Probing Proton versus Electron Heating and Energization during Magnetic Reconnection

TL;DR

This paper addresses how magnetic reconnection partitions energy between electrons and protons, focusing on why protons often dominate heating and energization observed in space plasmas. The authors use the kglobal macro-scale model, with varying mass ratios and upstream temperatures, to isolate the dependencies of energy gain on initial conditions and reconnection-driven Fermi reflection. They find that the first entry into the reconnection exhaust yields proton energy gains on the order of $m_i C_A^2$, while electrons gain $\sim (\beta_{e0} m_e/m_i)^{1/2} m_i C_A^2$, and that later energy growth continues via Fermi reflection, producing extended power-law tails with protons carrying more energy. The results help explain magnetotail observations and the energetic content in solar flares, while acknowledging model limitations such as collisionless assumptions, missing particle escape, and the potential effects of multi-scale dynamics.

Abstract

The mechanisms controlling the relative heating and energization of electrons and protons during magnetic reconnection are explored. Simulations are carried out with the kglobal model, which produces bulk heating and the extended powerlaw distributions of both species that have been documented in observations. The simulations have been carried out with a range of proton-to-electron mass ratios and upstream temperatures to isolate the factors that control energy gain. The simulations reveal that when the upstream temperatures of the two species are equal, the proton heating and energization exceeds that of electrons and that this is a consequence of the much larger energy gain of protons on their first entry into the reconnection exhaust. The effective energy gain of protons on exhaust entry scales as $m_iC_A^2$ since the protons counterstream at the Alfvén speed $C_A$ while the initial electron energy gain is smaller by the factor $(β_{e0}m_e/m_i)^{1/2}$. Since Fermi reflection during flux rope merger dominates energy gain in large-scale reconnecting systems and the rate of energy gain is proportional to energy, protons continue to gain energy faster than electrons for the duration of the simulations, leading to temperature increments of protons exceeding that of electrons and the non-thermal energy content of protons also exceeding that of electrons.

Figures (7)

• Figure 1: The temporal evolution of particle electron (lsft) and proton (right) temperatures in the $x\text{--}y$ plane for $B_g/B_0 = 0.25$. Six time steps are shown: $t/\tau_A = 0, 2, 3, 7, 13,$ and $21$. In each panel, magnetic field lines are overlaid in black.
• Figure 2: Time evolution of the energy spectra of electrons (dashed lines) and protons (solid lines) during the simulation with equal initial temperatures. The initial spectrum （proton/electron）is shown as a black solid line. Subsequent spectra are plotted at $t/\tau_A = 2$, 3, 6, and 18. A black dashed line line with a slope of -3.6 matches the slope of the late-time proton spectrum. The late time powerlaw index of the electrons is around -3.2.
• Figure 3: Temperature evolution and late-time profile for a simulation with $T_{e(\mathrm{initial})} = T_{i(\mathrm{initial})}$ and $m_i/m_e = 25$. Panels (a) and (b) show the space–time diagrams of the parallel electron and proton temperatures, respectively, along the center of the current sheet. The horizontal axis represents position $X/L_0$, and the vertical axis shows time normalized to the Alfvén time $\tau_A$. The color scale indicates normalized temperature, with redder regions corresponding to higher temperatures. Panel (c) presents a late-time 1D cut through the center of the current sheet, showing parallel electron temperature $T_e$ (orange) and proton temperature $T_i$ (blue). The red dashed line marks the initial temperature for both species, $T_0 = 0.0625\,m_i C_{A0}^2$.
• Figure 4: Early phase evolution of particle temperatures during magnetic reconnection for the simulation in Figure \ref{['fig:evo']}. In (a) and (b) the space–time evolution of the parallel electron and proton temperatures, respectively, along the center of the current sheet. Panel (c) shows cuts of the parallel temperature of protons (blue) and electrons (orange) along the center of the current sheet at $t = 2.5\,\tau_A$. The red dashed line marks the initial temperature.
• Figure 5: Parallel electron temperature profiles at the current sheet at $t = 2.2\,\tau_A$ for three different mass ratios: $m_i/m_e = 25$ (orange), $m_i/m_e = 100$ (green), and $m_i/m_e = 400$ (blue), with the ion mass $m_i$ held constant in all simulations. The red dashed line denotes the initial electron temperature, $T_{e0} = 0.0625\, m_i C_{A0}^2$. The horizontal axis is the spatial coordinate $x$, normalized to the characteristic length scale $L_0$. Results show that heavier electrons (i.e., lower mass ratios) gain more energy than lower mass electrons during their initial injection into reconnection exhausts, consistent with Fermi reflection scaling where energy gain is proportional to square-root of the electron mass (see Eq. (\ref{['eqn:fermi_electron']})).