Astronomical Easter Comparison & Calculation
by OP Armstrong
10.25/15
Table-A Calculation Formula for Astronomical Easter Calculation, Gregorian Year JD# Method A B C D E F YR pEpact JD#1Jan JD#1Jan JD#JanNM JD#PFM1 JD#Eqnx
G JD#PFM
H WkDy
JD# Easter
-400 -4000 0 24.3 24.3 260090.45 260113.8 260187.8 260170.2
260187.8
7
260188.8
2019
2458563.8
5
Yr
6 .0
I
2458566.8 IF(H=1,G+7,8pEPCT pEPCT JD#Jan JD#Jan1.3 1.3 (B-1)+C 74.02+D F.3 or F.7 F.7 IF(E>=F,E,E+29.5) f.6(G) INT(H)+G) 2458484.8 2458489.8 2458563.8 2458563.4
J D#ck
1 1 f.6(i)
The above Table-A is a tabulation of calculation of calculation steps steps for finding Astronomical finding Astronomical date date of of Easter. The example calculations use formula-2 of Table-B to determine moment of January of January New Moon New Moon for selected year, expressed as astronomical Julian Day number, JD#. Alternatively one could use any number of resources to arrive at a value for column B. Just be sure to express the moment as an Astronomical an Astronomical formatted formatted Julian Day Number, JD#. Julian calendar or Gregorian calendar dates may be verified verified by the day of week. week. The numbers numbers in column column F, F, JD#.Equ JD#.Equino inox, x, are for equinox for equinox moment by formula 4 of Table B. The month number and day-of-month may be determined in a spreadsheet by adding two more columns and using formula 8 and 9,Table 9,Table B, given that Year, Yr , is stated as input in column-A. Alternatively, the Easter JD# may be converted back to Gregorian Date by several free several free programs programs.. This routine was compared by 70 dates. If using JMT in place of GMT (UT), then add an offset of 0.098 days to step “I” and “I” and “J” “J”.. A check check was made against the WCC the WCC Easter Easter dates table dates table for for years 2025 to 2001. Against that Table this method using formula 2 and 4 reproduced their result. Other checked Other checked instances returned result results s that that matched matched either either the Catholic Easter Easter dates or alternative alternative astronomica astronomicall calculatio calculation n results. However the data of Ovidiu Ovidiu Vaduvescu did Vaduvescu did not confirm the astronomical values set forth in the WCC document nor results of this calculation. If Equinox and PFM dates are closer than one 1 (1) day, day, verification by a more precise routine precise routine is is advised on slide-9 on slide-9.. NEXT NEXT N3 N3
Astronomical Easter Comparison & Calculation Name (Nu) pEpact.Cassidy.f1
TABLE-B
pEpact.Lunation#.f2
9.5/15
Excel Astronomical Name Formula: Yr Yr-year; -year; JD#-Julian Day
29.09-MOD(MOD(Yr,19)*11-INT(( Yr-1502.57-12*MOD(Yr,19))/228),29.983)
pEpact.Cassidy.0.f1b
by OP Armstrong
29.5-MOD(MOD(Yr,19)*11-INT(( Yr-1584-12*MOD(Yr,19))/228),30)
IF((1+MOD((365.242454*(-4006- Yr)),29.5306))>=30,((1+MOD((365.242454*(-4006)),29.5306))-30),(1+MOD((365.242454*(-4006.242454*(-4006- Yr)),29.5306))) Yr)),29.5306))-30),(1+MOD((365
JD# Jan1.f3
257898.52-365.242454*(-4006- Yr)
JD# Equino JD# Equinox.f4 x.f4 1st Page
(2457102.448+(Yr-2015)*365.2422)+((-0.0005947871)*(( Yr-2015)/1000)^4+(0.00392591)*((Yr-2015)/1000)^3+(0.013808751)*(( Yr2015)/1000)^2+(0.1590901)*((Yr-2015)/1000))
March 1st Moon.f5
JD#.Jan1 + pEpact + 59
Day of Week.f6
(1+INT(MOD((1.5+JD#),7))) ),7) ))
JD#21March.f7
one is Sunday and 7 is Saturday, etc
257978.00-365.242454*(-4006- Yr)
Day of Month.f8
1+INT(MOD(((INT(MOD(((INT(JD#+0.5)+(37+INT(0.5+0.75*INT((INT(JD#+0.5)-4479.5)/36524.25))))59.25),365.25)))+0.5),30.6))
Month Number.f9 , 3=March, 4=April
