This document discusses hour angles and Aries as they relate to celestial navigation. It defines key terms like declination, Greenwich hour angle (GHA), local hour angle (LHA), and sidereal hour angle (SHA). It explains that the Nautical Almanac contains tables with the geographical position of celestial bodies by the second. It also notes that SHA is measured from the point of Aries instead of Greenwich for stars to make the almanac thinner. Diagrams are used to illustrate the relationships between an observer's meridian, Greenwich, Aries, and the hour angles of bodies.

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Hour Angles and Aries
 A Brief By Lance GrindleyA Brief By Lance Grindley

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Hour Angles and AriesHour Angles and Aries
Nautical Almanac:Nautical Almanac:
- in section 6 it was shown that line of position (LOP) can be- in section 6 it was shown that line of position (LOP) can be
determined from a sextant angle (hs) and geographical positiondetermined from a sextant angle (hs) and geographical position
(GP) of body(GP) of body
- GP constantly changes as earth rotates as does GMT- GP constantly changes as earth rotates as does GMT

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this section deals with determination of GP of body at exactthis section deals with determination of GP of body at exact
GMT of sextant observationGMT of sextant observation
- data to make this determination is found in Nautical Almanac- data to make this determination is found in Nautical Almanac
- Nautical Almanac is printed every calendar year and is- Nautical Almanac is printed every calendar year and is
available in fall of previous yearavailable in fall of previous year

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- Nautical Almanac contains tables that observer can obtainNautical Almanac contains tables that observer can obtain
GP of 4 planets, sun, moon, 57 navigation stars and Polaris forGP of 4 planets, sun, moon, 57 navigation stars and Polaris for
every second during yearevery second during year
- before determining GP of body, definitions of declination- before determining GP of body, definitions of declination
(DEC) and hour angle necessary(DEC) and hour angle necessary

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Declination (DEC):Declination (DEC):
- on earth’s surface, position of vessel described in terms of latitude and- on earth’s surface, position of vessel described in terms of latitude and
longitudelongitude
- in celestial navigation, instead of latitude, declination describes whether- in celestial navigation, instead of latitude, declination describes whether
body is north or south of equatorbody is north or south of equator

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Declination:Declination:
- stars have fairly constant DEC- stars have fairly constant DEC
- DEC of sun, moon and 4 planets change rapidly from day to- DEC of sun, moon and 4 planets change rapidly from day to
day and significantly from minute to minuteday and significantly from minute to minute

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Hour Angle:Hour Angle:
hour angles are roughly celestial navigation’s equivalent of longitude; therehour angles are roughly celestial navigation’s equivalent of longitude; there
are 3 hour angles that will be introduced:are 3 hour angles that will be introduced:
- Greenwich hour angle (GHA)- Greenwich hour angle (GHA)
- local hour angle (LHA)- local hour angle (LHA)
- sidereal hour angle (SHA), generally associated with stars- sidereal hour angle (SHA), generally associated with stars

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Hour Angle:Hour Angle:
- unlike longitude, hour angles are always measured westward- unlike longitude, hour angles are always measured westward
through 360through 360ºº
- hour angle of celestial body when measured from Greenwich- hour angle of celestial body when measured from Greenwich
meridian is known as Greenwich Hour Angle (GHA)meridian is known as Greenwich Hour Angle (GHA)

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Hour Angle:Hour Angle:
E - up to 180º GHA corresponds to Lo; over 180º, Lo = 360º –E - up to 180º GHA corresponds to Lo; over 180º, Lo = 360º –
GHAGHA
EXAMPLES:EXAMPLES:
GHA 90º, Lo = 90º WGHA 90º, Lo = 90º W
GHA 210º, Lo = 360º - 210º = 150º EGHA 210º, Lo = 360º - 210º = 150º E
GHA 350º, Lo = 360º - 350º = 10ºGHA 350º, Lo = 360º - 350º = 10º

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Hour Angle:Hour Angle:
- now necessary to relate position of body relative to observer’s position, not- now necessary to relate position of body relative to observer’s position, not
GreenwichGreenwich
- hour angle of celestial body when measured from observer’s meridian is- hour angle of celestial body when measured from observer’s meridian is
known as Local Hour Angle (LHA)known as Local Hour Angle (LHA)
- if observer’s Lo and GHA of body are known, then LHA can be calculated- if observer’s Lo and GHA of body are known, then LHA can be calculated

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Hour Angle:Hour Angle:
- if Lo is west, LHA = GHA – Lo- if Lo is west, LHA = GHA – LoWW
- if Lo is east, LHA = GHA + Lo- if Lo is east, LHA = GHA + LoEE
- meridian or great circle passing through a body and poles of celestial- meridian or great circle passing through a body and poles of celestial
sphere is known as hour circle of bodysphere is known as hour circle of body
- hour circles of bodies move with time while celestial meridians remain- hour circles of bodies move with time while celestial meridians remain
fixed with respect to earthfixed with respect to earth

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in this examplein this example
of Loof LoWW ::
LHALHAstarstar = GHA –= GHA –
LoLoWW observerobserver
LHALHAmoonmoon==
GHA – LoGHA – LoWW
observerobserver
Lo

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in this examplein this example
of Loof LoEE ::
LHALHAstarstar = GHA += GHA +
LoLoEE observerobserver
LHALHAmoonmoon==
GHA + LoGHA + LoEE
observerobserver Greenwich
Observer’s
GH
A
Lo

