha and Sukra, the mean positions of the planets themselves are considered and not as their sighrocca. As pointed out earlier, the mean sun is now the sighrocca for all the planets; (iii) the Surya siddhanta has taken the moon's mandocca at the Kali beginning as 9 0 ° while our modem computation for the (nirayana) mandocca brings it to 113° 6' 28".
Table 14.26 provides the mandoccas (nirayana apogees/aphelions) at the epoch, the beginning of the Kaliyuga. 14.6 Revolutions of bodies in a Kalpa In Table 14.2, we have provided the numbers of revolutions (rounded off to the nearest integer) executed by the heavenly bodies and the special points in the course of a Mahayuga of 432 x 10 years. However, to get more accurate values for the mean positions of the bodies we have listed the numbers of revolutions in a Kalpa (432 x 10^ years) based on the modem known rates of (sidereal) motion of these bodies in Table 14.27. In this table the revolutions of the (sidereal) mandoccas of all the planets, including Uranus, Neptune and Pluto, are also provided. Table 14.27: Revolutions of bodies in a Kalpa Grahas Revolutions of Grahas Revolutions of Mandoccas 48,81,25,074 Candra 57.75,29,85,910 Ravi 38,777 4,32,00,00,000 19,134 Budha 17,93,70,33,867 1,439 7,02,22,60,402 Sukra 53,367 Kuja 2,29,68,76,453 25,671 Guru 36,41,95,066 67,486 14,66,56,219 Sani 10,607 Uranus 5,14,16,997 3,424 2,62,19,242 Neptune Pluto 1,73,90,083 214 23,22,68,618 — Rahu
Suggested Bijas (Corrections) for Ranets' Positions
195
14.7 Conclusion In the preceding pages we have presented our work on a comparative study of the three traditional Indian astronomical texts with special reference to (i) computations of mean and true positions of the sun, the moon and the taragrahas and (ii)
computations of lunar and solar eclipses.
The results are compared with the corresponding modern values. The relevant algorithms are computerized and their source codes Clistings') are provided at the end of the present chapter. We propose to pursue the follow-up work in refining the proposed bija (corrections) to the parameters and improve further the procedures for eclipses by incorporating the latest available astronomical data. For this task we propose to take up many other Kararia texts and investigate the extents of the accuracy of their results. Particularly it will be relevant to use our improved algorithms to verify the correctness of the epochal positions of planets given in these texts for their comtemporary periods. AH the algorithms, provided in the ancient and medieval Indian astronomical texts as well as our suggested improvements (with bijas) will be computerised. Our results, as and when ready, will be presented in our future works for the benefit of students and researchers of siddhantic astronomy.
COMPUTER PROGRAMS
10 20
C L S : K E Y OFF : R E M * P R O G A M 7.1 :"SSRAMOON"* PRINT TAB (23); "**************************"
30 40 50
PRINT TAB (23); "* SURYA SIDDHANTA *" PRINT TAB (23); "* POSITIONS OF SUN, MOON A N D R A H U *" PRINT TAB (23); "**************************"
60 70
PI = 3.141592653589793# LOCATE 5,16:PRINT "(CHRISTIAN) DATE :":LOCATE 5,40:LINE INPUT 'YEAR:";YE$:LOCATE 5.53:LINE INPUT "MONTH:";MO$: LOCAIE 5,65:LINE INPUT 'DAIE:"a)A$:Y=VAL(YE$)MM=VAL(MO$)-Dl=VAL(DA$)
80
120
LOCATE 6,16:PRINT "TIME (AFTER MIDNIGHT):";:LOCATE 6,40:LINE INPUT "HOURS:" ;HR$:LOCATE 6,55:LINE INPUT "MINUTES:"; MIN$:H1=VAL(HR$):MI=VAL(MIN$) LOCATE 7,16:PRINT " N A M E OF THE P L A C E : " ;:LOCATE 7,40:LINE INPUT PLACES LOCATE 8,16:PRINT "LONGITUDE (- ve for West): ";:LOCATE 8,45:LINE INPUT "DEG:";LD$:LOCATE 8,60:LINE INPUT "MIN: ";LM$ LOCATE 9,16:PRINT "LATITUDE (- ve for South): ";:LOCATE 9,45:LINE INPUT "DEG:";PD$:LOCATE 9,60:LINE INPUT "MIN: ";PM$ LD=VAL(LD$):LM=VAL(LM$):PD=VAL(PD$):PM=VAL(PM$)
130
IF LD<0 THEN LAM=LD-LM/60:GOTO
140
LAM=LD-i-LM/60
150
IF PD<0 THEN PHI=PD-PM/60:GOTO 180
160
PHI=PD+PM/60
170
R E M ***UJJAYINI:LONG.75.75E, LAT23.18N ***
90 100 110
150
180
ULAM=75.75-LAM: R E M ***LONG w.r.t. UJJAYINI***
190
TC = INT((Y - 1900)/100)
200
T = Y-100*INT(Y/100)
210
IF TC<-4 THEN E=13
220
IF T C = ^ A N D Y < 1582 THEN E=13
230
IF TC=-4 AND Y > 1582 THEN E=3
240
IF TC>-4 AND TC<=0 THEN E=-TC
250
IF T O O THEN E=-(TC-1)
Computer Programs 260
197
270
Q=-(T MOD 4) DD=0
280 290 300
DATA 0,31,28,31,30,31,30,31,31,30,31.30 RESTORE 280:FOR 1=1 TO M M READ X
310 320
DD=DD+X NEXT I
330 340
IF (Y>1600) AND (Y/100=INT(Y/100)) A N D (Y/400<>INT(Y/400)) THEN GOTO 350 IF Y/4=INT(Y/4) A N D (MM=1 OR MM=2) THEN DD=DD-1
350
JJ=(TC* 100+T)*365.25+DD+D1 •fE+(Q/4)+2415020!
360
LET K A L I = JJ-588466!
370
WD=JJ-7*INT(JJ/7)
380
RESTORE 410
390
FORF=0TOWD
400
READ X$
410
DATA MONDAY,TUESDAY,WEDNESDAY,THURSDAY,FRIDAY,SATURDAY,
420 430 440 450
SUNDAY NEXTF PRINT TAB(22);"********WEEK D A Y : " ; X $ ; " * * * * * * * * " PRINT"KALI Y U G A DAYS ELAPSED (EPOCH:Feb.l7/18,3102 B.C.): " ; K A L I
460
R E M **4 320 000 REVOLNS IN 1 577 917 828 CIVIL DAYS **
470
DAILY=2.737785151635919D-03:REM ** R E V O L N
480
MRAVI#=KALI*DAILY:REM ** REVOLNS SINCE K A L I EPOCH
490
REV=INT(MRAVI#):PRINT TAB( 15) "REVOLNS SINCE K A L I EPOCH: ";REV
500
MRAVI = 360*(MRAVI#-REV) :REM ** DEGREES
510
PRINT " M E A N RAVI AT UJJAYINI
MIDNIGHT";
520
L=MRAVLGOSUB
530
R E M ** DESHANTARA COR. **
1550
540
DAILY= .9856026545889309#:REM * DEG *
550
GOSUB
1460
560
KAALA=(Hl+MI/60)*DAILY/24
570
PRINT "MOTION FOR ";H1; "HRS ";MI; "MIN:";
580
L=KAALA:GOSUB
590
PRINT " M E A N RAVI AT THE GIVEN L O C A L TIME";
600
MRAVI=MRAVI+DESH+KAALA:L=MRAVI:GOSUB
610
R E M ****** SUN'S M A N D O C C A *******************
1550 1550
198
Ancient Indian Astronomy
620
CIVIL=1577917828#:REM * 387 REVNS IN 1577917828*1000 GIVE. DAYS *
630
SMA=77.13+(KALI*387*360/(CIVIL* 1000))
640
PMA1=SMA-MRAVI:REM *** RAVI'S M A N D A K E N D R A ********
650
IF P M A U O THEN PMA1=PMA 1+360
660
K1=14/(2*PI):K2=20/(60*2*PI)
670
PMA=SMA : SMK=PMA1 :REM *SUN'S A N O M A L Y *
680
GOSUB 1610
690
SEQ=PEQ
700
TRAVI=MRAVI+SEQ
710
IF TRAVI<0 THEN TRAVI=TRAVI+360
720
IF TRAVI>360 THEN TRAVI=TRAVI-360
730
PRINT"
740
PRINT "TRUE R A V L " ;
"
750
L=TRAVI:GOSUB 1550
760
PRINT"
770
LOCATE 23,60:PRINT "PRESS A N Y K E Y TWICE"
780
A$=INPUT$(2)
790
PRINT :PRINT
800
PRINT "******************* C H A N D R A SPHUTA ********************"
810
R E M ** NO. OF REVNS:57 753 336 for 1 577 917 828 CIVIL DAYS **
820
R E M *** DAILY M E A N MOTION:790'34"52"'3.8"" ***
830
DAILY= 3.660097818477782D-02:REM ** R E V O L N
840
MOON#=KALI*DAILY
850
REV=INT(MOON#):PRINT TAB(15)"REV0LNS SINCE K A L I EPOCH: ";REV
860
M O O N = 360*(MOON#-REV).
870
PRINT " M E A N M O O N AT UJJAYINI MIDNIGHT:";
880
L=MOON:GOSUB 1550
"
890
DAILY=360*DAILY:GOSUB 1480
900
KAALA=(Hl+MI/60)*DAILY/24
910
PRINT "MOTION FOR ";H1;"HRS ";MI;" MIN:";
920
L=KAALA:GOSUB 1550
930
PRINT " M E A N MOON AT GIVEN TIME AT";PLACE$;
940
MOON=MOON+DESH+KAALA:L=MOON:GOSUB 1550
950
MMA=.25+(KALI*488203!/CIVIL):REM
960
MMA=
970
PMA1= M M A-MOON
980
IF PMA1 <0 THEN PMA1 =PMA1 +360
990
K1=32/(2*PI):K2=20/(60*2*PI)
360*(MMA-INT(MMA))
** MOON'S M A N D O C C A **
199
Computer Programs 1000
P M A = M M A :MMK=PMA1 :REM * M M K : M O O N ' S A N O M A L Y *
1010 1020
GOSUB 1610 TMOON=MOON+PEQ
1030
R E M *** BHUJANTARA CORRECTION ***
1040
BHUJ=SEQ/27
1050
PRINT "BHUJANTARA CORRECTION: ";
1060
IF BHUJ>0 THEN PRINT TAB(39); "-";
1070 1080
IF BHUJ<0 THEN PRINT TAB(39); "+"; L=BHUJ:GOSUB 1550
1090 1100
TMOON=TMOON+BHUJ IF TMOON<0 THEN TMOON=TMOON+360
1110
IF TMOON>360 THEN TMOON=TMOON-360
1120
PRINT"
1130
PRINT "TRUE MOON :";
"
1140
L=TMOON:GOSUB 1550
1150 1160 1170 1180 1190 1200 1210 1220 1230 1240 1250 1260 1270
PRINT" _ " PRINT "******************** R A H U SPHUTA **********************" R E M ** NO. OF REVNS. -232 238 REVOLNS IN 1 577 917 828 CIVIL DAYS** DAILY=-1.4718003426982D-04 :REM ** R E V O L N MRAHU=.5+KALI*DAILY:REM ** REVOLNS* 180 DEG AT EPOCH** REV=INT(MRAHU)+1 :PRINT TAB( 15) "REVLNS SINCE KALI EPOCH: ";REV MRAHU=360*(MRAHU-REV) MRAHU=360+MRAHU PRINT" " PRINT " M E A N RAHU AT UJJAYINI MIDNIGHT: "; L=MRAHU:GOSUB 1550 R E M *** DAILY MOTION: 3' 10.745" =0.05298481616778111 DEG *** DAILY=-1.4718003426982D-04 *360:GOSUB 1480:REM ** DEG
1280
KAALA=(Hl+MI/60)*DAILY/24
1290
PRINT "MOTION FOR ";H1; "HRS ";MI; "MIN:"';
1300
IF KAALA<0 THEN PRINT TAB(39) "-";
1310 1320 1330 1340
K=KAALA;GOSUB 1550 PRINT " " PRINT " M E A N R A H U AT GIVEN TIME AT";PLACES; MRAHU=MRAHU+DESH+KAALA:L=MRAHU:GOSUB 1550
1350 1360
PRINT" VRK=TMOON-TRAVI:IF VRK<0 THEN VRK=VRK+360
"
200
Ancient Indian Astronomy
1370 1380 1390
IF VRK<12 OR VRK>348 THEN PRINT TAB(34); "NEW M O O N DAY":ZZ=2 IF VRK>168 AND VRK<192 THEN PRINT TAB(28);" FULL MOON DAY":ZZ^1 IF VRK>12 A N D VRK<168 THEN PRINT TAB(28); "NOT NEW OR F U L L MOON D A Y "
1400
LOCATE 24,10:INPUT "DO COMPUTATlONS(E / P)";A$
1410
IF Z Z o l A N D ZZ<>2 A N D (A$="E" OR A$="e") THEN PRINT TAB(30) "ECLIPSE NOT POSSIBLE":END IF ZZ=1 A N D (A$="E" OR A$="e") THEN T0=H1+MI/60:CHAIN "SSLEC"„ALL
1420
OR
A$="e")
ECLIPSE
THEN
OR
PLANETS
1430
IF ZZ=2 A N D "SSSEC"..ALL
1440
IF A$="P" OR A$="p" THEN CHAIN "SSPLA"„ALL
1450
IF A$<>"E" A N D A$<>"e" A N D A$<>"P" A N D A$<>"P" THEN END
1460
R E M ***DESHANTARA COR.***
1470
(A$="E"
Y O U WANT
TO=H1+MI/60:CHAIN
DESH=ULAM*DAILY/360
1480
PRINT "DESHANTARA CORRECTION: ";
1490
IF DAILY<0 A N D ULAM>0 THEN PRINT TAB(39) "-";:GOTO 1530
1500
IF DAILY<0 A N D ULAM<0 THEN PRINT TAB(39) "+";:GOTO 1530
1510
IF ULAM>0 THEN PRINT TAB(39) "+";
1520
IF ULAM<0 THEN PRINT TAB(39) "-";
1530
L=DESH:GOSUB 1550
1540
RETURN
1550
IF L<0 THEN L=ABS(L)
1560 1570 1580
DEG=INT(L):MIN=(L-DEG)*60:SEC=INT((MIN-INT(MIN))*60+.5)
1590 1600 1610 1620 1630
IF DEG>=360 THEN DEG=DEG-360 IF SEC=60 THEN SEC=0:MIN=MIN+1 :IF MIN=60 THEN =DEG+1:IF DEG>=360 THEN DEG=DEG-360 PRINT TAB(40);DEG;"°";INT(MIN);"'";SEC;""" RETURN R E M *** EQUATION OF CENTRE *** IF Z=0 THEN PRINT "MANDOCCA:";:L=PMA:GOSUB 1550 PRINT " M A N D A ANOMALY:";
1640
L=PMA1:G0SUB 1550
1650 1660
PRINT " M A N D A EQUATION :"; PMA1=PMA1*PI/180 :REM ** RADIANS **
1670 1680
SN=SIN(PMA1) PEQ = (K1-K2*ABS(SN))*SN : R E M / * PLANET'S EQN. OF CENTRE **
MIN=0:DEG
Computer Programs 1690
IF PMA1
1700
IF PMA1>PI THEN PRINT TAB(39)
1710
L=PEQ :GOSUB 1550
1720
RETURN
201
202
Ancient Indian Astronomy 10
CLSrKEY OFF:REM * PROGRAM 7.2 : " K K R A M O O N "
20 30 40 50 60 70
PRINT TAB(23); "* POSITIONS OF SUN, MOON AND R A H U *" PRINT TAB(23);'. > t * « * 4c * « I k >|t i<< * PI=3.141592653589793#
>l< * >l< III >t< 4<« * sk >•<
Ik * >l<-It * lie >l< s|< l< >l< >l< * * •!< H< >l< >k >l<' >
80
LOCATE 6,18:PRINT "(CHRISTIAN) DATE ;":LOCATE 6,40:LINE INPUT "YEAR:":YE$:LOCATE 6,53:LINE INPUT "MONTH:";Mp$:LOCATE 6,65: LINE INPUT "DATE:";DA$:Y=VAL(YE$):MM=VAL(M0$):6l=VAL(DA$)
90
LOCATE 7,18:PRINT "TIME (IST):";:LOCATE 7,33:LINE INPUT "HOURS:"; HO$:LOCATE 7,47:LINE INPUT "MINS:";MIN$:H1=VAL(H0$):MI=VAL(MIN$)
100
LOCATE 8,18:PRINT " N A M E OF THE PLACE: ";:LOCATE 8,45:LINE INPUT PLACES
110
150
LOCATE 9,18:PRINT "LONGITUDE (- ve for West): ";:LOCATE 9,45:LINE INPUT "DEG:";LD$:LOCATE 9,60:LINE INPUT "MIN:";LM$ LOCATE 10,18:PRINT "LATITUDE (- ve for South): ";:LOCATE 10,45:LINE INPUT "DEG:";PD$:LOCATE 10,60:LINE INPUT "MIN: ";PM$ LD=VAL(LD$):LM=VAL(LM$):PD=VAL(PD$):PM=VAL(PM$)
140
IF LD<0 THEN LAM=LD-LM/60:GOTO
120
150 160
160
LAM=LD+LM/60 IF PD<0 THEN PHI=PD-PM/60:GOTO 180
170
PHI=PD+PM/60
180
R E M *** UJJAYINI:LONG.75.75E, LAT.23.18N ***
190
ULAM=75.75-LAM :REM ** LONG, w.r.t. UJJAYINI ***
200
TC = INT((Y-1900)7100)
210
T = Y-100*INT(Y/100)
220
IF TC<-4 THEN E=13
230
IF TC=-4 A N D Y<1582 THEN E=13
240
IF TC=-4 A N D Y>1582 THEN E=3
250
IF TC>-4 A N D TC<=0 THEN E=-TC
260
IF TC>0 THEN E=-(TC-1)
270
Q=-(TM0D4)
280
DD=0
290
DATA 0,31,28,31,30,31,30,31,31,30,31,30
300
RESTORE 290:FOR 1=1 TO M M
310
READ X
320
DD=DD+X
330
NEXT I
Computer Programs
203
340
IF (Y>1600) A N D (Y/100=INT(Y/100)) A N D (Y/400<>INT(Y/400)) GOTO 360
350
IF Y/4=INT(Y/4) A N D (MM=1 OR MM=2) THEN DD=DD-1
360
JJ=(TC* 100+T)*365.25+DD+DI +E+(Q/4)+2415020!
370
THEN
WD=JJ-7*INT(JJ/7)
380
RESTORE 410
390
FOR F=0 TO WD
400
READ X$
410
DATA MONDAY,TUESDAY,WEDNESDAY,THURSDAY.FRIDAY,SATURDAY, SUNDAY
420
NEXTF
430
PRINT TAB(32);"WEEK DAY: ";X$
440
PRINT "JULIAN DAYS (EIapsed):";JJ;"(GREENWICH NOON)"
450
LET K A L I = JJ-588466!
460
PRINT " K A L I Y U G A DAYS Elapsed(Epoch:Feb 17/18,3102 B.C.):";KALI
470
AHG=JJ-1964031!
480
PRINT " A H A R G A N A (Khandakhaadyaka Epoch: March 23,665 A.D.): ";AHG; "(Preceding Midnight)"
490 500
PRINT "******************** RAVI SPHUTA ************************" MRAVI#=((AHG*800)+438)/292207!
510
MRAVI = 360*(MRAVI#-INT(MRAVI#))
520
R E M ***
DESHANTARA
COR.
**
SUN'S
DAILY
MOTION:59'8"
=0.985555555 DEG *** 530
PRINT *'MEAN RAVI AT UJJAYINI MIDNIGHT: ";
540
L=MRAVI:GOSUB 1640
550
DAILY=.985555555#:GOSUB 1570
560
PRINT " M E A N RAVI AT L O C A L
570
MRAVI=MRAVI+DESH:L=MRAVI:GOSUB 1640
580
PRINT "MOTION FOR ";H1 ; "HRS.";MI; "MIN.:";
M I D N I ' G H T : ";
590
PRINT TAB(34) "+";
600
K A A L A = (H1+MI/60)*DAILY/24:L=KAALA:GOSUB 1640
610
PRINT " M E A N RAVI AT GIVEN TIME: ";
620
MRAVI=MRAVI+KAALA:L=MRAVI:GOSUB 1640
630
SMA=80: R E M ***** SUN'S M A N D O C C A *****************
640
SMK=MRAVI-80:REM *** RAVI'S M A N D A KENDRA ********
650
IF SMK<0 THEN SMK=SMK+360
660
PRINT " M E A N ANOMALY: ";
670
L=SMK:GOSUB 1640
204
Ancient Indian Astronomy
680
R E M *** RAVI'S M A N D A P H A L A (EQN. OF CENTRE):(-SEQ) ***
690 700 710
SEQ = 134*(SIN(SMK*PI/180))/60 :REM ** SUN'S EQN. OF CENTRE ** PRINT "EQN. OF CENTRE : "; IF SMK< 180 THEN PRINT TAB(34) "-";
720 730
IF SMK>180 THEN PRINT TAB(34) "+"; L=SEQ:GOSUB 1640
740 TRAVI=MRAVI-SEQ 750
IF TRAVI<0 THEN TRAVI=TRAVI+360
760 770
IF TRAVI>360 THEN TRAVI=TRAVI-360 PRINT"
780
PRINT "TRUE RAVI: ";
790
L=TRAVI:GOSUB 1640
800
LOCATE 22,55:PRINT "PRESS A N Y K E Y TWICE"
810
"
A$=INPUT$(2)
820 CLS 830 PRINT "***************** CHANDRA SPHUTA *********************" 840
MOON#=((AHG-*600)+417.2)/16393:REM ** REVNS. **
850 860
COR#=-AHG/4929:REM ** MINUTES ** M O O N = 360*(MOON#-INT(MOON#))+COR#/60
870 880
PRINT PRINT " M E A N MOON AT UJJAYINI MIDNIGHT: ";
890 900
L=MOON:GOSUB 1640 R E M *** DAILY M E A N MOTION: 790'31" = 13.175278DEG ***
910 920
DAILY=13.175278#:GOSUB 1570 PRINT " M E A N MOON AT L O C A L MIDNIGHT: ";
930
MOON=MOON+DESH:L=MOON:GOSUB 1640
940 950
PRINT "MOTION F0R";H1; "HRS.";MI; "MIN.:"; K A A L A = (HI+MI/60)*DAILY/24:L=KAALA:GOSUB 1640
960
PRINT " M E A N MOON AT GIVEN TIME: ";TAB(34) "+";
970 980
MOON=MOON+KAALA:L=MOON:GOSUB 1640 MMA=(AHG^53.75)/3232+AHG/(39298!*60*360):REM**MOON'S
990
M M A = 360*(MMA-INT(MMA))
1000
PRINT "MOON'S APOGEE: ";
1010 1020
L=MMA:GOSUB 1640 M M K = MOON-MMA
1030 1040
IF MMK<0 THEN MMK=MMK+360 PRINT "MOON'S ANOMALY: ";
1050
L=MMK:GOSUB 1640
APOGEE **
Computer Programs
205
1060 1070
MEQ=296*SIN(MMK*PI/180)/60 PRINT "EQN. OF CENTRE:";
1080
IF MMK<180 THEN PRINT TAB(34) "-";
1090 1100
IF MMK>180 THEN PRINT TAB(34) "+"; L=MEQ:GOSUB 1640
1110
TMOON=MOON-MEQ
1120
R E M •** BHUJANTARA CORRECTION ***
1130 1140 1150 1160
BHUJ=SEQ/27 PRINT "BHUJANTARA CORRECTION:"; IF BHUJ>0 THEN PRINT TAB(34); "-"; L=BHUJ:GOSUB 1640
1170 1180
TMOON=TMOON-BHUJ IF TMOON<0 THEN TMOON=TMOON+360
1190 1200
IF TMOON>360 THEN TMOON=TMOON-360 PRINT"
1210
PRINT "TRUE MOON :";
1220 1230 1240
L=TMOON:GOSUB 1640 PRINT" PRINT "****************** R A H U SPHUTA
1250 1260
MRAHU=(AHG-372)/6795 + AHG/(514656!*360) MRAHU=360*(MRAHU-INT(MRAHU))
1270 1280
MRAHU=360-MRAHU PRINT " M E A N R A H U AT UJJAYINI MIDNIGHT ";
1290 1300
L=MRAHU:GOSUB 1640 R E M *** DAILY MOTION: 3'10" = 0.052777777 DEG ***
1310 1320 1330
DAILY=-0.0527777777#:GOSUB 1570 PRINT " M E A N R A H U AT L O C A L MIDNIGHT: "; MRAHU=MRAHU+DESH:IF MRAHU>360 THEN MRAHU=MRAHU-360
1340
IF M R A H U < 0 THEN MRAHU=MRAHU+360
1350 1360 1370
L=MRAHU:GOSUB 1640 PRINT "MOTION F0R";H1;"HRS.";MI;"MIN.:";TAB(34)"-"; K A A L A = (HI+MI/60)*DAILY/24:L=KAALA:GOSUB 1640
1380 1390
PRINT " M E A N R A H U AT GIVEN TIME: "; MRAHU=MRAHU+KAALA:IF MRAHU>360 THEN MRAHU=MRAHU-360
1400
IF M R A H U < 0 THEN MRAHU=MRAHU+360
1410
L=MRAHU:GOSUB 1640
1420
R E M *** L A L L A ' S CORRECTION ***
1430
LC=-(Y-499)*96/(250*60)
"
" ************************"
206
Ancient Indian Astronomy
1440 1450
PRINT " L A L L A ' S CORRECTION: ";TAB(34)"-"; L=LC:GOSUB 1640
1460
PRINT " R A H U WITH L A L L A ' S CORRECTION: ";
1470 1480 1490 1500
MRAHU=MRAHU+LC:IF M R A H U > 360 THEN MRAHU=MRAHU-360 IF M R A H U < 0 THEN MRAHU=MRAHU+360 L=MRAHU:GOSUB 1640 LOCATE, 19:INPUT "DO Y O U WANT ECLIPSE / PLANETS' COMPUTATION (E/P)";Y$
1510 1520
IF Y$="P" OR Y$="p" THEN CHAIN " K K P L A " , , A L L IF Y$="E" OR Y$="e" THEN VRK=TMOON-TRAVI:IF VRK<0 THEN VRK=VRK+360
1530
IF VRK<12 OR VRK>348 THEN PRINT TAB(25)"NEWMOON D A Y :TRY SOLAR ECLIPSE":TO=H1+MI/60:CHAIN "KKSEC"„ALL IF VRK>168 A N D VRK<192 THEN PRINT TAB(25);"FULLMOON DAY":CHAIN " K K L E C " , , A L L IF VRK> 12 A N D VRK< 168 THEN PRINT "NOT F U L L OR NEW MOON DAY ":END IF VRK>192 A N D VRK<348 THEN PRINT "NOT F U L L OR NEW MOON DAY ":END R E M *** DESHANTARA CORRECTION ***
1540 1550 1560 1570
1580 DESH=ULAM*DAILY/360 1590
PRINT "DESHANTARA CORRECTION: ";
1600
IF DESH<0 THEN PRINT TAB{34) "-";
1610
IF DESH>0 THEN PRINT TAB(34) "+";
1620
L=DESH:GOSUB 1640
1630
RETURN
1640
L=ABS(L)
1650
DEG=INT(L):MIN=(L-DEG)*60:SEC=INT((MIN-INT(MIN))'*60+.5)
1660 1670
IF DEG>=360 THEN DEG=DEG-360 IF SEC=60 THEN SEC=0:MIN=MIN+1:IF MIN=60 THEN =DEG+1:IF DEG>=360 THEN DEG=DEG-360
1680
PRINT TAB(35);DEG;"° ";TAB(42);INT(MIN);""';TAB(47);SEC;"""
1690
RETURN
MIN=0:DEG
Computer Programs
207
10
20 30 40 50 60 70 80
90
100 110
LOCATE 6,18:PRINT "(CHRISTIAN) DATE : ":LOCATE 6,40:LINE INPUT " Y E A R : ";YE$:LOCATE 6,53:LINE INPUT "MONTH: ";MO$:LOCATE 6, 65:LINE INPUT "DATE: ";DA$:Y=VAL(YE$):MM=VAL(MO$):Dl=VAL(DA$) LOCATE 7,18:PRINT "TIME (AFTER SUNRISE):";:LOCATE 7,40:LINE INPUT "HOURS:";HO$:LOCATE 7,53:LINE INPUT "MINS:";MIN$: Hl=VAL(HO$):MI=VAL(MIN$) LOCATE 8,18:PRINT " N A M E OF THE PLACE: ";:LOCATE 8,45:LINE INPUT PLACES
i 30
LOCATE 9.18:PRINT "LONGITUDE (-ve for West): ";:LOCATE 9,45:LINE INPUT "DEG:";LD$:LOCATE 9,60:LINE INPUT "MIN: ";:LM$ LOCATE 10,18:PRINT "LATITUDE (-ve for South): ";:LOCATE 10,45:LINE INPUT "DEG:";PD$:LOCATE 10,60:LINE INPUT "MIN: ";PM$ LD=VAL(LD$):LM=VAL(LM$):PD=VAL(PD$):PM=VAL(PM$)
140
IF LD<0 THEN LAM=LD-LM/60:GOTO 160
150 160 170
LAM=LD+LM/60 IF PD<0 THEN PHI=PD-PM/60:GOTO 200 PHI=PD+PM/60
180 190
R E M *** UJJAYINI:LONG.75.75E, LAT.23.18N *** ULAM=75.75-LAM :REM ** LONG, w.r.t. UJJAYINI ****
200 210 220
TC = INT((Y-1900)/100) T = Y-100*INT(Y/100) IF TC<-4 THEN E=13
230
IF TC=-4 A N D Y<1582 THEN E=13
240 250
IF TC=-4 A N D Y>1582 THEN E=3 IF TC>-4 A N D TC<=0 THEN E=-TC
260 270
IFTC>0 THEN E=-(TC-1) Q=-(T MOD 4)
280 290 300 310 320
DD=0 DATA 0,31,28,31,30,31,30,31,31,30,31,30 RESTORE 290:FOR 1=1 TO M M READ X DD=DD+X
120
208
Ancient Indian Astronomy
330
NEXT I
340
IF (Y>1600) A N D (Y/100=INT(Y/100)) A N D (Y/400oINT(Y/400)) THEN GOTO 360
350
IF Y/4=INT(Y/4) AND (MM=1 OR MM=2) THEN DD=DD-1
360
JJ=(TC* 100+T)*365.25+DD+D 1 +E+(Q/4)+2415020!
