Coordinate Systems

More than X marks the spot

 The Sky from Here
From the ground, the sky looks like a big dome above us.
Both the “zenith” and horizon are locally defined.
 The Celestial Sphere
It is impossible to tell how far away anything is, or
whether there is any depth to the “celestial sphere”.
 The Celestial Sphere
We project the Earth into the sky, and its rotation
appears reflected there. The “diurnal” (daily) motion of
the sky is just due to the spinning Earth.
 Celestial Coordinates
To “map” a given point in the sky, you can specify how high it is, and
in what direction (altitude and azimuth). Or you can project latitude
(declination) and longitude into the sky, but since the Earth rotates,
we must use “right ascension” which is fixed on the stars.
 CELESTIAL SPHERE
 Vast expanse of space that surrounds the earth is
called the celestial sphere.
 The sky has the appearance of an inverted bowl, so
that the stars & other heavenly bodies irrespective of
their actual distances from the earth appear to be
situated on the inside of a sphere of immense radius
described about the earth as the center.
 It has no limits and goes on endlessly.
 The radius is infinite and therefore it is convenient to
assume that the celestial sphere is concentric with the
center of the earth. An observer can only see half of
the sphere at any one time and it is called the celestial
concave.
Apparent Motion of the Celestial Sphere
 Celestial Coordinate System

Terms
 Celestial sphere: an
imaginary sphere,
with Earth at the
center, having an
infinitely large radius.
 All stars and celestial
objects are projected
onto the inside
surface of this great
sphere.
 Right Ascension and Declination
 North celestial pole (NCP)
and south celestial pole
(SCP): where Earth’s
polar axis, if extended to
the celestial sphere, would
intersect.
 Celestial equator: a plane
from Earth outward to a
line around the celestial
sphere halfway between
the NCP and the SCP.
 Ecliptic: the plane in which
the Earth orbits the Sun,
or the Sun appears to
orbit the Earth.
 The First Point of Aries
 The ecliptic cuts the
equinoctial in two points. The
one through which the sun
passes on about the 21st of
March on its journey from
South to North is called the
First Point of Aries or the
‘Vernal Equinox’ & is denoted
by the Rams horn in the sign
of the Zodiac.
 First Point of Libra
 The point through which the
sun passes on or about the
23rd of Sept’ on its journey
from North to South is called
the First Point of Libra or the
‘Autumnal Equinox & is
denoted by the sign of the
Scales. The word equinox
means equal & on Vernal
equinox & Autumnal equinox
all places on the earth will
experience equal periods of
day & night.
 Libra (♎)
 Pole of the Ecliptic
 A point 900 removed
from any great circle is
called its pole. Therefore
the Ecliptic, being a
great circle, has its own
pole removed from the
ecliptic. This point is
called the ‘pole of the
ecliptic’. This point does
not appear on the
earth’s surface, since
the ecliptic does not
appear on the earth.
 Obliquity of the ecliptic
 The angle at which
the ecliptic crosses
the equinoctial is
called the ‘Obliquity
of the ecliptic’. It
makes an angle of
230 27’ due to the
inclination of the
earth’s axis of
rotation.
 Celestial Meridians

 The celestial
meridian is the
line on the
celestial sphere
joining the
observer's zenith
(i.e. the point
directly overhead)
with the north and
south celestial
poles.
 Position of Heavenly Bodies
in the Celestial Sphere
 For ordinary purposes the
position of a heavenly
body can be fixed in the
celestial sphere in
relation to the celestial
equator and a particular
meridian. The meridian
selected for reference for
heavenly bodies in the
celestial hemisphere is
the one through the First
Point of Aries.
 Sidereal Hour Angle (SHA)
 It is the angle of a heavenly
body between the meridian
through the First Point of Aries
& the meridian through the
heavenly body measured
Westwards from the former.
Therefore it is the angle or the
angular distance along the
celestial equator. Unlike the
longitude on the earth which is
measured both east & west of
the prime meridian of
Greenwich, S.H.A. is measured
westwards only, & will increase
from 0 degrees to 360
degrees.
 Right Ascension
Right Ascension:
 When the angle between the
meridian of the First Point of
Aries & the meridian of the
body is measured Eastwards
from Aries it is known as the
Right Ascension ( RA) of the
body.
 It is normally expressed in
units of time (hours minutes
& seconds).
 RA = 3600 - SHA. It is not
used in navigation but used
by astronomers.
 Declination
Declination:
- Similar to latitude.
- Measured in degrees,
perpendicular to the
celestial equator
(north-south direction).
- North is positive, south
is negative.

