Relativistic Shock Reflection using Integral Conservation Laws

TL;DR

This paper develops a new integral conservation law approach in the lab frame to analyze relativistic shock wave reflection, especially in super-luminal regimes where traditional steady-state methods fail, with implications for astrophysics.

Contribution

It introduces a novel method using integral conservation laws in the lab frame to study relativistic shock reflection, including super-luminal regimes, unifying sub-luminal and super-luminal analysis.

Findings

Bounded parameter space for weak and strong shock solutions.

Existence of critical lines (detachment and luminal) defining solution regimes.

Strong shock solutions only exist within specific incidence angle ranges.

Abstract

Shock wave reflection from a rigid wall has been thoroughly studied in the Newtonian limit, simplifying the problem by analyzing it in a steady-state frame, $S'$, where the point $P$ of the shock's intersection with the wall is at rest. However, a "super-luminal" regime emerges when the velocity of point $P$ ($v_p$) exceeds the speed of light ($v_p>c$), where no steady-state frame $S'$ exists. It occurs predominantly in the relativistic regime, relevant in astrophysics, where it encompasses nearly all of the shock incidence angles. To study this regime, we introduce a new approach. We formulate integral conservation laws in the lab frame $S$ (where the unshocked fluid is at rest) for regular reflection (RR), using two methods: a. fixed volume analysis and b. fixed fluid analysis. We show the equivalence between the two methods, and also to the steady-state oblique shock jump conditions in frame $S'$ in the sub-luminal regime ($v_p<c$). Applying this framework, we find that both the weak and strong shock RR solutions are bounded in the parameter space by the detachment line on the higher incidence angles side. The strong shock solution is also bounded by the luminal line on the lower incidence angles side, and exists only between these two critical lines, in the sub-luminal attachment region.