370
WD=JJ-7*INT(JJ/7)
380
RESTORE 410
390
FORF=0TOWD
400
READXS
410
DATA MONDAY,TUESDAY,WEDNESDAY,THURSDAy.FRIDAY,S ATURD AY.
480 490
SUNDAY NEXTF PRINT TAB(32); "WEEK DAY: ";X$ LET K A L I = JJ-588466! AHG1=KALI-1687850! CHAKRA=INT(AHG1/4016):AHG = AHGI - (4016*CHAKRA) PRINT TAB(18) " C H A K R A S : " ; C H A K R A ; " A H A R G A N A : " A H G ; "[EPOCH: 19-3-1520(1)]" PRINT :PRINT PRINT "******************** RAVI SPHUTA ***********************"
500
R E M **** D H R U V A K A : 1-49-11 DEG * * KSHEPAKA: 349-41 DEG ****
510
MRAVI = AHG-AHG/70-AHG/(150*60)-CHAKRA
520
PRINT " M E A N RAVI AT UJJAYINI SUNRISE: ";
530
L=MRAVI:GOSUB
420 430 440 450 460 470
* 1.8197222# +349.6833333333#
1780
540
MRAVI=L
550
REM
560 570 580 590 600 610 620 630 640 650 660
59'8"=0.985555555 DEG *** DAILY=.985555555#:GOSUB 1710 PRINT " M E A N RAVI AT L O C A L SUNRISE: "; MRAVI=MRAyi+DESH:L=MRAVI:GOSUB 1780 PRINT "MOTION FOR ";H1; "HRS.";MI; "MIN.:"; KAALA=(Hl+MI/60)*DAILY/24 :IF KAALA<0 THEN PRINT TAB(34); "-"; L=KAALA:GOSUB 1800 PRINT " M E A N RAVI AT GIVEN TIME: "; MRAVI=MRAVI+KAALA:L=MRAVI:GOSUB 1780 R E M ***** RAVI'S M A N D A P H A L A (EQN OF CENTRE) ******* S M A = 78 : R E M ** RAVI'S M A N D O C C A ** S M K = S M A - MRAVI: R E M ** RAVI'S M A N D A K E N D R A (ANOMALY)**
***
DESHANTARA
COR.
**
SUN'S
DAILY
MOTION:
Computer Programs
209
670 IF S M K < 0 THEN S M K = S M K + 360 680 IF S M K ^ 90 THEN BHUJA = S M K 690 IF S M K > 90 A N D S M K < 180 THEN BHUJA = 180 - S M K 700 IF S M K > 180 A N D S M K < 270 THEN BHUJA = S M K - 180 710 IF S M K > 270 A N D S M K < 360 THEN BHUJA = 360 - S M K 720 X X = BHUJA / 9 730 N U M = X X * (20 - X X ) 740 D E N = 57 - (1/9)*(XX*(20 - XX)) 750 SMP = N U M / D E N 760 PRINT " M E A N ANOMALY: "; 770 L=SMK:GOSUB 1780 780 PRINT " M A N D A P H A L A : "; 790 IF SMK<180 THEN PRINT TAB(34)"+"; 800 IF SMK>180 THEN SMP = - SMP:PRINT TAB(34)"-"; 810 L=SMP:GOSUB 1800 820 TRAVI=MRAVI+SMP 830 IF TRAVI<0 THEN TRAVI=TRAVI+360 840 IF TRAVI>360 THEN TRAVI=TRAVI-360 850 P R I N T "
"
860 PRINT TRUE RAVI: "; 870 L=TRAVI:GOSUB 1800 880 P R I N T "
"
890 LOCATE 23,55:PRINT "PRESS A N Y K E Y TWICE" 900
A$=INPUT$(2)
910 CLS:PRINT "****************** CHANDRA SPHUTA *****************" 920 M00N=AHG*14 - 14*AHG/17 - (AHG/140)/60 - (CHAKRA * 3.769722#) + 349.1 930 R E M ** DHRUVAKA:3-46-ll : KSHEPAKA: 11-19-6 ** 940 PRINT 950 PRINT " M E A N MOON AT UJJAYINI SUNRISE: "; 960 L=MOON:GOSUB 1780 970 R E M *** DAILY M E A N MOTION: 790' 35" = 13.17638888888883 DEG*** 980 DAILY = 13.17638888888883#:GOSUB 1710 990 PRINT " M E A N MOON AT L O C A L SUNRISE: '; 1000 MOON=MOON+DESH:L=MOON:GOSUB 1780 1010 PRINT "MOTION FOR ";H1;"HRS.";MI;"MIN.:"; 1020 K A A L A = (H1+MI/60)*DAILY/24:IF KAALA<0 THEN PRINT TAB(34); "-"; 1030 L=KAALA:GOSUB 1800
210
Ancient Indian Astrononuj
1040
PRINT " M E A N M O O N AT GIVEN TIME:";
1050
MOON=MOON+KAALA:L=MOON:GOSUB
1060
MOON=L
1070
R E M **MOON'S M A N D O C C A **
1080
MMA=AHG/9 -(AHG/70)/60 - (272.75*CHAKRA) + 167.55
1090
R E M ** DHRUVAKA:9-2-45 * *KSHEPAKA:5-17-33 **
1100
PRINT "MOON'S MANDOCCA:";:L = MMA:GOSUB 1780
1110
MMA=L
1120
MMK=
1130
IF MMK<0 THEN M M K = M M K + 360
1140
PRINT "MOON'S M A N D A K E N D R A (ANOMALY:)"; '
1780
MMA-MOON
1150
L=MMK:GOSUB 1780
1160
! F M M K < 90 THEN BHUJA = M M K
1170
IF M M K > 90 A N D M M K S 180 THEN BHUJA = 180 - M M K
1180
IF M M K > 180 A N D M M K <. 270 THEN BHUJA = M M K - 180
1190
IF M M K > 270 A N D M M K < 360 THEN BHUJA = 360 - M M K
1200
YY=BHUJA/6
1210
N U M = (30-YY)*YY
1220
D E N = 56 - (1/20) • N U M
1230
MMP = N U M / D E N
1240
PRINT " M A N D A P H A L A (EQN OF CENTRE):";
1250
IF M M K > 180 THEN M M P = - M M P : PRINT TAB(34) "-";
12flO IF M M K < 180 THEN PRINT TAB(34) "+"; 1270
U M M P : G O S U B 1800
12S0
TMOON = M O O N + M M P
1290
REM
1300
BHUJ=SMP/27
1310
PRINT "BHUJANTARA CORRECTION: ";
1320
IF BHUJ<0 THEN PRINT TAB(34); "-";
1330
L=BHUJ:GOSUB 1800
1340
BHUJANTARA CORRECTION ***
TMOON-TMOON+BHUJ
1350
IF TMOON<0 THEN TMOON=TMOON+360
1360
IF TMOON>360 THEN TMOON=TMOON-360
1370
PRINT"
1380
PRINT "TRUE M O O N :";
1390
L=TMOON:GOSUB 1780
1400
TMOON=L
1410
PRINT"
"
- "
Computer Programs
211
1420
PRINT
1430
PRINT "**•**•***•**•**•*•* RAHU.SPHUTA ************•*******"
1440
M R A H U = 360 -((AHG/19 + (AHG/45)/60))-(212.83333#*CHAKRA) + 27.633333#
1450
PRINT " M E A N R A H U AT UJJAYINI SUNRISE: ";
1460
L=MRAHU:GOSUB 1780
1470
MRAHU = L
1480
R E M *** DAILY MOTION: 3'10" = 0.0527777777 DEG ***
1490
DAILY = -.0527777777#:GOSUB 1710
1500
PRINT " M E A N R A H U AT L O C A L SUNRISE: ";
1510
MRAHU=MRAHU+DESH:IF M R A H U > 360 THEN MRAHU=MRAHU-360
1520
IF M R A H U < 0 THEN MRAHU=MRAHU+360
1530
L=MRAHU:GOSUB 1780
1540
K A A L A = (Hl+MI/60)*DAILY/24
1550
PRINT "MOTION F0R";H1;"HRS.";MI;"MIN.:";
1560
IF KAALA<0 THEN PRINT TAB(34) "-";
1570
L = K A A L A . G O S U B 1800
1580
MRAHU=MRAHU+KAALA:IF M R A H U > 360 THEN MRAHU=MRAHU-360
1590
PRINT"
1600
PRINT " M E A N R A H U AT GIVEN TIME: ";
1610
L=MRAHU:GOSUB 1780
1620
MRAHU=L
1630 1640
PRINT "_, ; " LOCAHe, 15:INPUT "DO Y O U WANT ECUPSE / PLANETS' COMPUTATION? (E/P)";Y$
1650
IF Y$="E" OR Y$="e" THEN T0=6+Hl+MI/60 : VRK=TMOON-TRAVI:IF VRK<0 THEN VRK=VRK+360 IF Y$="P" OR Y$="p" THEN CHAIN "GLPLA"„ALL IF VRK>168' A N D VRK<192 THEN PRINT TAB(35)"FULLMOON DAY":CHAIN "GLLEC"„ALL IF VRK<12 OR VRK>348 THEN PRINT TAB(35)"NEWMOON DAY ":CHAIN "GLSEC",. A L L IF VRK>I2 A N D VRK<168 THEN PRINT TAB(30)"NOT N E W / F U L L MOON DAY":END IF VRK>192 A N D VRK<348 THEN PRINT TAB(30)"NOT NEW/FULL MOON DAY":END R E M *** DESHANTARA CORRECTION ***
1660 1670 1680 1690 1700 1710 1720
DESH=ULAM'*DAILY/360
1730
PRINT "DESHANTARA CORRECTION: ";
1740
IF DESH<0 THEN PRINT TAB(34) "-";
"
212
Ancient Indian Astronomy
1750
IF DESH>0 TOEN PRINT TAB(34) "+";
1760
L=DESH:GOSUB 1800
1770
RETURN
1780 1790
IF L>360 THEN L=L-360*INT(L/360) IF L<0 THEN L=L+360*ABS(INT(L/360))
1800 1810
IF L<0 THEN L=ABS(L) DEG=INT(L):MIN=(L-DEG)*60:SEC=INT((MIN-INT(MIN))*60+.5)
1820 1830
IF DEG>=360 THEN DEG=DEG-360 IF SEC=60 THEN SEC=0:MIN=MIN+1:IF MIN=60 THEN MIN=0:DEG =DEG+1 :IF DEG>=360 THEN DEG=DEG-360
1840
PRINT TAB(35);DEG;"' ";TAB(42)INT(MIN);""';TAB(47)SEC;"""
1850
RETURN
1860
IF M<0 THEN M=ABS(M)
1870
MIN=INT(M):SEC=INT((M-NDN)*60+.5)
1880
IF SEC=60 THEN SEC=0:MIN=MIN+1
1890
PRINT TAB(35);MIN;"' ";SEC;"""
1900
RETURN
Computer Programs
213
10 C L S : K E Y OFF : R E M * PROGRAM 9.1 : "SSLEC" * 20 PRINT TAB(18)"***************************************************" 30 40
PRINT TAB(18) "* LUNAR ECUPSE ACCORDING TO SURYA SIDDHANTA *" PRINT TAB(18)"***************************************************"
50
PI=3.41592653589793#
60
R E M * TRUE RAVI,MOON,RAHU AT ISHTAKAALA TO IN GHATIS *
80
PRINT " A T ' ;GH ;"GH";VIG;"VIG. F R O M MIDNIGHT 0N";D1;"/";MM;"/";Y
90 100 110
PRINT "TRUE RAVI:";TRAVI;" TRUE MOON";TMOON;"RAHU:";MRAHU R E M * SMK:SUN'S ANOMALY; M M K : M O O N ' S A N O M A L Y * R E M * SUN'S M E A N DAILY MOTION: 59'8.17" ; MOON'S 790' 34.866" *
120
SDM=59.136*(1-(14/360)'*COS(PI*SMK/180)):REM MOTION *
130
230 240 250
MDM=790.581#-(31/360)*783.898*COS(PI*MMK/180):REM'*MOON'S TRUE DAILY MOTION * NDM=191/60 : R E M * R A H U ' S M E A N DAILY MOTION (MIN) * R E M * 783.898 = M E A N DAILY MOTION OF (MOON - MOON'S APOGEE) * SOPP=(TRAVI-TMOON)+180 IF SOPP>360 THEN SOPP=SOPP-360 IF SOPP<0 THEN SOPP=SOPP+360 PRINT "MOON'S DISTANCE FROM OPPN:";:L=SOPP:GOSUB 990 OPPT=SOPP*60/((MDM-SDM)/60) PRINT "INSTANT OF OPPN. AFTER MIDNIGHT: ";:H=(T0+OPPT)*2/5: GOSUB 1040 PRINT TAB(23) "•*** AT THE INSTANT OF OPPOSITION ***" OPSUN=TRAVI+OPPT*SDM/(60*60) PRINT "TRUE RAVI AT OPPN.:";:L=OPSUN:GOSUB 990
260 270
OPMOON=TMOON+OPPT*MDM/(60*60) PRINT "TRUE CHANDRA AT OPPN.:";:L=OPMOON:GOSUB 990
140 150 160 170 180 190 200 220
280
* SUN'S TRUE DAILY
OPNODE=MRAHU-190.745*OPPT/(3600*60)
290
PRINT " R A H U AT OPPN.:";:L=OPNODE:GOSUB 990
300 310
SCDIA=SDM*6500/59.136 . R E M * SUN'S COR.DIAMETER (YOJANAS) * SDIAY=SCDIA*4320000!/57753336# : R E M * SUN'S DIAMETER IN YOJANAS * SDIA=SDIAY/15 : R E M * SUN'S DIAMETER IN A R C (MIN)* PRINT "SUN'S A N G U L A R DIAMETER:";:M=SDIA:GOSUB 1090 MDIAY=MDM*480/790.581 : R E M * MOON'S DIAMETER IN YOJANAS * MDIA=MDIAY/15 : R E M * MOON'S D L \ M E T E R IN A R C (MIN) * PRINT "MOON'S A N G U L A R DIAMETER:";:M=MDIA:GOSUB 1090
320 330 340 350 360
214
Ancient Indian Astronomy
370
E C D I A = M D M * 1600/790.581 : R E M * E A R T H ' S C O R - D I A M E T E R ( Y O J A N A S ) *
380
S C D E D I A = S C D I A - 1600:REM * S U N ' S C O R . D I A - E A R T H ' S M E A N D I A *
390
SHDIAY=ECDIA-SCDEDIA*480/6500:REM*SHADOW'S D I A M E T E R (YOJANAS)*
400
S H D I A = SHDIAY/15 : R E M* S H A D O W D I A M E T E R I N A R C (MIN)*
410
PRINT "SHADOW'S A N G U L A R
420 430
DIAMETER:";:M=SHDIA:GOSUB1090
MLAT=270*SIN(PI*(OPMOON-OPNODE)/180):MLATI=MLAT PRINT
"MOON'S
LATITUDE
A T OPPN.:";:IF
MLAT<0
THEN
PRINT
TAB(34): "-"; 440 450 460
M = M L A T : ( J O S U B 1090 GRASA=.5*(MDIA+SHDIA)-MLAT PRINT
"MOON'S
OBSCURED PORTION
(GRASA):";:M=GRASA:GOSUB
1090 470
IF G R A S A > 0 A N D G R A S A < M D I A T H E N P R I N T TAB(29) " L U N A R E C L I P S E IS P A R T I A L " : Z Z = 1
480
IF
GRASA<0
THEN
PRINT
TAB(29)
"LUNAR
ECLIPSE
NOT
POSSIBLE":END 490
I F G R A S A > = M D I A T H E N P R I N T TAB(29) " L U N A R E C L I P S E IS T O T A L "
500
P R I N T T A B ( 6 0 ) " P R E S S A N Y K E Y " : A $ = I N P U T $ ( 1)
510
CLS:REM * HALF-DURATIONS OF ECLIPSE A N DTOTALITY *
520
HDUR=60*SQR(((SHDIA+MDIA)/2)^2 -MLAT'^2)/(MDM-SDM)
530
PRINT " H A L F - D U R A T I O N O F T H EECLIPSE:";TAB(35);HDUR; " N A A D I S "
540
I F Z Z = 1 T H E N G O T O 570
550
HDURT=60*SQR(((SHDIA-MDIA)/2)'^2 -MLAT'^2)/(MDM-SDM)
560
PRINT
"HALF-DURATION
OF
THE
T O T A L I T Y : ";TAB(35);
HDURT;
"NAADIS" 570 580
R E M * ATT H EE N D OF T H E ECLIPSE * TMOON=OPMOON+HDUR*MDM/3600: MRAHU=OPNODE-HDUR*NDM/3600 :REM *DEG*
590
MLAT=270*SIN(PI*(TMOON-MRAHU)/180):IF TAB(34) "-";
600
P R I N T " M O O N ' S L A T I T U D E A T T H E E N D : " ; : M = M L A T : G O S U B 1090
610
HDUR2=60*SQR(((SHDIA+MDIA)/2)^2 -MLAT'^2)/(MDM-SDM)
620
PRINT
"COR.SECOND
HALF-DURN.
