Observations of Energized Electrons in the Martian Magnetosheath

As the supersonic solar wind plasma encounters an obstacle, it is first slowed down to subsonic speeds and then diverted around the object. At the shock wave ahead of a planet, called the planet’s “bow shock”, individual electrons are accelerated by and electric field within the shock. These energized electrons move quickly along the local magnetic field from one side of the bow shock to the other. Downstream of the bow shock, the two electron populations moving in opposite directions along the magnetic field line should then have crossed the bow shock at the two locations where the field line meets the shock. Since the amount of energy gained by electrons is in general different at the two crossing locations, the two streaming electron populations observed downstream of the bow shock are expected to be energized by different amounts. On the contrary, this study identifies that away from the shock the two populations appear to have been energized very similarly. This may imply an additional acceleration downstream of the bow shock is required. This paper suggests two viable mechanisms that could explain the observations.

When the solar wind encounters an obstacle, the bulk flow is decelerated at the bow shock to become subsonic. The bulk flow is then decelerated further and diverted behind the bow shock so that the magnetic field lines—which are approximately frozen in to the fluid—“drape” around the obstacle. This effect is observed at comets [Koenders \BOthers. (\APACyear2016)] and downstream of planetary bow shocks such as at Earth [Spreiter \BOthers. (\APACyear1966)] and Mars [Nagy \BOthers. (\APACyear2004), C. Mazelle \BOthers. (\APACyear2004)].

The bow shock is the location where the solar wind goes from supersonic to subsonic. As the solar wind slows down, i.e. the bulk speed decreases, individual electrons are accelerated to higher energies by the cross-shock potential. The degree of electron energization is dependent on position at the shock surface: the electron kinetic temperature at the subsolar point in the sheath may reach $\sim$100 eV, while the energization is much less pronounced at the flanks of the shock. This may be compared to the ambient solar wind electron temperature $\sim$10 eV at Mars.

Earth studies of collisionless shocks (such as bow shocks) are applicable to Mars since the physics of shocks is universal. In a planetary bow shock, electrons are energized in a very thin region ($\lesssim$1 km wide) near the shock, and the electron distributions have a “flat-top” shape [Montgomery \BOthers. (\APACyear1970)]. A kinetic model was developed in \citeAmitchell14 to predict the form of this relatively isotropic feature in Earth’s magnetosheath, by propagating electrons along field lines that employed Rankine-Hugoniot jump conditions, described in e.g. \citeAkivelson95. The distributions are not perfectly isotropic however, as noted e.g. in \citeAmitchell12. Comparison of the field-parallel and perpendicular temperatures has been used to suggest that anisotropic heating might also take place [Feldman, Anderson, Bame, Gosling\BCBL \BOthers. (\APACyear1983)]. The electron energization is generally believed to be caused by an ambipolar potential [Scudder \BOthers. (\APACyear1986), Lefebvre \BOthers. (\APACyear2007)], which owes to the strong electron pressure gradient across the shock. However, other explanations of electron energization at shocks, such as via turbulent dissipation [Sagdeev (\APACyear1966), Galeev (\APACyear1976)] have been developed.

The region inside the bow shock, known as the sheath, is where the shocked solar wind diverts around the object and further deceleration of the advecting flux tubes takes place. At the lowest altitudes in the sheath, a transition region separates the external decelerating solar wind ions from an internal region where plasma processes are controlled by the planetary plasma environment. The transition from the sheath to the planetary plasma occurs over a region identifiable by multiple observational signatures. In this transition region, one can find for example the “magnetic pileup boundary” or “MPB” [Acuna \BOthers. (\APACyear1998)], Ion Composition Boundary (e.g., \citeAhalekas19), and Induced Magnetospheric Boundary [Lundin \BOthers. (\APACyear2004)]. The exact location of the boundary is not crucial for the outcome of this paper, so we will adopt the empirical position of the MPB reported in \citeAvignes00 to locate this transition region.

In the Martian sheath near the transition region that includes the MPB, the sheath electron distributions were found to be “eroded” [Crider \BOthers. (\APACyear2000)]; i.e., the phase space density of energetic electrons (at a given energy $\sim$100eV) sharply decreased over this region, by up to 2 orders of magnitude as compared to higher altitudes in the sheath. In that study the erosion was explained by the presence of the neutral Martian corona, which reaches well into the Martian sheath. It was suggested that sheath electrons collide with the neutral gas, and the resulting process of electron impact ionization (a process that has also been reported independently in the Martian foreshock, e.g. \citeAmazelle18) causes the electrons to lose energy. This suggestion was critiqued in \citeAschwartz19, where it was argued that sheath electrons spend too little time at the highest neutral densities for this process to be of importance. Therefore a collisionless kinetic model was developed describing electrons flowing along a solar wind magnetic flux tube as it drapes around Mars. The model accounted for the non-uniformity of the flux tube deceleration, and also distinguished between electrons that pass through the system and those that are temporarily trapped inside the bow shock. The different electron histories were evaluated, which resulted in eroded distributions that compared favorably with electron distributions observed by the MAVEN spacecraft. We may infer from this recent work that to a first approximation electrons evolve collisionlessly in the Martian sheath.

