Celestial Sphere

1. Celestial Sphere

2. 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

3. 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. Great Circles P A B O X Q

4. 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. Spherical Angles P A B O X Q

5. 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 Triangles A c b B C a

6. 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 Small Circles P y q A B S E C D O F Q

7. Spherical angle y is defined by APE. Same as CPF Matches COF AS and ES parallel CO and FO. ASE = y AE = y sinq Small Circle Arc P y q A B S E C D O F Q

8. Spherical polar coordinates are a 3-D vector. r, q, y Reduce to q, y on unit sphere Polar Coordinates Z y R S S A q X O B Y

9. Set A at a pole and AB on a great circle. Spherical Trigonometry A c b B a C

10. 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. f = 90° – NX Latitude N q X E f S

11. The prime meridian is at right angles to the equator. Defined at Greenwich Observatory, NGKS Longitude is the angle l = GNX. -180° < l < -180° Longitude N G X E O K l S

12. 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 Projection P X E O ϒ a Q

13. Declination is the celestial equivalent of latitude. d = 90° – PX Right ascension is the celestial equivalent of longitude. a = ϒPX Declination and Right Ascension P X E O d ϒ a Q

14. 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

15. Stellar coordinates use right ascension and declination. X(a,d) Displacement is measured as a difference of coordinates. X’(a + da, d + dd) Stellar Coordinates P X’ X E ϒ a Q

16. 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 Alt-Azimuth Z P X N S O W Q

17. Rising Star • Stars are visible to an observer when z > 90°. • Tables of rising and setting objects are computed for z = 90°.

18. 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 Hour Angle Z equator P X N S O W Q

19. Declination remains the same. d = 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. Circumpolar Z equator P X N S O W Q

20. 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 Sidereal time is defined by the hour angle. Relative Time N G X E O K l S

21. 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

22. 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

23. 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.

24. 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.