Extracting the Temperature Analytically In Hydrodynamics Simulations with Gas and Radiation Pressure

TL;DR

The paper addresses the temperature extraction problem in hydrodynamics simulations where gas and radiation pressure are in equilibrium, reducing to the quartic $T^4 + \beta^3 T - \gamma^4 = 0$ with $\beta^3$ and $\gamma^4$ defined from the density and internal energy. It provides a closed-form analytic solution by factoring the quartic into quadratics and solving a cubic to obtain the real root $T = - \frac{a}{2} + \sqrt{\frac{a^2}{4} - b^2}$, along with explicit expressions for the intermediate coefficients. Additionally, rapid Taylor expansions in the limits $\epsilon = \beta/\gamma \ll 1$ and $\epsilon \gg 1$ yield accurate temperature estimates with minimal computation. The authors discuss the trade-offs between analytic and iterative (e.g., Newton-Raphson) approaches, highlighting robustness, absence of a required initial guess, and platform-dependent performance, thereby providing practical tools for temperature extraction in gas–radiation hydrodynamics simulations across diverse astrophysical regimes.

Abstract

Numerical hydrodynamics simulations of gases dominated by ideal, nondegenerate matter pressure and thermal radiation pressure in equilibrium entail finding the temperature as part of the evolution. Since the temperature is not typically a variable that is evolved independently, it must be extracted from the the evolved variables (e.g. the rest-mass density and specific internal energy). This extraction requires solving a quartic equation, which, in many applications, is done numerically using an iterative root-finding method. Here we show instead how the equation can be solved analytically and provide explicit expressions for the solution. We also derive Taylor expansions in limiting regimes and discuss the respective advantages and disadvantages of the iterative versus analytic approaches to solving the quartic.

Figures (1)

• Figure 1: A comparison of the exact temperature (in units of $\gamma$) with two expansions in opposite limits of $\beta / \gamma$. The (black) solid line shows the exact value (\ref{['T_exact']}), the (blue) dashed line the expansion (\ref{['T_expand_1']}) for $\beta \ll \gamma$ and the (red) dotted line the expansion (\ref{['T_expand_2']}) for $\beta \gg \gamma$. Evidently, the two expansions show excellent agreement with the exact value except in a small region around $\beta / \gamma \simeq 1.2$. In the bottom panel we show the relative errors between exact and expanded values. The dark-colored lines show results using the (black) leading-order terms in the top lines in expansions (\ref{['T_expand_1']}) and (\ref{['T_expand_2']}) only, while the lighter-colored lines include the additional (gray) terms in the remaining lines.