Such collisionless evolution of electrons has been investigated in the context of Earth’s magnetosheath, in \citeAmitchell12, mitchell13, mitchell14. These studies emphasized the non-locality of electron kinetics in the sheath. By non-locality we mean the following: since guiding centers of moving electrons are expected to propagate along the magnetic field lines (which in turn advect with the bulk flow), and moreover because the electrons are transported collisionlessly at speeds much greater than the bulk flow speed, the distribution function $f(\mathbf{v})$ observed at a given point in the sheath will in general be a convolution of electrons that crossed the bow shock at different locations. This communication between distant bow shock locations was termed “electron cross talk”.

In \citeAmitchell12, using Cluster and THEMIS B data it was shown that the electron distributions can exhibit appreciable field-parallel anisotropy. The authors argued that this asymmetry arises because field-parallel and anti-parallel electrons cross the shock at two different locations along the field line, with different cross-shock potentials. Because the magnetic fields at Mars are similarly draped and thread the local bow shock, we may expect cross talk to generate Martian field-parallel electron distributions with this same systematic anisotropy.

This study investigates if the sheath electrons at Mars carry information from the bow shock via “cross talk”. The mission and the data set from MAVEN’s SWEA electrostatic analyzer is first presented in section 3. A statistical study of the energization of the electrons in the sheath is presented in section 4. This study will show the asymmetry of the electron distribution that may be caused by cross talk to be smaller than expected. Processes that could cause these more symmetric electron distributions are suggested in section 5. The paper is summarized in section 6. A detailed presentation of the distribution mapping (used to infer the energization) and error evaluation are provided in the supplementary material to this paper.

In a collisionless plasma, the evolution of the distribution function $f(\mathbf{v},\mathbf{x},t)$ obeys Liouville’s theorem. If the electric and magnetic fields are known along a particle path, one can perform a “Liouville mapping” [Schwartz \BOthers. (\APACyear1998)] to predict how the distribution will vary with position along the path. Conversely, if the particle distributions and magnetic fields at various points along an expected particle path are measured, the electric field along the path can be estimated. Since the variation of magnetic field strength does not influence the pitch angles of particles whose velocities are exactly field-aligned, the field-aligned cuts of the electron distribution are only influenced by the electric field. For particles with a significant perpendicular velocity component, the magnetic field gradients should also be considered when performing a Liouville mapping. This methodology is commonly applied assuming the conservation of magnetic moment, steady-state fields and particle distributions, and the absence of collisions, as in e.g. \citeAlefebvre07.

In the process of Liouville mapping, one must take care to distinguish between the “passing” and “reflected” populations. Both electric fields and magnetic field gradients can reflect particles, which may lead the distributions to develop a loss cone. The term “loss cone” usually refers to particles of certain pitch angles but also there are also regions in energy which are excluded. Therefore, when implementing Liouville mapping only the part of the distributions that can be observed at the two locations should be considered; only the portions outside the excluded in pitch angles and energies should be evaluated.

The sheath is populated by energized solar wind electrons. When they cross the bow shock, these electrons receive a net acceleration that can be attributed to the frame-invariant ambipolar component of the cross-shock potential. The size of the potential depends primarily on the solar wind conditions and the angle of the solar wind flow vector relative to the shock normal. The low-energy region of the electron distribution typically exhibits a “flat top” ($f=$const.) shape in Mars’s magnetosheath [Crider \BOthers. (\APACyear2000)]; electron distributions in Earth’s magnetosheath exhibit a similar feature [Feldman, Anderson, Bame, Gary\BCBL \BOthers. (\APACyear1983)]. The energy at which the flat-top “breaks” may be used to roughly estimate the degree of energization.

