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We consider a dilute granular gas of hard spheres colliding inelasticallywith coefficients of normal and tangential restitution $\alpha$ and $\beta$,respectively. The basic quantities characterizing the distribution function$f(\mathbf{v},\bm{\omega})$ of linear ($\mathbf{v}$) and angular($\bm{\omega}$) velocities are the second-degree moments defining thetranslational ($T^\text{tr}$) and rotational ($T^\text{rot}$) temperatures. Thedeviation of $f$ from the Maxwellian distribution parameterized by$T^\text{tr}$ and $T^\text{rot}$ can be measured by the cumulants associatedwith the fourth-degree velocity moments. The main objective of this paper isthe evaluation of the collisional rates of change of these second- andfourth-degree moments by means of a Sonine approximation. The results aresubsequently applied to the computation of the temperature ratio$T^\text{rot}/T^\text{tr}$ and the cumulants of two paradigmatic states: thehomogeneous cooling state and the homogeneous steady state driven by awhite-noise stochastic thermostat. It is found in both cases that theMaxwellian approximation for the temperature ratio does not deviate much fromthe Sonine prediction. On the other hand, non-Maxwellian properties measured bythe cumulants cannot be ignored, especially in the homogeneous cooling statefor medium and small roughness. In that state, moreover, the cumulant directlyrelated to the translational velocity differs in the quasi-smooth limit$\beta\to -1$ from that of pure smooth spheres ($\beta=-1$). This singularbehavior is directly related to the unsteady character of the homogeneouscooling state and thus it is absent in the stochastic thermostat case.

Sonine approximation for collisional moments of granular gases of inelastic rough spheres