Kepler's limit

The three body problem in its restricted form has two degrees of freedom and one constant of motion, the Jacobi integral. It describes the motion of a test particle P with mass under the gravitational influence of two bodies S and J. The assumptions are (i) $ m$ is so small that P has no effect on S and J, (ii) the Kepler motion of S and J is circular, and (iii) P moves in the same plane with S and J. We think of S, the ``Sun'', as being the main body with mass , and J, a fictitious ``Jupiter'', as the second body with mass . Using the constant distance $ R$ between S and J for scaling lengths, $T_{j}/2\pi = 1/\omega_{j}$ as the unit of time (where is the Kepler period of S and J), as the unit of energy, and $m\omega_{j}R^2$ as the unit of angular momentum, the equations of motion contain only one parameter, the relative mass $\mu := m_{j}/(m_{s}+m_{j})$ , with range . For the second main body has no influence on the test body's motion except that we view it from a rotating frame of reference; we call this the Kepler limit . It is to be distinguished from the Hill limit where becomes arbitrarily small and a neighborhood of size around J is considered. S is then effectively displaced to infinity and only the quadrupole component of its potential taken into account. This was in fact Newton 's approximation when he considered the lunar motion. Hill revolutionized perturbation theory when he detected that the moon stays close to a periodic orbit of this dynamical system, and analyzed the variational equations about this (stable) orbit. Poincaré developed his Méthodes Nouvelles from there, but stayed with small . The case of equal main masses was the subject of extensive numerical studies by Strömgren and collaborators (1925) and hence is called the