Propagation of weak shocks in cool-core galaxy clusters in two-temperature magnetohydrodynamics with anisotropic thermal conduction

We solve the equations of two-temperature magnetohydrodynamics joined by a heat-flux term that describes anisotropic electron thermal conduction in a magnetic field and a term that describes electron-ion equilibration as follows

where

where $\rho$ is the mass density (taken to be the ion mass density), $\mn@boldsymbol{v}$ the fluid velocity (assumed same for both species), $p_{i}$ and $p_{e}$ are the ion and electron pressures, $\mn@boldsymbol{B}$ the magnetic field, $s_{e}$ the electron entropy, and $Q$ the heat flux along the magnetic field lines whose direction is set by unit vectors $\mn@boldsymbol{b}$. Both electron and ion components are described by an ideal equation of state $p_{e/i}=n_{e/i}T_{e/i}$ (temperature is in energy units), where $n_{e}=n_{i}=\rho/m_{i}$ are the particle number densities (equal by plasma quasineutrality), and adiabatic index $\gamma=5/3$. For the parallel thermal conductivity, we use the Spitzer value (Spitzer & Härm, 1953) $\kappa_{\parallel}\approx 0.9n_{e}\lambda_{e}v_{{\rm th}e}$, where $n_{e}$ is the electron density, $v_{{\rm th}e}=(2T_{e}/m_{e})^{1/2}$ is the electron thermal speed, and

is the Coulomb electron mean free path in a hydrogen plasma expressed using characteristic parameters of the Perseus core in the vicinity of the observed weak shock. The two terms on the right-hand side of equation (5) describe dissipation due to electron-ion and electron-electron collisions.

Numerically, the dissipation terms are treated separately at the end of each time (sub-)step. First, the system of conservation laws for ($\rho$, $\rho\mn@boldsymbol{v}$, $\mn@boldsymbol{B}$, $E$, $s_{e}$) described by equations (1)–(5) with the RHS set to zero is evolved by a second-order van Leer integrator using the HLLD flux (Miyoshi & Kusano, 2005) modified to include electron pressure. The magnetic field is integrated via the constrained transport technique (Stone & Gardiner, 2009). Second, electron thermal conduction is added by solving anisotropic heat transfer equation

using Runge-Kutta-Legendre super-time-stepping (Meyer et al., 2012, 2014) with a monotonized central limiter applied to transverse temperature gradients to avoid temperature oscillations (Sharma & Hammett, 2007, 2011). Finally, electron-ion equilibration is accounted for by implicit integration (this might be a numerically stiff problem when the equilibration is fast) of equation

where

is the thermal equilibration rate (Book, 1983). The new electron temperature is then used to update the electron entropy $s_{e}$ (which corresponds to adding the two dissipative terms on the RHS of equation 5) and total energy density $E$ (only due to thermal conduction).

We also perform a series of single-fluid hydrodynamic runs with or without thermal conduction, where the same system of equations is solved with $p=2p_{i}=2p_{e}=2n_{e}T_{e}$ excluding the induction equation (4) and equation (5) for the electron entropy, as well as runs with isotropic thermal conduction, where instead of equation (9), the heat flux is $Q=-\kappa_{\parallel}\nabla T_{e}$.

As we are not concerned about the distribution of energy released during an AGN outburst between the radio bubble and the weak shock, it is sufficient to produce a spherical blast wave by the simplest suitable initial condition. The only requirement is that the shock needs to have Mach number $M\approx 1.2$ as it reaches radius $r\approx 15$ kpc to match observations of the weak shock in the Perseus cluster. Therefore, for the initial conditions, we produce an overpressurized (by a factor of $\sim 100$) region within $r<1.5$ kpc, large enough to avoid pixelation. The weak shock produced in the outburst travels outward unimpeded, while the hot gas shell formed at the center is of no relevance for our problem. Following the outburst, we observe how the shock is modified by thermal conduction and electron-ion equilibration as it propagates to $r=15$ kpc. Our 3D numerical box has size $L=40$ kpc and is centered around the shock. We choose to use Cartesian coordinates because of the easier implementation of runs with tangled magnetic fields.

