On the first Sonine correction for granular gases

We consider the velocity distribution for a granular gas of inelastic hard spheres described by the Boltzmann equation. We investigate both the free of forcing case and a system heated by a stochastic force. We propose a new method to compute the first correction to Gaussian behavior in a Sonine polynomial expansion quantified by the fourth cumulant $a_2$. Our expressions are compared to previous results and to those obtained through the numerical solution of the Boltzmann equation. It is numerically shown that our method yields very accurate results for small velocities of the rescaled distribution. We finally discuss the ambiguities inherent to a linear approximation method in $a_2$.