Electron and Proton Heating in Transrelativistic Magnetic Reconnection

To illustrate the time evolution of the reconnection layer, we show in Figure 2 a few snapshots of density from a representative simulation (A[0] in Table 1) with ${\beta }_{{\rm{i}}}=0.0078$ and ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=0.1$. We plot the 2D profile of the number density in units of the initial value, $n/{n}_{0}$. In each panel, we show only a small fraction of the domain in the y direction (we present only the region closest to the current sheet) and the full extent of the domain in x. White lines with arrows show magnetic field lines.

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Panels (a)–(c) show the time evolution of the system over ∼1 Alfvénic crossing time. By removing by hand the plasma pressure at the center of the current sheet ($x\sim 0$), we trigger a local collapse of the layer, forming an X-point. After the formation of the central X-point, two reconnection "wavefronts" are pulled outward in the $\pm \hat{{\boldsymbol{x}}}$ directions by the magnetic tension of the field lines, and the fronts recede from the center at close to the Alfvén speed. In panels (a)–(c), the wavefronts are located at $x\approx \pm 400,1100,$ and $1800c/{\omega }_{\mathrm{pe}},$ respectively, corresponding to the innermost (i.e., closest to x = 0) locations of the large semicircular red/yellow density blobs.

The fronts carry away the hot particles initialized in the current sheet. With periodic boundary conditions, this leads to the formation of a primary island at the boundary of the simulation domain (in Figure 2(c), located at $x\approx \pm 2200c/{\omega }_{\mathrm{pe}}).$ The primary island continues to accrete plasma as the system evolves, but eventually it grows so large that further accretion of magnetic flux into the layer is inhibited and reconnection stops.

The primary island shows the hottest electron temperatures. Here, electron heating might be due in part to reconnection, but also in part to weak shocks at the interface between the reconnection outflow and the island. In addition, the plasma conditions in the island are sensitive to our arbitrary choice for the current sheet initialization. For these reasons, we choose not to focus on the heating physics in the primary island.

In this paper, we focus exclusively on the outflow (i.e., before the the plasma reaches the primary island; see also Shay et al. 2014, in the context of nonrelativistic reconnection), shown by the green region between the two wavefronts in Figure 2. In Section 3.2, we detail the steps we take to avoid contamination of our temperature measurements by the primary island.

As the two reconnection fronts recede from the center, plasma flows into the reconnection layer and particles are heated and accelerated as a bulk, flowing along $\pm \hat{{\boldsymbol{x}}}$ toward the domain boundaries. The dense (green) region in between the two wavefronts is the reconnection outflow. A key feature of low-${\beta }_{{\rm{i}}}$ simulations is the formation in the reconnection exhausts of secondary islands due to the secondary tearing instability, e.g., Figure 2(c) at $x\approx 300\,c/{\omega }_{\mathrm{pe}}$ and $x\,\approx -900\,c/{\omega }_{\mathrm{pe}}$ (Daughton & Karimabadi 2007; Uzdensky et al. 2010). Between each pair of secondary islands, there is a secondary X-point, e.g., at $x\approx -1000c/{\omega }_{\mathrm{pe}}$. We discuss the structure of the reconnection layer as a function of ${\beta }_{{\rm{i}}}$ in Section 4.1.

We now describe how we determine which computational cells in the simulation domain belong to the upstream (or inflow) and downstream (or outflow) regions. We identify downstream cells by using a particle mixing criterion between the two sides of the current sheet. Particles that originate above y = 0 (top of the domain) are tagged, to distinguish them from particles originating below y = 0 (bottom of the domain).

In Figure 3, we show the ratio of top to total number density. Away from the current sheet, i.e., in the blue and red regions, there is no mixing between the two populations. Particles from the two sides of the current sheet get mixed as they enter the reconnection layer; the region with the greatest amount of mixing is shown in white/light yellow. We compute the ratio of top particle density ${n}_{\mathrm{top}}$ to total particle density ${n}_{\mathrm{tot}}=n$ (including particles from both top and bottom) in each cell. If this ratio in a given cell exceeds a chosen threshold ${r}_{\mathrm{down}}$ and is below the complementary threshold, i.e.,

$$\begin{eqnarray}&&{r}_{\mathrm{down}}\lt \displaystyle \frac{{n}_{\mathrm{top}}}{{n}_{\mathrm{tot}}}\lt 1-{r}_{\mathrm{down}},\end{eqnarray} \tag{ 7 }$$

then the cell is counted as one where plasma has reconnected (i.e., the cell belongs to the reconnection downstream). This technique is similar to that used in Daughton et al. (2014). In our analysis, we choose ${r}_{\mathrm{down}}=0.3,$ but we have verified that the identification of the reconnection region, and therefore the heating efficiencies that we extract, does not significantly depend on this choice. For ${r}_{\mathrm{down}}$ in the range 0.1–0.3, the heating efficiencies typically differ only by ∼15%. The choice ${r}_{\mathrm{down}}=0.3$ is restrictive enough to exclude contamination by the upstream region. This is especially important for high ${\beta }_{{\rm{i}}}$, where, even if the electron gyrocenter is located in a cell that is safely part of the downstream, if the cell is close to the interface between downstream and upstream, the particle gyro-motion may temporarily lead this "downstream" electron to the upstream side. If ${r}_{\mathrm{down}}$ were to be too small, the region where the electron motion extends into the upstream might be incorrectly counted as part of the downstream, biasing our temperature estimates toward lower values. Our choice of ${r}_{\mathrm{down}}$ is to some extent arbitrary, but we have found that a relatively large value like ${r}_{\mathrm{down}}=0.3$ is suitable for identifying the genuine reconnection downstream.

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In Figure 4, we show 1D plots of the density fraction of tagged particles and the temperature profiles along the y direction, in a slice located at $x\approx 1000c/{\omega }_{\mathrm{pe}}.$ In panel (a), we show the profiles of the ratio of top and bottom density to total density, denoted by solid and dashed lines, respectively, at time $t\approx 8400\,{\omega }_{\mathrm{pe}}^{-1}\approx 0.60\,{t}_{{\rm{A}}}$. Between the two vertical dotted lines, the ratio of top to total density ranges between 0.3 and 0.7, as required to satisfy our mixing criterion. As shown in panel (b), both the electron (blue) and the proton (red) temperature in the region between the vertical lines are remarkably uniform, proving that our mixing criterion can confidently capture the reconnection downstream.

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The upstream region is identified via

$$\begin{eqnarray}&&\left(\displaystyle \frac{{n}_{\mathrm{top}}}{{n}_{\mathrm{tot}}}\lt {r}_{\mathrm{up}}\right)\ \mathrm{or}\ \left(\displaystyle \frac{{n}_{\mathrm{top}}}{{n}_{\mathrm{tot}}}\gt 1-{r}_{\mathrm{up}}\right),\end{eqnarray} \tag{ 8 }$$

and we choose ${r}_{\mathrm{up}}=3\times {10}^{-5}$. As before, this definition avoids contamination of the upstream region by any "downstream" particles that leak out of the current sheet. In practice, a buffer zone with a width on the order of a few tens of $c/{\omega }_{\mathrm{pe}}$ is established between the regions we identify as upstream and downstream.

While Equation (7) (Equation (8)) identifies the whole reconnection outflow (inflow), we enforce an additional constraint on the downstream and upstream regions that we employ to extract our heating efficiencies. We select downstream regions far enough from the central X-point that the electron and proton outflow bulk velocities have saturated, and also that the electron and proton temperatures have reached their asymptotic values. At the same time, we select these regions to be far enough from the boundaries to avoid contamination from the material inside the primary island, and only capture the genuine reconnection outflow. The downstream region that satisfies these constraints (identified by the green contours in Figure 3) varies for different simulations: for ${\beta }_{{\rm{i}}}\lt 2$ it is located at a distance of $\sim 630\,c/{\omega }_{\mathrm{pe}}$ from the center, whereas for ${\beta }_{{\rm{i}}}=2$ it is at $\sim 350\,c/{\omega }_{\mathrm{pe}}$ from the center (as we show below, the primary island tends to be larger at higher ${\beta }_{{\rm{i}}}$). The extent of the downstream region across the layer (i.e., along y) is determined by the mixing criterion in Equation (7), while the length along the layer is fixed at $\sim 170\,c/{\omega }_{\mathrm{pe}}$ (see the green contours in Figure 3). The corresponding upstream values are measured at the same distance from the center of the layer, within the black contours in Figure 3. Their extent along the y direction does not significantly affect our results.

In this section, we describe our assessment of particle heating. First, in Section 3.3.1, we describe our calculation of rest-frame internal energy and temperature. Next, in Section 3.3.2, we define ratios that characterize the total amount of heating. Finally, in Section 3.3.3, we provide a more detailed analysis of the heating physics by isolating the effect of a genuine entropy increase (which we call "irreversible heating") from the contribution of adiabatic compression (giving "adiabatic heating").

