Wednesday, March 16, 2016
A Perihelion Problem
As shown previously, given the lengths of the seasons one can find an approximate value for the eccentricity, e, of the Earth's elliptical orbit.
One can also estimate the time and position of the Earth's perihelion by using Kepler's equation to compute true anomalies of the angular distance of the Earth from perihelion. All we know are differences in positions and times so we start with the time it takes for the Earth to go from the Vernal Equinox to the Summer Solstice, an angle of 90° in 92.753 days. We can assume a value for the time it will take to go from perihelion to the equinox, then use Kepler's equation to compare values obtained for a single step with that it takes for two steps by plotting their difference. The two values should be equal so their difference should be zero.
The data can be put in tabular form and we can use interpolation to when this difference is actually zero.
A solve block confirms the time it takes to go from perihelion to the equinox and we can calculate the corresponding true anomaly. We find it takes 76.146 days to move through an angle of 76.911°. The position of perihelion is then 283.089° for 2015 from which we can determine its date and time since those of the equinox are known.
The computed time of perihelion is off by a day from that given in HM Planetary & Lunar Coordinates 2001-2020 and one has some doubts about the complete accuracy of Kepler's elliptical orbits. Gravitational perturbations by the Sun, Moon and planets might be an explanation. One can see the difficulties that Ptolemy might have had with his determination of the lengths of the seasons if they varied that much. And, the ellipses are probably more accurate than his eccentric circle.
Supplemental (Mar 18): There's a round off error of ±½ minute in the times of the equinoxes and solstices that can affect results somewhat. It's not just perturbations.
Supplemental (Mar 18 23:48): The computed perihelion time is very close to the perihelion time of the Earth-Moon barycenter. The Moon tugs the Earth about a little altering the point in its path that is closest to the Sun.