Spectral curvature and breaks from Fermi acceleration at oblique shocks

Abstract

A major attraction of diffusive shock acceleration is the prediction of power-law spectra for energetic particle distributions. However, this property is not fundamental to the theory. We demonstrate that for planar shocks with an oblique magnetic field the generation of power-law spectra critically requires the particles' scattering rate to be both directly proportional to their gyro radius (Bohm scaling) and spatially uniform. Non-Bohm scaling results in curved spectra at oblique shocks, while abrupt changes in the spatial profile of the scattering upstream introduces spectral breaks. Using the publicly available code Sapphire++, we numerically explore the magnitude of these effects, which are particularly pronounced at fast shocks, as expected in active galactic nuclei and microquasar jets, or young supernova remnants.

Figures (6)

• Figure 1: Particle spectrum at the shock and far downstream, accelerated at a shock with magnetic obliquity $\theta_n = 60^\circ$ and $u_{\rm sh} = 0.1 c$. The scattering rate follows Kraichnan scaling, i.e. $\nu/\omega_{\rm g}= \sqrt{p/p_{\rm Bohm}}$. The scattering rate meets the Bohm limit, i.e. $\nu = \omega_{\rm g}$, at $p_{\rm Bohm} = 10^4 mc$.
• Figure 2: Variation of the spectral index ($q = -{\partial \ln f}/{\partial \ln p}$) at $x=0$ ($q_0$) and far downstream, at $x=1.9\times10^4 r_{\rm g,0}$ ($q_\infty$), for the same simulation as in Fig. \ref{['fig:1']}. For comparison with Bohm scaling simulations (Blue crosses), we use $\nu/\omega_{\rm g} = \sqrt{p/p_{\rm Bohm}}$ for the $x$-axis. The theoretical curve for $q_{\infty}$ line is calculated using the simulation result for $q_0$ with eq. \ref{['eq:q_infty']}.
• Figure 3: A comparison of a solution for a shock with $u_{\rm sh} = 0.1$ and $\theta_n=85^\circ$ in the presence of a strong scattering precursor of thickness $20 r_{\rm g,0}$ upstream of the shock. In both cases $\nu_{\rm up} = 0.03 \omega_{\rm g,up}$ far upstream, and $\nu_{\rm pre} = 0.3 \omega_{\rm g,up}$ in the precursor. For the $\nu$ fixed curve, $\nu$ takes the same value in the downstream ($\nu_{\rm down} = \nu_{\rm pre}$), while for $\eta$ fixed, $\nu$ jumps by a factor $\nu_{\rm down} = (\omega_{\rm g,down}/\omega_{\rm g,up}) \nu_{\rm pre}$.
• Figure 4: Magnitude of the spectral index $q$ at the shock for different shock obliquities, using the best fit to $f_{0}(0)\propto p^{-q}$. Here $\cos \theta = |B_x|/B$ is the angle between the shock normal and the magnetic field. $\nu =0.1 \omega_{\rm g}$.
• Figure 5: Spatial profiles of particles with $p=20 mc$ at a shock with $u_{\rm sh}= 0.1 c$ and Bohm-scattering rate ${\nu}/{\omega_{\rm g}}=0.1$ (Top) Plot of $f_{0}$ for different angles of shock obliquity. (Bottom) Spatial profiles of coefficients $f_{lms}$ for $\theta = 60^\circ$. Shock profile is also shown for scale. All curves are normalized with respect to the value of $f_0$ at $x=0$.