Accounting for non-ideal mixing effects in the hydrogen-helium equation of state

Supplemental materials of Howard & Guillot (2023, A&A). Abstract: Context. The equation of state for hydrogen and helium is fundamental to study stars and giant planets. It has been shown that, because of interactions at atomic and molecular levels, the behaviour of a mixture of hydrogen and helium cannot be accurately represented by considering these elements separately. Aims. This paper aims at providing a simple method to account for interactions between hydrogen and helium in interior and evolution models of giant planets. Methods. Using on the one hand ab initio simulations which involve a system of interacting hydrogen and helium particles and pure equations of state for hydrogen and helium on the other, we derive the contributions in density and entropy of the interactions between hydrogen and helium particles. Results. We show that relative variations of up to 15% in density and entropy arise when accounting for non-ideal mixing. These non-ideal mixing effects must be considered in giant planets interior models based on accurate gravity field measurements, particularly in the context of variations of the helium-to-hydrogen ratio. They also affect the mass-radius relation of exoplanets. We provide a table which contains the volume and entropy of mixing as a function of pressure and temperature. This table is to be combined with pure hydrogen and pure helium equations of state to obtain an equation of state which self-consistently includes mixing effects for any hydrogen and helium mixing ratio and may be used to model the interior structure and evolution of giant planets to brown dwarfs. Conclusions. Non-linear mixing must be included in accurate calculations of equations of state of hydrogen and helium. Yet, uncer- tainties on the equation of state still exist. Ab initio calculations of the behaviour of the hydrogen-helium mixture in the Mbar regime for various compositions should be performed in order to gain in accuracy