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 $$r$$ above the base does not affect the pressure or density below that height. By contrast, in a spherical symmetric self-gravitating star, heating at a radius $$r$$ will lead to a pressure and density decrease at the center in addition to the expansion of the outer layers (see Schwarzchild & Harm, 1965, ApJ, 146, 855).

We apply a heating function of the form:

(302)$\Hext = H_0 \exp \left [-(r-r_0)^2 / W^2 \right ] ,$

with $$r_0 = 4\times 10^7$$ cm, $$W = 10^7$$ cm, and $$H_0 = 1\times 10^{16}$$ erg g $$^{-1}$$ s $$^{-1}$$. This is the same functional form as used in the first test of paper II, but with a lower amplitude. Still, this heating rate is far higher than what is expected during the convective phase of Type Ia SNe. The heating term is added to the enthalpy equation in the low Mach number equations in the same fashion as described in paper II. In this test, we do not consider reactions. Since this is a one-dimensional test all perturbational quantities, as well as $$\Ubt$$, are zero, so we are directly testing the computation of $$w_0$$ as and the base state update as described in the Advect Base procedure defined above. Both the PPM and low Mach calculation use 768 zones in a domain $$5\times 10^8$$ cm high.

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 $$5\times 10^6$$ K. The PPM simulation required 13488 steps and the low Mach (CFL $$=0.5$$) calculation needed 203. Over the course of the simulation, the Mach number of the flow remained less than $$0.35$$, with the maximum Mach number occurring at the surface of the star. This Mach number pushes the limits of validity of the low Mach number model; a smaller perturbation amplitude would result in a smaller Mach number.

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.