24. Transformations of coordinates
(l, b) (, )

Slide3:

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 )
+ sin (90 N) sin (90 ) cos ( N)
sin b sin N sin + cos N cos cos ( N) (1)

Slide4:

Also Hence 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. (2)

Slide5:

(b) (, ) (, )

Slide6:

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)

Slide7:

or cos cos cos cos . (3)

Slide8:

(i) (, ) (, )
Use (1) to obtain .
Then find from (3) i.e. (ii) (, ) (, )
Use (2) to obtain .
Then find from (3) i.e.

Slide9:

Rotation of the Earth
a) Evidence for Earth rotation:
Diurnal E to W motion of celestial bodies.
Rotation of plane of oscillation of Foucault’s
pendulum (Paris, 1851).
Coriolis force on long-range ballistic projectiles.
Rotation of surface winds (cyclones and anticyclones).
Variation of g with latitude gequ = 9.78 m s-2;
gpoles ≃ 9.83 m s-2.

Slide10:

(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.

Slide11:

Rotation of the Earth Left: zones on the Earth resulting from the obliquity of
the ecliptic
Right: Polar motion or Chandler wobble of the Earth
on its rotation axis

Slide12:

(c) Changes in Earth rotation rate
(i) Periodic variations – mainly annual
P become ~0.001s longer in March, April and ~0.001s
shorter in Sept., Oct, than average day.
Cumulative effects of up to 0.030s fast or slow
at different seasons of year.
Caused by changes in moment of inertia due
to differing amounts of water, ice in polar regions.

Slide13:

Universal time (= Greenwich mean solar time)
UT0 uncorrected time based on Earth rotation
UT1 corrected for polar motion but not for changes
in rotation rate.
Discovery of periodic variations in UT1 by Stoyko (1937).
Define t as
t UT1 + TDT
TDT: terrestrial dynamical time
(a uniform time scale based on planetary orbits).

Slide14:

(ii) Irregular variations
Irregular variations in length of day of up to about
0.003 s.
The timescale for significant changes in LOD is
a few years to several decades.
Thus 1850 – 1880 day was shorter by several ms
1895 – 1920 LOD was longer by up to 4 ms
1950 – 1990 LOD was longer by up to 2 ms

Slide15:

(ii) Irregular variations in LOD (continued)
Cumulative errors of up to t ~ 30 s in UT1 over
last 200 yr.
(When LOD is longer, UT1 falls behind, t increases,
goes negative to positive.)
Irregular variations first suggested by Newcomb (1878);
confirmed by de Sitter (1927) and Spencer Jones (1939).

Slide16:

(iii) Secular variations
Earth’s rotation rate is steadily slowed down because
of tidal friction.
LOD is increasing, t is decreasing.
Angular momentum of Earth-Moon system is being
transferred to the Moon, causing an increase of
Earth-Moon distance and of lunar sidereal period.
Cumulative effect is ~3¼ h over 2000 yr.
Ancient data from lunar and solar eclipse records
(whether timed or untimed), going back to 700 BC
(Chinese, Babylonian and Arabic records).
Modern data from star transit timings.
Discovered by JC Adams (1853).

Slide17:

angular velocity of Earth
o present value of ( 86400 s/d)
angular deceleration rate ( is positive, in s/d2)
o t
ot ½t2
LOD (length of day) =
dynamical time (TDT) based on ot
UT1 based on ot ½t2
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

Slide18:

In one day ½t2
½ (if t 1 d)
2.2 10-8 s = 22 ns
increase in length of each day.

Slide21:

Orbital motion of the Earth
Evidence that Earth orbits Sun
(and not Sun orbiting the Earth).
(a) Annual trigonometric parallax of stars:
Nearby stars show small displacements relative to
distant stellar backgrounds due to Earth’s orbital motion.
A star as near as 3.26 light years at ecliptic pole
describes circular path of radius 1 arc second.
(Discovered 1837.)

Slide22:

The trigonometric parallax of stars causes a small annual
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.

Slide23:

(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.

Slide26:

Precession
(a) Discovery: Hipparchus in 150 B.C.
The phenomenon is a slow westwards rotation of
the direction of the rotation axis of the Earth,
thereby describing a cone whose axis is the ecliptic
pole.
Equator is defined by Earth’s rotation axis, so equator
also changes its orientation as a result of precession.
(c) Precessional period 25800 years for one complete
precessional cycle, or 50.2 arc seconds/year.

Slide29:

(d) The equinox defines the First Point of Aries
(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.

Slide30:

Example:
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.

Slide31:

Change in direction of the NCP and in the orientation
of the equatorial plane as a result of precession

Slide34:

(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.

Slide35:

(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.

Slide36:

Presession of the equinoxes 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.

Slide37:

Change in ecliptic coordinates (of a fixed star)
as a result of precession
Ecliptic longitude increases at rate of 50.2/yr.
Ecliptic latitude is unchanged by precession.
Thus (t) o + p t
p precessional constant 50.2/ tropical year.
(t) o

Slide38:

Changes in equatorial coordinates of a star
as a result of precession
sinδ = cosε sinβ + sinε cosβ sinλ
(see section 24(b) equn. (1))
23 27 obliquity of ecliptic (a constant)
ecliptic latitude, a constant (unaffected by precession)
0 + p t

Slide39:

(see section 24(b) equn (3)) (t in years) (n = psinε = 19.98 arcsec/yr.)

Slide40:

where
n 50.2 sin(2327)/yr
20.04/yr (see section 24(b) equn. (2)) constant (unaffected by precession)

Slide41:

(p cos + p sin tan sin ) t .
Let m p cos 3.07 s/yr
and n p sin 1.34 s/yr.
Then
where t is in tropical years.

Slide42:

End of sections 24 to 27

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