overlap between orbitting planets

Question

If two objects are orbiting around a common center point with different velocities and at different radius. The point in time in which they are overlapping can be found out using their angular velocities and calculating for a time at which the faster object is exactly 2pi ahead of the slower one. Is there any way possible to calculate this time of overlap for three, four, five, n different objects orbitting at different velocities and different radius? think of it like the planets in the solar system in a 2D plane

Answer 1

Let's say we have three objects in orbit around a common center, at different angular velocities. The first two will line up with the center at some time; without loss of generality, let's call that time zero. They will line up again at times $p$, $2p$, $3p$, and so on, for some nonzero "period" $p$. The first and third objects will line up at some time $t$, and then again at $t+q$, $t+2q$, $t+3q$, and so on, for some period $q$. For all three to line up, we need integers $m$ and $n$ such that $$mp=t+nq$$ or to put it another way, $mp-nq=t$. So a necessary (but not sufficient) condition for the three objects to line up is that $t$ has to be in the field generated (over the rationals) by $p$ and $q$. But this does not happen, for example, if $p=\sqrt2$, $q=\sqrt3$, $t=\pi$. Indeed, there's a well-defined sense in which the probability of $t$ being in that field is zero. So, unless the angular velocities and starting times are exceedingly special, the three objects will never line up.

And what goes for three objects goes a fortiori for four or more objects.