Power-Law Injection Spectrum from an Ensemble of Reconnecting Current Sheets

A collisionless, low-$\beta$ plasma undergoing strong sub-Alfvénic turbulence is expected to form a broad hierarchy of reconnecting current sheets immersed in a strong guide magnetic field. If the reconnecting field strength of a sheet scales with its length as a power law of index $\kappa$, and the number of sheets per unit interval of sheet length decreases as a power law of index $x$, the current-sheet ensemble injects nonthermal particles with a power-law energy spectrum $\propto {\gamma }^{-s}$, where $s=1+(x-2)\mathrm{/}(1+\kappa )$. The spectrum is hard ($s<2$) when the longest sheets deliver most of the injected energy, which requires $x<3+\kappa$. Past numerical studies have measured $2\lesssim x\lesssim 3.3$, and the scalings of intense current sheets support $\kappa \gtrsim 0$, resulting in $1\lesssim s\lesssim 2.3$. This injection may yield flat or mildly steep radio spectra in optically thin sources and hard ion spectra in the coronae of active galactic nuclei.