Coordinates and time

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Coordinates and time Sections 24 27
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24. Transformations of coordinates (l, b) ?? (?,
?)
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• ?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)

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Also
? ?
Hence
(2)
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.
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(b) (?, ?) ?? (?, ?)
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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)
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or cos ? cos ? ? cos ? cos ?. (3)
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(ii) (?, ?) ? (?, ?) Use (2) to obtain ?. Then
find ? from (3) i.e.

• (i) (?, ?) ? (?, ?)
• Use (1) to obtain ?.
• Then find ? from (3) i.e.

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• Rotation of the Earth
• a) Evidence for Earth rotation
• Diurnal E to W motion of celestial bodies.
• Rotation of plane of oscillation of Foucaults
• 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.

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(b) Variation of ? for fixed points on Earths
surface Position of poles on surface show
roughly circular paths, diameter 20 m, period
14 months, from observations of photographic
zenith tubes (PZT). But Earths rotation axis
stays fixed in space, so far as the latitude
variation is concerned. Discovered by Küstner
(1884). Also know as Chandler wobble, after
Chandlers (1891) explanation of effect in terms
of polar motion.
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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
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• (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.

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• 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).

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• (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

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• (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).

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• (iii) Secular variations
• Earths 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).

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? ? angular velocity of Earth ?o ? present value
of ? (? 86400 s/d) ? ? angular deceleration rate
(? is positive, in s/d2)
? ? ?o ? ?t
? ? ?ot ? ½??t2 LOD (length of day)
dynamical time (TDT) based on ?? ? ?ot UT1
based on ? ? ?ot ? ½?t2 ?? ? ?? ? ? ? ½?t2
(? ?t ? TDT ? UT1) Thus ?t ? 3¼ h 11700
s (? 48?75) in 20 centuries
(t ? ?730500
days) ?? ? s/d2 ?
4.4 ? 10-8 s/d2
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In one day ?? ? ½?t2 ? ½? (if t ? 1 d) ? 2.2
? 10-8 s 22 ns ? increase in length of each
day.
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• 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 Earths
  orbital motion.
• A star as near as 3.26 light years at
  ecliptic pole
• describes circular path of radius 1 arc
  second.
• (Discovered 1837.)

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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.
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(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.
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• 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 Earths 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.

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(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.
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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.
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Change in direction of the NCP and in the
orientation of the equatorial plane as a result
of precession
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(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 Earths 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.
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(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.
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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.
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• 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

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• Changes in equatorial coordinates of a star
• as a result of precession
• sind cose sinß sine cosß sin?
• (see section 24(b) equn. (1))
• ? ? 23? 27? ? obliquity of ecliptic (a constant)
• ? ? ecliptic latitude, a constant (unaffected by
  precession)
• ? ? ?0 p ?t

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(see section 24(b) equn (3))
(?t in years)
(n psine 19.98 arcsec/yr.)
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where n ? 50.2 sin(23?27?)??/yr
? 20.04??/yr
(see section 24(b) equn. (2))
? ? constant (unaffected by precession)
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?? ? (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.
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End of sections 24 to 27