# How to determine how often a planet's center will be collinear with its moon's?

Expanding on the question: Let's say I have a planet orbiting some star (earth-like, sun-like, for the sake of example). If this planet has two moons, M1 with orbital period of 30 earth-days and M2 with orbital period of 50 earth-days, how often will the center of these moons be collinear with the planet's center? What about if there were a third moon, with 70 earth-days orbit?

I tried solving for the intersection between the line formed by the two moons and the planet, but failed in simplifying the resulting equation.

• Can we assume that the moon orbits are perfectly circular and coplanar? – PM 2Ring Jul 22 at 3:40
• The key term for this kind of problem is "synodic period". See, eg astronomy.stackexchange.com/q/18380/16685 For your problem, you need half the synodic period, because the moons are collinear with the planet when their angular separation is 0° or 180°. – PM 2Ring Jul 22 at 3:52