Principles of Navigation

Principles of Navigation

Great Circles & The centre of a great circle cuts through the sphere.

'mall Circles & The centre of a small circle does not cut through the sphere.

A circle drawn on the sphere such that’s its plane passes through the centre of the sphere is called a Great Circle. The radius of a great circle is the same as the radius of the sphere.

The pole of a great circle circle is the point on the sphere sphere 900 from any point on that great great circle. very great great circle has two poles! poles! one at either end of the a"is perpendicular to the centre of the plane of the great circle.

Principles of Navigation Any two points on the sphere can (e )oined (y a great circle! and the smallest arc of a great circle is the shortest distance (etween the two points. The arc of a great great circle is measured measured in degrees degrees (y the angle it su(tends at the centre of the sphere. Thus distance may (e e"pressed as an angular measure. Circle * +,00

#0 * ,0’ -of arc

#’ * ,0/ -of arc

#00 1’ +,/can (e #00 1.,’ #00 1’

can (e #0.10

'pherical Angle2 β

A spherical angle is formed (y the intersection of two great circles circles on the surface of the sphere! and is e3uivalent to the plane angle (etween the tangents of the great circle at the point of intersection. 'pherical Triangle2 A (

A spherical triangle may only (e formed (y the intersection of three great circles.

c

C

4 a

A spherical triangle is formed formed (y the intersection of three great circles on the surface of a sphere. 'mall Circle2 A circle drawn on a sphere such that its plane does not pass through the centre of the sphere sphere is called a small small circle. A small circle circle cannot form part of a spherical triangle.

Principles of Navigation Any two points on the sphere can (e )oined (y a great circle! and the smallest arc of a great circle is the shortest distance (etween the two points. The arc of a great great circle is measured measured in degrees degrees (y the angle it su(tends at the centre of the sphere. Thus distance may (e e"pressed as an angular measure. Circle * +,00

#0 * ,0’ -of arc

#’ * ,0/ -of arc

#00 1’ +,/can (e #00 1.,’ #00 1’

can (e #0.10

'pherical Angle2 β

A spherical angle is formed (y the intersection of two great circles circles on the surface of the sphere! and is e3uivalent to the plane angle (etween the tangents of the great circle at the point of intersection. 'pherical Triangle2 A (

A spherical triangle may only (e formed (y the intersection of three great circles.

c

C

4 a

A spherical triangle is formed formed (y the intersection of three great circles on the surface of a sphere. 'mall Circle2 A circle drawn on a sphere such that its plane does not pass through the centre of the sphere sphere is called a small small circle. A small circle circle cannot form part of a spherical triangle.


The Terres Terrestrial trial 'phere -The arth The Poles2 The arth spins on its its a"is once per day! and the the point where the the a"is meets the surface of the earth are called the North 5 'outh Geographical Poles. Poles. North Pole

6eridians

3uator

Parallel Parallel of 8atitude

'outh Pole

The 3uator2 3uator2 6idway (etween the poles lies a great circle called the e3uator. e3uator. 7t divides the earth into two e3ual hemispheres! hemispheres! North 5 'outh! and all 0 points on it are 90 from the poles. Parallels of 8atitude2 'mall Circles lying parallel to the e3uator and diminishing in sie with increasing distance from it are :nown as parallels of latitude. The e3uator and the parallels parallels run east west and and cut all meridians at 0 90 . 6eridians2

Principles of Navigation 8ines of longitude or meridians are semi great circles )oining the poles. 6eridians run due due North ; 'outh. The one passing through through the meridian instrument instrument at the site of the Greenwich Greenwich <(servatory in 8ondon is ta:en as the Prime 6eridian for the purpose of measuring longitude.

8atitude2 The latitude of a point on the earth’s earth’s surface is the arc arc of a meridian measured measured North or 'outh from the e3uator to the parallel of latitude through the point. 7ts =alue ranges from 00 at the e3uator to 900 at the poles. 8ongitude2 The longitude of a point is the arc of the e3uator e3uator or the angle at the poles (etween the Greenwich 6eridian and the meridian passing through the point measured #%00 ast and >est from Greenwich. N.4. Assuming the earth to (e a perfect sphere! the arc of the great circle de?ning latitude and longitude are e3ual to the angles su(tended (y them at the centre of the earth.

$0

$0 $00 $00

Greenwich 6eridian

@istance
Principles of Navigation The earth is commonly commonly regarded regarded as a sphere! sphere! (ut is in fact slightly Battened at the poles and (ulged at the e3uator. e3uator. 7t is an o(late spheroid or ellipsoid. 3uatorial diameter Polar diameter

9 miles 900 miles

3uatorial ci circumference Polar circumference

190 mi miles 1%,0 miles

The 'tatute 6ile2 6easures $%0 feet -#,09 metres! Not used in Navigation -e"cept Canada. The Geographical 6ile2 The length of arc o off the e3uator su(tending su(tending at the centre of of the # earth an angle of # minute of arc - ;,0 degrees. The Nautical 6ile2 The length of arc of of the meridian su(tending su(tending at its centre of of curvature an angle angle of # minute. 7f the earth were were a perfect sphere! sphere! the Nautical 6ile would e3ual the Geographical 6ile. owever (ecause of the earth’s o(lateness the length of the Nautical 6ile is not the same in all latitudes! (eing greater at the poles than at the e3uator. e3uator. The unit used in Navigation is the 7nternational Nautical 6ile -#%$ metres! which is the average length of # minute of arc of a meridian in latitude 1$0. The unit of speed is the :not! which is # Nautical 6ile Per our. Dhum( 8ine2 7f a curve is drawn on a sphere so that it cuts all the meridians at the same angle! it is called a rhum( line. A rhum( line is a curve spiralling upwards towards towards the poles (ut never 3uite reaching them. >hen a ship follows a rhum( line (etween any two points! the true course remains constant. The rhum( line distance is not the shortest distance (etween two points e"cept where where the rhum( rhum( line coincides with the great circle. As on the e3uator or meridian. The 'olar 'ystem2 The solar system consists of the 'un and and many smaller (odies (odies which revolve around it. These include2 #. The 6a)or Planets & 6ercury! 6ercury! =enus! =enus! arth! arth! 6ars! 6ars! Eupiter! 'aturn! Franus! Neptune and Pluto.

