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A comment on NASA predicts asteroid to make one of closest approaches to Earth ever recorded. Parent comment

Re:NASA predicts asteroid to make one of closest approaches ... by graderjs on Thursday January 26, 2023 @09:44PM (#34541837)
Sounds like a nice heuristic but I don’t think that covers every case—and it’s not an equation. Sorry, wrong answer!!
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Re:NASA predicts asteroid to make one of closest approaches ... by lamacase on Friday January 27, 2023 @09:47AM (#34546613)
It does cover every case though. The solution for the orbit of a mass in a 2-body problem is always an ellipse! (Or a parabola/hyperbola for an escape trajectory). You can find the derivation here [1], it's not too complicated.

There is no way for an asteroid and the earth to interact gravitationally to change the asteroid's orbit from what it was coming in. Non-gravitational interactions (like hitting the earth/atmosphere) can do it.

Also, over many interactions and a long time you can have orbital capture in many-body situations, but there is no general equation for this (look up 3 body problem). This is how you get objects accumulating at Lagrange points for example.

TLDR: The equation you're asking for does not exist. Sorry, wrong question!!

https://en.m.wikipedia.org/wiki/Orbit

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