The equation is: if it isn't already in orbit, and it doesn't hit the atmosphere, then it flies past.
For an incoming object to start orbiting earth, it has to decelerate through active propulsion or aerobraking or something like that.
For an incoming object to start orbiting earth, it has to decelerate through active propulsion or aerobraking or something like that.
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!!