celestial sphere. local view on earth objects are usually viewed in flat euclidean geometry. from...
TRANSCRIPT
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.