The observed part of the spherical weak shock in the Perseus cluster is located $\approx 25$ kpc away from the cluster center in the north-east direction (its geometrical center is $\approx 10$ kpc off the cluster center). In the vicinity of the shock, the ICM has temperature $T_{0}\approx 4$ KeV and the radial temperature profile is flat (e.g., Graham et al., 2008; Zhuravleva et al., 2015). The ICM density in this region is $n_{0}\approx 0.04~{}{\rm cm}^{-3}$. In our simulations, we choose to use uniform background temperature and density for several reasons. First, both radial profiles within the shock sphere are relatively flat (e.g., Zhuravleva et al., 2015). Second, there exists uncertainty about the initial density and temperature distribution at the moment of the AGN outburst that launched the shock, and it is likely that the initial distribution was different from the one currently observed. More importantly for our particular problem, the structures we are interested in, i.e., distortions of the shock due to thermal conduction, develop quickly (compared to the time it takes for the shock to reach its current position) and locally, within $\sim 5$ kpc from the shock front. Therefore, it is sufficient to initialize the background density and temperature to those currently observed near the shock, as long as the simulated shock has the right Mach number $M\approx 1.2$ at radius $r\approx 15$ kpc.

There is currently a lot of uncertainty about the structure of magnetic field in the ICM, mainly due to the complicated physics of turbulent magnetized plasmas at high ratios of thermal to magnetic pressures and, on the other hand, limited possibilities to infer a 3D spectrum of magnetic fluctuations from radio observations (e.g., Carilli & Taylor, 2002; Govoni & Feretti, 2004; Feretti et al., 2012; Vogt & Enßlin, 2005; Bonafede et al., 2010; Kuchar & Enßlin, 2011). The magnetic spectrum is likely anisotropic, meaning the length scale of change of the field along itself is different (normally larger) than that transverse to itself (e.g., Schekochihin & Cowley, 2006). The parallel correlation length is expected to be set by large-scale hydrodynamic motions (on those scales the field is too weak to have any dynamic effect on the fluid). But the perpendicular correlation scale depends on the intricacies of plasma turbulence and ranges from Larmor to injection scales depending on the numerical experiment and its interpretation (for MHD/Braginskii turbulence, see Haugen et al. 2004; Schekochihin et al. 2004; Maron et al. 2004; Cho & Ryu 2009; Beresnyak & Lazarian 2009; Beresnyak 2012; Santos-Lima et al. 2014; St-Onge et al. 2020; for kinetic turbulence, see Rincon et al. 2016; St-Onge & Kunz 2018). Rotation measure (meaning rotation of the polarization plane of synchrotron radiation as it travels through a magnetic field) observations of some clusters and groups of galaxies give estimates of the perpendicular scale of order several kpc (Vogt & Enßlin, 2005; Bonafede et al., 2010; Kuchar & Enßlin, 2011). In any case, for the problem of anisotropic thermal conduction in a static (on time scales of shock propagation) tangled magnetic field, the more relevant correlation length is parallel. This is because heat diffusion is not affected by field reversals: diffusion operates in layers of antiparallel field lines the same way as in a uniform field. Then for qualitative purposes, it is sufficient to use a statistically isotropic field on scales characteristic of gas motions induced by AGN activity or gas sloshing in the Perseus cluster. These scales likely range from a few to several tens kpc corresponding to the size of structures (radio bubbles, filaments, sloshing-induced spiral patterns) seen in Chandra X-ray images (Fabian et al., 2006). For this reason, we model the magnetic field simply as a random isotropic solenoidal 3D field with coherence length $l_{B}\approx 10$ kpc. The field is set to be very weak, so it does not have a dynamic feedback on the plasma. Such coherence length is somewhat larger than the width of conductive weak shocks in our simulations ($\sim 5$ kpc).

Thermal conduction along the magnetic field lines operates via heat flux (9) with electron mean free path (10). To reduce the unnecessary computational load, we turn off thermal conduction in the central 6-kpc region that quickly becomes dominated by the hot shell. Initially, the ion and electron temperatures are set equal.

