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AST 443/ PHY 517 Astrometry Astrometry • Your detector records events at posi=ons X,Y (detector coordinates) • You need to transform this to sky coordinates (α, δ or equivalent) on the sky • Your detector is (probably) flat • The focal plane is (usually) curved • Everything moves Celes=al Reference Points • The reference point is γ, the first point of Aries, at (α, δ) = (0,0) • γ is the ascending node of the eclip=c, where it crosses the celes=al equator. • The celes=al coordinate system is actually defined by a grid of extragalac=c point radio sources. • The official reference is the Interna=onal Celes=al Reference System (ICRS). • The posi=on of the celes=al poles and the origin of right ascension are fixed to within 20 micro-­‐arcsec with reference to this grid of quasars. Conver=ng X,Y to α, δ : Tangent Plane Geometry • The sky , a spherical surface, is curved • Images of the sky are flat • The measured X,Y posi=ons on a flat image do not translate linearly into angular offsets on the sky • The math is straigh]orward Tangent Plane Geometry RA,DEC to X,Y: • X = -­‐ [cos(δ) sin(α-­‐a)] / [sin(δ) sin(d) + cos(δ) cos(d) cos (α-­‐a)] /S • Y = [sin(δ) cos(d) -­‐ cos(δ) sin(d) cos (α-­‐a)] / [sin(δ) sin(d) + cos(δ) cos(d) cos (α-­‐a)] /S • • • • • X,Y to RA/DEC: θ = tan(-­‐X*S/Y*S) φ = tan(sqrt[(X*S)2+(Y*S)2]) δ = arcsin(sin(d) cos(φ) + cos(d) sin(φ) cos(θ) α = a + arcsin((sin(φ) sin(θ) / (cos(d) cos(φ) -­‐ sin(d) sin(φ) cos(θ))) – a, d are the RA, DEC of the tangent point -­‐ the place where the image is tangent to the sky (generally the center of the image). – S is the plate scale (radians per mm, or equivalent). Distor=ons • All telescope op=cs introduce distor=ons into the images. – Coma and as=gma=sm do not directly affect posi=ons. – But they introduce systema=c posi=onal offsets unless the proper posi=on-­‐dependent point-­‐spread-­‐func=on is used. – Pincushion and barrel distor=ons directly affect posi=ons. – Higher order aberra=ons introduce other distor=ons. • In prac=ce, one measures these distor=ons by observing a dense grid of stars of known posi=ons (e.g., a globular cluster), and determining the empirical correc=on between X and Y on the plate and Δα, Δδ on the sky. • This correc=on is ususally carried to third order (i.e., terms to xny3-­‐n, including cross terms). HST imagers: residual distor=ons ager subtrac=ng second-­‐
order terms. (from the HST data handbook) Everything Moves • The Earth (your observing pla]orm) rotates (0.5 km/s), and orbits the Sun (30 km/s) • The Sun orbits the Galaxy (220 km/s) • The Galaxy moves rela=ve to the cosmic microwave background (600 km/s) Earth-­‐Induced shigs • Diurnal aberra6on: due to the Earth's rota=on. In celes=al coordinates: Δα =-­‐k cos(φ) cos(h) sec(δ) Δδ = -­‐k cos(φ) sin(h) sin(δ) – – – – k=0.3198" φ: geode=c la=tude of the observer h: hour angle of the target δ: declina=on of the target • Annual aberra6on: due to the Earth’s orbit. In eclip=c coordinates: Δλ = -­‐k cos(λ -­‐ λ0) sec(β) Δβ = -­‐k cos(λ -­‐ λ0) sin(β) • • – k = 20.496 arcsec – λ0: eclip=c longitude of the Sun Annual aberra=on is of order v⊕/C rad. It is a consequence of the finite speed of light. Over a year, the star traces an ellipse with semimajor axes k, k sin(β). At the eclip=c pole this reduces to a circle; on the eclip=c this is a line. Aberra=ons shig the en=re field, and so do not generally affect the rela=ve posi=ons of objects. You need to account for these aberra=ons if you need to point the telescope with arcsecond accuracy. In extreme cases, differen=al stellar aberra=on across the field is a significant effect: Differen=al distor=on due to the difference in stellar aberra=on across the HST/ACS field can amount to 0.04~arcsec (0.8~pixels) Differen=al Refrac=on dα = -­‐tan(α0) dn • α0: angle of incidence on plane parallel atmosphere (90o-­‐zenith distance) • With a large field of view, α0 can vary enough that dα is measurably different across the field • LSST will have a 3.5o FoV – At zd=45o, dα(46.5o)-­‐dα(43.5o) = 6.2” Proper Mo=on • The peculiar mo6on of an object rela=ve to some local standard, either the heliocenter or the local standard of rest (LSR). • The proper mo6on μ = v/D – v is the peculiar velocity of the object – D is its distance. • Units are usually expressed in arcsec/year. • Solar peculiar velocity: 19.5 km/s towards 18.0h, 30o. • Correc=on for the Solar peculiar velocity is called reduc6on to the local standard of rest. Proper Mo=on: Nearby neutron star RX J1856-­‐3754: 2 years of proper mo=on HST ACS/HRC Pixel size: 0.027” Wobble due to parallax Annual Parallax • Annual Parallax π: due to the orbit of the earth around the Sun. – At a distance of 1 parsec, π = 1 arcsec. – The mo=on is a circle at the eclip=c poles and a line on the eclip=c. – The shape of the ellipse is completely defined by the target’s coordinates, only the scaling (π) is a free parameter – Distance in parsecs = 1/π Diurnal Parallax Diurnal parallax: the difference in zenith distance seen by the observer (on the Earth's surface) and the zenith distance seen from the geocenter. • The maximum diurnal parallax is (r/d) cos l – r: radius of the Earth – d: distance to the object in Earth radii – l: observer's la=tude. • The maximum diurnal parallaxes of the Moon: 57' ; Sun: 8.8" Mo=on with PM subtracted Parallac=c Ellipse Probability Parallax probability distribu=on π (mas) What can you measure? • You can easily centroid to ~ 1/10 the PSF. – At Mt Stony Brook: ~0.1” (1/π=10 pc) – With CTIO 1.3m/Andicam: ~.04” (1/π=25 pc) – With HST/ACS: ~0.005” ” (1/π=200 pc) • Proper mo=on is cumula=ve. Be pa=ent. How do you measure? • Differen=ally. • Use lots of reference stars. – Uncertain=es ~ 1/√ n • Solve for the plate scale • Parallax + Proper mo=ons: – 3 free parameters (π, μα, μδ) – Need >3 observa=ons – the more, the be„er – Best to observe at extrema of the parallac=c ellipse Cookbook • Correct for plate distor=ons – xobs ⇒ xtrue – yobs⇒ ytrue • It is ogen easier to work in x,y rather than α, δ • xi=x0 + μx ti + Πx(DoYi)π • yi=y0 + μy ti + Πy(DoYi)π – Π is the parallac=c offset for the day of year at 1 pc – Π is cyclic – DoY is the day of year (ti modulo 365) • Convert μx, μy to μα, μδ Space Velocity • Proper Mo=on μ is ⊥ to the line of sight – Transverse velocity vT = μ/π (arcsec /pc /yr) – vT = 4.78 μ(“/yr) d(pc) km/sec • Radial velocity RV is along the line of sight • Space velocity v=sqrt(vT2 + RV2) Precession • The Vernal equinox undergoes a retrograde mo=on on a 25725 year cycle (the Platonic year). Eclip=c longitudes increase monotonically. • Lunisolar precession, due to the Moon and Sun: – P0 = 50.3878" + 0.000049" T (T is the number of tropical years since 1900). • Planetary precession: P1 = -­‐0.1055" + 0.000189" T • General precession: P = P0 + P1 cos (ε) = 50.2910" + 0.000222" T The change in α, δ: Pα = m + n sin(α) tan(δ) Pδ = n cos(α) m = 46.124" + 0.000279" t, n = 20.043" -­‐ 0.000085" t • Consequences of precession: – in 2115 CE, Polaris will be 28' from the NCP. – In 12,000 years, Vega will be within 1 degree of the NCP. • Nuta6on refers to small changes in the rate of precession. – The NCP traces a 9.2" ellipse on an 18.6 year period because of the inclina=on of the lunar orbit to the Earth's equator. Astrometric Catalogs • The Hipparcos catalog – Astrometry and photometry of 118218 stars to 9th magnitude. – Milli-­‐arcsec precision. • The USNO B1.0 catalog: – Posi=ons, magnitudes, and proper mo=ons for > 109 objects • The USNO-­‐A2.0: – posi=ons and magnitudes for 526,230,881 stars • FK5: the Figh Fundamental Catalog of 1535 stars.