OF
MLAT<0
THEN
PRINT
ECLIPSE:";TAB(35);HDUR2;
"NAADIS" 630
R E M * AT T H E BEGINNING O F T H E ECLIPSE *
640
TMOON=OPMOON-HDUR*MDM/3600
: MRAHU=OPNODE+HDUR*NDM/
3600 : R E M * D E G * 650
MLAT=270*SIN(PI*(TMOON-MRAHU)/180):IF TAB(34) "-";
MLAT<0
THEN
PRINT
Computer Programs 660 670 680
PRINT "MOON'S L A T AT BEGIN OF ECLIPSE:";:M=MLAT:GOSUB 1090 HDURl=60*SQR(((SHDIA+MDIA)/2)^2 -MLAT'^2)/(MDM-SDM) PRINT "COR. H R S T HALF-DURN. OF ECLIPSE:";TAB(35);HDUR1; "NAADIS"
690
IF ZZ=1 THEN GOTO 820
700
R E M * AT THE END OF THE TOTALITY *
710
TMOON»OPMOON+HDURT'*MDM/3600: MRAHU=OPNODE-HDURT*NDM/3600 :REM *DEG* MLAT=270*SIN(PI*(TMOON-MRAHU)/180):IF MLAT<0 TAB(34) "-";
720
THEN
PRINT
730 740 750 760
PRINT "MOON'S L A X AT END OF TOTALITY:";:M=MLAT:GOSUB 1090 HDURT2s60*SQR({{SHDIA-MDIA)/2)^2 -MLAT'^2)/(MDM-SDM) PRINT "COR.SECOND HALF-DURN.OF TOTALm':";TAB(35)J©UR2; "NAADIS" R E M * AT THE BEGINNING OF THE TOTALITY *
770
TMOON=OPMOON-HDURT*MDM/3600 : MRAHU=OPNODE+HDURT* NDM/3600 : R E M *DEG* MLAT=270*SIN(PI*(TMOON-MRAHU)/180):IF MLAT<0 T H E N PRINT TAB(34) "-"; PRINT "MOON'S L A T AT BEGIN OF TOTALITY:";:M=MLAT:GOSUB 1090
780 790 800 810 820 830 840
HDURTl=60*SQR(((SHDIA-MDIA)/2r2-MLAT^2)/(MDM-SDM) PRINT "COR.FIRST HALF-DURN.OF T0TALITY:";TAB(35);HDURT1; "NAADIS"
850
BEG=(T0+OPPT-HDURl)*2/5 :REM * BEGINNING OF ECLIPSE IN HRS *
860 870 880 890 900 910 920 930 940 950 960 970 980 990
PRINT "BEGINNING OF THE ECLIPSE:";:H=BEG:GOSUB 1040 IF ZZ=1 T H E N GOTO 900 BEGT=(T04OPPT-HDURTl)*275 :REM * BEGINNING OF TOTALITY IN HRS * PRINT "BEGINNING OF THE TOTALITY:";:H=BEGT:GOSUB 1040 MID=(T0+OPPT)*2/5 :REM * MIDDLE OF THE ECLIPSE IN HRS * PRINT "MIDDLE OF THE ECLIPSE:";:H=MID:GOSUB 1040 IF ZZ=1 T H E N GOTO 950 ENDT=(TO+OPPT+HDURT2)*2/5 :REM * END OF TOTALITY IN HRS * PRINT "END OF THE TOTALITY:";:H=ENDT:GOSUB 1040 ENDE=(T0+OPPT+HDUR2)*2/5 :REM * END OF ECLIPSE IN HRS * PRINT "END OF THE ECLIPSE:";:H=ENDE:GOSUB 1040 END R E M * CONVERSION INTO DEG.MIN.SEC OF A R C * IF L<0 THEN L=ABS(L)
216
Ancient Indian Astronomy
1000
DEG=INT(L):MINT=(L-DEG)*60:MIN=INT(MINT):SEC=INT((MINT-MIN)* 60+.5)
1010
IF SEC=60 THEN SEC=0:MIN=MIN+1:IF MIN=60 THEN MIN=0: DEG=DEG+1 PRINT TAB(35);DEG;""';MIN;""';SEC;""" RETURN R E M * CONVERSION TO HRS,MIN,SEC OF TIME * HRS=INT(H):MNT=(H-HRS)*60:MIN=INT(MNT):SEC=INT((MNT-MIN)*60+.5) IF SEC=60 THEN SEC=0:MIN=MIN+1:IF MIN=60 THEN MIN=0: HRS=HRS+1 PRINT TAB(35)HRS;"H-";MIN;"M-";SEC;"S" RETURN
1020 1030 1040 1050 1060 1070 1080 1090 1100 1110 1120
R E M * CONVERSION TO M I N A N D SEC OF A R C * MIN=INT(M):SEC=INT((M-MIN)*60+.5):IF SEC=60 THEN MIN+1 PRINT T A B ( 3 5 ) ; M I N ; " ' " ; S E C : ' RETURN
SEC=0:MIN=
Computer Programs
217
10 CLS: R E M * PROGRAM 9.2 : " K K L E C " 20
PRINT TAB(23);"*********************************************"
30
PRINT TAB(23);"*
L U N A R ECLIPSE
*"
40
PRINT TAB(23):"*
ACCORDING TO
*"
50
PRINT TAB(23)"'*
KHANDA-KHAADYAKA
*"
60
PRINT TAB(23);"**********************************************"
70
R E M ** True Sun (TRAVI), True Moon (TMOON), Node (MRAHU)
80
PRINT "AT MIDNIGHT ** TRUE SUN: ";TRAVI;" TRUE MOON: ";TMOON:
at midnight **
"NODE: " ; M R A H U 90 100
R E M ** S M K :Sun's anomaly; M M K :Moon's anomaly ** SDM=59.1333*(1-14*COS(SMK*PI/180)/360):REM * SUN'S TRUE DAILY MOTION *
110
PRINT "SUN'S TRUE DAILY MOTION: ";:M=SDM:GOSUB 480
120
MDM=790.5016-31*783.9*COS(MMK*Pyi80)/360:REM DAILY MOTION *
* MOON'S TRUE
130
R E M **** 783.9 = M E A N DAILY MOTION OF (MOON - MOON'S APOGEE)
140
PRINT "MOON'S TRUE DAILY MOTION: ";:M=MDM:GOSUB 480
150
SOPP= 180+(TRAVI-TMOON):IF SOPP<0 THEN SOPP=SOPP+360
160
IF SOPP>360 THEN SOPP=SOPP-360
170
PRINT "MOON'S DISTANCE FROM OPPN.: ";:L=SOPP:GOSUB 420
180
OPPT=SOPP'^24/(MDM-SDM)/60)
190
PRINT "INSTANT OF OPPN. AFTER MIDNIGHT: ";:H=OPPT:GOSUB 570
200
R E M ** AT THE OPPOSITION **
210 220 230 240 250 260 270
OPSUN=TRAVI+OPPT*SDM/(60*24) PRINT "TRUE SUN AT OPPN.:";:L=OPSUN:GOSUB 420 OPMOON=TMOON+OPPT*MDM/(60"»24) PRINT "TRUE MOON AT OPPN.:";:L=OPMOON:GOSUB 420 OPNODE=MRAHU-190.7*OPPT/(3600*24) PRINT "NODE AT OPPN.:";:L=OPNODE:GOSUB 420 MLAT=270*SIN(PI*(OPMOON-OPNODE)/180)
280
PRINT "MOON'S LATITUDE AT OPPN.:";:IF MLAT<0 THEN PRINT TAB(33)"-";
290
M=MLAT:GOSUB 530
300
MDIA=10*MDM/247:REM * MOON'S A N G U L A R DIAMETER *
310
PRINT "MOON'S A N G U L A R DIAMETER: ";:M=MDIA:GOSUB 480
320
SHDIA=(8*MDM-25"*SDM)/60:REM * SHADOW'S A N G . DIAMETER *
330
PRINT "SHADOW'S A N G U L A R DIAMETER: ";:M=SHDIA:GOSUB 480
340
0BS=.5*(MDIA+SHDIA)-ABS (MLAT)
350
PRINT "MOON'S OBSCURED PORTION (GRAASA):";:M=OBS:GOSUB 480
218
Ancient Indian Astronomy
360
IF OBS<0 THEN PRINT:PRINT TAB(20) "* ECLIPSE NOT POSSIBLE *"
370
IF OBS>MDIA THEN PRINT:PRINT TAB(20) "*LUNAR ECUPSE IS TOTAL*"
380
IF OBS":A$=INPUT$( 1) GOSUB 610 END L=ABS(L)
390 400 410 420 430 440 450 460 470 480 490 500 510 520
DEG=INT(L):MIN=(L-DEG)*60:SEC=INT((MIN-INT(MIN))*60+.5) IF SEC=60 THEN SEC=0:MIN=MIN+1:IF INT(MIN)=60 INT(MIN)=0:DEG=DEG+1 IF DEG>=0 THEN DEG=DEG-360 PRINT TAB(35);DEG;TAB(42);INT(MIN); " ' ";TAB(47);SEC; RETURN M=ABS(M) MIN=INT(M):SEC=INT(M-MIN)*60+.5) IF SEC=60 THEN SEC=0:MIN=MIN+1 PRINT TAB(35);MIN; " ' ";SEC; " " " RETURN
530 540
PRINT FOR N=l TO 10
550 560
HDUR(N)=24*SQR(.25*(SHDIA+MDIAr2-MLAT'^2)/(MDM-SDM) MOON 1 =OPMOON-HDUR(N)*MDM/(24*60)
570
R E M * CONVERSION INTO HRS,MIN,SEC *
580
THEN
HRS=INT(H):M=60*(H-HRS):MIN=INT(M):SEC=INT((M-MIN)*60+.5)
590 600
PRINT TAB(35);HRS; "H-";MIN; "M-";SEC; "S" RETURN
610 620
DBET=SIN(4.5*PI/180)*COS((OPMOON-OPNODE)*PI/180)*829.91833# A=(MDM-SDMr2+DBET'^2
630 B=2*MLAT*DBET 640 DEL 1 =(SHDIA+MDIA)/2 :DEL2=(SHDIA-MDIA)/2 650
C1=MLAT^2-DEL1^2
660
C2=MLAT^2-DEL2^2
670 680
DISCR1=B'^2-4*A*C1 DISCR2=B'^2-4*A*C2
690 700
PRINT R E M *** HRST-HALF DURATION OF THE ECLIPSE ***
710
T1=24*ABS((-B-SQR(DISCR1))/(2*A))
Computer Programs
219
720 730 740 750
PRINT "HRST-HALF DURATION OF ECLIPSE: ";:H=Tl:GOSUB 570 R E M *** SECOND-HALF DURATION.OF THE ECLIPSE *** T4=24*(-B+SQR(DISCR1))/(2*A) PRINT
760
PRINT "SECOND-HALF DURATION OF ECLIPSE: ";:H=T4:G0SUB 570
770 780
IF OBS
790 800
T2=24*ABS((-B-SQR(DISCR2))/(2*A)) PRINT
810 820
PRINT "FIRST-HALF DURATION OF TOTALITY: ";:H=T2:GOSUB 570 R E M *** SECOND-HALF DURATION OF MAX.OBSCURITY ***
830 840
T3=24*(-B+SQR(DISCR2))/(2*A) PRINT
850 860 870
PRINT "SECOND-HALF OF M A X . 0BSCURITY";:H=T3:G0SUB 570 PRINT PRINT TAB(5) "******************************************"
880
PRINT TAB(5) "**
890
PRINT TAB(5)
S U M M A R Y OF THE ECLIPSE
**"
"******************************************"
900
PRINT
910
PRINT TAB(35) " L O C A L M E A N TIME "
920
PRINT TAB(5) "BEGINNING OF THE ECLIPSE: ";:H=0PPT-T1 :GOSUB 570
930 940
IF OBS
950
PRINT TAB(5) "CENTRE OF THE ECLIPSE: ";:H=OPPT:GOSUB 570
960 970 980 990
IF OBS
1000
RETURN
220
Ancient Indian Astronomy
10 20 30
PRINT TAB(23);"*
L U N A R ECLIPSE
40
PRINT TAB(23):"*
ACCORDING TO
50
PRINT TAB(23)' "*
GRAHA LAGHAVAM
60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340
PRINT R E M * True Sun (TRAVI), True Moon (TMOON), Node (MRAHU) at TO HRS.FROM MIDNIGHT * PRINT "AT";TO; "HRS*** TRUE SUN: 'TTRAVI;" TRUE MOON: ";TMOON:" NODE: " ; M R A H U PRINT R E M ** S M K :Sun's anomaly; M M K :Moon's anomaly ** SDM=59.1333*(1-14*COS(SMK*PI/180)/360):REM * SUN'S TRUE DAILY MOTION * PRINT "SUN'S TRUE DAILY MOTION: ";:M=SDM:GOSUB 1070 MDM=790.5666-31*783.9*COS(MMK*PI/180)/360:REM * MOON'S TRUE DAILY MOTION * R E M **** 783.9 = M E A N DAILY MOTION OF (MOON - MOON'S APOGEE) PRINT "MOON'S TRUE DAILY MOTION: ";:M=MDM:GOSUB 1070 SOPP=180+(TRAVI-TMOON):IF SOPP<0 THEN SOPP=SOPP+360 IF SOPP>360 THEN SOPP=SOPP-360 PRINT "MOON'S DISTANCE FROM OPPN.: ";:L=SOPP:GOSUB 1010 OPPT=SOPP*24/((MDM-SDM)/60) OPPTG=OPPT*5/2 PRINT "TIME OF OPPN. AFTER MIDNIGHT(LMT): ";:H=T0+OPPT:GOSUB 970 PRINT R E M ** AT THE OPPOSITION ** OPSUN=TRAVI+OPPT*SDM/(60*24) PRINT "TRUE SUN AT OPPN.:";:L=OPSUN:GOSUB 1010 OPMOON=TMOON+OPPT*MDM/(60*24) PRINT "TRUE MOON AT OPPN.:";:L=OPMOON:GOSUB 1010 OPNODE=MRAHU-190.7*OPPT/(3600*24) PRINT "NODE AT OPPN.:";:L=OPNODE:GOSUB 1010 MLAT=270*SIN(PI*(OPMOON-OPNODE)/180) MLAT1=MLAT PRINT "MOON'S LATITUDE AT OPPN.:";:IF MLAT<0 THEN PRINT TAB(33)"-"; M=MLAT:GOSUB 1070
Computer Programs
221
350
PRINT
360 370
MDIA = M D M / 74 : R E M * MOON'S A N G . DIAMETER * PRINT "MOON'S DIAMETER (in Angulas): ";TAB(35);MDIA
380 390
SHDIAKS-^MDIA/l 1) + (3*MDIA) - 8 : REM * SHADOW'S ANG. DIAMETER * PRINT "SHADOW'S DIAMETER (in Angulas): ";TAB(35);SHDIA
400 410 420
R E M ** POSSIBILITY OF L U N A R ECLIPSE ** V R A = OPSUN - OPNODE :REM * V I R A H V A R K A IF V R A < 0 THEN V R A = V R A + 360
430 440
R E M ** V Y A G U = BHUJA OF (RAVI - RAHU) ** IF V R A < 90 THEN V Y A G U = V R A
450
IF V R A > 90 A N D V R A < 180 THEN V Y A G U = 180 - V R A
460 470
IF V R A > 180 AND V R A < 270 THEN V Y A G U = V R A - 180 IF V R A > 270 A N D V R A < 360 THEN V Y A G U = 360 - V R A
480 490
IF V Y A G U < 14 THEN PRINT TAB(30) "ECLIPE IS POSSIBLE" IF V Y A G U > 14 THEN PRINT TAB(30) "ECLIPE IS NOT POSSIBLE":END
500 510
SHARA = 11 ^ V Y A G U / 7 IF V R A > 180 THEN PRINT "(SOUTHERN) ";
520 530
IF V R A < 180 THEN PRINT "(NORTHERN) "; PRINT "SHARA (In Angulas):";TAB(35)SHARA
540 550
M K D = .5 * (SHDIA + MDIA) : R E M * M A N A I K Y A K H A N D A R D H A * GRASA = M K D - SHARA
560 570
PRINT "GRASA (in Angulas) : ";TAB(35) GRASA IF GRASA < 0 THEN PRINT TAB(27) "THERE WILL B E NO ECLIPSE" :END
580 590
IF GRASA < MDIA THEN PRINT TAB(20) "* LUNAR ECLIPSE IS PARTIAL *" IF GRASA > MDIA THEN PRINT:PRINT TAB(20) "* L U N A R ECLIPSE IS TOTAL *" LOCATE 23,60:PRINT "":A$=INPUT$(1) SQRT= SQR(GRASA * (MKD t SHARA) * 10) MSTH = (SQRT - SQRT/6) / MDIA :REM * M A D H Y A STHITI (Gh.)* PRINT " M A D H Y A STHITI (in Ghatis):";TAB(35) M S T H PALAS = 2 * V Y A G U IF V R A > 90 A N D V R A < 180 THEN SPSTH = MSTH-PALAS/60 :MKSTH = MSTH + PALAS/60
600 610 620 630 640 650 660 670 680
IF V R A > 270 A N D VAR < 360 THEN SPSTH = MSTH-PALAS/60 :MKSTH = MSTH + PALAS/60 IF V R A < 90 THEN SPSTH = MSTH+PALAS/60 :MKSTH = MSTH - PALAS/60 IF V R A > 180 A N D V R A ^ 270 THEN SPSTH = MSTH+PALAS/60 :MKSTH = MSTH - PALAS/60
222
Ancient Indian Astronomy
690 700 710 720
PRINT "SPARSHA STHITI (in Gh.):";TAB(35)SPSTH PRINT "MOKSHA STHITI (in Gh.):";TAB(35)MKSTH IF GRASA < MDIA THEN GOTO 840 K H G R A S A = GRASA - MDIA
730 740 750
SQRT = SQR(KHGRASA * ((.5*(SHDIA-MDIA)+SHARA) * 10)) PRINT " K H A G R A S A (in Angulas):";TAB(35) K H G R A S A M A R D A = (SQRT - SQRT/6) / MDIA :REM * H A L F DURN. OF TOTALITY *
760
PRINT " M A R D A (in Ghatis):";TAB(35) M A R D A
770
IF V R A > 90 A N D VRA<180 THEN SMAR = MARDA-PALAS/60 :UMAR = MARDA+PALAS/60
780
IF VRA>270 A N D VRA<360 THEN SMAR = MARDA-PALAS/60 :UMAR = MARDA+PALAS/60
790
IF V R A < 90 THEN MARDA-PALAS/60
800 810 820 830 840 850 860
IF VRA>180 A N D VRA<270 THEN SMAR = MARDA+PALAS/60 :UMAR = MARDA-PALAS/60 SAMMIT = OPPTG - SMAR : UNMIT = OPPTG + U M A R PRINT " S A M M I L A N A M A R D A (in Gh.):";TAB(35)SMAR PRINT " U N M I L A N A M A R D A (in Gh.):";TAB(35)UMAR SPART = OPPTG - SPSTH : MOKST = OPPTG + M K S T H PRINT PRINT "***********************************************"
870 880
PRINT"*** S U M M A R Y OF THE L U N A R ECLIPSE ; ***" PRINT "***********************************************"
890 900 910 920 930 940 950 960 970 980 990 1000
SMAR
=
MARDA+PALAS/60
.UMAR
=
PRINT TAB(3) "AFTER MIDNIGHT PRECEDING ";D 1;"/" ;MM;"/";Y;TAB(50) " L O C A L M E A N TIME" PRINT "SPARSHA (BEGINNING) TIME:";:H=TO+2*SPART/5:GOSUB 970 IF GRASA > MDIA THEN PRINT "SAMMILANA(START OF TOTALITY):;: H=T0+2*SAMMIT/5:GOSUB 970 PRINT " M A D H Y A (MIDDLE) OF ECL.:";:H=T0+OPPT:GOSUB 970 IF GRASA>MDL\ THEN PRINT " U N M I L A N A (END OF TOTALITY):";: H=T0+2*UNMIT/5:GOSUB 970 PRINT "MOKSHA (ENDING) TIME:";:H=T0+2*MOKST/5:GOSUB 970 PRINT "********************************************************" END R E M ** CONVERSION INTO HRS,MIN,SEC ** HRS=INT(H):M=60*(H-HRS):MIN=INT(M):SEC=INT((M-MIN)*60) PRINT TAB(48) HRS;"H-";MIN;"M-";SEC;"S" RETURN
Computer Progrcuns
223
1010
IF L<0 THEN L=ABS(L)
1020 1030 1040 1050
DEG=INT(L):MIN=(L-DEG)*60:SEC=INT((MIN-INT(MIN))*60+.5) IF SEC=60 THEN SEC=0:MIN=MIN+1:IF MIN=60 THEN MIN=0:DEG=DEG+1 IF DEG>=360 THEN DEG=DEG-360 PRINT T A B ( 3 5 ) ; D E G ; ; I N T ( M I N ) ; " ' ";SEC;" " "
1060
RETURN
1070 1080 1090
MIN=INT(M):SEC=INT({M-MIN)*60+.5):IF SEC=60 THEN SEC=0:MIN= MIN+1 PRINT TAB(35);MIN;"'";SEC;""" RETURN
224
Ancient Indicm Astronomy
10 20 30 PRINT TAB(23); "• 40 PRINT TAB(23); "* 50 PRINT TAB(23); "* 60
SOLAR ECLIPSE ACCORDING TO SURYA SIDDHANTA
70
PRINT :PI=3.14159256#:DTR=PI/180 : RTD=1/DTR :REM * DEG TO RAD & R A D TO DEG *
140
R E M ** Tnie Sun (TRAVI), True Moon (TMOON), Node (MRAHU) at TO HRS **
150 160
PRINT :PRINT TAB(25) "Ar';TO;"HRS ON ";D1;"/";MM;"/";Y T=(Y-1900+(MM-l)/12+Dl/365)/100 :REM '* JULIAN CENTURIES SINCE 1/1/1900 *
170 180 190
AYA=22.14604222#+( 1.39604*T*T6#*T)+(1.111/3600*T*T)+(.00011 *T*T/3600) PRINT :PRINT TAB(25);"AYANAMSA: ";:L=AYA:GOSUB 2130 PRINT :PRINT " * * TRUE SUN: ";TRAVI;" TRUE MOON: ";TMOON; "NODE: " ; M R A H U PRINT R E M ** S M K :Sun's anomaly; M M K :Moon's anomaly ** SDM=59.13333*(1-14*COS(SMK*PI/180)/360) :REM * SUN'S TRUE DAILY MOTION * PRINT "SUN'S TRUE DAILY MOTION: ";:M=SDM:GOSUB 2080 MDM=790.5666-31*783.9*COS(MMK*PI/180)/360:REM * MOON'S TRUE DAILY MOTION * R E M ***"*783.9 = M E A N DAILY MOTION OF (MOON - MOON'S APOGEE)**** PRINT :PRINT "MOON'S TRUE DAILY MOTION: ";:M=MDM:GOSUB 2080 SCON=(TRAVI-TMOON):IF SCON<0 THEN SCON=SCON+360 PRINT :PRINT "MOON'S DISTANCE FROM CONJN.: ";:L=SCON:GOSUB 2130
200 210 220 230 240 250 260 270 280 290 300
TCON=SCON*24/((MDM-SDM)/60) Hl=T0+TCON:H(0)=Hl
310
PRINT :PRINT "TIME OF CONJ. AFTER MIDNIGHT";:N=H1*5/2:G0SUB 2200
320
H=H1:G0SUB 1120
330
PRINT :PRINT TAB(60) "PRESS A N Y KEY":A$=INPUT$(1)
350
FOR 1=1 TO 20
360
CLS:PRINT TAB(15) "* AT THE TIME OF CONJUNCTION (APPROXN";I;")*"
370 380
IF l o l O THEN GOTO 390 CLS:PRINT TAB(12) "* AT THE TIME OF APPARENT CONJN. (AFTER";I; "ITERATIONS *"
Canputer Pro^xuns 390
225
CSUN1=TRAVI+TCON*SDM/(60*24)
400 PRINT "TRUE LONG. OF SUN :";:L=CSUN1:G0SUB 2130 410
CMOON1=TMOON+TCON*MDM/(60*24)
420 PRINT "TRUE LONG. p F MOON .";:L=CMOONl:GOSUB 2130 430 CNODE1=MRAHU=-190.7*TCON/(3600*24) 440 PRINT "LONG. OF NODE •.";:L=CN0DE1 :GOSUB 2130 450 TSUN1=CSUN1+AYA:IF TSUN1>360 THEN TSUN1=TSUN1- 360 460 IF T S U N K O THEN TSUN1=TSUN 1+360 470 PRINT :PRINT "SAYANA RAVI (TROFSUN):";:L=TSUNl-.GOSUB 2130 480 IF 1=10 THEN GOTO 840 490 PRINT "ORIENT ECLIPTIC PT.(LAGNA):";:T=H(I-1):G0SUB 1660 500 OREC=TLAG : L=OREC:GOSUB 2130 510 ORSIN = 3438*SIN(24*DTR)*SIN(OREC*DTR)/COS(PHI*DTR)
: REM *
ORIENT SINE : UDAYA JYA * 520 PRINT "ORIENT SINE (UDAYA JYA):";TAB(35);ORSIN;"'" 530 PRINT :PRINT " M E R I D L \ N ECLIPTIC POINT (M.C.):";:L=MC:GOSUB 2130 540 MERSIN=3438*SIN(24*DTR)*SIN(MC*DTR): R E M * MERIDIAN SINE * 550 PRINT "MERIDIAN SINE (MADHYA JYA):";:M=MERSIN:GOSUB 2080 560 X=MERSIN /3438:GOSUB 880:REM * DECLINATION OF M C (in Deg) * 570 PRINT "DECLINATION OF THE MERIDIAN:"; 580 IF DEC<0 THEN PRINT TAB(34); "-"; 590 L=DEC:GOSUB 2130 600 M E R Z E N = DEC - PHI : R E M • PHI=PD+PM/60 : LAT OF THE P L A C E * 610 PRINT "MERIDIAN ZENITH DISTANCE:";:IF MERZEN<0 THEN PRINT TAB(34); "-"; 620 L=MERZEN:GOSUB 2130 630
SNMERZ=3438*SIN(MERZEN*DTR)
640 OSMZ = ORSIN * SNMERZ/3438 650 DRKSHEPA = SQR(SNMERZ'^2 - 0SMZ''2) 660 PRINT "SINE OF ECL.ZEN.DIST.(DRKKSHEPA):";:M=DRKSHEPA:GOSUB 2080 670 SNECALTi=SQR(3438'^2-DRKSHEPA'^2) 680 PRINT "SINE OF ECL.ALTITUDE (DRGGATI):";:M=SNECALT:GOSUB 2080 690 CHEDA=1719^2/SNECALT : R E M • RSIN (30 Deg)=1719 * 700 PRINT "DIVISOR (CHEDA):";:M=CHEDA:GOSUB 2080 710 LAMBANA=3438*(SIN((MC-TSUN1 )*DTR))/CHEDA 720 PRINT " ( P A R A L L A X IN LONG.)LAMBANA:"; 730 IF LAMBANA<0 THEN PRINT TAB(34); "-";
226
Ancient Indian Astronomy
740
N=LAMBANA:GOSUB 2200
750 760
H=LAMBANA*2/5:IF H<0 THEN PRINT GOSUB 1120
770
H(I)=Hl+LAMBANA*2/5
780
PRINT :PRINT "COR.TIME OF APPARENT CONJN.:";:N=H(I)*5/2:GOSUB 2200
790 795
H=H(I):GOSUB 1120 PRINT "H(";I;")";H(I)
800 TCON=TCON+2*LAMBANA/5 810
PRINT :PRINT TAB(60);"PRESS A N Y K^Y":A$=INPUT$(1)
820 830
IF 1=20 OR ABS(H(I)-H(I-1))<.1 THEN 10=1: GOTO 380 NEXT I
840 850 860
NATI=(731.45/15)*DRKSHEPA/3438 PRINT " P A R A L L A X IN LATITUDE (NATI):";:IF NATI<0 THEN TAB(34);"-"; M=NATI:GOSUB 2080
870
GOTO 900
880 DEC=RTD*ATN(X/SQR(1-X*X)) 890
RETURN
900 910
MLAT1=270*SIN(DTR*(CMOON1-CNODE1)) PRINT "MOON'S LATITUDE AT APPNT CONJN.:";:IF M L A T R O THEN PRINT TAB(34);"-";
920
M=MLAT1:G0SUB 3080
930 APLAT=MLAT1+NATI 940 950 960 970 980 990 1000 1010 1030 1040 1050 1060 1070
PRINT "MOON'S APPARENT L A T AT CONJN.:";:IF APLAT<0 THEN PRINT TAB(34)"-"; M=APLAT:GOSUB 2080 SCDIA=SDM*6500/59.136 : R E M * SUN'S COR.DIAMETER (YOJANAS) * SDIAY=SCDIA*4320000!/57753336# : R E M * SUN'S DIAMETER IN YOJANAS * SDIA=SDIAY/15 : R E M SUN'S D L \ M E T E R IN A R C (MIN) * PRINT : PRINT "SUN'S A N G U L A R DIAMETER:";:M=SDIA:GOSUB 2080 MDIAY=MDM*480/790.581 : R E M * MOON'S DIAMETER IN YOJANAS * MDIA=MDIAY/15 : R E M * MOON'S DIAMETER IN A R C (MIN) * 0BS=.5*(SDIA+MDIA)-ABS(APLAT) PRINT :PRINT "OBSCURN.AT APPRNT CONJN (GRAASA):";: M=OBS: GOSUB 2080 PRINT IF OBS<0 THEN PRINT TAB(30)"ECLIPSE NOT VISIBLE":END IF OBS
Computer Programs
227
1080 1090 1100 1110 1120
IF OBS>=SDIA THEN PRINT TAB(31)"ECLIPSE IS TOTAL" PRINT rPRINT TAB(20)"MAGNITUDE OF THE ECLIPSE:";OBS/SDIA LOCATE 22.60:PRINT "":A$=INTPUT$(1) PRINT:IF ZZ=1 THEN GOTO 1170 R E M •* CONVERSION INTO HRS,MIN,SEC **
1130 1140 1150 1160
H=ABS(H):HRS=INT(H):M=60*(H-HRS):MIN=INT(M):SEC=INT((M-MIN)*60) PRINT TAB(54) HRS;"H-";MIN;"M-";SEC;"S" RETURN R E M * HALF-DURATIONS *
1170 1180 1190 1200 1210 1220 1230 1240 1250 1260 1270 1280 1290 1300 1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 1410 1420 1430 1440
DBET=SIN(4.5*DTR)*COS((CMOONl-CNODEl)*DTR)*829.91833# A=(MDM-SDMr2+DBET'^2 B=2*APLAT*DBET DELl=(SDIA+MDIA)/2 :DEL2=(SDIA-MDIA)/2 C1=APLAT'^2-DEL1'^2 C2=^PLAT'^2-DEL2'^2 DISCR1=B'^2-4*A*CI DISCR2=B'^2-4*A*C2 CLS R E M *** HRST-HALF DURATION OF THE ECLIPSE *** T1=60*ABS(-B-SQR(DISCR1))/(2*A) PRINT "HRST-HALF DURATION OF ECLIPSE: ";:N=Tl:GOSUB 2200 H=T1*2/5:G0SUB 1120 R E M *** SECOND-HALF DURATION OF THE ECLIPSE *** T4=60*(-B+SQR(DISCR1))/(2*A) PRINT PRINT "SECOND-HALF DURATION OF ECLIPSE: ";:N=T4:GOSUB 2200 H=T4*2/5:GOSUB 1120 IF ZZ=1 THEN GOTO 1460 * R E M *** HRST-HALF DURATION OF TOTALITY *** T2=60*ABS(-B-SQR(DISCR2))/(2*A) PRINT PRINT "HRST-HALF OF TOTALITY: ";:N=T2:G0SUB 2200 H=T2*2/5:GOSUB 1120 R E M *** SECOND-HALF DURATION OF TOTALITY *** T3=60*(-B+SQR(DISCR2))/(2*A) PRINT PRINT "SECOND-HALF OF TOTALITY: ";:N=T3:G0SUB 2200