Principles of Navigation . The 6inor Planets or Asteroids which consist of thousands of small roc:y o()ects or(iting mainly (etween 6ars and Eupiter. +. Comets 5 6eteors. ith the e"ception of Pluto! the or(its of the planets are very nearly in the same plane. The planets rotate on their a"es from west to east! e"cept for =enus which is thought to (e retrograde. All the planets revolve around the 'un from west to east.

epler’s 8aws of Planetry 6otion #. All Planets revolve in elliptical or(its with the sun as a focal point of the ellipse.

Perihelion

Aphelion

. ach planet revolves so that the line )oining it to the 'un sweeps out e3ual areas in e3ual times. +. The s3uare of the time ta:en from a planet to complete one revolution is proportional to the cu(e of its mean distance from the 'un. The arth’s
Principles of Navigation -Deal motion of the arth around the 'un. Perihelion 5 Aphelion refer to any (ody in or(it around the

Aphelion

Perihelion

9.900.000 miles & mean distance

arly Euly 'lower or(ital speed.

arly Eanuary Iaster or(ital speed.

An ellipse of small eccentricity with the 'un at the focus of the ellipse. The 'easons2 'pring ; =ernal 3uino" 6arch #st 'ummer 'olstice Eune nd

>inter 'pring

Aphelion

Perihelion

Autumn

'ummer

>inter 'olstice @ecem(er nd

Autumnal 3uino" 'eptem(er +rd The arth’s a"is is inclined at +H 0 to the perpendicular to the plane of its or(it! and it points in the same ?"ed direction is space. Thus each pole is tipped alternately towards and away from the 'un for half the year to give rise to the changing seasons. The inclination of the a"is determines the (oundaries of the climatic ones.

North Pole Irigid Arctic Circle

,,H North


Temperate Tropic of Cancer

+H North Torrid Jone +H 'outh

Tropic of Capricorn Temperate

,,H 'outh

Arctic Circle Irigid 'outh Pole

=ariation in the 8ength of @ay 5 Night2 Continuous @aylight

Continuous @ar:ness

'ummer 'olstice

>inter 'olstice

' F N 8 7 G  T

Continuous @ar:ness

Continuous @aylight

The only time of the year that there is an e3ual day and night in each hemisphere! is at the time of the e3uino"es. This occurs at all latitudes. There is e3ual day and night at all times of the year at the e3uator.

Principles of Navigation The Celestial 'phere2

North Celestial Pole

Jenith

Parallel of @eclination

NP <(server K

3

3

K

Celestial 3uator 'P Terrestial 3uator

Nadir 33 & Terrestrial 3uator N3' & <(servers 6eridian 6eridian

Celestial ; Dational orion KK & Celestial 3uator '.C.P. NK' & <(servers Celestial

Celestial Poles2 The arth’s a"is of rotation e"tended in (oth directions meets the celestial sphere at two points called the celestial poles. The pole a(ove the o(servers horion is called the elevated pole. 'ince the arth rotates from west to east! the celestial and all the (odies on it are apparently moving from east to west at an hourly rate of #$0. Celestial 3uator2 A great circle corresponding to the arth’s e3uator e"panded outwards onto the celestial sphere. All points on it Are ninety degrees from the celestial poles. our Circle2 A semiLgreat circle )oining the celestial poles! and cutting the celestial e3uator at right angles. Parallel of @eclination2 A small circle on the celestial sphere whose plane is parallel to the plane of the celestial e3uator. @eclination2

Principles of Navigation The declination of a celestial (ody is the arc of the hour circle passing through the (ody measured north or south from the celestial e3uator. 7ts value ranges from 00 at the celestial e3uator to 900 at the celestial poles. Polar @istance2 The arc of the hour circle passing through a (ody measured from the elevated celestial pole to the parallel of declination through the (ody. P<8AD @7'TANC * 90° ± @C87NAT7
Apparent Annual Path of the 'un MThe cliptic2 The arth’s e3uator is inclined at +° ’ to the plane of its or(it. The arth really revolves around the 'un! (ut to an o(server on the arth! the 'un traces out an annual path in the form of a great circle around the celestial sphere! inclined at +° ’ to the celestial e3uator.

Path of the 'un2

Eune nd 

'ept +rd 

<(li3uity of the ecliptic +°’

Celestial 3uator



6arch #st cliptic @ec nd 


 

Iirst point of 8i(ra Cancer Iirst point of Capricorn



Iirst point of



Iirst point of

Aries

Iirst point of Aries The point where the 'un crosses the celestial e3uator from North to 'outh! on or a(out 6arch #st is called the ?rst point of Aries and is a most important point in Navigation or Astronomy. 'idereal our Angle2 7s the angle at the celestial pole or arc of the celestial e3uator measured westwards from the ?rst point of Aries to the hour circle passing through the (ody starting at 00 at  through +,00! (ac: to  again. Dight Ascension2 7s the angle at the celestial pole or arc of the celestial e3uator measured eastwards from the ?rst point of Aries to the hour circle passing through the (ody starting at 00 at  through +,00! (ac: to  again. Astronomers use Dight Ascension. 'A O DA * +,00

Coordinate system without reference to the <(server2 'A 5 @eclination Algol

CNP

'A +#10 @ec 1#0N

@eclination

#%00

00

900 00



'A


NCP

'irius 'A $#0 @ec #0' 

Coordinates with reference to the <(server2

Jenith

Jenith @istance

Altitude

> Dational or Celestial orion

'

North point of the horion


Amplitude

Nadar

Jenith2 7s the point on the celestial sphere directly overhead the o(server. Nadar2 7s the point on the celestial sphere directly (elow the o(server. Dational -Celestial orion2 A great circle midway (etween the enith and the nadar is the Dational or Celestial orion. All points on it are 900 from the enith.

