ELECTRON INJECTION BY WHISTLER WAVES IN NON-RELATIVISTIC SHOCKS

Quasi-perpendicular shocks are characterized by the presence of the so-called shock foot. The foot is defined by the existence of a beam of ions reflected by the shock, whose bulk velocity in the upstream frame is close to v_sh and has comparable x- and z-components. The foot region can be seen in panels (a) and (b) of Figure 1 (between x = 410 and 470 c/ω_{p, e}), which depicts the phase space distribution functions of the ions. This region covers a distance comparable to the ion Larmor radius, R_{L, i} (calculated with the upstream magnetic field B₀), and is located right in front of the shock density jump or "ramp," as shown in panel (h) of the same figure. An important feature of the shock foot is the presence of oblique electromagnetic waves that grow on scales of ∼10 c/ω_{p, e}, forming an angle of ∼45° with the shock normal. These waves can be seen in panels (e)–(g) of Figure 1 (between x = 410 and 470 c/ω_{p, e}), which show the magnetic field along the x-direction, B_x, the magnetic field along the z-direction, B_z, and the ion density, n_i, respectively. The oblique modes have a right-handed circular polarization, and their phase velocity is comparable to the speed of the shock. All these features allow them to be identified as electron whistler waves propagating obliquely with respect to the background magnetic field.

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The whistler waves have been studied in the context of shock structure evolution, using one-, two-, and three-dimensional PIC simulations, as well as hybrid codes. The two-dimensional studies show that whistler waves can play an important role by suppressing the self-reformation of perpendicular and quasi-perpendicular shocks (Hellinger et al. 2007; Lembege et al. 2009; Yuan et al. 2009). This effect, however, would be less important in the fully three-dimensional geometry, as shown by a recent PIC study (I. Shinohara et al. 2011, in preparation).

The exact generation mechanism of these whistlers is still subject to debate. One candidate mechanism is the so-called modified two-stream instability (MTSI), driven by the relative cross-field velocity between the electrons and the ions in the foot of the shocks (Wu et al. 1983; Matsukiyo & Scholer 2003, 2006). Since in quasi-perpendicular shocks a significant fraction of the ions are reflected into the upstream, the MTSI can be driven by the relative motion between electrons and either the incoming or reflected ions. These two possibilities are usually referred to as MTSI1 and MTSI2, respectively. The analytic dispersion relation calculations show that the MTSI1 and MTSI2 will grow if cos (θ) ≳ 4(1 − r)M_A/(m_i/m_e)^1/2 and cos (θ) ≳ 4rM_A/(m_i/m_e)^1/2, respectively, where θ is the angle between the magnetic field and the wave vector of the waves, and r is the fraction of reflected ions, with typical values of r ∼ 0.2 (Matsukiyo & Scholer 2003). Independently of which of these two varieties of the MTSI is more dominant, these conditions show that the smaller is the ratio M_A/(m_i/m_e)^1/2 the larger is the range of θ where the excitation of whistler waves in the foot of shocks would be possible.

Another possibility for whistler generation is that these waves can be an intrinsic component of oblique quasi-perpendicular shocks. Indeed, Krasnoselskikh et al. (2002) proposed an analytical model for the structure of these shocks where the shock ramp is treated as a nonlinear whistler wave. This model shows that if $M_A \lesssim M_w\equiv |\cos (\theta _{Bn})|(\sqrt{m_i/m_e})^{1/2}/2$, where θ_Bn is the angle between the shock normal and the upstream magnetic field, $\vec{B}_0$, the shock precursor would contain a stationary whistler wave train. If M_w is exceeded, these whistler waves would become nonlinear and would be rather confined to the shock ramp. But if $M_A \gtrsim M_{nw}\equiv |\cos (\theta _{Bn})|(\sqrt{m_i/m_e})^{1/2}/2^{1/2}$ a stationary solution for the shock structure would no longer be possible, which would set the condition for shock reformation. In particular for run 2D-3 (θ ≈ 30° and θ_Bn = 75°) the condition for the whistler generation due to the MTSI (in any of its two varieties) appears to be less stringent than for the model proposed by Krasnoselskikh et al. (2002). This is in general the case for the simulations presented in this paper. But, even for runs where cos (θ_Bn) ≳ cos (θ), the growth of whistler waves in the foot of quasi-perpendicular shocks appears to be favored for small values of the M_A/(m_i/m_e)^1/2 ratio.

