# Electron Heating in Low Mach Number Perpendicular shocks.

II. Dependence on the Pre-Shock Conditions

###### Abstract

Recent X-ray observations of merger shocks in galaxy clusters have shown that the post-shock plasma is two-temperature, with the protons being hotter than the electrons. In this work, the second of a series, we investigate by means of two-dimensional particle-in-cell simulations the efficiency of electron irreversible heating in perpendicular low Mach number shocks. We consider values of plasma beta (ratio of thermal and magnetic pressures) in the range and sonic Mach number (ratio of shock speed to pre-shock sound speed) in the range , as appropriate for galaxy cluster shocks. As shown in Paper I, magnetic field amplification — induced by shock compression of the pre-shock field, or by strong proton cyclotron and mirror modes accompanying the relaxation of proton temperature anisotropy — can drive the electron temperature anisotropy beyond the threshold of the electron whistler instability. The growth of whistler waves breaks the electron adiabatic invariance, and allows for efficient entropy production. We find that the post-shock electron temperature exceeds the adiabatic expectation by an amount (here, is the pre-shock temperature), which depends only weakly on the plasma beta, over the range which we have explored, and on the proton-to-electron mass ratio (the coefficient of is measured for our fiducial , and we estimate that it will decrease to for the realistic mass ratio). Our results have important implications for current and future observations of galaxy cluster shocks in the radio band (synchrotron emission and Sunyaev-Zel’dovich effect) and at X-ray frequencies.

###### Subject headings:

galaxies: clusters: general — instabilities — radiation mechanisms: thermal — shock waves## 1. Introduction

Cluster merger shocks — collisionless low Mach number shocks (, where is the ratio of shock speed to pre-shock sound speed) generated by infalling subclusters — are routinely observed in the radio and X-ray bands. X-ray measurements can quantify the density and temperature jumps between the unshocked (upstream) and the shocked (downstream) plasma (e.g., Markevitch et al., 2002; Finoguenov et al., 2010; Russell et al., 2010; Ogrean et al., 2013; Eckert et al., 2016; Akamatsu et al., 2017). The existence of shock-accelerated electrons is revealed by radio observations of synchrotron radiation (e.g., van Weeren et al., 2010; Lindner et al., 2014; Trasatti et al., 2015; Kale et al., 2017). Recently, the pressure jump associated with a merger shock has been measured through radio observations of the thermal Sunyaev-Zel’dovich (SZ) effect (Basu et al., 2016).

Since all observational diagnostics are based on radiation emitted by electrons, the proton properties (in particular, their temperature) are basically unconstrained.
One usually makes the simplifying assumption that the electron temperature equals the proton temperature (and so, the mean gas temperature). This assumption is unlikely to hold in the vicinity of merger shocks, since most of the pre-shock energy is carried by protons and there is no obvious reason why protons should efficiently share with electrons the thermal energy they gain in passing through the shock.^{1}^{1}1Shocks in supernova remnants and the heliosphere are known to be two-temperature, with protons hotter than electrons (e.g., Ghavamian et al., 2013).
While Coulomb collisions will eventually drive electrons and protons to equal temperatures, the collisional equilibration timescale (Spitzer, 1962) for typical conditions in the intracluster medium (ICM) is as long as yrs.
In fact, X-ray observations by Russell et al. (2012) have shown that the electron temperature just behind a merger shock in Abell 2146 is lower than the mean gas temperature expected from the Rankine-Hugoniot jump conditions, and thus lower than the proton temperature. On the other hand, Markevitch (2006) found that the temperatures across the shock in 1E 0657-56 (the so-called “Bullet cluster”) are consistent with instant shock-heating of the electrons.

What is the mechanism responsible for electron heating at collisionless shocks, and how does the heating efficiency depend on the pre-shock conditions? This fundamental question can be answered only through a detailed plasma physics analysis, since the fluid-type Rankine-Hugoniot relations only predict the jump in the mean plasma temperature across the shock, without specifying how the shock-generated heat is distributed between the two species. To understand electron heating in collisionless shocks, fully-kinetic simulations with the particle-in-cell (PIC) method (Birdsall & Langdon, 1991; Hockney & Eastwood, 1981) are essential to self-consistently capture the role of electron and proton plasma instabilities in particle heating.

So far, most PIC studies of electron heating in shocks have focused on the regime of high sonic Mach number (, where is the ratio of the upstream flow speed relative to the shock to the upstream sound speed) and low plasma beta (, where is the ratio of thermal and magnetic pressures) appropriate for supernova remnants (Dieckmann et al., 2012; Matsukiyo & Scholer, 2003; Matsukiyo, 2010). In the first paper of this series (Guo et al., 2017, hereafter, Paper I), we investigated by means of analytical theory and two-dimensional (2D) PIC simulations the physics of electron heating in low Mach number perpendicular shocks, a regime so far unexplored. As we summarize in Section 2, we found that, while most of the electron heating is adiabatic — induced by shock-compression of the upstream magnetic field — the passage of electrons through the shock is also accompanied by entropy increase, i.e., by the production of irreversible electron heating. In analogy to the so-called “magnetic pumping” mechanism (L. Spitzer & Witten, 1953; Berger et al., 1958; Borovsky, 1986), we found that two basic ingredients are needed for electron irreversible heating: (i) the presence of a temperature anisotropy, induced by field amplification coupled to adiabatic invariance; and (ii) a mechanism to break the adiabatic invariance. We found that the growth of whistler waves — triggered by the electron temperature anisotropy induced by field amplification — was responsible for the violation of adiabatic invariance, and efficient entropy production.

