Modifications for a Spherical Self-Gravitating Star
In papers II and III, we calculated the hydrostatic expansion of the base state in plane-parallel geometry under the assumption that the weight of the material above (or below) any given fluid parcel does not change during hydrostatic expansion. This assumption holds when the gravitational acceleration is independent of location. Here we discuss the modifications to the algorithm in paper III required to treat a spherical self-gravitating star.
One-dimensional Results
To test the spherical base state expansions, we inject heat at a steady rate into a one-dimensional white dwarf model. This is similar to the first test in paper II, except now in spherical coordinates. As in that test, the compressible method with which we compare the low Mach number method is the FLASH code’s implementation of the piecewise-parabolic method (PPM) in a one-dimensional spherical geometry. The initial conditions for the white dwarf are those described in Section 4.1 of paper III for the central region.
In the expansion of a plane-parallel atmosphere, heating at a
height
We apply a heating function of the form:
with
Figure [[fig:spherical768]](#fig:spherical768){reference-type=”ref” reference=”fig:spherical768”} shows the structure of the star after
heating for 10 s. The gray line is the initial star before any
heating.
We see that the compressible and low Mach number models
agree extremely well. Both capture the decrease in the density and
pressure at the center of the star and the considerable expansion in
radius. Only at the surface of the star do the temperatures differ slightly.
In all calculations, we set the minimum temperature to
Future improvements to the overall spherical base-state adjustment algorithm will address the expansion in a simulation where the medium outside the star is not brought down to arbitrarily low densities, but instead a “cutoff density” is applied, as in the case of the plane-parallel results presented in this paper. However, we expect the changes to the overall method shown here to be small.
Add a figure showing that we retain the correct solution
even when we place higher density material outside the star.
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