=ertical Circle2 'emiLgreat circles )oining the enith and the nadar are called vertical circles. They cut the rational horion at 900. The one that runs North ; 'outh through the enith coincides with the o(servers celestial meridian. The one running east ; west through the enith is called the Prime =ertical Circle. Amplitude2 7s the angle contained at the enith! or arc of the rational horion! (etween the prime vertical circle and the vertical circle passing through a (ody rising and setting. LLLLLLLLLLLLL The position of a (ody on the celestial sphere may (e indicated (y two coLordinates having reference to the o(server. These are altitude and aimuth.


Altitude2 The true altitude of a (ody is the angular distance along the vertical circle from the rational horion to the (ody. 7ts value ranges from 00 at the rational horion to 900 at the enith. Jenith @istance2 This is the angular distance from the enith to the (ody measured along a vertical circle from the enith to the (ody. 7ts value ranges from 00 at the enith to 900 at the rational horion. Jenith @istance * 900 L True Altitude. Aimuth2 7s the arc of the rational horion from the north point cloc:wise to the vertical circle passing through the (ody! from 00 at the north point through +,00 . 'uperior 6eridian2 The semiLgreat circle )oining the north 5 south celestial poles through the <(server’s enith is called the <(server’s 'uperior 6eridian. An o()ect on the o(server’s superior meridian is said to (e at upper meridian passage or upper transit and its aimuth is either 0000 or #%00. The altitude of any (ody is greatest at upper transit. The reverse applies to the inferior meridian. Circumpolar 'tars2 A star (ecomes circumpolar whenever the polar distance is less than the o(server’s latitude -P"  Pn. A circumpolar (ody is visi(le at upper transit -" and also when it is on the meridian (elow the pole! i.e. lower transit.

Jenith

Circumpolar & Never 'ets

<(server’s 'uperior 6eridian " NCP "

"

'etting 'etting

'etting


North point of the horion

'

Dising Dising

Dising

'CP

<(server’s 7nferior 6eridian

The latitude of an o(server is the altitude of the elevated celestial pole.

6easurement of 6ovement of Celestial 4odies Greenwich our Angle2 7s the angle at the elevated celestial pole! or arc of the celestial e3uator measured westward from the Greenwich Celestial 6eridian to the hour circle passing through the (ody. 7ts value is 0000 when the (ody is on the Greenwich celestial meridian and #%00 when the (ody is on the inferior Greenwich celestial meridian.

PN

Celestial 3uator Celestial 3uator

'A



GA



'A






GA



GA GA







P'

GA

 *

'A

 O

GA

8ocal our Angle2 8A is the angle at the elevated celestial pole or arc of the celestial e3uator measured westwards from the o(server’s celestial meridian to the hour circle through the (ody.

PN

PN

<(server

Greenwich 8A



'A

8ong >



8ong 

Greenwich

<(server

PN

Greenwich Celestial 6eridian

Celestial 3uator

<(server GA



8ong 

8A


8A of Aries2 The angle at the elevated pole or arc of the celestial e3uator measured westward from the o(server’s celestial meridian to the hour circle passing through the ?rst point of Aries.

8A



* 'A

 O

"ample2 Iind the GA of 'pice on the 0th 6ay #9,% Q 0,h 1$m 0$s G6T GA  Q 0,h 7ncr. -1$m 0$s 'A  GA 

+0 $.1’ ##0 #%.#’ #$90 0.0’ #+%0 .$’ @eclination #00 $9.9'

"ample2 Iind the 8A of 4etelgeuse on the #th Eanuary #9,% Q #,h m +1s G6T for an o(server in longitude ++0 #.0’. 'tate whether east or west of the o(servers meridian. GA  Q #,h 7ncr. -m +1s 'A  GA  8ong  8A 

+$#0 #+.#’ $0 +9.1’ #0 +%.’ ,%0 +0.’ @eclination 0 1.N ++0 #.0’ +0#0 1. 4ody ast of the 6eridian.

Principles of Navigation 4odies on the 6eridian Consider the 'un on the 0th 6ay #9,% at 0,hrs G6T seen (y an o(server on his meridian at a true altitude of ,00 and aimuth #%00. 8atitude Jenith R

'unset

Jenith @istance

@ec.

PN

K @iurnal path of the 'un.

Altitude

8atitude

>

N Rational Horizon

'



Celestial Equator

'unrise

GA  Q 0,h GA  Q 0,h 'A  KJ -8at 8atitude 8atitude

00 $+.,’ @eclination 00 N +0 $.1’ +00 $,.’

* KR O JR * 00 O +00 * $0N

8A



*

8A  L 'A  +,00 & +00 $1.’ 8A  * $0 0+.%’

Consider the 'un on the $th August #9,% at 0hrs G6T seen (y an o(server on his meridian at a true altitude of 1$0 and aimuth 0000. 8atitude Jenith

K @ec.

P' R 'unset

8atitude >

Altitude


KJ -8atitude

* JR L KR * 1$ L #0 * %0' 0

8A  'A  8A  *

* +,00 * 10 19.,’ #+$0 #0.1’

Consider the 'un on the #0th Eanuary #9,% at #hrs G6T seen (y an o(server on his meridian at a true altitude of 100 and aimuth #%00. 8atitude K @ec.

R

Jenith


PN

'unset Altitude

>

8atitude



'unrise

Celestial Equator

@iurnal path of the 'un.

* #++0 0%.’ ,10 .’ * ,%0 1#.0’

GA  GA  'A 

*

8A  'A  8A  *

* +,00 * ,%0 1#.0’ 9#0 #9.0’

KJ -8atitude

@eclination 0N

* JR L KR * $0 L 0 * %0 N 0

Consider the 'tar Degulus on the #st 6ay #9,% seen (y an o(server on his meridian at a true altitude of 00 and aimuth 0000.