In Paper I, we validated our model for a shock with Mach number and plasma beta , which we took as representative of merger shocks in galaxy clusters. In this work, we extend our investigation to a wide range of plasma beta () and sonic Mach number (). We quantify how the efficiency of electron heating and the post-shock electron-to-proton temperature ratio depend on and . We focus on perpendicular shocks (i.e., where the pre-shock field is orthogonal to the shock direction of propagation). The choice of a perpendicular magnetic field geometry is meant to minimize the role of non-thermal electrons, which are self-consistently accelerated in oblique configurations, as we have shown in Guo et al. (2014a, b). Because of the absence of shock-accelerated electrons returning upstream, the shock can settle down to a steady state on a shorter time, thus allowing us to focus on the steady-state electron heating physics. However, we expect that the results presented in this paper will also apply to quasi-perpendicular configurations, as long as the non-thermal electrons are energetically sub-dominant.

We find that the dependence on of the electron irreversible heating efficiency can be cast in a simple form: the post-shock electron temperature exceeds the adiabatic expectation by an amount that scales with Mach number as , where is the pre-shock temperature. This depends only weakly on plasma beta (in the regime explored in this work) and on the proton-to-electron mass ratio (which we vary from 49 to 200).

The rest of the paper is organized as follows. In Section 2, we summarize the results of Paper I, where we found that the field amplification required for efficient entropy production can be induced either by shock compression of the upstream field, or by growth of proton cyclotron and mirror modes accompanying the relaxation of proton temperature anisotropy. With periodic box experiments meant to reproduce these two scenarios, Section 3 (Section 4, respectively) investigates the dependence of the electron heating efficiency on Mach number and plasma beta, in a controlled setup where only the first mechanism (the second, respectively) is allowed to operate. The reader primarily interested in the implications of our study for shocks can skip Sections 3 and 4 and proceed directly to Section 5, where we explore how the degree of electron irreversible heating depends on and , in full shock simulations (where the two processes discussed above generally co-exist). We present our key findings in Section 6 and conclude with a summary in Section 7.

## 2. The Physics of Electron Heating

In this section, we summarize the main results of Paper I. As electrons pass through the shock, they experience a density compression, which results in adiabatic heating. In addition, irreversible processes operate, which further increase the electron temperature. In Paper I, we found that efficient entropy production relies on the presence of two basic ingredients: (i) a temperature anisotropy; and (ii) a mechanism to break adiabatic invariance.
The change in electron entropy can then be written in two equivalent forms as:^{2}^{2}2Our analysis applies equally to electrons and protons, even though below we focus only on electron entropy.

(1) | |||||

(2) |

where is the electron density, the large-scale magnetic field strength (by “large-scale,” we mean the magnetic field on scales much larger than the electron Larmor radius and at frequencies much lower than the electron gyration frequency), and and are the electron temperature parallel and perpendicular to the local magnetic field. The term on the right hand side of Equations (1) and (2) represents the total energy per particle transferred to waves, including magnetic, electric and bulk kinetic contributions (in practice, we found that the magnetic term always dominates).^{3}^{3}3If the equations above were to be applied to protons, the corresponding term should include not only the energy residing in proton-driven waves, but also the energy lost by performing work on the electron plasma (see Paper I).

Note that the CGL double adiabatic theory of Chew et al. (1956) predicts that, for adiabatic perturbations, and , which follow from the conservation of the first and second adiabatic invariants. The first square bracket on the right hand side of Equations (1) and (2) explicitly shows that a mechanism to break the adiabatic invariance is needed for entropy production. In most cases, it is the temperature anisotropy (see the second square bracket on the right hand side of Equations (1) and (2)) that provides the free energy for generating the waves responsible for breaking the adiabatic invariance.

### 2.1. Our Reference Shock

In Paper I, we validated the heating model summarized above by performing PIC simulations with the electromagnetic PIC code TRISTAN-MP (Buneman, 1993; Spitkovsky, 2005) for a representative shock with sonic Mach number and plasma beta . The Mach number

(3) |

is defined as the ratio between the upstream flow velocity in the shock frame and the upstream sound speed . Here, is the upstream temperature (the same for both species, so ), is the Boltzmann constant, is the adiabatic index for an isotropic non-relativistic gas, and is the proton mass. The upstream magnetic field strength is parametrized by the plasma beta

(4) |

where is the number density of the incoming protons and electrons. Alternatively, one could quantify the magnetic field strength via the Alfvénic Mach number .

In Paper I, we considered a reference perpendicular shock with and and showed that equations (1) and (2) were in excellent agreement with the measured increase in electron entropy per particle (or specific entropy) across the shock, which can be computed directly from the electron distribution function as

(5) |

In the reference shock, we found that efficient electron entropy production occurs at two major sites: at the shock ramp, where density compression coupled to flux freezing leads to field amplification (we call this scenario “case A”); and farther downstream, where long-wavelength magnetic waves (more specifically, proton cyclotron and mirror modes) accompanying the relaxation of the temperature anisotropy of post-shock protons can also contribute to magnetic field growth (we call this scenario “case B”). At both locations, field amplification coupled to adiabatic invariance drives the electrons to a large degree of temperature anisotropy, exceeding the threshold of the electron whistler instability. The resulting electron whistler waves — whose presence is one of the common denominators at the two sites mentioned above — cause efficient pitch angle scattering, which leads to violation of the electron adiabatic invariance and allows for entropy increase.

In Paper I, we studied case A and case B in detail by employing controlled periodic box experiments meant to reproduce the shock conditions at the two major sites of entropy production. In particular, the shock physics in the ramp (case A) can be replicated in a periodic box where the PIC equations are modified to allow for a continuous large-scale compression, as in Sironi & Narayan (2015); Sironi (2015). While in Paper I we studied this scenario only for our reference case with and , in Section 3 of the present paper we extensively explore a range of Mach numbers and plasma betas. In Paper I we also investigated the physics of electron heating via anisotropy-driven proton waves (case B) by means of a periodic box initialized with anisotropic protons, with a degree of anisotropy inspired by our reference case with and . In Section 4, we extend the same analysis to a wide range of flow conditions.

