effect of tidal dissipation on the motion of celestial bodies

Open Access
Author:
Ai , Chong
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
August 20, 2012
Committee Members:
  • Mark Levi, Dissertation Advisor
  • Mark Levi, Committee Chair
  • Sergei Tabachnikov, Committee Member
  • Diane Marie Henderson, Committee Member
  • Milton Walter Cole, Committee Member
Keywords:
  • tidal effect
  • tidal dissipation
  • 3-body problem
  • dynamical systems
  • celestial bodies
  • stability
Abstract:
Tidal effects in celestial bodies manifest themselves in many ways. Tides cause periodic changes in sea and ground levels, they affect the length of day, and even volcanic activity. Tides cause effects on the scale larger than that of an individual body, affecting entire orbits of planets and moons. In this thesis we focus on the effect of tides on the dynamics of orbits, leaving aside internal effects of tides on planets. This thesis addresses a gap in the literature. On the one hand, the mathematical theory of celestial mechanics is a classical subject going back to Newton, and it reached a high level of development by people like Legendre, Lagrange, Laplace, Jacobi, Poincare, Moser, Arnold and others. Without exception (to our knowledge) this theory treats planets as point masses subject to Newtonian gravitational attraction, and without account for tidal effects. On the other hand, astronomers take more realistic models of the planets, but get few if any rigorous results. In this thesis we study problems which fall in the gap between these two approaches: they do include tidal dissipation on the one hand, making them more realistic than the classical system which completely ignores them, but we make this dissipation simple enough to be tractable mathematically. To build dissipation into the equations of motion, we use the Routh method of introducing dissipation into Lagrangian equations of motion. According to this method, to write the equations of motion one only needs, in addition to the Lagrangian of the system, also the so--called Routh dissipation function: the power dissipated as a function of generalized coordinates and generalized velocities of the system. We choose a simple class of dissipation functions, leaving more general questions for future work. In this thesis we study tidal dissipation in two problems: the Kepler problem, and the restricted three--body problem, and ask the question of the long--term behavior of these problems with dissipation. There are two main results. First, we show that all the negative energy solutions of Kepler's problem approach circular motion, and do so with an additional interesting feature. The second main result of this thesis deals with the restricted three body problem with dissipation. We show that the Lagrangian equilateral configurations become unstable due to tidal dissipation. This is a rather surprising result of dissipation causing instability. In addition, we show that almost all (in the Lebesgue sense) motions end either in a collision or an escape to infinity. The restricted three body problem, which is infinitely delicate in the classical conservative case, thus admits an essentially complete analysis if one introduces an arbitrarily small dissipation.