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The Pennsylvania State University The Graduate School EFFECT OF TIDAL DISSIPATION ON THE MOTION OF CELESTIAL BODIES A Dissertation in Mathematics by Chong Ai c 2012 Chong Ai Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2012 The dissertation of Chong Ai was reviewed and approved∗ by the following: Mark Levi Professor of Mathematics Department Dissertation Advisor, Chair of Committee Sergei Tabachnikov Professor of Mathematics Department Diane Henderson Professor of Mathematics Department Milton Cole Distinguished Professor of Physics Department Yuxi Zheng The Head of Mathematics Department ∗Signatures are on file in the Graduate School. 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 develop- ment by people like Legendre, Lagrange, Laplace, Jacobi, Poincaré,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 com- pletely 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 La- grangian 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. iii 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. iv Table of Contents List of Figures vii List of Tables viii Chapter 1 Tidal Effect 1 1.1 Introduction . 1 1.2 The properties of the tidal effect . 6 Chapter 2 Tidal Dissipation in General 11 2.1 General dissipation formulation . 11 2.2 Hamiltonian equations with dissipation under canonical transfor- mations . 13 2.3 LaSalle's invariance principle . 14 Chapter 3 Tidal Dissipation for Kepler Problem 16 3.1 Assumptions on the tidal dissipation . 16 3.2 Effect of dissipation on the evolution of the dynamics of the 2-body problem . 17 3.3 One choice of dissipation function . 20 Chapter 4 Tidal dissipation in the restricted planar 3-body problem 26 4.1 Introduction . 26 4.2 A model in rotating frame . 26 4.3 Assumptions about the tidal effects . 29 v 4.4 The asymptotic behavior of the dissipative system . 30 4.5 Effect of dissipation on stability . 33 4.5.1 Stability of the dissipative problem at relative equilibria . 33 4.5.2 A particular dissipation function . 36 4.5.3 Motion of the spectrum . 37 Bibliography 40 vi List of Figures 1.1 the bulge of Earth . 7 1.2 the dumbbell model . 8 1.3 the spin lag . 9 3.1 the graph of h(r) ............................ 19 3.2 Pictures of Kepler Problem with dissipation " = 10−5 . 21 3.3 Symmetry of the system . 24 4.1 the neighborhood, U .......................... 32 4.2 Hill's regions and a level set of E ................... 32 4.3 How dissipations causes instability µ = 0:01 and " = 0:01. Both figures are in configuration space. 37 vii List of Tables 1.1 Data from wikipedia about Jupiter's moons . 2 1.2 Data from wikipedia about Roche Limits . 4 1.3 Estimation of tidal forces . 8 viii Chapter 1 Tidal Effect 1.1 Introduction The word "tides" is a generic term used to define the alternating rise and fall in sea level with respect to the land, produced by the gradient of the gravitational attraction of the moon and the sun. Tidal forces affect not only the shape of the celestial bodies, but also their orbits. Tides on Earth Tides on Earth are due to both the Moon and the Sun. The Moon pull away the closest water body of oceans from the solid shell of the earth and also the solid shell of the earth from the farthest water body of the ocean. So two symmetric high tides always simultaneously appear on opposite positions on the earth. And on certain position there are two high tides every day. Because the tidal effect of the Sun is the smaller than that of the Moon, we usually consider that the tidal effect of the Sun enforce or diminish the tides by the Moon. The twice-monthly highest tides, called spring tides, reach at syzygy(full Moon and new Moon), when the three bodies staying in a straight line. And the lowest tides, called neap tides, occur when the Sun and the Moon form a right 2 angle with the Earth. Tidal Locking and Orbital Resonances In a time scale of thousands and millions of years, tidal forces tend to synchro- nize in integer ratios, the self-spin of satellites and orbital rotation of them. It is not an accident that the same side of the Moon always faces the Earth. The phe- nomenon, called synchronous rotation causes by the tidal force. When the Moon was spinning at some other speed, Earth had exert torque on the Moon, slowing or speeding its spin until this spin is locked with orbital period, at which point the bulge faced the Earth. Particularly when the masses of the primary and the satellite differs not much, both bodies will be tidally locked to each other, as Pluto and Charon do. Tidal locking is so widespread that, as Veverka in 1987 indicates, almost all satellites close enough to the primary conduct synchronous rotation. A good example is the moons of Jupiter, Callisto, which itself is, separates tidally locked moons from non-tidally locked moons. Also, the first three Galilean, Io, and Europa and Ganymede, in a 4:2:1 orbital resonance (some data from wiki about these moons in table (1.1)), give an example of higher-order resonances. Mass Semi-major Orbit period Eccen- Incli- (kg) axis(km) (day) (relative) tricity nation(◦) Io 8:93 × 1022 421,800 1.769 (1) 0.0041 0.050 Europa 4:80 × 1022 671,100 3.551 (2) 0.0094 0.471 Ganymede 1:48 × 1023 1,070,400 7.155 (4) 0.0011 0.204 Callisto 1:08 × 1023 1,882,700 16.69 (9.4) 0.0074 0.205 Table 1.1: Data from wikipedia about Jupiter's moons The satellites may reach an orbital resonance rather than tidally locked when the orbit of it is eccentric. It was thought that Mercury was tidally locked with the Sun [6]. But, by observations of Doppler radar, people proved in 1965 that Mercury has, in fact, a 3:2 spin-orbit resonance, spinning three times for every two 3 revolutions around the Sun. More recently Correia and Laskar [7] explained the 3:2 orbital resonance by using the equations for an oscillating damped pendulum. Tidal Heating Besides changing the orbits and the spins, tidal forces can also internally heat a satellite. A satellite spins on its center at a constant speed but, in an eccentric orbit, its orbital speed varies with the distance from the primary, according to Kepler's 2nd law. Thus the tidal bulge of a tidally locked satellite cannot always point precisely at its primary: most of time it will fail to do so and the primary will exert a small pull on the bulge, causing the deformation. The resulting internal friction heats the interior of the body. With over 400 active volcanoes, Io's internal heat mainly come from tidal heating resulting from the pulls by Jupiter and other Galilean moons. The height of Io's tidal bulge varies as much as 100 meters over its course. Peal et al [8] explained the tidal heat as follows. While all Galilean moons' orbits are nearly circular and the orbits of Ganymede, Europa, and Io are in a 1:2:4 resonance, Io frequently encounters Europa and Ganymede, causing frequent changes of tidal deformations on Io, in turn generating significant friction and in extreme case volcanic activities.

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