Numerical simulations of isothermal collapse and the relation to steady-state accretion
Date
2015-05
Authors
Herbst, Rhameez Sheldon
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Abstract
In this thesis we present numerical simulations of the gravitational collapse of isothermal
clouds of one solar mass at a temperature of 10K. We will consider two types of initial
conditions – initially uniform spheres and perturbed Bonnor-Ebert spheres. The aim
of the performed numerical simulations is to investigate the core bounce described by
Hayashi and Nakano [1]. They reported that if strong enough, the shock wave would be
capable of ionizing the gas in the collapsing cloud.
The simulations are performed using two numerical methods: the TVD MUSCL scheme
of van Leer using a Roe flux on a uniform grid and the TVD Runge-Kutta time-stepping
using a Marquina flux on a non-uniform grid. These two particular methods are used
because of their differences in numerical structure. Which allows us to confidently make
statements about the nature of the collapse, particularly with regards to the core bounce.
The convergence properties of the two methods are investigated to validate the solutions
obtained from the simulations. The numerical simulations have been performed only in
the isothermal regime by using the Truelove criterion [2] to terminate the simulation
before central densities become large enough to cause artificial fragmentation.
In addition to the numerical simulations presented in this thesis, we also introduce new,
analytical solutions for the steady-state accretion of an isothermal gas onto a spherical
core as well as infinite cylinders and sheets. We present the solutions and their properties
in terms of the Lambert function with two parameters, γ and m. In the case of spherical
accretion we show that the solution for the velocity perfectly matched the solutions of
Bondi [3]. We also show that the analytical solutions for the density – in the spherical
case – match the numerical solutions obtained from the simulations. From the agreement
of these solutions we propose that the analytical solution can provide information about
the protostellar core (in the early stages of its formation) such as the mass.
Description
A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy
in the Faculty of Science School of Computational and Applied Mathematics.
May 2015.