The declination of stars changes very slowly & like their SHA may be
constant up to one month. These declinations are tabulated in the
Nautical Almanac adjacent to their SHA. The declination of the sun
however changes from 23 1/20 North to 23 1/20 S and back again during
12 months. The declination of the navigational planets Venus, Jupiter,
Saturn & Mars and the moon also vary between wide limits. Therefore
these are tabulated in the Nautical Almanac for every hour of GMT.
 Polar Distance
 This is the angular distance
of a body from the elevated
pole, ie, the pole above the
observer’s horizon. It is PX
in diagram3, if the
observer is in the North
latitudes. When the
elevated pole & the
declination have the same
names the polar distance is
900 minus declination. (900
– dec). When they have
opposite names it is 900 +
declination
 Geographical Position
 - The Geographical
Position of a celestial
body it the point on the
earth's surface at
which the body is
directly overhead. This
position will normally be
expressed in terms of the
Greenwich Hour Angle
and the Declination of the
celestial body.
 Zenith and Meridian

 Zenith: Straight
overhead
 Meridian: Line
from north to
south, passing
through the
overhead zenith
 Meridian plane
 It is defined as the
plane passing
through any point A
on the surface and
the North and South
Poles.
 Equatorial plane is the
plane passing through
the Earth's center of
mass and perpendicular
to the rotation axis. And
equator is the
intersection of the
surface with the
equatorial plane.
 The Plane of the Celestial
Meridian
 The intersection of this
plane with Earth's
surface is the
geographical meridian,
and the intersection of
the plane with the
celestial sphere is the
celestial meridian for that
location and time. ... The
observer's upper
meridian passes through
the zenith while the
lower meridian passes
through the nadir.
 Celestial Horizon ( Rational
Horizon )
 Also called rational
horizon, geometrical
horizon, true horizon):
the plane, through the
center of the earth,
that is perpendicular
to a radius of the
earth that passes
through the point of
observation on the The great circle on the celestial
sphere, every point which is 900 from
earth's surface; or, the the observer’s zenith is known as the
intersection of that celestial horizon or the rational
plane with the celestial horizon. The celestial horizon divides
the celestial sphere into two equal
sphere. halves.
 Vertical circles
 All great circles
passing through
the observer’s
zenith are
necessarily
perpendicular to
the celestial
horizon and are
known as
vertical circles.
 The Prime Vertical
 The particular
vertical circle
passing through
the East & West
points is called
the prime
vertical circle.
 The Principal Vertical circle
 The observer’s
meridian is sometimes
called the principal
vertical circle because
it provides a fixed
direction in the
celestial sphere just as
the observer’s
terrestrial meridian
provides one on the
earth’s surface.
 The Azimuth of a Heavenly
body
 This is the angle at
the zenith between
the observer’s
meridian and the
vertical circle through
the heavenly body & it
is measured East or
West from his
meridian from 0
degrees to 180
degrees & named
North or South from
the elevated pole
 Altitude & Zenith Distance
 decide the
heavenly body’s
position by a
bearing from the
meridian and an
altitude from the
horizon.
 True Altitude of a Heavenly
Body
 This is the
angular distance
of the body above
the celestial
horizon measured
along the vertical
circle through the
body and the
observer’s zenith.
 Zenith Distance
 Once the true altitude of
a heavenly body has
been obtained by the use
of appropriate
corrections, the distance
of the body from the
observer’s zenith can be
found.
 It is 900 – altitude
 This distance ZX is the
zenith distance.
The observer’s sea, or visible
horizon
 This is the horizon-
usually the small
circle on the
earth’s surface
where the sea &
sky appear to
meet- above which
the observer
actually measures
the altitude of a
heavenly body.
 Geographical Position
 The Geographical
Position of a celestial
body it the point on
the earth's surface
at which the body is
directly overhead.
This position will
normally be expressed
in terms of the
Greenwich Hour Angle
and the Declination of
the celestial body.
 The Hour Angle
 The angular
distance west of a
celestial meridian
or hour circle; the
arc of the celestial
equator, or the
angle at the
celestial pole.
 The Hour Angle of a Heavenly
Body
 The angular distance west of a
celestial meridian or hour
circle; the arc of the celestial
equator, or the angle at the
celestial pole, between the
upper branch of a celestial
meridian or hour circle and the
hour circle of a celestial body
or the vernal equinox,
measured westward through
360°. At any instant, hour
circles are coincident with
particular celestial meridian.
 Effect of the Earth’s Rotation
 The earth’s steady rotation from West to East
results in an apparent and equally steady
rotation of the celestial sphere from East to
West. During one rotation of the earth the
hour angle of a body that is fixed in the
celestial sphere will increase from 00 when
the heavenly body is on the observer’s
meridian to 3600, when it returns to his
meridian. The hour angle of a heavenly body
thus increases steadily throughout the day.
 The Greenwich Hour Angle
 Greenwich Hour
Angle, abbreviated
GHA, is the
angular measure
of the celestial
body from the
Greenwich
Meridian (also
known as the
Prime Meridian)
along the celestial
equator.
 Local Hour Angle
 Local Hour Angle,
abbreviated LHA, is
the angular
measure of the
celestial body
from the
observers
Meridian along
the celestial
equator. Towards
West
 Sidereal hour angle (SHA)
 Abbreviated SHA The
angle, measured
westward through
360°, between the
meridian passing
through the vernal
equinox (first point of
Aries) and the
meridian of a celestial
body.
 In the case of an observer east of the
Greenwich meridian(at) K in fig
 GHA (Greenwich hour angle)
= GX (measured clockwise)
 Longitude = GK (measured
anti clockwise)
 LHA (local hour angle) =
KGX (measured clockwise
 KGX (LHA) = GX (GHA) +
GK (Long)

