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More of an alternative suggestion than an answer to your parameterized ellipse approach. I've recreationally programmed a few similar many-body simulations myself (e.g., click my profile, then my homepage, then click on that voronoi link -- the points are moving under a mutual force law). Just model the force (gravitational in your case) directly, and the ...

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According to Newton's theory of gravity, the orbit would be an ellipse, which could take the form of a perfect circle. However, Einstein's theory of General Relativity tells us that this elliptical orbit would very gradually decay due to the emission of gravitational waves, and perhaps also precess. The Wikipedia page on the two-body problem in general ...

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The stable orbits around a star are given by the Kepler's laws oft planetary motion. In general these are ellipses with the center star in one of the two foci. Circular orbits are the special case when there is only one focus. For a orbit with a given radius, there is only one speed which allows a circular orbit. If you have a starting condition where the ...

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The local escape velocity is $$v_{esc} = \sqrt{\text{2 G M}/\text{r}}$$ At infinty you observe that velocity slower by a factor of $$1-\text{r}_s/\text{r}$$ so at infinity you observe $$\text{v}_{esc} = \sqrt{\text{2 G M}/\text{r}} \cdot (1-\text{r}_s/\text{r})$$ because of gravitational length contraction radial to the mass and gravitational time ...

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