Slices of theoretical astrophysics: Solar system dynamics and relativistic explosions

This thesis presents studies in two distinct areas of theoretical astrophysics: dynamics of planetary systems and relativistic fluid flows from shocks emerging from stellar envelopes. The first pertains to the early solar system, planet formation, and extrasolar planets; the second is related to extreme explosions like gamma-ray bursts and supernovae.;We present two investigations of the dynamics and population evolution of solar system bodies. First, we explore the dynamics of mean-motion resonances for a test particle in a highly eccentric long-period orbit in the restricted circular planar three-body problem---a scenario relevant to the scattered Kuiper belt and the formation of the Oort cloud. We find infinitely many analogues to the Lagrange points; an explanation for the presence of asymmetric librations in particular mean-motion resonances; and a criterion for the onset of chaos at large semimajor axes.;Second, we study the size distribution of Kuiper belt objects (KBOs), which is observed to be a broken power law. We apply a simple mass conservation argument to the KBO collisional cascade to get the power-law slope for KBOs below the break in the distribution; our result agrees well with observations if KBOs are held together by self-gravity rather than material strength. We then explain the location and time evolution of the break.;We also present investigations of the flow that results when a relativistic shock propagates through and breaks out of a stellar envelope with a polytropic density profile. This work informs predictions of the speed of and energy carried by the relativistic ejecta in supernovae and perhaps in gamma-ray bursts. We find the asymptotic solution for the flow as the shock reaches the star's edge and find a new self-similar solution for the flow of hot fluid after breakout. Since the post-breakout flow acclerates by converting thermal energy into bulk kinetic energy, the fluid eventually cools to nonrelativistic temperatures. We derive a second new self-similar solution that includes the cooling portions of the flow. This second solution gives an exact relation between the terminal Lorentz factor of each fluid element and the Lorentz factor it acquired upon being shocked before breakout.