Position of a satellites satellite depending on orbital periods

Question

I have three celestial bodies, S, E and M. Let's assume, that S is in a fixed position and E rotates around S on a perfectly circular orbit. M rotates around E in the same plane and also on a perfectly circular orbit. (There are also no speed fluctuations involved or anything, just simple circular motions)

Now, E has a orbital period of 1 year, while M has an orbital period of 1/4 years. Now at time zero S, E and M are defined to be at positions (0,0), (10,0) and (11,0) respectively.

After 1/4 years E will of course be positioned at exactly (0,10), however my question is if orbital period is defined in a way, that considers the parents rotation around its parent and so forth, with M positioned at (0,11), or whether it is seen kind of like in a global coordinate system, and its position should be (1,10).

Basically the question is whether the orbit of M rotates depending on Es position in its own orbit. (With respect to the definition of the orbital period)

Thanks in advance for your responses.

Answer 1

There is not a unique definition of "orbital period." One could define a period in which M returns to the same geometrical alignment with E and S. Or one could chose a fixed star far outside the ESM system and return the same geometrical alignment with star/M/E or even star/M/S.

Consider our own Moon and Sun. The orbital period of the Moon to have the same alignment with the stars (the sidereal period) is 27.3 days. The period to have the same geometrical alignment with Earth and Sun (the synodic period, which goes from new moon to new moon, or full to full, etc) is 29.5 days.