1450
H=T3*2/5:GOSUB 1120
228
Ancient Indicun Astronomy
1460 1470 1480 PRINT TAB(5) "** S U M M A R Y OF THE SOLAR ECLISPE **" TAB(5)"************************************************' 1490 PRINT 1500 PRINT TAB(25);" ON";Dl"/";MM;"r;Y;" AT';PLACE$ 1510 PRINT TAB(35)"LOCAL M E A N TIME" 1520 PRINT TAB(5) "BEGINNING OF T H E E C U P S E : ";:N=5*H(I)/2-Tl :GOSUB 2200 1530 H=:N*2/5:GOSUB 1120 1540 1550 1560 1570 1580 1590 1600 1610 1620 1630 1640 1645 1650
IF ZZ=1 THEN GOTO 1570 PRINT TAB(5) "BEGINNING OF TOTALITY: ";:N=5*H(I)/2-T2:GOSUB 2200 H=N*2/5: GOSUB 1120 PRINT TAB(5) "MIDDLE OF THE ECUPSE:";:N=5*H(I)/2:GOSUB 2200 H=H(I):GOSUB 1120 IF ZZ=1 THEN GOTO 1620 PRINT TAB(5) "END OF TOTALITY: ";:N=5*H(I)/2+T3:GOSUB 2200 H=N*2/5:GOSUB 1120 PRINT TAB(5) "END OF THE ECLIPSE: ";:N=5*H(I)/2+T4:GOSUB 2200 H=N*2/5: GOSUB 1120 LOCATE ,24:INPUT "DO Y O U WANT ANOTHER TRIAL (Y/N)";Y$ IF Y$=" Y " OR Y$="y" THEN CHAIN "SSRAMOON" ELSE END
1660 R E M *** ORIENT ECLIPTIG POINT ( SAYANA L A G N A ) *** 1670 R E M * T : TIME IN 1ST FOR WHICH L A G N A & M C A R E REQUIRED • 1675 1680 1690 1700 1710 1720 1730 1740 1750 1760 1770 1780 1790 1800
PRINT "T=";T G1 =6.63627+6.570982*.01 *(JJ-2443144!) TS=Gl-INT(Gl/24)*24 IF LD>=0 THEN L=LD+(LM/60) IF LD<0 THEN L=LD-(LM/60) S=I715+TS+(T-5.5)/1436*4 IF S>24 THEN S=S-24 IF S<0 THEN S=S+24 ST=S T=T-5.5 H=ST+T:IF PD<0 THEN H=H+12 IF H>24 THEN H=H-24 IF H<0 THEN H=H+24 GOSUB 1810:GOTO 1830
Cennputer Programs 1810 1820
229
MIN={H-INT(H))*60:TrY=INT((MIN-INT(MIN))*60) RETURN
1830 IF PD+PM/60>0 THEN PHI=PD+PM/60 1840 IF PD+PM/60<0 THEN PHI=PD-PM/60 1850 S=H:S=S*15 1860
A=S+90:A=A*DTR:W=23.45*DTR
1870 GOSUB 1970 1880 S=H:S=S*15 1890 B=ATN(TAN(A)*COS(W)) 1900 T=ATN(COS(A)*TAN(W)) 1910 E1=ATN(SIN(A)*SIN(W)*TAN(ABS(PHI*DTR+T))) 1920 L=(B+E1)*RTD 1930 IF PD<0 THEN L=180+L 1940 IF L<0 THEN L=L+360 1950 IF S<180 THEN L=L+180 1960 TLAG=L 1970 IF PD<0 THEN S=S-180:IF S<0 THEN S=S+360 1980 TRS=S*DTR 1990 IF S=90 THEN MC=90 -.GOTO 2030 2000 IF S=180 THEN MC=180:GOTO 2030 2010 IF S=270 THEN MC=270 :GOTO 2030 2020 A2=RTD*ATN(TAN(TRS)/C0S(W)) 2030 IF S>90 A N D S<180 THEN
MC=180+A2
2040 IF S>180 A N D S<270 THEN MC=180 +A2 2050 IF S<90 OR S>270 THEN MC=A2 2060 IF MC<0 THEN MC=MC+360 2070
RETURN
2080 M=ABS(M) 2090
MIN=INT(M):SEC=INT((M-MIN)*60+.5)
2100 IF SEC=60 THEN SEC=0:MIN=MIN+1 2110 PRINT TAB(35);MIN;"' ";SEC;' 2120
RETURN
2130 IF L<0 THEN L=ABS(L) 2150
DEG=INT(L):MIN=(L-DEG)*60:SEC=INT((MIN-INT(MIN))*60+.5)
2160 IF DEG>=360 THEN DEG=DEG-360 2170 IF SEC=60 THEN SEC=0:MIN=MIN+1 :IF MIN=60 THEN MIN=0:DEG=DEG+1: IF DEG>=360 THEN DEG=DEG-360 2180 PRINT TAB(35);DEG;"' ";INT(MIN);"'";SEC;"""
230
Ancient Indian Astronomy
2190
RETURN
2200
N=ABS(N)
2210 NADI=INT(N):VIN=INT((N-NADI)*60+.5):IF NADI+1
VIN=60 THEN VIN=0:IMADI=
2220 PRINT TAB(35);NADI;TAB(39);"na. ";TAB(43);VIN;"vin."; 2230 RETURN
Computer Programs
231
10 CLS:REM * P R O G R A M 10.2 : "GLSEC" 20 PRINT TAB(23); "*********************************************" 30 PRINT TAB(23); "*
SOLAR ECLIPSE
*"
40 PRINT TAB(23); "* 50 PRINT TAB(23); " •
ACCORDING TO GRAHA LAGHAVAM
*' *"
60 PRINT TAB(23);"**********************************************" 70 PRINT :PI=3.1415926#:DTR=PI/180 : RTD=1/DTR :REM * DEG TO RAD & R A D TO D E G • 80 PRINT TAB(10);"AT ";6+Hl ;"HRS ";MI;"MIN. O N ";D1;"/";MM;"/";Y;"AT" +PLACE$ 90 PRINT "** TRUE SUN:";TRAVI; "TRUE MOON:";TMOON; "NODE:"; M R A H U ; "**" 100 Al=(Y-522)/60: R E M * A Y A N A M S A * 110 PRINT "AYANAMSA:";:L=A 1 :GOSUB 1810 120 R E M * INSTANT OF N E W M O O N • 130 CONJ=TRAVI-TMOON 140 SDM=59.9*(1-14*COS(SMK*DTR)/360):REM * SUN'S TRUE DAILY MOTION* 150 PRINT "SUN'S TRUE DAILY MOTION: ";:M=SDM:GOSUB 1890 160 MDM=790.5666-31*783.9*COS(MMK*DTR)/360:REM * MOON'S TRUE DAILY MOTION * 170 PRINT "MOON'S TRUE DAILY MOTION: ";:M=MDM:GOSUB 1890 180 CONJT=24*CONJ/((MDM-SDM)/60):REM * INTERVAL FOR CONJN.(HRS) * 190 PRINT "INTERVAL TO NEWMOON:;TAB(35):H=CONJT:GOSUB 1940 200 PRINT "INTERVAL OF NEWMOON:'';TAB(35):H=6+CONJT:GOSUB 1940:NMH=H 210 R E M * AT THE INSTANT OF NEWMOON * 220
NMSUN=TRAVI+CONJT*SDM/(60*24)
230 PRINT "TRUE RAVI AT NEWMOON:";:L=NMSUN:GOSUB 1810 240
NMMOON=TMOON+CONJT*MDM/(60*24)
250 PRINT "TRUE CHANDRA AT NEWMOON:";:L=NMMOON:GOSUB 1810 260
NMNODE=MRAHU-190.7*CONJT/(3600*24)
270 PRINT "NODE AT NEWMOON:":L=NMNODE:GOSUB 1810 280 MLAT=270*SIN((NMMOON-NMNODE)*DTR) 290 VRHK=NMSUN-NMNODE:IF VRHK<0 THEN VRHK=VRHK+360 300 PRINT " V I R A A H V A R K A (RAVI-RAHU)";:L=VRHK:GOSUB 1810 310 IF VRHK<90 A N D VRBHUJA=VRHK 3.20 IF VRHK>90 A N D VRHK<180 THEN VRBHUJA=180-VRHK 330 IF VRHK>180 A N D VRHK<270 THEN VRBHUJA=VRHK-180 340 IF VRHK>270 A N D VRHK<360 THEN VRBHUJA=360-VRHK
232
Ancient Indian Astronomy
350
SHARA=VRBHUJA*17/7:IF VRHK>180 THEN SHARA =-SHARA
360 370
PRINT "SHARA:";:ANG=SHARA:GOSUB 1780 IF VRBHUJA < 14 THEN PRINT TAB(25) "*SOLAR ECUPSE IS POSSIBLE *"
380 390 400 410
IF VRBHUJA > 14 THEN PRINT "•SOLAR ECLIPSE NOT POSSIBLE *":END GOSUB 1400:REM • L A G N A AND TOIBHONA LAGNA * GOSUB 410:GOTO 590 R E M ** KRAANTI (DECLINATION) OF TRIBHONALAGNA *•
420 430
R E M * BHUJA OF SAYANA TRIBHONA L A G N A • IF TBL<90 THEN BHUJA=TBL
440 450
IF TBL>90 A N D TBL^ISO THEN BHUJA = 180-TBL IF TBL>180 A N D TBL<270 THEN BHUJA =TBL-180
460 470
IF TBL>270 A N D TBL<360 THEN BHUJA = 360-TBL PRINT "BHUJA OF SAYANA TRIBHONA LAGNA:";:L=BHUJA:GOSUB 1810
480 490
KRNK=INT(BHUJA/10) RESTORE 1980
500 510
FOR K=0 TO KRNK+1 READ KRN.IF K=KRNK THEN KRN1=KRN
520 530 540 550
N E X T K:KRN2=KRN PRINT " A N K A = ";KRNK;"KRN1=";KRN1;" KRN2=";KRN2 KRANTI=(KRN 1 +(KRN2-KRN 1 )*(BHUJA-10*KRNK)/10)/l 0 IF TBL>180 THEN KRANTI=-KRANTI
560
PRINT "KRANTI OF SAYANA TRIBHONA LAGNA:";:IF KRANTI<0 THEN PRINT TAB(34)"-";
570
L=KRANTI:GOSUB
580 590
RETURN LOCATE 23,65:PRINT "PRESS A N Y K E Y "
600 610 620
A$=INPUT$(1) R E M ** NATAAMSA ** CLS:PRINT
630 640
NATAMSA=KRANTI-P*180/PI PRINT "NATAAMSA=";TAB(34);NATAMSA
650
SQ=(NATAMSA/22)^2
660
IF SQ>2 THEN HARA=SQ+(SQ-2)^+12:GOTO 680
670 680 690 700
HARA=SQ+12 PRINT " H A A R A (DEN.)=";TAB(34);HARA VRKTBL=ABS(NMSUN-NTBL) NUM=(14-VRKTBU10)*VRKTBL/10:PRINT .TAB(34);NUM
1830
"NUM.
FOR
LAMBANA:";
Computer Programs
233
710 720
LMBN=NUM/HARA : R E M * L A M B A N A IN GHATIKAS * IF NTBL
730
PRINT " L A M B A N A TAB(34);"-";
740
H=2*LMBN/5:GOSUB
750
MIDDLE=NMH+2*LMBN/5 : R E M * MIDDLE OF ECLIPSE IN HOURS *
760 770 780
PRINT "MIDDLE OF THE ECUPSE:";:H=MIDDLE:GOSUB 1940 R E M ** STHITI S A D H A N A M ** SDIA=SDM*2/11 :MDIA=MDM/74: R E M * DIAMETERS OF SUN & MOON (ANGULAS) *
790 800
GRASA=.5*(SDIA+MDIA) - SHARA :REM * IN A N G U L A S * PRINT " G R A A S A : ";:ANG=GRASA:GOSUB 1780
810
PRINT " P R A M A N A M (MAGNITUDE:";TAB(35):GRASA/SDIA
820 830
NUM1=(NATAMSA/10)*(18 - (NATAMSA/10)):REM * IN MINUTES * PRINT " N U M . FOR NATI:";TAB(34);NUM1
840 850 860 870 880
DENl=6.3-NUMl/60 PRINT "DEN. FOR NATL";TAB(34);DEN1 NATI=NUM1/DEN1 :REM * N U M l IS CONSIDERED AS DEGREES * IF NATAMSA<0 THEN NATI=-NATI PRINT " NATI:";TAB(34);NATI
890
SPSHARA=SHARA + NATI:REM * SPASHTA SHARA *
:
";LMBN;"Gh.=";:IF
LMBN<0
THEN
PRINT
1940
900
PRINT "SPASTHA SHARA:";:ANG=SPSHARA:GOSUB 1780
920
SDIA=2*SDM/11 : R E M * SURYABIMBA (SUN'S DIAMETER)
930 940 950
PRINT "SURYA BIMBA:";:ANG=SDIA:GOSUB 1780 MDIA=MDM/74 : R E M * CHANDRABIMBA (MOON'S DIAMETER) PRINT "CHANDRA BIMBA:";:ANG=MDIA:GOSUB 1780
960
MKDRD=(SDIA+MDIA)/2
970 980
PRINT " M A A N A I K Y A KHANDAARDHA:";:ANG=MKDRD:GOSUB 1780 SGR=MKDRD - SPSHARA : R E M * SURYAGRAASA *
990
PRINT "SURYA GRAASA:";:ANG=SGR:GOSUB 1780
: REM * MAANAIKYA KHANDAARDHA *
1000
STHITI=SQR(10*(MKDRD+SPSHARA)*SGR)"^(5/6)/MDIA
1010 1020 1030
PRINT "STHITI:";STHITI;"GH. =";:H=STHITI*2/5:GOSUB 1940 R E M * SPARSHAKAALA L A M B A N A * SKTL=TBL - 6'^STHm
1040
PRINT "SPARSHAKAALA TRIBHONALAGNA:";:L=SKTL:GOSUB 1810
1050 1060
TBL=SKTL:GOSUB 410 NATAMSA=KRANTI-P*180/PI
234
Ancient Indian Astronomy
1060
N A T A M S A = K R A N n - P * l 80/PI
1070
SKSUN=NMSUN-SDM*STHITI/60
1080 1090 1100 1110 1120
PRINT "SPARSHAKALA RAVI:";:L=SKSUN:GOSUB 1810 NUM=(SKSUN-SKTL)/10 * (14-(SKSUN-SKTL)/10) HAARA=(NATAMSA/22r2 - 2y2 + (NATAMSA/22)^2 +12 :REM * DENOMINArOR* SKLAMB = N U M / HAARA PRINT "SPARSHAKAALA L A M B A N A : " ; SKLAMB;"GH";:IF SKLAMB<0 THEN PRINT TAB(34);"-"; H=SKLAMB*2/5 :GOSUB 1940 . PRINT TAB(60);"PRESS A N Y KEY":A$=INPUT$( 1) CLS:REM * M O K S H A K A A L A LAMB/VNA * MOKTL=TBL + 6*STHrri PRINT "MOKSHA K A A L A TRIBHONALAGNA:";:L=MOKTL:GOSUB 1810 TBL=MOKTL:GOSUB 410 NATAMSA=KRANTI-P*180/PI MOKSUN=NMSUN+SDM*STHITI/60 PRINT " M O K S H A K A A L A RAVI:";:L=MOKSUN:GOSUB 1810 NUM=(MOKSUN-MOKTL)/10 * (14-(MOKSUN-MOKTL)/10) HAARA=((NAT\MSA/22r2 - 2y2+(NAT\MSA/22r2 +12 :REM * DENOMINAIOR* MOKLAMB = N U M / HAARA PRINT " M O K S H A K A A L A LAMBANA:";MOKLAMB;"GH.";:IF MOKLAMB<0 THEN PRINT TAB(34):"-"; H=MOKLAMB*2/5 :GOSUB 1940 B E C L = MIDDLE - STHITI + SPLAMB EECL = MIDDLE + STHITI + M O K L A M B PRINT
1130 1140 1150 1160 1170 1180 1190 1200 1210 1220 1230 1240 1250 1260 1270 1280 1290 1300 1310 1320 1330 1340 1350
PRINT :PRINT "BEGINNING OF ECLIPSE";TAB(21);BECL;"GH."; H=6+BECL*2/5:GOSUB 1940 PRINT :PRINT "MIDDLE OF ECLIPSE:";TAB(21);MIDDLE:"GH.";
1360 H=6+MIDDLE*2/5:GOSUB 1940 1370 PRINT :PRINT "END OF ECLIPSE:";TAB(21);EECL:"GH."; 1380 H=6+EECL*2/5:GOSUB 1940 1390 END 1400
R E M ********** L A G N A & TRIBHONA L A G N A **********
1420
T=NMH:REM * INSTANT OF NEWMOON *
Computer Programs
235
1430 G1 =6.63627+6.570982*.01 *(JJ-2443144!) 1440 TS=Gl-INT(Gl/24)*24 1450 IF LD>=;0 THEN L=LD+(LM/60) 1460 IF LD<0 THEN L=LD-(LM/60) 1470
S=I715+TS+(T-5.5)/1436*4
1480 IF S>24 THEN S=S-24 1490 IF S<0 THEN S=S+24 1500
ST=S:T=T-5.5
1510 H=S+TIF PD<0 THEN H=H+12 1520 IF H>24 THEN H=H-24 1530 IF H<24 THEN H=H+24 1540 IF PD>0 THEN P=PD+FM/60 1550 IF PD<0 THEN P=PD-PM/60 1560 P=P*PI/180 1570
S=H:S=S*15
1580 A=S+90:A=A*Pyi80:W=23.45*PI/180 1590 B=ATN(TAN(A)*COS(W)) 1600 T=ATN(COS(A)*TAN(W)) 1610
E1=ATN(SIN(A)*SIN(W)*TAN(P+T))
1620 L=(B+E1)*180/PI 1630 IF PD<0 THEN D=180+((B+E1)*RTD) 1640 IF L<0 THEN L=L+360 1650 IF S<180 THEN L=L+180 1660 IF L>360 THEN L=L-360 1670 SLAGNA=L :REM * SAYANA L A G N A * 1680 PRINT "SAYANA LAGNA:";:L=SLAGNA:GOSUB 1810 1690 TBL=SLAGNA-90 :REM * SAYANA TRIBHONA L A G N A * 1700 IF TBL<0 THEN TBL=TBL+360 1710 PRINT "SAYANA TRIBHONA LAGNA:";:t=TBL:GOSUB 1810 1720 R E M * A Y A N A M S A : A l * 1730 NTBL=TBL-A1 : R E M * NIRAYANA TRIBHONA L A G N A * 1740 IF NTBL<0 THEN NTBL=NTBL+360 1750 IF NTBL>360 THEN NTBL=NTBL-360 1760 PRINT "NIRAYANA TRIBHONA LAGNA:";:L=NTBL:GOSUB 1810 1770
RETURN
1780 AN=INT(ANG): PRA=INT((ANG-AN)*60+i) :IF PRA=60 THEN PRA=0:AN=AN+1 1790 PRINT TAB(35);AN;"Ang.";PRA;"Prat." 1800
RETURN
Ancient Indian Astronomy
236 1810
I F L>=360 T H E N
1820
IF L<0 T H E N
1830
IF L<0 T H E N L=ABS(L)
1840
L=L-360*INT(L/360)
L=L+360*ABS(INT(I7360))
DEG=INT(L):MIN=(L-DEG)*60:SEC=INT((MIN-INT(MIN))*60+.5)
1850
I F DEG>=360 T H E N D E G = D E G . 3 6 0
1860
IF
SEC=60
THEN
SEC=0:MIN=MIN+1:IF
MIN»60
THEN
MIN=0:DEG
=DEG+I:IF DEG>=360 T H E N DEG=DEG-360 1870
P R I N T TAB(35);DEG;"° ";TAB(42)INT(MIN);"'";TAB(47)SEC;"""
1880
RETURN
1890
M=ABS{M)
1900
MIN=INT(M):SEC=INT((M-MIN)*60+.5)
1910
I F SEC=60 T H E N
SEC=0:MIN=MIN+1
1920
PRINT TAB(35):MIN;"' ";SEC;
1930
RETURN
1940
R E M ** C O N V E R S I O N I N T O H R S , M I N , S E C **
1950
H=ABS(H):HRS=INT(H):M=60*(H-HRS):MIN=INT(M):SEC=INT((M-MIN)*60)
1960
P R I N T TAB(35);HRS;TAB(39);"H-";MIN;TAB(45);"M-";SEC;"S"
1970
RETURN
1980
D A T A 0,40,80,117,151,181,206,224,236,240
1990
R E M * K R A A N T I (DECLINATION) T A B L E *
Computer Programs
237
10 C L S : R E M * P R O G R A M 10.3 : " K K S E C " 20 PRINT TAB(23);
" * * * * * * * * * * * * * « * * « * * * * * * * * * * * * * * * * * * * * > i < > i ' * * * * * # * * "
30
PRINT TAB(23); "*
SOLAR ECLIPSE
*"
40
PRINT TAB(23);
50
PRINT TAB(23): "*
ACCORDING TO
60
PRINT TAB(23);"****************************'*''''''''''*''''''''i''<''<'''''*''*<'*''<''*''''**'''''
70
PRINT :PI=3.14159256#
KHANDA - KHAADYAKA
*"
110
R E M ** Tiue Sun (TOAVI), True Moon (TMOON). Node (MRAHU) at TO HRS **
120
PRINT TAB(25) "Ar';TO;"HRS O N ";D1;"/";MM;"/";Y
130
PRINT " *• TRUE SUN: ";TRAVI;" TRUE MOON: ";TMOON;" NODE: ";MRAHU
140
PRINT
150 160
R E M ** S M K rSun's anonialy; M M K :Moon's anomaly ** SDM=59.13333*(1.14*COS(SMK*PI/180)/360) :REM * SUN'S TRUE DAILY MOTION *
170 180
PRINT "SUN'S TRUE DAILY MOTION: ";:M=SDM:GOSUB 1960 MDM=790.5666-31*783.9*COS(MMK*PI/I80)/360:REM •* MOON'S TRUE DAILY MOTION *
190
R E M ****783.9 = M E A N DAILY MOTION OF (MOON - MOON'S APOGEE) ****
200
PRINT "MOON'S TRUE DAILY MOTION: ";:M=MDM:GOSUB 1960
210
SCON=(TRAVI-TMOON):IF SCON<0 THEN SCON=SCON+360
220
PRINT "MOON'S DISTANCE F R O M CONJN.: ";:L=SCON:GOSUB 2020
230
TCON=SCON*24/((MDM-SDM)/60)
240
PRINT "INSTANT OF CONJUNCTION (LMT)"";:H=T0+TCON;GOSUB 1080
250
DIMTdO)
260
PRINT
270
R E M ** AT THE CONJUNCTION **
280
CSUN1=TRAVI+TCON*SDM/(60*24)
290
PRINT ' T R U E SUN AT CONJN.:";:L=CSUNl:GOSUB 2020
300 310 320
CMOON1=TMOON+TCON-»MDM/(60'*24) PRINT "TRUE MOON AT C0NJN.:";:L=CM00N1:G0SUB 2020 CNODE1=MRAHU-190.7*TCON/(3600*24)
330
PRINT "NODE AT C0NJN.:";:L=CN0DE1:G0SUB 2020
340
AYA = 22.4604222 # + (1.3960416#*TC) + (3.0875E-04*TC*TC) + (.0006175*TC'*T/10O)+(0.139604 *T)+(9.44E-06 *Q)+(.0003086*T*T/10000) +(3.80555E-05*E)+(3.82215E-05*(DD+D))
350
PRINT " A Y A N A M S A : ";:L=AYA:GOSUB 2020
360
TSUN1=CSUN1+AYA:IFTSUN1>360
370
IF TSUN1 <0 THEN TSUN1 =TSUN 1 +360
THEN TSUN1=TSUN 1-360
238
Ancient Indian Astronomy
380 PRINT "SAYANA RAVI (TRORSUN):";:L=TSUNI:GOSUB 2020 390 X=SIN(24*PI/180)*SIN(TSUN1*PI/180) 400 SDEL=180*ATN(X/SQR(1-X*X))/PI 410 PRINT "SUN'S DECLINATION:";:IF SDEL<0 THEN PRINT TAB(33)"-"; 420 L=SDEL:GOSUB 2020 430 M L A T l =270*SIN(PI*(CMOON 1-CNODEl)/180) 440 PRINT "MOON'S LATITUDE AT CONJN.: ";:IF M L A T l <0 THEN PRINT TAB(33) "-"; 450 M=MLAT1:G0SUB 1960 460 PRINT • 470 MDIA=10*MDM/247:REM * MOON'S A N G . DIAMETER * 480 PRINT "MOON'S A N G U L A R DIAMETER: ";:M=MDIA:GOSUB 1960 490 SDIA=11*SDM/20:REM * SUN'S ANG.DIAMETER • 500 PRINT "SUN'S A N G U L A R DIAMETER: ";:M=SDIA:GOSUB 1960 510 LOCATE 23, 55:PRINT "PRESS A N Y K E Y TWICE" 520 A$=INPUT$(2) 530 540 550 560 570 580
T(I)=T0+TCON FOR 1=1 TO 10 IF 1=4 THEN LOCATE 23.55:PRINT "PRESS A N Y K E Y TWICE": A$=INPUT$(2) PRINT TAB(32) "APPROXIMATON (";I;")" T=T(I):PRINT "T(I)=";T(I) GOSUB 1520
590 PRINT "SAYANA L A G N A (ORIENT ECLIPTIC PT):";:L=TLAG:GOSUB 2020 600 NONA=TLAG-90:IF NONA<0 THEN NONA=NONA+360 610 PRINT "NONAGESIMAL ( TRIBHONA ):";:L=NONA:GOSUB 2020 620 Y=SIN(24*Pyi80)*SIN(NONA*PI/180) 630 640 650 660 670 680 690 700
NDEL=180*ATN(Y/SQR(1-Y*Y))/PI PRINT "DECL.OF NONAGESIMAL:";:IF NDEL<0 THEN PRINT TAB(33)"-"; L=NDEL:GOSUB 2020 R E M ** ZENITH DIST. OF NONAGESIMAL ** CSUN=CSUNl+(HEQAC*SDM/(60*24)) CMOON=CMOON 1 +(HEQAC*MDM/(60*24)) CNODE=CNODEl-(HEQAC*.05277/24) MLAT=270*SIN(PI*(CMOON-CNODE)/180)
710 PRINT "MOON'S LATITUDE:";:IF MLAT<0 THEN PRINT TAB(34)"-"; 720 M=MLAT:GOSUB 1960 730 ZEN=NDEL+MLAT/60-PHI
Computer Programs
239
740
PRINT "ZENITH DIST OF NONAGESIMAL:";:IF ZEN<0 THEN PRINT TAB(34)"-";
750 760
L=ZEN:GOSUB 2020 R E M ** EQN OF APPARENT CONJN. **
770
TSUN=CSUN+AYA:IF TSUN>360 THEN TSUN=TSUN-360
780
IFTSUN
790 800
EQAC=-4*COS(PI*ZEN/180)*SIN(PI*(TSUN-NONA)/180):REM * GHATIKAS * PRINT "EQN OF APP CONJN.:";:EQAC;"Gh.";
810 820
HEQAC=2*EQAC/5:IF HEQAC<0 THEN PRINT TAB(34)"-"; H=ABS(HEQAC):GOSUB 1080
830 840 850
(I+1)=TO+TCON+HEQAC IF I>6 OR ABS(T(I)-T(I-1))<.01 THEN PRINT TAB(5)"INSTANT OF APPCONJN.:";:H=T(I):GOSUB 1080:GOTO 860 NEXT I
860
0BS=.5*(SDIA+MDIA)-ABS(MLAT)
870 880 890 900
PRINT " M A X . OBSCURATION:";:M=OBS:GOSUB 1960 IF OBS<0 THEN PRINT TAB(30)"ECLIPSE NOT VISIBLE":END IF OBS=SDIA THEN PRINT TAB(31)"ECLIPSE IS TOTAL"
910
PRINT TAB(20)"MAGNITUDE OF THE ECLIPSE: ";OBS/SDIA
920 930 940
LOCATE 23.60:PRINT "":A$=INPUT$(1) GOTO 1120 NEXTN
950 960
LOCATE 24,60:PRINT "":A$=INPUT$(l) PRINT
970
PRINT "********** SECOND H A L F OF THE ECLIPSE ****•**•**"
980
PRINT
990
N=1:PRINT
1000 10)0 1020 1030 1040
1050 1060
TSUN=TSUN+360
"SECOND
HALF-DURATION • (APPR0XN.1):";:H=HDUR1:
GOSUB 1080 PRINT IF N=l THEN GOTO 1300 FOR N=2 TO 10 HDUR(N)=24*SQR(.25*(SDIA+MDIA)'^2-APLAT'^2)/(MDM-SDM) IF N>1 A N D ABS(HDUR(N)-HDUR(N-1 ))<.0002 THEN PRINT :PRINT "SECOND HALF-DURATION OF THE ECL.: ";:H=HDUR(N):GOSUB 1080:GOTO 1400 PRINT PRINT "SECOND HALF-DURATION (APPROXN.";N; ")";:H=HDUR(N): GOSUB 1080
240
Ancient Indian Astronomy
1070
PRINT
1080
R E M ** CONVERSION INTO HRS. MIN. SEC **
1090 1100 1110
HRS=INT(H):M=60*(H-HRS):MIN=INT(M):SEC=INT((M-MIN)*60) PRINT TAB(35) HRS; "H-";MIN; "M-";SEC; "S" RETURN
1120 DBET=SIN(4.5*PI/810)*COS((OPMOON-OPNODE)*PI/180)*829.91833# 1130 A=(MDM-SDM)^2+DBET'^2 1140 B=2*MLAT*DBET 1150 D E L l=(SDIA+MDIA)/2 :DEL2=(SDIA-MDIA)/2 1160 C l = M L A T ^ 2 - D E L r 2 1170 C2=MLAT'^2-DEL2'^2 1180 1190 1200
DESCR1=B'^2-4*A*C1 DESCR2=B'^2-4*A*C2 PRINT
1210
R E M *** HRST-HALF DURATION OF THE ECLIPSE ***
1220
T1=24*ABS((-B-SQR(DESCR1 ))/(2*A))
1230
PRINT "FIRST-HALF DURATION OF THE ECLISPE: ";:H=T1:G0SUB 1080
1240
R E M *** SECOND-HALF DURATION OF THE ECLIPSE ***
1250
T4=24*(-B+SQR(DISCR1 ))/(2*A)
1260 1270 1280
PRINT PRINT "SECOND-HALF DURATION OF THE ECLISPE: ";:H=T4:GOSUB 1080 IF ZZ=1 THEN GOTO 1370
1290
PRINT "HRST-HALF DURATION OF TOTALITY: •**"
1300
T2=24*ABS((-B-SQR(DISCR2))/(2*A))
1310
PRINT
n20
PRINT "HRST-HALF OF TOTALITY: ";:H=T2:G0SUB 1080
1330 R E M *** SECOND-HALF DURATION OF TOTALITY *** 1340
T3=24*(-B+SQR(DISCR2))/(2*A)
1350
PRINT
1360
PRINT "SECOND-HALF OF TOTALITY: ";:H=T3:GOSUB 1080
1370
PRINT
1380
PRINT TAB(5) "***********************************************"
1390
PRINT TAB(5) "**
1400
PRINT TAB(5) "***********************************************"
1410
PRINT
1420
PRINT TAB(32)" INDIAN STANDARD TIME"
1430
PRINT TAB(5)"BEGINNING OF THE ECLISPE: ";:H=T(I)-Tl:GOSUB 1080
S U M M A R Y OF THE ECLIPSE**"
Computer Programs
241
1440
IF ZZ=1 THEN GOTO 1460
1450
PRINT TAB(5)"BEGINNING OF M A X OBSCURITY: ";:H=T(I)-T2:G0SUB 1080
1460 1470
PRINT TAB(5)"MIDDLE OF THE ECLIPSE:";:H=T(I):GOSUB 1080 IF ZZ=1 THEN GOTO 1490
1480
PRINT TAB(5)"END OF M A X 0BSCURITY:";:H=T(I)+T3:G0SUB
1490 1500 1510
PRINT TAB(5)"END OF THE ECLIPSE: ";:H=T(I)+T4:G0SUB 1080 PRINT TAB(5)"************************************************" END
1520
R E M *** ORIENT ECLIPTIC POINT ( SAYANA L A G N A ) ***
1080
1530
PRINT "T(I)=";T(I)
1540
T=T(I)-(LAM-82.5)/15:REM * CONVERSION F R O M L M T TO 1ST *
1550
G1 =6.63627+6.570982*.01 *(JJ-2443144!)
