A critical shock Mach number for particle acceleration in the absence of pre-existing cosmic rays: $M=\sqrt 5$

TL;DR

This paper establishes a fundamental Mach number threshold of sqrt(5) for particle acceleration at shocks without pre-existing cosmic rays, highlighting the energetic constraints involved.

Contribution

It introduces a critical shock Mach number for particle acceleration in the absence of pre-existing cosmic rays, considering magnetic field effects and plasma conditions.

Findings

Shocks cannot accelerate particles if Mach number <= sqrt(5).

Magnetic field pressure increases the Mach number threshold, especially in low plasma beta conditions.

Pre-existing cosmic rays lower the Mach number limit for acceleration.

Abstract

It is shown that, under some generic assumptions, shocks cannot accelerate particles unless the overall shock Mach number exceeds a critical value M > sqrt(5). The reason is that for M <= sqrt(5) the work done to compress the flow in a particle precursor requires more enthalpy flux than the system can sustain. This lower limit applies to situations without significant magnetic field pressure. In case that the magnetic field pressure dominates the pressure in the unshocked medium, i.e. for low plasma beta, the resistivity of the magnetic field makes it even more difficult to fulfil the energetic requirements for the formation of shock with an accelerated particle precursor and associated compression of the upstream plasma. We illustrate the effects of magnetic fields for the extreme situation of a purely perpendicular magnetic field configuration with plasma beta = 0, which gives a minimum Mach number of M = 5/2. The situation becomes more complex, if we incorporate the effects of pre-existing cosmic rays, indicating that the additional degree of freedom allows for less strict Mach number limits on acceleration. We discuss the implications of this result for low Mach number shock acceleration as found in solar system shocks, and shocks in clusters of galaxies.