The advantage of the periodic domains is twofold: (i) they allow for more direct control of the relevant physics; and (ii) due to less demanding computational requirements, they permit to extend our investigation up to the realistic mass ratio. In Paper I we were able to ascertain that the electron entropy increase has only a weak dependence on mass ratio (less than a drop, as we increase the mass ratio from up to ).

## 3. Electron Heating by Shock-Compression of the Upstream Field

In this section, we focus on case A, i.e., we investigate the efficiency of electron heating (and its dependence on the flow conditions) when the field amplification that induces the electron anisotropy — which in turn leads to electron whistler waves, and then to entropy increase — is due exclusively to the large-scale density compression occurring in the shock ramp.

### 3.1. Simulation Setup

The simulation setup parallels the one employed in Section 5 of Paper I, which we summarize here for completeness. We set up a suite of compressing box experiments, using the method introduced in Sironi & Narayan (2015); Sironi (2015), that re-defined the unit length of the axes such that a particle subject only to compression stays at fixed coordinates in the primed system. Then, compression with rate is accounted for by the diagonal matrix

(6) |

which has been tailored for compression along the axis, perpendicular to the uniform ordered magnetic field initialized along the direction (in analogy to the shock setup that we will discuss in Section 5). Maxwell’s equations in the primed coordinate system automatically account for flux freezing (i.e., the field grows in time as , in the same way as the density ), and the form of the Lorentz force in the primed system guarantees the conservation of the first and second adiabatic invariants.

Since in Paper I we have shown that the wavevector of the whistler mode is nearly aligned with the field direction (i.e., along ), we employ 1D simulations with the computational box oriented along . Yet, all three components of electromagnetic fields and particle velocities are tracked. In 1D simulations, we can employ a large number of particles per cell (we use 1600 particles per species per cell) so we have adequate statistics for the calculation of the electron specific entropy from the phase space distribution function, as in Equation (5).

As a result of the large-scale compression encoded in Equation (6), both electrons and protons will develop a temperature anisotropy, and we should witness the development of both electron and proton anisotropy-driven modes. However, as we have done in Paper I, our goal is to isolate the role of the large-scale field amplification (as expected in the shock ramp) in generating electron irreversible heating, regardless of the presence of proton-driven modes (which will be the focus of Section 4). For this reason, in our compressing box runs, we artificially inhibit the update of the proton momentum (effectively, this corresponds to the case of infinitely massive protons). A similar strategy, but in the case of field amplification driven by shear, rather than compression, has been employed by Riquelme et al. (2017).

The compression rate (which we cast in units of , i.e., the proton Larmor frequency in the initial field ) is measured directly from our shock simulations described in Section 5. In fact, we can quantify the profile of electron density as a function of the co-moving time of the electron fluid

(7) |

where is the electron fluid velocity in the shock frame, and the integral goes from the upstream to the downstream region.^{4}^{4}4As described in Section 5, the shock propagates along in our simulations.
Figure 1 shows the electron density profile as a function of for a suite of shock simulations with and varying Mach number (as indicated in the legend, solid lines), which will be discussed in Section 5. The faster rise seen for higher values of is driven by the fact that the density jump across the shock monotonically increases with Mach number, for two reasons. First, the density jump from upstream to downstream as derived from the Rankine-Hugoniot relations is a monotonically increasing function of . Second, the density overshoot in the shock ramp also increases with (Leroy, 1983, see also
Section
5). Since the thickness of the shock is nearly independent of Mach number (and always of order of the proton Larmor radius), a larger density jump for higher corresponds in

a faster compression rate. In fact, if we model the density compression as a linear function of time (dashed lines in

the resulting compression rate steadily increases with . The values of adopted in the periodic simulations discussed in this section are given in Table 1 in units of the proton Larmor frequency (all the runs presented in this section employ a reduced mass ratio ). We remark that the value of the shock Mach number enters the compressing box experiments presented here only via the compression rate (i.e., should be meant as the Mach number of the shock simulation that corresponds to a given choice of for the compressing box).

We summarize our physical and numerical parameters in Table 1 (the run names have the suffix “c” to indicate that we employ compressing boxes). The first four runs explore the dependence on Mach number (or equivalently, compression rate) for fixed , whereas the last four simulations investigate the dependence on for a fixed compression rate , as appropriate for a shock with . In all the runs, we initialize a population of isotropic electrons with temperature . We resolve the electron skin depth

(8) |

with 10 cells, so the Debye length is marginally resolved. The box extent along the direction is fixed at for , which is sufficient to capture several wavelengths of the electron whistler instability. For and 64, we increase the box length roughly as , since the wavelength of whistler waves increases with plasma beta (see Appendix B, where we study the linear dispersion properties of the whistler instability).

run name | |||||
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### 3.2. Dependence on

Figure 5 compares the results of compressing box simulations (runs , , , in Table 1) with the same and different compression rates (or equivalently, different Mach numbers of the corresponding shock simulations). We present the evolution of the whistler wave energy (panel (a)), the electron temperature anisotropy (panel (b)), the rate of breaking adiabatic invariance (panel (c)) and the electron entropy increase (panel (d)) when varying the compression rate from up to 4.0 (from blue to red, see the legend in the first panel). For mass ratio , the compression rate in units of the electron gyration frequency is , i.e., compression occurs slowly as compared to the electron gyration time (so, our choice of fulfills the requirement expected for the realistic mass ratio).

As a result of the large-scale compression, the electron perpendicular and parallel temperatures are expected to scale as and const, according to the double adiabatic theory. In fact, the electron anisotropy at early times grows as (Figure 5(b)), thus, at a faster rate for higher (or equivalently, in higher shocks).

