The equation of state (EOS) of ionized matter is a key ingredient in models of stars, gas giant planets, accretion disks, and many other astrophysical systems. These applications span many orders of magnitude in both density and temperature, and include both low density systems that are thermally ionized (e.g., stellar atmospheres) and high-density ones that are pressure–ionized (e.g., planetary interiors). Moreover matter can have many different compositions, ranging from pure hydrogen to exotic mixtures of heavy metals. As a result, approximations to nature’s EOS of ionized matter must capture a wide variety of physics (Figure 1) including relativity, quantum mechanics, electron degeneracy, pair production, phase transitions, and chemical mixtures.
Despite these challenges, several different equations of state have been introduced for ionized matter. In 1990, Chabrier introduced an EOS for non-relativistic ionized hydrogen, incorporating sophisticated quantum and electron screening corrections. Improvements then led to the PC EOS. PC allows for arbitrary compositions and incorporates relativistic ideal electrons as well as modern prescriptions for electron screening and multi-component plasmas. Later, Potekhin & Chabrier extended the PC EOS to include the effects of strong magnetic fields such as those found in neutron stars. One of the distinguishing features of the PC EOS is the use of analytic prescriptions to capture non-ideal physics.
One of the limitations of the PC EOS is that it does not capture the effects of electron-positron pair production at high temperatures, which is important for the pair instability in massive stars. The treatment of electron degeneracy and the ideal quantum electron gas is also approximate, based on fitting formulas which approximate the relevant Fermi integrals. These limitations are addressed by the HELM EOS. While HELM does not include the sophisticated non-ideal corrections which are a defining strength of PC, it provides a tabulated Helmholtz free energy treatment of an ideal quantum electron-positron plasma, obtained by highprecision evaluation of the relevant Fermi-Dirac integrals. As such, HELM accurately and efficiently handles relativistic effects, degeneracy effects, and high-temperature pair production.
Now, Jermyn and colleagues presented a new equation of state called “Skye”. This EOS designed to handle density and temperature inputs over the range 10¯12 g cm¯3 < ρ < 1013 g cm3 and 103 K < T < 1013 K (Figure 1). Skye assumes material is fully-ionized, so the suitability of the result is subject to the (composition-dependent) constraint that material is either pressure-ionized (ρ ≳ 103 g cm¯3) or thermally-ionized (T ≳ 105 K). Further limits to Skye’s suitability can arise due to violations of its other physics assumptions. Building on HELM, researchers used the full ideal equation of state for electrons and positrons, accounting for degeneracy and relativity. Ions are assumed to be a classical ideal gas. They then added non-ideal classical and quantum corrections to account for electron-electron, electron-ion, and ion-ion interactions following a multi-component ion plasma prescription. These corrections are generally similar to those used by the PC EOS, though they have used updated physics prescriptions in some instances.
Thermodynamic quantities in Skye are derived from a Helmholtz free energy to ensure thermodynamic consistency. Automatic differentiation machinery allows extraction of arbitrary derivatives from an analytic Helmholtz free energy, allowing Skye to provide the high-order derivatives needed for stellar evolution calculations. Researchers further leverage this machinery to make the EOS easily extensible: adding new or refined physics to Skye is as easy as writing a formula for the additional Helmholtz free energy. The often painstaking and error-prone process of taking and programming analytic first, second, and even third derivatives of the Helmholtz free energy is eliminated. In this way Skye is a framework for rapidly developing and prototyping new EOS physics as advances are made in numerical simulations and analytic calculations. They emphasized that Skye is not tied to a specific set of physics choices; Skye in 10 years is unlikely to be the same as Skye as described by them in their paper.
In addition to being a single EOS which can be used at both high temperatures, like HELM, and high densities, like PC, Skye currently includes two significant physical improvements. First, whereas PC fixes the location of Coulomb crystallization of the ions, Skye picks between the liquid and solid phase to minimize the Helmholtz free energy. This enables a self-consistent treatment of the phase transition, albeit one currently without chemical phase separation, and means that the Helmholtz free energy is continuous across the transition. Secondly, they introduced the technique of thermodynamic extrapolation, which provides a principled way to extend Helmholtz free energy fitting formulas beyond their original range of applicability and thus enables comparisons of the liquid and solid phase Helmholtz free energies.
“Because Skye is a framework for developing new EOS physics we expect future work to bring several key improvements.”— told Jermyn, first author of the study.
First, and most pressing, is handling of partial ionization and neutral matter. With that Skye could be used across the entire range of densities and temperatures which arise in stellar evolution calculations. This could be done in a Debye-Huckle-Thomas-Fermi formalism or other approaches in the physical picture, or else via free energy minimization in the chemical picture. The key constraint in each of these approaches is that Skye needs to remain fast enough to use in practical stellar evolution calculations.
“Our hope is that the flexibility afforded to Skye by its automatic differentiation machinery will allow us to rapidly prototype and test these various possibilities.“— told Jermyn, first author of the study
Along similar lines, Skye could be made to support phase separation by minimizing the free energy with respect to the compositions of the liquid and solid phases. The major bottleneck to supporting this is the current lack of Fortran compiler support for parameterized derived types. Once this compiler challenge is resolved, phase separation physics should not be difficult to implement. More broadly, they make Skye openly available with the hope that it will grow into a community resource to use automatic differentiation to explore analytic free energy terms that captures improvements in existing physics and development of new or not yet considered physics.
Skye is distributed as part of the eos module of the MESA stellar evolution software instrument. It is also available as a standalone package from https://github.com/adamjermyn/Skye, and the version used here is available from Jermyn et al. (2021a). Compilation is supported on the GNU Fortran compiler version 10.2.0.
Featured image: The fraction of Skye used in the MESA EOS is shown as a function of density and temperature. © Jermyn et al.
Reference: Adam S. Jermyn, Josiah Schwab, Evan Bauer, F. X. Timmes, Alexander Y. Potekhin, “Skye: A Differentiable Equation of State”, Astrophysical Journal, pp. 1-27, 2021. https://arxiv.org/abs/2104.00691
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