Sunday, May 17, 2009

Equation of Time

The equation of time is the difference at any moment deduced from the current position of the Sun and time as read from a regulated clock set to the local mean time. The equation of time varies over the course of a year, in way that is almost exactly reproduced from one year to the next. It can be ahead (fast) by as much as 16 minutes 33 seconds (around November 3) or fall behind by as much as 14 minutes 6 seconds (around February 12). It is caused by irregularity in the path of the Sun across the sky, due to a combination of the obliquity of the Earth's rotation axis and the eccentricity of its orbit. The equation of time is the east or west component of the analemma, a curve representing the angular offset of the Sun from its mean position on the celestial sphere as viewed from Earth.

The equation of time was used historically to set clocks. One of two common land based ways to set clocks was by observing the passage of the sun across the local meridian at noon. The moment the sun passed overhead, the clock was set to noon, offset by the number of minutes given by the equation of time for that date. The second method did not use the equation of time, it used stellar observations to give sidereal time, in combination with the relation between sidereal time and solar time.The equation of time values for each day of the year, compiled by astronomical observatories, were listed in almanacs and ephemerides.

Other planets will have an equation of time too. On Mars the difference between sundial time and clock time can be as much as 50 minutes, due to the considerably greater eccentricity of its orbit.

The Earth revolves around the Sun. As such it appears that the Sun makes one rotation around the Earth in one year. If the Earth orbited the Sun with a constant speed, in a circular orbit in a plane perpendicular to the Earth's axis, then the Sun would culminate every day at exactly the same time, and be a perfect time keeper, except for the very small effect of its slowing rotation. But the orbit of the Earth is an ellipse, and its speed varies between 30.287 and 29.291 km/s, according to Kepler's laws of planetary motion, and its angular speed also varies, and the Sun appears to move faster at perihelion (currently around 3 January) and slower at aphelion a half year later. At these extreme points, this effect increases (respectively, decreases) the real solar day by 7.9 seconds from its mean. This daily difference accumulates over a period. As a result, the eccentricity of the Earth's orbit contributes a sine wave variation with an amplitude of 7.66 minutes and a period of one year to the equation of time. The zero points are reached at perihelion (at the beginning of January) and aphelion (beginning of July) while the maximum values are in early April (negative) and early October (positive).

Even if the Earth's orbit were circular, the motion of the Sun along the celestial equator would still not be uniform. This is a because of the tilt of the Earth's rotation with respect to its orbit, or equivalently, the tilt of the ecliptic (the path of the sun against the celestial sphere) with respect to the celestial equator. The projection of this motion onto the celestial equator, along which "clock time" is measured, is a maximum at the solstices, when the yearly movement of the Sun is parallel to the equator and appears as a change in right ascension, and is a minimum at the equinoxes, when the Sun moves in a sloping direction and appears mainly as a change in declination, leaving less for the component in right ascension, which is the only component that affects the duration of the solar day. As a consequence of that, the daily shift of the shadow cast by the Sun in a sundial, due to obliquity, is smaller close to the equinoxes and greater close to the solstices. At the equinoxes, the Sun is seen slowing down by up to 20.3 seconds every day and at the solstices speeding up by the same amount.

The equation of time was mean minus apparent solar time in the British Nautical Almanac and Astronomical Ephemeris. Earlier, all times in the almanac were in apparent solar time because time aboard ship was determined by observing the Sun. In the unusual case that the mean solar time of an observation was needed, the extra step of adding the equation of time to apparent solar time was needed. Since 1834, all times have been in mean solar time because by then the time aboard most ships was determined by marine chronometers. In the unusual case that the apparent solar time of an observation was needed, the extra step of adding the equation of time to mean solar time was needed, requiring all differences in the equation of time to have the opposite sign.

As the daily movement of the Sun is one revolution per day, that is 360° every 24 hours, and the Sun itself appears as a disc of about 0.5° in the sky, simple sundials can be read to a maximum accuracy of about one minute. Since the equation of time has a range of about 30 minutes, the difference between sundial time and clock time cannot be ignored. In addition to the equation of time, one also has to apply corrections due to one's distance from the local time zone meridian and summer time, if any.