Due to their high speeds the electrons will approximately follow trajectories along the instantaneous draped magnetic field. A kinetic theory of electrons in the Martian magnetosheath was developed in \citeAschwartz19; we note that in that study the cone angle $\theta_{c}$ of magnetic field was assumed to be exactly $90^{\circ}$. The cone angle is defined here as follows:

In this study the observed field-parallel anisotropy of the electron distribution function will be parametrized by the quantity $\Delta\Phi$:

The MAVEN (Mars Atmosphere and Volatile Evolution) mission’s primary focus is to study the Martian atmosphere [Jakosky \BOthers. (\APACyear2015)]. As a result the satellite includes a comprehensive suite of instruments capable of providing high-quality measurements of the space plasma environment near Mars. In this paper, we will focus on measurements of the electron velocity distribution provided by MAVEN’s Solar Wind Electron Analyzer (SWEA) instrument [D\BPBIL. Mitchell \BOthers. (\APACyear2016)]. The SWEA instrument has an energy resolution $\Delta\mathcal{E}/\mathcal{E}$ (FWHM) = 17%, providing 79% coverage of the sky at $\sim$$7^{\circ}$$\times$22.5^∘ angular resolution. Magnetic field observations are made by the MAG magnetometer [Connerney \BOthers. (\APACyear2015)]. The moment information from the Solar Wind Ion Analyzer (SWIA) onboard MAVEN [Halekas \BOthers. (\APACyear2015)] is used to get the solar wind speed.

In this study, we will consider pitch angle distributions (PADs) computed onboard the MAVEN satellite. These “survey” data were regularly sampled by the SWEA instrument, with a time cadence of $\sim$2 seconds with 32 distinct energy steps. These energies are given in the spacecraft frame, which for the fast-moving ($\gtrsim$)30 eV electrons considered here is nearly identical to the Mars rest frame—the frame assumed in our calculations, see supplementary document for details. At each energy, an automated algorithm chooses 16 different angular positions in phase space (azimuth+elevation pairs) that were sampled by the detector during the accumulation period; these angular positions are so chosen as to lie roughly on a great circle that intersects the local instantaneous magnetic field direction determined by the MAG instrument. The pitch angle is calculated onboard, by comparison with the contemporaneously measured magnetic field provided by MAG. For the study here most of the time at least one sector was within $\sim$15^∘ of the magnetic field.

The data considered here cover the time range January 1, 2015 to May 15, 2019. The MAVEN spacecraft orbits Mars in an inclined ellipse with a nominal periapsis altitude targeting a pressure corridor at 150-180 km and an apoapsis altitude of 6220 km [Jakosky \BOthers. (\APACyear2015)], resulting in an orbit period of 4.5 hours. The subsolar point of the bow shock is located approximately at 2200 km, well within MAVEN’s orbit. However, over the Martian year the apoapsis moves from being in front of the planet in the solar wind to deep into the tail of the planet. Consequently there are time periods where MAVEN never crosses the bow shock into the solar wind. For this study only orbits where the satellite reaches well into the solar wind are included.

To estimate the energization of the sheath electrons, the electron distributions in the sheath are compared to distributions in the solar wind via Liouville mapping. The simplest approach is to only use the field-aligned (or anti-aligned) portion of the particle distributions to conduct the mapping, yielding the quantities $\Phi_{\parallel}$ and $\Phi_{\downarrow}$ (which appear, e.g., in Eq. 2). Therefore, for this part of the analysis only distributions where observations fell within $30^{\circ}$ of the field are used to represent field-aligned (or anti-aligned) electrons. At low energies photoelectrons can dominate the spectrum, motivating the application of an energy cutoff at 30 eV. The maximum acceleration is expected to be $<$500 eV, which is selected to be the upper energy range for the comparison. Above that energy range, count rates of the instrument are often too low for our purposes and errors in measured strahl can result in an incorrect determination of the energization. So, only energy bins in the 30-500 eV energy range and only energy bins which register $\geq$5 counts are included in the comparison. Examples of two electrons distributions, one from the sheath and a reference spectrum from the solar wind, are presented in Figure 1(a). These distributions are from September 22, 2015—the sheath distribution is measured at the nominal time 12h16m55s and the solar wind reference distribution is averaged over a 10-minute period centered on 11h27m44s.

The solar wind reference distribution, an example of which is presented in Figure 1(a), is derived in the same way for each orbit as follows. A time period where the MAVEN spacecraft is located in the solar wind is first identified based on the empirical bow shock position [Vignes \BOthers. (\APACyear2000)]. Then a 10-minute interval is selected, which is centered on the time in the orbit where the satellite is radially farthest from the empirical bow shock position. The SWEA energy spectra are then averaged over this 10-minute interval to derive one solar wind reference spectrum. For each orbit, all other electron distributions will be compared to this reference spectrum.

