[Paper Review] Diffusive Shock Acceleration by Multiple Weak Shocks

Hyesung Kang|arXiv (Cornell University)|Jun 16, 2021

Astrophysics and Cosmic Phenomena16 references4 citations

TL;DR

This paper investigates diffusive shock acceleration (DSA) by multiple weak shocks in the intracluster medium (ICM), showing that repeated re-acceleration of cosmic rays (CRs) through shocks with Mach numbers M ≤ 3 flattens the CR proton spectrum and enhances DSA efficiency. The key result is that the final CR spectrum slope approaches p⁻³ due to cumulative re-acceleration, independent of individual shock compression ratios, with efficiency increasing significantly over three shock passages.

ABSTRACT

The intracluster medium (ICM) is expected to experience on average about three passages of weak shocks with low sonic Mach numbers, $M\lesssim 3$, during the formation of galaxy clusters. Both protons and electrons could be accelerated to become high energy cosmic rays (CRs) at such ICM shocks via diffusive shock acceleration (DSA). We examine the effects of DSA by multiple shocks on the spectrum of accelerated CRs by including {\it in situ} injection/acceleration at each shock, followed by repeated re-acceleration at successive shocks in the test-particle regime. For simplicity, the accelerated particles are assumed to undergo adiabatic decompression without energy loss and escape from the system, before they encounter subsequent shocks. We show that in general the CR spectrum is flattened by multiple shock passages, compared to a single episode of DSA, and that the acceleration efficiency increases with successive shock passages. However, the decompression due to the expansion of shocks into the cluster outskirts may reduce the amplification and flattening of the CR spectrum by multiple shock passages. The final CR spectrum behind the last shock is determined by the accumulated effects of repeated re-acceleration by all previous shocks, but it is relatively insensitive to the ordering of the shock Mach numbers. Thus multiple passages of shocks may cause the slope of the CR spectrum to deviate from the canonical DSA power-law slope of the current shock.

Motivation & Objective

• To investigate how multiple shock passages in the intracluster medium (ICM) affect cosmic ray (CR) proton spectra through repeated re-acceleration.
• To assess the impact of shock Mach number variations and adiabatic decompression on CR spectrum evolution in the test-particle regime.
• To determine whether cumulative effects from multiple shocks can explain discrepancies between X-ray and radio-derived Mach numbers in radio relic shocks.
• To extend semi-analytic models of DSA to include decompression and injection at each shock, based on Liouville’s theorem and prior plasma simulations.

Proposed method

• Uses a semi-analytic approach to model CR proton spectra after multiple shock passages, assuming test-particle regime and momentum-independent injection.
• Applies Liouville’s theorem to account for adiabatic decompression in postshock regions, with momentum scaling p′ = Rp, where R = (D/r)¹ᐟ³.
• Models injection of suprathermal particles with p ≥ 3.8p_th at each shock, using injection momentum p_inj,k and normalization f_o,k specific to each shock.
• Considers both identical shocks (M = 3) and variable Mach number sequences (e.g., M = 1.7, 2.0, 3.0) to test ordering effects on final spectra.
• Computes DSA efficiency η and CR energy ratio E_CR,k / E_th,k to assess nonlinearity and back-reaction potential.
• Compares results across cases with and without decompression (D = 1 vs. D = 0.5) to isolate expansion effects on spectrum shaping.

Experimental results

Research questions

• RQ1 How does the power-law slope of the CR proton spectrum evolve after multiple passages through weak shocks with M ≤ 3?
• RQ2 Does the order of shocks with different Mach numbers affect the final CR spectrum or its slope?
• RQ3 To what extent does adiabatic decompression between shocks weaken the flattening and amplification of the CR spectrum?
• RQ4 Can cumulative re-acceleration by multiple shocks explain the observed discrepancy between X-ray Mach number (M_X) and radio Mach number (M_radio) in radio relics?
• RQ5 How does DSA efficiency (η) and CR energy gain scale with the number of shock passages in the ICM?

Key findings

• The final CR proton spectrum after three identical shocks with M = 3 and D = 1 flattens to f(p) ∝ p⁻³, independent of individual shock compression ratios, consistent with prior theoretical expectations.
• DSA efficiency η increases from 7.1×10⁻³ (after one shock) to 0.13 (after three shocks), indicating significant CR energy gain through repeated acceleration.
• The ratio E_CR,k / E_th,k increases from 9.6×10⁻³ to 0.17 over three shocks, confirming that the test-particle assumption remains valid (η < 0.1) for shocks with M ≤ 3.
• The final CR spectrum is nearly independent of the order of shocks with different Mach numbers, and the power-law slope q_sum(p) remains nearly constant for p/m_p c ≫ 1.
• Decompression in postshock regions weakens spectrum flattening and amplification, with D = 0.5 cases producing lower CR energy than D = 1 cases due to reduced pre-shock density and momentum scaling.
• The cumulative effect of multiple shocks governs the final spectrum slope, which can differ significantly from the DSA slope q_DSA of the most recent shock, suggesting that M_radio may reflect integrated shock history rather than a single shock.