Singular small-power piston emission time
= Singular small-power piston emission time
For $X_p(t)=-t^{1+\alpha}/(1+\alpha)$ with $0<\alpha\ll1$, a point on the ray $x=(c_0-a)t$, $0<a<(\gamma+1)/2$, receives the characteristic emitted at the exponentially small time
$$
s\sim\left(\frac{2a}{\gamma+1}\right)^{1/\alpha}.
$$
Although $s\to0$, the piston speed $-s^\alpha$ has the finite limit $-2a/(\gamma+1)$, producing the centered-rarefaction profile of the singular limit $\alpha=0$.