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Aries:Aries:
- Nautical Almanac tabulates GHA for sun, moon and planets- Nautical Almanac tabulates GHA for sun, moon and planets
for every hour in yearfor every hour in year
- if it tabulated every hour for stars, almanac would be 10 times- if it tabulated every hour for stars, almanac would be 10 times
thickerthicker
- tabulating every hour not necessary for stars which are slow- tabulating every hour not necessary for stars which are slow
movingmoving

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Aries:Aries:
- instead of measuring star hour angles from Greenwich, they are measured- instead of measuring star hour angles from Greenwich, they are measured
from point in sky known as Ariesfrom point in sky known as Aries ( )( )
- Aries is hour circle projected on earth’s surface of celestial position- Aries is hour circle projected on earth’s surface of celestial position
occupied by sun at exact instant sun crosses equator, moving north,occupied by sun at exact instant sun crosses equator, moving north,
approximately March 21approximately March 21stst
each year (Vernal Equinox)each year (Vernal Equinox)

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Aries:Aries:
- this point was in constellation Aries 2000 years ago but due to presession- this point was in constellation Aries 2000 years ago but due to presession
motion of earth spinning on polar axis, it has moved 30motion of earth spinning on polar axis, it has moved 30º westwardº westward
(Bowditch page 230)(Bowditch page 230)
- while this point is still known as first point of Aries, currently is in- while this point is still known as first point of Aries, currently is in
constellation of Pisces and will make complete circle in 25,800 yearsconstellation of Pisces and will make complete circle in 25,800 years

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Aries:Aries:
- selection of this point was arbitrary as was selection of- selection of this point was arbitrary as was selection of
Greenwich meridian for defining longitudeGreenwich meridian for defining longitude
- instead of Almanac tabulating GHA for every hour for all 57- instead of Almanac tabulating GHA for every hour for all 57
stars, hour angle known as sidereal hour angle (SHA) isstars, hour angle known as sidereal hour angle (SHA) is
measured from Ariesmeasured from Aries
- Almanac just tabulates GHA of Aries- Almanac just tabulates GHA of Aries

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Aries:Aries:
-- SHA of a star rarely varies by more than 0.1’ in a week andSHA of a star rarely varies by more than 0.1’ in a week and
Almanac tabulates stars every 3 daysAlmanac tabulates stars every 3 days
GHAGHAstarstar = GHA= GHAAriesAries + SHA+ SHAstarstar
- GHA and LHA can be applied to all celestial bodies and- GHA and LHA can be applied to all celestial bodies and
reference point Aries; SHA routinely used for starsreference point Aries; SHA routinely used for stars

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Aries:Aries:
- SHA of navigation planets is shown in Almanac generally to assist in use- SHA of navigation planets is shown in Almanac generally to assist in use
of Star Finder 2102-Dof Star Finder 2102-D
- by definition, SHA of sun is exactly zero at Vernal Equinox- by definition, SHA of sun is exactly zero at Vernal Equinox

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Hour Angle Diagrams:Hour Angle Diagrams:
-- to avoid perspective drawings, convention is to use flat circle,to avoid perspective drawings, convention is to use flat circle,
viewed from south poleviewed from south pole
- circumference- circumference
of circle isof circle is
equatorequator
- hour angles- hour angles
measuredmeasured
westward, shownwestward, shown
anticlockwiseanticlockwise

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Hour Angle Diagrams:Hour Angle Diagrams:
- top of circle “M” represents observer’s meridian with solid line to centre of- top of circle “M” represents observer’s meridian with solid line to centre of
circlecircle
- this solid line- this solid line
known as upperknown as upper
branchbranch

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Hour Angle Diagrams:Hour Angle Diagrams:
dashed line drawn at 180dashed line drawn at 180º to M, is calledº to M, is called
lower branchlower branch
and labeled “m”and labeled “m”

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Hour Angle Diagrams:Hour Angle Diagrams:
- after M located arbitrarily at top of diagram, next locate Greenwich “G”- after M located arbitrarily at top of diagram, next locate Greenwich “G”
meridianmeridian
- if M has Lo =- if M has Lo =
7575ºE, GreenwichºE, Greenwich
will be 75º westwill be 75º west
or anti-or anti-
clockwise fromclockwise from
M; draw as solidM; draw as solid
line to centreline to centre

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Hour Angle Diagrams:Hour Angle Diagrams:
- draw dashed line opposite from centre of circle at 180- draw dashed line opposite from centre of circle at 180º to Gº to G
- known as “g”- known as “g”
or lower branchor lower branch
of Gof G
- g represents- g represents
InternationalInternational
Date LineDate Line

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Hour Angle Diagrams:Hour Angle Diagrams:
- for star diagram, add Aries by using Nautical Almanac GHA for the correct- for star diagram, add Aries by using Nautical Almanac GHA for the correct
time, measuringtime, measuring
from Gfrom G
- if diagram for- if diagram for
sun, moon orsun, moon or
planets, get GHAplanets, get GHA
from Almanacfrom Almanac
and draw as forand draw as for
AriesAries

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Hour Angle Diagrams:Hour Angle Diagrams:
for star diagram, after add Aries, get star’s SHA from Nautical Almanac forfor star diagram, after add Aries, get star’s SHA from Nautical Almanac for
correct day,correct day,
measuring frommeasuring from
AriesAries
SHA 200º

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HourHour Angle Diagrams:Angle Diagrams:
likewise, same technique is used for a second star with, e.g., SHA = 60likewise, same technique is used for a second star with, e.g., SHA = 60ºº