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.

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 *l ^{II},b^{II}* - but
by now simply

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

cos xx′ | cos yx′ | cos zx′ |

cos xy′ | cos yy′ | cos zy′ |

cos xz′ | cos yz′ | cos zz′ |

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 |

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
**ξ _{0} = (23042.53 + 139.75 τ) Δ T + (30.23 -0.27 τ)
Δ T^{2} + 18.00 Δ T^{3}**
where values are in arcseconds,

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.

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 α_{0}, δ_{0}. 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 - α _{0})] /
[sin δ_{0} sin δ + cos δ_{0} cos δ
cos (α - α_{0})]**
and

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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