Is there a formula for calculating synodic/siderial periods?

Question

I'm in a history of science class and our teacher gave us this question as an example of what might be on our upcoming test :
On a planet where the Sidereal period of the Sun is 205 days, the Sidereal period of the Moon is 15 days, and the Synodic period of the Moon is 20 days-using a Solar calendar, how many days and months are in a year?

We weren't given any formulas, can someone tell me how to calculate this?

Answer 1

The sidereal period of an object is the amount of time it takes for the object to move once around the sky and return to the same position with respect to the stars. The synodic month is the time for it to return to the same position with respect to the sun. This diagram will help clarify things: http://www.sumanasinc.com/webcontent/anisamples/astronomy/sidereal.html .

You can find the length of the year from the sun's sidereal period. If the sidereal period is 205 days, that's the length of a year. A "month" means a synodic month, so the length of the month in your example is 20 days. You can compute the number of months in a year from:
synodic period of moon / sidereal period of sun
which in your example will be 10 1/4 months.

Good luck on your exam. Hope that helps.

Answer 2

Um, this must be the history part, because it's not scientific! The difference between the sidereal and synodic month is the extra bit of time the moon has to revolve to catch up to the planet as it moves around the sun. Therefore, there must be exactly one more sidereal month than synodic month in a year. In your example, someone has arbitrarily chosen the length of the synodic month, which is nonsense. You might (gently) point this out to your teacher.

For calendrical purposes, cultures that have lunar calendars use the synodic month (period between identical phases), so maybe you're just meant to ignore the sidereal month. The sidereal period of the sun is approximately one year, the difference being caused by precession of the equinoxes. This question apparently having been concocted by a scientific illiterate, I suppose you're meant to assume no precession.