Astrodynamics/The Kepler Problem

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The Kepler Problem[edit]

While Kepler's equation is easy to solve for time, there is no general solution for the reverse problem. To determine eccentric anomaly (and thus spacecraft position) at a given time, generally an iterative numerical method is used, such as Newton's method:


The iteration takes the form

For most elliptical orbits an initial guess of E0 = M is sufficient; for orbits with eccentricities greater than 0.8, E0 = π may be used. Better initial guesses are possible, but are not generally required. The iterative process is repeated until required accuracy conditions are achieved, for example:

Where δ is the desired accuracy.

Numerical Solution[edit]