LHA = GHA + Longitude

 When the observer is west of the Greenwich
meridian (at H)
 GHA (Greenwich hour angle)
= GX (measured clockwise)
 Longitude = GH (measured
clockwise)
 LHA = HGX (measured
clockwise)
 HGX = HG + GX
 HG = 3600 _ GH

 HGX = (3600- Longitude) +

GHA
 LHA = GHA – Longitude
 (since addition of 3600 does
not affect the actual angular
distance)
Thus to find the local hour angle of heavenly body from its
Greenwich hour angle, east longitude is always added, &
3600 subtracted from the resulting sum when necessary;
west longitude is always subtracted, 3600 being added
when necessary. If for example in fig 12 the GHA of the
heavenly body is 62039’, and the longitude of H is1640 47’
W & that of K is 121013’E then:

At H (long.W) At K (long. E)
GHA = 620 39’ + 3600 GHA = 620 39’
4220 39’ Long = 121013’ E
Long = 1640 47’ (W) LHA = GHA + Long
LHA = GHA – Long LHA = 620 39’ + 121013’ E
LHA = 4220 39’ - 1640 47’ (W) LHA = 183052’
LHA = 2570 52’
 The Zodiac Belt
 The zodiac is a belt-
shaped region of the
sky that extends
approximately 8° north or
south (as measured in
celestial latitude) of the
ecliptic, the apparent path
of the Sun across the
celestial sphere over the
course of the year. The
paths of the Moon and
visible planets are within
the belt of the zodiac.
 The GHA of a star
is made up of two
components: the
GHA of Aries and
the SHA of the star.
Aries is a
benchmark in the
sky that “circles”
the Earth at the
same speed as the
stars. The SHA of a
star is fairly
constant. It
changes only a few
minutes each year.
 Zenith
Meridian Polaris (NCP)

The declination
of my zenith is
equal to my
latitude!
WEST
SOUTH

EAST NORTH
 Thank You
 END
 Defines ‘rational horizon’,
‘zenith’ and ‘nadir’
 Celestial Horizon ( Rational Horizon ): The great circle on the
celestial sphere, every point which is 900 from the observer’s
zenith is known as the celestial horizon or the rational horizon.
The celestial horizon divides the celestial sphere into two equal
halves. The upper one containing the zenith is called the visible
hemisphere & all the heavenly bodies on this half are visible to the
observer. It is the datum for measuring the sextant altitudes of
heavenly bodies along the vertical circle.

 Observer’s Zenith: This is the point where a straight line from

the earth’s center passing through the observer’s position cuts the
celestial sphere. (Z)

 Nadir : The point diametrically Opposite to zenith is call Nadir

 Defines ‘vertical circle’ ,‘prime vertical
circle’ and ‘Principle vertical circle’
 Vertical circles: All great circles passing through the
observer’s zenith are necessarily perpendicular to the
celestial horizon and are known as vertical circles.

