Approximate Sidereal Time

Sidereal time is a system of timekeeping based on the rotation of the Earth with respect to the fixed stars in the sky. More specifically, it is the measure of the hour angle of the vernal equinox. If the hour angle is measured with respect to the true equinox, apparent sidereal time is being measured. If the hour angle is measured with respect to the mean equinox, mean sidereal time is being measured. When the measurements are made with respect to the meridian at Greenwich, the times are referred to as Greenwich mean sidereal time (GMST) and Greenwich apparent sidereal time (GAST).

Given below is a simple algorithm for computing apparent sidereal time to an accuracy of about 0.1 second, equivalent to about 1.5 arcseconds on the sky. The input time required by the algorithm is represented as a Julian date (Julian dates can be used to determine Universal Time.)

Let JD be the Julian date of the time of interest.  Let JD0 be the Julian date of the previous midnight (0h) UT (the value of JD0 will end in .5 exactly), and let H be the hours of UT elapsed since that time. Thus we have JD = JD0 + H/24.

For both of these Julian dates, compute the number of days and fraction (+ or -) from 2000 January 1, 12h UT, Julian date 2451545.0:

D = JD - 2451545.0
D0 = JD0 - 2451545.0

Then the Greenwich mean sidereal time in hours is

GMST = 6.697374558 + 0.06570982441908 D0 + 1.00273790935 H + 0.000026 T2

where T = D/36525 is the number of centuries since the year 2000; thus the last term can be omitted in most applications. It will be necessary to reduce GMST to the range 0h to 24h. Setting H = 0 in the above formula yields the Greenwich mean sidereal time at 0h UT, which is tabulated in The Astronomical Almanac.

The following alternative formula can be used with a loss of precision of 0.1 second per century:

GMST = 18.697374558 + 24.06570982441908 D

where, as above, GMST must be reduced to the range 0h to 24h. The equations for GMST given above are adapted from those given in Appendix A of USNO Circular No. 163 (1981).

The Greenwich apparent sidereal time is obtained by adding a correction to the Greenwich mean sidereal time computed above. The correction term is called the nutation in right ascension or the equation of the equinoxes. Thus,

GAST = GMST + eqeq.

The equation of the equinoxes is given as eqeq = Δψ cos ε where Δψ, the nutation in longitude, is given in hours approximately by

Δψ ≈ -0.000319 sin Ω - 0.000024 sin 2L

with Ω, the Longitude of the ascending node of the Moon, given as

Ω = 125.04 - 0.052954 D,

and L, the Mean Longitude of the Sun, given as

L = 280.47 + 0.98565 D.

ε is the obliquity and is given as

ε = 23.4393 - 0.0000004 D.

The above expressions for Ω, L, and ε are all expressed in degrees.

The mean or apparent sidereal time locally is found by obtaining the local longitude in degrees, converting it to hours by dividing by 15, and then adding it to or subtracting it from the Greenwich time depending on whether the local position is east (add) or west (subtract) of Greenwich.

If you need apparent sidereal time to better than 0.1 second accuracy on a regular basis, consider using the Multiyear Interactive Computer Almanac, MICA. MICA provides very accurate almanac data in tabular form for a range of years.

NOTES ON ACCURACY

The maximum error resulting from the use of the above formulas for sidereal time over the period 2000-2100 is 0.432 seconds; the RMS error is 0.01512 seconds. To obtain sub-second accuracy in sidereal time, it is important to use the form of Universal Time called UT1 as the basis for the input Julian date.

The maximum value of the equation of the equinoxes is about 1.1 seconds, so if an error of ~1 second is unimportant, the last series of formulas can be skipped entirely. In this case set eqeq = 0 and GAST = GMST, and use either UT1 or UTC as the Universal Time basis for the input Julian date.