Sidereal Time - Exact Duration and Its Variation

Exact Duration and Its Variation

A mean sidereal day is about 23 h 56 m 4.1 s in length. However, due to variations in the rotation rate of the Earth, the rate of an ideal sidereal clock deviates from any simple multiple of a civil clock. In practice, the difference is kept track of by the difference UTC–UT1, which is measured by radio telescopes and kept on file and available to the public at the IERS and at the United States Naval Observatory.

Given a tropical year of 365.242190402 days from Simon et al. this gives a sidereal day of 86,400 ×, or 86,164.09053 seconds.

Aoki et al., defined UT1 such that the observed sidereal day at the beginning of 2000 would be ⅟1.002737909350795 times a UT1 day of 86,400 seconds, which gives 86,164.090530833 seconds of UT1. For times within a century of 1984, the ratio only alters in its 11th decimal place. This web-based sidereal time calculator uses a truncated ratio of ⅟1.00273790935.

Because this is the period of rotation in a precessing reference frame, it is not directly related to the mean rotation rate of the Earth in an inertial frame, which is given by ω=2π/T where T is the slightly longer stellar day given by Aoki et al. as 86,164.09890369732 seconds. This can be calculated by noting that ω is the magnitude of the vector sum of the rotations leading to the sidereal day and the precession of that rotation vector. In fact, the period of the Earth's rotation varies on hourly to interannual timescales by around a millisecond, together with a secular increase in length of day of about 2.3 milliseconds per century, mostly from tidal friction slowing the Earth's rotation.