>How does the asteroid lose enough kinetic energy to be captured by Earth's gravitational field?
The asteroid is currently in *solar* orbit. A near encounter with Earth would slightly deflect the asteroids orbit, by gravitationally tugging it to one side. It would still be in a solar orbit, just a slightly different one. However, the change in orbit cannot be accurately calculated until we know *exactly* how close it will pass by the Earth/Moon system (the moon will *also* tug slightly on the asteroid).
And with timescales of 30+ years, you also really need to factor in Jupiters gravitational effect as well.
Orbital mechanics is based on some pretty simple equations... when you only have two bodies. But in this circumstance, there are at least 5 that need to be accounted for. Extremely complex, requiring lots of computer power and *very* precise observations.
Agreed. A many-body problem has no analytic solution - it requires zillions of calcs to find a best solution.
But by 2029, your wristwatch will be able to do those calcs.
Do you need to include the 11-year solar maximum cycle as well? This would add a certain degree of uncertainty to the calculation, if it is significant.