20 70 120 170 M/m 0 2 4 T$_{eff}$/T$_{L}$ FIG. 6: (Color online) Ratio of the effective temperature, T$_{eff}$ , defined in Eq. (12) to the tem- perature of the gas in the vicinity of the piston, T$_{L}$ , as a function of the mass ratio for the same systems as in Fig. 5. difference is observed in the behavior of the two temperature ratios for a given constant coefficient of restitution α . While T$_{eff}$/T$_{L}$ exhibits a strong dependence on M/m and does not seem to tend to a well defined limit as it increases, the dependence of T$_{eff}$/T$_{P}$ on M/m is very weak, being only appreciable for the least inelastic cases and when the mass ratio is small. Thus it is concluded that for large enough mass ratio M/m , the effective temperature is proportional to the temperature parameter of the piston with a coefficient of proportionality that is independent of the mass of the piston, i.e. where a possible dependence of the coefficient on α$_{P}$ has been included. To identify the dependence of the coefficient b on α , in Fig. 8 the plateau values of T$_{eff}$/T$_{P}$ , reached upon increasing the value of the mass ratio M/m , are plotted versus 1 − α $^{2}$. The points are very well fitted by a straight line, indicating that T$_{eff}$/T$_{P}$ grows T$_{eff}$ = b ( α, α$_{P}$ ) T$_{P}$, (13) 15