1560
TS=Gl-INT(Gl/24)*24
1570
IF LD>=0 THEN L=LD+(LM/60)
1580 1590 1600
IF LD<0 THEN L=LD.(LM/60) S=L/15+TS+(T-5.5)/1436*4 IF S>24 THEN S=S-24
1610
IF S<0 THEN S=S+24
1620
ST=INT(S*100)/100
1630 1640
T=T-5.5 H=ST+T:IF PD<0 THEN H=H+12
1650 1660
IF H>24 THEN H=H-24 IF H<0 THEN H=H+24
1670 1680
GOSUB 1680:GOTO 1700 MIN=(H-lNT(H))*60:TrY=INT((MIN-INT(MIN))*60)
1690 1700
RETURN IF PD+PM/60>0 THEN P=PD+PM/60
1710 1720
IF PD+PM/60<0 THEN P=PD-PM/60 P=P*PI/180
1730
S=H:S=S*15
1740
A=S+90:A=A*PI/180:W=23.45*PI/180
1750
GOSUB 1850
1760 1770 1780
S=H:S=S*15 B=ATN(TAN(A)*COS(W)) T=ATN(COS(A)*TAN(W))
1790 1800
E1=ATN(SIN(A)*SIN(W)*TAN(ABS(P+T))) L=(B+E1)*180/PI
242
Ancient Indian Astronomy
1810
IF PD<0 THEN L=180+L
1820
IF L<0 THEN L=L+360
1830 1840
IF S<180 THEN l^L+180 TLAG=L
1850
IF PD<0 THEN S=S-180:IF S<0 THEN S=S+360
1860 1870
TRS=S*PI/180 IF S=90 THEN MC=90 :GOTO 1910
1880 1890
IF S=180 THEN MC=180 :GOTO 1910 IF S=270 THEN MC=270 :GOTO 1910
1900 1910
A2=(l 80/PI)*ATN(TAN(TRS)/COS(W)) IF S>90 A N D S<180 THEN MC=180+A2
1920 1930
IF S>180 A N D S<270 THEN MC=l80+S2 IF S<90 OR S>270 THEN MC=A2
1940 1950
IF MC<0 THEN MC=MC+360 RETURN
1960 1980
IF M<0 THEN M=ABS(M) MIN=INT(M):SEC=INT((M-MIN)*60+.5)
1990
IF SEC=60 THEN SEC=0:MIN=MIN+1
2000 PRINT TAB(35);MIN;""';SEC;""" 2010 RETURN 2020 IF L<0 THEN L=ABS(L) 2040 DEG=INT(L):MIN=(L-DEG)*60:SEC=INT((MIN-INT(MIN))*60+.5) 2050 IF DEG>=360 THEN DEG=DEG-360 2060 IF SEC=60 THEN SEC=0:MIN=MIN+1:IF MIN=60 THEN MIN=0:DEG =DEG+1:IF DEG>=360 THEN DEG=DEG-360 2070 PRINT TAB(35);DEG;"* ";INT(MIN);""';SEC;""" 2080 RETURN
Computer Programs
243
10 CLS:REM ** PROGRAM 12.1 : "SSPLA" 20 PRINT TAB(23)"***********************************************" 30 PRINT TAB(23)"* PLANETS'POSITIONS ACCORDING TO *" 40 PRINT TAB(23)"'^ SURYASIDDHANTA 50 PRINT TAB(23) "*************************•********************" 60 PI=3.141592653589793# 70 PRINT " " 80 PRINT "•*************•*****•**** KUJA SPHUTA*******""****'****'''*" 90 100 110 120 130 140 150 160 170 180 190 200
REM ** NO. OF REVNS. 2 296 832 REVOLNS IN 1 577 917 828 CIVIL DAYS ** DAILY = 1.455609385509776D-03:REM ** R E V O L N MKUJA=KALI*DAILY.REM ** R E V O L N REV=INT(MKUJA):PRINT TAB( 15)"REV0LNS SINCE K A L I EPOCH: ";REV MKUJA=360*(MKUJA-REV) PRINT " M E A N KUJA AT UJJAYINI MIDNIGHT: "; L=MKUJA:GOSUB 1380 R E M *** DAILY MOTION: 3r26" = 0.523888888 D E G *** DAILY=360*DAILY:GOSUB 1290 KAALA=(GH+VIG/60)*DAILY/60 PRINT "MOTION FOR ";GH;" G H ";VIG;" VIG:"; L=KAALA:GOSUB 1380
210
PRINT " M E A N KUJA AT GIVEN TIME AT ";PLACE$
220 230
MKUJA=MKUJA+DESH+KAALA:L=MKUJA:GOSUB 1380 K3=235/(2*PI):K4=3/(2*PI):MPLANET=MKUJA:SHIGHROCCA=MRAVI
240 250 260 270 280
PMA=(129.96/360)+(KALI*204/(CIVIL*1000)):REM ** KUJA'S MANDOCCA ** PMA=360*(PMA-INT(PMA)) P M A 1 =PMA-MKUJA:IF P M A 1 <0 THEN P M A 1 =PMA 1+360 P$="KUJA";K1=75/(2*PI):K2=3/(2*PI) GOSUB 1600
290 300
LOCATE 23,60:PRINT "PRESS A N Y K E Y TWICE" A$=INPUT$(2)
310
PRINT "********************BUDHA SPHUTA*********************"
320 330 340 350
R E M ** 17 937 060 REVNS IN 1 577 917 828 CIVIL DAYS ** DAILY=1.136755012314874D-02:REM * R E V N * MBUDHA=KALI*DAILY :REM * R E V N * REV=INT(MBUDHA):PRINT TAB(15)"REV0LNS SINCE K A L I ";REV
360 370
MBUDHA=360*(MBDUHA-REV) PRINT "BUDHASHIGHRA AT UJJAYINI MIDNIGHT : ";
EPOCH:
244
Ancient Indian Astrononxy
380
L=MBUDHA:GOSUB 1380
390 400
R E M *** BUDHA'S DAILY MOTION: 245'32" = 4.092222222 DEC *** DAILY=360'*DAILY:GOSUB 1290:REM * DEG *
410 420 430
KAALA=(GH+VIG/60)*DAILY/60 PRINT "MOTION FOR" ;GH;" G H ";VIG;" VIG:"; L=KAALA:GOSUB 1380
440 450
PRINT "SHIGHROCCA AT GIVEN TIME A r ' ; P L A C E $ ; MBUDHA=MBUDHA+DESH+KAALA:L=MBUDHA:GOSUB 1380
460
PRINT " M E A N B U D H A (i.e. RAVI):";:L=MRAVI:GOSUB 1380
470 480
K3=133/(2*PI):K4=1/*2*PI) SHIGHROCCA=MBUDHA:MPLANET=MRAVI
490 500 510 520
PMA=(220.32/360)+(KAU*368/(CIVIL*1000)):REM ** B U D l U ' S MANDOCCA ** P M A = 360*(PMA-INT(PMA)) P M A 1 =PMA-MPLANET:IF P M A 1 <0 THEN P M A 1 =PMA 1 +360 P$="BUDHA":K1=30/(2*PI):K2=2/(2*PI)
530 540
GOSUB 1600 LOCATE 23,60:PRINT "PRESS A N Y K E Y TWICE"
550 560 570
A$=INPUT$(2) PRINT " " PRINT "**********************GURU SPHUTA*""*"'****''"'"'"'"'"''*''"''**"
580
R E M ** 364 220 REVOLNS IN I 577 917 828 CIVIL DAYS **
590 600 610 620
DAILY=2.308231731316746D-04:REM * R E V N * MGURU=KALI*DAILY:REM * REVNS * REV=INT(MGURU):PRINT TAB( 15)"REVOLNS SINCE K A L I EPOCH ";REV MGURU=360*(MGURU-REV)
630
PRINT " M E A N G U R U AT UJJAYINI MIDNIGHT : ";
640 650
L=MGURU:GOSUB 1380 R E M *** GURU'S DAILY MOTION: 4'59" = 0.083055555 DEG ***
660
DAILY=360*DAILY:GOSUB
670
1290
KAALA=(GH+VIG/60)*DAILY/60
680
PRINT "MOTION FOR ";GH;" ;GH ";VIG;" VIG:";
690
L=KAALA:GOSUB 1380
700 710 720
PRINT " M E A N G U R U AT GIVEN TIME AT ";PLACE$; MGURU=MGURU+DESH+KAALA:L=MGURU:GOSUB 1380 K3=70/(2*PI):K4=-2/(2*PI):MPLANET=MGURU:SHIGHROCCA=MRAVI
730
PMA=(171/360)+(KALI*900/(CIVIL*1000)):REM ** GURU'S MANDOCCA **
740 750
P M A = 360*(PMA-INT(PMA)) PMA1=PMA-MGURU:IF P M A K O THEN
PMAl=PMAl+360
Computer Programs
245
760 770
P$="GURU":K1=33/(2*PI):K2=1/(2*PI) GOSUB 1600
780 790 800
LOCATE 23,60:PRINT "PRESS A N Y K E Y TWICE" A$=INPUT$(2) PRINT"
810 820
PRINT "**********************SHUKRA SPHUTA*******************" R E M ** 7 022 376 REVNS IN 1 577 917 828 CIVIL DAYS **
830
DAILY=4.450406653241768D-03:REM * R E V N *
840
MSHUKRA=KALI*DAILY:REM * REVNS *
850 860
REV=INT(MSHUKRA):PRINT TAB(15)"REV0LNS SINCE KALI EPOCH ";REV MSHUKRA=360*(MSHUKRA-REV)
870
PRINT " M S H U K R A SHIGHRA AT UJJAYINI MIDNIGHT : ";
880
L=MSHUKRA:GOSUB 1380
890 900
R E M *** SHUKRA'S DAILY MOTION: 96'7"43'" 37.3"" *** DAILY=360*DAILY:GOSUB 1290:REM * DEG *
910 920
KAALA=(GH+VIG/60)*DAILY/60 PRINT "MOTION FOR ";GH;" ;GH ";VIG:" VIG:";
930
L=KAALA:GOSUB 1380
940
PRINT "SHIGHROCCA AT GIVEN TIME AT ";PLACE$;
950
MSHUKRA=MSHUKRA+DESH+KAALA:L=MSHURA:GOSUB 1380
960
PRINT " M E A N S H U K R A (i.e.RAVI):";:L=MRAVI:GOSUB
970 980
K3=262/(2*PI):K4=2/(2*PI) SHIGHROCCA=MSHUKRA:MPLANET=MRAVI
990
PMA=(79.65/360)+{KAU*535/(CIVIL*1000)):REM ** SHUKRA'S MANDOCCA **
"
1380
1000
PMA =
1010
PMA1=PMA-MPLANET:IF PMA1<0 THEN
1020 1030
P$="SHUKRA":K1=12/(2*PI):K2=1/(2*PI) GOSUB 1600
1040 1050 1060
LOCATE 23,60:PRINT "PRESS A N Y K E Y TWICE" A$=INPUT$(2) PRINT "**********************SHANI SPHUTA ******************"
1070 1080
R E M ** 146 568 REVNS IN 1577197828 CIVIL DAYS ** DAILY=9.28869662280031D-05:REM * R E V N *
1090 1100
MSHANI=KALI*DAILY:REM * REVNS * REV=INT(MSHANI):PRINT TAB( 15)"REV0LNS SINCE K A L I EPOCH ";REV
1110
360*(PMA-INT(PMA)) PMAl=PMAl+360
MSHANI=360*(MSHANI-REV)
1120
PRINT " M E A N SHANI AT UJJAYINI MIDNIGHT : ";
1130
L=MSHANI:GOSUB
1380
246
Ancient Indian Astronomy
1140
R E M ** S H A M ' S DAILY MOTION: 2'0"22"' 53.4""**
1150
DAILY=360*DAILY:GOSUB 1290:REM * DEG *
1160
KAALA=(GH+VIG/60)*DAILY/60
1170 1180
PRINT "MOTION FOR ";GH;" ;GH ";VIG;" VIG:"; L=KAALA:GOSUB 1380
1190
PRINT " M E A N SHANI AT GIVEN TIME AT ";PLACE$;
1200
MSHANI=MSHANI+DESH+KAALA:L=MSHANI:GOSUB 1380
1210
K3=40/(2*PI):K4=1/(2*PI):MPLANET=MSHANI:SHIGHROCCA=MRAVI
1220
PMA=(236.61/360)+{KALI*39/(CIVIL* 1000)):REM ** SHANI'S MANDOCCA **
1230
P M A = 360*(PMA-INT(PMA))
1240
PMA1=PMA-MSHANI:IF P M A K O THEN
1250 1260 1270 1280
P$="SHANI":K 1 =49/(2*PI):K2= 1/(2*PI) GOSUB 1600 LOCATE 23,60:PRINT " END OF THE PROGRAM " END
1290
R E M *** DESHANTARA CORRECTION ***
1300
DESH=ULAM*DAILY/360
1310 1320
PRINT "DESHANTARA CORRECTION: "; IF DAILY<0 A N D ULAM>0 THEN PRINT TAB(39) "-";:GOTO 1360
1330 1340
IF DAILY<0 A N D ULAM<0 THEN PRINT TAB(39) "+";:GOTO 1360 IF ULAM>0 THEN PRINT TAB(39) "+";
1350 1360
IF ULAM<0 THEN PRINT TAB(39) "-"; L=DESH:GOSUB 1380
1370
RETURN
1380
IF L<0 THEN L=ABS(L)
1390
DEG=INT(L):MIN=(L-DEG)*60:SEC=INT((MIN-INT(MIN))*60+.5)
1400 1410
IF DEG>=360 THEN DEG=DEG-360 IF SEC=60 THEN SEC=0;MIN=MIN+1:IF MIN=60 THEN MIN=0:DEG=DEG +1:IF DEG>=360 THEN DEG=DEG-360 PRINT TAB(40);DEG;"° ";INT(MIN);""';SEC;""" RETURN R E M *** SHIGHRA EQNS. *** IF P$<>"BUDHA" A N D P $ o " S H U K R A " AND Z=0 THEN PRINT "SHIGHROCCA: ";:L=SHIGHROCCA:GOSUB 1380
1420 1430 1440 1450 1460
PRINT "SHIGHRA ANOMALY: ";
1470
L=PMK:GOSUB 1380
1480
PI=3.14159256#
1490
PMK=PMK*PI/180:REM * RADIAN *
PMAl=PMA-l+360
Computer Programs
247
1500
SN=SIN(PMK):CS=COS(PMK)
1510
K=(PI/180)*(K3-K4*ABS(SN)):REM ** COS.SHIGHRA RADIUS IN RADIANS **
1520
DPL=K*SN:REM ** DOHPHALA **
1530
SKR=SQR(K*K+2*K*CS+1):REM ** SHIGHRAKARNA **
1540
SE=(180/PI)*ATN(ABS(DPUSQR(SKR*SKR-DPL*DPL))):REM ** SHIGHRA EQN **
1550
IF PMK>PI THEN SE=-SE
1560
PRINT "SHIGHRA EQN: ";:IF PMK
1570
IF PMK>PI THEN PRINT TAB(39) "-";
1580
L=SE :GOSUB 1380
1590
RETURN
1600
R E M *** M A N D A and SHIGHRA CORRECTIONS ***
1610
P M K = SHIGHROGCA-MPLANETIF PMK<0 THEN PMK=PMK+360:REM **
1620 1630 1640 1650 1660
SHIGHRA A N O M A L Y ** Z=0:GOSUB 1440 PI = MPLANET+SE/2 :REM *** PLANETS LONG. AFTER 1st OPERATION *** PRINT "LONG, after 1st COR.( SE/2)"; L=P1:G0SUB 1380 PMA1=PMA-P1 : R E M ** PLANET'S M A N D A A N O M A L Y AFTER 1st OPERN. **
1670
IF PMA1<0 THEN PMAl=PMAl+360
1680
GOSUB 1900
^
1690
PRINT "LONG.after 2nd C0R.(ME/2):";
1700
P2 = PI + PEQ/2 : R E M ** PLANET'S LONG. AFTER 2nd OPERATION **
1710
L=P2:G0SUB 1380
1720
PMA2=PMA-P2 :IF PMA2<0 THEN PMA2=PMA2+360
1730
PRINT " COR. ";:Z=Z+1
1740
PMA1=PMA2:G0SUB 1900:REM ** PALNET'S EQN.OF CENTRE **
,
1750
PRINT "LONCafter 3rd COR.(ME):";
,
1760
P3=MPLANET+PEQ:IF P3>360 THEN P3=P3-360
'
1770
IF P3
|
1780
L=P3:G0SUB 1380
1790
P M K = SHIGHROCCA - P3 :IF PMK<0 THEN PMK=PMK+360
1800
PRINT "COR.";
1810
GOSUB 1440
1820
PRINT "LONG.after 4th COR.(SE):";
1830
PTL=P3+SE
1840
L=PTL:GOSUB 1380
[ f
248
Ancient Indian Astronomy
1850
PRINT"
"
1860
PRINT P$; "'S TRUE LONGITUDE:";
1870
L=PTL.GOSUB 1380
1880
PRINT"
,
1890
RETURN
1900
R E M **** EQUATION OF CENTRE **•
1910
IF Z=0 THEN PRINT "MANDOCCA:";:L=PMA:GOSUB 1380
1920
PRINT " M A N D A ANOMALY:";
1930
L=PMA1:G0SUB 1380
"
1940
PRINT " M A N D A EQUATION :";
1950
PMA1=PMA1*PI/180 :REM ** RADIANS **
1960
SN=SIN(PMA1)
1970
PEQ = (K1-K2*ABS(SN))*SN :REM ** PLANETS'S EQN.OF CENTRE **
1980
IF P M A K P I THEN PRINT TAB(39) "+";
1990
IF PMA1>PI THEN PRINT TAB(39) "-";
2000 L=PEQ :GOSUB 1380 2010
RETURN
Computer Programs
249
10 C L S . K E Y OFF: R E M ** PROGRAM 12.2 : " K K P L A " 20 PRINT TAB(23);"*************************************************" 30 40 50
PRINT TAB(23);"* PLANETS'POSITIONS ACCORDING TO *" PRINT TAB(23);"* KHANDA - KHAANDYAKA PRINT TAB(23); "*************************************************"
60
PRINT
70
PRINT "**********************KUJA SPHUTA *******************"
80
MKUJA=(AHG-496+.25)/687+AHG/(174259!*60*360)
90 MKUJA=360"*(MKUJA-INT(MKUJA)) 100
PRINT
110
PRINT " M E A N KUJA AT UJJAYINI MIDNIGHT ";
120
L=MKUJA:GOSUB 1180
130
R E M *** DAILY MOTION: 31'26" = 0.523888888 D E G ***
140
DAILY= 0.523888888#:GOSUB 1110
150
PRINT " M E A N KUJA AT L O C A L MIDNIGHT: ";
160
MKUJA=MKUJA+DESH:L=MKUJA:GOSUB 1180
170
PRINT "MOTION F0R;H1;"HRS.";MI;"MIN.:";TAB(34);"+";
180
K A A L A = (HI +MI/60)*DAILY/24
190
L=KAALA:GOSUB 1180
200
PRINT " M E A N KUJA AT GIVEN TIME: ";
210
MKUJA=MKUJA+KAALA:L=MKUJA:GOSUB 1180
220 EM=360:MP=234:MPLANET=MKUJA:SHIGHROCCA=MRAVI 230
M A N D O C C A = 110 :P$="KUJA";K=5:REM ** K:MULTIPLIER OF SUN'S EQN.OF CENTRE
240
GOSUB 1420
250
IF PMK>164 A N D PMK<196 THEN PRINT TAB(60) "RETROGRADE"
260
LOCATE 23,55:PRINT "PRESS A N Y K E Y TWICE"
270
A$=INPUT$(2)
280
CLS:PRINT "****************** B U D H A SPHUTA *****************"
290
MBUDHA=(AHG* 100-2181 )/8797+AHG/(71404!*60*360)
300 MBUDHA=360*(MBUDHA-INT(MBUDHA)) 310
PRINT
320
PRINT " B U D H A SIGHROCCA (UJJAYINI MIDNIGHT):";
330
L=MBUDHA:GOSUB 1180
340
R E M *** BUDHA'S DAILY MOTION: 245'32" = 4.092222222 DEG ***
350
DAILY=4.092222222#:GOSUB
360
PRINT "BUDHA SIGHROCCA (LOCAL MIDNIGHT):";
370
MBUDHA=MBUDHA+DESH:L=MBUDHA:GOSUB 1180
1110
250
Ancient Indian Astronomy
380 PRINT "MOTION F0R";H1;"HRS"MI;
"MIN.:";TAB(34);"+";
390 K A A L A = (Hl+MI/60)*DAILY/24 : L = K A A L A : GOSUB 1180 400 M B U D H A = M B U D H A + K A A L A 410 PRINT " B U D H A SIGHRA AT GIVEN TIME:"; 420 L=MBUDHA:GOSUB 1180 430 PRINT " M E A N RAVI:";:L=MRAVI:GOSUB 1180 440 ES=360:SM=132 450
EM=ES:MP=SM:SHIGHROCCA=MBUDHA:MPLANET=MRAVI
460 MANDOCCA = 220 J^"BUDHA":K=2:REM ** K M U L U P L E R OF SUN'S EQN. ** 470 GOSUB 1420 480 IF PMK>146 A N D PMK<214 THEN PRINT TAB(60) "RETROGRADE" 490 LOCATE 23,55:PRINT "PRESS A N Y K E Y TWICE" 500 A$=INPUT$(2) 510 CLS:PRINT "*****•******•***•* G U R U SPHUTA *****************" 520 MGURU=(AHG-2113+.2)/4332-AHG/( 162621 !*360) 530
MGURU=360*(MGURU-INT(MGURU))
540 PRINT " M E A N G U R U AT UJJAYINI MIDNIGHT :"; 550 L=MGURU:GOSUB 1180 560 R E M *** GURU'S DAILY MOTION: 4'59" = 0.083055555 DEG *** 570 580 590 600
DAILY=0.083055555#:GOSUB 1110 PRINT " M E A N G U R U AT L O C A L MIDNIGHT: "; MGURU=MGURU+DESH:L=MGURU:GOSUB 1180 PRINT "MOTION F0R";H1;"HRS";MI; "MIN.:";TAB(34);"+";
610 K A A L A = (H1 +MI/60)*DAILY/24:L=KAALA:GOSUB 1180 620 PRINT " M E A N G U R U AT GIVEN TIME:"; 630 MGURU=MGURU+KAALA:L=MGURU:GOSUB 1180 640 EM=360:MP=72.5:MPLANET=MGURU:SHIGHROCCA=MRAVI 650 M A N D O C C A = 160 :P$="GURU":K=2*(1+1/7):REM ** K:MULTIPLIER OF SUN'S EQN. ** 660 GOSUB 1420 670 IF PMK>125 A N D PMK<235 THEN PRINT TAB(60) "RETROGRADE" 680 LOCATE 22,55:PRINT "PRESS A N Y K E Y TWICE" 690 A$=INPUT$(2) 700 CLS:PRINT "****************** S H U K R A SPHUTA ****************" 710 MSHUKRA=(AHG-37.25)* 10/2247+(AHG-712)/(77043!*360) 720
MSHUKRA=360*(MSHUKRA-INT(MSHUKRA))
730
PRINT
740 PRINT "SHUKRA SIGHRA (UJJAYINI MIDNIGHT):";
Corrq>uter Programs
251
750
L=MSHUKRA:GOSUB 1180
760
R E M *** SHUKRA'S DAILY MOTION: 96'7"= 1.601944444 DEG ***
770
DAILY=1.601944444#:GOSUB 1110
780
PRINT "SHUKRASIGHRA (LOCAL MIDNIGHT):";
790
MSHUKRA=MSHUKRA+DESH:L=MSHUKRA:GOSUB 1180
800
PRINT " M E A N RAVI:";:L=MRAVI:GOSUB 1180
810
PRINT "MOTION
F0R";H1;"HRS.";MI;"MIN.:";TAB(34);"+";
820
K A A L A = (Hl+My60)*DAILY/24:L = K A A L A : G O S U B 1180
830
PRINT "SHUKRA SIGHROCCA AT GIVEN TIME:";
840
MSHUKRA=MSHUKRA+KAALA:L=MSHUKRA:GOSUB 1180
850
ES=360:SM=260
860
EM=ES :MP=SM:SHIGHROCCA=MSHUKRA:MPLANET=MRAVI
870
MANDOCCA = 80 :P$="SHUKRA":K=1 .REM ** R M U L U P U E R OF SUN'S EQN. **
880
GOSUB 1420
890
IF PMK>167 A N D PMK<193 THEN PRINT TAB(60) "RETROGRADE"
900
LOCATE 23,55:PRINT "PRESS A N Y K E Y TWICE"
910
A$=INPUT$(2)
920
CLS:PRINT "****************** SHANI SPHUTA ****************"
930
MSHANI=(AHG-2491.5)/l0766-AHG/80450!*60*360)
940
MSHANI=360*(MSHANI-INT(MSHANI))
950
PRINT
960
PRINT " M E A N SHANI AT UJJAYINI MIDNIGHT:";
970
L=MSHANI:GOSUB
980
R E M *** SHANI'S DAILY MOTION: 2'00" = 0.033333333 DEG ***
990
DAILY=0.033333333#:GOSUB 1110
1180
1000
PRINT " M E A N SHANI AT L O C A L MIDNIGHT: ";
1010
MSHANI=MSHANI+DESH:L=MSHANI:GOSUB
1020
PRINT "MOTION
1030
K A A L A = (H1+MI/60)*DAILY/24:L = K A A L A : G O S U B 1180
1040
PRINT " M E A N SHANI AT GIVEN TIME:";
1050
MSHANI=MSHANI+KAALA:L=MSHANI:GOSUB
1060
1180
F0R";H1;"HRS.";MI;"MIN.:";TAB(34);"+";
1180
EM=360:MP=40:MPLANET=MSHANI:SHIGHROCCA=MRAVI
1070
M A N D O C C A = 240:P$="SHANI":K=4*( 1+1/4)
1080
GOSUB 1420
1090
IF PMK>114 A N D PMK<246 THEN PRINT TAB(60) "RETROGRADE"
1100
END
1110
R E M *** DESHANTARA CORRECTION ***
1120
DESH=ULAM*DAILY/360
252
Ancient Indian Astronomy
1130
PRINT "DESHANTARA CORRECTION: ";
1140
IF DESH<0 THEN PRINT TAB(34) "-"';
1150
IF DESH>0 THEN PRINT TAB(34) "+";
1160 1170
L=DESH:GOSUB 1180 RETURN
1180 1190
IF L<0 THEN L=ABS(L) DEG=INT(L):MIN=(L-DEG)*60:SEC=INT((MIN-INT(MIN))*60+.5)
1200
IF DEG>=360 THEN DEG=DEG.360
1210
IF SEC=60 THEN SEC=0:MIN=MIN+1:IF MIN=60 T H E N MIN=0:DEG=DEG +1:IF DEG>=360 THEN DEG=DEG-360
1220
