Solution for timelike orbits and precession
where and e is the eccentricity of the orbit [ if e=0 it is circular ].
We can obtain an approximate solution to the exact orbit equation [ valid for ], if we substitute the Newtonian solution into the term quadratic in U, that is if we solve
where we have put [ for timelike orbits ]. This equation is solved by choosing a particular integral of the form [ Assignment 7 ]:
The most important term is the one that is linear in because it is the only one which in the course of time [ with many revolutions of the planet ] becomes larger and larger. We therefore ignore the other corrections to and obtain
But we have so , and this simplifies our solution to
The orbit of the planet is thus only approximately an ellipse. The solution for U is still a periodic function, but no longer with a period . The point at which the orbit is closest to the sun is reached again only after an additional rotation through the angle