University:
California State University, Northridge
Course:
ASTR 300 | Introductory Astrophysics
Academic year:

2024
Views:

241

Pages:

2
Author:

Beatrice Ayers

Kepler's Equation Kepler's equation connects the mean and the eccentric anomalies and is given by (23) To derive this equation, recall that the elliptic orbit is given by the formula Differentiating this with respect to time we get: (24) On the other hand we have that and hence Differentiating this term we get (25) Because of the equal area law we have mean angular velocity , where denotes the . Using this we get from the last equation (25) Comparing this with (24), we get (26) However, from the last section we know that and therefore since , this becomes Integration over time yields However, since at , the constant of integration and (27) which is Kepler's equation. Kepler's equation makes it easy to compute the mean anomaly, whenever the eccentric anomaly is given, however, the other way round is difficult. The equation is a transcendental equation and astronomers have devised many numerical and analytic schemes to solve this equation. When the eccentricity is very small, the left hand side of (27) can be reasonably well approximated by itself, and therefore . In the next section we investigate a few methods that will give numerical solutions of Kepler's equation.

Kepler's Equation

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