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ASTR211: COORDINATES AND TIME Coordinates and time Prof. John Hearnshaw Sections 24 – 27 ASTR211: COORDINATES AND TIME Prof. John Hearnshaw 24. Transformations of coordinates (l, b) (, ) ASTR211: COORDINATES AND TIME 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) Prof. John Hearnshaw N +27 08 N 12 h 51m Coordinates of NGP are (N, N) 123 (a constant that specifies gal. centre direction) ASTR211: COORDINATES AND TIME Also sin(360 θ ) sin(α α N ) sin(90 δ) sin(90 b) sin(θ ) sin(α α N ) cosδ cos b cos sin(θ+ )= sin(α α N ) cosb 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) Prof. John Hearnshaw Hence ASTR211: COORDINATES AND TIME (, ) (, ) Prof. John Hearnshaw (b) ASTR211: COORDINATES AND TIME cos (90 ) cos (90 ) cos + sin (90 ) sin cos (90 + ) sin cos sin + sin cos ( sin ) sin cos sin sin cos sin (2) Prof. John Hearnshaw cos (90 ) cos cos (90 ) + sin sin (90 ) cos (90 ) sin cos sin + sin cos sin . (1) ASTR211: COORDINATES AND TIME sin 90 β sin 90 δ sin 90+α sin 90 λ cosβ cosδ cosα cosλ cos cos cos cos . (3) Prof. John Hearnshaw or (i) (, ) (, ) (ii) (, ) (, ) Use (1) to obtain . Then find from (3) i.e. Use (2) to obtain . Then find from (3) i.e. cosβ cosλ cos α= cosδ cosα cos δ cos λ= cosβ Prof. John Hearnshaw ASTR211: COORDINATES AND TIME ASTR211: COORDINATES AND TIME 25. Rotation of the Earth • 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. Prof. John Hearnshaw a) Evidence for Earth rotation: ASTR211: COORDINATES AND TIME (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). Discovered by Küstner (1884). Also know as Chandler wobble, after Chandler’s (1891) explanation of effect in terms of polar motion. Prof. John Hearnshaw But Earth’s rotation axis stays fixed in space, so far as the latitude variation is concerned. ASTR211: COORDINATES AND TIME 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 Prof. John Hearnshaw Rotation of the Earth ASTR211: COORDINATES AND TIME (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. Prof. John Hearnshaw • ASTR211: COORDINATES AND TIME 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. • Define t as t UT1 + TDT • TDT: terrestrial dynamical time (a uniform time scale based on planetary orbits). Prof. John Hearnshaw • Discovery of periodic variations in UT1 by Stoyko (1937). ASTR211: COORDINATES AND TIME • 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 Prof. John Hearnshaw (ii) Irregular variations ASTR211: COORDINATES AND TIME • 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). Prof. John Hearnshaw (ii) Irregular variations in LOD (continued) ASTR211: COORDINATES AND TIME • 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). Prof. John Hearnshaw (iii) Secular variations 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) 2θ 2 11700 2 4.4 10-8 s/d2 2 s/d 2 t (730500) Prof. John Hearnshaw ASTR211: COORDINATES AND TIME ASTR211: COORDINATES AND TIME Prof. John Hearnshaw In one day ½t2 ½ (if t 1 d) 2.2 10-8 s = 22 ns increase in length of each day. Prof. John Hearnshaw ASTR211: COORDINATES AND TIME Prof. John Hearnshaw ASTR211: COORDINATES AND TIME ASTR211: COORDINATES AND TIME 26. Orbital motion of the Earth Evidence that Earth orbits Sun (and not Sun orbiting the Earth). Prof. John Hearnshaw (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.) 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. Prof. John Hearnshaw ASTR211: COORDINATES AND TIME ASTR211: COORDINATES AND TIME (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). Constant of aberration, K v/c radians 206265 v/c arc s 20.5 arc s. Prof. John Hearnshaw 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. Prof. John Hearnshaw ASTR211: COORDINATES AND TIME Prof. John Hearnshaw ASTR211: COORDINATES AND TIME ASTR211: COORDINATES AND TIME (a) Discovery: Hipparchus in 150 B.C. (b) 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. Prof. John Hearnshaw 27. Precession Prof. John Hearnshaw ASTR211: COORDINATES AND TIME Prof. John Hearnshaw ASTR211: COORDINATES AND TIME ASTR211: COORDINATES AND TIME (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). Both (right ascension) and (declination) are affected by precession. Prof. John Hearnshaw The drift in equator and equinox means that the coordinates of stars change slowly with epoch. ASTR211: COORDINATES AND TIME Example: (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. Prof. John Hearnshaw Canopus ( Carinae): (, ) (1900.0) 6h 21m 44s, 52 38 (, ) (2000.0) 6h 23m 57s, 52 41 Change in direction of the NCP and in the orientation of the equatorial plane as a result of precession Prof. John Hearnshaw ASTR211: COORDINATES AND TIME Prof. John Hearnshaw ASTR211: COORDINATES AND TIME Prof. John Hearnshaw ASTR211: COORDINATES AND TIME ASTR211: COORDINATES AND TIME (f) Cause of precession: (luni-solar precession) The Earth is non-spherical, in fact an oblate spheroid. The torque (couple) on a spinning object results in precession – cf. the precession of a spinning top inclined to vertical. Prof. John Hearnshaw 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). ASTR211: COORDINATES AND TIME (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. 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. Prof. John Hearnshaw Difference 20 m 27 s ASTR211: COORDINATES AND TIME 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. Prof. John Hearnshaw Presession of the equinoxes ASTR211: COORDINATES AND TIME (h) 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. (t) o + p t p precessional constant 50.2/ tropical year. (t) o Prof. John Hearnshaw Thus ASTR211: COORDINATES AND TIME (i) Changes in equatorial coordinates of a star as a result of precession sinδ = cosε sinβ + sinε cosβ sinλ (see section 24(b) equn. (1)) 0 + p t p 50.2 arcsec/yr Prof. John Hearnshaw 23 27 obliquity of ecliptic (a constant) ecliptic latitude, a constant (unaffected by precession) ASTR211: COORDINATES AND TIME cos δ δ = sin ε cos β cos λ λ sin ε δ = cosβ cos λ λ cos δ sin ε = cos α cos δ λ cos δ (see section 24(b) equn (3)) δ = (p sin ε) cos α t n cos α t (t in years) (n = psinε = 19.98 arcsec/yr.) Prof. John Hearnshaw δ = sin ε cos α λ = t ASTR211: COORDINATES AND TIME where n 50.2 sin(2327)/yr 20.04/yr sin cos sin sin cos sin (see section 24(b) equn. (2)) 0 cos cos sin sin sin sin cos cos Prof. John Hearnshaw constant (unaffected by precession) ASTR211: COORDINATES AND TIME (cos cos sin sin sin ) sin cos cos cos cos sin sin sin sin cos sin cos cos (cos sin tan sin ) t Let m p cos 3.07 s/yr and n p sin 1.34 s/yr. Then ( m n tan sin )t where t is in tropical years. Prof. John Hearnshaw (p cos + p sin tan sin ) t . ASTR211: COORDINATES AND TIME Prof. John Hearnshaw End of sections 24 to 27