Scaling of Particle Heating in Shocks and Magnetic Reconnection

Abstract Particles are heated efficiently through energy conversion processes, such as shocks and magnetic reconnection, in collisionless plasma environments. While empirical scaling laws for the temperature increase have been obtained, the precise mechanism of energy partition between ions and electrons remains unclear. Here we show, based on coupled theoretical and observational scaling analyses, that the temperature increase, Δ T , depends linearly on three factors: the available magnetic energy per particle, the Alfvén Mach number (or reconnection rate), and the characteristic spatial scale L . Based on statistical data sets obtained from Earth’s plasma environment, we find that L is on the order of: (1) the ion gyroradius for ion heating at shocks; (2) the ion inertial length for ion heating in magnetic reconnection; and (3) the hybrid inertial length for electron heating in both shocks and magnetic reconnection. With these scales, we derive the ion-to-electron ratios of temperature increases as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo mathvariant="normal">Δ</mml:mo> <mml:msub> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">i</mml:mi> </mml:mrow> </mml:msub> <mml:mo>/</mml:mo> <mml:mo mathvariant="normal">Δ</mml:mo> <mml:msub> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:msub> <mml:mrow> <mml:mi>β</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">i</mml:mi> </mml:mrow> </mml:msub> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">i</mml:mi> </mml:mrow> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> </mml:math> for shocks and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo mathvariant="normal">Δ</mml:mo> <mml:msub> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">i</mml:mi> </mml:mrow> </mml:msub> <mml:mo>/</mml:mo> <mml:mo mathvariant="normal">Δ</mml:mo> <mml:msub> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">i</mml:mi> </mml:mrow> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> </mml:math> for magnetic reconnection, where β i is the ion plasma beta and m i and m e are the ion and electron particle masses, respectively. We anticipate that this study will serve as a starting point for a better understanding of particle heating in space plasmas, enabling more sophisticated modeling of its scaling and universality.