First, we analyze 1D hydrodynamic runs with a thermal conductivity at full Spitzer level. This models a configuration in which the magnetic field is radial and does not impede conduction. Fig. 1 shows the effects of thermal conduction and a finite electron-ion equilibration time relative to pure single-fluid hydrodynamics. In pure hydrodynamics (solid line), the density and temperature jumps agree with the Rankine-Hugoniot relations for a shock with Mach number $M\approx 1.18$:

Activating thermal conduction leads to formation of a temperature and, to a lesser extent, pressure and density precursors (dashed line in Fig. 1). Its length is consistent with the simple estimate of the distance $l_{\rm diff}$ that the electrons are able to diffuse in front of the shock based on equating the diffusion time and the time it takes for the shock to travel the diffusion length. The former is $t_{\rm diff}\sim l_{\rm diff}^{2}/(\lambda_{e}v_{\mathrm{th}e})$, the latter $t_{\rm sh}\sim l_{\rm diff}/(Mv_{\mathrm{th}i})$. Then $l_{\rm diff}\sim(m_{i}/m_{e})^{1/2}\lambda_{e}$. Recalling that $\lambda_{e}\approx 0.1$ kpc (equation 10), we get $l_{\rm diff}\sim 4$ kpc, in agreement with Fig. 1. In comparison with ideal hydrodynamics, the temperature jump becomes smaller due to thermal conduction, and because the gas becomes closer to isothermal, the shock starts to slightly lag behind.

Enabling different temperatures (dash-dotted line in Fig. 1) for the electrons and ions leads to little changes of the electron temperature profile, but the density profile becomes much closer to the hydrodynamic case. This happens because the electrons diffusing ahead of the shock do not have sufficient time to heat the ions (the ion temperature is shown in the lower panel of Fig. 1 by the dotted line). Because now the ions are mostly adiabatic, the speed of sound rises relative to the model with instant equilibration, and, therefore, the shock lags less.

Our single-temperature models produce weak shock profiles qualitatively similar to Graham et al. (2008) (see their Fig. 7; note that we do not match the shock front locations for different models in Fig. 1). Even though these authors use a more elaborate model of the outburst and introduce a smooth background density gradient, the weak shock itself is not sensitive to the initial conditions, and its profile is mainly determined by the intensity of thermal conduction across the shock. Our two-temperature model, however, is significantly different. While Graham et al. (2008) approximate the decoupling of electrons and ions by doubling the effective thermal conductivity, we solve a separate equation for the electron temperature, which is coupled to the ion temperature by a collision operator. This leads to noticeably different density and temperature profiles: namely, the ions tend to preserve their density and temperature jumps much better compared to the single-fluid model with conduction, while the electron temperature behaves the same as the gas temperature in the single-fluid model (compare this with the blue dotted line in Fig. 7 of Graham et al. 2008).

We can use the simulated density and temperature profiles in combination with large-scale radial profiles observed in the Perseus cluster to produce mock surface brightness maps and compare them with X-ray observations of the weak shock in the Perseus cluster. Given its spatial resolution, the most appropriate comparison data are from the Chandra Advanced Camera for Imaging Spectroscopy (ACIS). We obtained all ACIS observations of the core of Perseus that did not employ the diffraction gratings, totalling 900ks of good data spread across 21 separate observations (ObsIDs). Each of separate ObsIDs were reprocessed up to the calibration version 4.8.1 using CIAOv4.10, and then were merged into a single events file from which we extract a 0.7–4keV band image. We analyze a $40\times 40$ kpc region of the Perseus core that contains the weak shock. The final image, shown in Fig. 3, has been divided by an appropriate $\beta$-model with central density $n_{c}=0.05~{}{\rm cm}^{-3}$, core radius $r_{c}=29$ kpc, and $\beta=0.53$ (Zhuravleva et al., 2015) and smoothed with a Gaussian filter of size $\sigma\approx 1^{\prime\prime}\approx 0.4$ kpc. Its bottom right corner coincides with the geometrical center of the shock sphere. In our energy range, the emissivity of an optically thin plasma is almost independent of temperature observed by the ACIS-S detector on Chandra (e.g., Forman et al., 2007; Zhuravleva et al., 2015). Therefore, surface brightness is simply an integral of the electron density squared along the line of sight.

