On tidal energy in Newtonian two-body motion

Abstract

In this work, which is based on an essential linear analysis carried out by Christodoulou, we study the evolution of tidal energy for the motion of two gravitating incompressible fluid balls with free boundaries obeying the Euler-Poisson equations. The orbital energy is defined as the mechanical energy of the two bodies' center of mass. According to the classical analysis of Kepler and Newton, when the fluids are replaced by point masses, the conic curve describing the trajectories of the masses is a hyperbola when the orbital energy is positive and an ellipse when the orbital energy is negative. The orbital energy is conserved in the case of point masses. If the point masses are initially very far, then the orbital energy is positive, corresponding to hyperbolic motion. However, in the motion of fluid bodies the orbital energy is no longer conserved because part of the conserved energy is used in deforming the boundaries of the bodies. In this case the total energy $\tilde{\mathcal{E}}$ can be decomposed into a sum $\tilde{\mathcal{E}}:=\widetilde{\mathcal{E}_{{\mathrm{orbital}}}}+\widetilde{\mathcal{E}_{{\mathrm{tidal}}}}$, with $\widetilde{\mathcal{E}_{{\mathrm{tidal}}}}$ measuring the energy used in deforming the boundaries, such that if $\widetilde{\mathcal{E}_{{\mathrm{orbital}}}} 0$, then the orbit of the bodies must be bounded. In this work we prove that under appropriate conditions on the initial configuration of the system, the fluid boundaries and velocity remain regular up to the point of the first closest approach in the orbit, and that the tidal energy $\widetilde{\mathcal{E}_{{\mathrm{tidal}}}}$ can be made arbitrarily large relative to the total energy $\tilde{\mathcal{E}}$. In particular under these conditions $\widetilde{\mathcal{E}_{{\mathrm{orbital}}}}$, which is initially positive, becomes negative before the point of the first closest approach.