Celestial Sphere

Local ViewOn 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 objectsEarths rotation

Great CirclesAny plane through the center of a sphere intersects the sphere in a great circle.AXBPAQBPoints are opposite if for any great circle that passes through one it passes through both.PQAXOB

Spherical AnglesThe angle APX projects onto the plane of a great circle AOX.Defines angle APXPAX right angle

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

TrianglesThree 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 beforebcAaCB

Small CirclesA parallel circles have centers on the same axis. AB and CDArc AP = qAS = AO sin(AOS)

Pick E on AB.Great circle PEFPE = qPQACOBFESqyD

Small Circle ArcSpherical angle y is defined by APE. Same as CPFMatches COF

AS and ES parallel CO and FO.ASE = yAE = y sinqPQACOBFESqyD

Polar CoordinatesSpherical polar coordinates are a 3-D vector.r, q, yReduce to q, y on unit sphere

ZRXOSSqyABY

Spherical TrigonometrySet A at a pole and AB on a great circle.

bcAaCB

LatitudeOrient 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 NXNSXqfE

LongitudeThe prime meridian is at right angles to the equator. Defined at Greenwich Observatory, NGKS

Longitude is the angle l = GNX. -180 < l < -180NSOXKGlE

ProjectionProject the earth outward into space. North and south celestial poles P, QCelestial equator EEast orientation is defined by the suns position at vernal equinox.Crosses equator from S to NMarch 21PQOXaE

Declination and Right AscensionDeclination is the celestial equivalent of latitude. d = 90 PX

Right ascension is the celestial equivalent of longitude. a = PXPQOXaEd

Heavenly TimeRight ascension is not measured in degrees.Degrees are converted to time.24 hours = 3601h = 151 = 4m1m = 15'1' = 4s1s = 15'' 1'' = 1/15 s

Stellar CoordinatesStellar coordinates use right ascension and declination. X(a,d)

Displacement is measured as a difference of coordinates.X(a + da, d + dd)

PQXaEX

Alt-AzimuthThe alt-azimuth system is fixed to an observer on earth.Zenith distance is measured from vertical.z = ZXAltitude a = 90 - zAzimuth is measured west of north.A = PZXPQOXSZNW

Rising Star

Stars are visible to an observer when z > 90.

Tables of rising and setting objects are computed for z = 90.

Hour AngleAlt-azimuth moves with the stars.PZ was fixed by the transformation.Hour angle is measured from zenith and celestial north.HA = ZPX to the westPZSQ is the observers meridianPQOXSZNWequator

CircumpolarDeclination remains the same. d = 90 PXThe small circle through X is a parallel of declination.

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

Relative TimeProject points from Greenwich G and an observer X onto the celestial sphere.Hour angle at Greenwich GHAObserver hour angle is HA = GHA + l

Sidereal time is defined by the hour angle.NSOXKGlE

Sidereal TimeSidereal time is defined by the hour angle.Moves with the starsLST = HA + RA

A sidereal day is shorter than a solar day.23 h 56 m

Universal TimeThe sidereal and solar time scales depend on the earths rotation.Irregular on short time scalesSlowing 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 ETDefines the second based on the year 1900Replaced 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.