Tuesday, February 16, 2016

Keplerian Elements From Ptolemy's Anomaly for the Sun

  Ptolemy gave a table of the Sun's anomaly that was accurate to about 1 arc minute in his Almagest along with other data from which we can make an estimate of the Sun's "orbit."
We start with a figure of the equinoxes and solstices and the positions of perigee and apogee in the plane of the ecliptic. In Toomer, Ptolemy's Almagest, p. 153, Ptolemy states that the apogee was 24.5° in advance of the summer solstice which would place it at Gem 5.5° in his twelve 30° divisions of the ecliptic measured from the vernal equinox. Similarly the perigee is at Sag 5.5°.

The positions of perigee and apogee were determined by the nonuniform rate of motion of the Sun being slowest at apogee and fastest at perigee. He also gives Hipparchus' values for  the time it took to go from the vernal equinox to the summer solstice (94½ days) and the time to go from the summer solstice to the autumnal equinox (92½ days). He says that his observations gave similar values and from a fit of his anomaly we see that this is so.

The time it takes to traverse a quadrant is greatest for the quadrant containing the apogee and least in the quadrant containing perigee. The value for λ0 was chosen so that the autumnal equinox occurs at 132 Sep 25 2pm. Ptolemy's anomaly only has sin terms in it and is accurate to about 1 arc minute as the residuals for the fit show.

We can use Ptolemy's ecliptic longitude to compute an "orbit" for the Sun in the same way that we did using the longitude obtained with the MICA data. The epoch chosen was the beginning of the year, 132 Jan 1, noon, and we can use the the longitude to estimate the angle and time of perigee, then calculate the true anomaly, ν, at intervals of one day.

Using the linear relation between the cosine of ν and the square root of its rate of change we can estimate σ and τ which in turn allows us to estimate the eccentricity, e, and the other Keplerian elements. The residuals for fit of the formula are a few minutes of arc.

One can find Ptolemy's value for obliquity of the ecliptic on p. 63 of Toomer's book. The summary of the estimated Keplerian elements is as follows.

Most of the changes in orientation of the elliptical orbit are due to the rotation of the equatorial reference frame in the plane of the ecliptic. One would expect some of the difference in the value for the eccentricity e to be due to Ptolemy's longitude and the use of an eccentric circle but some of it may be due to perturbations also. The astronomical unit, AU, used here is based on the elliptical orbit for Ptolemy's data but a change in the mean motion would suggest a change in the mean radius since the areal velocity is fairly constant.

edit (Feb 17): Corrected the date and time for the autumnal equinox but the formula assumed for the longitude above proved to be incorrect. See the next blog for more details.

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