Note that the electrons measured in the sheath by MAVEN will not generally be found on a flux tube that connects to the position where the solar wind reference spectrum is sampled. It is here assumed that the reference spectrum approximates the source distribution of electrons entering the sheath during a given orbit.

To evaluate whether the field-parallel sheath electrons can be viewed as the solar wind population accelerated by a parallel electric field, a Liouville mapping is performed. To this end the reference spectrum is shifted by a constant energy to best match each electron distribution of the orbit, yielding the energization ($\Phi_{\parallel}$ or $\Phi_{\downarrow}$). For the mapping to be valid, only distributions moving in the same direction with the respect to the magnetic field are compared (i.e. with the same orientation in the solar wind and sheath). However, since the orbital period is long the solar wind magnetic field orientation might change between the times of the sheath and solar wind measurements; say, if the planet encounters a new flux tube in the interim. This fact is accounted for by identifying the orientation of the electron strahl population (if it is significant enough to be identified), which is either aligned or anti-aligned with the magnetic field. The strahl component is required to maintain the same orientation with respect to the field across both the sheath and solar wind in order for the Liouville mapping to be performed—this accounts for some of the natural variability of the interplanetary conditions.

The Liouville mapping is implemented as a least-squares fit, that calculates the energy the solar wind spectrum would need to be shifted by in order to match the sheath spectrum. This energy is denoted as either $\Phi_{\parallel}$ or $\Phi_{\downarrow}$, respectively, dependent on whether the fit is conducted between two field-parallel or anti-parallel energy spectra. The results of such a fit for a field-parallel energy spectrum can be seen in Figure 1(b). In the example, the derived $\Phi_{\parallel}$ is 54$\pm$4 eV. The Liouville mapping process involves comparing the energies of two spectra at common values of the phase space density; this comparison is made possible by linearly interpolating the discretely sampled data between the solar wind and sheath spectra. The fitting is done by weighting each energy bin appropriately by the count rate. The solar wind distribution effectively maps to the sheath distribution at energies $\gtrsim$$\Phi_{\parallel}$, suggesting the sheath electrons originated from the solar wind. The details of the fitting and the calculation of uncertainties are provided in the supplementary material.

Figure 2 shows the same fitting procedure applied to the anti-parallel electrons. These have entered the sheath through the shock at the opposite end of the field line, where the shock potential may be different. The fitting procedure yields $\Phi_{\downarrow}$=62$\pm$3 eV, similar to the measurement of $\Phi_{\parallel}$ at the same time.

The energy spectra are not corrected for the spacecraft potential $\phi_{sc}$ that arises from spacecraft charging, and this omission introduces systematic error in the estimates of $\Phi_{\parallel}$ and $\Phi_{\downarrow}$ of the order $\lesssim$10 eV. In order to more accurately estimate the quantities $\Phi_{\parallel}$, we may want to correct for the spacecraft potential difference, $\Delta\phi_{sc}$, as measured between the locations of the two spectra: for example ${\Phi_{\parallel}\rightarrow(\Phi_{\parallel}+\Delta\phi_{sc})}$ and ${\Phi_{\downarrow}\rightarrow(\Phi_{\downarrow}+\Delta\phi_{sc})}$. However, this correction is not of great importance for the present work, as we are interested primarily in measuring the difference $\Delta\Phi$=${(\Phi_{\parallel}-\Phi_{\downarrow})}$, i.e. the spacecraft potential correction cancels out with the subtraction.

The energization is calculated via Liouville mapping in this manner for every electron distribution in the $>$4-year data set, enabling the statistical study of $\Phi_{\parallel}$ and $\Phi_{\downarrow}$. The statistical average of the observed parallel energization $\Phi_{\parallel}$ is shown in Figure 3, revealing the spatial structure of the electron energization. As expected $\Phi_{\parallel}$ is nearly zero in the solar wind and increases dramatically near the bow shock location. Near the subsolar point at the bow shock the average $\Phi_{\parallel}$ is $\sim$100 eV.

A focus of this study is the physical configuration where a magnetic flux tube is connected at two ends to the bow shock and these two bow shock locations possess different cross-shock potentials. In such a situation the observed electron distributions in the sheath should be asymmetric due to cross-talk. The energies of the two field-aligned electron distributions should show different amounts of acceleration, i.e. one expects $\Phi_{\parallel}\neq\Phi_{\downarrow}$. Assuming that the cross-shock potential is roughly cylindrically symmetric about the $x_{mse}$ axis, one expects that the difference between $\Phi_{\parallel}$ and $\Phi_{\downarrow}$ would be most suppressed at cone angles $\theta^{\circ}\approx 90^{\circ}$. Likewise the effect would be most stark at the smallest angles. From the Parker spiral model one may quickly estimate a typical cone angle of $\theta_{c}\approx 60^{\circ}$ at Mars. Under these conditions one expects a maximum difference in energization $\Delta\Phi\approx 60$eV at a location behind the bow shock just offset from the subsolar point (see supplementary material for details).