3.3.1. Temperature Calculation

We measure the total particle energy density in the simulation frame and then extract the corresponding fluid-frame internal energy and temperature, by employing the perfect, isotropic fluid approximation, i.e.,

$$\begin{eqnarray}&&{T}^{\mu \nu }=(e+p){U}^{\mu }{U}^{\nu }-{{pg}}^{\mu \nu },\end{eqnarray} \tag{ 9 }$$

where ${T}^{\mu \nu }$ is the stress–energy tensor of the fluid, $e$ is the rest-frame energy density, $p$ is the pressure, ${U}^{\mu }$ is the fluid dimensionless four-velocity, and ${g}^{\mu \nu }$ is the flat-space Minkowski metric. The rest-frame energy density is the sum of rest-mass and internal energy densities, i.e.,

$$\begin{eqnarray}&&e=\overline{n}{{mc}}^{2}+u\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&=\,\overline{n}{{mc}}^{2}+\displaystyle \frac{p}{\hat{\gamma }-1},\end{eqnarray} \tag{ 11 }$$

where $\overline{n}$ is the rest-frame particle number density, $u$ is the internal energy density, and $\hat{\gamma }$ is the adiabatic index. The dimensionless internal energy per particle in the fluid rest frame ${\upsilon }_{s}$ may be expressed as

$$\begin{eqnarray}&&{\upsilon }_{s}=\displaystyle \frac{({T}_{s}^{00}/{n}_{s}{m}_{s}{c}^{2}-{{\rm{\Gamma }}}_{s}){{\rm{\Gamma }}}_{s}}{1+{\hat{\gamma }}_{s}({{\rm{\Gamma }}}_{s}^{2}-1)},\end{eqnarray} \tag{ 12 }$$

where T⁰⁰_s is the total energy density in the simulation frame, n_s is the lab-frame particle number density, ${{\rm{\Gamma }}}_{s}$ is the Lorentz factor corresponding to the local fluid velocity, ${\hat{\gamma }}_{s}$ is the adiabatic index, and the subscript $s={\rm{e}},{\rm{i}}$ refers to the particle species.

To make use of Equation (12), we need to express the adiabatic index ${\hat{\gamma }}_{s}$ as a function of the internal energy per particle, so that the equation may be solved iteratively. For a plasma described by a Maxwell–Jüttner distribution with dimensionless temperature ${\theta }_{s}$,

$$\begin{eqnarray}&&{f}_{\mathrm{MJ}}(\gamma ,{\theta }_{s})\propto \gamma \sqrt{{\gamma }^{2}-1}\exp (-\gamma /{\theta }_{s}),\end{eqnarray} \tag{ 13 }$$

where γ denotes the particle Lorentz factor, the dimensionless internal energy is given by

$$\begin{eqnarray}&&{\upsilon }_{s}=\displaystyle \frac{{\int }_{1}^{\infty }\gamma {f}_{\mathrm{MJ}}(\gamma ,{\theta }_{s})d\gamma }{{\int }_{1}^{\infty }{f}_{\mathrm{MJ}}(\gamma ,{\theta }_{s})d\gamma }-1.\end{eqnarray} \tag{ 14 }$$

We have numerically evaluated the integral on the right-hand side for a range of temperatures and thereby produced interpolating tables for ${\hat{\gamma }}_{s}({\upsilon }_{s})$ and ${\theta }_{s}({\upsilon }_{s})$, to be used for finding ${\upsilon }_{s}$ in Equation (12).

Equations (9) and (12) assume that the stress–energy tensor is diagonal and isotropic in the fluid frame. We have explicitly tested this assumption by measuring all the components of the stress–energy tensor in our computational domain. By boosting into the local fluid frame, we can calculate all the components of the pressure tensor. We find that the off-diagonal components are generally negligible. With regard to the diagonal components, we quantify the degree of anisotropy with the temperature ratios ${T}_{{xx}}/{T}_{\mathrm{tot}},{T}_{{yy}}/{T}_{\mathrm{tot}},$ and ${T}_{{zz}}/{T}_{\mathrm{tot}},$ where ${T}_{\mathrm{tot}}=({T}_{{xx}}+{T}_{{yy}}+{T}_{{zz}})/3$. For an isotropic fluid, ${T}_{{xx}}/{T}_{\mathrm{tot}}\,={T}_{{yy}}/{T}_{\mathrm{tot}}={T}_{{zz}}/{T}_{\mathrm{tot}}=1$. For electrons in the reconnection downstream, we find that these ratios typically lie in the range ${T}_{{yy}}/{T}_{\mathrm{tot}}\approx {T}_{{zz}}/{T}_{\mathrm{tot}}\approx 0.9$–0.95 and ${T}_{{xx}}/{T}_{\mathrm{tot}}\approx 1.2$–1.1 (see Appendix E for further discussion, including the dependence of the anisotropy on ${\beta }_{{\rm{i}}}$ and ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}$). We find greater anisotropy along the outflow direction $\hat{{\boldsymbol{x}}}$ than either $\hat{{\boldsymbol{y}}}$ or $\hat{{\boldsymbol{z}}}$. This is in qualitative agreement with the findings of Shay et al. (2014), who demonstrated that the electron pressure tensor in the immediate reconnection exhausts is anisotropic, with the component parallel to the local magnetic field larger than the perpendicular component.

As an additional test, we have also measured the temperature and internal energy via an explicit boost of the stress–energy tensor into the fluid rest frame, and we compared the results to those computed by employing the perfect-fluid approximation as described above. We find that the disagreement between the two methods is only of order $\sim 1 \% ,$ providing a posteriori justification for the perfect-fluid assumption.

3.3.2. Total Heating

The main focus of our investigation is particle heating by reconnection, and how the heating efficiency depends on the upstream parameters. From each simulation, we extract a dimensionless ratio ${M}_{u{\rm{e}},\mathrm{tot}},$ which we define as

$$\begin{eqnarray}&&{M}_{u{\rm{e}},\mathrm{tot}}\equiv \displaystyle \frac{{\upsilon }_{{\rm{e}},\mathrm{down}}-{\upsilon }_{{\rm{e}},\mathrm{up}}}{{\sigma }_{{\rm{i}}}{m}_{{\rm{i}}}/{m}_{{\rm{e}}}}.\end{eqnarray} \tag{ 15 }$$

The numerator is the difference in dimensionless internal energy per electron between downstream and upstream, while the denominator represents (apart from a factor of two) the available magnetic energy per electron in the upstream, in units of the electron rest-mass energy ($={B}_{0}^{2}/4\pi {n}_{0}{m}_{{\rm{e}}}{c}^{2}$). The ratio ${M}_{u{\rm{e}},\mathrm{tot}}$ is then a measure of the efficiency of reconnection in converting available magnetic energy to electron heating. Alternatively, the efficiency parameter may be phrased in terms of the dimensionless temperature,

$$\begin{eqnarray}&&{M}_{T{\rm{e}},\mathrm{tot}}\equiv \displaystyle \frac{{\theta }_{{\rm{e}},\mathrm{down}}-{\theta }_{{\rm{e}},\mathrm{up}}}{{\sigma }_{{\rm{i}}}{m}_{{\rm{i}}}/{m}_{{\rm{e}}}},\end{eqnarray} \tag{ 16 }$$

as in Shay et al. (2014). We define analogous ratios for protons as

$$\begin{eqnarray}&&{M}_{u{\rm{i}},\mathrm{tot}}\equiv \displaystyle \frac{{\upsilon }_{{\rm{i}},\mathrm{down}}-{\upsilon }_{{\rm{i}},\mathrm{up}}}{{\sigma }_{{\rm{i}}}}\end{eqnarray} \tag{ 17 }$$

and

$$\begin{eqnarray}&&{M}_{T{\rm{i}},\mathrm{tot}}\equiv \displaystyle \frac{{\theta }_{{\rm{i}},\mathrm{down}}-{\theta }_{{\rm{i}},\mathrm{up}}}{{\sigma }_{{\rm{i}}}}.\end{eqnarray} \tag{ 18 }$$

For the results presented below, we average the dimensionless internal energy and temperature appearing in the above equations over time, starting at ∼0.3 Alfvénic crossing times (or equivalently, $\sim 4500\ {\omega }_{\mathrm{pe}}^{-1}$), when the two reconnection wavefronts—and with them, the particles initialized in the current sheet—have moved beyond the region that we use for our computations (green and black boxes in Figure 3). We typically time-average our results over an interval of ∼0.3 Alfvénic crossing times.

3.3.3. Adiabatic and Irreversible Heating

When gas is adiabatically compressed, its internal energy increases while its entropy remains constant. The reconnecting plasma may experience such adiabatic heating, since the downstream region is denser than the upstream (see Figure 2). However, adiabatic heating is not a genuine signature of the conversion of field energy into particle energy. We isolate the irreversible heating generated by magnetic field dissipation by subtracting out the adiabatic heating from the total particle heating.

