Tuesday, February 2, 2016
Finding the Mean Motion by Averaging
One does not have to use least squares to find the mean motion of the Sun along the ecliptic. The 3 years of data for the ecliptic longitude obtained from MICA give a fairly good estimate of the mean motion, n. One can interpolate to find the time it takes to move an integral multiple of 360° then divide to estimate the mean motion.
Using this estimate of the mean motion it's a simple task to find the nonuniform motion.
The longer the interval of time the better the estimate of the mean motion as can be seen in the following plot. Notice that the difference between the successive averages and the original estimate returns to zero at the end of the years.
Judging by Ptolemy's value of the mean motion found in Toomer, Ptolemy's Almagest, p. 140, it hasn't changed much in 2000 years.
Ptolemy's ecliptic longitude was measured from the Vernal Equinox. To find the mean motion in a fixed reference frame like J2000.0 one has to subtract the precession rate of the Equatorial reference frame. A similar numerical value is used to define the astronomical unit (AU) which is the radius of a circular orbit that has this value for its mean motion. The AU is a trifle smaller than the Earth mean distance from the Sun.