1+MOD((2+INT(((INT(MOD(((INT(JD#+0.5)+(37+INT(0.5+0.75*INT((INT(JD#+0.5)-4479.5)/36524.25))))59.25),365.25)))+0.5)/30.6)),12) 2
Astronomical Easter Comparison & Calculation Why the Sunday after the Paschal Full Moon? Take a look at either the proleptic tic Catho tholic or Astronomical Easters for years 30 and 33AD. For these dates, the proleptic Gregorian Easter dates are April 7 and and Apri Aprill 3, resp respec ectiv tively ely.. Next Next dete determ rmin ine e the the Luna Lunar r Conjunction (astronomical new moon) age of 17.9 and 17.2 days, respectively, at 4pm Sunday. Given that Jesus died before Friday sunset and was buried before sunset, so 1 day back is, Saturday 4pm and another day back is Friday 4pm. Thus at time of Jesus death, the Lunar days are 15.9 and 15.2 respectively. Given that the visible new crescent or sighted moon is some where about 1 day after conjunction. So on Good Friday the lunar sighted moon of these years is 14 days in age. Thus Good Friday of years 30AD and 33AD would either be on Nissan 14 or 15. However, if this skip week, “after” week, “after” were were not used, then sometim sometimes es the Easte Easterr would would come come early early to the lunar lunar calendar. Why the Postponement Rule? From From this this it is seen seen that that reck reckon onin ing g East Easter er by these these methods speaks of "Christ our Passover", Jn18:39, Passover", Jn18:39, 19:14, 19:14, 1Cor5:7,, Heb11:28 1Cor5:7 Heb11:28,, Num9:13 Num9:13,, Ex12:14 Ex12:14.. Since Since the the Cath Cathol olic ic Easte Easterr conc concur urs s the the Astr Astron onom omic ical al Easter in about than 90% of the dates, keeping Good Friday and Easter are keeping "Christ our Passover" as "a memorial; and .... a feast to the LORD throughout your generations; ... feast by an ordinance for ever." Ex12:14. The The post postpo pone neme ment nt rule rule impo importa rtanc nce e is illu illust stra rate ted d by calculation. NEXT
by OP Armstrong
9.5/15
Look at 135 differe different nt years between between 4000BC 4000BC and and 2038AD 2038AD.. The The averag average e lunar lunar conjun conjuncti ction on age at 4pm Easter Sunday was about 19. days, with a minimum of 15.5 days and a max of 22.2 days. Without the postponement rule, then some Easter Sundays would land before a lunar age of 13.5 day This gives a very early Lunar Good Friday of just 12.5 12.5 days. ays. Appl Appliicati catio on of the the rul rule "Sun "Sund day following the Paschal Moon" keeps Good Friday in better alignment with the 14th lunar day. For at the minim minimum um found found above, above, Good Good Frida Friday y falls falls more more th th closely on the 14 or 15 lunar day. Recapping the "raised up on 3rd day", Good Friday was day one by inclusive counting as Jesus was laid in grave before sunset on the Sabbath. Then day 2 was Friday after sunset to sunset Saturday, and and day day 3 incl inclus usiv ive e coun countt was was Satu Saturd rday ay afte after r suns sunset et unto unto abou aboutt sunri sunrise se Sund Sunday ay.. Thus Thus the the reason for sunrise Easter service is to memorialize "Christ our Passover", "a feast by an ordinance for ever." To the end that others may wish to abolish this ".- feast by an ordinance for ever", they neglect the blessings of the Almighty and sadly invite the fires of judgment to blot their memory from off the earth! UTC to Jerusalem Jerusalem Meridian Meridian Time Time Adjustment to Jerusalem ti time can be accomplished by adding 2.33 hours or 0.098 days to the “if” the “if” statements statements and day of week calculations.
