Plebeian said:
Can't you just treat it as a one planet system with a sun whose mass is equal to the mass of the Earth and the mass of the planet is equal to the mass of the moon?
No, you can't. Gravity doesn't work that way in general. You can treat objects with a spherical mass distribution (objects whose density varies only with radial distance from the center of mass of the object) as point masses, but in general you cannot treat an arbitrary collection as a point mass.
Besides, that is not the question raised in the original post. That asked what the period of the Moon's orbit about the Earth would be in the absence of the Sun. That is an easy calculation to make. Since the OP won't do it, here it is:
P = 2\pi\,\sqrt{\frac {a^3}{G(M_e+M_m)}} = 2\pi\,\sqrt{\frac {a^3}{GM_e(1+M_m/M_e)}}
Plugging in the numbers,
a=384,399 km, M
m/M
e=0.01230004, GM
e=398,600.442 km
3/s
2, yields a period of 27.284500 days. The Moon's synodic period is 27.321582 days, about 53.40 minutes longer than the value calculated assuming an isolated Earth-Moon system.
That 53.40 minute discrepancy is about 0.136% of the true synodic period. Rather small, but very much observable. So what explains this discrepancy? Simple: This calculation ignores the effects of the Sun, which are obviously of a perturbative nature given that the simplistic calculation comes pretty close to the mark.
A frame with its origin at the Earth-Moon barycenter is not an inertial frame. The Earth and Moon are orbiting the Sun. We can pretend it is an inertial frame by accounting for the acceleration of the frame toward the Sun as a fictitious force. With this, the gravitational force exerted by the Sun becomes a tidal gravitational force (physics term) or a third body effect (solar system dynamics term).
The resultant forces will be quite similar to those depicted in this diagram from the wikipedia article on tides. Interpret the arrow as pointing to the Sun and interpret the circle as representing the Moon's orbit about the Earth:
Note that the tidal forces are directed outward when the Moon is new and when it is full. The forces are directed inward when the Moon is at first and last quarters, but the magnitude of this inward force is about half that of the outward force when the Moon is new or full. The net effect is outward, equivalent to a small reduction in the combined mass of the Earth and the Moon. This makes the Earth-Moon period a bit longer than that calculated assuming a two body problem value.