To illustrate the variation of the electron energy near the bow shock under typical conditions, a study of a single orbit is now presented. Orbit 1907 on September 22, 2015 was selected because the orbital geometry and IMF B angle conditions are such that maximum $\Delta\Phi$ may be expected to be large (i.e. on the order of the expected $\sim$60eV mentioned above). During the interval, the (10-minute avg.) solar wind magnetic field had the value $\mathbf{B}$=(-0.93, 2.06, 0.34)nT in MSO cartesian coordinates. This corresponds with a cone angle $\theta_{c}$$\approx$66^∘, which is within 10% to the typical Parker spiral value. This magnetic field is used to calculate the spacecraft vector position in the MSE frame ($\mathbf{x_{mse}}$). The spacecraft’s traversal of the sheath takes place over a range of positions satisfying $|z_{mse}|$$\lesssim$0.5$R_{m}$ ($R_{m}$ denotes the Martian radius), appropriate for this study since the simple bow shock model assumes ${z_{mse}\approx 0}$ (see supplementary material). Likewise, the spacecraft crosses the shock the near the subsolar point, which is of interest because this is where the strongest signal $\Delta\Phi$ is expected according to the model. Although the exact location of the maximum $\Delta\Phi$ depends on the solar wind cone angle, this expectation may be roughly explained by the fact that in our model the cross-shock potential (which sets $\Phi_{\parallel}$) peaks at the sub-solar point. This is where the incident ram energy along the vector normal to the bow shock is maximal.

The results of the fits from MAVEN’s 1907^th orbit on September 22, 2015 are presented in Figure 4(a). The $\sim$30-minute time interval during which the spacecraft crossed into the sheath is divided into 30 subintervals of $\sim$1-minute duration, and averages of $\Phi_{\parallel}$ and $\Phi_{\downarrow}$ within those subintervals are plotted. The central time within each subinterval is displayed by the color. The standard deviation of the $\Phi_{\parallel}$ and $\Phi_{\downarrow}$ data within each subinterval (the scatter) is displayed as error bars. Recall the quantities $\Phi_{\parallel}$ and $\Phi_{\downarrow}$ represent the derived net energization that the two electron populations (moving along and against the field) have each experienced. The spacecraft location for the selected orbit is presented in Figures 4(b)-(d) with the same color coding.

As discussed above, for the conditions of the selected orbit one may expect $\Delta\Phi$$\sim$60 eV just downstream of the bow shock. But, $\Phi_{\parallel}$ and $\Phi_{\downarrow}$ for this orbit lie along the unity line (solid line) in Fig. 4(a), suggesting that the magnitudes of the energizations are actually quite similar. The time where the spacecraft crosses the bow shock (07h49m20s) has been identified by manually looking at the data and is marked by the “+” sign in Figure 4. Note that this shock crossing is detected at a lower altitude than that of the average bow shock [Vignes \BOthers. (\APACyear2000)], indicating that the shock happened to be relatively compressed during this orbit. The largest $\Delta\Phi$ should be observed just behind the bow shock, near the subsolar point. But no such systematic difference is observed, as the points in Fig. 4(a) adhere to the unity line throughout the time interval. The scatter in the data is slightly larger at the bow shock (where $\Phi_{\parallel}$, $\Phi_{\downarrow}$$\sim$100 eV). Although there seems to be a slight bias $\Phi_{\parallel}$$>$$\Phi_{\downarrow}$ at this time, the magnitude of $\Delta\Phi$ is only $\sim$10 eV (i.e. $\ll$60 eV). This figure therefore suggests that electrons in the sheath cannot have been accelerated at the bow shock alone.

To see if the observed trend $\Phi_{\parallel}\approx\Phi_{\downarrow}$ holds generally for other orbits, a statistical evaluation of the two energies $\Phi_{\parallel}$, $\Phi_{\downarrow}$ and their difference $\Delta\Phi$ is presented in Figures 5(a)-(c). Again only times for which the cone angle satisfied 50^∘$<$$\theta_{c}$$<$70^∘ are considered in the averages. The individual $\Phi_{\parallel}$ and $\Phi_{\downarrow}$ values that go into the averages are calculated as already described in this section (following the same selection criteria). The $>$4 years of data are aggregated by calculating the spatial averages of these quantities in a dynamic coordinate system.

