Wednesday, January 27, 2016

Derivation of the Ellipse Fit Formula

  One can derive the linear formula for the angular velocity fit by starting with the first two of Kepler's laws. The constant coefficients can be solved for using linear least squares.

The error that results from fitting the ecliptic longitude with the linear function plus some Fourier terms (δ below) is mainly due to the motion of the Earth about the Earth-Moon barycenter. Using the barycenter as the frame of reference would probably result in less error. The plot covers 3 Julian years and one sees there is a little more than 12 months per year per Julian year.

The coefficients for the λ fit in radians are as follows.

Looking at the error in the computed longitude can see how perturbation theory might have gotten its start.

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