The spectrum of Cosmic Rays escaping from relativistic shocks

We derive expressions for the time integrated spectrum of Cosmic Rays (CRs) that are accelerated in a decelerating relativistic shock wave and escape ahead of the shock. It is assumed that at any given time the CRs have a power law form, carry a constant fraction of the energy E_tot of the shocked plasma, and escape continuously at the maximal energy attainable. The spectrum of escaping particles is highly sensitive to the instantaneous spectral index due to the fact that the minimal energy, E_min ~ \Gamma^2 m_pc^2 where \Gamma is the shock Lorentz factor, changes with time. In particular, the escaping spectrum may be considerably harder than the canonical N(E)\propto E^-2 spectrum. For a shock expanding into a plasma of density n, a spectral break is expected at the maximal energy attainable at the transition to non relativistic velocities, E ~ 10^19 (\epsilon_B/0.1)(n/1 cm^-3)^(1/6)(E_tot/10^51 erg)^(1/3) eV where \epsilon_B is the fraction of the energy flux carried by the magnetic field. If ultra-high energy CRs are generated in decelerating relativistic blast waves arising from the explosion of stellar mass objects, their generation spectrum may therefore be different than the canonical N(E)\propto E^-2.

💡 Research Summary

The paper presents an analytical treatment of the time‑integrated spectrum of cosmic rays (CRs) that escape ahead of a decelerating relativistic shock. The authors adopt three central assumptions: (i) at any instant the CR distribution behind the shock follows a power‑law N(E,t)=K(t)E^{-p}, where p is the instantaneous spectral index; (ii) CRs carry a fixed fraction η of the shocked plasma’s total energy E_sh(t), so that ∫E N(E,t)dE = η E_sh(t); and (iii) particles escape continuously as soon as they reach the instantaneous maximum attainable energy E_max(t).

A key physical insight is that the minimum CR energy is set by the shock Lorentz factor Γ as E_min≈Γ² m_p c². During the relativistic phase Γ drops rapidly (from thousands to a few), causing E_min to decline sharply with time. Because the escaping flux is dominated by particles at E_max, the cumulative escaped spectrum N_esc(E) differs from the instantaneous downstream spectrum. By integrating over the shock evolution the authors derive N_esc(E)∝E^{-p_esc}, where the effective index p_esc depends on both the instantaneous index p and the time‑dependence of Γ. In particular, if p<2, the reduction of Γ² makes p_esc<2, yielding a spectrum that is considerably harder than the canonical N(E)∝E^{-2}. This “minimum‑energy‑driven hardening” is a distinctive feature of relativistic shocks and does not appear in non‑relativistic cases where E_min is essentially constant.

The analysis also identifies a spectral break associated with the transition to non‑relativistic velocities (Γ≈1). At this point the maximum energy attainable by the shock drops, and the authors estimate the break energy as

E_break ≈ 10¹⁹ eV · (ε_B/0.1) · (n/1 cm⁻³)^{1/6} · (E_tot/10⁵¹ erg)^{1/3},

where ε_B is the fraction of the shock’s energy flux carried by magnetic fields, n is the ambient density, and E_tot is the total explosion energy. Below E_break the escaped spectrum reverts to the familiar E^{-2} form, while above it the spectrum remains harder (p_esc<2).

The authors discuss the implications for ultra‑high‑energy cosmic ray (UHECR) sources. If stellar‑mass explosions (e.g., gamma‑ray bursts or supernovae with relativistic jets) generate relativistic blast waves that decelerate in the interstellar medium, the escaping CR spectrum could differ substantially from the standard expectation. The break energy depends sensitively on ε_B and n, suggesting that variations in magnetic‑field amplification efficiency and ambient density will imprint observable features on the UHECR spectrum.

Finally, the paper acknowledges limitations: the model assumes a simple power‑law downstream distribution, neglects non‑linear feedback between CRs and the shock, and treats magnetic field amplification in a parametrized way. Future work is proposed to (1) test the analytic predictions with kinetic or magnetohydrodynamic simulations, (2) compare the predicted escaped spectra with actual UHECR observations, and (3) explore a broader parameter space of ambient conditions and explosion energies. By highlighting how the evolving minimum energy in a relativistic shock can harden the escaped CR spectrum, the study provides a fresh perspective on the origin of the highest‑energy particles in the universe.