The dynamical coordinate system is developed as follows. For each orbit, the location of the bow shock is identified from among times where the derived quantity $\overline{\Phi}\equiv(\Phi_{\parallel}+\Phi_{\downarrow})/2$ is in the 99^th percentile for that orbit. This simple criterion is used because the electrons are known to be strongly energized at the shock. From among these times, the shock crossing is designated as the location where the spacecraft altitude is maximal. Once the location of the bow shock has been specified, this information is used to estimate the local scale of the bow shock relative to the nominal shock size [Vignes \BOthers. (\APACyear2000)]. The spacecraft’s position in MSE coordinates is normalized by the local shock size (evaluated each orbit) before conducting the spatial averages presented in Figures 5(a)-(c). These normalized MSE positions are denoted by the vector components $x^{\prime}$, $y^{\prime}$, $z^{\prime}$. Normalizing in this way minimizes the effects of natural variance of the system. For instance, the signature of the electron energization near the shock is less blurred out by the time-varying size of the shock, and the sheath and solar wind populations are well-separated before averaging.

The presented statistical maps should only include data with similar cone angle $\theta_{c}$, so as not to not mix electrons originating from different locations along the bow shock. The typical $\theta_{c}$ at Mars is about 60^∘, as predicted by the Parker spiral model. For Figure 5 we therefore only include data where the cone angle at the nominal time of the solar wind reference spectrum’s measurement satisfied 50^∘$<$$\theta_{c}$$<$70^∘. Analogous, nearly identical plots (not shown) may be produced for the cases when the magnetic field had the same orientation with opposite polarity ($\theta_{c}$=120^∘).

In the averages presented in Figure 5, only data from the regions nominally occupied by the solar wind and sheath are included [Vignes \BOthers. (\APACyear2000)]. Also, only data where the spacecraft position satisfied $|z_{mse}|$$<$0.3$R_{m}$ are included. This reflects the fact that the parallel asymmetry, if it exists, should be most stark in the plane $z_{mse}$=0. The (unscaled) surface of Mars and the empirical boundary locations of the bow shock and the transition region where the MPB is located are presented by the black lines in the figure. Also, a drawing of the magnetic field with $\theta_{c}$=60^∘ illustrates how the magnetic field encounters the system; this is shown as a dashed line in the sheath region because the draped field is actually curved there. It is quite obvious that the statistical result of Figures 5(a)-(b) is similar to Figure 4—that is, $\Phi_{\parallel}$ and $\Phi_{\downarrow}$ are similar in magnitude and the largest $\Phi$ are seen close to the subsolar point.

Because of the selected range of cone angles, $50^{\circ}$$<$$\theta_{c}$$<$$70^{\circ}$, the energization of the electron distributions may be expected to be asymmetric. That is, we may expect ${\Phi_{\parallel}\neq\Phi_{\downarrow}}$, and some difference between Figure 5(a) and 5(b) should be observed. Therefore $\Delta\Phi$ is first calculated for each individual point before deriving the average, which is presented in Figure 5(c). The difference is close to zero throughout the sheath, which is an unexpected result in this study. Near the bow shock there is a slight trend in the $\Delta\Phi$ data, with $\Delta\Phi\gtrsim 0$ in the region $y_{mse}>0$ and $\Delta\Phi\lesssim 0$ in the region $y_{mse}<0$. This systematic signal is strongest at the flanks, with a maximum strength of 10-20 eV. The region where the maximum $\Delta\Phi$ may be expected is outlined by a dashed trapezoid—although the model predicts $\Delta\Phi$$\sim$60eV in this region (see next paragraph), the actual signal varies within the range -21eV$<$$\Delta\Phi$$<$16eV. In other words, the observed $\Delta\Phi$ is only about $\lesssim$25% of the predicted value in the region where the signal is expected to be strongest.

For comparison, Figure 5(d) shows a model prediction of $\Delta\Phi$ just downstream of the bow shock in the region $z_{mse}$=0, assuming a cone angle of 60^∘ and a peak cross-shock potential of 100 eV at the subsolar point (see supplementary material for details). For such conditions the expected $\Delta\Phi$ is estimated to be 61 eV, as calculated from the difference in cross-shock potentials at two ends of a flux tube. In the model, ${\Delta\Phi=0}$ near where the downstream magnetic field is tangent to the shock surface ${\mathbf{B}\cdot\hat{\mathbf{n}}}$=0.

