# Equations of motion for Earth and Moon

One of the simplest versions of Newton's equations of motion for the Sun, Earth and Moon can be obtained by making the approximation that the three bodies are perfect spheres. In this approximation, the pair potential between body i and j is $$\frac{-G m_i m_j}{ r_{ij}},$$ where $r_{ij}$ is the distance between the bodies: one can solve the equations numerically and get the orbits.

This simple model does not include multiple, additional factors, such as

• the fact that the Earth is not perfectly spherical (equatorial bulge),
• Earth's tides,
• the influence of other celestial bodies,

and others.

Amongst all these factors, what is the dominant one as for predicting the orbit of the Earth? In other words, what is the leading factor that should be included in the simple model above in order make its predictions for the Earth's orbit closer to the observations?

• You may have to specify "dominant". The non-spherical Earth probably has a big effect on the moon, but not much on the sun. Over the long term, the cumulative effects of other planets may add up whereas the tides average out, even though tidal forces are much stronger. – James K Jul 28 '17 at 15:26
• Without actually doing any research, I'd suspect it's the influence of other celestial bodies, particularly Jupiter. – user21 Jul 28 '17 at 15:28
• James K, you are right, I revised the question. I am interested in the prediction of the Eart's orbit, e.g., the position of its center of mass, rotation speed as a function of time, and direction of axis of rotation. – James Jul 28 '17 at 15:53
• one can solve the equations The general three body problem has no solution of this type. This is a detailed history of solution methods for the Earth-Moon-Sun system (it's 50 pages). – StephenG Jul 28 '17 at 16:31
• Obviously, I was talking about numerical solutions, for which my statement is true. – James Jul 28 '17 at 20:06