The predicted internal energy per particle in the downstream resulting from adiabatic compression alone (which we shall call ${\upsilon }_{s,\mathrm{down}}^{\mathrm{ad}}$ for species s) is calculated from the upstream internal energy per particle ${\upsilon }_{s,\mathrm{up}}$, the upstream rest-frame number density ${\bar{n}}_{s,\mathrm{up}}$, and the downstream rest-frame number density ${\bar{n}}_{s,\mathrm{down}}$ using the second law of thermodynamics for constant entropy,

$$\begin{eqnarray}&&{{dU}}_{s}=-{p}_{s}{dV}.\end{eqnarray} \tag{ 19 }$$

From the ideal gas equation of state, the pressure is ${p}_{s}={\bar{n}}_{s}{k}_{{\rm{B}}}{T}_{s}=({\hat{\gamma }}_{s}-1){u}_{s}$. Using the relation ${U}_{s}/V={u}_{s}\,={\upsilon }_{s}{\bar{n}}_{s}{m}_{s}{c}^{2}$, we can integrate Equation (19) to obtain

$$\begin{eqnarray}&&{\int }_{{\upsilon }_{s,\mathrm{up}}}^{{\upsilon }_{s,\mathrm{down}}^{\mathrm{ad}}}\displaystyle \frac{1}{(\hat{\gamma }({\upsilon }_{s})-1){\upsilon }_{s}}d{\upsilon }_{s}-\mathrm{log}\left(\displaystyle \frac{{\bar{n}}_{s,\mathrm{down}}}{{\bar{n}}_{s,\mathrm{up}}}\right)=0.\end{eqnarray} \tag{ 20 }$$

We compute the argument of the logarithm in Equation (20) as the ratio of downstream to upstream rest-frame density, spatially averaged over the regions marked in Figure 3. The lower bound of the integral ${\upsilon }_{s,\mathrm{up}}$ is computed as a density-weighted spatial average in the selected upstream region. The adiabatic index ${\hat{\gamma }}_{s}({\upsilon }_{s})$ is tabulated as discussed above. We numerically solve Equation (20) for the predicted downstream dimensionless internal energy per particle ${\upsilon }_{s,\mathrm{down}}^{\mathrm{ad}}$ resulting from adiabatic compression. We refer to the corresponding dimensionless temperature as ${\theta }_{s,\mathrm{down}}^{\mathrm{ad}}$. We call the difference between the initial and the predicted dimensionless temperature or internal energy per particle due to adiabatic compression, i.e., ${\rm{\Delta }}{\theta }_{s,\mathrm{ad}}\equiv {\theta }_{s,\mathrm{down}}^{\mathrm{ad}}-{\theta }_{s,\mathrm{up}}$ and ${\rm{\Delta }}{\upsilon }_{s,\mathrm{ad}}={\upsilon }_{s,\mathrm{down}}^{\mathrm{ad}}-{\upsilon }_{s,\mathrm{up}}$, the "adiabatic" component of heating.

The irreversible heating, which is associated with a genuine increase in entropy, is the residual between the total heating and the adiabatic heating:

$$\begin{eqnarray}&&{\rm{\Delta }}{\theta }_{s,\mathrm{irr}}=({\theta }_{s,\mathrm{down}}-{\theta }_{s,\mathrm{up}})-{\rm{\Delta }}{\theta }_{s,\mathrm{ad}},\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{\upsilon }_{s,\mathrm{irr}}=({\upsilon }_{s,\mathrm{down}}-{\upsilon }_{s,\mathrm{up}})-{\rm{\Delta }}{\upsilon }_{s,\mathrm{ad}}.\end{eqnarray} \tag{ 22 }$$

As in Section 3.3.2, we introduce efficiency ratios to characterize the irreversible and adiabatic heating of electrons,

$$\begin{eqnarray}&&{M}_{T{\rm{e}},\mathrm{irr}}\equiv \displaystyle \frac{{\rm{\Delta }}{\theta }_{{\rm{e}},\mathrm{irr}}}{{\sigma }_{{\rm{i}}}{m}_{{\rm{i}}}/{m}_{{\rm{e}}}},\qquad {M}_{T{\rm{e}},\mathrm{ad}}\equiv \displaystyle \frac{{\rm{\Delta }}{\theta }_{{\rm{e}},\mathrm{ad}}}{{\sigma }_{{\rm{i}}}{m}_{{\rm{i}}}/{m}_{{\rm{e}}}},\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{M}_{u{\rm{e}},\mathrm{irr}}\equiv \displaystyle \frac{{\rm{\Delta }}{\upsilon }_{{\rm{e}},\mathrm{irr}}}{{\sigma }_{{\rm{i}}}{m}_{{\rm{i}}}/{m}_{{\rm{e}}}},\,\qquad {M}_{u{\rm{e}},\mathrm{ad}}\equiv \displaystyle \frac{{\rm{\Delta }}{\upsilon }_{{\rm{e}},\mathrm{ad}}}{{\sigma }_{{\rm{i}}}{m}_{{\rm{i}}}/{m}_{{\rm{e}}}},\end{eqnarray} \tag{ 24 }$$

and define analogous ratios for protons,

$$\begin{eqnarray}&&{M}_{T{\rm{i}},\mathrm{irr}}\equiv \displaystyle \frac{{\rm{\Delta }}{\theta }_{{\rm{i}},\mathrm{irr}}}{{\sigma }_{{\rm{i}}}},\qquad {M}_{T{\rm{i}},\mathrm{ad}}\equiv \displaystyle \frac{{\rm{\Delta }}{\theta }_{{\rm{i}},\mathrm{ad}}}{{\sigma }_{{\rm{i}}}},\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{M}_{u{\rm{i}},\mathrm{irr}}\equiv \displaystyle \frac{{\rm{\Delta }}{\upsilon }_{{\rm{i}},\mathrm{irr}}}{{\sigma }_{{\rm{i}}}},\qquad {M}_{u{\rm{i}},\mathrm{ad}}\equiv \displaystyle \frac{{\rm{\Delta }}{\upsilon }_{{\rm{i}},\mathrm{ad}}}{{\sigma }_{{\rm{i}}}}.\end{eqnarray} \tag{ 26 }$$

The physics of reconnection shows a marked difference between low- and high-${\beta }_{{\rm{i}}}$ regimes. In Figures 5 and 6, we present various fluid quantities for representative low- and high-${\beta }_{{\rm{i}}}$ simulations, respectively (${\beta }_{{\rm{i}}}=0.0078$ in Figure 5 and ${\beta }_{{\rm{i}}}=2$ in Figure 6). In both cases, ${\sigma }_{w}=0.1$, ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=0.1$, and ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=25$. At $t=11250\,{\omega }_{\mathrm{pe}}^{-1}\approx 0.75\,{t}_{{\rm{A}}}$, we show 2D plots of (a) the total density in the simulation frame in units of the initial density, $n/{n}_{0};$ (b) the dimensionless electron temperature, ${\theta }_{{\rm{e}}};$ (c) the magnetic energy fraction, ${\varepsilon }_{B}\,={B}^{2}/8\pi {n}_{0}{m}_{{\rm{i}}}{c}^{2};$ (d) the inflow velocity, ${v}_{\mathrm{in}}/{v}_{{\rm{A}}}={\boldsymbol{v}}\cdot \hat{{\boldsymbol{y}}}/{v}_{{\rm{A}}}$ (${v}_{{\rm{A}}}$ is the Alfvén speed); and (e) the outflow velocity, ${v}_{\mathrm{out}}/{v}_{{\rm{A}}}\,={\boldsymbol{v}}\cdot \hat{{\boldsymbol{x}}}/{v}_{{\rm{A}}}$.

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A striking difference between the simulations shown in Figures 5 and 6 is that, while the reconnection outflow at high ${\beta }_{{\rm{i}}}$ is nearly homogeneous, a number of secondary magnetic islands appear at low ${\beta }_{{\rm{i}}}$ (see Figure 5(a)). The secondary islands are overdense, and at their center they can reach temperatures a few times larger than the bulk of the outflow (Figure 5(b)). They also correspond to peaks in magnetic energy (Figure 5(c)).

The difference in electron temperature between inflow and outflow regions is more pronounced in the low- than in the high-${\beta }_{{\rm{i}}}$ case (compare Figures 5(b) and 6(b)). However, as we demonstrate in Section 4.3, the fraction of available magnetic energy converted into total electron heating is roughly comparable between the two cases.

The inflow velocity ${v}_{\mathrm{in}}/{v}_{{\rm{A}}}={\boldsymbol{v}}\cdot \hat{{\boldsymbol{y}}}/{v}_{{\rm{A}}}$ is shown in panel (d). For low ${\beta }_{{\rm{i}}}$, the inflow velocity is $| {v}_{\mathrm{in}}| /{v}_{{\rm{A}}}\approx 0.08.$ It is nearly uniform in the upstream, with the exception of the regions ahead of the secondary islands, where the velocity reverses its sign relative to the ambient inflow (see, e.g., Figure 5(d) at $x\approx -1100c/{\omega }_{\mathrm{pe}}).$ This reversal occurs as the secondary island moves along the outflow direction, pushing aside the inflowing plasma. For high ${\beta }_{{\rm{i}}},$ the plasma inflow is remarkably uniform, with $| {v}_{\mathrm{in}}| /{v}_{{\rm{A}}}\approx 0.04,$ which is half the value of the low-${\beta }_{{\rm{i}}}$ case. The inflow velocity at high ${\beta }_{{\rm{i}}}$ shows no reversals near the reconnection exhausts, as there are no secondary islands.

The outflow velocity ${v}_{\mathrm{out}}/{v}_{{\rm{A}}}={\boldsymbol{v}}\cdot \hat{{\boldsymbol{x}}}/{v}_{{\rm{A}}}$ is shown in panel (e). For low ${\beta }_{{\rm{i}}}$ the outflow speed nearly reaches the Alfvén limit, $| {v}_{\mathrm{out}}| /{v}_{{\rm{A}}}\approx 1$, whereas for high ${\beta }_{{\rm{i}}}$ it approaches a smaller value, $| {v}_{\mathrm{out}}| /{v}_{{\rm{A}}}\approx 0.6.$ For both low and high ${\beta }_{{\rm{i}}},$ the outflow velocity is nearly uniform in the reconnection exhausts, but it drops close to the periodic boundaries at $x\approx \pm 2200,$ as the outflowing plasma accretes onto the primary island.

