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Solar time is a calculation of the passage of time based on the Sun's position in the sky. The fundamental unit of solar time is the day. Two types of solar time are apparent solar time (sundial time) and mean solar time (clock time).
Fix a tall pole vertically in the ground; at some instant on any sunny day the shadow will point exactly north or south (or disappear, if the Sun is directly overhead). That instant is local apparent noon: 12:00 local apparent time. About 24 hours later the shadow will again point north/south, the Sun seeming to have covered a 360-degree arc around the Earth's axis. When the Sun has covered exactly 15 degrees (1/24 of a circle, both angles being measured in a plane perpendicular to the Earth's axis), local apparent time is 13:00 exactly; after 15 more degrees it will be 14:00 exactly.
The problem is that in September the Sun takes less time (as measured by an accurate clock) to make an apparent revolution than it does in December; 24 "hours" of solar time can be 21 seconds less or 29 seconds more than 24 hours of clock time. As explained in the equation of time article, this is due to the eccentricity of the Earth's orbit (i.e. the Earth's orbit is not perfectly circular, meaning that the Earth-Sun distance varies throughout the year), and the fact that the Earth's axis is not perpendicular to the plane of its orbit (the so-called obliquity of the ecliptic).
The effect of this is that a clock running at a constant rate – e.g. completing the same number of pendulum swings in each hour – cannot follow the actual Sun; instead it follows an imaginary "mean Sun" that moves along the celestial equator at a constant rate that matches the real Sun's average rate over the year. This is "mean solar time", which is still not perfectly constant from one century to the next but is close enough for most purposes. Currently a mean solar day is about 86,400.002 SI seconds.
The two kinds of solar time (apparent solar time and mean solar time) are among the three kinds of time reckoning that were employed by astronomers until the 1950s. (The third kind of traditional time reckoning is sidereal time, which is based on the apparent motions of stars other than the Sun.) By the 1950s it had become clear that the Earth's rotation rate was not constant, so astronomers developed ephemeris time, a time scale based on the positions of solar system bodies in their orbits.
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