Jenith 8atitude

P' K

Principles of Navigation 8atitude 'etting

@ec.

>

R Altitude

Celestial Equator

 Dising

@iurnal path of the 'tar

'A Degulus #0 0.$' 8A  8A  * KJ -8atitude

*

0%0 #9.’

@eclination

* +,00 #$#0 10.+’ * JR L KR * 0 L #0 0.$' * $0 $.$’' 0

Consider the 'tar Achemar on the #1th Euly #9,% seen (y an o(server on his meridian at a true altitude of ,$0 and aimuth #%00. @ec. R

Jenith

Altitude

8atitude

K @iurnal path of the 'un.

P'


Celestial Equator

'etting

>

8atitude

Dising

'A Achemar $0 +.1’' 8A  8A  * KJ -8atitude



*

++$0 $#.’

@eclination

* +,00 10 0%.+’ * KR L JR * $ +.1’' L $0 * +0 +.1’' 0

Consider the 'tar @u(he on the $th Novem(er #9,% seen (y an o(server on his meridian at a true altitude of $00 and aimuth 0000. 8atitude Jenith K

R @ec.

Altitude

@iurnal path of the 'tar

PN

Principles of Navigation 'etting

>

8atitude

 Dising

Celestial Equator

'A @u(he 8A  8A  *

#910 +.9’ * +,00 #,$0 .#’

KJ -8atitude

* KR L JR * ,# $$’ N L 100 * #0 $$’ '

@eclination #,0 $$’ N

*

0

Altitudes2

Jenith

Apparent Altitude 'ensi(le orion

O

'

=isi(le orion =

'# <(served Altitude =#


@ip

True Altitude Dational orion D

C

D#

=isi(le orion2 Dange of vision is limited (y the curvature of the earth. To an o(server at S<’ the small circle ==# represents this and is called the =isi(le orion. 'ensi(le orion2 This is a plane through the o(server’s eye -' < '# at right angles to he arth’s radius. Dational orion2 A plane -D C D# parallel to the sensi(le horion passing through the earth’s centre.
Principles of Navigation True Altitude2 The altitude of a (ody’s centre measured at the earth’s centre! i.e. the altitude of the (ody’s centre a(ove the rational horion. Corrections to Altitude2 #. @ip & Deduces the altitude from the visi(le horion to the sensi(le horion. 7t is the angle at the o(server’s eye (etween the sensi(le horion and the line tangential to the visi(le horion. 7t is ta(ulated against height of eye. -Always su(tracted. . 7nde" rror & An instrument error applied directly to the se"tant altitude. -
Ialse Position True Position

<

Nautical Almanac corrections are (ased on standard atmospheric conditions of #00 and #0#0m(. There is a ta(le for non standard corrections at low altitudes. 7n practice (odies of altitudes less than #00 should (e avoided for the purposes of position ta:ing unless there is nothing else availa(le.

Principles of Navigation 1. 'emi @iameter & The 'un and the 6oon are (odies that present a ?nite diameter. The Navigator should measure the altitude of the upper or lower limits. To o(tain the altitude of the (ody’s centre a correction for semiLdiameter must (e applied -Fpper 8im( -L! 8ower 8im( -O. The 'un’s apparent diameter is greatest when the earth is at perihelion in early Eanuary! and smallest when at aphelion in early Euly. $. Augmentation of the 6oon’s 'emiL@iameter &

"tra @istance

6oon

As the moon rises the distance from the o(server decreases and hence the semiLdiameter increases. This only applies to the moon due to its relative pro"imity -10!000 miles 7t is always positive! (ut is only needed for greater accuracy. 6a"imum value is 0.+!

J

,. Paralla" &

Paralla" oriontal Paralla" <

'ensi(le orion


Dational orion

Caused (y the (ody (eing o(served a(ove the sensi(le horion instead of the rational horion. 7t is only eective on (odies relatively close and therefore only applies to (odies within the solar system. oriontal Paralla"2 The angle su(tended at the (ody’s centre (y the earth’s radius when the (ody’s centre is on the o(server’s sensi(le horion. 6a"imum value of paralla". Paralla" in Altitude2 The angle su(tended at the (ody’s centre (etween the o(server and the earth’s centre. The shape of the earth causes a reduction in paralla" due to polar compression. The amount of reduction is measura(le for the moon only and is given in Norries Ta(les! (ut its ma"imum value is o.. 7t is hardly large enough to of any particular importance. 'ine Paralla" * 'ine P " 'ine -900 O Altitude 'ine Paralla" * 'ine P " Cosine Apparent Altitude As Paralla" and oriontal Paralla" are small! therefore 'ine P 5 'ine P and P 5 P are also small! Paralla" * P Cos Apparent Altitude.

'ummary2 'e"tant Altitude #. 7nde" rror . L @ip +. L Defraction

Principles of Navigation 1. O Paralla" $. 'emiL@iameter ,. O Augmentation of 6oon’s '.@. Corrections Corrections Corrections Corrections

to to to to

the the the the

6oon 'un L Planets 'tars L

L #&, #&$ L #&1 #&+

4ac: Angles2 >hen the horion (elow the (ody is o(scured! it may (e necessary to measure the altitude to the opposite point of the horion. To correct a (ac: angle2 #. Apply 7! @ip and '@. . 'u(tract from #%00 +. Apply paralla" and refraction. "amples2 The se"tant angle of @ene( is #$0 .,’! 7nde" rror #.#’ on the arc. eight of eye $ feet.

@ene(

Norries Ta(les

Nautical Almanac

0

'e"tant Altitude

0

#$ .,’

7nde" rror

L#.#’ 0

<(served

#$ .,’ L#.#’ 0

#$ #.$’

#$ #.$’

Altitude @ip

L.0%’ 0

Apparent

L.0’ 0

#$ #1.1’

#$ #1.$’

Altitude Defraction

L+.1$’ 0

True Altitude

#$ #0.9’

L+.$’ 0

#$ ##.0’


The se"tant angle of the 'un’s lower lim( is +10 ,.’! 7nde" rror #.$’ o the arc. eight of eye % feet on Eanuary +rd #9,%.