We show in Figure 7 a direct comparison between one low-${\beta }_{{\rm{i}}}$ and one high-${\beta }_{{\rm{i}}}$ simulation. The left column in Figure 7 refers to ${\beta }_{{\rm{i}}}=0.0078$ (the same as in Figure 5), whereas ${\beta }_{{\rm{i}}}=0.5$ for the right column. In both cases, ${\sigma }_{w}=0.1$, ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=0.1$, and ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=25$. In the top row, we show the profile along x of the outflow velocity, for protons (red) and electrons (blue). We find that electrons move slightly faster than protons in the vicinity of the central X-point, but at larger distances the speeds of the two species are the same, and they saturate at a fixed fraction of the Alfvén limit. We show in the middle row of panels the x-profile of the dimensionless electron temperature ${\theta }_{{\rm{e}}}$, in the upstream (magenta) and downstream (green). The secondary magnetic islands present in the low-${\beta }_{{\rm{i}}}$ simulation (panel (c)) are correlated with spikes in the downstream electron temperature (see the peak at $x\,\approx -500\,c/{\omega }_{\mathrm{pe}}$ in Figure 7(b)). Aside from the temperature spikes at low ${\beta }_{{\rm{i}}}$, the two panels in the middle row of Figure 7 demonstrate that, far enough from the central X-point, the electron temperature is nearly uniform.

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To estimate the reconnection heating efficiency, we measure the downstream temperature in the two slabs delimited by the vertical dotted lines in Figures 7(b) and (e) (more precisely, within the green contours in Figure 3). The time evolutions of the total electron heating efficiency ${M}_{T{\rm{e}},\mathrm{tot}}$, of the adiabatic contribution ${M}_{T{\rm{e}},\mathrm{ad}}$, and of the irreversible component ${M}_{T{\rm{e}},\mathrm{irr}}$ are shown in Figure 8 with black, dashed blue, and dashed red lines, respectively. The top panel refers to a low-${\beta }_{{\rm{i}}}$ simulation with ${\beta }_{{\rm{i}}}=0.0078$, whereas the bottom panel refers to the high-${\beta }_{{\rm{i}}}$ case ${\beta }_{{\rm{i}}}=2$. In both cases, ${\sigma }_{w}=0.1$, ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=0.1$, and ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=25$. The horizontal axis in the figure starts from $t=5000\,{\omega }_{\mathrm{pe}}^{-1}$, which corresponds to the time when the two reconnection wavefronts pass beyond the region that we employ for calculating the downstream quantities (as discussed above, after this time the measurements are no longer affected by our choice of initialization of the current sheet).³ While the heating efficiencies are nearly constant in time for high ${\beta }_{{\rm{i}}}$ (bottom panel), the temporal profiles at low ${\beta }_{{\rm{i}}}$ (top panel) present quasi-periodic modulations. They mark the passage of secondary islands—whose temperature is typically hotter than the bulk outflow—through the region used for our computations. To minimize the temperature variations associated with secondary islands, we average the heating efficiencies over time, as described in Section 3.3.2. In doing so, the results we obtain are a reliable assessment of the steady-state heating physics in reconnection.

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Panels (a) and (b) in Figure 8 also demonstrate that the fractional contributions of adiabatic and irreversible heating to the total electron heating significantly depend on ${\beta }_{{\rm{i}}}$, as we further discuss in Section 4.3. In the low-${\beta }_{{\rm{i}}}$ regime, adiabatic heating is unimportant as compared to the irreversible part, whereas the two components are comparable at high ${\beta }_{{\rm{i}}}$.

In Figure 9, we show the dependence on ${\beta }_{{\rm{i}}}$ and ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}$ of various fluid quantities, from a suite of simulations with fixed ${\sigma }_{w}=0.1$ and ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=25$. We present (a) the inflow velocity normalized to the Alfvén speed, $| {v}_{\mathrm{in}}| /{v}_{{\rm{A}}};$ (b) the outflow velocity normalized to the Alfvén speed, $| {v}_{\mathrm{out}}| /{v}_{{\rm{A}}};$ (c) the ratio of inflow to outflow velocity, $| {v}_{\mathrm{in}}| /| {v}_{\mathrm{out}}| ;$ (d) the downstream rest-frame density in units of the initial density in the upstream, ${\bar{n}}_{\mathrm{down}}/{n}_{0};$ and (e) the width of the reconnection region at a distance of $430\,c/{\omega }_{\mathrm{pe}}$ from the center of the layer. Blue, green, and red points denote simulations with upstream temperature ratios ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=0.1,0.3,$ and $1,$ respectively.

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As described in Section 3.3.2, the quantities we extract are time-averaged, typically over 0.3 Alfvénic crossing times, corresponding to $\sim 4500\,{\omega }_{\mathrm{pe}}^{-1}.$ The points in Figure 9 represent the time averages, with vertical error bars indicating one standard deviation. At low ${\beta }_{{\rm{i}}},$ the inflow velocity is $| {v}_{\mathrm{in}}| /{v}_{{\rm{A}}}\approx 0.08,$ independent of the upstream temperature ratio (panel (a)). In the high-${\beta }_{{\rm{i}}}$ case, the inflow speed is smaller, $| {v}_{\mathrm{in}}| /{v}_{{\rm{A}}}\approx 0.04$, and shows a weak dependence on the temperature ratio, with higher temperature ratios attaining lower values of $| {v}_{\mathrm{in}}| /{v}_{{\rm{A}}}$.

The outflow velocity (panel (b)) nearly saturates the Alfvén limit at low ${\beta }_{{\rm{i}}}$ (the Alfvén limit is indicated with the horizontal dashed black line), whereas for high ${\beta }_{{\rm{i}}}$ it is sub-Alfvénic, $| {v}_{\mathrm{out}}| /{v}_{{\rm{A}}}\approx 0.75$. For low values of ${\beta }_{{\rm{i}}}$, i.e., ${\beta }_{{\rm{i}}}\lesssim 0.1,$ the outflow velocity is nearly independent of the temperature ratio, whereas at high ${\beta }_{{\rm{i}}}$ it shows a weak dependence on ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}$, with higher temperature ratios corresponding to greater outflow speeds.

The dependence of the reconnection rate $| {v}_{\mathrm{in}}| /| {v}_{\mathrm{out}}|$ on ${\beta }_{{\rm{i}}}$ and ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}$(panel (c)) follows from the variations in inflow speed and outflow velocity that we have just discussed. At low ${\beta }_{{\rm{i}}},$ the reconnection rate is $| {v}_{\mathrm{in}}| /| {v}_{\mathrm{out}}| \approx 0.08$ regardless of the temperature ratio, whereas at high ${\beta }_{{\rm{i}}}$, and specifically for ${\beta }_{{\rm{i}}}=2,$ the reconnection rate at ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=1$ is $| {v}_{\mathrm{in}}| /| {v}_{\mathrm{out}}| \approx 0.04,$ only half that of the ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=0.1$ case. So, in the high-${\beta }_{{\rm{i}}}$ regime reconnection proceeds slower for hotter upstream electrons.

As ${\beta }_{{\rm{i}}}$ increases, the plasma is less prone to be compressed during the reconnection process. As shown in Figure 9(d), the downstream-to-upstream density ratio decreases as ${\beta }_{{\rm{i}}}$ increases. The value of ${\overline{n}}_{\mathrm{down}}/{n}_{0}$ is nearly independent of the upstream temperature ratio. Though the ratio ${\overline{n}}_{\mathrm{down}}/{n}_{0}$ approaches unity at high ${\beta }_{{\rm{i}}},$ this does not necessarily imply that the fractional contribution of adiabatic heating to total heating is negligible at high ${\beta }_{{\rm{i}}}$ (we demonstrate this in Section 4.3).

Lastly, in panel (e) we show the ${\beta }_{{\rm{i}}}$ dependence of the reconnection layer width ${\delta }_{\mathrm{rec}},$ in units of the electron skin depth $c/{\omega }_{\mathrm{pe}}$. We measure the width across the reconnection layer, as identified by Equation (7), at a distance $\sim 430\,c/{\omega }_{\mathrm{pe}}$ downstream of the central X-point. The width shows strong variability in time at low ${\beta }_{{\rm{i}}}$, as secondary islands pass through the region employed for our measurements (note the large error bars). Despite the uncertainty in the measurement, panel (e) shows a consistent trend of increasing reconnection layer width ${\delta }_{\mathrm{rec}}$ with ${\beta }_{{\rm{i}}}$. The measured values of ${\delta }_{\mathrm{rec}}$ lie in the range 25—$50\,c/{\omega }_{\mathrm{pe}},$ which is apparently close to the chosen current sheet thickness at initialization, ${\rm{\Delta }}=40\,c/{\omega }_{\mathrm{pe}}.$ However, we demonstrate in Appendix F that the measured reconnection layer width is independent of our choice of the initial sheet thickness. It follows that our measurement leads to a reliable assessment of the opening angle of the reconnection outflow, $\sim {\delta }_{\mathrm{rec}}/(430\,c/{\omega }_{\mathrm{pe}})\sim 0.1$.

In Figure 10, we show the ${\beta }_{{\rm{i}}}$ and ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}$ dependence of electron (panel (a)) and proton (panel (b)) dimensionless temperature and the ratio of proton-to-electron skin depth (panel (c); see Equation (5)). In each panel, solid and dashed lines indicate downstream and upstream quantities, respectively. As in Figure 9, blue, green, and red points refer to electron-to-proton temperature ratios ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=0.1,0.3$, and 1, respectively. The upstream electron dimensionless temperatures lie in the range 10⁻³ to 10, as in Table 1; for protons, the dimensionless temperature in the upstream ranges from 4 × 10⁻⁴ to 0.4.