In order to model the observed scaled X-ray surface brightness by integrating the emissivity along the line of sight, we need to know the large-scale ICM density distribution. In our simulations, we have used a uniform density and temperature background (see Section 2.2). This is justified because the $\beta$-model in the vicinity of the shock (its width including the precursor is $\lesssim 10$ kpc) is smooth. Then we can factorize the large-scale electron density as $n_{e}(x,y,z)=n_{e0}(r)\times n_{e,{\rm sim}}(r_{s})$, where $n_{e0}(r)=n_{c}[1+(r/r_{c})^{2}]^{-3/2\beta}$ is the smooth background $\beta$-model and $n_{e,{\rm sim}}(r_{s})$ is the simulated electron density (shown in the upper panel of Fig. 1). The former is a function of the distance from the center of the cluster $r=(x^{2}+y^{2}+z^{2})^{1/2}$, where $x$, $y$ are the sky coordinates relative to the cluster center and $z$ is the distance along the line of sight. The latter is a function of the distance from the center of the shock sphere ($x_{0s},y_{0s}$): $r_{s}=((x-x_{0s})^{2}+(y-y_{0s})^{2}+z^{2})^{1/2}$. Then the mock surface brightness divided by the $\beta$-model is

In Fig. 3, the cluster center is located at $x\approx 26$ kpc, $y\approx-8$ kpc.

In Fig. 3, we compare the mock surface brightness profiles for the three 1D models of the weak shock with surface brightness profiles obtained from the X-ray data by averaging over the three different sectors in Fig. 3. The locations of the simulated shocks have been matched to facilitate the comparison. While it is rather difficult to distinguish between the pure hydrodynamic model and the one with both thermal conduction and ion-electron equilibration enabled, we see no indication of a conductive precursor, except for the third region, where variations of surface brightness before the shock preclude simple assessment. Deviations of the electron density from the $\beta$-model manifested in the significant variation of the observed profiles in the three chosen regions, particularly in the upstream region where precursors could be produced, also make further matching with our models problematic.

In addition, we can calculate the projected electron temperature $T_{e,{\rm proj}}(x,y)$ using the simulated data $T_{e,{\rm sim}}(r_{s})$ (shown in the lower panel of Fig. 4) by weighting the temperature with the emissivity and dividing it by the smooth background as follows

where

is the background temperature profile with $T_{ec}=4$ keV and $r_{c}=58$ kpc taken from Zhuravleva et al. (2015). We have omitted the coordinate $(x,y,z)$-dependence in equation (17) for brevity. We have factorized the large-scale temperature analogously to the density. The projected temperature profiles as functions of the distance from the center of the outburst are demonstrated in the upper panel of Fig. 4. Both models with thermal conduction lead to formation of a $\sim 4$-kpc long temperature precursor, and the difference between these models and the hydrodynamic model is much more prominent than that seen in the surface brightness maps.

Finally, it is possible to measure variations of electron pressure in cluster cores by employing the Sunyaev-Zeldovich (SZ) effect (Sunyaev & Zeldovich, 1972), which manifests itself as distortions of the cosmic microwave background (CMB) radiation in the direction of galaxy clusters due to Compton scattering of the CMB photons off the hot ICM electrons. The magnitude of the SZ effect is proportional to the integral of electron pressure along the line of sight. Therefore, we can produce the SZ map divided by the background model from our simulations:

The result is shown in the lower panel of Fig. 4: as one may have expected, the conductive precursor is now less prominent than in the projected temperature, but stronger than in the surface brightness profiles.

Let us now switch to the 3D runs with anisotropic thermal conduction. For these simulations, we initialize a weak random isotropic magnetic field with coherence length $l_{B}\approx 10$ kpc (as justified in Section 2.2). Anisotropic conduction imprints the characteristic correlation scale of the magnetic field onto the density and temperature structure of the shock front, as seen clearly in Fig. 5. When ion-electron equilibration is instant, both temperature and density jumps of the shock become distorted. The distortion is caused by two effects. First, the jumps are smoothed in regions where the magnetic field is largely radial, while they stay sharper where the field is close to tangential. Second, the shock front becomes corrugated because strong thermal conduction along radial field lines reduces the temperature jump (as seen in Fig. 1 for the 1D models), the shock becomes more isothermal and slows down, while if the field is tangential, it propagates at the adiabatic speed of sound as in ideal hydrodynamics. A finite temperature equilibration time attenuates the link between the ions and electrons, allowing the former to retain the sharpness of the density jump. The shock front is also less corrugated because the ions now preserve their temperature jump. The electron temperature map, however, stays qualitatively unchanged.