The predicted signal Figure 5(d) differs from the observed signal Figure 5(c) in a number of important respects. Note that the model predicts $\Delta\Phi$$\geq$0 throughout the sheath, whereas the actual signal skews negative in the region $y_{mse}<0$, as mentioned above. Also the observed $\Delta\Phi$ is very close to zero near the subsolar point, whereas this is where the model predicts $\Delta\Phi\approx$60 eV. In fact, the observations show no systematic signal of strength $\sim$60 eV anywhere in the sheath, and in the statistical map the maximum and average values of $|\Delta\Phi|$ observed are about 20 eV and 8 eV respectively.

We have assumed so far that the electrons in the sheath have evolved collisionlessly, and moreover that the magnetic moment is a conserved quantity. If other processes such as collisions and nonlinear wave acceleration are present in the flux tube, the modification of the distributions will be clearly observable in the particles with large pitch angles. Therefore an investigation is made to see if the angular distributions of sheath electrons can be described as solar wind electron distributions that have been exposed to a quasi-static electric field (ignoring nonlinear effects). In investigating electrons with large pitch angles, magnetic field gradients must also be considered. This investigation is based on Liouville mapping; first the field-aligned distribution is used to identify the energization (parametrized by $\Phi_{\parallel}$, $\Phi_{\downarrow}$) due to the electric field and then conservation of magnetic moment is applied to calculate the mapped angular distribution. Note that in the presence of a spatially-varying magnetic field, the pitch angles of particles will change in order to conserve the magnetic moment.

Two pitch angle distributions, one from the solar wind and another from the sheath, are presented in Figure 6(a). The data are from 5 August, 2016 where the sheath was measured at 00h02m48s and the solar wind reference spectrum is derived from a 10-minute window centered at 01h45m28s. The sheath distribution was measured by SWEA at energy 132 eV. The solar wind pitch angle distribution displayed in Fig. 6(a) is taken from the solar wind reference as usual, but interpolated to the appropriate energy. That is, the field-aligned part ($\theta$$<$90^∘) of the solar wind distribution is interpolated to the energy $\mathcal{E}_{\parallel}$, so that the total energy of the electrons (after they migrate from the solar wind to the sheath location) would match the sheath distribution; i.e., $\mathcal{E}_{\parallel}+\Phi_{\parallel}$=132 eV. An analogous interpolation process, using $\Phi_{\downarrow}$, is used to construct the rest of the distribution (where $\theta$$>$90^∘). In this way, a solar wind pitch angle distribution is constructed that may be mapped to the sheath distribution.

The results of the Liouville mapping are presented in Fig. 6(b). In this panel the solar wind distribution from panel 6(a) has been mapped to new pitch angles—as the distribution would appear once the electrons had migrated to the selected location in the sheath. The change in pitch angle $\theta$ can be predicted by accounting for the gain in parallel energy ($\Phi_{\parallel}$ and $\Phi_{\downarrow}$, derived as shown in Fig. 1) and the conservation of magnetic moment. That is, given an electron’s initial velocity components $v_{\parallel,1}$, $v_{\perp,1}$, the ratio of the magnetic fields in sheath and solar wind $B_{sw}/B_{sh}$, and the gain in energy $\Phi_{\parallel}$ (or $\Phi_{\downarrow}$), the final velocity of a particle can be computed. The relevant equations for computing the final (sheath) velocity components are described in section 1.1 of the supplementary document.

For the purposes of the pitch angle mapping, the magnetic field values in the solar wind and sheath locations were measured as $B_{sw}$=4.6 nT and $B_{sh}$=9.5 nT. The parallel and anti-parallel energization of the solar wind electrons were $\Phi_{\parallel}$=22.6eV and $\Phi_{\downarrow}$=30.3eV, respectively. The position of the MAVEN spacecraft at the two locations, in MSO coordinates, is shown in Figure 6(c).

For this example, no mapping is conducted at pitch angles $|\theta$$-$$90^{\circ}|$$\lesssim$$40^{\circ}$ because the ratio of magnetic fields dictates that any such particles from the solar wind would have been reflected (the mirror condition) before they reached the location of the sheath observation. The pitch angles corresponding to the mirror condition, for the particular magnetic fields used in the mapping, are shown as vertical lines in Fig. 6(a). Note that these boundaries are only valid if a strong magnetic field doesn’t exist between the sheath and solar wind locations—such a field would change the domains in phase space of the “passing” and “reflected” populations.

