Properties of the first-order Fermi acceleration in fast magnetic reconnection driven by turbulence in collisional magnetohydrodynamical flows

Fast magnetic reconnection may occur in different astrophysical sources, producing flare-like emission and particle acceleration. Currently, this process is being studied as an efficient mechanism to accelerate particles via a first-order Fermi process. In this work we analyse the acceleration rate and the energy distribution of test particles injected in three-dimensional magnetohydrodynamical (MHD) domains with large-scale current sheets where reconnection is made fast by the presence of turbulence. We study the dependence of the particle acceleration time with the relevant parameters of the embedded turbulence, i.e., the Alfv\'en speed $V_{\rm A}$, the injection power $P_{\rm inj}$ and scale $k_{\rm inj}$ ($k_{\rm inj} = 1/l_{\rm inj}$). We find that the acceleration time follows a power-law dependence with the particle kinetic energy: $t_{acc} \propto E^{\alpha}$, with $0.2 < \alpha < 0.6$ for a vast range of values of $c / V_{\rm A} \sim 20 - 1000$. The acceleration time decreases with the Alfv\'en speed (and therefore with the reconnection velocity) as expected, having an approximate dependence $t_{acc} \propto (V_{\rm A} / c)^{-\kappa}$, with $\kappa \sim 2.1- 2.4$ for particles reaching kinetic energies between $1 - 100 \, m_p c^2$, respectively. Furthermore, we find that the acceleration time is only weakly dependent on the $P_{\rm inj}$ and $l_{\rm inj}$ parameters of the turbulence. The particle spectrum develops a high-energy tail which can be fitted by a hard power-law already in the early times of the acceleration, in consistency with the results of kinetic studies of particle acceleration by magnetic reconnection in collisionless plasmas.