The increasing temperature anisotropy leads to the exponential growth of the electron whistler instability. When the energy in whistler waves reaches a fraction of the compressed background field energy (

)), the waves are sufficiently strong to scatter the electrons in pitch angle, breaking their adiabatic invariance and decreasing the electron anisotropy. In fact, the peak in panel (c), i.e., the time when the electron adiabatic invariance is most violently broken, always corresponds to the time when the electron anisotropy in panel (b) shows the sharpest decrease. This occurs earlier for higher (if time is measured in , as in

since electrons are driven sooner to large levels of anisotropy. As a result of efficient pitch angle scattering, the electron anisotropy is reduced to the marginal stability threshold of the electron whistler instability (Gary, 2005)

(9) |

(dotted lines in Figure 5(b), with the same color coding as in the legend of panel (a)), which is nearly the same for all the runs, as they start with the same and maintain a similar value of . Here, is the electron plasma beta measured with the parallel temperature .

Near the end of the exponential growth of whistler waves, the electron entropy shows a rapid increase (panel (d)). Here, the electron anisotropy is still large, and at the same time whistler waves are sufficiently powerful to provide effective pitch-angle scattering. In other words, both terms in the square brackets of either Equation (1) or Equation (2) are large. The resulting entropy increase is a monotonic function of , for the following reason. First, larger values of allow the electrons to reach higher levels of peak anisotropy (

)). This can be understood from the competition between the large-scale compression rate (which increases the electron anisotropy) and the growth rate of whistler waves (that try to reduce the anisotropy via pitch angle scattering). Since the whistler growth rate depends on how much the anisotropy exceeds the whistler threshold in Equation (9), a higher anisotropy is needed for larger (in fact,

ows that the whistler growth rate is higher for larger ). Second, since whistler waves are sourced by the free energy in electron anistropy, the wave energy at saturation will be larger for higher (Figure 5(a)). Third, stronger whistler waves will be more efficient in breaking the electron adiabatic invariance (Figure 5(c)). The combination of these effects explains the monotonic trend in electron entropy observed in

) near the end of the exponential growth of whistler waves (i.e., at the time of sharp increase in the curves of panel (d)).

After the exponential growth, electron whistler waves enter a secular phase where the wave energy (normalized to the compressed background field energy) stays almost constant in time. At this point, the whistler wave energy is also nearly independent of (panel (a)), and the same will hold for the rate of violation of electron adiabatic invariance (panel (c)). The electron anisotropy settles around the threshold of marginal stability (compare solid and dotted lines in panel (b) at late times), which is independent of at fixed . It follows that the rate of increase of electron entropy during the secular phase will be the same regardless of , as indeed confirmed by panel (d) (see the slow growth at late times, at a rate independent of ). We remark that the overall increase of electron entropy as measured directly from our simulations using Equation (5) is in excellent agreement with our heating model of Equation (1) (compare solid and thin dashed lines in panel (d), respectively).

From

), we can infer how the entropy increase in the shock ramp should scale with Mach number, if field amplification is induced by shock-compression of the upstream field (case A). Since the compression in the shock ramp lasts about one proton gyration time, we compare the entropy curves at , as indicated by the vertical dotted black line in panel (d). We then find that the efficiency of electron heating in the shock ramp is expected to be larger at higher (corresponding to faster compressions in

at fixed . We will confirm this trend in our shock simulations presented in Section 5. However, we anticipate that in shocks with high Mach number, proton-driven waves already appear near the shock ramp. In this case, large-scale field compression and proton-driven waves (as discussed in Section 4) co-exist and co-contribute to efficient electron entropy production in the shock transition region.

### 3.3. Dependence on

In this subsection, we study the dependence of the electron heating efficiency on plasma beta , by means of compressing box experiments (as appropriate for case A). We fix the compression rate at , as appropriate for a shock with , and vary the initial plasma beta from to (runs , , , , ).

Figure 13 compares the results of our runs. Initially, the electron temperature anisotropy increases as , regardless of (panel (b)). Eventually, this leads to the development of the whistler instability, whose exponential growth needs to balance the large-scale compression (and so, its growth rate is nearly independent of , as confirmed by panel (a)). As we discuss in Appendix B, a given growth rate of the whistler mode requires a lower degree of electron anisotropy for higher plasma beta, which explains the trend in peak anisotropy of

). In turn, a lower level of peak anisotropy corresponds to a smaller amount of free energy available to be converted into whistler waves, whose amplitude is indeed weaker at higher (see the inset in

), where we normalize the whistler wave pressure with respect to the electron parallel pressure ). Weaker whistler wave activity at higher explains why the breaking of electron adiabatic invariance is less violent at higher beta (panel (c)). Since both the electron anisotropy and the violation of adiabatic invariance are weaker for larger , the resulting entropy increase near the end of the exponential phase of whistler growth is smaller for higher plasma beta, as shown in

) (solid lines; compare the values attained during the early phase of rapid growth).

In the secular phase of the electron whistler instability, when efficient pitch angle scattering has brought the anisotropy down to the threshold of marginal stability (dotted lines in

), with the same color coding as the solid lines), the electron entropy keeps increasing, yet at a slower rate. In the secular phase, the trend discussed above still holds (i.e., weaker entropy production at higher plasma beta), primarily because the degree of electron anisotropy stays smaller at higher (panel (b)), due to the monotonic dependence on of the threshold of marginal stability, see Equation (9). As we have emphasized in the previous subsection, the increase of electron entropy in both exponential and secular phases, as measured directly from our simulations (Equation (5)), is in excellent agreement with our heating model of Equation (1) (compare solid and thin dashed lines in panel (d), respectively).