'un

Norries Ta(les

Nautical Almanac

0

'e"tant Altitude

0

+1 ,.’

7nde" rror

+1 ,.’

L#.$’

L#.$

0

<(served

0

+1 1.’

+1 1.’

Altitude @ip

L%.,,’

L%.,,’

0

Apparent

0

+1 ,.’

+1 ,.#’

Altitude Defraction

L#.+9’

'emi @iameter

O#,.9’

Paralla"

O#1.%’

O0.#’ 0

True Altitude

U

#$ +#.0,’

0

+1 +0.9’

The se"tant angle of the 6on’s lower lim( is 0 #.$’! 7nde" rror #.’ on the arc. eight of eye % feet on
6oon

Norries Ta(les 0

 #.$’ 'e"tant Altitude

Nautical Almanac 0 #.$’


7nde" rror

O#.’

O#.’

0

<(served

0

 0.+’

 0.+’

Altitude @ip

L%.%’

L%.%’

0

Apparent

0

 ##.1+’

 ##.1+’

Altitude Defraction

L0.+’

O%.0’

U 'emi @iameter

O#,.+,’

Paralla" in

O $.0’

O#%.0+’

Altitude 0

True Altitude

0

 1$.$’

'@ Augmentation Corrected '@

#,.#’ 0.,’

 1$.+’

P $9.0’ Deduction L0.01’ #,.+,’ Corrected P $%.9,’

Paralla" in Altitude * P Cos Apparent Altitude * #%.0+’

Pro Iorma 6eridian

@ate

@D 8atitude

8A +,00 ; 0000 8ong GA Ne"t 8ess Demainder s G6T m s

d

h m

h

@D 8ongitude

4ody

'e"tant Altitude 7nde" rror <(served Altitude @ip Apparent Altitude Total Correction True Altitude 0900 00.0’

@eclination

Jenith @istance

Principles of Navigation @-  @eclination

@eclination 8atitude

Answer P;8 0900 ; 00 through 8atitude

8ongitude

Time of 6eridian Passage #. The 8A of a (ody on the meridian is 0000! (ut when the longitude is east it is more convenient to call it +,00. . Fse altitude corrections from the Nautical Almanac. +. Jenith @istance is named opposite to the aimuth of the (ody. 1. >hen the JR 5 @eclination have the same names! add for latitude. $. >hen the JR 5 @eclination have opposite names! su(tract for latitude. ,. All aimuths! courses 5 position lines must (e three ?gure notation. . 7nformation regarding the stars for every month of the year is located at the (ac: of the Nautical Almanac.

"ample2 the se"tant angle of the 'un’s lower lim( when on the meridian was $10 +9’. 7 O.$  +0ft. @ate 01.0#.,% 8A 8ong

@D 8atitude 0000 9, 0 +0’>

@D 8ongitude 9,0 +0’>

'e"tant Altitude 7nde" rror

4ody 'un $10 +9’ O .$’

Principles of Navigation GA Ne"t 8ess #%h Demainder 1%s G6T 1%s @eclination @-  @eclination

9,0 +0’ %%0 1%’ 0 1’

1d

<(served Altitude @ip

$10 1#.$’ L $.+’

+0m

Apparent Altitude

$10 +,.’

1d #%h +0m 0 1,’ 0.’  1$.%’ 0

Total Correction True Altitude

Jenith @istance @eclination 8atitude

Answer P;8 090 0 ; 00 through 8atitude Time of 6eridian Passage

#0 .,’ N

1d #%h +0m 1%s G6T

Pro Iorma Pole

O #$.$’ $10 $#.’ 0900 00.0’ +$0 0%.+’ 0 1$.%’ #0 .,’ N 8ongitude 9, 0 +0’>

Principles of Navigation @ate

@D 8atitude

8ocal Time 8ongitude Appro"imate G6T

@D 8ongitude

'e"tant Altitude 7nde" rror <(served Altitude @ip Apparent Altitude Total Correction True Altitude

Chronometer rror G6T

L #0 GA  incr GA  8ongitude 8A 

A0 A# A 8atitude N Aimuth

P;8

Through 8atitude

8ongitud e

#. 7f the chronometer time is given! then apply the correction to o(tain G6T. . 7t must always (e (orne in mind that the date at Greenwich may dier from the date at the ship. The date at Greenwich must (e o(tained. 8ocal time ± longitude * Appro"imate G6T. +. The direction of the position line is always 900 either side of the (odies aimuth.


"ample2 ! the se"tant angle of Polaris was $00 #.+’ 7 &#.+.  19ft. G6T 02$#2. @ate

#+;09;, %

@D 8atitude


190 +’N

@D 8ongitude

#+d 0$h 00m 0h $#m #+d 0h $#m


#+d 0h $#m s

GA  incr GA  8ongitude 8A 

90 0.0’ #0 $+.9’ ##00 #+.9’ 10 $0’> ,0 +.9

'e"tant Altitude 7nde" rror <(served Altitude @ip Apparent Altitude Total Correction True Altitude A0 A# A 8atitude N

10 $0’> $00 #.+’ L#.+’ $00 #,.0’ L,.%’ $0 09.’ 0

L0.% $00 0%.1’ L #0 #,.9’ 0.,’ 0.’ 190 ,.#’ N

Aimuth P;8

0%9. 0

,9. Through 0 8atitude

190 ,.#’ N

8ongitud e

1 0 $0’>


Pro Iorma @ate

@D 8atitude

8ocal Time 8ongitude Appro"imate G6T Chronometer rror G6T @eclination @- @eclination

@D 8ongitude

GA  incr GA  8ongitude 8A  A 4 C True Aimuth Compass Aimuth rror =ariation @eviation

#. 7f the 8A e"ceeds +,00! su(tract +,00. . >hen the 8A is (etween 0000 5 #%00! the (ody is west of the o(server. +. >hen the 8A is (etween 0#%0 5 +,00! the (ody is east of the o(server. 1. GA  * GA  O 'A  $. >hen a (ody is rising or setting it is an amplitude. ,. Name the compass error (y2 Compass 4est rror >est or CA@T2 Compass to True Add ast.


