Singular small-power piston emission time

For $X_p(t)=-t^{1+\alpha }/(1+\alpha )$ with $0<\alpha \ll 1$, a point on the ray $x=(c_0-a)t$, $0<a<(\gamma +1)/2$, receives the characteristic emitted at the exponentially small timeAlthough $s\to 0$, the piston speed $-s^{\alpha }$ has the finite limit $-2a/(\gamma +1)$, producing the centered-rarefaction profile of the singular limit $\alpha =0$.