PRINT TAB(35);DEG;"° ";TAB(42);INT(MIN);""';TAB(47);SEC;"""
1230
RETURN
1240
IF M<0 THEN M=ABS(M)
1260
MIN=INT(M):SEC=INT((M-MIN)*60+.5)
1270
IF SEC=60 THEN SEC=0:MIN=MIN+1
1280
PRINT TAB(35);MIN;"' ";SEC; " " "
1290
RETURN
1300
R E M *** SHIGHRA EQNS. ***
1310
PI=3.14159256#
1320
BHUJ=PMK
1330
IF P M K > 180 THEN BHUJ = 360-PMK
1340
BHUJ = BHUJ * PV.m
1350
E = BHUJ/2 - ATN (MULT * TAN(BHUJ/2))
1360
E=E*180/PI
1370
IFPMK>180THENE=-E
1380
PRINT "SHIGHRA EQN.: ••;:IF E<0 THEN PRINT TAB(34) "-";
1390
IF E>0 THEN PRINT TAB(34) "+";
1400
L=E :GOSUB 1180
1410
RETURN
1420
R E M *** M A N D A and SHIGHRA CORRECTIONS ***
1430
P M K = SHIGHROCCA - M P L A N E T :IF PMK<0 THEN PMK=PMK+360
1440
MULT=(EM-MP)/(EM+MP):GOSUB 1300
1450
M P L A N E T l = MPLANET+E/2 : R E M *** M E A N PLANET AFTER 1st OPERATION *** PRINT " M E A N L O N G . A F I E R 1st OPERN.(Pl):"; L=MPLANET1:G0SUB 1180 PMA1=MPLANET1-MANDOCCA: R E M ** PLANET'S M E A N A N O M A L Y AFTER 1st OPERN.**
1460 1470 1480
Computer Programs 1490 1500 1510 1520 1530
253
PRINT "EQUATION OF CENTRE:"; PEQ= -K*134*(SIN(PMAl*PI/180)y60 :REM ** PALNET'S EQN.OF CENTRE ** IF PEQ < 0 THEN PRINT TAB(34) IF PEQ > 0 THEN PRINT TAB(34) "+"; L=PEQ :GOUSB 1180
1540 1550
PRINT " M E A N LONG.AFTER 2nd 0PERN.(P2):"; MPLANET2 = M P L A N E T l + PEQ/2 : R E M ** M E A N P L A N E T AFTER 2nd OPERATION ** 1560 L=MPLANET2:G0SUB 1180 1570 PMA2=MPLANET2-MANDOCCA 1580 PRINT "EQN.OF CENTRE AFTER 2nd OPERN:"; 1590 PEQ = - K*I34*(SIN(PMA2-*Pyi80))/60 :REM ** PLANET'S EQN. OF CENTRE ** 1600 IF PEQ < 0 THEN PRINT TAB(34) "-"; 1610 IF PEQ > 0 THEN PRINT TAB(34) "+"; 1620 L=PEQ:GOSUB 1180 1630 PRINT "HELIOCENTRIC LONGITUDE (P3): "; 1640 PHL=MPLANET+PEQ:IF PHL>360 THEN PHL=PHL-360 1650 IF PHL <0 THEN PHL=PHL+360 1660 L=PHL:GOSUB 1180 1670 P M K = SHIGHROCCA - PHL :IF PMK<0 THEN PMK=PMK+360 1680 GOSUB 1300 1690 P R I N T " : " 1700 PRINT P$;"'S GEOCENTRIC L O N G (P4):"; 1710 PGL=PHL+E 1720 L=PGL:GOSUB 1180 1730 P R I N T " " 1740
RETURN
254 10
Ancient Indian Astronomy C L S : K E Y O F F : R E M * P R O G R A M 12.3 : " G L P L A "
20
PI=3.141592653589793#
30
PRINT
40
P R I N T TAB(23);"**
50
P R I N T TAB(23);"*
60
PRINT
70
G O T O 280
80
R E M
90
DESH=ULAM'*DAILY/360
XAB(23);"*************************************************" P L A N E T S ' POSITIONS A C C O R D I N G T O
*"
GRAHA LAGHAVAM
*"
XAB(23);"***'*'***'******'****'*'**'*'**'*'*'*'*'*'*'**'*'*'**'**'*'*'**'*'**'**'''*
D E S H A N T A R A C O R R E C T I O N ***
100
P R I N T " D E S H A N T A R A C O R R E C T I O N : ";
110
I F D E S H < 0 T H E N P R I N T TAB(34) "-";
120
IF DESH>0 T H E N P R I N T TAB(34)
130
L = D E S H : G O S U B 170
140
RETURN
150
I F L>=360 T H E N L=L-360'*INT(U360)
"+";
160
IF L<0 T H E N L=L+360*ABS(INT(U360))
170
IF L<0 T H E N L=ABS(L)
180
DEG=INT(L):MIN=(L-DEG)"*60:SEC=INT((MIN-INT(MIN))*60+.5)
190
IF DEG>=360 T H E N
200
IF SEC=60
THEN
DEG=DEG-360 SEC=0:MIN=MIN+1:IF
DEG+1:IF DEG>=360 T H E N
MIN=60
THEN
MIN=0:DEG=
DEG=DEG-360
210
P R I N T TAB(35);DEG;"°
";TAB(42)INT(MIN);""';TAB(47)SEC;"""
220
RETURN
230
IF M<0 T H E N M=ABS(M)
240
MIN=INT(M):SEC=INT((M-MIN)'*60+.5)
250
IF SEC=60 T H E N SEC=0:MIN=MIN+1
260
P R I N T TAB(35);MIN; "' ";SEC;"""
270
RETURN
280
PRINT :PRINT "*************I«KUJA SPHUTA
290
D H R = 5 5 . 5 3 3 3 3 3 3 3 # : K S P = 307.133333333333#
300
MKUJA=
310
PRINT " M E A N K U J A A T UJJAYINI SUNRISE:";
320
L = M K U J A : G O S U B 150:MKUJA=L
330
D A I L Y = . 5 2 3 8 8 8 9 : R E M * * D A I L Y = 3 1 ' 2 6 " **
340
G O S U B 80 : R E M * D E S H A N T A R A *
******************"
10*AHG/19-10*AHG/(73*60)-CHAKRA*DHR+KSP
350
PRINT " M E A N K U J A A T L O C A L SUNRISE:";
360
M K U J A = M K U J A + D E S H : L = M K U J A : G O S U B 150
370
PRINT " M O T I O N F0R";H1;"HRS.";MI;"MIN.:";
Computer Programs
255
380 390 400
K A A L A = (H1 +My60)*DAILY/24:IF KAALA<0 THEN PRINT TAB(34);"-"; L=KAALA:GOSUB 170 PRINT TAB(34);" "
410
PRINT " M E A N KUJA AT GIVEN TIME:";
420 430 440
MKUJA=MKUJA+KAALA:L=MKUJA: GOSUB 150 PRINT TAB(34);" " R E M •* TRUE KUJA ***
450 460
PRINT :PRINT "SIGHROCCA (MEAN RAVI) ;";:L=MRAVI:GOSUB 150 SK=MRAVI-MKUJA:IF SK<0 THEN SK=SK+360
470
L=SK:PRINT "SIGHRA KENDRA:";:GOSUB 150
480 490 500
IF SK<180 THEN SK1=360-SK ELSE SE1=SK L=SK1:PRINT "SIGHRA K E N D R A BHUJA:";:GOSUB 150 IF SK > 180 THEN SK1=360-SK ELSE SK1=SK
510 520
SKN=INT(SK1/15):REM ** SIGHRAANKA ** RESTORE 1120:FOR K=0 TO SNK+1
530 540
READ SPH:IF K=SNK THEN SPH1=SPH N E X T K:SPH2=SPH
550
PRINT "SNK=";SNK; "SPH1=";SPH1; "SPH2=;SPH2
560 570
SGP=(SPH2-SPH1)*(SK1-15*SNK)/15 SGRPH=(SPH1+SGP)/10:IFSK>180 THEN SGRPH=-SGRPH:PRINT TAB(34)"-";
580
PRINT "SIGHRAPHALA:";
590 600
IF SGRPH<0 THEN PRINT TAB(34);"-"; L=SGRPH:GOSUB 170
610 620
SGRPHl=SGRPH/2 PRINT "SIGHRAPHALAARDHA;";
630
IF SGRPH1<0 THEN PRINT TAB(34);"-";
640
L=SGRPH1:G0SUB 170
650 S1KUJA=MKUJA+SGRPH1 660 PRINT :PRINT TAB(34);"
"
670 680
PRINT "1st SIGHRA C0R.KUJA:";:L=S1KUJA:G0SUB 150 PRINT TAB(34);" "
690 700
LOCATE 22,67:PRINT "PRESS A N Y K E Y " A$=INPUT$(1)
710
R E M ** M A N D A SPASHTA KUJA **
720
CLSrPRINT "**** M A N D A SPASHTA KUJA ****"
730 740
MK=120 - S1KUJA:IF MK<0 THEN MK=MK+360 L=MK:PRINT .PRINT " M A N D A KENDRA:";:GOSUB 150
750
IF MK<90 THEN M K B = M K : R E M ** BHUJA OF M A N D A K E N D R A **
Ancient Indian Astronomy
256
760
IF MK>90 A N D MK<180 THEN MKB=180-MK
770 780
IF MK>180 A N D MK<270 THEN MKB=MK-180 IF MK>270 A N D MK<360 THEN MKB=360-MK
790
L=MKB:PRINT " M A N D A KENDRA BHUJA:";:GOSUB 150
800
MNK=INT(MKB/15)
810 820
RESTORE 1130 FOR K=0 TO MNK+1
830 840
READ M P H : IF K=MNK THEN MPH1=MPH N E X T K: MPH2=MPH
850
PRINT "MNK=";MNK;" MPH1=";MPH1;" MPH2=";MPH2
860 870 880
MNP=(MPH2-MPH 1 )*(MKB-15*MNK)/15 MNDP=(MPH1+MNP)/10:IF MK>180 THEN MNDP=-MNDP PRINT "MANDAPHALA:";:IF MNDP<0 THEN PRINT TAB(34);"-";
890 900
L=MNDP.GOSUB 170 PRINT TAB(34);"
910 920
PRINT " M A N D A CORRECTED KUJA:"; MNKUJA=MKUJA+MNDP:L=MNKUJA:GOSUB 150
930 940
PRINT TAB(34);" " R E M *• SECOND SIGHRA CORRECTION **
950
PRINT " **** SECOND SIGHRA CORRECTION ****"
960 970 980 990 1000 1010 1020 1030 1040 1050 1060 1070
SK2=SK-MNDP PRINT "2nd SIGHRA KENDRA:";:L=SK2:GOSUB 150 PRINT " SIGHRA KENDRA BHUJA:";:L=SKB:GOSUB 150 IF SK2>180 THEN SKB=360-SK2 ELSE SKB=SK2:REM * SIGHRA KENDRA BHUJA * SNK2=INT(SKB/15):REM * SIGHRAANKA * RESTORE 1120 :FOR K=0 TO SNK2+1 READ SPH:IF K=SNK2 THEN SPH1=SPH NEXT K: SPH2=SPH PRINT "SNK=";SNK;" SPH1=";SPH1;" SPH2="SPH2 SGP=(SPH2-SPH 1 )*(SKB-15 *SNK2)/15 SGRPH=(SPH1+SGP)/10:IF SK2>180 THEN SGRPH=-SGRPH PRINT "2nd SIGHRAPHALA:";:IF SGRPH<0 THEN PRINT TAB(34);"-";
1080
L=SGRPH:GOSUB 170
1090 1100
PRINT TAB(34);" " PRINT "TRUE KUJA:";:L=MNKUJA+SGRPH:GOSUB 150
1110 1120
PRINT TAB(34);" " DATA 0,58,117,174,228,279.325,365,393,400,368,209,0
"
Computer Programs
257
1130 1140 1150
DATA 0,29,57,85,109,124,130 LOCATE 20,67:PRINT "PRESS A N Y K E Y " A$=INPUT$(1)
1160
CLS:PRINT "********************* BUDHA SPHUTA **************-
1170
DHR=123.45:KSP=269.55
1180 1190 1200 1210 1220 1230 1240 1250 1260 1270 1280 1290 1300 1310
SK=3*AHG+3*AHG/28-AHG/(38*60)-CHAKRA*DHR+KSP PRINT :PRINT "SIGHRA KENDRA AT UJJAYINI SUNRISE:"; L=SK:GOSUB 150 DAILY=3.106667:REM * DAILY MOTION 186*24" * GOSUB 80:REM * DESHAANTARA * PRINT "SIGHRA KENDRA AT GIVEN TIME:"; SK = SK + DESH : L = SK : GOSUB 150 PRINT "MOTION F0R",H1;"HRS.",MI;"MIN.:"; K A A L A = (Hl+My60)*DAILY/24:IF KAALA<0 THEN PRINT TAB(34);"-"; L=KAALA:GOSUB 170 PRINT TAB(34);" " PRINT "SIGHRA KENDRA AT GIVEN TIME:"; SK = SK + K A A L A : L = SK : GOSUB 150:SK=L PRINT TAB(34);" ;"
1320 1330
R E M ** BUDHA'S 1st SIGHRA CORRECTION ** IF SK > 180 THEN SK1=360-SK ELSE SK1=SK:PRINT "SK1=";SK1
1340
L=SK1:PRINT "SIGHRA KENDRA BHUJA:";:GOSUB 170
1350
SNK=INT(SK1/15):REM ** SIGHRAANKA **
1360 1370
RESTORE 1950:FOR K=0 TO SNK+1 READ SPH:IF K=SNK THEN SPH1=SPH
1380 1390
N E X T K:SPH2=SPH PRINT "SNK=";SNK;" SPH1=";SPH1;" SPH2=";SPH2
1400 1410 1420
SGP=(SPH2-SPH1)*(SKM5*SNK)/15 PRINT "SIGHRAPHALA:"; SGRPH=(SPH1+SGP)/10:IF SK>180 THEN SGRPH=-SGRPH:PRlNT TAB(34)"-";
1430
L=SGRPH:GOSUB 170
1440 1450
SGRPHl=SGRPH/2 PRINT "SIGHRAPHALAARDHA:";
1460
IF SGRPHUO THEN PRINT TAB(34);"-";
1470
L= SGRPHLGOSUB 170
1480 1490
S1BUDHA=MRAVI+SGRPH1 PRINT TAB(34);"
1500
PRINT "1st SIGHRA COR.BUDHA:";:L=SlBUDHA:GOSUB 150
"
258
Ancient Indian Astronomy
1510
PRINT TAB(34);"
1520
LOCATE 20,67:PRINT "PRESS A N Y K E Y "
1530
"
A$=INPUT$(1)
1540
R E M ** M A N D A SPASHTA B U D H A **
1550
CLS:PRINT " **** M A N D A SPASHTA BUDHA ****"
1560
MK=210 - SlBUDHArlF MK<0 THEN MK=MK+360
1570
L=MK:PRINT :PRINT " M A N D A KENDRA:";:GOSUB 150
1580
IF MK<90 THEN M K B = M K : R E M ** BHUJA OF M A N D A K E N D R A **
1590
IF MK>90 A N D MK<180 THEN MKB=180-MK
1600
IF MK>180 A N D MK<270 THEN MKB=MK-180
1610
IF MK>270 A N D MK<360 THEN MKB=360-MK
1620
L=MKB:PRINT " M A N D A KENDRA BHUJA:";:GOSUB 150
1630
MNK=INT(MKB/15)
1640
RESTORE 1960
1650
FOR K=0 TO MNK+1
1660
READ M P H : IF K=MNK THEN MPH1=MPH
1670
N E X T K: MPH2=MPH
1680
PRINT "MNK=";MNK;" M P H 1 =";MPH 1;" MPH2=";MPH2
1690
MNP=(MPH2-MPH 1 )*(MKB-15 *MNK)/15
1700
MNDP=(MPH1+MNP)/10:IF MK>180 THEN MNDP=-MNDP
1710
PRINT "MANDAPHALA:";:IF MNDP<0 THEN PRINT TAB(34);"-";
1720
L=MNDP:GOSUB 170
1730
PRINT TAB(34);"
1740
PRINT " M A N D A COR. BUDHA:";
1750
MNBUDHA=MRAVI+MNDP;L=MNBUDHA:GOSUB 150
1760
PRINT TAB( 34);"
1770
R E M ** SECOND SIGHRA CORRECTION **
1780
PRINT:PRINT " **** SECOND SIGHRA CORRECTION ****"
1790 1800 1810
"
"
1820
SK2=SK-MNDP PRINT:PRINT "2nd SIGHRA KENDRA:";:L=SK2:G0SUB 150 IF SK2>i 80 THEN SKB=360-SK2 ELSE SKB=SK2:REM * SIGHRA KENDRA BHUJA t PRINT " S I G H R A KENDRA BHUJA:";:L=SKB:GOSUB 150
1830
SNK2=INT(SKB/15):REM * SIGHRAANKA *
1840
RESTORE 1950 :FOR K=0 TO SNK2+1
1850
READ SPH:IF K=SNK2 THEN SPH1=SPH
1860
N E X T K: SPH2=SPH
1870
PRINT "SNK=";SNK; "SPH1=";SPH1; "SPH2="SPH2
Con^uter Programs
259
1880 SGP=(SPH2-SPH 1 )*(SKB-15 *SNK2)/15 1890 SGRPH=(SPH1+SGP)/10:IF SK2>180 THEN SGRPH=-SGRPH 1900 PRINT "2nd SIGHRAPHALA:";:IF SGRPH<0 THEN PRINT TAB(34);"-"; 1910 L=SGRPH:GOSUB 170 1920 PRINT TAB(34);"
"
1930 PRINT "TRUE BUDHA:";:L=MNBUDHA+SGRPH:GOSUB 150 1940 PRINT TAB(34);" " 1950 1960 1970 1980
DATA 0,41,81,117,150,178,199,212,212,195,155,89,0 DATA 0,12,21,28,33,35,36 LOCATE 20,67:PRINT "PRESS A N Y K E Y " A$=INPUT$(1)
1990 CLS:PRINT "******************* GURU SPHUTA ******************" 2000 DHR=26.3: KSP = 212.26667* 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100
M G U R U = AHG/12-AHG/(70*60)-CHAKRA*DHR+KSP PRINT :PRINT " M E A N G U R U AT UJJAIN SUNRISE:''?. L=MGURU:GOSUB 150 DAILY = 8.333335E-02 : R E M ** DAILY = 5'0" ** GOSUB 80 :REM * DESHANTARA * PRINT " M E A N G U R U AT L O C A L SUNRISE:"; M G U R U = M G U R U + DESH : L = M G U R U : GOSUB 150 K A A L A = (HI+My60)*DAILY/24 PRINT "MOTION F0R";H1;"HRS.";MI;"MIN.:"; L=KAALA:GOSUB 170
2110 P R I N T " 2120 PRINT " M E A N G U R U AT GIVEN TIME:"; 2130 M G U R U = M G U R U + K A A L A : L = M G U R U : GOSUB 150 2140 R E M ** 1st SIGHRA CORRECTION * 2150 PRINT :PRINr'*** 1st SIGHRA CORRECTION ***" 2160 L=MRAVI:PRINT :PRINT " M E A N RAVI:";:GOSUB 150 2170 L=MGURU:PRINT " M E A N GURU:";:GOSUB 150 2180 SK=MRAVI-MGURU:IF SK<0 THEN SK=SK+360 2190 L=SK:PRINT "SIGHRA KENDRA:";:GOSUB 150:SK=L 2200 IF SK >180 THEN SK1=360-SK ELSE SK1=SK 2210 PRINT "SIGHRA K E N D R A BHUJA:";.L=SK1:G0SUB 150 2220 SNK=INT(SK1/15):REM ** SIGHRAANKA ** 2230 RESTORE 2740:FOR K=0 TO SNK+1 2240 READ SPH:IF K=SNK THEN SPH1=SPH 2250 N E X T K:SPH2=SPH
"
260
Ancient Indian Astronomy
2260 PRINT "SNK=";SNK;" SPHl=";SPHr;"SPH2=";SPH2 2270
SGP=(SPH2-SPH1)*(SK1-15*SNK)/15
2280 SGRPH=(SPH1+SGP)/10:IF SK>180 THEN SGRPH=-SGRPH 2290 L=SGRPH:PRINT "SIGRAPHALA:";:IF SGRPH<0 THEN PRINT TAB(34);"-"; 2300 GOSUB 170 2310
SGRPHl=SGRPH/2
2320 PRINT "SIGHRAPHALAARDHA:"; 2330 IF SGRPHUO THEN PRINT TAB(34);"-"; 2340 L=SGRPH1:G0SUB 170 2350 S1GURU=MGURU+SGRPH1 2360 PRINT TAB(34);"
"
2370 PRINT "1st SIGHRA C0R.GURU:";:L=S1GURU:G0SUB 150? 2375 S1GURU=L 2380 PRINT T A B ( 3 4 ) ; " _
"
2390 LOCATE 22,67:PRINT "PRESS A N Y K E Y " 2400
A$=INPUT$(i)
2410 R E M ** M A N D A SPASHTA G U R U ** 2420 CLS:PRINT "**** M A N D A SPASHTA GURU •***" 2430 MK=180 - Si.<}URU:IF MK<0 THEN MK=MK+360 2440 L=MK:PRINT :PRINT " M A N D A KENDRA:";:GOSUB 150 2450 IF MK<90 THEN M K B = M K 2460 IF MK>90 A N D MK<180 THEN MKB=180-MK 2470 IF MK>180 A N D MK<270 THEN MKB=MK-180 2480 IF MK>270 A N D MK<360 THEN MKB=360-MK 2490 L=MKB:PRINT " M A N D A KENDRA BHUJA:";:GOSUB 150 2500 MNK=INT(MKB/15) 2510 RESTORE 2830 2520 FOR K=0 TO MNK+1 2530 READ M P H : IF K=MNK THEN MPH1=MPH 2540 N E X T K: MPH2=MPH 2550 PRINT "MNK=";MNK;" M P H 1=";MPH1;"MPH2=";MPH2 2560 MNP=(MPH2-MPH 1 )*(MKB-15*MNK)/15 2570 MNDP=(MPH1+MNP)/10:IF MK>180 THEN MNDP=-MNDP 2580 PRINT "MANDAPHALA:";:IF MNDP<0 THEN PRINT TAB(34);"-"; 2590 I =MNDP:GOSUB 170 2600 PRINT TAB(34) "
"
2610 PRINT " M A N D A COR.GURU:"; 2620- MNGURU=MGURU+MNDP:L=MNGURU:GOSUB 150
Computer Programs 2630 PRINT TAB(34) "
261
"
2640 R E M ** 2nd SIGHRA CORRECTION ** 2650 PRINT :PRINT "**** 2nd SIGHRA CORRECTION ****" 2660 SK2=SK-MNDP 2670 L=SK2:PRINT :PRINT "2nd SIGHRA KENDRA:";:GOSUB 150 2680 IF SK2>180 THEN SKB=360-SK2 ELSE SKB=SK2:REM *BHUJA * 2690 PRINT "SIGHRA K E N D R A BHUJA:";:L=SKB:GOSUB 150 2700 SNK2=INT(SKB/15):REM ** SIGHRAANKA ** 2710 RESTORE 2820:FOR K=0 TO SNK2+1 2720 R E A D SPH.IF K=SNK2 THEN SPH1=SPH 2730 N E X T K: SPH2=SPH 2740 PRINT "SNK=";SNK2;" SPH1=";SPH1;" SPH2="SPH2 2750 SGP=(SPH2-SPH1 )*(SKB-15*SNK2)/15 2760 • SGRPH=(SPH1+SGP)/10:IF SK2>180 THEN SGRPH=-SGRPH 2770 PRINT "2nd SIGHRAPHALA:";:IF SGRPH<0 THEN PRINT TAB(34);"-"; 2780 L=SGRPH:GOSUB 170 2790 PRINT TAB(34);"
" •
2800 PRINT "TURE GURU:";:L=MNGURU+SGRPH:GOSUB 150 2810 PRINT TAB(34);" " 2820 DATA 0,25,47,68,85,98,106,108,102,89,66,36,0 2830 DATA 0,14,27,39,48,55,57 2840 LOCATE 21,67:PRINT "PRESS A N Y K E Y " 2850 A$=INPUT$(1) 2860 CLS:PRINT "******************SUKRA SPHUTA ******************" 2870 DHR=44.03333: KSP = 230.15 2880 SK = 3*AHG/5+3*AHG/181-CHAKRA*DHR+KSP 2890 PRINT :PRINT "SIGHRA K E N D R A AT UJJAIN SUNRISE:"; 2900 L=SK:GOSUB 150 2910 DAILY = .6166667 : R E M ** DAILY = 37' **
^
2920 GOSUB 80 :REM * DESHANTARA * 2930 PRINT "SIGHRA K E N D R A AT L O C A L SUNRISE:"; 2940 SK = SK + DESH : L = SK : GOSUB 150 2950 K A A L A = (Hl+MI/60)*DAILY/24 2960 PRINT "MOTION F0R";H1;"HRS.";MI;"MIN.:"; 2970 L=KAALA:GOSUB 170 2980 PRINT "SIGHRA K E N D R A AT GIVEN TIME:"; 2990 SK = SK + K A A L A : L = SK : GOSUB 150:SK=L 3000 PRINT: PRINT "**** SUKRA'S 1st SIGHRA CORRECTION ***"
262
Ancient Indian Astronomy
3010 IF SK >180 THEN SK1=360-SK ELSE SK1=SK 3020 L=SK1:PRINT :PRINT "SIGHRA K E N D R A BHUJA:";:GOSUB 170 3030 SNK=INT(SK1/15):REM ** SIGHRAANKA ** 3040 RESTORE 3610:FOR K=0 TO SNK+1 3050 R E A D SPH:IF K=SNK THEN SPH1=SPH 3060 N E X T K:SPH2=SPH 3070 PRINT "SNK=";SNK;" SPH1=";SPH1;"SPH2=";SPH2 3080 SGP=(SPH2-SPH1)*(SK1-15*SNK)/15 3090 PRINT "SIGHRAPHALA:"; 3100 SGRPH=(SPH1+SGP)/10:IFSK>180THEN SGRPH=-SGRPH:PRINTTAB(34)"-"; 3110 L=SGRPH:GOSUB 170 3120 SGRPHl=SGRPH/2 3130 PRINT "SIGHRAPHALAARDHA:"; 3140 IF SGRPH1<0 THEN PRINT TAB(34);"-"; 3150 L=SGRPH1:G6$UB 170 3160 S1SUKRA=MRAVI+SGRPH1 3170 PRINT TAB(34);"
"
3180 PRINT "1 St SIGHRA COR.SUKRA:";:L=S 1 SUKRA:GOSUB 150:S 1 SUKRA=L 3190 PRINT TAB(a4);"
"
