In an elliptical kepler orbit there is an easy recipe to describe the motion/position of a satellite at time $t$. One just follows the following steps - an important detail for me is that the numerical part has always the same error, the error doesn't increase with time, nor is it based on the timestep. (Which it would if one would solve a differential equation).
Calculate mean motion $n$ (given a semi major axis $a$, and central mass $M$, this is basically the 2 pi divided by the period): $$n = \sqrt{\frac{GM}{a^3}}$$
Now the definition of mean anomaly:
$$M(t) = M_0 + n \cdot t$$
And one can (numerically) solve the following equation, to get the eccentric anomaly $E$:
$$M = E - \varepsilon \sin(E)$$
And then simple geometry allows one to find the true anomaly $\theta$.
Now this is all fine and good working. I'm at a loss as how to to apply this to non-elliptical orbits. The main problems I have are that there's no notion of semi major axis (or when using a negative for hyperbolic orbits- it is infinite for parabolae, which would result in a mean motion of 0.).
Nor is there an meaning for the eccentric anomaly, non elliptical orbits don't have a geometric center. (The true anomaly would still be defined as the focal point).
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