The phase space distribution functions for electrons are depicted in panels (c) and (d) of Figure 1. Although the electrons are mainly heated at the shock, significant electron energization also occurs in the foot region. This is seen from the electron spectra shown in Figure 2, which are measured at several positions in the shock region. The positions are marked by the vertical lines in Figure 1(h), with colors matching the ones of the corresponding spectra in Figure 2. We see that the two energy distributions measured in the downstream (red and black lines) show a high-energy power-law tail with spectral index α ≃ 3.6 (for comparison a Maxwellian distribution corresponding to the downstream electron temperature is shown in black dotted line). The two spectra measured in the foot and the beginning of the ramp (blue and green lines, respectively) do not show a power-law tail, but a bump in the high-energy part of the distribution. This high-energy bump is also observed further upstream, showing that some accelerated electrons also outrun the shock as they move along the upstream magnetic field. The maximum Lorentz factor of the electrons (Γ_e ≃ 50) is consistent with the electron Larmor radii being comparable to that of the ions, i.e., Γ_e ≈ v_sh(m_i/m_e)/c. Also, two downstream ion spectra, measured at the locations of the vertical black and red lines in panel (h) of Figure 1, are depicted by dashed lines. These spectra show an incomplete ion thermalization, and essentially no ion acceleration, with only a small, exponentially decreasing bump at energies a few times above thermal. Note that the downstream electron temperature is close to the mean kinetic energy of the ions (for m_i/m_e = 100), implying energy equilibration between both species. Finally, the spectra depicted in Figure 2 are fully evolved, in the sense that they do not show significant variations in time. This will be the case in all the simulations shown in this paper after a time of ∼6ω⁻¹_{c, i}.

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In this section, we illustrate the physics of the electron acceleration process by focusing on the evolution of a typical non-thermal electron. In the three panels of Figure 3 we plot the energy of the electron, represented by the solid, black line. In order to identify the source of the energy gain, we also plot the accumulated energy gain, _j, due to the work done by the electric field along the j-axis (red, green, and blue lines represent j = x, y, and z, respectively). This energy gain is calculated as _j ≡ ∫v_{e, j}E_je/m_edt, where v_{e, j} and E_j correspond to the electron velocity and electric field along the j-axis, respectively. Since these cumulative energies are shown on the logarithmic scale, we plot their absolute values, and use dotted lines when they correspond to negative quantities. In each panel we also highlight a small time interval, whose starting and final points are marked by yellow and red circles, respectively. The particle trajectories corresponding to each of these intervals are shown on top of Figures 5 and 6 by a solid, black line. The black circle on top of the highlighted intervals marks the time of each field snapshot. The initial increase in the electron energy happens in the first highlighted interval, shown in Figure 3(a), with the particle location depicted in Figures 5(a) and 6(a). We can see that this initial energization happens in the shock foot as the particle moves through the region of amplified whistler waves. The energy gain is initially due to increase in _y, with _x contributing most of the energy gain after tω_{p, e} ≈ 1960. Although before the initial energy growth (the one dominated by _y) _x has already increased significantly, its contribution to the total energy of the electron is almost exactly compensated by a decrease in _z (see rising dotted blue and solid red lines between tω_{p, e} = 1737 and 1920 in Figure 3(a)). Indeed, the particle motion until this point was dominated by the $\vec{E}\times \vec{B}$ drift, which does not allow any net work to be done by the electric field. However, as soon as electric field fluctuations on scales comparable to the electron Larmor radius appear, this almost perfect cancellation stops, and a net energy gain becomes possible. Since the y-axis is quasi-parallel to the initial field $\vec{B}_0$, the initial energization due to E_y already suggests an important feature of the whistlers waves: their electric field has a non-negligible component along the magnetic field direction. This parallel electric field component can be explained by the obliquity of the waves with respect to the initial magnetic field, which makes the projection of whistler electric field on $\vec{B}_0$ have a magnitude ${\sim} {\rm sin} (\theta)|\vec{E}|$. The energy gain due to the parallel electric field can be verified from Figure 4, where we have plotted in red and green the energy gain due to the electric field perpendicular and parallel to $\vec{B}$ (_⊥ and _||, respectively). Indeed, we can see that _|| is what dominates the initial energy gain of the electron, confirming that the presence of the electric field parallel to the magnetic field is the essential component in the initial electron energization. This feature makes this process fundamentally different from fast Fermi acceleration where the shock acts as a magnetic mirror, energizing the electrons due to the electric field component perpendicular to the magnetic field. Finally, we note that this initial energization is described in the frame where the downstream plasma is at rest (usually called normal incidence frame). The picture, however, should be essentially the same if described from any other frame like, for instance, the de Hoffman–Teller frame. In particular, the initial energization will continue to be dominated by the electric field component parallel to the magnetic field. This is due to the relativistic invariance of $\vec{E} \cdot \vec{B}$, which would make $\vec{E} \cdot \vec{B}/|\vec{B}|$ a nearly invariant quantity under any non-relativistic frame transformation.