3200 LOCATE 20,67:PRINT "PRESS A N Y K E Y " 3210 A$=INPUT$(1) 3220 CLS:PRINT "*** M A N D A SPHASHTA SUKRA ***" 3230 MK=90 - S1SUKRA:IF MK<0 THEN MK=MK+360 3240 L=MK:PRINT :PRINT " M A N D A KENDRA:";:GOSUB 150 3250 IF MK<90 THEN M K B = M K : R E M ** BHUJA OF M A N D A K E N D R A ** 3260 IF MK>90 A N D MK<180 THEN MKB=180-MK 3270 IF MK>180 A N D MK<270 THEN MKB=MK-180 3280 IF MK>270 A N D MK<360 THEN MKB=360-MK 3290 L = M K : 3 : P R I N T " M A N D A K E N D R A BHUJA:";:GOSUB 150 3300 MNK=INT(MKB/15) 3310 RESTORE 3620 3320 FOR K i O TO MNK+1 3330 READ M P H : IF K=MNK THEN MPH1=MPH 3340 N E X T K: MPH2=MPH 3350 PRINT "MNK=";MNK;" MPH1=";MPH1;"MPH2=";MPH2 3360 MNP=(MPH2-MPH 1 )*(MKB-15 *MNK)/15 3370 MNDP=(MPH1+MNP)/10:IF MK>180 THEN MNDP=-MNDP 3380 PRINT "MANDAPHALA:";:IF MNDP<0 THEN PRINT TAB(34);"-";
Computer Programs
263
3390 L=MNDP:GOSUB 170 3400 PRINT TAB(34) "
"
3410 PRINT " M A N D A CORRECTED SUKRA:"; 3420 MNSUKRA=MRAVI+MNDP:L=MNSUKRA:GOSUB 150:MNSUKRA=L 3430 PRINT TAB(34) " " 3440 PRINT "**SECOND SIGHRA CORRECTION**" 3450 SK2=SK-MNDP 3460 PRINT :PRINT "2nd SIGHRA KENDRA:";:L=SK2:GOSUB 150: SK2=L 3470 IF SK2>180 THEN SKB=360-SK2 ELSE SKB=SK2:REM *SIGHRA K E N D R A BHUJA * 3480 PRINT "SIGHRA K E N D R A BHUJA:";:L=SKB:GOSUB 150 3490 SNK2=INT(SKB/15):REM * SIGHRAANKA * 3500 RESTORE 3610:FOR K=0 TO SNK2+1 3510 R E A D SPH:IF K=SNK2 THEN SPH1=SPH 3520 N E X T K: SPH2=SPH 3530 PRINT "SNK=";SNK;" SPH1=";SPH1;" SPH2="SPH2 3540 SGP=(SPH2-SPH1)*(SKB-15*SNK2)/15 3550 SGRPH=(SPH1+SGP)/10:IF SK2>180 THEN SGRPH=-SGRPH 3560 PRINT "2nd SIGHRAPHALA:";:IF SGRPH<0 THEN PRINT TAB(34);"-"; 3570 L=SGRPH:GOSUB 170 3580 PRINT TAB(34);"__
"
3590 PRINT "TRUE SUKRA:";:L=MNSUKRA+SGRPH:GOSUB 150 3600 PRINT TAB(34);" _ " 3610 DATA 0,63,126,186,246,302,354,402,440,461,443,326,0 3620 DATA 0,6,11,13,14,15,15 3630 LOCATE 21,67:PRINT "PRESS A N Y K E Y " 3640 A$=INPUT$(1) 3650 CLS'PRINT "******************SANI SPHUTA ******************" 3660 DHR=225.7: KSP = 285.35 3670 MSANI =
AHG/30+AHG/(156*60)-CHAKRA*DHR+KSP
3680 PRINT :PRINT " M E A N SANI AT UJJAIN SUNRISE:"; 3690 L=MSANI:GOSUB 150 3700 DAILY = 3.333334E-02 : R E M ** DAILY = 2' ** 3710 GOSUB 80 :REM * DESHANTARA * 3720 PRINT " M E A N SANI AT L O C A L SUNRISE:"; 3730 MSANI = MSANI + DESH : L = MSANI : GOSUB 150 3740 K A A L A = (HI+MI/60)*DAILY/24 3750 PRINT "MOTION F0R";H1;"HRS.";MI;"MIN.:";
264
Ancient Indian Astronomy
3760 L=KAALA:GOSUB 170 3770 3780 3790 3800 3810 3820 3830 3840
PRINT" _" PRINT " M E A N SANI AT GIVEN TIME:"; MSANI = M S A N I + K A A L A : L = MSANI : GOSUB 150:MSANI=L PRINT" " R E M ** 1st SIGHRA CORRECTION • L=MRAVI:PRINT " M E A N RAVI:"::GOSUB 150 L=MSANI:PRINT " M E A N SANI:";:GOSUB 150 SK=MRAVI-MSANI:IF SK<0 THEN SK=SK-»-360
3850 L=SK:PRINT "SIGHRA KENDRA:";:GOSUB 150 3860 IF SK >180 THEN SK1=360-SK ELSE SK1=SK 3870 PRINT "SIGHRA KENDRA BHUJA:";:L=SKl:GOSUB 150 3880 SNK=INT(SK1/15):REM ** SIGHRAANKA ** 3890 RESTORE 4370:FOR K=0 TO SNK+1 3900 READ SPH:IF K=SNK THEN SPH1=SPH 3910 N E X T K:SPH2=SPH 3920 PRINT "SNK=";SNK;" SPH1=";SPH1;"SPH2=";SPH2 3930 SGP=(SPH2-SPH1 )*(SK1 -15*SNK)/15 3940 SGRPH=(SPHls-SGP)/10:IF SK>180 THEN SGRPH=-SGRPH 3950 PRINT "SIGHRAPHALA:";:IF SGRPH<0 THEN PRINT TAB(34);"-"; 3960 L=SGRPH:GOSUB 170 3970 SGRPHl=SGRPH/2 3980 PRINT "SIGHRAPHALAARDHA:";:IF SGRPHl<0 THEN PRINT TAB(34);"-"; 3990 L=SGRPH1:G0SUB 170 4000 S1SA^^=MSANI+SGRPH1 4010 PRINT TAB(34);" " 4020 PRINT "1st SIGHRA COR.SANI:";:L=SlSANI:GOSUB 150:S1SANI=L 4030 PRINT TAB(34);"
"
4040 LOCATE 21.67:PRINT "PRESS A N Y K E Y " 4050
A$=INPUT$(1)
4060 CLS:PRINT "*** M A N D A SPASHTA SANI ***" 4070 MK=24U- S1SANI:IF MK<0 THEN MK=MK+360 4080 L=MK:PRINT :PRINT " M A N D A KENDRA:";:GOSUB 150 4090 IF MK<90 THEN M K B = M K 4100 IF MK>90 A N D MK< 180 THEN MKB= 180-MK 4110 IF MK> 180 A N D MK<270 THEN MKB=MK-180 4120 IF MK>270 A N D MK<360 THEN MKB=360-MK 4130 L=MKB:PRINT " M A N D A K E N D R A BHUJA:";:GOSUB 150
Computer Progrcuns
265
4140 MNK=INT(MKB/15) 4150 RESTORE 4460 4160 FOR K=0 TO MNK+1 4170 R E A D M P H : IF K=MNK THEN MPH1=MPH 4180 N E X T K: MPH2=MPH 4190 PRINT "MNK=";MNK;" M P H 1 =";MPH 1 ;"MPH2=";MPH2 4200 MNP=(MPH2-MPH 1 )*(MKB-15*MNK)/15 4210 MNDP=(MPH1+MNP)/10:IF MK>180 THEN MNDP=-MNDP 4220 PRINT "MANDAPHALA:";:IF MNDP<0 THEN PRINT TAB(34);"-"; 4230 L=MNDP:GOSUB 170 4240 PRINT TAB(34) "
"
4250 4260 4270 4280
PRINT " M A N D A CORRECTED SANI:"; MNSANI=MSANI+MNDP:L=MNSANI:GOSUB 150:MNSANI=L PRINT TAB(34) " _ _ " PRINT ** 2iid SIGHRA CORRECTION **
4290 4300 4310 4320
SK2=SK-MNDP L=SK2:PRINT :PRINT "2nd SIGHRA KENDRA:";:GOSUB 150 IF SK2>180 THEN SKB=360-SK2 ELSE SKB=SK2:REM *BHUJA * PRINT "SIGHRA K E N D R A BHUJA:";:L=SKB:GOSUB 150
4330 SNK2=INT(SKB/15):REM • SIGHRAANKA * 4340 RESTORE 4450:FOR K=0 TO SNK2+1 4350 READ SPH:IF K=SNK2 THEN SPH1=SPH 4360 N E X T K: SPH2=SPH 4370 PRINT "SNK=";SNK2;" SPH1=";SPH1;" SPH2="SPH2 4380 SGP=(SPH2-SPH1)*(SKB-15*SNK2)/15 4390 SGRPH=(SPH1+SGP)/10:IF SK2>180 THEN SGRPH=-SGRPH 4400 PRINT "2nd SIGHRAPHALA:";:IF SGRPH<0 THEN PRINT TAB(34);"-"; 4410 L=SGRPH:GOSUB 170 4420 PRINT TAB(34);"
"
4430 PRINT "TRUE SANI:";:L=MNSANI+SGRPH:GOSUB 150 4440 PRINT TAB(34);" " 4450 DATA 0,15,28,39,48,54,57,57,53,45,33,18,0 4460 DATA 0,19,40,60,77,89,93 4470 END
APPENDICES APPENDIX - 1 : PRECESSION OF EQUINOXES A - 1.1 Phenomenon of precession We saw earlier in the text that the equinoctial points yand Q, the points of intersection of the celestial equator and the ecliptic, are not fixed. If fact, these two points "precede" i.e. move backwards along the ecliptic. This phenomenon is called "precession of the equinoxes", and it is important to know this to appreciate the difference between the frameworks of the western and the Indian astronomical systems. In Fig A - 1.1 let S be the position of a celestial body on the celestial sphere. Then 55| is the celestial latitude of S measured along the great circle KSSi passing through the pole K of the ecliptic. The celestial longitude of S is Y 5, measured from y along the ecliptic.
Ecliptic Cel. Equator
The ecliptic is a fixed great circle on the celestial sphere with reference to the background of the stars. The celestial equator keeps on moving, though slowly, in such a way '"'9- ^-1.1: Precession of equinoxes that the first point of Aries y moves backwards (precedes) along the ecliptic, say to a position Yj at a future time at an average annual rate of about 50".2. Further, if the change in the obliquity of the ecliptic € is considered negligible, the celestial north pole P of the celestial equator describes a small circle of which the pole is K with the angular radius 6. It is clear from Fig.A-1.1 that yy, is equal to the angle PkPi which gives the amount by which the equinoctial point y has moved, F] being the pole of the changed position of the celestial equator (shown with a broken line arc in Fig. A-1.1) which intersects the ecliptic at y^
A - 1.2 Ancient Indian reference to the precession The phenomenon of the precession of the equinoxes appears to have been familiar to Indian astronomers for thousands of years. Next, comes the question of the so-called "zero-year" when the zero point of the Indian zodiac, considered as fixed one, coincided with the zero point of the moving
Appendices
267
zodiac viz., the vernal equinox. In other words, when did the first point of Mesa coincide with the first point of Aries last? Indian astronomers have differed on the rates of precession during different periods as also in respect of the "zero year". The SUrya siddhanta takes the rate of precession as 54" per year. The accumulated amount of precession starting from the "zero-year" upto any given time is called ayanamsa. The zero-year according to different Indian astronomical texts is given below. Siddhantic text (i) SUryasiddhdnta (ii) Laghumdnasam (Munjala)
year of zero ayanamsa 499 A. D. 527 A. D.
(iii) Grahalaghavam (Ganesa Daivjna)
522 A. D.
(iv) Bhatatulyam (Damodara)
420 A. D.
A-1.3 Effects of precession on celestial longitude The vernal equinox has precession rate of about 50".2 per year. Therefore, it takes about 71.7 years for y to move by 1°. The equinox completes one full revolution (i.e. 360°), moving backwards, along the ecliptic in about 25,800 years. Since the celestial longitudes of all the heavenly bodies are measured along the ecliptic starting from the vernal equinox y as the reference point and this point of reference itself precedes along the ecliptic, the celestial longitudes of all these bodies increase by a constant amount of about 50".2 per year. However, the latitudes are unaffected by the precession of the equinoxes since the plane of the ecliptic remains fixed. A-1.4 Tropical (Sdyana) and sidereal (Nirayana) longitudes In modern astronomy, the zodiac is divided into twelve signs viz., Aries, Taurus etc. Starting from the vernal equinoctial point called "the first point of Aries"as noted eariier. The celestial longitudes of heavenly bodies measured along such a "moving zodiac" are referred to as tropical (or sdyana) longitudes. In Sanskrit sdyana means "with motion". However, in Indian astronomy, the celestial longitudes are measured starting from a fixed point of reference, the end of the Revati constellation (identified as Zeta Piscium). Thus a "fixed zodiac" is used in this system where the celestial longitudes are measured with reference to fixed stars. Hence the longitudes are referred to as sidereal (or nirayana); the word nirayana means "without motion". Now, due to precession, the first point of Aries moves backwards constantly as compared to the first point of Mesa. Once in about 25,800 years, the period of complete revolution of y, the first points of the moving and the fixed zodiacs coincide when the longitudes of heavenly bodies according to the sdyana and the nirayana systems will be the same.
268
Ancient Indian Astronomy
However, there is a divergence of opinion as to when y coincided with the first point of Mesa last—the year of "zero precession". Different Indian astronomers have been taking different years for this. On the recommendation of the Indian Calendar Reform Committee, the Government of India has adopted 285 A.D. as the year of zero-precession. The accumulated amount of precession of the equinox from the zero-precession year is called ayanamsa. For example, according to the Indian Astronomical Ephemeris the true Ayanamsa as on January 1, 1997 is 23° 48'56".l.
APPENDIX - 2 : LAGNA (ASCENDANT) A-2.1 Intervals of rising of rasis According to tiie Khandakhddyaka, the durations of the rising of Mesa, Vrsabha and Mithuna at Latika are respectively 278,299 and 323 vinadis. These diminished by the vinadis of the local ascensional difference (cara) are the durations of the risings of these three rasis at one's own place. The figures written in the reverse order increased by the ascensional difference (cara) are the durations of the risings of the next three signs at the observer's place. We have sin (cara) = tan (j> tan 8 where 0 is the latitude of the place and 5 the declination corresponding to the ending of the Rasi viz. for longitudes = 30°, >^ = 60° and X3 = 90°. Also with latitude P = 0 (for the points on the ecliptic) we have sin 8 = sin X sin Now, for X,, = 30°, X2 = 60° and
6
(e = 24°)
= 90°, with the obliquity of the ecliptic 6 being
taken as 24°, we get the corresponding values of the declination : 8, = sin- ' (sin 30° sin 24°) = 11°.734 82 = s i n - ' (sin 60° sin 24°) = 20°.624646
83 = s i n - ' (sin 90° sin 24°) = 24° The tabular differences of cara (ascensional difference) for a place of latitude $ are given by the successive differences of R tan (t) tan S,- (asus) where R = 3438' and 6 asus = 1 vinddikd; 21,600 asus = 1 day Therefore, corresponding to the declination 8; of the endings of the first 3 rasis, the tabular differences of cara, are given by the successive differences of (R tan <) tan 8,)/6 (vinddikds) Thus, for example, for Bangalore (
tan 13° (tan ll°.734) x 3438/6 = 27.5
vin.
(ii)
tan 13° (tan 20°.625) x 3438/6 = 49.79 vin.
(iii)
tan 13° (tan 24°) x 3438/6 = 58.90 vm.
270
Ancient Indian Astronomy
Therefore, the tabular differences of the cara for the first three rasis at Bangalore are respectively, 27.5, (49.79 - 27.5), (58.9 - 49.79) i.e., 27.5,22.29 and 9.11 vinadis. The durations of the risings of the twelve rasis at Bangalore calculated according to the Khandakhddyaka, are given in Table A-2.1. Table A-2.1: Durations of risings of rasis at Bangalore Rasis
Durations of risings at Lanka (vinadis)
Ascensional difference (tabular diff). Bangalore
Durations of risings at Bangalore (vinadis)
Mesa
278
-27.5
250.5
Vrsabha
299
-22.29
276.71
Mithuna
323
- 9 . 11
313.89
Karkataka
323
+ 9.11
332.11
Simha
299
+ 22.29
321.29
Kanya
278
+ 27.5
305.5
Tula
278
+ 27.5
305.5
Vrscika
299
+ 22.29
321.29
Dhanus
323
+ 9. 11
332.11
Makara
323
- 9 . 11
313.89
Kumbha
299
- 22. 29
276.71
Mina
278
-27.5
250.5
Note :
(1) The tabular differences are additive for the rasis from Karkataka (X, = 90°) to the end of Dlianus ( \ = 270°) and negative otherwise. (2) The total of the durations of risings of all the twelve rasis is 60 nadis.
A - 2.2 Determination of Lagna at a given time and place The sun's longitude increased proportionately from the time in ghatikds (or nddikds) elapsed since sunrise, at the given place on the given day. by means of local time durations for the risings of the rasis becomes the lagna (or orient ecliptic point or the ascendant). Again, conversely, by making the longitude of the sun equal to the orient ecliptic point by the local time intervals for the risings of the rasis, we get the time elapsed since sunrise. Example : Find the Lagna of 5 ghatikds elapsed since sunrise at Bangalore given the longitude of the sun as 11' 19° 46' 36" at that instant.
Appendix-2
271
The sun is in Mina rasi with remainder = 3 0 ° - (19° 46'36") = 10° 13'24" = 613'.4 The duration of the rising of Mina is 250.5 vinadis (Table A-2.1) at Bangalore. Therefore, the time taken for the rising of 613'.4 is (613'.4) (250.5)/(30 x 60) = 85.365 vinadis The given time elapsed since sunrise
= 5gh. = 300 vinadis
Now, out of 300 vinadis subtract 85.365 vinadis for the residue of Mina; we get (300-85.365) = 214.635 vin. The duration of the rising of the next rasi (Mesa) is 250.5 vm.(see Table A-2.1). Therefore, 214.635 vw. corresponds to (214.635 / 250.5) x 30° = 25° 42' 17" of Mesa. Therefore, we have Lagna = Mesa 25° 42' 17" = 0^25° 42'17".
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274
Ancient Indian Astronomy
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Sarasvati Amma T.A., Geometry in Ancient & Medieval India, Motilal Banarsidass, Delhi, 1979. Saraswati T.A., The Development of Mathematical Series in India after Bhaskara II, Bulletin of the National Inst, of Sci. in India, 21, pp. 320-43, 1963. Sarma K.V., A History of the Kerala School of Hindu Astronomy (in perspective), Vishveshvaranand Inst., Hoshiarpur, 1972. Sen, S.N., Bag A . K . and Sarma S.R., A Bibliography of Sanskrit Works on Astronomy and Mathematics, Part I, National Inst, of Sciences of India, New Delhi, 1966. Sen, S.N., and Shukla K.S., eds.. History of Astronomy in India, I.N.S.A., New Delhi, 1985. Sengupta P.C, Aryabhata, the Father of Indian Epicyclic Astronomy, J of Dept. of letters, Uni. of Calcutta, 1929, pp. 1-56. Somayaji D.A., A Critical Study of Ancient Hindu Astronomy, Kamataka University, Dharwar, 1971. Srinivas M.D., Indian Approach to Science: The Case of Jyotisdstra, P.P.S.T. Bulletin, Nos, 19-20, . -le, Madras, 1990. Srinivasiengar C.N., The History of Ancient Indian Mathematics, The World Press Private Ltd., Calcutta, 1967. Sriram M.S., Man and the Universe (Draft of a Text Book on Astronomy for Classes IV - X), Dept. of Theoretical Physics, Uni. of Madras, Guindy Campus, Madras, 1993. Subbarayappa B.V. and Sarma K.V., Indian Astronomy A Source-Book, Nehru Centre, Bombay, 1985.
