Hydrogen-Helium Equation of State for Giant Planet Interior Models
Derived from Ab Initio Computer Simulations

Figure 12 in Ref. [1]

Using density functional molecular dynamics simulations, we determine the equation of state (EOS) for hydrogen-helium mixtures spanning density-temperature conditions typical of giant planet interiors. In our manuscript [1], a comprehensive equation of state table with 391 density-temperature points is constructed and the results are presented in form of two-dimensional free energy fit for interpolation. This web page was setup togive everyone access to our EOS points, the coefficients of the free energy fit, and our free energy interpolation code, which is currently written in Fortran 77. A C++ version could be constructed if there is sufficient demand.

We found that our DFT-MD simulations led to significant EOS corrections (for details see [1]) to the well-known, semi-analytical Saumon-Chabrier-van Horn EOS, which gave rise to a recalibration of mass-radius relationship (left figure) in solar and extrasolar giant planets.

Three files are available:

• EOS points
• Free energy fit coefficients
• Free energy interpolation code (Fortran 77)

Temperature-density conditions were DFT-MD simulations were performed (figure 1 in Ref. [1])

Comments: This table contains the pressure, internal energy, entropy, and free energy from our ab initio simulations a hydrogen-helium mixture of a solar-type composition with 18 helium in 220 hydrogen atoms. The helium mass fraction is Y=0.245.

The density parameter r_s is given by 4/3*pi*r_s³=V/N_e where N_e is total number of electrons (bound and free) in volume V. The number density of electrons is n=N_e / V.

The fit uses atomic units:
  r_s is given Bohr radii = 5.29177208607e–11 m
  F and E are given in Hartree/electron = 4.35974380267e–18 J
  S is given in k_b/electron where k_b is Boltzmann's constant.
Here is an example for an energy conversion.

The zero of energy was set to state of isolated, neutral atoms at rest, which is constistent with the convention in the VASP simulation code. The groundstate energy is the isolate hydrogen molecule using PBE including zero-point energy then becomes -0.2386100 Hartree per molecule.

Limitations: In density, the fit ranges from r_s=3.581 to 0.526 [0.06696 ... 19.97757 g/cc, log₁₀(density)=–1.1742 ... 1.3005]. So an alternative EOS table is needed to model the outer envelop of a giant planet where the H-He mixture is at a lower density.
The temperature range of the fit is 500 ... 120,000 K [log₁₀(T)=2.6990 ... 5.0792] but only 1,000 ... 80,000 K is based on data from DFT-MD simulations. Elsewhere we use an extrapolation.