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The range of temperatures in the downstream is smaller than in the upstream (compare the solid and dashed lines in Figures 10(a) and (b)). At low ${\beta }_{{\rm{i}}}$, the available magnetic energy is large compared to the particle thermal content in the upstream, so dissipation of the magnetic field leads to electron and proton temperatures in the downstream that are nearly independent of ${\beta }_{{\rm{i}}}$. At high ${\beta }_{{\rm{i}}}$, the energy transferred from the fields to the particles is much smaller than the initial particle thermal content, giving a minor increase of temperature from upstream to downstream. Even if the fractional increase in temperature is extremely small at high ${\beta }_{{\rm{i}}}$, the fraction of available magnetic energy being converted into particle heating might still be as large as at low ${\beta }_{{\rm{i}}}$. The rest of the section addresses this question.

We show the plasma-${\beta }_{{\rm{i}}}$ and temperature ratio ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}$ dependence of electron and proton heating in Figure 11. The simulations presented here are those referenced in Table 1, for which ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=25$ and ${\sigma }_{w}=0.1$. We indicate the total, adiabatic, and irreversible heating by ${M}_{T{\rm{e}},\mathrm{tot}}$ (Equation (16)), ${M}_{T{\rm{e}},\mathrm{ad}}$, and ${M}_{T{\rm{e}},\mathrm{irr}}$ (Equations (23)) for electrons and by ${M}_{T{\rm{i}},\mathrm{tot}}$ (Equation (18)), ${M}_{T{\rm{i}},\mathrm{ad}}$, and ${M}_{T{\rm{i}},\mathrm{irr}}$ (Equations (25)) for protons. Blue, green, and red points indicate simulations with upstream electron-to-proton temperature ratios of $0.1,0.3,$ and 1, respectively. As in Section 4.2, filled points are the time-averaged results of our simulations, and vertical error bars indicate one standard deviation from the mean. The top, middle, and bottom rows show heating fractions of electrons, protons, and the overall fluid, respectively, which we now discuss in turn.

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In Figure 11(a), we show the dependence of the total electron heating efficiency ${M}_{T{\rm{e}},\mathrm{tot}}$ on ${\beta }_{{\rm{i}}}$ and ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}$. Although the initial plasma ${\beta }_{{\rm{i}}}$ spans more than two orders of magnitude, and the initial temperature ratio an order of magnitude, even the most extreme values of ${M}_{T{\rm{e}},\mathrm{tot}}$ differ by no more than a factor of ∼4. The value of ${M}_{T{\rm{e}},\mathrm{tot}}$ in our ${\beta }_{{\rm{i}}}\lesssim 0.5$ simulations, for which electrons stay nonrelativistic both in the upstream and in the downstream, is $\sim 0.04,$ which is consistent with the results of nonrelativistic reconnection by Shay et al. (2014) for mass ratio ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=25$.⁴ As shown by Shay et al. (2014), the electron heating efficiency in nonrelativistic reconnection is expected to decrease with increasing mass ratio; in Sections 4.4 and 4.6, we present the dependence of the electron and proton heating fractions in transrelativistic reconnection on ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}$, up to the physical value.

The total electron heating fraction ${M}_{T{\rm{e}},\mathrm{tot}}$ is decomposed into adiabatic and irreversible components in panels (b) and (c). By comparing the two panels, we see that most of the heating in the low-${\beta }_{{\rm{i}}}$ regime comes from irreversible processes, i.e., it is accompanied by a genuine increase in entropy, while heating at high ${\beta }_{{\rm{i}}}$ mostly results from adiabatic compression.

The electron adiabatic heating efficiency increases with the inflow temperature ratio ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}$ (Figure 11(b)). The dependence is most apparent at high ${\beta }_{{\rm{i}}},$ where adiabatic heating represents a significant contribution to the total heating. The dependence of adiabatic heating on temperature ratio can be simply understood through the adiabatic law $T/{\overline{n}}^{\hat{\gamma }-1}=\mathrm{const}.$ As electrons get compressed from upstream to downstream, the adiabatic heating fraction can be written as⁵

$$\begin{eqnarray}&&{M}_{T{\rm{e}},\mathrm{ad}}=\displaystyle \frac{1}{2}{\beta }_{{\rm{i}}}\displaystyle \frac{{T}_{{\rm{e}}}}{{T}_{{\rm{i}}}}\left[{\left(\displaystyle \frac{{\overline{n}}_{\mathrm{down}}}{{n}_{0}}\right)}^{{\hat{\gamma }}_{{\rm{e}}}-1}-1\right].\end{eqnarray} \tag{ 27 }$$

As shown in Figure 9(d), the ratio of downstream to initial upstream density ${\overline{n}}_{\mathrm{down}}/{n}_{0}$ is nearly independent of the upstream temperature ratio, so that ${M}_{T{\rm{e}},\mathrm{ad}}\propto {T}_{{\rm{e}}}/{T}_{{\rm{i}}}$ at fixed ${\beta }_{{\rm{i}}}$. Equation (27) also provides insight into the ${\beta }_{{\rm{i}}}$ dependence of adiabatic heating. It shows that, for a given temperature ratio, the adiabatic heating efficiency would scale linearly with ${\beta }_{{\rm{i}}}$, if the compression ratio ${\overline{n}}_{\mathrm{down}}/{n}_{0}$ were to be fixed. As shown in Figure 9(d), the downstream-to-upstream density ratio decreases with ${\beta }_{{\rm{i}}}$, approaching unity at high ${\beta }_{{\rm{i}}}$. However, the decrease of ${\overline{n}}_{\mathrm{down}}/{n}_{0}$ with ${\beta }_{{\rm{i}}}$ is quite shallow, and insufficient to counteract the linear dependence on ${\beta }_{{\rm{i}}}$ in Equation (27). It follows that at low ${\beta }_{{\rm{i}}}$ the effect of adiabatic heating is negligible, while at high ${\beta }_{{\rm{i}}}$ the role of adiabatic heating can be more important than that of irreversible heating.

This statement can be further justified by considering electron energization in the diffusion region as the main source of irreversible electron heating, following Le et al. (2016). In the diffusion region, the electron energy will increase by ${{eE}}_{\mathrm{rec}}{{\ell }}_{{\rm{e}}}$, where ${E}_{\mathrm{rec}}\sim 0.1({v}_{{\rm{A}}}/c){B}_{0}$ is the reconnection electric field (assuming a reconnection inflow rate of $\sim 0.1\,{v}_{{\rm{A}}}/c$; see Figure 9(a)) and ${{\ell }}_{{\rm{e}}}$ is the distance traveled by electrons along the electric field (along z, in our geometry). For the sake of simplicity, let us now assume that ${\beta }_{{\rm{i}}}$ is sufficiently small that $w\sim {n}_{0}{m}_{{\rm{i}}}{c}^{2}$ and so ${\sigma }_{w}\sim {\sigma }_{{\rm{i}}}$ (this is the case for ${\beta }_{{\rm{i}}}\lesssim 0.1$, at our reference magnetization ${\sigma }_{w}=0.1$). The corresponding irreversible heating efficiency can be written in the case ${\sigma }_{w}\sim {\sigma }_{{\rm{i}}}\lesssim 1$ as

$$\begin{eqnarray}&&{M}_{T{\rm{e}},\mathrm{irr}}\sim 0.1\displaystyle \frac{{{\ell }}_{{\rm{e}}}}{c/{\omega }_{\mathrm{pi}}},\end{eqnarray} \tag{ 28 }$$

which does not depend explicitly on ${\beta }_{{\rm{i}}}$. It follows that, as long as ${{\ell }}_{{\rm{e}}}$ is a weak function of ${\beta }_{{\rm{i}}}$, the adiabatic heating efficiency in Equation (27), which scales linearly with ${\beta }_{{\rm{i}}}$, will be unimportant at low ${\beta }_{{\rm{i}}}$, whereas it will dominate over irreversible heating at high ${\beta }_{{\rm{i}}}$.

We remark that Equation (28) does not capture a number of important effects. First, by tracking individual particle orbits, we have found that the in-plane components of the electric field, which we have neglected above, can provide a significant contribution to the total electron energization (a comprehensive discussion of the physics of electron heating will be presented elsewhere). Second, we have neglected the ${\beta }_{{\rm{i}}}$ dependence of the reconnection rate. Third, we have assumed $w\sim {n}_{0}{m}_{{\rm{i}}}{c}^{2}$, which is incorrect at high ${\beta }_{{\rm{i}}}$. Fourth, we do not have a direct measure of ${{\ell }}_{{\rm{e}}}$, which would assess its dependence on the flow conditions. For these reasons, it is likely that the electron irreversible heating will be dependent on ${\beta }_{{\rm{i}}}$.

In fact, the irreversible electron heating efficiency (shown in Figure 11(c)) systematically decreases with ${\beta }_{{\rm{i}}}$, and the trend is largely independent of the initial temperature ratio, apart from the case with ${\beta }_{{\rm{i}}}=2$ (rightmost points in Figure 11(c)). Here, the irreversible heating fraction reaches ${M}_{T{\rm{e}},\mathrm{irr}}\approx 0.03$ for ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=1$, a factor of ∼3 larger than for the ${\beta }_{{\rm{i}}}=2$ cases with lower temperature ratios, ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=0.1$ and 0.3.⁶ We attribute the peculiar behavior of this case to the fact that, among the ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=25$ simulations presented in Figure 11, the ${\beta }_{{\rm{i}}}=2,{T}_{{\rm{e}}}/{T}_{{\rm{i}}}=1$ case is the only one for which the scale separation $(c/{\omega }_{\mathrm{pi}})/(c/{\omega }_{\mathrm{pe}})$ between protons and electrons approaches unity (see Figure 10(c)). For the case ${\beta }_{{\rm{i}}}=2,{T}_{{\rm{e}}}/{T}_{{\rm{i}}}=1$ in Figure 11, this statement holds true for both the upstream and the downstream scale separation, since the reconnection process at high ${\beta }_{{\rm{i}}}$ does not appreciably change the plasma thermal content. However, as we further discuss in the next two subsections, where we investigate the dependence of our results on the mass ratio and the magnetization, we find that the necessary and sufficient condition for the electron and proton heating efficiencies to be comparable is that the downstream scale separation approaches unity. In retrospect, this is not surprising, since if $(c/{\omega }_{\mathrm{pi}})/(c/{\omega }_{\mathrm{pe}})\to 1$ in the downstream, the fluid effectively behaves like an electron-positron plasma.

