## Friday, January 11, 2013

### Sidereal Time

If a number of observers at different locations around the world want to track an NEO in real time relative to the background stars they need to adopt a common reference frame. J2000 geocentric equatorial coordinates would be a convenient choice since the positions of the stars in this frame are known. The positions of the observers need to be adjusted as a consequence of the Earth's rotation since they will affect the apparent position of the NEO due to its parallax.

The Earth's rotation about its axis determines a specific direction in space, the celestial pole, from which we get the equatorial plane. We need another point in space to act as a reference point for rotation and the place where the Sun crosses the Equator, the direction of the Equinox, is the customary direction. This allows us to use Right Ascension (RA) and Declination (Decl) as our astronomical coordiante system. The subsolar point on the Earth's surface at the time of the Equinox points to the hour angle of RA as the Earth rotates. This hour angle is used to define the mean sidereal time for the Greenwich meridian which changes at a steady rate as the Earth rotates. On January 1, 2000 12:00 (J2000.0) the Greenwich mean sidereal time (GMST) was 18h 41m 50.548s. The rotational period of the Earth relative to the stars is 23h 56m 4.091s. By just using these two facts we can get a fairly good estimate of the sidereal time for Greenwich. For other locations we have to convert the longitude of the location to hours by dividing by 15deg/hr and then add the result to the GMST to get the local sidereal time (LST). Some Mathcad functions that do this are shown in the image below. The Julian date (JDate) is the number of days and a fraction from the arbitrary beginning of the period. The astronomical day started noon which was more convenient for observations during the night. Astronomers now count years in Julian years of 365.25 days which is why J2000.0 converts to an whole number. The Julian date was used to measure the elapsed time in days.

This will work fairly well for short periods of time. One will find more complicated formulas for the sidereal time but these take into account the change in the position of the Vernal Equinox as a consequence of the Earth's precession and nutation. The shape of the Earth and the increase in the distance of the Moon from the Earth can also alter the Earth's rotational rate. For the short period of time that an NEO is near the Earth as it is being tracked these changes can be neglected if one is only interested in the NEO's relative motion.

Supplemental (Jan 13): The function for the GMST could be modified so that the result falls between 0h and 24h as was done for the LST. Both functions were originally parts of a single function for the LST that was split and I neglected to reduce the result for GMST.