In summary, when field amplification is due to compression alone (case A), the efficiency of electron entropy production decreases monotonically with increasing plasma beta. In particular, this holds after one proton gyration time (see the vertical dotted black line in panel (d)), which we have taken to be the characteristic compression time in the shock ramp. However, as well shall see in the next section, the dependence on is opposite when field amplification is provided by proton-driven waves. As a result of the two opposite trends, in our shock simulations of Section 5, where both compression and proton-driven waves contribute to field amplification, the dependence on is rather weak, in the range of plasma beta that we explore.

run name | ||||||||
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## 4. Electron Heating by Proton-Driven Waves

In this section, we focus on case B, i.e., we investigate the efficiency of electron irreversible heating (and its dependence on the flow conditions) when field amplification is due exclusively to proton-scale waves induced by the relaxation of proton temperature anisotropy. To study this effect, we employ periodic boxes (not compressing, so with ), where the degree of proton anisotropy is prescribed to mimic the conditions expected in the shock downstream. Since both the proton cyclotron instability, which dominates over the mirror mode in our runs (see Section 5), and the electron whistler instability have the fastest growing wavevector aligned with the background field, we employ 1D simulation domains aligned with the direction of the field.

### 4.1. Simulation Setup

We now describe how the initial conditions of our 1D periodic box experiments are set up to be representative of the downstream region of a shock with Mach number and plasma beta . We need to prescribe the initial proton temperatures perpendicular and parallel to the magnetic field ( and ), the initial electron temperature (; as we justify below, we consider isotropic electrons) and the initial electron plasma beta (). We employ the subscript “box” to distinguish them from the initial conditions of our shock simulations. In principle, these parameters can be set by measuring directly the flow conditions behind our simulated shocks. However, we show below that they can be simply prescribed using the Rankine-Hugoniot jump conditions. This has the advantage of showing explicitly the expected dependence on the shock Mach number and plasma beta .

Since the pre-shock flow moves along the direction, while the pre-shock magnetic field is oriented along (see Section 5), the post-shock protons will be promptly gyrotropic in the plane perpendicular to the pre-shock field, but generally anisotropic with respect to the direction parallel to the field. In the absence of instabilities that mediate efficient isotropization, the motions perpendicular and parallel to the field are effectively decoupled, and the post-shock protons behave as a plasma with two degrees of freedom (so, with adiabatic index ). This is demonstrated by the 2D out-of-plane and 1D shock simulations shown in Appendix A of Paper I. The density and temperature jumps across a shock with adiabatic index (as appropriate for the region just behind the shock) are

(10) |

(11) |

where is, more precisely, the jump in the perpendicular temperature for the overall fluid (the parallel one stays unchanged).

We then assume that the electron temperature jump in the perpendicular direction follows from the adiabatic law (in Section 5, we quantify the efficiency of electron irreversible heating; still, for the parameter regime we explore, most of the electron temperature increase comes from adiabatic compression). Coupling electron adiabatic invariance with flux freezing, this implies that the post-shock electron temperature is . From Equation (11), which prescribes the jump in perpendicular temperature for the overall (electron proton) fluid, we find that the post-shock proton temperature in the direction perpendicular to the field, which we take as the initial condition in our periodic boxes, will be

(12) |

The parallel proton and electron temperatures are the same as in the pre-shock region, and in particular

(13) |

As regard to the initialization of electrons, we assume that the electron anisotropy is rapidly erased due to the rapid development of electron-scale instabilities, as it indeed occurs in the shock ramp.^{5}^{5}5Despite the fact that we now assume isotropic electrons, the ansatz of a gas that we employed to calculate the jump conditions will still hold, since the post-shock pressure is mostly contributed by protons (see Section 5). If this happens without any energy exchange with the protons, the isotropic electron temperature is

(14) |

Finally, the initial electron plasma beta can be computed from flux freezing as

(15) |

The physical and numerical parameters used for the periodic box experiments of this section are summarized in Table 2 (the run names have the suffix “nc” to indicate that the boxes are not compressing). We indicate the Mach number and plasma beta of the corresponding shock simulations, the initialization values of , , and employed in the periodic boxes, the number of computational particles per cell () and the box length along the direction of the background large-scale field, in units of the proton skin depth . We employ a reduced mass ratio . From Table 2, it is apparent that the Mach number has a pronounced effect on the initial proton anisotropy. In fact, in the limit of , we have and , so that from Equations (12) and (13).

### 4.2. Dependence on

In this subsection, we compare the periodic box simulations , , , , that correspond to shocks with fixed but different ranging from to . Our results are shown in Figure 18. The case corresponding to is plotted with a dashed line to emphasize that its time axis has been reduced by a factor of two, to facilitate comparison with the other cases.

As we have discussed above (see also Table 2), runs with larger are initialized with a stronger degree of proton anisotropy (see the curves in panel (b) at the initial time). In response to the greater amount of free energy stored in proton temperature anisotropy, runs with larger develop stronger proton cyclotron waves (panel (a)). In addition, since the growth rate of the proton cyclotron instability is higher for larger temperature anisotropies, at fixed plasma beta (see the linear dispersion properties of the proton cyclotron instability in Appendix C), the proton cyclotron wave energy grows faster at higher (panel (a)). Since pitch angle scattering by the proton cyclotron modes is responsible for relaxing the temperature anisotropy, this explains why the proton anisotropy in

) drops faster for higher , despite starting from higher initial values.

The growth of proton cyclotron waves provides a source of field amplification that can perform work on the electrons. However, this does not automatically lead to electron irreversible heating. In fact,

ows that for the energy in proton cyclotron modes is so small (dashed blue line in panel (a)) that the electron temperature anisotropy (dashed blue line in panel (d)) never exceeds the threshold of the electron whistler instability (indicated in

) by the corresponding dotted blue line). In fact, in this case no whistler waves are observed to grow (no blue line appears in

), where we plot the magnetic pressure associated with whistler waves in units of the electron parallel pressure). In the absence of whistler waves, the electron evolution stays adiabatic (dashed blue line in panel (e)), and no electron entropy is generated (dashed blue line in panel (f)).

