How often does the configuration of the Trappist-1 system repeat

Question

My question is simply how long it takes for the 7 known planets in Trappist-1 to be in the same configuration in relation to their host star, and to each-other. Those seven planets are in a 2:3:4:6:9:15:24 resonance, which is just extraordinary. I thought that they would be configured in the same manner roughly every 36 days, but I am not sure I did it right. I basically thought, "if all their orbital ratios are perfect, then I only need to see how long it takes for one of the planets to go through the amount of orbits listed in said ratio, and then I'll get my answer." I know that the answer I am looking for won't be exactly perfect, I am just wanting to know when they will be in practically the same place.

The specific calculation I did was take the orbital period of the sixth planet, say that 1/12th of that orbit was a "day"(instead of using earth days) and then multiplied it by the number of orbits that it has in the ratio (3) and got my answer. (36) I know this doesn't answer it in earth days, but since I am using that "orbit of #6/12 as the daylength in the story I am writing in this world, I am using the measuring stick that matters to my situation. So, am I right? Does it take 36 "days" for Trappist-1 to reconfigure itself, or did I get the math terribly wrong?

Answer 1

If the resonances are 2:3:4:6:9:15:24, we need to find the smallest number that is a multiple of all those numbers, so each planet has made an integer number of revolutions, and is back where it started.

The most obvious way to do this is to work out the prime factors of those numbers. They are:

2: 2
3: 3
4: 2x2
6: 2x3
9: 3x3
15: 3x5
24: 2x2x2x3

For each prime, we take the largest number of copies needed for any number: so three 2s from 24, two 3s from 9 and one 5 from 15 and multiply them together to get the least common multiple, or 360.

So, it would take 360 units of time for the planets to be in the same place relative to the stars, where the orbit of the closest planet is two units of time.

The orbital period of the closest planet is 1.5 days, so our unit is 0.75 earth days, so it takes 0.75 x 360 earth days (about 9 months) for the planets to return to their initial configuration.