Astronomical Easter Comparison & Calculation Frequ Frequenc ency y of Variat ariation ion For For 24 year years, s, from from 2001 2001 to 2025 2025,, only only once once did the Astronomical Easter differ from the Catholic Easter. That being 2019, with Astro Easter on 24March, and Catholic Easter on 21April. Many of the tabulated dates of the table were evaluated for known difficult dates likely to offer discrepancy between the two methods. Even so, only 11 of 60 years show variance between the methods. So most likely, 90% of years will show the Catholic Easter and Lunar Easter are same. Other Feasts: Pentecost, Tabernacles, and The Lord's Supper
The other two of the three main feasts were Pentecost and Tabernacles. For Christians, Pentecost (7 weeks after Easter) celebrates the Holy Spirit. The feast of Tabernacles celebrates being freed from bondages of the flesh, the world, and eternal death, thru election by God the Father. By celebrating these 3 feasts, Passover (hope of a more better better resurrect resurrection ion thru Jesus Christ), Christ), Pentecost Pentecost (hope of continual continual renewal renewal by Holy Spirit), Spirit), and Taberna Tabernacles cles (hope of elec electi tion on and free freedo dom m from from bonda bondage ge thru thru grac grace e of Father God), the Holy Trinity can be honored by Christians. Beca Because use Jesu Jesuss initia initiate ted d the Holy Holy Comm Commun union ion,, then then communion on these days should be held in high esteem. The 1st of the first and seventh lunar months are seen as spec special ial days days or Sabb Sabbath ath's. 's. Also Also,, the the middl middle e of these these months are feasts of remembrance. Pente Pentecos costt is by the definitio definition, n, alway alwayss on a Sunday Sunday by virtue of the week count, 7th Sunday after Easter. Easter.
by OP Armstrong 10.25/15
"And ye shall count unto you from the morrow after the Sabbath, from the day that ye brought the sheaf of the wave offering; seven Sabbaths shall be complete: Even unto the morrow morrow after after the sevent seventh h Sabbat Sabbath h shall shall ye number number fifty days; and ye shall offer a new meat offering unto the LORD." Lev Lev 23: 23: 15/1 15/16. 6. The The Fift Fifty y days days wer were numb number ered ed from from a Saturday, Saturday, so the 50th day falls always on a Sunday, 49days 49days or 7 weeks after Easter Sunday For Christians, the Easter Sunday sets the precedent to find the Sunday in the 7th month. The earliest Easter being 22 March. By this the 7th lunar month starts around the last week of September September.. The latest latest Easter falls falls on 25 April, by th which the 7 lunar month looks to fall on last week of October October. The 1st Sunday after and a 3 rd Sunday would be a logical memorial for the Autumn Feasts. Effe Effect ct of using using Jerusa Jerusalem lem as Princ Princip iple le Meri Meridi dian an Exer Exerci cise se Cautio Caution n when when the Univ Univer ersal sal Time Time new new moon moon moment is on a Saturday night after 21:30 hours. As in 1998, when the Full Moon moment was Saturday at 22:23. The +2h21m offset between GMT and Jerusalem, gives a Paschal full moon on 00:44 Sunday 12 Apr. This astronomical Easter is dela delaye yed d unto unto the foll follow owing ing Sund Sunday ay.. Comp Compar ared ed to a Catholic and/or Universal Time Easter of 12April. 12April. This shows the principle difference in the two methods. The Catholic calculation is stepped in days and weeks. The Astronomical method is a moment defined calculation. Thus it depends upon upon the the deta detail ilss of complex astronomical astronomical calculations. calculations. NEXT 4