In Figure 11 (second row of panels), we also explore the ${\beta }_{{\rm{i}}}$ dependence of (d) total, (e) adiabatic, and (f) irreversible proton heating. As before, blue, green, and red points correspond to simulations with upstream ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}$ of $0.1,0.3,$ and $1,$ respectively (we change the temperature ratio by varying the electron temperature, while the proton temperature at a given ${\beta }_{{\rm{i}}}$ is kept fixed). While the initial dimensionless electron temperature in our simulations ranges from nonrelativistic to ultrarelativistic values, protons always stay at nonrelativistic or transrelativistic energies, ${\theta }_{{\rm{i}}}\approx 0.0004$–0.5 (this is true in both upstream and downstream cases). At low ${\beta }_{{\rm{i}}}$, protons are heated more efficiently than electrons, typically by a factor of 2–3 at mass ratio ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=25$ (compare panels (a) and (d), ${M}_{T{\rm{e}},\mathrm{tot}}\approx 0.05$ while ${M}_{T{\rm{i}},\mathrm{tot}}\approx 0.13$). At larger values of ${m}_{{\rm{i}}}/{m}_{{\rm{e}}},$ the ratio of proton to electron heating is even larger, as we discuss in Sections 4.4 and 4.6. Once again, the notable exception is the high-${\beta }_{{\rm{i}}}$ case with ${\beta }_{{\rm{i}}}=2$ and ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=1,$ where the electron and proton heating fractions are comparable, ${M}_{T{\rm{e}},\mathrm{tot}}\approx 0.06$ and ${M}_{T{\rm{i}},\mathrm{tot}}\approx 0.08$. Similar to electrons, the irreversible component of proton heating decreases with ${\beta }_{{\rm{i}}}$ and shows only weak dependence on the upstream temperature ratio ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}$ (panel (f)). As shown in panel (e), the fractional contribution of adiabatic heating to the total proton heating increases with ${\beta }_{{\rm{i}}}$, as for electrons.

Finally, we show the total particle (i.e., sum of electron and proton) heating, as well as the corresponding adiabatic and irreversible components, in Figures 11(g)–(i). Given that protons are heated more efficiently than electrons, the trends in the bottom row of Figure 11 are primarily controlled by protons (again, with the exception of the case ${\beta }_{{\rm{i}}}=2,$ ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=1$). Panel (g) shows that the total particle heating efficiency is $\sim 0.15$ across all simulations, with a weakly declining trend with increasing ${\beta }_{{\rm{i}}}$. Panels (h) and (i) show that, as discussed for electrons and protons individually, heating in the low-${\beta }_{{\rm{i}}}$ regime is associated with an increase in entropy, while at high ${\beta }_{{\rm{i}}}$ it is dominated by adiabatic compression.

While we cast the heating fractions in Figure 11 in terms of temperature differences between upstream and downstream, they may be expressed, alternatively, via differences in internal energy per particle; see Appendix G.

We have extended our results up to the physical mass ratio ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=1836$, and in this section we focus on the case with ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=1$ (runs with ${\sigma }_{w}=0.1$ and unequal temperature ratios are presented in Section 4.6). The separation between the electron scale $c/{\omega }_{\mathrm{pe}}$ and the proton scale $c/{\omega }_{\mathrm{pi}}$ is regulated by Equation (5). For nonrelativistic particles, the ratio of proton to electron skin depth is $\sqrt{{m}_{{\rm{i}}}/{m}_{{\rm{e}}}}\sim 40$, so that a large simulation domain is required to properly capture the proton physics. However, in the transrelativistic regime of our simulations, the particles can approach (or exceed, in the case of electrons) relativistic temperatures. Here, the effective increase in electron inertia can bring the ratio of proton to electron skin depth close to unity (see Equation (5)). This condition holds, for example, in simulations C[3], C[4], and B[4], when the mass ratio is increased to ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=1836$ at fixed ${\sigma }_{w}$ and ${\beta }_{{\rm{i}}}$.

We show in Figure 12 the dependence of (a) total, (b) adiabatic, and (c) irreversible electron heating on ${\beta }_{{\rm{i}}},$ for mass ratios ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=10,25,$ and $1836.$ We fix the magnetization ${\sigma }_{w}=0.1$ and the temperature ratio ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=1;$ the legend is shown in the upper part of panel (b). The points are colored according to the dimensionless temperature of upstream electrons (the corresponding color bar is to the right of panel (c)), ranging from nonrelativistic (${\theta }_{{\rm{e}}}\sim {10}^{-4}$) to ultrarelativistic (${\theta }_{{\rm{e}}}\sim {10}^{3}$) values. In agreement with earlier studies of nonrelativistic reconnection by Dahlin et al. (2014) and Le et al. (2016), we find that the total electron heating efficiency at low ${\beta }_{{\rm{i}}}$ is a decreasing function of mass ratio. For the realistic mass ratio, at low ${\beta }_{{\rm{i}}}$ the total heating fraction ${M}_{T{\rm{e}},\mathrm{tot}}\approx 0.016$ is in good agreement with the observed value in the magnetopause, ${M}_{T{\rm{e}},\mathrm{tot}}=0.017$ (Phan et al. 2013). At ${\beta }_{{\rm{i}}}=2,$ the electron heating efficiency is remarkably insensitive to the mass ratio, with ${M}_{T{\rm{e}},\mathrm{tot}}\approx 0.06$. As we have anticipated above, in this case the upstream and downstream skin depths of protons and electrons are comparable (once we account for the effects of relativistic inertia), so the physics should resemble that of an electron-positron plasma, regardless of the mass ratio. The adiabatic heating efficiency (panel (b)) shows only a weak dependence on mass ratio, in agreement with Equation (27). For realistic mass ratios, electron heating is governed by irreversible processes at low ${\beta }_{{\rm{i}}},$ adiabatic heating dominates at intermediate ${\beta }_{{\rm{i}}}\sim 0.1-1$, while the two components equally contribute at high ${\beta }_{{\rm{i}}}\sim 2$.

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We show the ${\beta }_{{\rm{i}}}$ dependence of the proton heating fractions ${M}_{T{\rm{i}},\mathrm{tot}},{M}_{T{\rm{i}},\mathrm{ad}},$ and ${M}_{T{\rm{i}},\mathrm{irr}}$ in panels (d)–(f). The points are colored according to the upstream dimensionless proton temperature, ${\theta }_{{\rm{i}}}$ (the scale is to the right of panel (f)). The upstream proton temperatures are nonrelativistic or transrelativistic, with ${\theta }_{{\rm{i}}}\lesssim 0.5$. At fixed ${\sigma }_{w}$ and ${\beta }_{{\rm{i}}}$, the initial proton temperature stays the same, when we vary the mass ratio (as opposed to the electron temperature, which increases with mass ratio). So, the proton heating efficiencies are expected to remain unchanged, as long as the box size L_x is sufficiently large (in units of the proton skin depth $c/{\omega }_{\mathrm{pi}}$) to capture the physics of proton heating. In the bottom row of Figure 12, the proton heating fractions ${M}_{T{\rm{i}},\mathrm{tot}},{M}_{T{\rm{i}},\mathrm{ad}},$ and ${M}_{T{\rm{i}},\mathrm{irr}}$ are nearly independent of the mass ratio, which demonstrates that even for the realistic mass ratio the box used here is sufficiently large to capture the physics of proton heating (and even more so, of electron heating). The results discussed in Section 4.3 for ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=25$ and ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=1$ are therefore still valid here: proton heating is dominated by irreversible processes at low ${\beta }_{{\rm{i}}},$ whereas irreversible and adiabatic components equally contribute at high ${\beta }_{{\rm{i}}};$ the irreversible heating efficiency of protons is a decreasing function of ${\beta }_{{\rm{i}}};$ protons are heated more efficiently than electrons (although the total proton-to-electron heating ratio for ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=1836$ is ∼7 at low ${\beta }_{{\rm{i}}}$, larger than the value measured for ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=25$, since the electron heating efficiency decreases with mass ratio); and the heating fractions of the two species approach comparable values at ${\beta }_{{\rm{i}}}=2,$ with ${M}_{T{\rm{i}},\mathrm{tot}}\approx 0.08$ and ${M}_{T{\rm{e}},\mathrm{tot}}\approx 0.06$.

