A211coords and time4

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Slide1: 

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 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)

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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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(i) (, )  (, ) Use (1) to obtain . Then find  from (3) i.e. (ii) (, )  (, ) Use (2) 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 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.

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(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.

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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 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).

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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 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.)

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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 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.

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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 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.

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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 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

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(see section 24(b) equn (3)) (t in years) (n = psinε = 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

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