Astronomical Techniques - Astrometry

This subfield deals with position measurements on the (imaginary) celestial sphere, from correction for errors due to distortions in the optics, atmospheric refraction, and aberration caused by the Earth's motion, to determination of positions in an inertial reference frame, coordinate transformations, and stellar parallaxes. We will encounter, in numerous guises, the basic equation of spherical trigonometry.

Coordinate systems

We are confined to two angular coordinates in celestial measurements, but several different systems of angular coordinates are of use for various applications. For any observer, there is an altazimuth coordinate system, defined by two angles: altitude above the horizon (or 90° - zenith distance) and azimuth, usually measured eastward from north. This is a natural local system, occurring in refraction and airmass calculations, and directly used in driving altazimuth mountings. As the Earth rotates, the vector to a distant object changes simultaneously on both altitude and azimuth; the angular rate has a singularity at the zenith.

The most-used coordinate system is equatorial, defined on the celestial sphere by right ascension and declination. The poles of this coordinate system coincide with the instantaneous poles of the Earth's rotation, and therefore precess with time; usually a reference direction (epoch) is specified, most commonly 1950.0 or by now 2000.0 (noting that there is a slight difference between Besselian B1950 and Julian J2000 epochs, so these two systems are not related solely by precession). Related concepts are sidereal time (right ascension currently crossing the observer's meridian) and hour angle (RA difference between an object and the sidereal time). Declination is defined purely by the Earth's equator and poles; right ascension requires an arbitrary zero point. This is set to occur at the point where the ecliptic (i.e. projection of the Earth's orbit) intersects the equator going northward, known as the first point of Aries though it has precessed away from that point. The normal units of declination are degrees, minutes, and seconds of arc, and for right ascension hours, minutes, and seconds of time (24 hours to the circle, or one hour = 15 degrees on the equator); for some purposes degrees or radians may be more convenient.

For problems in solar-system or galactic dynamics, it may be useful to employ ecliptic or galactic coordinates, tied to the ecliptic or galactic plane. The galactic plane makes an angle of 62.9° with the ecliptic, and galactic longitude is zero in the direction from the sun to the galactic center. This is the IAU's second try at galactic coordinates, so formally the coordinates are formally denoted lII,bII - but by now simply l,b are understood to be on the current system.

Transformations between these various systems are often needed, and the same mathematics can deal with precession. The traditional approach (see, for example, the treatment in Smart, Spherical Astronomy) is to exploit the spherical triangle. If (as in Smart, p. 34) we consider a sphere where the pole of a coordinate system is at P, some point of interest at Z, and another point of interest at X, the angular separation of Z from X is cos ZX = cos PZ cos PX + sin PZ sin PX cos ZPX where now ZPX is the interior angle formed between these points. Applied to calculating zenith distance z for a given hour angle h, declination δ, and latitude φ, this takes the form cos z = sin φ sin δ + cos φ cos δ cos h where some sines and cosines change roles because declination is defined from the equator rather than the poles. Similarly, the azimuth A can be calculated by taking an appropriate triangle to yield sin δ = sin φ cos z + cos φ sin z cos A where some care should be taken in inverting the trig functions so that the identity of the quadrant is kept (beware the principal-value conventions of a particular calculator or computer language).

Similar applications of spherical triangles can perform arbitrary coordinate transformations. A cleaner and more easily generalized approach uses the fact that coordinate rotations are equivalent to matrix multiplications of so-called direction cosines, and repeated rotations (say about different axes) to successive multiplications. Consider the transformation from spherical coordinates to Cartesian ones: (x,y,z) = (r sin c cos e, r sin c sin e, r \cos c) which is then equal to r(sin c cos e, sin c sin e, cos c). Here, r is a dummy parameter, since only the angular variables have meaning for objects nominally at infinite distance. Upon rotating from axes x,y,z to coordinates of the same point with respect to new axes x′,y′,z′, the unit vector i transforms to i′ = i cos xx′ + j cos yx′ + k cos zx′. For rotations about all three axes, we may define a matrix M which is

 cos xx′ cos yx′ cos zx′ cos xy′ cos yy′ cos zy′ cos xz′ cos yz′ cos zz′
(which is why these elements are called the direction cosines of the transformations). An arbitrary vector X transforms under this rotation of coordinates according to X′ = M X. M also has the property that its inverse is its transpose; that is, the reverse transformation is just the same with change of sign for the elements. Note that this whole transformation can be expressed as the product of successive trivial rotations about the coordinate axes: M = R11) R22) R33). Each R has the same properties as above; each gives an odd function.

To apply this formalism, consider the equatorial-to-altazimuth transformation above. If the x-axis is taken to be westward, the whole rotation is in the y,z-axes, where they are rotated upward (toward the zenith) by an amount equal to the latitude φ. The elements of the unit direction vector to the object are, in equatorial coordinates, (cos δ cos α, cos δ sin α, sin δ) and the transformation matrix has elements

 1 0 0 0 cos φ sin φ 0 -sin φ cos φ
since the mixed xz and xy elements remain zero. Looking at the new vector elements, we recover the altitude (zenith distance) and azimuth as above, since the new vector components give the components of tan A and finally the altitude. The same formalism can handle any coordinate transformation once the appropriate matrix elements are worked out (which requires knowing the pole and zero-point of one system as given in the other).

