Information on resource 'Computation of GAST, GMST, and ERA'

Computation of Greenwich Apparent Sidereal Time, Greenwich Mean Sidereal Time, and the Earth Rotation Angle

Intermediate and apparent places of stars are usually required for geodetic purposes, like determining a position (longitude, latitude on the Earth's surface) or an azimuth (which is the angle between the geographic north direction and an arbitrary terrestrial object visible from the observation site). A variety of corresponding tasks is treated in any textbook of positional geodesy and cannot be addressed here.

Nevertheless, many of these tasks make use of the "nautical triangle": this can be imagined as a spherical triangle located on the celestial sphere. The nautical triangle is formed by three points:

  1. the zenith - i.e. the point where the projection of the theodolite's plumb line intersects the celestial sphere -,
  2. the north pole of the celestial sphere - now called Celestial Intermediate Pole (CIP) -, and
  3. a selected star entering via its intermediate or apparent place.

One parameter of the nautical triangle is immediately given by the declination of the star (or its complement to 90 degree, respectively).

The local hour angle "t" is frequently determined as second parameter; "t" is the angle between these two lines of the triangle which emanate from the CIP.

The local hour angle using right ascensions referred to the "Celestial Intermediate Origin (CIO)" can be determined by:

t = ERA + longitude - intermediate right ascension

where ERA stands for the Earth Rotation Angle and longitude means the geographic east longitude of the observer. Traditionally, the local hour angle has been determined by

t = GAST + longitude - apparent right ascension

where GAST stands for Greenwich Apparent Sidereal Time, and with apparent right ascensions referred to the equinox. Both procedures yield the same hour angle, apart from negligible differences which are of the order of one milliarcsecond or below. BUT: either ERA and INTERMEDIATE (CIO-based) right ascensions or GAST and APPARENT (equinox-based) right ascensions have to be used consistently.

An additional line or angle has yet to be provided by the observer as third parameter in the nautical triangle. This may be a zenith distance or an azimuth. Then the nautical triangle is completely determined and can be applied for the specific geodetic task.

Note the different time arguments for intermediate and apparent places (Terrestrial Time TT) and UT1 for ERA and GAST. UT1 and UTC do not differ by more than 1 second. The actual difference UT1 - UTC is published by the International Earth Rotation Service (IERS) - Bulletin D - and can be accessed via the IERS web-pages where also a useful glossary can be found. For detailed information we recommend the printed version of the Astronomical Almanac, or the Astronomical Almanac Online.

GAST and GMST are consistent with IAU 2006 precession.

Software Routines from the IAU SOFA Collection were used. Copyright © International Astronomical Union Standards of Fundamental Astronomy (; actually, we are using ERFA in the meantime, but SOFA credit is still due.

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