"ample2 Euly #0th at appro"imately 0%200 86T in @D +0 #$’N +#0 +,’>. =ariation #00>. 'un (earing 09+0C. Chronometer #0h0m01s! slow (y $m 0s. @ate

#0;0;, %

@D 8atitude

+0 #$’N


0%h 00m 0h 0,m #0h 0,m


#0h 0m 01s 0$m 0s #0h 0m 1s

@eclination @- @eclination

0 #.,’ N 0 0 #.,’ N

@D 8ongitude

+#0 +,’>

GA  incr GA  8ongitude 8A 

+%0 1#.0’ #0 $#.0’ ++00 +.0’ +# 0 +,’> 9%0 $,.0’

A 4 C True Aimuth Compass Aimuth rror =ariation @eviation

0.1 ' 0.1 N 0.+ N 0%.#0 09+.00 #1.9’ > #0’ > 1.9’ >


Pro Iorma 'ine Amplitude * 'ine @eclination 'ecant

@ate

@D 8atitude

@D 8ongitude


8og 'in @ec 8og 'ec 8at 8og 'in Amplitude

Chronometer rror

True Aimuth Compass Aimuth rror =ariation @eviation

G6T @eclination @- @eclination

Amplitude

Amplitudes are named east at sunrise! west at sunset and the same as declination. "ample2

Principles of Navigation @ecem(er +0th #9,% at 0h 0,m 86T. @D ++0 1’ N 00 +#’ . 'unrise #,0C =ariation #0 >. 8ocal Time 8ongitude Appro"imate G6T

+0d 0h 0,m #h m +0d 0$h 11m

8og 'in @ec 8og 'ec 8at 8og 'in Amplitude True Aimuth Compass Aimuth rror =ariation @eviation

+0 #0.#’ ' 0.#’ 0 + #0.0’ '

@eclination @- @eclination

#.$91%1 0.0%+9 #.,++ ##%.#0 #,.00 .90 > #0 > ,.90 >  %.#0 '

Amplitude

4odies at #h $1m ##.$s G6T on the nd August #9,%. To Iind2 True Altitude 5 Aimuth. Jenith

K

PN R >

Rational Horizon

our Circle

 =ertical Circle Celestial Equator


GA



,,0 +0.’

PJ * Co 8at -900 & 8at

#+0 +$.#’

7ncr GA



8ong > 8A 

PJ * Polar @istance

%00 0$.%’

JR * Jenith @istance -900 & True Alt

+0 0 00.0’> $00 0$.%’

'A



190 $1.’

8A



+000 00.0’

4ody  of 6eridian P * +,00 L 8A

4ody >est of 6eridian

J * Aimuth

J * +,00 L Aimuth

P * 8A

True Altitude2 av JR * av P V 'in PR V 'in PJ O av -PJ W PR 8og 8og 8og 8og

av ,00 'in 110 $0’ 'in ,00 av

#.+991 #.%1% #.9+$+ #.#%+,9

Nat av Nat av -W Nat av JR

0.#$,$ 0.0#1 0.#00

Jenith @istance * 1% 0 1.,’ Altitude * 90 0 & JR * 1#0 #.1’ Aimuth2 av J * -av P & av -PJWJRCosec PJ V Cosec JR Nat av 110 $0’ 0.#1$1 Nat av ##0 0.009,% #.1’ Nat av 0.#+$1

8og av 8og Cosec ,00 8og Cosec 1%0 1.,

#.#++ 0.0,1 0.#1#1 #.+#9+1


Aimuth * 0$1 0 #.+’ -T

Consider the star Alpherat! seen (y an o(server in 8atitude 0N on the #st Euly #9,%. 8A 1#0 1#.’ To Iind2 True Altitude 5 Aimuth. Jenith K

PN

R

 our Circle Rational Horizon

> =ertical Circle Celestial Equator


8A



1#0 1#.’

'A



+$%0 #%.%’

8A



10 0 00.0’

@eclination %0 $1.%’


av 'in 'in av

#.0,%#0 #.9,# #.11#% .91$


0.0919$ 0.00+,1 0.09%$9

Jenith @istance * +,0 +,’ Altitude * 900 & JR * $10 1’ Aimuth2 av J * -av P & av -PJWJRCosec PJ V Cosec JR Nat av Nat av Nat av

0.$%, 0.0+ 0.#%$01

8og av 8og Cosec 8og Cosec

#.,, 0.0+%+ 0.1$9 #.$1,%


Aimuth * %9 0 #%.’ -T

Consider the 'un! seen (y an o(server in 8atitude 10N at an instant when the 8A  was 010 +.$’. 'un’s 'A #00 .$’ @eclination 00 +0’' Iind2 True Altitude 5 Aimuth. Jenith Co 8at K

Jenith Jenith @istance

PN

Polar @istance

K

>

R

Rational Horizon 

our Circle Celestial Equator =ertical Circle


8A



010 +.$’

'A



#00 .$’

8A



+$0 00.0’

@eclination 00 +0.0’


av 'in 'in av

.9$,% #.9#$9 #.%#0 .9%91


0.,9#+ 0.0,91 0.++0

Jenith @istance * 0 0 .,’ Altitude * 90 0 & JR * #90 +.1’ Aimuth2 av J * -av P & av -PJWJRCosec PJ V Cosec JR Nat av Nat av Nat av

0.,$#0 0.0+,$ 0.,+1$

8og av 8og Cosec

#.%011+ 0.0$9%

Principles of Navigation 8og Cosec

0.#%9+ #.9$9+1

Aimuth * #1$ 0 #+.1’ -T

7n 8atitude 00' a star (ore 000. True Altitude 00 +0’. To Iind2 @eclination. Jenith Jenith K K

Jenith Jenith @istance Co 8at

Polar @istance

R >

our Circle

 Celestial Equator

=ertical Circle

Rational Horizon

PN


True Altitude2 av PR * av J V 'in PJ V 'in JR O av -PJ W JR 8og 8og 8og 8og

av 'in 'in av

#.%,+ #.999 #.9#$9 #.#+#

Nat av Nat av -W Nat av PR

0.$90, 0.0000 0.$90,1

Polar @istance * #00 0 ,.,’ @eclination * #00 ,.,’ N

Pro Iorma 7ntercept

@ate

@D 8at

@D

4ody

8ong

86T

@ec

'. A.

