16.2 Gamma Law and Multigamma

FLASH uses the method of Colella & Glaz (1985) to handle general equations of state. General equations of state contain 4 adiabatic indices (Chandrasekhar 1939), but the method of Colella & Glaz parameterizes the EOS and requires only two of the adiabatic indices. The first is necessary to calculate the adiabatic sound speed and is given by

$$\gamma_1 = \frac{\rho}{P}\frac{\partial P}{\partial \rho} \; .$$ (16.1)

The second relates the pressure to the energy and is given by

$$\gamma_4 = 1 + \frac{P}{\rho\epsilon} \; .$$ (16.2)

These two adiabatic indices are stored as the mesh-based variables GAMC_VAR and GAME_VAR. All EOS routines must return $\gamma_1$, and $\gamma_4$ is calculated from (16.2).

The gamma-law EOS models a simple ideal gas with a constant adiabatic index $\gamma$. Here we have dropped the subscript on $\gamma$, because for an ideal gas, all adiabatic indices are equal. The relationship between pressure $P$, density $\rho$, and specific internal energy $\epsilon$ is

$$P = \left(\gamma - 1\right)\rho\epsilon .$$ (16.3)

We also have an expression relating pressure to the temperature $T$

$$P = \frac{N_a k}{\bar{A}} \rho T ,$$ (16.4)

where $N_a$ is the Avogadro number, $k$ is the Boltzmann constant, and $\bar{A}$ is the average atomic mass, defined as

$$\frac{1}{\bar{A}} = \sum_{i}\frac{X_{i}}{A_{i}} ,$$ (16.5)

where $X_i$ is the mass fraction of the $i$th element. Equating these expressions for pressure yields an expression for the specific internal energy as a function of temperature

$$\epsilon = \frac{1}{\gamma - 1} \frac{N_a k} {\bar{A}} T .$$ (16.6)

The relativistic variant of the ideal gas equation is explained in more detail in Sec:RHD.

Simulations are not restricted to a single ideal gas; the multigamma EOS provides routines for simulations with several species of ideal gases each with its own value of $\gamma$. In this case the above expressions hold, but $\gamma$ represents the weighted average adiabatic index calculated from

$$\frac{1}{\left(\gamma - 1\right)} = \bar{A}\sum_{i}\frac{1}{\left(\gamma_{i} - 1\right)}\frac{X_{i}}{A_{i}} .$$ (16.7)

We note that the analytic expressions apply to both the forward (internal energy as a function of density, temperature, and composition) and backward (temperature as a function of density, internal energy and composition) relations. Because the backward relation requires no iteration in order to obtain the temperature, this EOS is quite inexpensive to evaluate. Despite its fast performance, use of the gamma-law EOS is limited, due to its restricted range of applicability for astrophysical problems.

16.2.1 Ideal Gamma Law for Relativistic Hydrodynamics

The relativistic variant of the ideal gas equation is explained in more detail in Sec:RHD.