# What mathematical function would describe the relationship between two moons?

I have a planet with two moons. I want them on opposite sides during the equinoxes and together during the solstices. The seasons are a little lopsided, with spring and autumn having 90 days and the other seasons having 126 days. Thus, there are 108 days between equinox and solstice.

What function would describe this relationship?

• A function explains the relationship between several variables. What are the variables that you want a relationship for? Anyway, either you take some orbital mechanics equations and use them to design some realistic moon data, or you decide your moon data regardless of orbital mechanics and forget about variables, waiving how it happens. Unless you are extremely lucky you cannot have it both ways. – SJuan76 Dec 13 '15 at 3:35
• I read the last word as "morons" at first and was really excited by what answers I'd find here. – BlackVegetable Feb 21 '19 at 0:41

To describe two functions which are equal every 216 days and opposite every 108 days after being equal, you simply need two sinusoidal functions properly scaled and shifted.

Moon 1: $\sin({{1}\over{216}}\pi \times days)$
Moon 2: $\sin({{1}\over{216}}\pi \times days - \pi)$

Overlaid, the functions look like this (shifting sine by $-\pi$ is the same as −sine): Starting at day zero, the moons are at the same point in the function, 108 days later they are the maximum distance apart, after another 108 days (216 total) they are back to being at the same point.

If as SJuan76 points out, you want to have more realistic orbital periods (non-identical, but still line up as requested) you can just use a multiple period, like this:

Moon 1: $\sin({{1}\over{216}}\pi \times days)$
Moon 2: $\sin({{1}\over{72}}\pi \times days)$ • If the moons have the same period they are both at the same distance of the planet (Kepler Laws), which leads to a highly unstable situation. – SJuan76 Dec 14 '15 at 20:49
• @SJuan76 Naturally. But this does answer the question of what functions would describe the requested effect, obviously integer multiples of periodic functions work too, I'll add an example. – Samuel Dec 14 '15 at 21:10