8ong

@-

7..

G6T

@ec

<. A. @ip

Chro n

A

A. A.


rror

4

T n

Corr G6T

C

T. A.

T Aimuth

JR

GA

av JR * av 8A V Cos 8at V Cos @ec O av -8 W @

7ncr

8og av 8A

GA

8og Cos 8at

8ong

8og Cos @ec

8A

8og av

Nat av Nat av W Nat av JR C JR T JR 7ntercept

#. 8atitude 5 @eclination & 'ame Names & 'u(tract. . 8atitude 5 @eclination & @ierent Names & Add. +. 8ongitude >est & GA 4est 5 =isa =ersa. 7f 8ongitude is greater than #%00! su(tract from +,00 and change the sign. 1. Correction for 7ntercept & Aimuth as course! and intercept as distance. Named the same as aimuth when towards and opposite when away. $. Calculated JR (est & Towards. Calculated JR least & Away.

T;A


Pro Iorma 7ntercept 8ongitude (y

@ate

@D 8at

@D

4ody

8ong

86T

@ec

'. A.

8ong

@-

7..

G6T

@ec

<. A. @ip

Chro

A

A. A.

4

T

n rror

n

Corr G6T

C

T. A.

T Aimuth

JR

GA

av 8A * -av JR & av -8W@ 'ec 8at V 'ec @ec

7ncr

Nat av JR

GA

Nat av W

8ong

Nat av

8og av

W

8og 'ec

8A

8at 8og 'ec


@ec 8og av 8A 8A GA 8ong

P;8

;

through 8atitude

;>

8ongitude

Appro"imate 8A indicates if the (ody is east ; west of the o(server. 7f east use the (ottom of the haversine ta(les to ?nd calculated 8A! if west use the top.

"ample2 At 0920 86T on the th the se"tant angle of the 'un’s lower lim( was 1$0 0.#’.  1$ft! 7 #.#’ o the arc. Chronometer ,h m %s slow (y #m 0s. 6arc 't illaire2 0

@ate

;#0;+

@D

0 #,’N

0

@D 8ong

#+ #’>

4ody

'un -88

8at

0

86T

9h 0m

@ec

8ong

%h 19m

@-

0

$ +%.9’ ' O0.#’

'. A. 7..

0

G6T

d #%h 9m

@ec

$ +9.0’ '

1$ 0.#’ O #.#’ 0

<. A. @ip

1$ #.’ L ,.$

Principles of Navigation 0

Chro

,h m %s

A

L 0.1

A. A.

4

L 0.#

T

1$ #1.’

n rror

O #m

O #$.0 n

0s

Corr 0

G6T

d #%m

C

0.,1 '

T. A.

1$ 9.’

%m 1%s 0

T

0

#0.9

JR

11 +0.+’

Aimuth

0

GA

9+ 0+.+’

av JR * av 8A V Cos 8at V Cos @ec O av -8 W @

0

7ncr

 #.’

8og av

#.00#0

8A 0

GA

9$ #$.+’ #+0 #’>

8ong

8og Cos 8at

#.9$

8og Cos

#.99%9

@ec 0

8A

++ 0+.+’

8og av

.9#%1

Nat av

0.09+

Nat av W

0.0$0%

Nat av

0.#1100

JR 0

C JR

11 +,.# 0

T JR 7ntercept

11 +0.+ $.%’ T


0

@ate

;#0;+

@D 8at 0 #,’N

0

@D

#+ #’>

4ody

'un -88

8ong

0

86T

9h 0m

@ec

0

$ +%.9’

'. A.

1$ 0.#’

' 8ong

%h 19m

@-

7..

O #.#’

O0.#’ 0

G6T

d #%h

@ec

9m

0

$ +9.0’

<. A.

1$ #.’

' @ip

L ,.$ 0

Chro

,h m

n

%s

rror

O #m

A

L 0.1

A. A.

4

L 0.#

T

1$ #1.’

O #$.0 n

0s

Corr 0

G6T

d #%m

C

0.,1 '

T. A.

1$ 9.’

%m 1%s 0

T Aimuth #0.9

0

JR

11 +0.+’

0

GA

9+ 0+.+’

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Nat av JR

0.#1+1

Principles of Navigation 0

GA

9$ #$.+’

Nat av W

0.0$0%

Nat av

0.09+#+

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8ong

#+

8og av

.9,90%

0.00#

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through 8atitude 0 #,’N 8ongitude #+ 0+.%’>

Iigure @rawing on the Plain of the <(server’s Dational orion Construct a ?gure on the plane of the <(server’s Dational orion for 8atitude 100N! @eclination 00'. 4y measurement ?nd the Jenith @istance -JR of a (ody when the Aimuth is #+00 -T. N


Rational Horizon

P

Polar @istance -our Circle

Prime =ertical Circle

>

J

 =ertical Circle

Celestial Equator K d#

d @iurnal Circle ; Parallel of @eclination

R

@ 8

8ocus 8ine

'

# @raw a circle of convenient radius which can (e divided into 1 e3ual parts. The centre of the circle is the <(server’s Jenith! and the circle is the Dational orion.  @raw NJ'! the <(server’s 6eridian! as shown! and perpendicular to this line >J! the Prime =ertical. + 6a:e NP 5 KJ e3ual to the latitude of the o(server. Northern latitude! K will (e (elow J! and visa versa for southern latitude. 1 Fsing the centre of NJ'! draw an arc K>! the celestial e3uator. $ Ii" point @ such that K@ * declination. , Fse the latitude and declination to ?nd the amplitude and ?" points d@d. -d and d are the positions in which the (ody rises and sets. >ith the centre on NJ'! draw the arc d@d  the diurnal circle or parallel of declination.  Aimuth is measured from J. =ertical circles appear as straight lines from J. The (ody S"’ is at the intersection of the vertical circle and the diurnal circle.

