17.3 Helmholtz

The Helmholtz EOS provided with the FLASH distribution contains more physics and is appropriate for addressing astrophysical phenomena in which electrons and positrons may be relativistic and/or degenerate and in which radiation may significantly contribute to the thermodynamic state. Full details of the Helmholtz equation of state are provided in Timmes & Swesty (1999). This EOS includes contributions from radiation, completely ionized nuclei, and degenerate/relativistic electrons and positrons. The pressure and internal energy are calculated as the sum over the components

$$P_{\rm tot} = P_{\rm rad} + P_{\rm ion} + P_{\rm ele} + P_{\rm pos} + P_{\rm coul}$$ (17.8)

$$\epsilon_{\rm tot} = \epsilon_{\rm rad} + \epsilon_{\rm ion} + \epsilon_{\rm ele} + \epsilon_{\rm pos} + \epsilon_{\rm coul} \; .$$ (17.9)

Here the subscripts “rad,” “ion,” “ele,” “pos,” and “coul” represent the contributions from radiation, nuclei, electrons, positrons, and corrections for Coulomb effects, respectively. The radiation portion assumes a blackbody in local thermodynamic equilibrium, the ion portion (nuclei) is treated as an ideal gas with $\gamma = 5/3$, and the electrons and positrons are treated as a non-interacting Fermi gas.

The blackbody pressure and energy are calculated as

$$P_{\rm rad} = {a T^4 \over 3}$$ (17.10)

$$\epsilon_{\rm rad} = { 3 P_{\rm rad} \over \rho}$$ (17.11)

where $a$ is related to the Stephan-Boltzmann constant $\sigma_B = a c/4$, and $c$ is the speed of light. The ion portion of each routine is the ideal gas of (Equations Eqn:eos2b - Eqn:eos2a) with $\gamma = 5/3$. The number densities of free electrons $N_{\rm ele}$ and positrons $N_{\rm pos}$ in the noninteracting Fermi gas formalism are given by

$$N_{\rm ele} = {8 \pi \sqrt{2} \over h^3} m_{\rm e}^3 c^3 \beta^{3/2} \left[ F_{1/2}(\eta,\beta) + F_{3/2}(\eta,\beta) \right]$$ (17.12)

$$N_{\rm pos} = {8 \pi \sqrt{2} \over h^3} m_{\rm e}^3 c^3 \b... ... + \beta F_{3/2} \left( -\eta - 2 /\beta, \beta \right) \right] \enskip ,$$ (17.13)

where $h$ is Planck's constant, $m_{\rm e}$ is the electron rest mass, $\beta \: = \: k T / (m_{\rm e} c^2)$ is the relativity parameter, $\eta \: = \: \mu / k T$ is the normalized chemical potential energy $\mu$ for electrons, and $F_{k}(\eta,\beta)$ is the Fermi-Dirac integral

$$F_{k}(\eta,\beta) = \int\limits_{0}^{\infty} {x^{k} (1 + 0.5 \beta x)^{1/2} dx \over \exp(x - \eta) + 1 } .$$ (17.14)

Because the electron rest mass is not included in the chemical potential, the positron chemical potential must have the form $\eta_{{\rm pos}} = -\eta - 2/\beta$. For complete ionization, the number density of free electrons in the matter is

$$N_{\rm ele,matter} = {\bar{Z} \over \bar{A}} N_a \rho = \bar{Z} N_{\rm ion} ,$$ (17.15)

and charge neutrality requires

$$N_{\rm ele,matter} = N_{\rm ele} - N_{\rm pos} .$$ (17.16)

Solving this equation with a standard one-dimensional root-finding algorithm determines $\eta$. Once $\eta$ is known, the Fermi-Dirac integrals can be evaluated, giving the pressure, specific thermal energy, and entropy due to the free electrons and positrons. From these, other thermodynamic quantities such as $\gamma_1$ and $\gamma_4$ are found. Full details of this formalism may be found in Fryxell et al. (2000) and references therein.

The above formalism requires many complex calculations to evaluate the thermodynamic quantities, and routines for these calculations typically are designed for accuracy and thermodynamic consistency at the expense of speed. The Helmholtz EOS in FLASH provides a table of the Helmholtz free energy (hence the name) and makes use of a thermodynamically consistent interpolation scheme obviating the need to perform the complex calculations required of the above formalism during the course of a simulation. The interpolation scheme uses a bi-quintic Hermite interpolant resulting in an accurate EOS that performs reasonably well.

The Helmholtz free energy,

$$F = \epsilon - T S$$ (17.17)

$$dF = -S dT + {P \over \rho^2} d\rho \enskip ,$$ (17.18)

is the appropriate thermodynamic potential for use when the temperature and density are the natural thermodynamic variables. The free energy table distributed with FLASH was produced from the Timmes EOS (Timmes & Arnett 1999). The Timmes EOS evaluates the Fermi-Dirac integrals (17.14) and their partial derivatives with respect to $\eta$ and $\beta$ to machine precision with the efficient quadrature schemes of Aparicio (1998) and uses a Newton-Raphson iteration to obtain the chemical potential of (17.16). All partial derivatives of the pressure, entropy, and internal energy are formed analytically. Searches through the free energy table are avoided by computing hash indices from the values of any given $(T,\rho \bar{Z}/\bar{A})$ pair. No computationally expensive divisions are required in interpolating from the table; all of them can be computed and stored the first time the EOS routine is called.

We note that the Helmholtz free energy table is constructed for only the electron-positron plasma, and it is a 2-dimensional function of density and temperature, i.e. $F(\rho,{\rm T})$. It is made with ${\bar {\rm A}} = {\bar {\rm Z}} = 1$ (pure hydrogen), with an electron fraction $Y_{\rm e} = 1$. One reason for not including contributions from photons and ions in the table is that these components of the Helmholtz EOS are very simple (Equations Eqn:eos4a - Eqn:eos4b), and one doesn't need fancy table look-up schemes to evaluate simple analytical functions. A more important reason for only constructing an electron-positron EOS table with $Y_{\rm e} = 1$ is that the 2-dimensional table is valid for any composition. Separate planes for each $Y_{\rm e}$ are not necessary (or desirable), since simple multiplication by $Y_{\rm e}$ in the appropriate places gives the desired composition scaling. If photons and ions were included in the table, then this valuable composition independence would be lost, and a 3-dimensional table would be necessary.

The Helmholtz EOS has been subjected to considerable analysis and testing (Timmes & Swesty 2000), and particular care was taken to reduce the numerical error introduced by the thermodynamical models below the formal accuracy of the hydrodynamics algorithm (Fryxell, et al. 2000; Timmes & Swesty 2000). The physical limits of the Helmholtz EOS are $10^{-10} < \rho < 10^{11} ({\rm g cm}^{-3})$ and $10^{4} < T < 10^{11}$ (K). As with the gamma-law EOS, the Helmholtz EOS provides both forward and backward relations. In the case of the forward relation ($\rho, T$, given along with the composition) the table lookup scheme and analytic formulae directly provide relevant thermodynamic quantities. In the case of the backward relation ( $\rho, \epsilon$, and composition given), the routine performs a Newton-Rhaphson iteration to determine temperature. It is possible for the input variables to be changed in the iterative modes since the solution is not exact. The returned quantities are thermodynamically consistent.