Precession

An especially important coordinate-transformation application is in accounting for precession. The nonspherical shape of the Earth means that the solar and lunar tidal forces exert a net torque, which results in a precession of the Earth's spin axis about the perpendicular to the mean plane of perturbation (the ecliptic). The has a period about 25,750 years, and amounts to a steady rotation in ecliptic coordinates (modulated by changes in the Earth's orientation elements, such as the angle between orbit and equator, known as the obliquity of the ecliptic, plus the 18.6-year nutation produced by the regression of nodes of the Moon's orbit). We will examine lunisolar precession, which dominates effects due to other planets. The rate and exact direction of precession are known from observation and celestial mechanics, and can be approximated for fairly long periods by time series in the quantities ξ0 = (23042.53 + 139.75 τ) Δ T + (30.23 -0.27 τ) Δ T2 + 18.00 Δ T3 where values are in arcseconds, Δ T = T - T0, τ = T0 - 1.900 and times are in millenia. Further, z = ξ0 + (79.27 +0.06 τ) Δ T2 + 0.32 Δ T3 and J = (20046.85 - 85.33 τ - 0.37 τ2) Δ T - (42.67 + 0.37 τ) Δ T2 -41.80 Δ T3 with numerical quantities still in arcseconds. Here, ξ0 is the rotation in the equatorial plane, z is the polar shift, and J is the inclination of the transformation. In terms of rotations about unit vectors, the precession transformation is R3(-(90° - ξ0)) R1(-J) R3(90° +z) which is finally a form useful for doing the calculation.

Aberration

Observations from a moving platform (all observations) suffer aberration in the arrival direction of starlight, due to the finite speed of light (a.k.a. the umbrella effect). To high accuracy, if we look at an angle θ to the instantaneous motion with respect to some constant reference frame (say the Sun's motion), the displacement is δ θ = v sin θ /c. The amplitude of this annual aberration is 30 km/s × 206264.8 arcsec / c or 20 arcseconds in each direction. A given star then sweeps out an apparent ellipse of this semimajor axis each year. There also exists diurnal aberration, caused by the Earth's rotation; its amplitude is much smaller at 0.32 arcsecond. Differential aberration across the field of view is actually an issue for HST observations; one doesn't want to pick the wrong instrument as the primary for certain observations as that will induce PSF blurring in one far from the optical axis.

Narrow-field astrometry

Most high-precision astrometry uses differential measures across a small field, using some set of local standard stars (an exception is the Hipparcos global solution). Here, we define some mapping from celestial to image coordinates, and determine the constants of the mapping by using coordinates of well-known stars in the same image. This determination as known historically as a plate solution. The reference stars must finally tie back into sets of fundamental stars, measured using transit or zenith instruments fixed to the Earth. Such sets include the FK3 and FK4, Perth-70, and at lower accuracy but larger numbers, the SAO and HST-GSC catalogs. The USNO catalog is a significant improvement over the GSC.

An especially important approximation for narrow-field astrometry is the tangent-plane mapping. This considers (conceptually) the projection of part of the celestial sphere outward onto a plane tangent to it at a reference point α0, δ0. The distance of a star located on the sky at some angular distance θ from the reference point will be, in the focal plane, f tan θ. Normally one defines standard coordinates in units of the focal length f such that ξ = [cos δ sin (α - α0)] / [sin δ0 sin δ + cos δ0 cos δ cos (α - α0)] and η = [sin δ cos δ0 - cos δ sin δ0 cos (α - α0] / [sin δ0 sin δ + cos δ0 cos δ cos (α - α0)] and use these as follows: take some assumed α0, δ0 and predict ξ, η on the plate, which are related to Cartesian x,y by x = f ξ, y = f η for some known reference stars. Use the real coordinates of the reference stars to update α0, δ0 and perhaps f, also allowing for the possibility that the x,y coordinates may be slightly skew to the ξ, η system or indeed might not be quite perpendicular. When this process has converged, the scatter in standard-star coordinates gives an estimator of how well the coordinate system is determined. Once these mapping constants are known (for a particular image), the reverse transformations cot δ sin (α - α0) = (ξ) / (sin δ0 + η cos δ0) and cot δ cos (α - α0) = (cot δ0 - η sin δ0) / (sin δ0 + η cos δ0) are used to derive the coordinates of desired targets. Suitable reference stars must be available, more than 3 if constants are to be determined. This may require a multistep transfer from an all-sky catalog to a local grid of faint stars, using for example measures from wide-field Schmidt plates. Additional constants may be needed to account for optical distortions or breakdown of the tangent-plane approximation, normally taking the form of radial distortions; functions up to fifth order have been used.

Additional effects may enter into these measurements. From plates, since the emulsion response is nonlinear, guiding errors may affect stars differently at different magnitudes. Thus it may be necessary to include some magnitude-dependent terms in the solution. Also, except near the zenith, there may be color-dependent terms, since stars of different color will have their mean wavelength within the passband differently refracted in the atmosphere. Observations from space with linear detectors are wonderful things.

Stellar applications of narrow-field astrometry include parallax and proper motion measurements.

Radio interferometers can measure source positions in declination with no outside reference except the observing latitude: relative right ascensions can be found, but a zero point still needs to be defined. To match the optical and radio coordinate systems, active galactic nuclei and radio-loud stars are important. The match is still less accurate than is either system internally.

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