We now use the 3D runs to produce surface brightness maps using the same approach as before (equation 16). The results are demonstrated in Fig. 6. The model with instant ion-electron equilibration expectedly leads to noticeable distortions of the shock front, even though projection effects blur the density variations seen more prominently in Fig. 5. Introducing gradual ion-electron equilibration yields a surface brightness map hardly distinguishable from pure hydrodynamics. The current Perseus data (right panel of Fig. 6) does not allow one to distinguish between the two models.

Analogously, we obtain projected temperature maps by applying equation 17 to the 3D simulated temperature. The result can be seen in the upper row of Fig. 7. Distortions of the shock front are now seen in both models because electron temperature is relatively unaffected by a finite equilibration time. The maps of electron pressure integrated along the line of sight are demonstrated in the lower row of Fig. 7.

The above results stay qualitatively the same as long as the coherence length of the magnetic field is appreciably larger than the shock width ($\sim 5$ kpc). If the field is tangled on smaller scales (which, however, appears unlikely because the field is naturally expected to be produced by turbulent gas motions driven by the jet-bubble dynamics on scales $\sim 10$ kpc), the distortions of the shock front associated with the topology of the field become averaged out (even more so by the projection effects). In this case, one can introduce an effective isotropic thermal conductivity as a fraction of the Spitzer value determined by the statistics of the field lines. Due to the current lack of the complete theory of high-$\beta$ MHD turbulence, the exact value of such effective conductivity cannot be calculated. One way to provide its estimate is to apply the theory of strong MHD turbulence, based on the assumption of the so-called critical balance (Goldreich & Sridhar, 1995), locally to calculate how quickly magnetic field lines diverge and, thus, how quickly thermal electrons spread in space (Narayan & Medvedev, 2001). Such calculation leads to effective conductivities of order $1/5$–$1/3$ of the Spitzer value. Then, the case of small-scale magnetic fields is simply reduced to a 1D model with a smaller conductivity. The opposite case of a very large-scale magnetic field on scales of tens of kpc is trivial and described by the 1D model at full Spitzer conductivity along the direction of the field, and ideal hydrodynamics (assuming the field is weak) in the perpendicular plane.

The 900 ks Chandra/ACIS imaging spectroscopy of weak shocks in the core of the Perseus cluster currently give us our best view of bulk transport properties on scales approaching the electron mean-free-path. Even this dataset, however, is not able to provide tight constraints on the presence or absence of conductive precursors once finite electron-ion equilibration is taken into account. Significantly deeper Chandra/ACIS observations (bringing the core exposure to $\sim 3$ Ms) would permit sufficiently accurate measurements of (projected) temperature profiles across the shock to address suppression of electron thermal conduction parallel to the magnetic field, providing an important complement to existing limits on perpendicular heat transport at cold fronts.

The equilibration issue can be sidestepped in our study of electron heat transport if we are able to measure electron and ion temperatures separately. This requires high-spectral resolution so that the (electron) bremsstrahlung can be cleanly separated from the ionic emission lines, and it requires high signal-to-noise so that the ion temperature can be determined from the ratio of charge states of given elements. With only modest spectral resolution ($E/\Delta E\sim 100$), Chandra/ACIS is unable to conduct such a study. Separate measurements of $T_{e}$ and $T_{i}$ were performed in the Perseus cluster by the micro-calorimeter on Hitomi satellite (Hitomi Collaboration et al., 2018), but the limited spatial resolution of these measurements meant that the spectrum encompassed essentially the entire inner core of the cluster. However, ATHENA (c2031, Barret et al. 2020) and Lynx (c2040, The Lynx Team 2018) will enable spatially resolved microcalorimeter resolution ($E/\Delta E\sim 1000$) spectroscopy, allowing us to map $T_{e}$ and $T_{i}$ separately. Lynx, with its subarcsecond spatial resolution, will be a particularly powerful observatory for probing the physics of transport on mean-free-path scales in ICM plasmas.

In our study, we have used the weak shock in the Perseus cluster as a fiducial model. We have done so for several reasons. First, the shock appears remarkably regular, without any significant distortions of the front, which allows one to put constraints on processes that can potentially deform the shock. Second, the Chandra X-ray observatory has provided a wealth of high-resolution data for the Perseus core, allowing us to limit the shock width down to $\sim 1$ kpc. Finally, because the shock is very weak ($M\sim 1.2$), nonlinear kinetic effects could be subdominant (but see Section 4.3), and it should be approximated by fluid models better. However, there is at least one more example of a prominent weak shock in a cluster cool core: the shock in the giant elliptical galaxy M87, located $\approx 13$ kpc from the cluster center, exhibits very similar qualities (regularity, low Mach number, and even the electron mean free path $\sim 0.1$ kpc), however the signal-to-noise ratio of its observations by Chandra is significantly lower than for Perseus. All our results should apply to M87, as well as any very weak shock in cluster cores where the electron mean free path is such that the size of the potential conductive precursor ($\sim 40$ mean free paths) is significant to distinguish between the different models.

