Equation of The Center - Analytical Expansions

Analytical Expansions

For small values of orbital eccentricity, the true anomaly, may be expressed as a sine series of the mean anomaly, . The following shows the series expanded to terms of the order of :

Related expansions may be used to express the true distance of the orbiting body from the central body as a fraction of the semi-major axis of the ellipse,

;

or the inverse of this distance has sometimes been used (e.g. it is proportional to the horizontal parallax of the orbiting body as seen from the central body):

.

Series such as these can be used as part of the preparation of approximate tables of motion of astronomical objects, such as that of the moon around the earth, or the earth or other planets around the sun, when perturbations of the motion are included as well.