 Vacuum formation at an accelerating withdrawing piston

An initially resting perfect gas with adiabatic index $\gamma >1$ next to a piston $X(t)=-f{t}^2/2$ develops a smooth rarefaction. The incoming invariant gives $c_p=c_0-(\gamma -1)ft/2$. Pressure falls as $p_p=p_0(1-t/t_v{)}^{2\gamma /(\gamma -1)}$ until $t_v$, when the piston speed is $2c_0/(\gamma -1)$. Thereafter the gas edge moves at that limiting constant speed while the piston accelerates away, opening a vacuum gap.