 The Prime Vertical: The particular vertical circle

passing through the East & West points is called the
prime vertical circle.

 The Principal Vertical circle: The observer’s

meridian is sometimes called the principal vertical circle
because it provides a fixed direction in the celestial
sphere just as the observer’s terrestrial meridian
provides one on the earth’s surface.
 Defines ‘elevated pole’ and
‘depressed pole
 The two points of
intersection of the
celestial sphere and
the extended axis of
the earth are called
celestial poles. The
celestial pole above
the horizon is called
the elevated pole; that
below the horizon the
depressed pole.
 Proves that the altitude of the elevated
pole is equal to the observer’s latitude

 In the northern
hemisphere, the
altitude of the
north celestial
pole is equal to
the observer's
latitude and the
altitude of the
south celestial pole
is equal to the
negative of the
observer's latitude.
 Proves that the altitude of the elevated
pole is equal to the observer’s latitude

 Theorem  Proof
 Let O - be an observer of the  Let z - denote the
celestial sphere. zenith distance of Pole.
 Let P – be the position of the  Let ψ - denote the
north celestial pole with
respect to O.
(terrestrial) colatitude of
Observer.
 Let a - denote the altitude of
Pole.  By definition we have:
 Let ϕ - denote the  a=90∘−z ϕ−90∘−ψ
(terrestrial) latitude of Then:
Observer.  z=ψ

Zenith Distance of North C
Then: elestial Pole equals
 a=ϕ Colatitude of Observer ⇝
90∘−z = 90∘−ψ ⇝
 a = ϕ Hence the result.
 Defines ‘true altitude’,
‘azimuth’ and ‘true zenith
distance’
 True Altitude of a Heavenly
Body
 This is the
angular distance
of the body above
the celestial
horizon measured
along the vertical
circle through the
body and the
observer’s zenith.
 True azimuth
 The Azimuth of a
celestial body is
the bearing of
the body from
your position,
as measured
clockwise from
true North.
 True zenith distance
 The angular distance
of a celestial body
from the zenith. The
zenith distance is 90 °
minus the body's
altitude above the
horizon (i.e. the
complement of the
altitude) and hence is
also known as
coaltitude.
 Altitude & Zenith Distance
 Although the position of a heavenly body is fixed in the
celestial sphere by its declination & sidereal hour angle,
these angles & distances are measured from theoretical axes,
and for observers own convenience .Observer will use own
meridian and the horizon, and will decide the heavenly
body’s position by a bearing from the meridian and an
altitude from the horizon. Then use of that altitude when
finding his own position.
 Observer is not only concerned with the altitude of the body
above the horizon which he actually sees, but also with its
altitude above the celestial horizon. Having measured the
altitude above the visible horizon he applies certain
corrections to obtain the true altitude above the celestial
horizon. These corrections, the Index error, Dip, Refraction,
Semi- diameter, and Parallax are covered in a separate
lecture titled – ‘corrections to sextant altitude’.
 Rising and Setting of Celestial Bodies

 As the Earth rotates on its axis from west to east, all

heavenly bodies appear to rise in the east, move westwards,
gaining in altitude until it is on the observer’s meridian
(transit the meridian). After transit, it continues to move
westwards decreasing in altitude till it sets over the western
horizon.

 Knowledge of rising & setting of heavenly bodies is essential

to the navigator because the times at which he can take his
star sights are governed by the times of sunrise & sunset,
and the times of beginning of twilight.
 Theoretical Rising & Setting
 This occurs when
the center of the
heavenly body is
on the observer’s
celestial or rational
horizon, East or
West of his
meridian. At these
times the True
Zenith Distance is
900.
 Visible Rising & Setting
 This occurs when the upper
limb of the heavenly body is
just appearing or disappearing
below the observer’s visible
horizon.
 It will be shown later that the
moon’s center lies practically
on the celestial horizon at
these moments;
 but when the sun’s center lies
there, the sun itself appears
appreciably above the
observer’s visible horizon.
 Sunrise & Sunset
 visible sunrise or
sunset occurs
when the sun’s
upper limb appears
on the visible
horizon. At this
moment the
observed altitude
of the sun’s upper
limb is 0° 00’
 True Altitude at Visible Sunrise & Sunset

