Coordinates and time Sections 24 – 27. 24. Transformations of coordinates ( l, b) ( , ). N +27 08 N 12 h 51m Coordinates of NGP are ( N , N ) 123 (a constant that specifies gal. centre direction) cos (90 b ) cos (90 N ) cos (90 )
Sections 24 – 27
If (, ) are known, use (1) to obtain b
(note that N, N are equatorial coordinates of
north galactic pole),
and then use (2) to find ( + l) and hence l.
(b) (, ) (, )
cos (90) cos cos (90)
+ sin sin (90) cos (90)
sin cos sin + sin cos sin . (1)
cos (90) cos (90) cos
+ sin (90) sin cos (90 + )
sin cos sin + sin cos ( sin )
sin cos sin sin cos sin (2)
or cos cos cos cos . (3)
(ii) (, ) (, )
Use (2) to obtain .
Then find from (3) i.e.
(b) Variation of for fixed points on Earth’s surface
Position of poles on surface show roughly circular paths,
diameter ~ 20 m, period ~ 14 months, from observations
of photographic zenith tubes (PZT).
But Earth’s rotation axis stays fixed in space, so far as
the latitude variation is concerned.
Discovered by Küstner (1884).
Also know as Chandler wobble, after Chandler’s (1891)
explanation of effect in terms of polar motion.
Left: zones on the Earth resulting from the obliquity of
Right: Polar motion or Chandler wobble of the Earth
on its rotation axis
angular velocity of Earth
o present value of ( 86400 s/d)
angular deceleration rate ( is positive, in s/d2)
LOD (length of day) =
dynamical time (TDT) based on ot
UT1 based on ot ½t2
(t TDT UT1)
Thus t 3¼ h = 11700 s ( 4875) in 20 centuries
(t 730500 days)
s/d2 4.4 10-8 s/d2
In one day ½t2
½ (if t 1 d)
2.2 10-8 s = 22 ns
increase in length of each day.
displacement of nearby stars measured relative to distant
ones, and of amplitude inversely proportional to the
distance of the nearby star. This is evidence for the orbital
motion of the Earth about the Sun.
(b) Aberration of starlight: (Bradley 1725)
All stars in given direction describe elliptical paths,
period one year, semi-major axis 20.5 arc s
(much greater than parallax even for nearest stars).
At ecliptic pole motion is circle but 3 months out of phase
with parallactic motion.
v 30 km/s speed of Earth in orbit
c 3 105 km/s speed of light.
Constant of aberration, K v/c radians
206265 v/c arc s 20.5 arc s.
(intersection of ecliptic and equator), and is the zero point
for ecliptic coordinates ( 0) and for equatorial
coordinates ( 0 h).
The drift in equator and equinox means that the
coordinates of stars change slowly with epoch.
Both (right ascension) and (declination) are
affected by precession.
Canopus ( Carinae):
(, ) (1900.0) 6h 21m 44s, 52 38
(, ) (2000.0) 6h 23m 57s, 52 41
(e) In the 2600 years since first Greek astronomers
(e.g. Thales), precession of equinox amounts to ≃ 30
along ecliptic. First Point of Aries was then in
constellation of Aries (hence the name). The N. Pole was
in 3000 B.C. near the star Draconis. It is now near Polaris
( UMa) (closest ~½ in 2100 A.D.) and will be near Vega
( Lyr) in 14000 A.D.
of the equatorial plane as a result of precession
(f) Cause of precession: (luni-solar precession)
The Earth is non-spherical, in fact an oblate spheroid.
Pull of Sun and Moon on spheroidal Earth applies a
weak couple on Earth (i.e. Sun tries to make Earth’s
rotation axis perpendicular to ecliptic).
The torque (couple) on a spinning object results
in precession – cf. the precession of a spinning top
inclined to vertical.
(g) Consequences of precession
Tropical year time for Sun to progress through
360 50.2 around ecliptic 365.2422 days.
Sidereal year time for Sun to progress through 360
around ecliptic 365.2564 days.
Difference 20 m 27 s
Note that the tropical year time between two successive
passages of Sun through March equinox. This is the time
interval over which the seasons repeat themselves, and
therefore the time interval on which the calendar is based.
Presession results in
the tropical year, which
governs the cycle of the
seasons, being 20 m 27 s
shorter than the sidereal
year, which is the orbital
period of the Earth.
(see section 24(b) equn (3))
(t in years)
(n = psinε = 19.98 arcsec/yr.)
n 50.2 sin(2327)/yr
(see section 24(b) equn. (2))
constant (unaffected by precession)
(p cos + p sin tan sin ) t .
Let m p cos 3.07 s/yr
and np sin 1.34 s/yr.
where t is in tropical years.