Astronomical Easter Comparison & Calculation
by OP Armstrong Armstrong st
th
10.25/15 10.25/15
The 3 Mosaic Feasts of 1 month, 7 Month Ex23:14 Ex23:14,, and Pentecost Pentecost Comp Compari arison son of Thre Three e Major Major Hebr Hebrew ew Feas Feasts ts Pentecost, The Table at right compares the three feasts set forth by Reference Nissan, Lev23 Lev23,, Nu28:16 Trumpets Trumpets -Tishari Du16:16-17 Moses. Two of these feast weeks are routinely routinely celebrated celebrated Day/ in most churches. The first is Easter Holy week. That starts Month Marc Ma rch/ h/Ap Apri rill May/ Ma y/Ju June ne Ex23 Ex23:14 :14 Sept Septem embe ber/ r/Oc Oct. t. date+\with Palm Palm Sunda Sunday y, follow followed ed by Good Friday Friday,, and lastly lastly,, Sabbath & blow Resurrection Sunday or Easter Sunday. The first of these 1st Day-A Sabbath, 1st Day of 1st 15th Nissan + 49D, Trumpets 1st of 7th, New Moon & sin two Holy Days of Palm Sunday and Good Friday, correspond correspond Sabbath Month Easter + 49 days offering approximately to the 10th and 15th days of a lunar month. A Sabbath Sabbath Lev23:8, Lev23:8, This provisioned that the 1st day of the lunar month falls 7th Day A Sabbath Ex12:6 th th upon a Friday, the 8 day being a Friday also, the 10 day is pick passover Lamb, 9th evening to 10 th day of atonement, th then then a Sund Sunday ay,, and and the the 15th day day being being a Frid Friday ay.. It is 10 Psalm Sunday evening fasting provisioned that Jesus died on a Friday and His empty Passover: sunset tomb found early Sunday; thus the t he term Holy Week. th 14 Deut16:6, kill lamb, The feast of Pentecost is celebrated in most churches 7 Ex12:6 weeks after Easter. These first two Mosaic Feasts being unleven bread 7 days, Tabernacles for 7 celebrated, begs the question? If two are celebrated, then 15th 15th is Good Friday & days, start with a th rd 17 Easter Sabbath why not the 3 and if so, then how so? If one remembers st 21 evening end unleven the words of the blessed Savior, “When you Fast” they nd 22 offering for 7 day were not in the permissive sense but in the imperative rd instance. instance. The 40 days of Lent could be taken taken as something something 23 Assembly dealing with fasting. However the Master was noted to Taking out the two and a fraction of weeks elapsed unto have kept the Feast of Tabernacles. So then does it not Easter, then 22 weeks after Easter corresponds to nearly the seem proper for churches to also remember this 3rd Feast? start of the seventh seventh lunar month. A comparison comparison of dates 22 dates 22 The Sunday following Passover is typically Catholic Easter weeks after Easter to 1 to 1 Tishrei shows Tishrei shows the two dates agree Sunday. Sunday. As Easter is the Sunday following the first full moon within a few days. The odd exceptions being when Hebrew after Spring Equinox, then the start of the lunar month is calend calendar ar postp postponem onement ent rules rules are are applied applied.. One possi possible ble about 2 weeks prior. church memorial to this time would be Eucharist 22 and 24 A luna lunarr mont month h is about about 28.5 days days in length length.. Six Six lunar lunar weeks after Easter. BACK NEXT months are then 171 days or o r 24.4 weeks.
