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

Context. The equation of state for hydrogen and helium is fundamental for studying 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 that involve a system of interacting hydrogen and helium particles and pure equations of state for hydrogen and helium on the other, we derived 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 non-ideal mixing is accounted for. These non-ideal mixing effects must be considered in interior models of giant planets based on accurate gravity field measurements, particularly in the context of variations in the helium-to-hydrogen ratio. They also affect the mass-radius relation of exoplanets. We provide a table that 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 that 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 the equations of state of hydrogen and helium. Uncertainties on the equation of state still exist, however. Ab initio calculations of the behaviour of the hydrogen-helium mixture in the megabar regime for various compositions should be performed in order to gain accuracy.