For higher , the proton waves are sufficiently strong to drive the electron anisotropy beyond the whistler threshold (compare solid and dotted lines of the same color in

)). Since whistler waves provide the pitch-angle scattering required to break adiabatic invariance, this leads to electron entropy production (

)). As shown in

), the increase in electron entropy is a monotonic function of . As we now discuss, this is related to the fact that the same trend holds separately for the two terms in square brackets of Equation (1) (or equivalently, Equation (2)).

In fact, the degree of violation of electron adiabatic invariance is larger for higher (

)), since whistler waves are more powerful (

)).^{6}^{6}6In order to obtain the magnetic energy in whistler-like fluctuations, we have applied a high-pass filter in frequency and wavenumber, as described in Paper I. The trend in electron anisotropy (

)) is less clear, with the green, orange and red curves showing comparable peak values. Unlike in the compressing box experiments of the previous section, where the degree of electron anisotropy was roughly uniform throughout the simulation domain, here proton-driven modes introduce large spatial variations in the electron properties (in analogy to Figure 10 of Paper I). It is then useful to complement the information on the box-averaged anisotropy (solid lines in

)) with shaded regions indicating the percentile of electron temperature anisotropy. This reveals that spatial variations in electron anisotropy are stronger at higher (or equivalently, the distance between the solid curve and the upper boundary of the corresponding shaded region increases with ), where proton modes are also more powerful. In addition, runs with larger present localized regions with systematically higher peak anisotropies (the peak of the shaded red region is higher than the orange one, which in turn is higher than the green one). On the one hand, this justifies the fact that the strength of whistler waves, which are seeded by electron anisotropy, increases with . On the other hand, when coupled with the dependence on of the violation of adiabatic invariance (

)), it fully justifies why runs with higher lead to more efficient electron entropy production.

While for low most of the entropy increase occurs near the end of the exponential phase of the proton cyclotron instability, at high a steady growth in electron entropy is observed during the secular stage. Here, the strong cyclotron modes can occasionally excite local patches of electron anisotropy that exceed the whistler threshold (e.g., see the peak in the red shaded region at in

)). The resulting whistler activity (see the corresponding peaks in the curves of panels (c) and (e)) can further increase the electron entropy.

Finally, we point out that, as we have also discussed in the case of compressing box simulations, the increase of electron entropy, as measured directly from our simulations (Equation (5)), is in excellent agreement with our heating model of Equation (1) (compare solid and thin dashed lines in panel (f), respectively).

### 4.3. Dependence on

In this subsection, we compare the periodic box simulations , , , , that correspond to shocks with fixed but different ranging from to . Our results are shown in Figure 32.

At fixed , the initial proton temperature anisotropy is nearly independent of (see

) at early times). From the dispersion properties of the proton cyclotron mode (see Appendix C), at fixed proton anisotropy the growth is faster at higher plasma beta, in agreement with the exponential phase in

) and with the drop in proton anisotropy in

). At fixed proton anisotropy, the free energy available to be converted into proton waves will be larger with increasing , just because the proton thermal content is higher. In fact, when normalized to the magnetic energy of the large-scale field , the magnetic energy of proton cyclotron waves at saturation is larger for higher plasma beta, as shown in

).

In all the cases that we investigate, the growth of proton cyclotron waves drives the electron anisotropy above the whistler threshold (Figure 32(d), compare solid and dotted lines). At higher , the electron whistler instability starts earlier (Figure 32(c)), due to the combination of two effects: the proton waves grow faster, and the whistler threshold is lower, so easier to be exceeded.

In addition, the stronger proton waves generated for higher will have the chance to perform more work onto the electrons. In particular, at higher we expect that electrons will be driven further into the unstable region of the electron whistler mode (i.e., beyond the marginal stability threshold, which is itself a function of plasma beta, see Equation (9)). This is suggested by the general trend seen in

), where, e.g., the orange solid curve (for ) lies at significantly above the corresponding marginal stability threshold (indicated by the dotted orange line), whereas the solid blue line (for ) at is only marginally above the corresponding threshold (dotted blue curve). In turn, a higher level of electron anisotropy (with respect to the baseline provided by the marginal stability threshold) results in stronger whistler wave activity (

)), and in more violent breaking of electron adiabatic invariance (

)). It follows that the electron entropy increase will be more pronounced for higher , as confirmed by

).

We conclude with two comments. First, once again, our heating model is in good agreement with the measured entropy increase (compare thin dashed and solid lines in

)). Second, while in the compressing box experiments of the previous section (meant to mimic field amplification by shock compression of the upstream field), higher values of plasma beta resulted in a weaker increase in electron entropy, the opposite trend is observed here, where field amplification is due to proton cyclotron waves. In the shock simulations presented in the next section, where the two processes will co-exist, we should expect a weak dependence on plasma beta, as indeed we will find for the parameter regime we explore.

## 5. Electron Heating in Shocks

In the previous two sections, we have investigated the efficiency of electron irreversible heating when the magnetic field amplification that induces electron anisotropy — which in turn sources the growth of whistler waves, and eventually results in entropy production — is due to two separate mechanisms: in Section 3, we have discussed case A, where field amplification is due to a large-scale density compression; in Section 4, we have focused on case B, where proton waves accompanying the relaxation of proton anisotropy can increase the magnetic field strength. In shocks, the two mechanisms co-exist, as we have already discussed in Paper I for our reference case with and . We now explore the dependence of irreversible electron heating in perpendicular shocks on Mach number and plasma beta. In Section 6, we summarize the key findings from the shock simulations and provide an empirical fit to our results, which can be used in comparing with the observations.

run name | ||||||
---|---|---|---|---|---|---|

### 5.1. Simulation Setup

We perform shock simulations using the 3D electromagnetic PIC code TRISTAN-MP (Buneman, 1993; Spitkovsky, 2005). Our setup parallels closely what we have employed in Paper I. We use a 2D simulation box in the plane, with periodic boundary conditions in the direction. Yet, all three components of particle velocities and electromagnetic fields are tracked. The shock is set up by reflecting an upstream electron-proton plasma moving along the direction off a conducting wall at the leftmost boundary of the computational box (). The interplay between the reflected stream and the incoming plasma causes a shock to form, which propagates along . In the simulation frame, the downstream plasma is at rest. The pre-shock magnetic field is initialized along the direction perpendicular to the shock direction of propagation (i.e., we focus on the case of a “perpendicular” shock). Our 2D setup allows to capture the physics of both electron (whistler) and proton (mirror and proton cyclotron) anisotropy-driven instabilities that are crucial for electron heating.