by OP Armstrong Armstrong
Astronomical Easter Comparison & Calculation Only one application application offers offers astronomical or ‘uniform’ Easter Easter calcul calculati ation. on. Kalen Kalendis dis has not been been port ported ed to mobi mobile le devi device ces. s. This Excel spread spreadshee sheett will find the Astronom Astronomica icall Easter Easter for years between 4007BC and 3027AD. In Table to right, are differences between this calc and others others Astro-East Astro-Easter er.. For these, these, exact values of full Moon full Moon and and Equinox Equinox are are input. The evaluated Uniform Easter dates by this excel sheet were the same day as those from Kalendis from Kalendis.. The below chart shows variations of my Full Moon Date to Kale Kalend ndis. is. The The norm normal al rang range e is +/+/- 1 hour hour for for about 7000 years. Beware, not all impl implem emen enta tati tion onss of mobi mobile le Exce Excell have have the the needed needed numeri numerical cal accur accuracy acy to evalua evaluate te these these complex expressions. Only IOS-Numbers and MSExcel Excel routines routines could accurately execute execute these complex calculations.
Easter Easter
97 YR's As tro v Catholic Easter Variants Variants 2038AD-1AD
YR
Astro
Cath
2038
28Mr
25Ap
2019
24Mr
1998
10.25/15
by
UTC
thi s astro
F u l l Mo on on
Eq ui ui n ox ox
2465503 503.59 .59 28-Mar 2465
24655 24655 03.03 03. 03
21Ap
24- Mar
2458563.57 2458563.57
2458563.42 2458563.42
19Ap
12Ap
19-Apr
2450915.43 2450915.43
2450893.33 2450893.33
1967
02Ap
26Mr
2- Apr
2439575.64 2439575.64
2439570.82 2439570.82
1962
25Mr
22Ap
25-Mar
2437744.83 2437744.83
2437744.60 2437744.60
1954
25Ap
18Ap
25-Apr
1943
28Mr
25Ap
28- Mar
2430805.42 2430805.42
2430805.00 2430805.00
1927
24Ap
17Ap
24-Apr 9-Apr
2406353.32 2406353.32
2406333.76 2406333.76
1876
16Ap
1845
23Mr
30Mr
30- Mar
2395014.35 2395014.35
2395011.24 2395011.24
1829
19Ap
26Ap
26-Apr
2389167.08 2389167.08
2389167.36 2389167.36
03Ap
10Ap
10-Apr
2387719.77 2387719.77
2387706.39 2387706.39
1825
UTC
09Ap
1818
22Mr
29Mr
29- Mar
2385151.09 2385151.09
2385149.70 2385149.70
1805
14Ap
21Ap
21-Apr
2380425.49 2380425.49
2380401.54 2380401.54
1802
18Ap
25Ap
25-Apr
2379333.61 2379333.61
2379305.83 2379305.83
1744
29Mr
05Ap
1724
09Ap
16Ap
9- Apr
1700
04Ap
11Ap
4- Apr
550
26Ap
19Ap
26-Apr
1922022.20 1922022.20
1922022.43 1922022.43
- 61
me 26Mr
N oC oCal c
29-Mar
26- Mar 1698859.401
1698859.395
Astronomical Easter Comparison & Calculation
by OP Armstrong
Excel Formula for Catholic Easter by Julian Day Number for Calendar Converter or direct C var .f10
P'
.f11
P
.f12 D.1
