Reduction of Coordinates

Star catalogues give coordinates for some standard epoch. In the following we give the formulas needed to reduce the coordinates to a given date and time. The full reduction is rather laborious, but the following simplified version is sufficient for most practical purposes.

We assume that the coordinates are given for the epoch J2000.0.

1. First correct the place for proper motion unless it is negligible.

2. Precess the coordinates to the time of the observation. First we use the coordinates of the standard epoch (a0, S0) to find a unit vector pointing in the direction of the star:

cos So sin ao sin S0 y

Precession changes the ecliptic longitude of the object. The effect on right ascension and declination can be calculated as three rotations, given by three rotation matrices. By multiplying these matrices we get the combined precession matrix that maps the previous unit vector to its precessed equivalent. A similar matrix can be derived for the nutation. The transformations and constants given here are based on the system standardized by the IAU in 1976. The precession and nutation matrices contain several quantities depending on time. The time variables appearing in their expressions are t = J - 2,451,545.0 ,

36,525

Here J is the Julian date of the observation, t the number of days since the epoch J2000.0 (i. e. noon of January 1, 2000), and T the same interval of time in Julian centuries.

The following three angles are needed for the precession matrix

Z = 2306.2181"T + 0.30188"T2 + 0.017998"T3, z = 2306.2181"T +1.09468"T2 + 0.018203"T3, 6 = 2004.3109"T - 0.42665"T2 - 0.041833"T3.

The precession matrix is now

P21 P22 P23 P31 P32 P33

The elements of this matrix in terms of the abovementioned angles are

The new coordinates are now obtained by multiplying the coordinates of the standard epoch by the precession matrix:

This is the mean place at the given time and date. If the standard epoch is not J2000.0, it is probably easiest to first transform the given coordinates to the epoch J2000.0. This can be done by computing the precession matrix for the given epoch and multiplying the coordinates by the inverse of this matrix. Inverting the precession matrix is easy: we just transpose it, i. e. interchange its rows and columns. Thus coordinates given for some epoch can be precessed to J2000.0 by multiplying them by it

'P11 P21 P31 P12 P22 P32

In case the required accuracy is higher than about one minute of arc, we have to do the following further corrections. 3. The full nutation correction is rather complicated. The nutation used in astronomical almanacs involves series expansions containing over a hundred terms. Very often, though, the following simple form is sufficient. We begin by finding the mean obliquity of the ecliptic at the observation time:

The mean obliquity means that periodic perturbations have been omitted. The formula is valid a few centuries before and after the year 2000. The true obliquity of the ecliptic, e, is obtained by adding the nutation correction to the mean obliquity:

The effect of the nutation on the ecliptic longitude (denoted usually by Af) and the obliquity of the ecliptic can be found from

C1 = 125°- 0.05295°t, C2 = 200.9° + 1.97129°t, Af = -0.0048° sin C1 - 0.0004° sin C2, Ae = 0.0026° cos C1 + 0.0002° cos C2.

Since Af and As are very small angles, we have, for example, sin Af ~ Af and cos Af ~ 1, when the angles are expressed in radians. Thus we get the nutation matrix

This is a linearized version of the full transformation. The angles here must be in radians. The place in the coordinate frame of the observing time is now

4. The annual aberration can affect the place about as much as the nutation. Approximate corrections are obtained from

- 18-8" cos a cos X , A8 = 20-5" cos a sin 8 sin X

+18-8" sin a sin 8 cos X - 8-1" cos 8 cos X , where X is the ecliptic longitude of the Sun. Sufficiently accurate value for this purpose is given by

G = 357-528° + 0-985600°t, X = 280-460°+ 0-985647°t

These reductions give the apparent place of the date with an accuracy of a few seconds of arc. The effects of parallax and diurnal aberration are even smaller.

Example. The coordinates of Regulus (a Leo) for the epoch J2000.0 are a = 10 h 8min 22-2 s = 10-139500 h , 8 = 11° 58' 02"= 11-967222° -

Find the apparent place of Regulus on March 12, 1995.

We start by finding the unit vector corresponding to the catalogued place:

- 0-86449829 0-45787318 v 0-20735204 )

The Julian date is J = 2,449,789-0, and thus t = -1756 and T = - 0-04807666. The angles of the precession matrix are Z = - 0-03079849°, z = -0-03079798° and 0 = -0-02676709°. The precession matrix is then

0-00046717 - 0-00000025 0-99999989

0-99999931 0-00107506 - 0-00107506 0-99999942 \ - 0-00046717 - 0-00000025 The precessed unit vector is

The angles needed for the nutation are Af = 0-00309516°, As =- 0-00186227°, s = 23-43805403°, which give the nutation matrix

/ 1 - 0-00004956 - 0-00002149^ 0-00004956 1 0-00003250

\0-00002149 - 0-00003250 1 The place in the frame of the date is - 0-86393578

0-45876618 0-20772230

To correct for the aberration we first find the longitude of the Sun: G =- 1373-2° = 66-8°, X = - 8-6°. The correction terms are then

Adding these to the previously obtained coordinates we get the apparent place of Regulus on March 12, 1995:

Comparison with the places given in the catalogue Apparent Places of Fundamental Stars shows that we are within about 3'' of the correct place, which is a satisfactory result.

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