•
GLOSSARY OF TECHNICAL TERMS IN INDIAN ASTRONOMY
I English to Sanskrit Altitude Unnata, Unnati Anomaly {Manda) Kendra or {Sighra) Kendra Mandocca Apogee or aphelion Apse line Ascending node of the moon Celestial equator
Celestial latitude Celestial meridian Co-azimuth Co-latitude Declination Ecliptic Epicycle Equation of centre Equation of conjunction
'Nicocca rekhd Rahu Visuvadvrtta Visuvadvalaya, r0divrtta Ksepa, Viksepa, sara Yamyottara mandala Digamsa Lamba
Equatorial horizon
Equinoctial shadow Aksabha, palabha Equinox Gnomon
Krdntipdta Sahku
Hemisphere Hypotenuse
Kapdla Karna, srava
Meridian zenith distance Orbit Parallax in latitude Parallax in longitude Perigee Polar longitude Pole of the cel. equator Precession of the equinoxes
Apakrama, Krdnti, apama Apama maridala, Krdntivrtta Ni coccavrtta, anuvrtta Solstice Zenith distance Mandaphala Zodiac Sighraphala
II Sanskrit to English Abda Abhijit Acala
Year Alpha Lyrae Successive approximation
Adhikamdsa
Additional lunar (intercalary) month in a lunar year
A dyantakdla Agastya Agni
Duration of ah eclipse Star Canopus Star Beta Taurii
Niraksa ksitija
Agra
Avanati Kaksd Nati, avanad Lambana Mandanica Dhruvaka Dhruva Ayanamsa, ayanacalana, krdntipdtagati Ayandnta Natdmsa Bhacakra
Amplitude i.e. the arc along the cel. horizon lying between the east point and the rising point of a heavenly body
Agrajyd {agrajivd) Rsine of the amplitude Number of civil days Ahargana elapsed on a given day since a chosen epoch; dinagana, dyugana Ahordtra viskambha
Diameter of the diumalcircle
Glossary of Technical Terms in Indian Astronomy Ahoratra vrtta
Diurnal circle
Asu
Aja
Mesa (Aries sign)
Avanati
Aksa (fiksdmsa)
terrestrial latitude
Ayanacalana
Aksabha (palabha) Equinoctial shadow Aksakoti
Aksavalana (valanamsa)
Amdvdsyd Amsa (amsaka) Angula
Antyaphala
Anurddhd Anuloma
Corlatitude; Rsine of colatitude or Rcosine of latitude Angle subtended at a heavenly body on the ecliptic by the arc joining the north pole of the cel. equator and the north point of the cel. horizon Newmoon day Degree, fraction
Apama
Declination
Apara Ardhakarna
West {pascima) Semi-diameter (i.e. radius) Midnight Calculations from the midnight Approximate; asphuta Dark half of the lunar month {Krsna paksa) Approximate; asanna. anityam
Asanna Asitapaksa Asphuta
Ayandnta
l/6th of a Vighati (i.e.. 4 secortds) Parallax in cel. latitude {nati) Precession of the equinoxes Amount of precession of equinoxes (in degrees) Solstice
Bdhu
Base of a right-angled triangle for an angle 9, its bhuja is G, 180°-0, e-180°, or 360°-e according as 0 is in I, II, III, IV quadrant; bhuja
Bhacakra
Zodiac (consisting of 27 naksatras or 12 rasis) Revolutions of a celestial body in a long period of time (like mahayuga); parydya
Approximately one inch; Bhagana 1/12 of the length of the standard gnomon (sanku). Maximum equation of centre {manda phala) or Bhagana kdla of conjunction {sighra Bhagola phala) Bhaga Delta Scorpionis Bhoga Direct motion of a planet Bhogya khanda (opp. of Vakra)
Apakrama Declination Apakrama mandala Ecliptic
Ardharatra Ardhardtrika
Ayanamsa
277
Bhogya mandaphala BhUcchdyd BhUgola Bhuja Bhujajyd Bhujdntara
BhUkarna
Sidereal period of a heavently body Celestical (Starry) sphere One degree of arc; amsa Portion Tabular difference of Rsine etc. yet to be covered Portion of equation of centre yet to be covered Earth's shadow Terrestrial globe See bdhu Rsine of bhuja; dorjya Correction for the equation of time due to the eccentricity of the orbit Diameter of the earth; bhuvyasa
Bhukta
Covered (or traversed) already
Ancient Indian Astronomy
278
Bhuktagati (-phaldmsa)
Increase in the last Chddaka sighrakendra covered in Chddya order to find the true Chandrakald sighra phala
Bhukti
Daily motion of a Chdya heavenly body; dinagati Chdydkarna Correction (to the parameters for cel. longitude, cel. latitude etc.) Disc or diameter of a body (particularly of the Cheda moon, the sun or the Daksina earth's shadow cone in Daksiridyana an eclipse) Semi-diameter (i.e. Darsa radius) of the disc of a heavenly body Desa
Bija
Bimba
Bimbdrdlia
Cakra Cakraliptd
Cakrdmsa Cala Candra Candra (vydsa) bimba Candrakarna Candramdsa Candranatajyd Cara
Cara samskdra
Caturyuga
Circle, cycle, 360° ' Desantara Minutes of arc in a circle; 360x60'=21,600' Degrees in a circle, 360° Dhana Variable Dhanurbhdga The Moon Dhruva Angular diameter of the Dhruvaka moon Distance of the moon from the earth's cenffe Digamsa Lunar month; candramdsa Rsine of the zenith Dina Dinagarm distance of the moon Arc on the cel. equator Dindrdha between 6'0 clock circle Disd and the hour circle of a heavenly body at rising Dorjyd Correction to the mean position of a heavenly body due to the difference between instants of midnight at Lanka and the given place Great age of 43,20,000 years; mahayuga
Drgguna Drglambana Dvdparayuga
Eclipser; Grdhaka Eclipsed body; Grdhya l/16th of the moon's disc Shadow Hypotenuse of the right angled triangle whose other two sides are gnomon {saiiku) and its shadow Division; denominator South, Ydmya Southern course of the sun Conjunction of the sun and the moon Place Difference in longitudes of a given place and the prime meridian (usually Ujjayini) Positive; excess Arc of 225' (j^) Pole star; fixed Polar logitude along the ecliptic at the foot of the circle from the pole Co-azimuth Day Civil days; Ahargana Half-day Direction Rsine of the base of a right angled triangle; bhujajyd Rsine of the zenith distance Parallax in zenith distance Third of the four yugas of a mahayuga; duration: 8,64,000 years
Glossary of Technical Terrrxs in Indian Astronomy Bodies having the same declination Gati Motion; usually daily motion Cube Ghana Ghatikd A unit of time; l/60th part of a day (24 minutes); nddi, nddikd; ghati Ghata Multiplication, product Gola Globe, sphere Graha Planet; a moving heavenly body Grahana Eclipse; upardga, upapluti Grahana madhya Middle of an eclipse Graha (yoga) yuti Conjunction of planets
Ekdyanagata
Grdhaka Grdhya Grdsa Grdsamdna Guna Harija Hard Inducca Ista Isu
Kadamba Kadambaprota Kaksd Kald
Eclipsing body, eclipser; chddaka Eclipsed body; chddya Measure of obscuration (eclipse) Magnitude of eclipse Multiple Parallax in longitude; lambana One hour; l/24th of a day Moon's apogee; mandocca of the moon Desired or given Rversinc i.e., R-R cosine of an angle where R is the radius of the deferant circle (/? = 3438' according to Aryabhatiyam) Pole of the ecliptic Secondary to the ecliptic Orbit (or path) of a heavenly body A unit of measurement of angle; l/60th of one
Kdla Kaliyuga
Kapdla Karna
Kararia
279
degree i.e. 1 minute of arc; lipid, liptikd Time Last of the yugas of a Mahdyuga; duration: 4,32,000 years Hemisphere Diameter; hypotenuse of a right angled triangle; srava Half of a lunar day (tithi)
Khagola
Anomaly of a heavenly body, angular distance from its mandocca (apogee) or sighrocca (apex point of conjunction) Descending node of the moon; 180°- Rahu Celestial sphere
Khamadhya
Zenith
Koria Koti
Angle Perpendicular
Kotijyd
Rcosine of an angle
Krdnti
Declination; apakrama
Krdntipdta
Equinoxes, points of intersection of the ecliptic and the cel. equator; visuva, visuvat Dark half of a lunar month First of the four yugas constituting a mahdyuga; A unit of time; 2 ghatis (according to SUrya siddhanta) An additive quantity; cel. latitude; Viksepa, sara Horizon Ascendant; orient ecliptic point; point of intersection of the ecliptic with the eastern horizon
Kendra
Ketu
Krsnapaksa Krtayuga Ksaria
Ksepa Ksitija Lagna
280
Lamba Lambana Lambdmsa Lanka
Lipta (liptikd) Madhyabha Madhya bhukti Madhyacchdyd Madhya dina Madhyagati Madhya graha Madhya grahana Madhyajya
Madhyakarna
Ancient Indian Astronomy Co-latitude of a place Parallax in longitude Co-latitude of a place in degrees Place on the earth's equator dirou^ which the prime meridian passes Minute of arc; kald Meridian ecliptic point Mean daily motion Midday shadow of the gnomon Mid-day Mean (daily) motion Mean longitude of a planet Middle of an eclipse Rsine of the zenith distance of the meridian ecliptic point Radius; vydsdrdha, kartidrdha
Madhya krdnti
Declination of the meridian ecliptic point Madhya lagna Meridian ecliptic point Madhya lambana Parallax in longitude at the middle of an eclipse Madhyama krdnti Declination obtained from mean longitude Madhya rdtra Midnight
Madhydhna paridhiCircumference of the epicycle at mid-day Mahdsatiku Mahdyuga
Tretd, Dvdpara and Kali of durations in the ratio 4:3:2:1. According to the Aryabhatiyam, the four yugas are of equal durations : 108 x 10"* years each Manda
Slow
Mandakarman
The process of finding out the equation of centre
Mandakendra
Madhydhna Mid-day Madhydhna cchdydMid-day shadow
Angular distance of a heavenly body from its Mandocca; anomaly fixxn apogee or aphelion
Mandala
Circle; revolution; vrtta, cakra
Mandanica
Perigee; 180° from mandocca
Mandanicocca
Manda epicycle; manddnu vrtta Equation of centre (due to the eccentricity of an orbit)
vrtta Mandaphala
Madhya rekhd
Cenu-al meridian of the earth passing through Lanka, Ujjayini Kuruksetra and Mount Meru; madhyasQtra Gnomon formed with Madhya sanku the sun crossing the meridian Madhya sthityardhaIntervals between the middle of an eclipse and the moments of contact and separation (end)
Rsine of altitude Caturyuga : Great age consisting of 432 x lO'* years; consists of four parts (yugas) viz; Krta,
Mandocca
Apex of the slowest motion; apogee or aphelion
Manvantara
Age of Manu : 308448000 solar years; Month
Mdsa Melaka
Conjunction of planets; yuti, yoga
Meru Mesa caradala
Earth's pole Ascensional difference at the end of Mesa rasi (the first sign of the zodiac)
Glossary of Technical Terms in Indian Astronomy Mithuna caradala Ascensional difference at the end of the Mithuna rdsi Moksa End i.e., the last contact of an eclipse Moksa lambana Mrgavyddha
Muhurta
Square-root
Nddikd (Nddi)
l/60th of a day; 24 minutes; ghatikd
Ndksatra dina
Sidereal day
Nata
Hour angle; zenith distance Rsine of the hour angle
Natakdla kotijyd
Rcosine of the hour angle
Nati
Parallax in cel. latitude; avanati
Nemi Nimesa
Pdda Paksa
Parallax in longitude at Pala the end of an eclipse Palabha Alpha Canis Major; Sinus; Lubdhaka; brighParama Krdnti test visible star in the Parama lambana sky A unit of time; 30 muhHrfas = 1 day; Pdramdrthika 1 muMirta = 2 ghatikds valana = 48 minutes
Mula
Natakdlajyd (Natajyd)
Oja
Total obscuration of eclipsed body
Niraksa
Earth's ^uator; zero latitude
Niraksa ksitija
Equatorial horizon
Nisd
Night
Nicoccarekhd Nicoccavrtta
Odd; usually w.r.t. 1st and 3rd quadrants A quarter; quadrant Fortnight; half of a lunar month A unit of time; vinddi Equinoctial shadow; aksabha Maximum declination Horizontal parallax; maximum parallax Algebraic sum of aksavalana and ayana valana (numerical addition when both are north or both south and difference when they are of opposite directions)
Paridhi
Circumference; ««/nJ
Parilekhd
Projection Middle of an eclipse
Parvamadhya Parvanddi
Instant of conjunction or opposition of the sun and the moon in nddis.
Parvdnta Parydya
End of a pan>a Revolutions of a heavenly body in a given period (like mahdyuga); bhagana West
Circumference of a circle; paridhi Pascima A unit of time; acconling Pdta to the Siddhanta siromaiti, 1 day= 9,72,000 nimesas
Nimilana
281
Phala
Node of the moon (or a planet); point of intersection of the ecliptic with the orbit (kaksd) of the moon (or a planet) Result (like mandaphala and sighraphala)
Pragrdsa
Beginning of an eclipse; first contact of an eclipse
Line of apse
Prdhna
Forenoon; pUrvdhna
Epicycle; anuvrtta
Praksepa
Addition
282
Pram
Pratiloma Pratipada
Purnimd Purnimdnta Purva Purvdhna Rahu
Rdhumdna Rasi Rdsi kald
Ravikarria Ravi paramalambana Rekhd Rju igati) Rtu Saka
Satna Samamatidala Samdsavrtta
Ancient Indian Astronomy
A unit of time, 4 seconds of sidereal time; asu Retrograde; vdcra The first tithi (lunar day) of either bright or dark half of a lunar month Full moon; opposition of the sun and the moon End of the full-moon day East Forenoon Ascending node of the moon; pdta of the moon; tamas Angular diameter of the earth's shadow Zodiacal constellation; each of 30° extent Number of minutes of arc in a zodiacal sign; 30'X 60'= 1800' Distance of the sun from the earth's cenU'e Horizontal parallax of the sun Prime meridian Direct (motion), as opposed to retrograde Season; a year consists of six rtus.
Sankrdnti Sauradina Sauramdsa Sauravarsa Sdvanadina
Solar ingress into a sign of the zodiac Solar day Solar month Solar year Civil day; kudina. bhUdivasa
Sighra
Fast motion (as opposed to numda)
Sighrakarrruin
Finding out the equation of 'conjunction' {sighraphala)
Sighrakendra
Angular distance of a mean planet from its sighrocca (apex of 'conjunction')
Sighra nicocca vrtta Sighraphala
Epicycle of sighra
Sighrocca
Apex of fast {sighra) motion. In the case of superior planets the mean sun is the sighrocca while for Mercury and Venus two special points are defined as their sighrocca. However,
Era of King Salivahana Sparsakdla commencing in the year 78 A.D., Salivahana saka Sparsa lambana Even Prime vertical Spastagati Circle of radius equal to the sum of the radii of Spastagraha the eclipsing and the Sphutagati eclipsed bodies
Equation of conjunction
Nilakantha Somayaji maintains that the sun must be the sighrocca for all the planets Time of the first contact of an eclipse Parallax at the beginning of an eclipse True motion of a planet; sphutagati True planet; sphutagraha
Saiiku
Gnomon; kilaka
True motion of a planet; jpastagati
Sara
Celestial latitude; ksepa Sphutagraha
Tme planet; Spastagraha
Glossary of Technical Terms in Indian Astronomy Sphutaviksepa Srava Srdvistha Suklapaksa Tamas Tantra
Celestial latitude Udvrtta Hypotenuse; karria Beta Delphini; Dhanisthd Bright half of a lunar Unmilana month Unnati Ascending node of the tJnnatakdla moon; Rahu Indian astronomical texts which adopt the beginning of Kaliyuga as the Unnatdmsa epoch
283
Equinoctial colure; east and west hour circle; 6'0 clock circle End of totality of eclipse Altitude Distance from the horizon in time, unit; time elapsed after rising of a celestial body Complement of zenith distance
Taraka (Tdrd)
Star; asterism
Upapluti
Tdrdgraha
Star-planet, referring to Mars, Mercury, Jupiter, Venus, Saturn Delta Cancri; Pusya
upardga Urdhva yamyottara Celestial meridian vrtta
Tisya
Uttara Uttaragola
Tithi
Lunar day; l/30th of a lunar month
Trairdsika
Rule of three
Tretdyuga
Second of the four parts of a mahdyuga; duration : Vaidhrta 12.96,000 years
friohona lagna
Nonagesimal point: point of the ecliptic at the shortest distance from the zenith; 90° less less than Lagna.
Ucca
Apogee; mandocca Rising
Udaya Udayajyd
Udayaprdna
Uttardyana
Vakragati Varsa
Vara Orient sine; Rsine of the Vikala amplitude of lagna from Vikald the east Duration of the rising of Vikalagati the signs {rdsis) in prdria (or asu) unit of time; Vikrama samvat udaydsu
Udaydntara samskdra
The correction to the cel. longitude on account of the equation of time due to obliquity of the ecliptic; comfilementary to the bhujdntara
Viksepa Vilipta Vilomagati
Eclipse, grahana,
North Northern hemiphere; Uttara Kapdla Northern course of the sun along the ecliptic The "aspect" between the sun and the moon when the sum of their longitudes is 360° and their declinations {krdnti) are equal in magnitude but opposite in directions ReUrogression Year; abda Weekday; vdsara Remainder; sesa One second of arc; 1/3600th of a degree; viliptd Stationary motion Era named after King Vikrama starting fr . i 57-56 B.C. Polar latitude; cel. latitude; ksepa, sara One second of arc; vikala Retrograde motion; Vakragati
284
Ancient Indian Astronomy viskambha
Yimarda
Totality of an eclipse
Vinddi
l/60th of a imdU; 24
Vydsdrt^
Radius of a circle
seconds; pala, vighati l/60th of a pala Odd Diameter of a circle; ^ vydsa Equinox; Krdntipdta
Vyatipdta
The "aspect" of the sun and the moon when the sum of their longitudes is 180° and their declinations (krdnti) are same both in magnitude and direction South; daksina
Vipala Visama Viskambha Visuva (Visuvat) Vitasli Vitribha lagna Vrsacaradala
Vrtta Vyaksa Vydsa
A length of 12 angulas; Ydmya approximately one foot Ydmyagola Lagna - 90°; tribhona Yojana lagna; nonagesimal Ascensional difference at the end of Vrsabha (second sign of the Yoga tdrd zodiac) Circle; maridala, cakra Terrestrial equator; zero Yoga (Yuti) latitude; niraksa Yuga Diameter of a circle;
Yugma
Southern hemisphere A unit of distance; usually about 5 miles, but taken differendy by different authors Principal star in an asterism Conjunction; melaka An age (of long duration) Even; couple
INDEX I
Acuta Pisarati, 13 Adhikamasa, 4. 6, 25-29, 33 Ahargana, 10, 11, 13, 25-41,42, 51,69,106,119, 120, 122-24, 126, 143, 149 Altitude, 20-21 Amaraja, 12 Aiigira, 5 Annual equation, 63, 190-91 Apamandala, 17 Ardha-ratri, 7 Aryabhata I, 5-9, 10, 22, 43, 73, 172, 189 Aryabhatan system, 9, 170 Aryabhatiyam, 5,6,9, 10,12. 23, 50, 57,73,171, 172, Ascendant, 269-71 Asus, 269 Atri, 5 Autumnal equinox, 17 Ayanamsa, 103, 108, 115, 193-94, 267-68 Azimuth, 20-21 Bhaskara 1, 9, 10, 12 Bhaskara II, 9, JO, 12, 43,44, 56, 57, 61, 63, 172, 189 Bhattotpala, 12 Bhrgu, 5 Bhucchaya, 79 Bhujaiitara, 61-62, 143 Bija, 50, 160, 195 Bijas, 14, 170-95 Bijasarnskaras, 13 Bimba viyogardham, 161, 164 Bimba yogardham, 161 Brahma, 5 Brahmagupta, 7-10, 12-13, 28, 29, 36, 43, 62, 63. 85. 113. 142, 172, 189 Braha, Tycho, 189-90 Brahmasphutasiddhanta, 9, 142 Brhat samhita, 73, 74
Brhat-tithi-cintamani, 171 Budha. 12, 119, 120, 121, 123, 125, 130-32, 147-48, 155-56, 191-92 Cakra, 29, 30-31, 51, 52, 106, 123, 124, 149 Calendar system, 25 Candra bimba, 93, 110, 160 Candragrahanam, 79 Candra-Grahanadhikara, 12 Cara. 269 Ceasar, Julius. 35 Celestial longitude. 15. 19-20. 74, 141, 267 Chandas, 1 Chaya bimba, 160 Christian date, 31, 33 Citrabhanu, 13 Copernicus, 13, 192 Computer program for lunar eclipse, 213-23 GLLEC-According to Graha laghavam, 220-23 KKLEC-According to Khanda-Khaadyaka, 217-19 SSLEC-According to Surya Siddhanta, 213-16 Computer program for planets position, 243-65 GLPLA-According to Graha Laghavam, 254-65 KKPLA-According to Khanda laghavam. 249-53 SSPLA-According to Surya Siddhanta, 243-48 Computer program for position of Sun, Moon and Rahu, 191-212 GLRA MOON-According to Graha Laghavam, 207-12 KKRA MOON-According to Khanda Khaadyaka. 202-06 SSRA MOON-According to Surya Siddhanta, 191-201 Computer program for Solar eclipse, 224-42 GLSEC-According to Graha Laghavam, 231-36 KKSEC-According to Khanda Khaadyaka, 237-42 SSS EC-According to Surya Siddhanta, 224-30 Copernicus, 13. 192 Cyavana, 5
286 Daily motions, 61-63, 67-72, 80, 84, 101, 102, 110, 114, 119, 120, 121, 124, 163, 172 Daivajiia, 64 Daksinayana, 20 Damodara, 267 Declination, 21, 107, 115, 116 Desantara correction, 47-54, 59, 62, 120-22, 124, 133, 138 Dhniva, 15-16 Dhmvaka, 51, 52, 123, 124 Dhruvakas, 123 Digamsa, 190-91 Digamsa samskara, 190-91 Dohpala, 128, 131, 132, 136 Drgiya. 166-67 Drggati, 103, 104-05 Drkkaraiia, 170 Drkksepa, 103-05 Earth's shadow, 73, 81, 92, 93, 160 Ecliptic, 6, 16-17, 61, 74, 160-69, 266 Ecliptic limits, 76-78 Ecliptic system, 19-20 Epicyclic theory, 55-66 Equation, 59-61, 115-16, 117-18, 189-91 Equation of centre, 60-63 Equatorial system, 19, 20 Equinoxes, 16-17, 266-68 Evection, 63, 189-91 Ganesa Daivajna, 10, J3, 64, 65, 91, 97, 171 Gargya, 5 Gatiphala, 71 Ghatika, 61, 93, 94, 102, 107, 109, 270 Graha laghavam, 10, 13, 29-35, 51, 53, 54, 70, 91, 97, 98, 106, 108, 123, 124, 149-55, 159-60 Grahana, 73, 93 Grasa, 79, 81, 87, 92-94 Greenwich meredian, 47, 48, 51, 121 Gregory, Pope, XIII, 36 Guru, 119-21, 124, 126-27, 145-46, 151-55 Heliocentric distance, 185 Horizontal parallax, 75-77, 84-85, 100-01, 167 Horizontal system, 19-21 Hour angle, 21 Jupiter, 119, 126 Jyesthadeva, 10, 13, 170 Jyotisa, 1 Kaksavrtta, 55, 127 Kali'epoch, 25-27. 43, 193-94 Kali Era, 7, 23, 171
Ancient Indian Astronomy Kalpa. 1, 6, 9, 14, 22-23, 42, 43, 58, 119, 126, 172, 194 Karatiakuthuhalam, 10 Kasyapa, 5 Ketu, 23, 74, 99 Khagola, 15 Khagrasa, 93, 94 Khandakhadyaka, 7, 10, 12-13, 27-29, 36, 43, 49, 51, 54, 57, 62-63, 69, 84-85, 113, 122, 142-45, 160, 172, 189, 269-70 Kotiphala, 128, 131, 132, 136 Kranti, 107, 110-112 Kranti vrtta, 17 Krta era, 24 Ksepaka, 51-52, 123-24 Kuppanna Sastri, T.S., 160 Laghumanasam, 170, 189 Lagna, 102-05, 166, 269i71 Lalla, 22, 50, 54 Lambana, 103-04, 107-09, 167 Lambert, 7 Libra, 17 Lindemann, 7 Lomasa, 5 Lunar eclipses, 7, 13, 72-76, 160-63 Lunar months, 25-27 Madhava, 3, 181 Madhyamadhikara, 11 Mahabharata, 23 Mahasiddhanta, 172 Mahayuga, 9, 14, 22-23, 25-26, 42-46, 80, H9, 172-73. 194 Makara, 158 Manaikyakhandardha, 92, 93, 110 Manasam, 171 Manda, 13-14, 125-27. 129, 133-34, 137-38, 144- 50, 152, 154-56, 172-75, 192 Mandakendrajya, 56, 60. 63, 64, 69-71, 190-91 Mandanica, 55, 56, 149 Mandaparidhi, 57 Mandaphala, 56, 57, 60. 63-66, 125, 127, 135, 145- 48, 151, 153-54, 156 Mandashputagraha, 127-28 Mandoccas,' 14, 46. 49, 50, 52. 55, 67, 69-70, 125-26, 135, 138, 144-50, 152, 154-55, 157, 171-72, 181-84, 190, 193-94 Manvantara, 22. 23 Mafijula, 10, 12, 63. 170-71, 189 Manu, 5 Marda, 94, 96
Index Marici, 5 Mars. 119, 140 Mean positions, 42-47, 149, 193 Mercury, 119, 192-93 Meridian system, 19, 21, 103 Mesa, 18, 23, 45, 141, 267-69 Moksa, 95 Moksa kala, 96, 111-13, 165 Moksa sthiti, 96 Moon's diameter, 77, 80, 84-86, 87, 91, 93, 100, 102, 105, 110, 115, 160, 163 Nadika, 4, 28 Naksatras, 2, 5, 166 Narada, 5 Natamsa, 105, 110-12 Natamsas, 107, 108 Nati, 110 Nicoccarekha, 55 Nicoccavrtta, 55, 127 Nilakantha, 12, 13 Nilakantha Somayaji, 7, 10, 12, 73, 171, 181, 192 Nirayana, 14, 104, 108, 123, 181, 194, 267 Nirukta, I Nonagesimal, 107, 108, 114, 116 Obliquity, 61 Paitamaha, 5 Paksika, 191 Paiica Siddhantas, 73 Parahita, 171 Parallax, 84, 85, 101, 103; 104, 107, 163 Parama grasa, 168 Paramesvara, 10, 12, 73, 170, 181, 192 Parasara, 5 Parvanta, 96 Paulisa, 5 Penumbra, 75, 99, 100 Peripheries, 57-58, 60, 68, 126, 130-32, 134, 136, 173-75 Pluto, 193 Pole star, 16 Post-Aryabhatan astronomers, 9-10 Pramanam, 79,163, 169 Prthudakasvamin, 12, 28 Ptolemy, 189 Rahu, 23, 43, 46, 47, 50, 53, 54, 74, 79, 80, 86, 87,91,92, 95, 97,99, 101, 104, 106-08, 113, 114, 193 Rahu Sphuta, 98 Ramasubramanian, 192 RasT, 17, 18, 269-70
287 Ravi bimba. 160, 164 Retrograde motion, 139-41 RevatT, 267 Revolutions, 43-46, 58, 119, 171-73, 194, 267 Rgveda, 1-3 Romaka, 5 Right ascension, 20 Rtus, 3 Salivahana saka era, 24, 108 Samanta, Chandra Sekhara Simha, 63, 190-91 Sammilanam, 96 Sani, 119, 120, 122, 124, 130-32, 134-36, 138-39, 143, 146-47, 153-55 Sankara Varman, 10 Saiiku, 166 Sara, 91-93, 108, 110, 111 Saros, 2 Saturn, 119 Saunaka, 5 Saura, 5 Saurasiddhanta, 57 Sayana, 14, 103-05, 107, 114-16, 267 Sengupta, P C , 29, 189 Siddhanta darpana, 190 Siddhantas, 5, 9,' 11-12, 102, 171, 189 Siddhanta siromani, 12, 43, 44, 172, 192 Sidereal, 4, 9, 26,' 102, 103, 166, 172, 181, 194. 267 Sidereal solar year, 4 sTghra correction, 123, 127-37, 139, 140, 142, 145-48, 150-54, 156-58, 192 sTghra epicycles. 14, 127, 129, 130, 142, 172, 184-89. sTghrakarna, 128, 129, 132, 137, 139 Sighrakendra, 123-24, 128. 140, 142, 144, 153. 155-57 Sighrankas, 149, 151-54, 156-58 Sighraphala, 128-29, 131, 132, 136, 150-52, 154-55, 157-58 Sighrocca, 119-23, 125, 127-29, 137, 140, 144, 147-48, 155, 191-92 Siksa, 1 _ Sisya-dhi-vrddhida, 50 Solar eclipse, 7, 13, 73, 99-118, 160, 163-69 Sparsa, 95, 96, 101, 165 Sparsa sthiti, 96 Spastadhikara, 11 Sphuta, 56, 128 Sphutakoti, 128, 131, 132, 136 Sripati, 61 Stationary point, 140, 141-43
286
288 Daily t II Daivajii Daksini Damodi Declina Desanta 13: Dhruva, Dhnival Dhruva! Digamsi Digamsi Dohpala Drgiya. Drggati, Drkkarai Drkksep Earth's s Ecliptic, Ecliptic Ecliptic Epicyclic Equation Equation Equatorii Equinox( Evection Ganesa I Gargya, . Gatiphah Ghatika, Graha la, 91, 159Grahana, Grasa, 7S Green wic Gregory, Guru, Hi Heliocent Horizonts Horizonta Hour ang Jupiter, 1 Jyesthade Jyotisa, I Kaksavrtt; Kali epoc Kali Era,
Ancient Indian Astronon
Sthiti, 94, 95, 111, 113 Sukra, 12. 119-24. 139. 148-49. 156. 191-92 Sun's diameter. 80, 85,91,92,101,102,105.110. 115.163 SiJrya. 5 Surya bimba. 110 Suryadera yajvan. 170 Surya-Grahanadhkara, 12 Suryagrasa. 110, 111 Surya siddhanta. 8, 9. 12. 13. 15. 25. 35. 43-45. 48. 54. 57. 58. 61. 64. 67. 99. 101. 119. 126. 127. 129. 133. 137, 140, 142, 160. 172. 181. 190-92. 267 Taittiriya samhita. 2 Tantrasahgraha. 10. 12. 171. 192 •raragrahas, 13. 45. 119. 122. 123, 124, 125-59, 195
nthis, 25-26 Tribhona lagna. 107-09. 111-12 Triparsnadhikara. 11-12 Tme daily motion. 68-71. 72. 85-87. 101, 106, 110-11, 114, 137-39, 163 Tula. 23 Tungantara, 191 Udayintara. 61 Ujjayini. 28. 36. 42. 45, 47, 48, 52, 53, 120-22, 124, 193 Umbra, 74, 75, 99 Unnmilanam, 96 Uparaga, 73
Uttara khanda khadyaka, 189 Uttarayana, 3, 20 Vakragati, 140 Vakrarambha, 143 Vakratyaga, 143 Varahamihira, 3, 5. 9, 10, 57, 73, 74 Variation, 63, 189-90 Vasistha, 5 Vateivara, 10. 22 Vedanga Jyotisa. 1. 3-5. 6. 22 Venketesa Ketkar, 10 Venus, 119 Vernal equinox, 18, 267 Vikrama era, 23-24 Viksepa, 91 Viksepam, 167 Viiihucandra, 161, 164 Visuvd vrtta, 15-16 Vyagu. 95, 108 Vyakarana, 1 Vyarkendu. 161, 163, 167 Vyasa, 5 Week days, 35-37 Yajurveda, 2, 3 Yavana, 5 Yojana, 6, 48. 80. 102 Yuga. 2. 4. 6. 22. 42 Yuga system. 22-24 Zeta piscium. 267 Zodiac, 5, 6, 13, 15-18, 45. 266-67