I'm not a big astronomy whiz or anything, nor am I great at math, but I need help. For some reason or other, I can't seem to figure this out. I have 2 moons for a planet I'm writing about, and I need to accurately figure out their sidereal months. The planet has 722 days a year. Moon #1's synodic period is rounded off to 48 days, and moon #2's synodic period is 80 days. To me, it's a simplistic, and probably lazy way to calculate periods between phases since those are multiples of 8. Never mind the distance from the planet each moon is (I really don't think it matters, but correct me if I'm wrong). I can't figure out their sidereal months for the life of me. Assuming one synodic month begins at "new" phase, I'd like to know the days and hours after one synodic month for each moon, after doing a complete 360° around the planet, to reach "new" again for its sidereal month. If anyone can figure this out for me, that would be great. However I'd like to know how you reached your conclusions so I can figure out future problems like this on my own. P.S. There's a rival planet that crosses my story's main planet ever 412 days (between the main planet and the sun), keeping in mind that, like the moons, both the rival planet and the main planet are revolving around the sun in the same direction. My question is, if the rival planet crosses the main planet every 412 days (that's 412 main planet days), how many days does the rival planet take to revolve around the sun? Thank you.