Kepler's equation

Remarks

For each $M \in \R$ and $e \in [0, 1)$ the equation has exactly one solution $E \in \R$, because $E \mapsto E - e \sin E$ is continuous, has derivative $1 - e \cos E \ge 1 - e > 0$, and tends to $\pm \infty$ as $E \to \pm \infty$. Example: at the apocentre $E = \pi$, and the equation gives $M = \pi$, so the body reaches the apocentre half a period after the pericentre, as symmetry requires. Example with numbers: for $e = 0.5$ and $M = \pi/2$, Newton's method started at $E_0 = M$ converges to $E \approx 2.0210$.

The definition fails for $e \ge 1$, where the orbit is not an ellipse. For a hyperbolic orbit, with $e > 1$, the analogous relation is $M = e \sinh H - H$ with a hyperbolic anomaly $H$.

The distance to the focus is $\norm{r} = a (1 - e \cos E)$, and the true anomaly $\theta$ satisfies $\tan \frac{\theta}{2} = \sqrt{\frac{1 + e}{1 - e}} \tan \frac{E}{2}$.