 By correcting this zero altitude, the true altitude & therefore the
true Zenith Distance of the sun’s center can be found. Then if the
observer is assumed to have no height of eye & the sun’s semi-
diameter is 16’

 Observed Altitude 0° 00.0’

 Refraction -34.0’
 -00 34.0’
 Semi-diameter (U.L.) - 16 .0’
 True Altitude - 0050.0’
 The true zenith distance is therefore taken as 900 50’. & the sun’s
center is about 10 below the celestial horizon, when the upper limb
is just visible. Therefore visible sunrise occurs before theoretical
sunrise & visible sunset occurs after theoretical sunset.
 CIRCUMPOLAR BODIES
 For a body to be
circumpolar, the body
should always be above
the rational horizon i.e
the body should not set.
Therefore, a circumpolar
body will have upper transit
(upper meridian passage)
which is above the elevated
pole and lower transit (lower
meridian passage) which is
below the elevated pole.
 A star can be circumpolar at one latitude, but not when
observed from another latitude. As you get nearer to the
poles, more stars are circumpolar. At the poles all stars are
circumpolar, but at the equator there are no circumpolar
stars.

The nearer a star is to one of the Celestial Poles, the more

likely it is to be circumpolar at any particular latitude. The
actual condition for a star to be circumpolar is that the
star’s declination (D) is greater than 90 degrees minus the
latitude (L) of the observer. The condition can thus be
written as D > 90 - L.
 Note: When a star is south of the celestial equator, its
declination is negative and you need to be careful with the
formula. The condition is the same, but you must use the
absolute value of D (that means you ignore the negative
sign of D). Using this method, southern latitudes are also
positive numbers.
 Describes the condition necessary for a body
to cross the prime vertical

 In other words, the prime

vertical is the vertical circle
perpendicular to the
meridian, and passes
through the east and west
points, zenith, and nadir of
any place. A heavenly
body is in or on the prime
vertical when it bears true
east or true west—when it
is at right angles to the
meridian.
 Declination and location
 Declination of zenith = latitude
 AND…
 Latitude = angle of Polaris above
northern horizon
 Sky Movement - Daily
Solar Day & Sidereal Day
 In a 24 hour solar day,
last night’s star
arrives at tonight’s
meridian 3 min 55.9
seconds earlier than
last night.
 Stars cross the
meridian 3 m 55.9 s
earlier each night.
 Sidereal day is 23 hr
56 min 4.1 sec long.
 Sky Movement - Daily
Solar Day & Sidereal Day
 Sky Movement - Annual

 As a consequence, each night the stars

appear about 59 arc-min (~10) farther
west at the same time as the night before.
 Over one year, the entire sky will
completely slip past.
 Therefore, what is up tonight depends not
only on what time of night it is, but on
which night of the year it is. Last month’s
stars are 2 hours farther west.
Sky Movement - Annual
9:00 pm, May 15, 2005
Sky Movement - Annual
9:00 pm, June 15, 2005
Sky Movement - Annual
9:00 pm, July 15, 2005
 Sky Movement - Annual

Manage both:
 time of night, and

 night of the year

with a Planisphere.
 Using your star wheel
 Set date and
time
 Hold overhead
 Align horizons
 Read
constellations
 Sky Movement – 26,000 years
 The Roman calendar, as established by
Julius Caesar, had problems:
• Started at Winter Solstice on January 1, 45 BC.
• Year was 365 ¼ days long, with leap year
corrections. Decimals had not been invented
yet;
• Year is actually 365.242199… days.

 So, after a few centuries, people noticed

the seasons (equinoxes) were moving
slowly in relationship to their calendar.
 Precession of the Equinoxes
 Earth’s drifting
polar axis causes
the North Celestial
Pole to move in a
circle around the
sky, once every
26,000 years.
 Thuban was the
pole star when the
Great Pyramid was
built in Egypt.
Summer Solstice, 400 BC
Summer Solstice, 2005 AD
Spring Equinox, 2005 AD
 Leap Years
 Gregorian solar year: 365d 5h 48m 20s; still has
25.967s/SY error.
 Summer solstice moves later on calendar each year.
Slightly over-corrected each leap year.
 Cumulative annual error is corrected each century by
not having a leap year, unless that year is divisible by
400 (1600 and 2000 were leap years).