Title: 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
$\widetilde{\mathcal{E}_{\mathrm{orbital}}}<-c<0$ for some absolute constant
$c>0$, 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.