We have studied the propagation of a weak shock with Mach number $M\sim 1.2$ in the ICM in a number of fluid models. In the Perseus cluster, the electron mean free path in the vicinity of the shock is $\sim 0.1$ kpc, which is 4 times smaller that the resolution of Chandra. Therefore we cannot infer from the data whether the shock is collisional or collisionless. Nevertheless, there are several Chandra observations of merger shocks at higher Mach numbers $M\sim 2$–3 on cluster outskirts where the mean free path reaches $\sim 20$ kpc. These observations indicate the presence of collisionless processes at play: e.g., Markevitch (2006) argued that the electron-ion equilibration time at the shock in the Bullet cluster is much shorter than the Spitzer value, while Russell et al. (2012) were able to measure the width of one of the two shocks in Abell 2146 at $\sim 6$ kpc, roughly 4 times smaller than the Coulomb mean free path.

Let us discuss the latter observation first. We note that even if the shock is collisionless, i.e., there is ion scattering by plasma instabilities that generates entropy at the shock, the electrons might still be able to transfer heat across the shock, even though they are compressed with the ions by the electric fields ensuring plasma quasineutrality, unless electron plasma instabilities develop as well. In this case, when the heat flux is still mediated by Coulomb interactions, a conductive precursor can develop in the upstream region identically to our simulations. However, this may not be the case, at least for higher Mach number $M\sim 2$–3 shocks, as we speculate below.

Markevitch (2006) showed that the electron temperature behind the shock in the Bullet cluster quickly becomes close to the ion post-shock temperature, suggesting the presence of a kinetic mechanism that heats the electrons above the adiabatic compression. Di Mascolo et al. (2019), however, used observations of the SZ effect by the Atacama Large Millimeter/submillimeter Array and Atacama Compact Array to argue that collisional electron-ion equilibration leads to a better agreement between the radio and Chandra X-ray data. A general discussion on the electron-ion temperature ratio at collisionless shocks can be found in Vink et al. (2015). Fundamentally, this process needs to be studied by means of particle-in-cell plasma simulations. However, the regime of low Mach numbers and high plasma beta relevant for the ICM was investigated only recently (Guo et al., 2014a, b, 2017, 2018; Ha et al., 2018; Kang et al., 2019). In particular, Guo et al. (2017) proposed a mechanism for electron heating at quasi-perpendicular shocks via betatron acceleration of electrons by the amplified magnetic fields (both due to the shock itself and the associated ion instabilities in the downstream region) combined with scattering by electron whistler waves that disrupts the conservation of electron magnetic moment, thermalizes particles, and generates entropy. For our problem, the heating itself is irrelevant, because for very weak shocks the difference between the shock and adiabatic heating is small. On the other hand, the potential generation of whistlers at the shock is important, because even for small shock compression ratios of $\sim 1.3$ the electron temperature anisotropy generated by the amplified field via adiabatic invariance may exceed the whistler instability threshold:

where $T_{\perp}$ and $T_{\parallel}$ are the perpendicular and parallel (to the local magnetic field) temperatures, and $\beta_{\parallel}$ is the ratio of the parallel electron pressure to the magnetic-energy density (Gary et al., 2005). From adiabatic invariance, the temperature ratio is roughly the amplification factor of the magnetic field (equal to the shock compression ratio), which is $\sim 1.3$ for Mach number $M\sim 1.2$, so the left-hand side of equation (20) is $\sim 0.3$. The electron plasma beta in cluster cores is typically $\sim 100$, which gives the right-hand side of equation (20) $\sim 0.01$. We see that the instability threshold is clearly exceeded even for very weak shocks. In case the whistlers are indeed important, our results can be reformulated as follows: if no precursors or deformations of the shock are observed, this could be a signature of electron scattering by whistler waves at the shock.