Principles of Navigation % To draw the hour circle through S"’! ?nd the point on NJ' which is the radius of >P. 8a(el this S8’! and draw a locus line through S8’ perpendicular to NJ'. All hour circles have their radius points on the locus line. 9 @raw the hour circle PR. #0 To measure the angle P in the triangle PJR! divide the celestial e3uator into convenient e3ual parts and measure the arc of the celestial e3uator from the meridians to the hour circle through S"’.

"ample2 8atitude 100 N @eclination 00 N True Altitude +$0 west of the meridian. Jenith @istance * 900 & True Altitude * $$0 N

Rational Horizon

P

Polar @istance -our Circle

d

d#

@iurnal Circle ; Parallel of @eclination

J

>




=ertical Circle

R

@ Celestial Equator

K

8

8ocus 8ine

'

An o(server in latitude 1,0N sees the 'un theoretically rising when its declination was #$0 #0’N. @raw a ?gure on the plane of the o(server’s rational horion and calculate the time dierence (etween theoretical sunrise and the 'un (earing 0900. N


@iurnal Circle ; Parallel of @eclination

Rational Horizon

P Polar @istance -our Circle

R

d

>


J R#

@#



=ertical Circle

@ Celestial Equator

K

8ocus 8ine

8

' J

R

PR

PJ

P

Cos P * CoTan PR V Cotan PJ

Cos P * CoTan PR V Tan PJ

8og CoTan PR 8og Cotan PJ 8og Cos P

8og Tan PR 8og Cotan PJ 8og Cos P

P0 -RPJ

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P0 -R#PJ

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8A

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8A

Time @ierence  hrs 1.1 mins Circumpolar 4odies J

R -Fpper Transit

PNKP' * <(server’s 'uperior 6eridian PNK#P' * <(server’s 7nferior

PN 8atitude

R# -8ower Transit

K

>


Parallel of @eclination Circumpolar

N

R# NR# * True Altitude -8ower Transit 4 NP * 8atitude PNR# * Polar @istance

A

AJ4 * 900 * 6a"imum Aimuth PN

R

( >



J

Celestial E uator

a

K

Rational Horizon

8

The term circumpolar refers to celestial o()ects which remain ' continuously a(ove the o(server’s horion and do not rise or set. 'uch (odies have no amplitude. A (ody is circumpolar when the polar distance is les then the latitude. PR  8atitude. @eclination X Co8atitude.

Principles of Navigation A circumpolar (ody has two visi(le transits! Fpper Transit or Fpper 6eridian Passage and 8ower Transit or 8ower 6eridian Passage. >hen on the o(server’s superior meridian at R it is then crossing from east to west and is a ma"imum altitude. The 8A * 0000 -or +,00! the aimuth is 0000 or #%00. >hen on the o(server’s inferior meridian at R# it is then crossing from west to east. The 8A * #%00! the aimuth is 0000 or +,00 in the Northern emisphere and #%00 in the 'outhern emisphere. The latitude of a stationary o(server may (e found (y ta:ing the mean of the true altitudes at upper and lower transits. 8atitude (y 8ower Transit <(servation NP * NR# O PR# 8atitude * True Altitude O Polar

6a"imum or 8imiting Aimuth >hen the declination is named opposite to the latitude the ma"imum aimuth occurs when the (ody is rising or setting. Irom the ?gure! the ma"imum aimuth occurs when the (ody is at Sa’ and S(’! such that the vertical circles JA and J4 are tangential to the (odies diurnal circle. >hen the declination is less than the latitude -same name such that the enith lies within the circle of declination! then there is no limiting aimuth. The conditions necessary for a (ody to cross the prime vertical are2 #. The declination and 8atitude must have the same name. . The declination must (e less than the o(served latitude.

Twilight Twilight is the period of Spart light’ occurring (efore sunrise and after sunset! caused (y the scattering of sunlight (y dust and

Principles of Navigation moisture in the arth’s atmosphere. 6orning twilight lasts from when the sun is #%0 (elow the horion until sunrise. vening twilight lasts from sunset to when the sun is #%0 (elow the horion.

,0

Rational Horizon

#0 Civil Twilight Y

 'ights

#%0 Nautical Twilight Astronomical twilight & A(solute @ar:ness

Civil twilight 7s said to (egin or end when the sun’s centre is ,0 (elow the rational horion. At that instant the sun’s JR is 9,0 and it is too dar: to read outside. Nautical Twilight 4egins or ends when the sun’s centre is #0 (elow the rational horion. At that instant the sun’s JR is #00. Astronomical Twilight 4egins or ends when the sun’s centre is #%0 (elow the rational horion. At that instant the sun’s JR is #0%0 and the s:y is a(solutely dar:. The Nautical Almanac ta(ulates the 86T of sunrise! sunset and the (eginning or ending of civil and nautical twilights. The interval of time (etween civil and nautical twilights is the (est time for ta:ing star sights. @uring the period of civil twilight very few stars will (e visi(le (ut the horion will (e clear. @uring Astronomical twilight many stars will (e visi(le (ut the horion will not (e well de?ned. owever during nautical twilight most navigational stars will (e visi(le together with a clear horion! under normal atmospheric conditions.

Principles of Navigation

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