Length of Tropical Year
An oversimplified definition of the tropical year would be the time required for the Sun, beginning at a chosen ecliptic longitude, to make one complete cycle of the seasons and return to the same ecliptic longitude. Before considering an example, the equinox must be examined. There are two important planes in solar system calculations, the plane of the ecliptic (the Earth's orbit around the Sun), and the plane of the celestial equator (the Earth's equator projected into space). These two planes intersect in a line. The direction along the line from the Earth in the general direction of the zodiac sign Aries (Ram) is the Northward equinox, and is given the symbol ♈ (the symbol looks like the horns of a ram).
The opposite direction, along the line in the general direction of the sign Libra, is the Southward equinox and is given the symbol ♎. Because of precession and nutation these directions change, compared to the direction of distant stars and galaxies, whose directions have no measurable motion due to their great distance (see International Celestial Reference Frame).
The ecliptic longitude of the Sun is the angle between ♈ and the Sun, measured eastward along the ecliptic. This creates a complicated measurement, because as the Sun is moving, the direction the angle is measured from is also moving. It is convenient to have a fixed (with respect to distant stars) direction to measure from; the direction of ♈ at noon January 1, 2000 fills this role and is given the symbol ♈0.
Using the oversimplified definition, there was an equinox on March 20, 2009, 11:44:43.6 TT. The 2010 March equinox was March 20, 17:33:18.1 TT, which gives a duration of 365 d 5 h 49 m 30s. (Astronomical Applications Dept., 2009) While the Sun moves, ♈ moves in the opposite direction . When the Sun and ♈ met at the 2010 March equinox, the Sun had moved east 359°59'09" while ♈ had moved west 51" for a total of 360° (all with respect to ♈0). (Seidelmann, 1992, p. 104, expression for pA)
If a different starting longitude for the Sun is chosen, the duration for the Sun to return to the same longitude will be different. This is because although ♈ changes at a nearly steady rate there is considerable variation in the angular speed of the Sun. Thus, the 50 or so arcseconds that the Sun does not have to move to complete the tropical year "saves" varying amounts of time depending on the position in the orbit.
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