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.– AXB

– PAQB

• Points are opposite if for any great circle that passes through one it passes through both.

P

Q

A

X

O B

Spherical Angles

• The angle APX projects onto the plane of a great circle AOX.– Defines angle APX

– PAX right angle

• The distance between two points is the angle between the points.

P

Q

A

X

O B

Triangles

• Three points not on the same great circle define a spherical triangle.– Defines a plane that

excludes the origin

• Each angle is less than 180°, but the sum exceeds 180°. – Triangle PAX from before

bcA

aC

B

Small Circles

• A parallel circles have centers on the same axis. – AB and CD

– Arc AP = – AS = AO sin(AOS)

• Pick E on AB.– Great circle PEF

– PE =

P

Q

A

C O

B

F

E

S

D

Small Circle Arc

• Spherical angle is defined by APE. – Same as CPF

– Matches COF

• AS and ES parallel CO and FO.– ASE = – AE = sin

P

Q

A

C O

B

F

E

S

D

Polar Coordinates

• Spherical polar coordinates are a 3-D vector.

– r

– Reduce to , on unit sphere

Z

R

X O

SS

A

BY

cossinx

sinsiny

cosz

Spherical Trigonometry

• Set A at a pole and AB on a great circle.

bc A

aC

B ccrB cos,0,sin

Acbcba cossinsincoscoscos

bAbAbrC cos,sinsin,cossin

1,0,0Ar

c

C

b

B

a

A

sin

sin

sin

sin

sin

sin

Latitude

• Orient the sphere of the earth with N, S poles.

• The equator is the great circle at 90° from N.

• The latitude is measured from the equator.– = 90° – NX

N

S

X

E

Longitude

• The prime meridian is at right angles to the equator. – Defined at Greenwich

Observatory, NGKS

• Longitude is the angle = GNX.– 180° < <°

N

S

OX

K

G

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

Q

OX

ϒ

E

Declination and Right Ascension

• Declination is the celestial equivalent of latitude.– = 90° – PX

• Right ascension is the celestial equivalent of longitude.– = ϒPX

P

Q

OX

ϒ

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. – X(,)

• Displacement is measured as a difference of coordinates.– X’(d, d)

P

Q

X

ϒ

E

X’

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

P

Q

OX

S

Z

N

W

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

P

Q

O

X

S

Z

N

W

equator

Circumpolar

• Declination remains the same.– = 90° – PX

• The small circle through X is a parallel of declination.

• A small circle that does not intersect the horizon does not set – circumpolar stars.

P

Q

O

X

S

Z

N

W

equator

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 +

• Sidereal time is defined by the hour angle.

N

S

OX

K

G

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.