Oblique Collision of a Relativistic Cold Shell with an Ideal Reflecting Wall

Abstract

Relativistic flows are common in astrophysics and often form shocks when different parts of the flow collide at relativistic relative velocities. Such collisions are often oblique, forming two shocks whose shocked fluids are separated by a contact discontinuity, which is treated here as an ideal reflecting ``wall'' where the flow on either side is modeled separately. The latter is modeled in the lab frame $S$ as a uniform cold planar shell propagating into vacuum at velocity $v_1=β_1c$ normal to its vacuum interface, colliding with the wall at an incidence angle $α_1$. The collision point $P$ moves along the wall at a velocity $v_p=v_{1}/\sinα_1$, and a boost along the wall at $v_p$ leads to a steady-state frame $S'$ where this problem is highly simplified. However, a ``super-luminal'' regime exists where $v_p>c\Leftrightarrow\tanα_1<Γ_{1}β_{1}=(1-β_{1}^2)^{-1/2}β_1$ and no steady-state frame $S'$ exists. It corresponds to only very small $α_1$ in the Newtonian regime, but nearly all $α_1$ in the relativistic regime. We solve this problem \textit{\textbf{fully analytically}} using integral conservation laws, in the attachmrnt region where point $P$ is attached to the wall. This region of parameter space is bound at high $α_1$ by the detachment line, which coincides with the sonic line for a cold initial shell. A weak-shock solution exist in all this region, while a strong-shock solution exists only in the sub-luminal attachment region -- between the luminal line and the detachment/sonic line where the two solutions coincide and beyond which point $P$ detaches from the wall and shocked fluid spills into the vacuum.