  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. This
paper aims at providing a simple method to account for interactions between
hydrogen and helium in interior and evolution models of giant planets. 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. 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. 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.