We vary the shock Mach number (defined in Equation (3)) and the pre-shock plasma beta (defined in Equation (4)) as listed in Table 3, where we summarize the physical and numerical parameters of our shock simulations. The values of and cover the regime of pre-shock conditions expected in galaxy cluster shocks.

We point out that, in setting up our simulations, we do not have direct control on the upstream velocity in the shock frame (which enters the definition of ), but only on the upstream velocity in the downstream frame . In other words, we have no direct way of prescribing . In order to obtain a given “target” Mach number , we iteratively solve the Rankine-Hugoniot jump conditions and select the value of that corresponds to the chosen . Since in the parameter regime covered by our simulations, the protons in the immediate post-shock region retain a significant degree of anisotropy, we solve the Rankine-Hugoniot relations assuming a 2D adiabatic index . It follows that the actual value of Mach number measured a posteriori in our simulations ( in Table 3) may differ from the Mach number targeted a priori. In practice, Table 3 shows that the two values differ at most by a few percent.

The pre-shock particles are injected at a “moving injector”, which recedes from the wall in the direction at the speed of light. For further numerical optimization, we allow the moving injector to periodically jump backward (i.e. in the direction), so that its distance ahead of the shock is always of order of a few proton Larmor radii (see Paper I for details). The pre-shock particles are initialized as a drifting Maxwell-Jüttner distribution with a temperature . We resolve the electron skin depth (defined in Equation (8)) with computational cells, so that the electron Debye length is appropriately captured. We use a time resolution of . The number of particles per cell (including both species) is in Table 3. Convergence checks as regard to spatial resolution and number of particles per cell have been performed in Paper I.

The shock structure is controlled by the proton Larmor radius

(16) |

where the Alfvén speed is . Similarly, the evolution of the shock occurs on a time scale given by the proton Larmor gyration period . The need to resolve the electron scales, and at the same time to capture the shock evolution for many , is an enormous computational challenge for the realistic mass ratio . Thus we adopt a reduced mass ratio for most of our runs, but we have tested that a higher mass ratio yields identical results (in Appendix A, we explore the dependence on Mach number of a few simulations with ). In Paper I, we have argued that for our reference shock with and the efficiency of electron irreversible heating is nearly insensitive to the mass ratio, up to . For , we choose the transverse size of the box to be , which is sufficient to capture the growth of proton instabilities in the downstream.

### 5.2. Dependence on

In this subsection, we compare the results of shock simulations with fixed and varying Mach number from up to (runs , , and in Table 3). We employ a reduced mass ratio of , but in Appendix A we show that identical results are obtained for a higher value of the mass ratio, .

Figure 42 shows the -averaged profiles of various quantities in the shock at time , as a function of the coordinate relative to the shock location , in units of the proton Larmor radius defined in Equation (16). Panel (a) shows the profiles of density (thick lines, see the legend in panel (d)) and magnetic field strength (thin lines with the same color coding as the thick lines). In agreement with the Rankine-Hugoniot relations, the density jump is larger for higher . As a result of flux freezing alone, one would expect that the magnetic field be , i.e., its spatial profile should be identical to the density profile. The fact that the field strength in

) exceeds the expectation from flux freezing is to be attributed to magnetic fluctuations. Since the deviation is more pronouced at high , whereas thin and thick lines overlap at low , we expect from panel (a) that the strength of magnetic field fluctuations should increase with Mach number.

This is confirmed in

here we present the 2D plots of for the same simulations as in

t is apparent that the strength of long-wavelength fluctuations steadily increases with Mach number (i.e., from top to bottom). Such waves — a combination of proton cyclotron modes and mirror modes — accompany the relaxation of the proton temperature anisotropy. As we have discussed in Section 4.1, protons are expected to be highly anisotropic in the immediate post-shock region, with a degree of anisotropy that scales as . The fact that shocks with higher lead to stronger proton anisotropy has two consequences: (i) the larger amount of free energy in proton anisotropy can generate stronger proton waves, as indeed observed in

d

); (ii) as predicted by linear theory (see Appendix C), the waves grow faster for higher levels of anisotropy (so, higher ). In fact,

ows that the peak of wave activity is located right at the shock for high Mach numbers ( and 5), and it shifts farther and farther downstream for lower and lower Mach numbers (it lies at for and at for ), due to the slower and slower wave growth.

For and 3, the wave pattern in the shock ramp is dominated by short-wavelength electron whistler waves, rather than by the long-wavelength proton waves appearing for and 5. In fact, the peak at in the green and blue lines of

) reflects the energy in whistler modes. For high Mach numbers, proton-driven modes are so strong that they dominate the wave energy right at the shock (orange and red curves in

)), hiding the presence of whistler waves in

),(d). Still, electron whistler modes remain active in the shock ramp (their pattern appears more clearly at higher mass ratios, where proton and electron scales are better separated, see Appendix A). There, they mediate efficient entropy production, as we discuss below.

