Sidereal time

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Sidereal time is a measure of the position of the Earth in its rotation around its axis, or time measured by the apparent diurnal motion of the vernal equinox, which is very close to, but not identical to, the motion of stars. They differ by the precession of the vernal equinox in right ascension relative to the stars. Earth's sidereal day also differs from its rotation period relative to the background stars by the amount of precession in right ascension during one day (8.4 ms).[1] Its J2000 mean value is 23h56m4.090530833s.[2]

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[edit] Definition

Sidereal time is defined as the hour angle of the vernal equinox. When the meridian of the vernal equinox is directly overhead, local sidereal time is 00:00. Greenwich Sidereal Time is the hour angle of the vernal equinox at the prime meridian at Greenwich, England; local values differ according to longitude. When one moves eastward 15° in longitude, sidereal time is larger by one hour (note that it wraps around at 24 hours). Unlike computing local solar time, differences are counted to the accuracy of measurement, not just in whole hours. Greenwich Sidereal Time and UT1 differ from each other by a constant rate (GST = 1.00273790935 × UT1).[3] Sidereal time is used at astronomical observatories because sidereal time makes it very easy to work out which astronomical objects will be observable at a given time. Objects are located in the night sky using right ascension and declination relative to the celestial equator (analogous to longitude and latitude on Earth), and when sidereal time is equal to an object's right ascension, the object will be at its highest point in the sky, or culmination, at which time it is best placed for observation, as atmospheric extinction is minimised.

[edit] Sidereal time and solar time

Sidereal time vs solar time. Above left: a distant star (the small red circle) and the Sun are at culmination, on the local meridian. Centre: only the distant star is at culmination (a mean sidereal day). Right: few minutes later the Sun is on the local meridian again. A solar day is complete.
Sidereal time vs solar time. Above left: a distant star (the small red circle) and the Sun are at culmination, on the local meridian. Centre: only the distant star is at culmination (a mean sidereal day). Right: few minutes later the Sun is on the local meridian again. A solar day is complete.

Solar time is measured by the apparent diurnal motion of the sun, and local noon in solar time is defined as the moment when the sun is at its highest point in the sky (exactly due south or north depending on the observer's latitude and the season). The average time taken for the sun to return to its highest point is 24 hours.

During the time needed by the Earth to complete a rotation around its axis (a sidereal day), the Earth moves a short distance (around 1°) along its orbit around the sun. Therefore, after a sidereal day, the Earth still needs to rotate a small extra angular distance before the sun reaches its highest point. A solar day is, therefore, around 4 minutes longer than a sidereal day.

The stars, however, are so far away that the Earth's movement along its orbit makes a generally negligible difference to their apparent direction (see, however, parallax), and so they return to their highest point in a sidereal day. A sidereal day is around 4 minutes shorter than a mean solar day.

Another way to see this difference is to notice that, relative to the stars, the Sun appears to move around the Earth once per year. Therefore, there is one less solar day per year than there are sidereal days. This makes a sidereal day approximately 365.24⁄366.24 times the length of the 24-hour solar day, giving approximately 23 hours, 56 minutes, 4.1 seconds (86,164.1 seconds).

[edit] Precession effects

The Earth rotation is not simply a simple rotation around an axis that would always remain parallel to itself. The Earth's rotation axis itself rotates about a second axis, orthogonal to the Earth orbit, taking about 25,800 years to perform a complete rotation. This phenomenon is called the precession of the equinoxes. Because of this precession, the stars appear to move around the Earth in a manner more complicated than a simple constant rotation.

For this reason, to simplify the description of Earth orientation in astronomy and geodesy, it is conventional to describe Earth rotation relative to a frame which is itself precessing slowly. In this reference frame, Earth rotation is close to constant, but the stars appear to rotate slowly with a period of about 25,800 years. It is also in this reference frame that the tropical year, the year related to the Earth's seasons, represents one orbit of the Earth around the sun. The precise definition of a sidereal day is the time taken for one rotation of the Earth in this precessing reference frame.

[edit] 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.[4] this gives a sidereal day of 86,400 × 365.242190402⁄366.242190402, or 86,164.09053 seconds.

According to Aoki et al.,[2] an accurate value for the sidereal day at the beginning of 2000 is 1⁄1.002737909350795 times a mean solar day of 86,400 seconds, which gives 86,164.090530833 seconds. 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 1⁄1.00273790935.

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.[2] 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,[5] together with a secular increase in length of day of about 2.3 milliseconds per century which mostly results from slowing of the Earth's rotation by tidal friction.[6]

[edit] See also

[edit] References

  1. ^ Seidelmann, p. 48.
  2. ^ a b c Aoki, S., B. Guinot, G. H. Kaplan, H. Kinoshita, D. D. McCarthy and P. K. Seidelmann: "The new definition of Universal Time". Astronomy and Astrophysics 105(2), 359-361, 1982.
  3. ^ Seidelmann, pp. 52 and 698.
  4. ^ Simon, J. L., P. Bretagnon, J. Chapront, M. Chapront-Touzé, G Francou and J. Laskar: "Numerical expressions for precession formulas and mean elements for the moon and the planets". Astronomy and Astrophysics 282, 663-683, 1994.
  5. ^ Hide, R., and J. O. Dickey: "Earth's variable rotation". Science 253 (1991) 629-637.
  6. ^ Stephenson, F.R. Historical eclipses and Earth's rotation. Cambridge University Press, 1997, 557pp.
  • P. Kenneth Seidelmann, ed., Explanatory supplement to the Astronomical Almanac, (Mill Valley, Cal.: University Science Books, 1992)

[edit] External links