Planetary Equations of State

Configuring SWIFT with the --with-equation-of-state=planetary and --with-hydro=planetary options enables the use of multiple equations of state (EoS). Every SPH particle then requires and carries the additional MaterialID flag from the initial conditions file. This flag indicates the particle’s material and which EoS it should use.

It is important to check that the EoS you use are appropriate for the conditions in the simulation that you run. Please follow the original sources of these EoS for more information and to check the regions of validity. If an EoS sets particles to have a pressure of zero, then particles may end up overlapping, especially if the gravitational softening is very small.

So far, we have implemented several Tillotson, ANEOS, SESAME, and Hubbard & MacFarlane (1980) materials, with more on the way. The material’s ID is set by a somewhat arbitrary base type ID (multiplied by 100) plus an individual value:

  • Ideal gas: 0
    • Default (\(\gamma\) set using --with-adiabatic-index, default 5/3): 0

  • Tillotson (Melosh, 2007): 1
    • Iron: 100

    • Granite: 101

    • Water: 102

    • Basalt: 103

  • Hubbard & MacFarlane (1980): 2
    • Hydrogen-helium atmosphere: 200

    • Ice H20-CH4-NH3 mix: 201

    • Rock SiO2-MgO-FeS-FeO mix: 202

  • SESAME (and similar): 3
    • Iron (2140): 300

    • Basalt (7530): 301

    • Water (7154): 302

    • Senft & Stewart (2008) water in a SESAME-style table: 303

  • ANEOS (in SESAME-style tables): 4
    • Forsterite (Stewart et al. 2019): 400

    • Iron (Stewart, zenodo.org/record/3866507): 401

    • Fe85Si15 (Stewart, zenodo.org/record/3866550): 402

The data files for the tabulated EoS can be downloaded using the examples/Planetary/EoSTables/get_eos_tables.sh script.

To enable one or multiple EoS, the corresponding planetary_use_*: flag(s) must be set to 1 in the parameter file for a simulation, along with the path to any table files, which are set by the planetary_*_table_file: parameters, as detailed in Equation of State (EoS) and examples/parameter_example.yml.

Unlike the EoS for an ideal or isothermal gas, these more complicated materials do not always include transformations between the internal energy, temperature, and entropy. At the moment, we have implemented \(P(\rho, u)\) and \(c_s(\rho, u)\) (and more in some cases), which is sufficient for the Planetary (Density-Energy, Multi-Material) SPH hydro scheme, but some materials may thus currently be incompatible with e.g. entropy-based schemes.

The Tillotson sound speed was derived using \(c_s^2 = \left. ( \partial P / \partial \rho ) \right|_S \) as described in Kegerreis et al. (2019). Note that there is a typo in the sign of \(du = T dS - P dV = T dS + (P / \rho^2) d\rho \) in the appendix; the correct version was used in the actual derivation.

The ideal gas uses the same equations detailed in Equations of State.

The data files for the tabulated EoS can be downloaded using the examples/EoSTables/get_eos_tables.sh script.

The format of the data files for SESAME, ANEOS, and similar-EoS tables is similar to the SESAME 301 (etc) style. The file contents are:

# header (12 lines)
version_date                                                (YYYYMMDD)
num_rho  num_T
rho[0]   rho[1]  ...  rho[num_rho]                          (kg/m^3)
T[0]     T[1]    ...  T[num_T]                              (K)
u[0, 0]                 P[0, 0]     c[0, 0]     s[0, 0]     (J/kg, Pa, m/s, J/K/kg)
u[1, 0]                 ...         ...         ...
...                     ...         ...         ...
u[num_rho-1, 0]         ...         ...         ...
u[0, 1]                 ...         ...         ...
...                     ...         ...         ...
u[num_rho-1, num_T-1]   ...         ...         s[num_rho-1, num_T-1]

The version_date must match the value in the sesame.h SESAME_params objects, so we can ensure that any version updates work with the git repository. The header contains a first line that gives the material name, followed by the same 11 lines printed here to describe the contents.