Using the Times for Quadrant Passage to Fit an Elliptical Orbit

  The times of occurrence of the Equinoxes and Solstices provide enough information to determine some of the Keplerian orbital elements. The Equinoxes and Solstices, which are the Cardinal Points of the Celestial Sphere, are separated by 90° and we can use Ptolemy's times for quadrant passage to illustrate the process. The eccentric circle (still used today) and the elliptical orbit are shown in the figure below as well as the arcs W, X, Y and Z for the quadrants and we need to find the unknown times it takes for the true anomaly, θ, to travel through arcs Y and Z. It is fairly easy to compute the eccentric anomaly, E, and the true anomaly for some arbitrary time if the eccentricity, e, is known so we can start by assuming e = 0.02 and see what happens.

We can use a trick found in Ptolemy to find the time, t_P, it takes for the Sun to travel from the Autumnal Equinox to perigee by searching for the value of t_P which makes the difference in the true anomaly between the Summer Solstice and the Vernal Equinox 90°. The time, t'_P, it takes to go from perigee to the Winter Solstice can be found by also setting the sum of the two perigee angles equal to 90°. The difference in angles can again be used to find the time, t_Z, it takes to travel through arc Z on the ellipse.

Ptolemy gives the time it takes to go from the Autumnal Equinox to the Vernal Equinox as 178¼ days. The assumed value of e does not necessarily make the sum of times  t_Y and t_Z equal to this but we can search for a value of e that does and e = 0.0206 works quite well. With the times known we can compute their correspond true anomalies and we get a value about ½° off from Ptolemy's result of 65½° for the angle between the Autumnal Equinox and perigee. We can use t_P to compute the date of perigee and that of apogee a half year later. Here is a compilation of the results and some checks.

The value of the eccentricity for this orbit is very close to that found using just the fitted formula for Ptolemy's anomaly table. So we have shown that a knowledge of the times of the Equinoxes and Solstices can be used to compute an orbit for the Sun and indirectly for the Earth. More generally the times and angular positions of any three points can be used to compute an orbit so the circumstances of solar and lunar eclipses can also be used.