Appendix: A The particle spectrum produced in decelerating ejections

It is straight forward to calculate the particle distribution produced by single pass shock acceleration in a decelerating ejection in the ultra-relativistic limit, assuming that one has knowledge of the single pass shock acceleration kernel at a given shock velocity $\Gamma_{\rm rel}$.

Take an ejection of initial mass M₀ and Lorentz factor $\Gamma_0$, which is sweeping up and shocking external matter. The total energy of the ejection and the swept up matter is

$$E \sim \Gamma c^2 \left(M_0 + \int_0^N {\rm d}N' \Gamma(N') m_{\rm p} \right)$$ (A.1)

where the integration is over the number N of swept up particles with mass $m_{\rm p}$ at a given ejection Lorentz factor $\Gamma$, and the extra factor of $\Gamma$ inside the integral takes account of the shock acceleration of the swept up particles.

Since E is conserved, we can take the derivative of Eq. (A.1) with respect to $\Gamma$, and arrive at

$$\frac{{\rm d}N}{\rm d\Gamma} \sim \frac{\left(\int_0^N {\rm d... ...ight)}{\Gamma^2 c^2} \sim \frac{E}{m_{\rm p} c^2} \Gamma^{-3}$$ (A.2)

for $1 \ll \Gamma \leq \Gamma_0$. For $\Gamma \sim 1$, the non-relativistic corrections reduce the amount of energy available, leading to a low energy cutoff at $\Gamma \sim 1$ and a high energy cutoff at $\Gamma_0$.

To arrive at the observed particle distribution ${\rm d}N/{\rm d}\gamma$, this must be convolved with the single shock acceleration kernel $\gamma(\Gamma)$, however, for a narrow kernel, such as assumed in this paper, the powerlaw approximation seems sufficient: ${\rm d}N/{\rm d}\gamma \propto \gamma^{-3}$.