As we have just described, the downstream proton waves are sourced by the proton anisotropy induced at the shock, which is expected to be stronger at higher . This trend is confirmed in the peak anisotropy of

) (compare the curves at ), with the exception of the red curve of . Here, anisotropy-driven proton instabilities are so violent that the proton anisotropy is not even allowed to reach its expected peak. Due to pitch angle scattering by the proton cyclotron and mirror modes, the proton anisotropy is reduced below the marginal stability condition

(17) |

which is indicated with dotted lines in

). Since proton-driven waves are stronger for higher , the relaxation toward the marginal stability threshold is faster for higher , in analogy to what we discussed in Section 4.2. The case of , where proton modes are the weakest, is the only one where the degree of proton anisotropy remains significant (see the blue line in

) in the far downstream).

Given that protons in the far downstream are generally isotropic, their temperature can be properly quantified by the isotropic-equivalent estimate

(18) |

which we present in

). The trend seen in the plot, i.e., increasing with , is driven by the fact that the temperature jump predicted by the Rankine-Hugoniot relations for the overall fluid is a monotonic function of , in combination with the fact that most of the post-shock fluid energy resides in protons (rather than electrons or proton-driven modes). At sufficiently high Mach numbers, we then expect that (i.e., as predicted by the Rankine-Hugoniot relations), a trend confirmed by

).

So far, we have mostly focused on the proton physics. As regard to electrons, we find that the post-shock electron temperature is a monotonically increasing function of (Figure 42(e)). This might just follow from the dependence on of the adiabatic heating efficiency, since the density compression increases with (Figure 42(a)).

However, we find that the efficiency of irreversible electron heating is also higher at larger . In panel (h) we present the electron entropy profile as measured directly from our simulations using Equation (5), while in panel (g) we show the excess of electron temperature beyond the adiabatic prediction appropriate for a 3D non-relativistic gas

(19) |

The fact that the efficiency of electron irreversible heating increases with Mach number can be promptly understood from the results obtained in Section 3 and Section 4. First of all, the electron fluid suffers a faster compression while passing through the ramp of a high- shock, as compared to a low- shock. As we have shown in Section 3.2 (case A), this drives the electrons to larger levels of anisotropy, resulting in stronger whistler waves and faster rates of adiabatic breaking, which leads to more efficient entropy production. In addition, the highly anisotropic protons present in high- shocks generate strong proton modes, as we have discussed in Section 4.2 (case B). The resulting field amplification provides another channel to induce electron anisotropy and ultimately leads to electron entropy increase. The stronger proton-driven modes at higher result in higher efficiencies of electron irreversible heating (Section 4.2).

While the first mechanism (case A) is present for all the values of Mach number that we investigate, the second (case B) does not operate for . Here, the post-shock proton anisotropy is weak, and the strength of the resulting proton modes is insufficient to drive the electrons above the threshold of the whistler instability. In the absence of pitch angle scattering to break the adiabatic invariance, the electron entropy (blue line in

)) does not change behind the shock ramp (where entropy production is induced by compression, as in case A). The same had been observed in Section 4.2.

In the shock with (green line in panel (h)), case A controls the entropy increase in the shock ramp, whereas proton modes (so, case B) are responsible for the additional entropy jump seen at . For and 5 (orange and red curves, respectively), proton-driven modes grow fast, and in the shock ramp the field amplification that they induce co-exists with the large-scale compression of the upstream field (i.e., case A and B are spatially coincident). This explains why most of the entropy increase for and 5 is localized in the shock transition region. However, since the strength of proton modes remains significant for several proton Larmor radii behind the shock, the proton waves can occasionally excite local patches of electron anisotropy that exceed the whistler threshold. The resulting whistler activity can further increase the electron entropy downstream from the shock, in analogy to what we have discussed in Section 4.2.

### 5.3. Dependence on

In this subsection we investigate the dependence of our results on plasma beta, at fixed Mach number. We vary from 4 up to 32, for two different values of Mach number: in Section 5.3.1 and in Section 5.3.2. We demonstrate that both choices of lead to similar conclusions.

#### 5.3.1

Figure 61 compares the results of runs , , and having a fixed Mach number . The density compression across the shock is only weakly dependent on (Figure 61(a)), as expected from the Rankine-Hugoniot relations in the limit of high beta. Similarly, the Rankine-Hugoniot jump conditions justify why the post-shock proton temperature (Figure 61(c)) and the proton anisotropy at the shock (Figure 61(d) at ) are nearly insensitive to .

Given the relatively weak proton temperature anisotropy attained for at the shock (, panel (d)), proton-driven modes grow rather slowly in the downstream, and the magnetic field fluctuations in the shock ramp are powered by the electron whistler instability, for all the values of we explore (see

the shock; electron whistler modes dominate the peak in magnetic energy seen in

) at the shock). In analogy to what we discussed in Section 4.3, the development of proton instabilities is faster at higher , since the marginal stability threshold is lower (see Equation (17)), and so easier to be exceeded, and because the growth rate is larger at higher , for a fixed degree of anisotropy. This explains why the magnetic energy in proton waves peaks closer to the shock at higher (in fact, it peaks at for the blue line of

), which refers to , and at for the red line, which refers to ). From

), we also see that proton modes are stronger for higher , if normalized to the flux-frozen field. This is just a consequence of the fact that the free energy in proton anisotropy available to source the waves is larger for higher , when compared to the magnetic energy of the background field (the degree of anisotropy is -independent, but the proton thermal content obviously increases with ).

Due to pitch angle scattering by the proton modes, the proton anisotropy drops at a faster rate for higher (

)), since in this case the waves grow faster and are also stronger. In the far downstream, the proton anisotropy settles to the marginal stability threshold of Equation (17), which is higher for lower (see the dotted lines in

)). It follows that low- shocks maintain an appreciable degree of proton anisotropy in the far downstream (see the blue line in

)). The fact that the resulting adiabatic index will be larger than the value