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After the initial energization driven by the whistlers, the acceleration due to E_z (the convective electric field) starts to play a more relevant role, as can be seen from Figure 3(b). We see that in the highlighted interval (marked by the two vertical, dashed lines) the energy increase is driven by a jump both in _z and _y. In Figures 5(b) and 6(b), we can see that during these energy increases the electron moves mainly along the +y-direction, and stays close to the shock ramp. The relevance of the energy gain due to E_z is more obvious at later times, when the electron Larmor radius has become comparable to a sizable fraction of the foot length. As shown in Figure 3(c), most of the energy is gained due to an increase in _z. The particle trajectory corresponding to the marked interval (between the two vertical, dashed lines) is plotted in Figures 5(c) and 6(c). We can see that, as it moves (mainly along the +y-direction), the particle jumps a couple of times from the upstream to the downstream and vice versa, gaining energy in a way similar to shock drift acceleration.

Apart from dominating the initial electron energization, the energy gain along the magnetic field, _||, also increases the particle's ability to move along the magnetic field lines. This is important because if an electron velocity along $\vec{B}_0$ becomes larger than v_sh/cos (θ_Bn), it will be possible for the particle to move along the +x-direction faster than the shock. When this happens, the electron will be able to escape upstream, increasing its chances to be further scattered by the whistlers waves. This qualitative picture already shows that having θ_Bn ≠ 90° is an important component in the acceleration process. This angle dependence of the acceleration is confirmed in the following section where the importance of the shock parameters: θ_Bn, m_i/m_e, v_sh, and M_A for electron acceleration is studied. Finally, it is important to point out that the evolution of thermal electrons is fundamentally different to the one of the non-thermal particles, which is what we described in this section. Thermal electrons gain some energy due to a random combination of the parallel and perpendicular electric field mainly in the ramp and shock transition region, and then are rapidly incorporated into the downstream medium.

We illustrate the importance of the angle θ_Bn for electron acceleration using Figure 7. The downstream spectrum of the electrons at tω_{p, e} = 10, 000 (tω_{c, i} = 10) is plotted for simulations that only differ in their angle θ_Bn. The rest of the parameters are the same as for simulation 2D-3 in Table 1. We can see that the hardest spectrum is reached for θ_Bn = 70° (red line), with α ≈ 3.6 while the softest spectrum is obtained for the θ_Bn = 90° case (purple line), which corresponds to a Maxwellian distribution with a small increase in the number of particles at energies a few times above thermal. This result confirms the picture suggested in Section 3.3, in which the electron energization strongly depends on the ability of these particles to propagate in front of the shock as they slide along the magnetic field lines. This condition implies that the most accelerated particles must move preferentially along the +y-axis. This is in fact what we noticed when tracking the trajectory of the accelerated electron analyzed in Section 3.3, which represents the typical behavior of the most energetic particles in our runs. For θ_Bn < 70°, the softer spectra can be explained by the less efficient confinement of the particles to the shock vicinity. A quasi-perpendicular configuration keeps the electrons close to the shock, not allowing them to easily escape upstream. From that perspective, it is reasonable to think that the most efficient acceleration will happen when cos (θ_Bn) is just large enough to allow the electrons to move at v_sh in the +x-direction, with decreasing efficiency for larger values of cos (θ_Bn).

