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. 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

4. 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

5. 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 bc A a C B

6. 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

7. 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

8. Polar Coordinates Spherical polar coordinates are a 3-D vector. –r  –Reduce to ,  on unit sphere Z R XO S S   A B Y

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

10. 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

11. Longitude The prime meridian is at right angles to the equator. –Defined at Greenwich Observatory, NGKS Longitude is the angle = GNX. –  180° <  <  ° N S O X K G E

12. 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 O X ϒ  E

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

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 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’

16. 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 O X S Z N W

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

18. 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

19. 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

20. 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 O X K G E

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.