Celestial Sphere
Local View
• On earth objects are usually viewed in flat
Euclidean geometry.
• From the earth the stars appear to be fixed on a
sphere that rotates.
– Great distance to objects
– Earth’s rotation
Great Circles
• Any plane through the
center of a sphere
intersects the sphere in a
great circle.
P
– AXB
– PAQB
• Points are opposite if for
any great circle that passes
through one it passes
through both.
A
O
X
Q
B
Spherical Angles
• The angle APX projects
onto the plane of a great
circle AOX.
P
– Defines angle APX
– PAX right angle
• The distance between two
points is the angle
between the points.
A
O
X
Q
B
Triangles
• Three points not on the
same great circle define a
spherical triangle.
– Defines a plane that
excludes the origin
c
• Each angle is less than
180°, but the sum exceeds
180°.
– Triangle PAX from before
A
B
a
b
C
Small Circles
• A parallel circles have
centers on the same axis.
– AB and CD
– Arc AP = q
– AS = AO sin(AOS)
• Pick E on AB.
– Great circle PEF
– PE = q
q
y
A
P
S
E
C
O
F
Q
B
D
Small Circle Arc
• Spherical angle y is
defined by APE.
– Same as CPF
– Matches COF
q
y
A
• AS and ES parallel CO
and FO.
– ASE = y
– AE = y sinq
P
S
E
C
O
F
Q
B
D
Polar Coordinates
• Spherical polar
coordinates are a 3-D
vector.
– Reduce to q, y on unit
sphere
x  sin q cosy
y  sin q sin y
z  cosq
Z
y
– r, q, y
R
S
A
X
B
q
Y
O
S
Spherical Trigonometry
• Set A at a pole and AB on
a great circle.

rA  0,0,1

rB  sin c,0, cos c

rC  sin b cos A, sin b sin A, cos b
cos a  cos b cos c  sin b sin c cos A
sin A sin B sin C


sin a sin b sin c
A
c
B
b
a
C
Latitude
• Orient the sphere of the
earth with N, S poles.
• The equator is the great
circle at 90° from N.
N
q
X
f
• The latitude is measured
from the equator.
– f = 90° – NX
S
E
Longitude
• The prime meridian is at
right angles to the equator.
– Defined at Greenwich
Observatory, NGKS
N
G
• Longitude is the angle l =
GNX.
– -180° < l < -180°
X
O
K
l
S
E
Projection
• Project the earth outward
into space.
– North and south celestial
poles P, Q
– Celestial equator E
• East orientation is defined
by the sun’s position ϒ at
vernal equinox.
– Crosses equator from S to N
– March 21
P
X
O
ϒ
a
Q
E
Declination and Right Ascension
• Declination is the celestial
equivalent of latitude.
– d = 90° – PX
P
• Right ascension is the
celestial equivalent of
longitude.
– a = ϒPX
X
O d
ϒ
a
Q
E
Heavenly Time
• Right ascension is not measured in degrees.
• Degrees are converted to time.
– 24 hours = 360°
– 1h = 15°
1° = 4m
– 1m = 15'
1' = 4s
– 1s = 15''
1'' = 1/15 s
Stellar Coordinates
• Stellar coordinates use
right ascension and
declination.
P
– X(a,d)
X’
• Displacement is measured
as a difference of
coordinates.
– X’(a  da, d  dd)
X
ϒ
a
Q
E
Alt-Azimuth
• The alt-azimuth system is
fixed to an observer on
earth.
• Zenith distance is
measured from vertical.
– z = ZX
– Altitude a = 90° - z
• Azimuth is measured west
of north.
– A = PZX
Z
P
N
X
S
O
W
Q
Rising Star
• Stars are visible to an observer when z > 90°.
• Tables of rising and setting objects are computed
for z = 90°.
Hour Angle
• Alt-azimuth moves with
the stars.
• PZ was fixed by the
transformation.
• Hour angle is measured
from zenith and celestial
north.
– HA = ZPX to the west
– PZSQ is the observer’s
meridian
Z
equator
P
X
N
S
O
W
Q
Circumpolar
• Declination remains the
same.
– d = 90° – PX
Z
• The small circle through X
is a parallel of declination.
• A small circle that does
not intersect the horizon
does not set – circumpolar
stars.
equator
P
X
N
S
O
W
Q
Relative Time
• Project points from
Greenwich G and an
observer X onto the
celestial sphere.
– Hour angle at Greenwich
GHA
– Observer hour angle is HA
= GHA + l
N
G
X
O
K
• Sidereal time is defined by
the hour angle.
l
S
E
Sidereal Time
• Sidereal time is defined by the hour angle.
• Moves with the stars
• LST = HA + RA
• A sidereal day is shorter than a solar day.
• 23 h 56 m
Universal Time
• The sidereal and solar time scales depend on the earth’s
rotation.
– Irregular on short time scales
– Slowing on long time scales
• Irregularities can be smoothed to get universal mean sun.
• Universal time is UT = 12 h + GHA (UMS).
– UTC uses leap seconds to coordinate
Dynamical Time
• A dynamical model of time replaced rotation based
systems in 1952.
– Ephemeris time ET
– Defines the second based on the year 1900
– Replaced by TA1 atomic clocks in 1972
• In 1976 this was replaced by Terrestrial Dynamical Time
to account for general relativity.
Atomic Time
• Absolute time measurement is based on the vibrational
period of the hyperfine lines in cesium.
• Absolute time is measured in Julian days beginning at
noon Jan 1, 4713 BC.
• Time is converted to earth-based time like UTC for use in
astronomy.