The solution of two dimensional problems in astrophysics
Fryxell, Bruce Alan
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https://hdl.handle.net/2142/25579
Description
Title
The solution of two dimensional problems in astrophysics
Author(s)
Fryxell, Bruce Alan
Issue Date
1978
Doctoral Committee Chair(s)
Arnett, D.
Department of Study
Physics
Discipline
Physics
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
two dimensional problems in astrophysics
astrophysics
spherically symmetric calculations
general hydrodynamic behavior
cylindrical coordiantes
Poisson's equation
Language
en
Abstract
Many important problems in astrophysics cannot be solved accurately using spherically symmetric calculations. Techniques for studying the general hydrodynamic behavior of problems in two dimensional cylindrical coordinates with rotational symmetry are discussed. These techniques involve the solution of the hydrodynamic equations and Poisson's equation for the gravitational potential on a rectangular Eulerian finite difference grid. Calculations of problems with known analytic solutions, which were performed to test the accuracy of the numerical methods, are described.
The results of two hydrodynamic calculations which have significant astrophysical implications are presented. The first of these is an investigation of the hydrodynamic behavior of off-center point explosions in stars. An important application of these calculations is the possible formation of high velocity pulsars. The dependence of the final velocity of the collapsed remnant on the location and energy of the explosion is computed. The largest remnant velocities result from explosions located at a mass fraction of about 0.5. An explosion energy fifty per cent greater than the binding energy of the star produces a 1.4 Me remnant with a velocity of about 400 km/sec. However, this energy must be generated in a very small region of the star in order to create the required asymmetry in the explosion. Because of this a specific energy of about 1020 ergs/gm is needed for the explosion. Nuclear reactions can produce no more than about
17
5xl0ergs/gm and it is unclear how the energy produced in gravitational
collapse models can be sufficiently localized. Unless a supernova mechanism can be found which is able to produce enough energy in a small region of the star, off-center explosions do not provide a satisfactory explanation for high velocity pulsars.
The second problem to be discussed is whether the impact of an expanding supernova shell on a binary companion star can disrupt the binary system. Three effects are considered. First, the completely inelastic,
collision of the shell with the companion strips off some mass from the outer layers of the star. Second, the impact of the shell on the companion will heat some of the matter near the stellar surface enough to cause mass ablation to occur. Finally, some of the matter in the supernova shell will stick to the star. The results of this calculation are compared to the results of previous attempts to solve the problem analytically by assuming that the stellar surface is planar. However, the curved surface of the star is found to have a significant effect on the momentum imparted to the companion. Much of the matter in the shell streams around the star and thus has very little effect. Furthermore, not all of the matter which is ablated from the star is ejected straight back toward the supernova as was assumed for the analytic calculations. In fact, mass ablation spreads over most of the stellar surface and becomes nearly spherically symmetric, further reducing the momentum given to the companion. As a result, the final momentum of the companion star is only 0.8 of the incident momentum of the shell, instead of the factor of about four greater than the incident momentum predicted by the analytic calculations. Therefore, it seems unlikely that the collision between the supernova shell and the companion star will have a significant effect on the orbit of the binary system except perhaps in a few extreme cases.
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