!. Single Input of Gregorian Year, ear, if BC then 1-Yr 1-Yr.BC & find Easter in 7 steps, i.e. 30 30AD=> AD=>
Excel Cell 30
=MOD(-8-11*MOD(Yr =MOD(-8-11*MOD(Yr,19)+INT((Yr-1600)/100)-INT(( ,19)+INT((Yr-1600)/100)-INT((Yr Yr-1600)/400)-INT((8*INT(( -1600)/400)-INT((8*INT((Yr Yr-1400)/100))/25),30) -1400)/100))/25),30)
14
=P' P'-IF( -IF(P' P'=29,1,IF((1+MOD( =29,1,IF((1+MOD(Yr Yr,19))>11,IF( ,19))>11,IF(P' P'= =28,1,0),0))
14
=118+INT(365.25*(Yr =118+INT(365.25*(Yr+4712))-INT(0.75*INT((( +4712))-INT(0.75*INT(((Yr Yr))/100)+49))
1732097
=D.1 D.1+ +P
1732111
=1+INT(MOD((1.5+D.2 =1+INT(MOD((1.5+D.2)),7))
Next page
5
.f13 D.2 .f6 D.3 .f14 D.4 end Easter
10.25/15
=IF(D.3 D.3=1, =1,D.2 D.2+7,8+7,8-D.3 D.3+ +D.2 D.2)&"Catholic )&"Catholic Easter as Julian Day Number"
1732114
use D.4 to get Day of Month, f.9, f.9, &Month#, f.8, &day-of- Week,f.6 as =D&f.n&… D7M4wkd1
Above is Catholic Easter formulation using Gregorian Year as inpu inputt to find find East Easter er Date Date as a Julian Julian Day Day Numb Number er.. This This simplifies several steps as compared to other methods. The resultant JD# may be used to rapidly find Pentecost Sunday or any number of Easter dependant days. The JD# can be rapidly changed to calendar dates via formula f.6, f.8, & f.9 for most, if not all, dates. Beware excel dates do not display for years prior to 1900. Thus the above work around. The Easter JD# can be ported to excel date system for years 1900 forward. Simply determine the offset to Excel Excel date number and JD#, then subtract subtract offset offset to other other dates. dates. Apply this method when a wide range of dates are to be reviewed.
Most Easter routines, but not this one, are valid for a few hundred years after 1901. This is substantial since Christ our Passover was slain from the foundation of the world. Thus there has been a perpetual Easter since the day this world was founded. That date was when Adam sinned, 4000 years prior to Jesus Baptism at river Jordan by Saint John John.. The The day day of resu resurr rrec ectio tion n was was hidd hidden en by God and revealed by Christ earthly Easter day in 30AD. The following list dates when Astronomical Easter is not same date as the Catholic Easter, Easter, about 1 in 12 years.
257973-365.242454*(-4006-Y)+MOD((365.242454*(-4006257973-365.242454*(-4006-Y )+MOD((365.242454*(-4006-Y Y)),29.5306)+CF Another estímate of astronomical PFM is given at left. This uses Delaunay arguments CF= -0.40614*SIN( l')+ 0.01614*SIN(2 l’) + 0.17302*SIN( l) - 0.17+ CFt for lunar lunar and sun anomal anomalie ies, s, expres expresse sed d in CFt = - C^3/999999.45 C^3/999999.45 - C^2/4028.335 - C/64.259 C/64.259 + 1/ 547.41 Julian century, century, C of J2000.
