Cambridge Journal of Mathematics

Volume 7 (2019)

Number 4

On tidal energy in Newtonian two-body motion

Pages: 469 – 585



Shuang Miao (School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei, China; and Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, Hubei, China)

Sohrab Shahshahani (Department of Mathematics and Statistics, University of Massachusetts, Amherst, Mass., U.S.A.)


According to the classical analysis of Newton the trajectory of two gravitating point masses is described by a conic curve. This conic curve is a hyperbola if the mechanical energy of the system is positive, and an ellipse if the mechanical energy is negative. The mechanical energy is a conserved quantity in an isolated two-body point mass system. If the point masses are initially very far, then the orbital energy is positive, corresponding to hyperbolic motion. In this work we consider the situation when the point masses are replaced by gravitating incompressible fluid balls with free boundaries obeying the Euler–Poisson equations. In this case the conserved total energy $\tilde{\mathscr{E}}$ can be decomposed as $\tilde{\mathscr{E}} =: \widetilde{\mathscr{E}_{\operatorname{orbital}}} + \widetilde{\mathscr{E}_{\operatorname{tidal}}}$, where $\widetilde{\mathscr{E}_{\operatorname{tidal}}}$ is the energy used in deforming the boundaries, and $\widetilde{\mathscr{E}_{\operatorname{orbital}}}$ takes the role of the mechanical energy in the point mass system and is such that if $\widetilde{\mathscr{E}_{\operatorname{orbital}}} \lt -c \lt 0$ for some absolute constant $c \gt 0$, then the orbit of the bodies must be bounded. In analogy with the point mass picture we consider the scenario where initially the two fluid balls are very far and their boundaries are unperturbed. In this case the initial orbital energy is equal to the total energy and is positive. 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 [4], based on a linear calculation Christodoulou conjectured that under appropriate conditions on the initial configuration of the system, the tidal energy $\widetilde{\mathscr{E}_{\operatorname{tidal}}}$ can become larger than the total energy $\tilde{\mathscr{E}}$ during the evolution. In particular under these conditions $\widetilde{\mathscr{E}_{\operatorname{orbital}}}$, which is initially positive, becomes negative before the point of the first closest approach. In this work we prove Christodoulou’s conjecture in [4] for the full nonlinear system. That is, we prove that for a family of initial configurations the fluid boundaries and velocity remain regular up to the point of the first closest approach in the orbit, and that the tidal energy $\widetilde{\mathscr{E}_{\operatorname{tidal}}}$ can be made arbitrarily large relative to the total energy $\tilde{\mathscr{E}}$. This reveals the possibility that the center of mass orbit, which is unbounded initially, may become bounded during the evolution. Since the initial distance of the bodies can be arbitrarily large relative to their distance at closest approach, the a priori estimates, which as in the framework of [16] are carried out in the language of Clifford analysis, are independent of the initial time and separation of the bodies.

This work is based on a linear analysis carried out by Demetrios Christodoulou [4]. We thank him for pointing us to this interesting direction and for many insightful discussions. We thank Sijue Wu for many illuminating discussions about free boundary problems for incompressible fluids. We also thank Lydia Bieri for many stimulating conversations. Part of this work was done while S. Miao was at École Polytechnique Fédérale de Lausanne. The support of the Swiss National Science Fund is gratefully acknowledged. Finally we thank the anonymous referee for carefully reading the manuscript and for many valuable suggestions.

Received 12 December 2018

Published 1 November 2019