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We now explore the ion to electron mass ratio m_i/m_e dependence of the electron acceleration. To do this, we use three runs in two dimensions with the same parameters M_A = 7, θ_Bn = 75°, and v_sh = 0.14c, but with m_i/m_e = 25, 100, and 400 (run 2D-7, 2D-8, and 2D-3 of Table 1, respectively). Their downstream spectra are shown by the solid lines in Figures 8(a)–(c). We can clearly see the role played by m_i/m_e in the electron acceleration. While for the mass ratios m_i/m_e = 25 and 100 cases, the spectral index α ≈ 5.4 and ≈4.7, respectively, the m_i/m_e  =  400 case shows the decrease of the spectral index to α ≈ 3.6.⁵ In order to show the correlation between this acceleration and the growth of the whistler modes, in Figure 9 we have plotted the electric field along $\hat{x}$, E_x, in the shock transition region of these three simulations (panels (a), (b), and (c) for m_i/m_e  =  25, 100, and 400, respectively). These plots show how the shock foot becomes dominated by large amplitude whistler waves as the mass ratio increases from m_i/m_e = 25 to 400. Also this figure confirms the length scale of the whistlers waves (∼10c/ω_{p, e}) determined in Section 3, which is consistent with previous dispersion relation calculations (Wu et al. 1983). In this study of mass ratio dependence, we also want to include the possibility of electron energization by "shock surfing" of electrons, due to electrostatic waves produced by the Buneman instability in the foot of the shocks (Amano & Hoshino 2009). These waves appear in the leading edge of the shock foot on scales of ∼c/ω_{p, e}, and are due to the relative velocity between the upstream electrons and the ions reflected by the shock (Buneman 1958). As electrons get scattered by these waves, they gain energy due to the work performed by the fluctuating electric field, combined with the convective field of the upstream plasma. Since the wave vector of the fastest growing Buneman mode, $\vec{k}_{{\rm bun}}$, is parallel to the ion beam, the growth of the waves and their effect on the electron acceleration are better resolved when the ion motion is parallel to the plane of the simulation (x–y plane). This is achieved if the magnetic field, $\vec{B}_0$, is quasi-perpendicular to the x–y plane, implying that the Buneman acceleration will be suppressed in our two-dimensional runs with the magnetic field in the simulation plane (Amano & Hoshino 2009). We used this behavior to distinguish the contribution of the "shock surfing" mechanism to electron acceleration by comparing the results of the two-dimensional simulations with analogous three-dimensional runs, where the $\vec{k}_{{\rm bun}}$ is resolved by adding a third dimension of a few c/ω_{p, e}. The corresponding downstream spectra of the three-dimensional runs with m_i/m_e = 25, 100, and 400 (runs 3D-1, 3D-2, and 3D-3 of Table 2, respectively) are depicted using dashed lines in Figures 8(a)–(c). We can see significant differences between the two- and three-dimensional spectra for m_i/m_e = 25, implying an important contribution to acceleration due to the shock surfing mechanism. Indeed, when analyzing the trajectories of the energetic particles of the three-dimensional run with m_i/m_e = 25, we find that most of their energy is gained as the electrons get scattered by Buneman waves in the leading edge of the foot. The difference between two- and three- dimensional runs, however, is substantially reduced for the case m_i/m_e = 100, and almost disappears for m_i/m_e = 400, showing that the relative importance of Buneman acceleration, compared to the energization due to whistler waves, decreases for more realistic mass ratios. Further details on this mass ratio dependence of the shock surfing mechanism are given in the Appendix, where the acceleration of electrons is studied for two-dimensional runs with $\vec{B}_0$ quasi-perpendicular to the plane of the simulation. The results presented in the Appendix also demonstrate that the acceleration due to whistler waves disappears if the direction along $\vec{B}_0$ is not resolved by the simulation, which makes the two-dimensional configuration used in the main part of this paper (i.e., with $\vec{B}_0$ in plane), the most suitable for the study the electron injection due to whistlers. Also, the convergence between the two- and three-dimensional runs depicted in Figure 8(c) shows that whistler acceleration is well modeled by our two-dimensional simulations, with no appreciable changes due to the additional third dimension of a few c/ω_{p, e}. This result, however, does not rule out significant three dimensional effects when a scale comparable to a few whistler scales (i.e., a few 10 c/ω_{p, e}) is resolved along the third dimension. We leave this possibility as a subject of future investigation.