In Figure 13(a), we directly compare the ${\beta }_{{\rm{i}}}$ dependence of electron and proton heating fractions ${M}_{T{\rm{e}},\mathrm{tot}}$ (solid blue), ${M}_{T{\rm{e}},\mathrm{irr}}$ (dashed blue), ${M}_{T{\rm{i}},\mathrm{tot}}$ (solid red), and ${M}_{T{\rm{i}},\mathrm{irr}}$ (dashed red) for ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=1836,{\sigma }_{w}=0.1,$ and ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=1$.⁷ As anticipated above, the proton and electron total and irreversible heating fractions differ roughly by a factor of ∼7 at low ${\beta }_{{\rm{i}}},$ but they approach a similar value at ${\beta }_{{\rm{i}}}=2$ ($\approx 0.03$ for the irreversible component and $\approx 0.06$ for the total). In Figure 13(b), we show the ${\beta }_{{\rm{i}}}$ dependence of the ratio of electron to overall total heating ratio (solid blue),

$$\begin{eqnarray}&&{q}_{T{\rm{e}},\mathrm{tot}}=\displaystyle \frac{{M}_{T{\rm{e}},\mathrm{tot}}}{{M}_{T{\rm{e}},\mathrm{tot}}+{M}_{T{\rm{i}},\mathrm{tot}}},\end{eqnarray} \tag{ 29 }$$

and similarly, the ratio of electron to overall irreversible heating ratio (dashed blue),

$$\begin{eqnarray}&&{q}_{T{\rm{e}},\mathrm{irr}}=\displaystyle \frac{{M}_{T{\rm{e}},\mathrm{irr}}}{{M}_{T{\rm{e}},\mathrm{irr}}+{M}_{T{\rm{i}},\mathrm{irr}}}.\end{eqnarray} \tag{ 30 }$$

At low ${\beta }_{{\rm{i}}},$ the electron-to-overall total heating ratio is ${q}_{T{\rm{e}},\mathrm{tot}}\approx 0.14$, and it increases with ${\beta }_{{\rm{i}}}$ up to ${q}_{T{\rm{e}},\mathrm{tot}}\approx 0.45$ at ${\beta }_{{\rm{i}}}=2.$ The corresponding ratio of the irreversible components ${q}_{T{\rm{e}},\mathrm{irr}}$ is comparable to ${q}_{T{\rm{e}},\mathrm{tot}}$ at both low ${\beta }_{{\rm{i}}}$ (where adiabatic heating is negligible) and ${\beta }_{{\rm{i}}}=2$ (where adiabatic and irreversible contributions are similar), but for intermediate ${\beta }_{{\rm{i}}}$ we find that ${q}_{T{\rm{e}},\mathrm{irr}}$ can be as low as 0.07, smaller than ${q}_{T{\rm{e}},\mathrm{tot}}$ by up to a factor of $\approx 3$.

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In the previous sections, we have focused on the case ${\sigma }_{w}=0.1;$ in Figure 14, we show the ${\beta }_{{\rm{i}}}$ dependence of the heating efficiencies for a suite of simulations with ${\sigma }_{w}=0.1,0.3,1,3,$ and 10.⁸ We fix the temperature ratio ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=1$ and the mass ratio ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=1836.$ The panels are similar to those in Figure 12: (a)–(c) show the electron heating fractions ${M}_{T{\rm{e}},\mathrm{tot}},{M}_{T{\rm{e}},\mathrm{ad}},$ and ${M}_{T{\rm{e}},\mathrm{irr}};$ (d)–(f) show the proton heating fractions ${M}_{T{\rm{i}},\mathrm{tot}},{M}_{T{\rm{i}},\mathrm{ad}},$ and ${M}_{T{\rm{i}},\mathrm{irr}}.$ The legend is in panel (b): green, purple, brown, magenta, and black curves connect the points having ${\sigma }_{w}=0.1,0.3,1,3,$ and 10, respectively, to guide the eye. The points of panels (a)–(c) are colored according to the upstream dimensionless electron temperature ${\theta }_{{\rm{e}}}$, as indicated by the color bar to the right of panel (c). Similarly, in panels (d)–(f) the points are colored according to the upstream dimensionless proton temperature ${\theta }_{{\rm{i}}}$, as indicated by the color bar to the right of panel (f). For fixed ${\beta }_{{\rm{i}}},{T}_{{\rm{e}}}/{T}_{{\rm{i}}},$ and ${m}_{{\rm{i}}}/{m}_{{\rm{e}}},$ an increase in magnetization leads to an increase in the upstream dimensionless temperature of both electrons and protons, which can be seen by comparing the colors of data points in panel (a) or panel (d) at fixed ${\beta }_{{\rm{i}}}$.

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We note that the data points in Figure 14 extend up to a maximum value of ${\beta }_{{\rm{i}}}$ that depends on ${\sigma }_{w}$. For our choice of defining the magnetization using the enthalpy density, rather than the rest-mass energy density, the ion ${\beta }_{{\rm{i}}}$ cannot exceed ${\beta }_{{\rm{i}},\max }\sim 1/4{\sigma }_{w}$. For each value of ${\sigma }_{w},$ the points with the highest value of ${\beta }_{{\rm{i}}}$ are also those for which the proton-to-electron scale separation ratio $(c/{\omega }_{\mathrm{pi}})/(c/{\omega }_{\mathrm{pe}})$ is the smallest (see Figure 15). We find that in the limit ${\beta }_{{\rm{i}}}\to {\beta }_{{\rm{i}},\max }$, the total electron heating efficiency shows a characteristic upturn (panel (a)), with a typical value ${M}_{T{\rm{e}},\mathrm{tot}}\approx 0.05$ that is nearly independent of ${\sigma }_{w}$. In the low-${\beta }_{{\rm{i}}}$ regime, the electron total heating efficiency approaches a ${\sigma }_{w}$-dependent plateau, with higher ${\sigma }_{w}$ yielding larger electron efficiencies (panel (a)). The opposite holds for protons: higher magnetizations give smaller proton heating efficiencies (panel (d)). Indeed, for ${\sigma }_{w}=10$ the electron and proton efficiencies are comparable in the whole range of ${\beta }_{{\rm{i}}}$ we have explored, in agreement with the results by Sironi et al. (2015).

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As anticipated in Section 4.3, we find that the necessary and sufficient condition for having comparable electron and proton heating efficiencies is that the separation between the electron and proton scales in the downstream be of order unity (or equivalently, that the two species be relativistically hot, with comparable temperatures). As shown in Figure 15, this can be achieved in two ways: (i) at high ${\sigma }_{w}$, regardless of ${\beta }_{{\rm{i}}}$, the reconnection process transfers so much magnetic energy to the particles that both species become relativistically hot, with comparable temperatures; (ii) at low ${\sigma }_{w}$ and in the limit ${\beta }_{{\rm{i}}}\to {\beta }_{{\rm{i}},\max }$, both electrons and protons already start relativistically hot in the upstream region (and more so, will be relativistically hot in the downstream).

Most of the ${\sigma }_{w}$ dependences that we have now presented for the total heating efficiencies ${M}_{T{\rm{e}},\mathrm{tot}}$ and ${M}_{T{\rm{i}},\mathrm{tot}}$ also apply to the irreversible components ${M}_{T{\rm{e}},\mathrm{irr}}$ and ${M}_{T{\rm{i}},\mathrm{irr}}$, since the adiabatic contribution is independent of the magnetization, at fixed ${\beta }_{{\rm{i}}}$(see Equation (27)). However, since the magnetization affects the efficiency of irreversible heating at fixed ${\beta }_{{\rm{i}}}$, while the adiabatic component remains the same, this can lead to a significant change in the relative contributions of irreversible and adiabatic heating. This can be seen, for example, at ${\beta }_{{\rm{i}}}\approx 0.5$. For ${\sigma }_{w}=0.1$, ${M}_{T{\rm{e}},\mathrm{irr}}/{M}_{T{\rm{e}},\mathrm{tot}}\approx 0.1,$ whereas at ${\sigma }_{w}=0.3$, we find ${M}_{T{\rm{e}},\mathrm{irr}}/{M}_{T{\rm{e}},\mathrm{tot}}\approx 0.5.$

To connect with the recent work of Werner et al. (2016), we show in Figure 16 the dependence of electron and proton heating on the magnetization ${\sigma }_{{\rm{i}}}$, defined with the rest-mass energy density (see Equation (2)). We fix temperature ratio ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=1$, mass ratio ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=1836,$ and ${\beta }_{{\rm{i}}}\approx 0.03$ (which is close to the upstream plasma ${\beta }_{{\rm{i}}}$ employed in Werner et al. (2016), ${\beta }_{{\rm{i}}}=0.01$). In panel (a), we show the ${\sigma }_{{\rm{i}}}$ dependence of the electron total (solid blue), electron irreversible (dashed blue), proton total (solid red), and proton irreversible (dashed red) heating fractions, phrased in terms of internal energy as in Werner et al. (2016), ${M}_{u{\rm{e}},\mathrm{tot}}$, ${M}_{u{\rm{e}},\mathrm{irr}}$, ${M}_{u{\rm{i}},\mathrm{tot}}$, and ${M}_{u{\rm{i}},\mathrm{irr}}$ (see Equations (15) and (17)). As ${\sigma }_{{\rm{i}}}$ increases, the downstream scale separation between protons and electrons gets reduced (see Figure 15), and the two species approach comparable heating efficiencies (whereas the two differ by a factor of ∼3 at low magnetization). This holds for both the total efficiencies ${M}_{u{\rm{e}},\mathrm{tot}}$ and ${M}_{u{\rm{i}},\mathrm{tot}}$ and the irreversible components ${M}_{u{\rm{e}},\mathrm{irr}}$ and ${M}_{u{\rm{i}},\mathrm{irr}}$, since the amount of adiabatic heating at fixed ${\beta }_{{\rm{i}}}$ does not depend on ${\sigma }_{w}$. This is further illustrated in Figure 16(b), where we show the ${\sigma }_{{\rm{i}}}$ dependence of the electron-to-overall total heating fraction, phrased in terms of internal energy (solid blue),

$$\begin{eqnarray}&&{q}_{u{\rm{e}},\mathrm{tot}}=\displaystyle \frac{{M}_{u{\rm{e}},\mathrm{tot}}}{{M}_{u{\rm{e}},\mathrm{tot}}+{M}_{u{\rm{i}},\mathrm{tot}}},\end{eqnarray} \tag{ 31 }$$

and the electron-to-overall irreversible heating ratio (dashed blue),

$$\begin{eqnarray}&&{q}_{u{\rm{e}},\mathrm{irr}}=\displaystyle \frac{{M}_{u{\rm{e}},\mathrm{irr}}}{{M}_{u{\rm{e}},\mathrm{irr}}+{M}_{u{\rm{i}},\mathrm{irr}}}.\end{eqnarray} \tag{ 32 }$$

