Why the average length of the tropical year is different for different starting points in the tropical year.

Brief answer:

A tropical year is the time it takes the sun to appear to travel around the sky from a given point of the tropical zodiac back to that same point in the tropical zodiac. This is, in the mean, a little less time than the period it takes the Earth to complete one dynamical circuit of its elliptical* orbit around the sun.

Expansion of the brief answer:

The average time it takes to complete one ellipse is called the anomalistic year (currently about 365.26 days or 365 days, 6 hours and some 14 minutes). The elliptical shape of the earth's orbit characterizes the varying speed with which the Earth travels. It travels faster while it is closer to the sun (near the perihelion of of its ellipse) and slower while it is farther from the sun (near the aphelion of its ellipse).

Because the ellipse or anomaly governs the dynamics of the Earth's motion around the sun, the mean anomalistic year will necessarily be the same wherever in the ellipse or anomalistic period we begin timing this kind of year. Wherever we begin an anomalistic year the Earth will go through the complete cycle of its varying speeds while completing an ellipse back to the starting position. It will cover the same angles (from the sun) in the same intervals of time as it would if the measurement were begun from some other point in the orbit; only the order of the varying angles in the varying intervals will be different.

A tropical year, however, is completed before the Earth recovers its same position and velocity in the ellipse. This is because the Earth's axis is not perfectly upright or perpendicular with respect to its path around the sun. The Earth thus wobbles** like a top, its axis of rotation slowly and steadily changing at a rate that (if held at its current value) would take some 21,000 years to complete a circuit (around a perpendicular through its orbital plane) with respect to the perihelion-to-aphelion axis.

A tropical year is fulfilled when the Earth's axis completes a full cycle of angles with respect to the line joining the center of the Earth to the sun. Because of the wobble*** this cycle is completed before an ellipse is completed.

The small angle that the Earth's axis has wobbled in one tropical year (about 0.0003 radians), and which the Earth must travel through (with respect to the sun and the perihelion-aphelion axis) in order to complete an ellipse, will require a travel-time which is dependent upon the Earth's position in the ellipse. This must be so because the Earth's position in the ellipse determines its angular speed. A mean tropical year can thus be thought of as a constant mean anomalistic year, less a small variable period of time (between 24 and 26 minutes of an hour), which depends for its exact duration upon the position in the ellipse from which we choose to begin our mean tropical year.

Summary of expanded answer:

Since, as we have said, the mean anomalistic year is 365 days 6 hours and 14 minutes, and the time taken to travel the final 0.0003 radians and complete the anomalistic year (once a tropical year has elapsed) is between 26 and 24 minutes, we can see by simple subtraction that a mean tropical year must have a duration between 365 days, 5 hours, 48 minutes and 365 days, 5 hours, 50 minutes. Furthermore the exact value between these two limits must depend on how long it takes to travel that last 0.0003 radians of angular distance and thus depend upon where in the ellipse we choose to begin measuring from. Q.E.D.

Notes:

* This argument is not logically dependent upon Kepler's demonstration of a sun-centered dynamics nor upon his discovery that planetary orbits are elliptical in shape. Astronomers were cognisant since Babylonian times of the varying speed of the sun through its annual zodiacal course. By Ptolemy's time it was known that the points in the zodiac where the sun moved fastest (our perihelion) and slowest (our aphelion) were themselves progressing through the tropical zodiac. All cosmologies, from Ptolemy on, implied a dynamics of the apparent solar movement dependent on an anomalistic reference frame. Kepler finally correctly characterised this frame as elliptical motion around the sun. A pre-Keplerian astronomer like John Dee, for instance, could have advanced the above argument by using such phrases as "cycle of anomaly" or "anomalistic orbit" wherever I have used "ellipse" or "elliptical orbit" etc. Similarily, earlier astronomers unfamiliar with a sun-centered framework, would have had no logical problem with the argument if it was rephrased with the solar and terrestrial positions swapped and with the references to wobbling tops omitted (the wobbling top imagery is included here only because it is commonly used in modern explanations of "precession").

** This wobbling phenomenon which underlies these complexities is usually alluded to in explaining "the precession of the equinoxes"*** and is given as a cycle with respect to the fixed stars (a retrogression around the constellations of the stellar zodiac) of about 26,000 years (at the current precession rate). Here however, we focus on "equinoctial precession with respect to the ellipse". These two precessions have slightly different rates because the ellipse is not quite fixed with respect to the distant stars. The ellipse too is rotating with respect to the fixed stars (around the stellar zodiac), but much more slowly than, and in the opposite direction to, the equinoxes.

*** As the Earth's North-South axis wobbles (around a perpendicular to the orbital plane through the center of the Earth) at a rate of 0.0003 radians per year with respect to the perihelion-aphelion axis of its elliptical orbit (or 0.00024 radians with respect to the fixed stars), it takes with it, the equinoxes and solstices of the tropical zodiac. This is because the equinoxes and solstices are determined by the intersection of the plane of the Earth's elliptical orbit and the plane of the Earth's equator. The plane of the equator is by definition exactly at right angles to the North-South axis and thus a wobble in the axis means that there is a movement of this equatorial plane and thus a rotation of the intersection of this plane with the zodiacal orbital plane.

Copyright © 1996 by Simon Cassidy, Emeryville Ca. U.S.A. All rights reserved.

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