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It is interesting to see how this electron acceleration may change if a different shock velocity is used. We test the shock velocity dependence by running simulations with the same parameters as for the two-dimensional simulations shown in Figure 8 but using v_sh = 0.042c, which corresponds to v_in = 0.03c (simulations 2D-18, 2D-19, and 2D-20 in Table 1). Note that in order to keep the same M_A = 7 the magnetic field also has to be reduced by a factor of 3.3. The resultant downstream electron spectra are shown in Figure 10 for m_i/m_e = 25, 100, and 400 (red, black, and green lines). These spectra essentially reproduce the ones corresponding to two-dimensional runs with v_sh = 0.14c, shown in Figure 8. The same spectral index α dependence on m_i/m_e is obtained, with a decrease in α when passing from m_i/m_e = 100 to 400. The overall normalization of the power-law tails, however, seems to drop by a factor of ∼3 with respect to the v_sh = 0.14c cases. This result suggests that the mechanism for electron acceleration due to growth of whistler waves has a rather weak dependence on the shock velocity, as long as M_A is kept constant. It is important to note, however, that these measured spectral indices are for the same inclination angles that produce the lowest α in the case of v_sh = 0.14c (θ_Bn = 75°). Thus, in this case we are assuming that the angle θ_Bn at which α is minimized does not depend on the shock velocity. This assumption needs to be confirmed by further exploration of the angle dependence of acceleration for different shock velocities. A thorough study aimed to clarify this point will be presented elsewhere.

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We test the effect of varying M_A by running two-dimensional simulations that are analogous to the ones shown in Figure 8, but also using the Mach numbers M_A = 3.5 and M_A = 14. We do this by changing the magnitude of the magnetic field while keeping the same shock velocity v_sh = 0.14c. We use the mass ratios m_i/m_e = 25, 100, 400, and 1600, in order to simultaneously take into account the mass ratio dependence for each of the studied Mach numbers. The parameters of the simulations are summarized in Table 1 (where for m_i/m_e = 25, 100, 400, and 1600 the M_A = 3.5 simulations are called, respectively, 2D-10, 2D-11, 2D-12, and 2D-13, the M_A = 7 simulations are called 2D-7, 2D-8, 2D-3, and 2D-9, and the M_A = 14 simulations are called 2D-14, 2D-15, 2D-16, and 2D-17).

The corresponding downstream electron spectra for M_A = 3.5, 7, and 14 are shown in panel (a), (b), and (c) of Figure 11 (with the red, black, green, and blue lines showing the spectra for m_i/m_e = 25, 100, 400, and 1600, respectively). We see that, apart from the hardening of the spectra due to the increase in the mass ratio, the spectral index α also drops as the Mach number goes from 14 to 3.5. This tendency is consistent with our finding that the electron acceleration is driven by the growth of whistler waves in the foot of quasi-perpendicular shocks. Indeed, our analysis presented in Section 3.1 showed that the growth of whistler waves requires M_A/(m_i/m_e)^1/2 ≲ 1 (Wu et al. 1983; Matsukiyo & Scholer 2003; Krasnoselskikh et al. 2002). In Figure 11, we see that when this condition is not satisfied (for instance, when M_A = 7 and m_i/m_e = 25, shown by the red line in plot (b)) we obtain a rather soft spectrum, with α ∼ 6. The acceleration efficiencies, in terms of the fraction of non-thermal particles, follow a similar trend, with the number of non-thermal particles being larger for smaller values of M_A/(m_i/m_e)^1/2. The typical fraction of non-thermal particles in our runs ranges between ∼2% and 10%, with corresponding energy fractions of ∼10%–30%.⁶

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For each Alfvénic Mach number, we explored the angle dependence of the electron acceleration by using several values for θ_Bn. Thus, the spectra presented in Figure 11 correspond to the hardest spectra for each M_A, whose angles are θ_Bn = 75° for M_A = 3.5, θ_Bn = 75° for M_A = 7, and θ_Bn = 60 for M_A = 14. However, these angles were determined using a single mass ratio m_i/m_e = 400. Thus, our results assume that the angle of the hardest spectral index depends only weakly on the used mass ratio. This assumption needs to be confirmed by further exploration of the angle dependence of this acceleration mechanism for different values of m_i/m_e. Despite this uncertainty, our results show, in general, a hardening of the electron spectra as the M_A/(m_i/m_e)^1/2 ratio decreases. This can also be seen from Figure 12, which shows a compilation of the spectral indices for the cases depicted in Figure 11 as a function of the mass ratio and Mach number. The yellow dashed line, which corresponds to M_A = (m_i/m_e)^1/2, clearly separates the regions of small α from regions of rather soft spectra. In the cases with realistic mass ratio (m_i/m_e = 1600), the spectral index goes from α = 2.7 to 4 as M_A changes from 3.5 to 14. We will discuss the consequences that this result has for electron acceleration in SNR shocks below.