Blue circles show the results of our simulations, and the black dotted line indicates the empirical formula suggested by Werner et al. (2016),

$$\begin{eqnarray}&&{q}_{u{\rm{e}},\mathrm{emp}}=\displaystyle \frac{1}{4}\left(1+\sqrt{\displaystyle \frac{{\sigma }_{{\rm{i}}}/5}{2+{\sigma }_{{\rm{i}}}/5}}\right).\end{eqnarray} \tag{ 33 }$$

We find reasonable agreement between this empirical formula and our simulations, for ${\beta }_{{\rm{i}}}\approx 0.03$. For low values of the magnetization, ${q}_{u{\rm{e}},\mathrm{tot}}\approx {q}_{u{\rm{e}},\mathrm{irr}}\approx 0.25$, but as ${\sigma }_{{\rm{i}}}$ increases toward the ultrarelativistic limit, ${q}_{u{\rm{e}},\mathrm{tot}}$ and ${q}_{u{\rm{e}},\mathrm{irr}}$ approach $\approx 0.5$, i.e., electrons and protons are heated with comparable efficiencies. However, Figure 14 shows that, at fixed magnetization, the heating efficiencies depend on ${\beta }_{{\rm{i}}}$, a trend that cannot be properly captured by the empirical formula of Werner et al. (2016).

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We then propose the following formula, which captures the dependence of the electron-to-overall heating ratio ${q}_{u{\rm{e}},\mathrm{tot}}$ on both magnetization ${\sigma }_{w}$ and proton ${\beta }_{{\rm{i}}}$:

$$\begin{eqnarray}&&{q}_{u{\rm{e}},\mathrm{fit}}=\displaystyle \frac{1}{2}\exp \left[\displaystyle \frac{-{(1-{\beta }_{{\rm{i}}}/{\beta }_{{\rm{i}},\max })}^{3.3}}{1+1.2\,{\sigma }_{w}^{0.7}}\right],\end{eqnarray} \tag{ 34 }$$

where ${\beta }_{{\rm{i}}}\leqslant {\beta }_{{\rm{i}},\max }=1/4{\sigma }_{w}.$ The formula in Equation (34) has two desirable, and physically motivated, features. First, for ${\beta }_{{\rm{i}}}\to {\beta }_{{\rm{i}},\max },$ the electron-to-overall heating ratio approaches $0.5,$ independently of the magnetization. Second, for ${\sigma }_{w}\gg 1,\ {q}_{u{\rm{e}},\mathrm{tot}}=0.5$, regardless of ${\beta }_{{\rm{i}}}$. In both these limits, the scale separation between electrons and protons in the downstream will be of order unity (as we have discussed above), which we have demonstrated is a necessary and sufficient condition for comparable heating efficiencies between electrons and protons.

In Figure 17, we compare Equation (34) to the results of simulations with ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=1836$ and ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=1$ (this is the same set of simulations presented earlier in this section, as well as in Section 4.4). In Figure 17(a), we show the ${\beta }_{{\rm{i}}}$ dependence of the electron-to-overall heating ratio ${q}_{u{\rm{e}},\mathrm{tot}}$ for a range of ${\sigma }_{w}$ (see the legend). The simulation results are shown by filled circles, while solid lines are based on Equation (34). The curves are plotted up to to the maximum allowed value of ${\beta }_{{\rm{i}}}$, namely, ${\beta }_{{\rm{i}},\max }=1/4{\sigma }_{w}$. The black dotted line at ${q}_{u{\rm{e}},\mathrm{tot}}=0.5$ shows the limit of comparable heating efficiencies for electrons and protons, which will be reached as ${\beta }_{{\rm{i}}}\to {\beta }_{{\rm{i}},\max }$, independently of ${\sigma }_{w}$. We find that both the simulation data and the fitting formula in Equation (34) asymptote to a constant value for ${\beta }_{{\rm{i}}}\ll {\beta }_{{\rm{i}},\max }$, with smaller heating ratios at lower ${\sigma }_{w}$. In the nonrelativistic limit ${\sigma }_{w}\ll 1$, our formula prescribes that ${q}_{u{\rm{e}},\mathrm{fit}}\to 0.18$, not very different from the value ${q}_{u{\rm{e}},\mathrm{fit}}\approx 0.22$ obtained for ${\sigma }_{w}=0.1$. This is consistent with the expectation that in the nonrelativistic regime ${\sigma }_{w}\ll 1$, the heating efficiencies will be independent from the magnetization.

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In Figure 17(b), we show the dependence of the electron-to-overall heating ratio on the magnetization ${\sigma }_{w}$, for a range of ${\beta }_{{\rm{i}}}$. The simulation results are shown by filled solid circles, which are colored according to the value of ${\beta }_{{\rm{i}}}$ in the upstream (the color scale is located to the right of Figure 17(b)). We select a few representative values of ${\beta }_{{\rm{i}}}$ and plot the corresponding predictions based on Equation (34) with the solid curves (see the legend in the plot). The curves are plotted up to ${\sigma }_{w,\max },$ which for a fixed ${\beta }_{{\rm{i}}}$ is given by ${\sigma }_{w,\max }\sim 1/4{\beta }_{{\rm{i}}}$. In summary, Figures 17(a) and (b) show that our proposed formula (Equation (34)) properly captures the magnetization and plasma ${\beta }_{{\rm{i}}}$ dependence of the electron-to-overall heating ratio over the whole range of ${\sigma }_{w}$ and ${\beta }_{{\rm{i}}}$ explored in this work.

In Figure 18, we present the dependence of electron and proton heating efficiencies on the proton beta ${\beta }_{{\rm{i}}}$ and the temperature ratio ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}$ for the realistic mass ratio ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}\,=1836$ (the figure layout is the same as in Figure 11, where we had employed a reduced mass ratio ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=25$). We fix ${\sigma }_{w}=0.1$. Even at the realistic mass ratio, the conclusions drawn in the reduced mass ratio case ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=25$ (see Section 4.3) still hold: electron and proton heating at low ${\beta }_{{\rm{i}}}$ is dominated by irreversible processes, while heating in the high-${\beta }_{{\rm{i}}}$ regime is mostly a result of adiabatic compression; the irreversible component of electron heating is independent of ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}$ at ${\beta }_{{\rm{i}}}\lesssim 1$ (Figure 18(c)); the proton irreversible heating shows only a weak dependence on temperature ratio (Figure 18(f)); and protons are heated more efficiently than electrons (compare the top and middle rows).

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For both electrons and protons, the adiabatic heating efficiencies for ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=1836$ (Figures 18(b) and (e)) are similar to those of the reduced mass ratio case. In fact, according to Equation (27), the adiabatic heating efficiency is independent of mass ratio.⁹ For protons, the adiabatic heating efficiency decreases at ${\beta }_{{\rm{i}}}\gtrsim 2;$ this is largely an effect of the decrease in the adiabatic index, as the protons transition from nonrelativistic to relativistic temperatures.

For ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=1836$, the irreversible heating of protons at low ${\beta }_{{\rm{i}}}$ is a factor of ∼5–7 greater than that of electrons; in the ${m}_{{\rm{i}}}/{m}_{{\rm{e}}}=25$ case, the ratio of proton-to-electron irreversible heating was smaller, ∼2–3. As in the reduced mass ratio case, the simulation with ${\beta }_{{\rm{i}}}=2$ and ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=1$ shows a sharp increase in irreversible electron heating as compared to the decreasing trend observed at lower ${\beta }_{{\rm{i}}}$(Figure 18(c)), and the heating efficiencies of the two species become comparable. As we argued in Section 4.5, the electron and proton heating efficiencies are about equal if and only if the downstream scale separation is of order unity. Even for the highest values of ${\beta }_{{\rm{i}}}$ that we can explore ($\approx 3.9$ for ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=0.1,$ and $\approx 4.6$ for ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=0.3.$), this condition is not realized for smaller temperature ratios ($(c/{\omega }_{\mathrm{pi}})/(c/{\omega }_{\mathrm{pe}})\gtrsim 3.2$ for ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=0.1,$ and $(c/{\omega }_{\mathrm{pi}})/(c/{\omega }_{\mathrm{pe}})\gtrsim 1.8$ for ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=0.3$), which explains why—despite the upturn in electron heating efficiency at high ${\beta }_{{\rm{i}}}$(Figure 18(c))—the ratio of irreversible proton to electron heating for ${T}_{{\rm{e}}}/{T}_{{\rm{i}}}=0.1$ and 0.3 remains larger than unity.