Astronomical Easter Comparison & Calculation Variant ariant Astro-Easte Astro-Easterr Years to Catholic Catholic
2049 2045 2038 2019 1981 1974 1967 1962 1954 1943 1927 1924 1923
1700 1693 1685 1666 1629 1622 1609 1598 1590 1582 1578 1571
145 1457 145 1453 144 1446 142 1429 142 1427 141 1419 140 1409 140 1408 140 1402 138 1389 137 1375 137 1370 1362
12 1206 11 1199 11 1182 11 1162 11 1155 11 1142 11 1135 11 1128 1111 11 1108 11 1101 10 1088 1081
915 895 881 861 854 846 837 827 817 810 783 736 729
543 536 519 516 509 496 489 482 474 469 438 411
254 242 235 218 216 216 191 191 188 171 168 168 144 137 124
This tabulation covers from years 2050 back to 60BC. A span of about 2100 years. The above Tabulation shows that some centuries have less discrepancy than others. The cycle cycle of 391-19 391-19 or or 372 years years can be seen in some skip sequences. The average is about 11 of 12 years agree and 1 in 12 years are not in agreement between between the two Easter Methods. However the pattern is not uniform. Some centuries have fewer concurrences than others. Next page
by OP Armstrong
10.25/15
Variant ariant Uniform Uniform Easter Easter Years to Catholic Catholic
1903 1900 1876 1873 1845 1829 1825 1818 1805 1802 1778 1744 1724
15 1558 15 1552 15 1551 15 1527 15 1514 15 1507 15 1503 14 1487 14 1484 14 1483 14 1473 14 1465 1463
135 1351 133 1332 133 1331 132 1328 131 1313 131 1311 130 1308 129 1294 128 1284 127 1277 123 1237 122 1226 1218
107 1074 106 1061 103 1030 100 1003 998 979 960 959 941 935 932 922
716 685 634 614 607 590 588 587 570 569 563 550
384 367 343 323 320 313 303 292 289 273 269 262
73 66 42 22 15 -5 -29 -49 -53 -56 1998
Given the primitive nature of physics and astronomy at the the time time of the origi origina nall form formul ulat ation ion of the the East Easter er Calendar, the Catholic method mostly agrees with the Astr Astron onomi omica call form formul ulat ation ion.. That That being being stat stated ed,, the Pasc Pascha hall Moon Moon of the the Catho Catholic lic Easter Easter is not not a true true moon moon.. In fact fact there there is simil similar arity ity to to ‘Molad’ of the the calculated Hebrew Calendar. The Molad event is timed to star startt aroun around d the autum autumn n equi equino noxx vs. arou around nd or about the Spring Equinox for Catholic Easter. Easter. 8
Astronomical Easter Comparison & Calculation Uniform Uniform Easter Spreadshee Spreadsheett Method Method: Find Find full full moon moon
using longitude routine and: Moon age = (Solar less Moon Longitudes)*29.5306/360 Since Full moon is 180 degree, then full moon age is half half of 29.530 29.530… … Three approximations approximations (February, (February, March, April) are used to find f ind Paschal full moon date. When solved yield average days between spring full moons for the year. The first guess when corrected, may not meet Paschal Criteria after correction. Thus 2 checks are used. Given the Equinox for a year, as EQ.jd, then the logics are: 1) IF(guess < EQ.jd, guess, then then new new is (old (old gues guess+ s+29 29.5 .5), ), Gues Guesss-1 1
is 257972.7-
365.242454*(-4006-Yr)+MOD((365.242454*(-4006-Yr)),29.5306)
from from whic which h gues guess, s, an age age is dete determ rmie ied. d. This This AGE AGE is converted to full moon correction, CF, for an improved guess as follows: follows: Cf i =29.5306/2 - IF(AGE>0, AGE, 29.5306+AGE)
Guess2 Guess2 = Guess1 + CF1, CF1, other iterations iterations are made as Guess3 = Guess2 +CF2, the 4th and final value, Guess4 = Guess3 + CF3 is taken as the Full Moon date. A final check is used to ensure the Full Moon date falls after the Spring Equinox. Equinox. Macro Macro loops loops are avoi avoide ded d by usin using g succ succes essi sive ve Full Full Moon Moon date dates: s: estimate, estimate +/- 29.5 days and logically select PFM. The day, 257973, is March FM of Gregorian year -4006. The file is embedded on page 6 of the MS document file. This file may be used to better see the method. A simplified method screened the prior table for validation by 70 term method. FM JD# Constants: April 258002.07, March257972.70, February 257943.37
by OP Armstrong
10.25/15
