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 v1=β1c normal to its vacuum interface, colliding with the wall at an incidence angle α1. The collision point P moves along the wall at a velocity vp=v1/sinα1, and a boost along the wall at vp leads to a steady-state frame S' where this problem is highly simplified. However, a ``super-luminal'' regime exists where vp>c⇔tanα1<Γ1β1=(1−β12)−1/2β1 and no steady-state frame S' exists. It corresponds to only very small α1 in the Newtonian regime, but nearly all α1 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 α1 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.
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BibTeX
@misc{granot2026oblique,
title = {{Oblique Collision of a Relativistic Cold Shell with an Ideal Reflecting Wall}},
author = {Jonathan Granot and Prasanta Bera and Michael Rabinovich and Paz Beniamini},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07417},
eprint = {2609.07417},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05759231v1},
}
RIS
TY - UNPB
TI - Oblique Collision of a Relativistic Cold Shell with an Ideal Reflecting Wall
AU - Jonathan Granot
AU - Prasanta Bera
AU - Michael Rabinovich
AU - Paz Beniamini
PY - 2026
DA - 2026/09/07
PB - HAL
DO - 10.48550/arxiv.2609.07417
UR - https://hal.science/hal-05759231v1
AB - 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⇔tanα_1<Γ_1β_1=(1−β_1^2)^−1/2β_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.
ER -