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:

 x_{k+1}=x_{k}-\frac{F(x_k)}{F'(x_k)}

Where

F(E) = E - e \sin(E) - M(t)

The iteration takes the form

E_{k+1}=E_{k} - \frac{ E_{k} - e \sin(E_{k}) - M(t) }{ 1 - e \cos(E_{k})}

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:

 |E_{k+1}-E_k|<\delta

Where δ is the desired accuracy.

Numerical Solution[edit]