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Finally, we emphasize that this acceleration mechanism does not show a significant dependence on the chosen value of β_e. This is verified by directly comparing run 2D-3 and 2D-21, which only differ in their β_e (β_e = 0.5 and 0.05, respectively) and show essentially no difference in their final electron spectra.

In this appendix, we explore the mass ratio dependence of the electron energization due to shock surfing acceleration (Amano & Hoshino 2009). This acceleration is driven by the presence of electrostatic waves, produced by the Buneman instability (Buneman 1958), in the foot of quasi-perpendicular shocks. The Buneman waves grow in the leading edge of the foot due to the relative velocity between the upstream electrons and the shock-reflected ions, and have a typical scale comparable to ∼c/ω_{p, e}. The small length scale of these waves, which can be comparable or even smaller than the electron Larmor radii, makes the Buneman waves a potentially important means for electron scattering and energization in the foot of quasi-perpendicular shocks. As the upstream electrons encounter these waves, they gain energy due to the work performed by the fluctuating electric field, combined with the convective field of the upstream plasma. The wave vector of the fastest growing mode is parallel to the velocity of the beam of returning ions, so these waves are best studied either by three-dimensional simulations or by two-dimensional runs where the magnetic field is quasi-perpendicular to the simulation plane, so that the gyrational motion of the ions is resolved on the plane.⁷

Using a two-dimensional simulation with shock parameters v_in = 0.2c (v_sh ≈ 0.3c), M_A = 14, θ_Bn = 90° (with $\vec{B}_0$ perpendicular to the simulation plane), and m_i/m_e = 25, Amano & Hoshino (2009) showed that an electron spectrum with index α ≈ 2–2.5 can be produced in the shock transition region due to this mechanism.

In this appendix we show, however, that the importance of this shock surfing acceleration decreases as the mass ratio increases to more realistic values. As an example, let us compare the downstream spectra of three simulations with the same shock parameters v_in = 0.1c (v_sh ≈ 0.14c), M_A = 7, θ_Bn = 75° (and the angle between $\vec{B}_0$ and $\hat{z}$ equal to 15°), but different mass ratios: m_i/m_e = 25, 100, and 400 (the rest of the parameters are specified in Table 3). Figure 14 shows the spectra at tω_{c, i} = 10 measured in the downstream medium. Although in all three cases a non-thermal component can be seen, the maximum energy reached by the electrons is about the same (Γ_e − 1 ≈ 1), despite the factor of 16 difference in the used mass ratios. In particular, in the m_i/m_e = 400 case, the closeness between the peak of the thermal part of the distribution and the maximum energy of the accelerated particles reduces significantly the relative fraction of non-thermal particles compared with m_i/m_e = 25 and 100 cases. Also, note that in these two-dimensional simulations the acceleration due to whistler waves gets suppressed. This is because these simulations only allow the wave vector of the whistlers to be quasi-perpendicular to the magnetic field, inhibiting their growth and their effect on the electron energization.

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In terms of the evolution of the Buneman waves, we also observe a decrease in their relative amplitude. This can be seen from Figure 15, where the electric field along the x-axis is depicted for the same simulations shown in Figure 14. The Buneman waves correspond to the fluctuations on ∼c/ω_{p, e} scale that appear in the leading edge of the shock (at x/c/ω_{p, e} = 115–120, 240–245, and 365–370, in panels (a), (b), and (c), respectively). The black dashed line in the three panels marks the position of the shock density peak (overshoot) in the three simulations. When compared to the convective electric field (E₀ = B₀v_in/c), we see that the maximum electric field of the Buneman waves is reduced by a factor of ∼4 when m_i/m_e passes from 25 to 400.

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The same maximum energy for the accelerated electrons is observed in similar simulations but using M_A = 21 (simulations 2Db-4 and 2Db-5). Also, when trying cases similar to the ones shown in Figures 14 and 15 but using v_sh = 0.042 (simulations 2Db-6 and 2Db-7), a decrease in the maximum energy to Γ_e − 1 ≈ 0.1 is seen, with no change by varying m_i/m_e.

Thus, in general, the maximum energy attainable by the shock surfing mechanism appears to depend mainly on the shock velocity, and shows no changes for different values of m_i/m_e. Therefore, as the mass ratio m_i/m_e is increased, the peak of the thermal part of the distribution gets closer to this maximum energy. This implies that the fraction of non-thermal electrons is reduced when m_i/m_e approaches more realistic values. We conclude that this mechanism would not contribute significantly to electron acceleration in the fully realistic case of m_i/m_e = 1836.