Astrodynamics/The Kepler Problem

Astrodynamics

The Kepler Problem

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.