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Computing 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. Specifically, it is the measure of the hour angle of the vernal equinox. If the hour angle is measured with respect to the mean equinox, mean sidereal time is being measured, and if the hour angle is measured with respect to the true equinox, apparent sidereal time is being measured. When the measurement is made with respect to the meridian at Greenwich, the time is referred to as Greenwich mean sidereal time (GMST) and Greenwich apparent sidereal time (GAST).

Below is an algorithm for computing apparent sidereal time to an accuracy of about 0.1 second of time, equivalent to about 1.5 arcseconds on the sky.

Sidereal time is a function of both the rotation of the Earth on its axis and its motion about the Sun. Because the Earth's rotation rate is not constant the input needs to be split into two parts: Terrestrial Time, TT, and Universal Time, UT1. The input times are represented as Julian dates.

Let JDTT be the TT Julian date of the time of interest, JDUT be the UT1 Julian date. The difference between TT and UT1 is small, ∼0.001 day, so for most applications JDTT = JDUT. Also let JD0 be the Julian date of the previous midnight (0h) UT1 (the value of JD0 will end in .5 exactly), and H be the hours of UT1 elapsed since that time. Thus we have JDUT = JD0 + H/24.

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

DTT = JDTT - 2451545.0
DUT = JD0 - 2451545.0

Also compute T the number of TT centuries since the year 2000.

T = DTT / 36525

Then the Greenwich mean sidereal time in hours is:

GMST = mod (6.697375 + 0.065707485828 DUT + 1.0027379 H + 0.0854103 T + 0.0000258 T2, 24) h

Or, assuming JDTT = JDUT :

GMST = mod (6.697375 + 0.065709824279 DUT + 1.0027379 H + 0.0000258 T2, 24) h

where mod(•,•) is the modulo function. The T2 term can be omitted in most applications. Setting H = 0 in the above formulae yields the Greenwich mean sidereal time at 0h UT1, which is tabulated in The Astronomical Almanac.

The following alternative formula can be used for approximate GMST at 0h UT with a loss of precision of 0.1 second per century:

GMST = mod (18.697375 + 24.065709824279 DUT, 24h)

The equations for GMST given above are adapted from those given in section 2.6.2 of USNO Circular No. 179 (2005).

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 DTT,

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

L = 280.47 + 0.98565 DTT.

ε is the obliquity and is given as

ε = 23.4393 - 0.0000004 DTT.

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

The local mean or apparent sidereal time is found by obtaining the local longitude in degrees (east longitudes positive), dividing by 15 to convert it to hours, and adding it to the Greenwich sidereal time.

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 in GAST resulting from the use of these formulae 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 transformation from GMST to GAST can be skipped. In this case use either UT1 or UTC as the Universal Time basis for the input Julian date, set eqeq = 0, and GAST = GMST.