Sohrab Shahshahani MSRI
Tidal energy in Newtonian two-body motion

 In this work, based on an essential linear analysis by
Christodoulou, we study the 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
center of mass of the two bodies. When the fluids are replaced by point
masses, according to the classical analysis of Kepler and Newton, the
conic curve describing the trajectories of the bodies is a hyperbola when
the orbital energy is positive and an ellipse when the orbital energy is
negative. If the point masses are initially very far, then the orbital
energy, which is conserved in the case of point masses, is positive
corresponding to hyperbolic motion. However, in the motion of fluid balls
the orbital energy is no longer conserved, as part of the conserved energy
is used in deforming the boundaries of the bodies. This energy is called
the tidal energy. If the tidal energy becomes larger than the total energy
during the evolution,
the orbital energy must change its sign, signaling a qualitative change in
the orbit of the bodies. We will show that under appropriate conditions on
the initial configuration this change of sign occurs. Our analysis relies
on an a-priori estimates which we establish up to the point of closest
approach. This is joint work with Shuang Miao from EPFL.