Understanding the Solar System with Numerical Simulations and Lévy Flights

Citation

Collins, Benjamin Forster (2009) Understanding the Solar System with Numerical Simulations and Lévy Flights. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/840K-TD87. https://resolver.caltech.edu/CaltechETD:etd-05292009-130440

Abstract

This thesis presents several investigations into the formation of planetary systems and the dynamical evolution of the small bodies left over from this process.

We develop a numerical integration scheme with several optimizations for studying the late stages of planet formation: adaptive time steps, accurate treatment of close encounters between particles, and the ability to add non-conservative forces. Using this code, we simulate the phase of planet formation known as "oligarchic growth." We find that when the dynamical friction from planetesimals is strong, the annular feeding zones of the protoplanets are inhabited by not one but several oligarchs on nearly the same semimajor axis. We systematically determine the number of co-orbital protoplanets sharing a feeding zone and the width of these zones as a function of the strength of dynamical friction and the total mass of the disk. The increased number of surviving protoplanets at the end of this phase qualitatively affects their subsequent evolution into full-sized planets.

We also investigate the distribution of the eccentricities of the protoplanets in the runaway growth phase of planet formation. Using a Boltzmann equation, we find a simple analytic solution for the distribution function followed by the eccentricity. We show that this function is self-similar: it has a constant shape while the scale is set by the balance between mutual excitation and dynamical friction. The type of evolution described by this distribution function is known as a Levy flight.

We use the Boltzmann equation framework to study the nearly circular orbits of Kuiper Belt binaries and the nearly radial orbits of comets during the formation of the Oort cloud. We calculate the distribution function of the eccentricity of Kuiper belt systems, like the moons of Pluto, given the stochastic perturbations caused by close encounters with other Kuiper belt objects. For Oort cloud comets, we find the distribution function of the angular momentum as it is excited by perturbations from passing stars in the Galaxy. Both systems evolve as Levy flights. This work unifies the effects of stochastic stellar encounters and the smooth torque from the Galactic potential on Oort cloud comets.

Thesis Files