Figure 6(b) demonstrates that there is good agreement between the solar wind reference distribution and the Liouville-mapped sheath distribution. If pitch angle scattering or perpendicular wave heating were present in the sheath, the Liouville mapping would fail and the sheath distributions would be expected to be more isotropic. The importance of these two processes is small, as can be seen from the significant asymmetry of the distribution, where the phase space density of the field-parallel beam ($\theta$=$0^{\circ}$) is about an order of magnitude greater than that of the anti-parallel electrons. This inferred lack of pitch-angle scattering provides a key to understanding the similarity between $\Phi_{\parallel}$ and $\Phi_{\downarrow}$ as presented in Figure 5.

In this study we have operated under the assumption that the Martian magnetosheath is a collisionless medium. We have further assumed that 1) the electrons observed in the sheath are sourced from the solar wind and are only energized by a parallel (or anti-parallel) electric field that increases the particle energy by an amount $\Phi_{\parallel}$ (or $\Phi_{\downarrow}$), and 2) the solar wind and sheath do not vary dramatically on timescales $\Delta t$$\sim$1 hour, allowing distributions measured at two points in the MAVEN orbit to be compared. This simple framework is sufficient to explain the field-aligned sheath distributions (e.g., Fig. 1), and accounting for the magnetic field can also explain the pitch angle distributions (Fig. 6). Therefore any process of collisions or plasma wave acceleration must have only a minor effect on the sheath electron distributions.

This study shows that the electron energization is more symmetric than expected from the electron “cross talk” picture, as demonstrated by Figures 4 and 5. This suggests that the model of local shock energization, quantified explicitly in the supplementary document, does not fully describe the electron distributions. Some additional acceleration must be introduced to explain the relative symmetry of the energization experienced by the sheath electrons. This leads us to consider two different explanations for why the observed $\Delta\Phi$ is not more significant, detailed below.

In this paper we analyzed the energization of electrons in the Martian magnetosheath. The $>$30eV electrons considered in this study move quickly enough to traverse the magnetosheath in about 1 second, so that during this time the field line along which an electron moves is essentially fixed. Due to the different cross-shock potentials at the two locations where the field line intersects the shock (under typical solar wind conditions), we may expect to see a significant difference (as much as $\sim$60 eV) between the derived quantities $\Phi_{\parallel}$ and $\Phi_{\downarrow}$. The absence of such a signature, as demonstrated for a single orbit (Fig. 4) and in a statistical average of the $z_{mse}$=0 plane (Fig. 5), indicates that our basic model of the Martian bow shock needs to be reconsidered.

We presented two possible resolutions for the discrepancy between our model and the observations of $\Delta\Phi$. In one case, we suggested that an ambipolar, (nearly) potential electric field distributed throughout the magnetosheath region could plausibly explain the observation $\Delta\Phi$$\sim$0. In another case, we considered that our predictions for $\Delta\Phi$ would change (and possibly agree better with the observations) if a more self-consistent model for the cross-shock potential $\Phi_{s}$ were applied. Further investigation of these two explanations is beyond the scope of the present paper, which is observational in its focus. But in any case, we may conclude that diffusive effects such as collisions and wave-particle interactions have a negligible effect on the electron distributions through most of the magnetosheath. This is based on the effectiveness of the Liouville mapping technique.

The study was motivated by the simple collisionless model of the sheath developed in \citeAschwartz19, which sought to explain the so-called “erosion” of the electron flux observed in the inner magnetosheath. This study suggests that additional acceleration inside the sheath is taking place, obscuring the observational signature that would otherwise be seen if electrons were solely energized at the shock. We are not concerned with the electron flux erosion here. However, we note that if a significant electrostatic field is present in the magnetosheath (as suggested above), incorporating this field’s effect may improve the \citeAschwartz19 model. Not incorporated into this study is the interaction of electrons with neutral hydrogen in the Martian foreshock [C\BPBIX. Mazelle \BOthers. (\APACyear2018)].

The significant energization of electrons observed at a planetary bow shock is not unique to the Martian system. We speculate that techniques similar to those employed here may be applicable to magnetosheaths at Venus and Earth, for instance. No two systems are identical, however, and we foresee that the conditions at other planets may contradict some assumptions applied here. For instance, at Earth one cannot assume that electrons flow along essentially fixed field lines due to the larger shock scale [J\BPBIJ. Mitchell \BOthers. (\APACyear2012)]. Though such details may complicate the observational analysis, it is nonetheless clear that Liouville mapping can be an effective technique for probing the electric field structure in planetary shocks and magnetosheaths elsewhere in the solar system.