Cococubed.com Stellar Equations Of State
Home Astronomy Research    2024 Radiative Opacity    2024 Neutrino Emission from Stars    2023 White Dwarfs & 12C(α,γ)16O    2023 MESA VI    2022 Earendel, A Highly Magnified Star    2022 Black Hole Mass Spectrum    2021 Skye Equation of State    2021 White Dwarf Pulsations & 22Ne    Software Instruments      Stellar equation of states      EOS with ionization      EOS for supernovae      Chemical potentials      Stellar atmospheres      Voigt Function      Jeans escape      Polytropic stars      Cold white dwarfs      Adiabatic white dwarfs      Cold neutron stars      Stellar opacities      Neutrino energy loss rates      Ephemeris routines      Fermi-Dirac functions      Polyhedra volume      Plane - cube intersection      Coating an ellipsoid      Nuclear reaction networks      Nuclear statistical equilibrium      Laminar deflagrations      CJ detonations      ZND detonations      Fitting to conic sections      Unusual linear algebra      Derivatives on uneven grids      Pentadiagonal solver      Quadratics, Cubics, Quartics      Supernova light curves      Exact Riemann solutions      1D PPM hydrodynamics      Hydrodynamic test cases      Galactic chemical evolution      Universal two-body problem      Circular and elliptical 3 body      The pendulum      Phyllotaxis      MESA      MESA-Web      FLASH      Zingale's software      Brown's dStar      GR1D code      Iliadis' STARLIB database      Herwig's NuGRID      Meyer's NetNuc AAS Journals    2024 AAS YouTube    2024 AAS Peer Review Workshops 2024 ASU Energy in Everyday Life 2024 MESA Classroom Outreach and Education Materials Other Stuff:    Bicycle Adventures    Illustrations    Presentations Contact: F.X.Timmes my one page vitae, full vitae, research statement, and teaching statement.	Before using the software instruments below, perhaps glance at the articles that describe them. The first law of thermodynamics $$ {\rm dE = T \ dS + {P\over \rho^2} \ d\rho} \label{eq1} \tag{1} $$ is an exact differential, which requires that the thermodynamic relations $$ \eqalignno { {\rm P} \ & = \ {\rm \rho^2 \ \dfrac{\partial E}{\partial \rho} \Biggm|_T \ + \ T \ \dfrac{\partial P}{\partial T} \Biggm|_{\rho} } & (2) \cr {\rm \dfrac{\partial E}{\partial T} \Biggm|_{\rho}} \ & = \ {\rm T \ \dfrac{\partial S}{\partial T} \Biggm|_{\rho} } & (3) \cr {\rm - \dfrac{\partial S}{\partial \rho} \Biggm|_T } \ & = \ {\rm {1 \over \rho^2} \ \dfrac{\partial P}{\partial T} \Biggm|_{\rho} } & (4) \cr } $$ be satisfied. An equation of state is thermodynamically consistent if all three of these identities are true. Thermodynamic inconsistency may manifest itself in the unphysical buildup (or decay) of the entropy (or temperature) during numerical simulations of what should be an adiabatic flow. When the temperature and density are the natural thermodynamic variables to use, the appropriate thermodynamic potential is the Helmholtz free energy $$ {\rm F = E - T \ S} \hskip 0.5in {\rm dF = -S \ dT + {P \over \rho^2} \ d\rho} \label{eq5} $$ With the pressure defined as $$ {\rm P \ = \ \rho^2 \ \dfrac{\partial F}{\partial \rho} \Biggm|_T } \label{eq6} \tag{6} $$ the first of the Maxwell relations (Eq. 2) is automatically satisfied, as substitution of Eq. (5) into Eq. (6) demonstrates. With the entropy defined as $$ {\rm S \ = \ -\dfrac{\partial F}{\partial T} \Biggm|_{\rho} } \label{eq7} \tag{7} $$ the second of the Maxwell relations (Eq. 3) is automatically satisfied, as substitution of Eq. (5) into Eq. (7) demonstrates. The requirement that the mixed partial derivatives commute $$ {\rm \dfrac{\partial^2 F}{\partial T \ \partial \rho} \ = \ \dfrac{\partial F}{\partial \rho \ \partial T} } \label{eq8} \tag{8} $$ ensures that the third of the thermodynamic identity (Eq. 4) is satisfied, as substitution of Eq. (5) into Eq. (8) shows. Consider any interpolating function for the Helmholtz free energy $F(\rho,{\rm T})$ which satisfies Eq. (8). Thermodynamic consistency is guaranteed as long as Eq. (6) is used first to evaluate the pressure, Eq. (7) is used second to evaluate the entropy, and finally Eq. (5) is used to evaluate the internal energy. In fact, this procedure is almost too robust! The interpolated values may be a horribly inaccurate but they will be thermodynamically consistent. Here then are bzip2 tarballs of six stellar interior equations of state: helmholtz.tbz nadyozhin.tbz iben.tbz weaver.tbz arnett.tbz timmes.tbz Also see Josiah Schwab's python-helmholtz for the Helmholtz equation of state, and Matt Coleman's port of the Helmholtz equation of state to python helmeos. The Skye EOS = an improved Helmholtz EOS for the non-interacting parts + an improved Potekhin & Chabrier EOS for the Coulomb plama parts + auto-differentiation. Its the bees knees for ionized plasmas as of 2021. Skye is avaliable at https://github.com/adamjermyn/Skye, and the article is described more on the thermodynamics research page. The Helmholtz EOS implements the formalism above on a grid, executes the fastest (memory is faster than cpu), displays perfect thermodynamic consistency, and has a maximum error on the default grid of 10$^{-6}$. Helmholtz is the stellar EOS of choice in the FLASH software instrument and a backplane of the EOS module in the MESA software instrument. The Helmholtz free energy data file provided spans 10$^{-12}$ ≤ density (g cm$^{-3}$) ≤ 10$^{15}$ and 10$^{3}$ ≤ temperature (K) ≤ 10$^{13}$ at 20 points per decade. The Nadyozhin EOS is the fastest of the analytic routines, has very good thermodynamic consistency, a maximum error of 10$^{-5}$, and is also avaliable in FLASH. The Timmes EOS is as slow as molasses during a North Dakota winter, but it computes the non-interacting electron-positron equation of state with no approximations, is exact to machine precision in IEEE double precision arithmetic, has excellent thermodynamic consistency, and serves as the reference point for comparisons to the other EOS routines. In fact, the Helmholtz free energy table used by the Helmholtz EOS is calculated from the Timmes EOS. There are times when a simpler cold fermi gas EOS is a wonderful thing. Such an EOS is in cold_fermi_gas.tbz. One can see this equation of state in action on this cold white dwarf page.
	Please cite the relevant references if you publish a piece of work that use these codes, pieces of these codes, or modified versions of them. Offer co-authorship as appropriate.