The Paschal Full Moon is defined as first moon, after the the Spri Spring ng or Verna ernall Equi Equino nox. x. This This spre spread adsh shee eett method of longitude was corrected and adapted after T. Alo Alonso nso Albi’s adap adapta tati tion on of P.Duf .Duffe fett but but with with additional terms. T. Albi’s original Albi’s original lambda terms were only about 24. Accuracy was improved by using more Lambda terms. The total Sine terms comes to about 70. Evaluation of Delaunay’s arguments Delaunay’s arguments from Chapront & Chapront ELP-S2001 was used to improve accuracy. accuracy. The ELP-S2001 4th order terms are reported to be within 10 arc seconds down to 1500BC, if using the full series of 100’s 100’s of of adjustments. Here only about 70 terms are employed. The original method does not account for Tidal Braking. Thus, an Equation of Time is used used to extend extend the soluti solution on validi validity ty.. The extend extended ed solution calculates Paschal Full moon dates to within an hour hour of NASA NASA or Kale Kalend ndis is for for year years s betw betwee een n 4000BC to 3000AD. Typically less than 60 minutes difference is found. The graph on on page 6 page 6 shows the variation for 110 points over a 7100 year span. It is incorrect to apply angle reduction, Mod360, for values less than zero. The spreadsheet solution is as fol follow lows: redu reduce ce angl angle e by A/36 A/360 0=rA =rA, then then fin find IF(ABS(rA)>1, 360*(ABS(rA)-INT(ABS(rA)))*SIGN(rA), 360*rA). This yields a correctly reduced angle for either positive possib ible le to impr improv ove e or negati negative ve input input values values.. It is poss accuracy by limiting the year range, to just a few century.
Astronomical Easter Comparison & Calculation New New Moon Moon Mome Moment nt by Spreadsheet The prior calculations were for finding a Full Moon moment by longitudes. Here a new moon moment is also calculated by longitudes. This improved improved the accuracy accuracy of of “moon age” cell in spreadsheet ‘Gregorian spreadsheet ‘Gregorian date date to/from Julian Day’ Julian Day’.. The prior prior calc calc used used lunat lunation ion numbe numberr. The The Sola Solarr to Luna Lunarr longitud longitudes es method method improv improves es accura accuracy cy when when used used with adaption. The adaptation was to tailor an EOT specific for this data set. In the first instance, Broomberg’s instance, Broomberg’s 4th order lunatio lunation n number number corre correcti ction on was adjust adjusted ed to minimum minimum error around J2000’ J2000’. The results of that fit are given in graph of this page. This gave +/- one hour error around the mean mean betwee between n 4000BC 4000BCE E and 3000AD. 3000AD. Next a 3rd order suppl supplem emen enta tall corr correc ectio tion n was was fitt fitted ed in term termss of Juli Julian an Millennia or c’/ or c’/10. 10. Where c’ Where c’ is is (JD#- 2451550.098)/36525. 2451550.098)/36525.
by OP Armstrong
Nov7/15
The new moon moment moment was found found by setting setting the longitude For 128 128 poin points ts,, the the Stan Standa darrd difference to zero. The EOT is required because Longitude deviation was about 3 minutes at methods do not account for tidal braking of earth rotation. a mean mean of about about zero. zero. The RMS RMS This method is known as a truncated series of the VSOP mean was 2 minutes. The analytical series, as compared to the gold standard direct integration on of DE200, DE200, as discussed discussed in USNO write ups. Take maximum/min maximum/minimum imum differen differences ces integrati wer were 10 and and -7 minu minute tes. s. This This as example March 30 AD: guess M15 and JD# is 1732091, nearly corresponds to the 12 min and EOT date is 1732095.950, and is used for second guess, gap of 2 times SD. This method the third and final calc gives a JD# of 1732096.244 for the expla xplain inss 98% of dete deterrmine mined d EOT corrected date when moon age is zero, this is Mar20 17:51HR vs. NASA table of 17:47 or 4 minute later. Likewise variance. Because the original equation is in hours, divide by 24 to for April 457BC , start midmonth at 1554614.00 to arrive at give days. This correction is then added to the the Broomberg Broomberg EOT JD# 1554617.311 or 19:28HR vs.19:25 NASA. NEXT term. The sum is then subtracted to yield a new moon BACK
