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:

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

Where

${\displaystyle F(E)=E-e\sin(E)-M(t)}$

The iteration takes the form

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

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:

${\displaystyle |E_{k+1}-E_{k}|<\delta }$

Where δ is the desired accuracy.

Numerical Solution