For an ideal gas, the notion of thermodynamic equilibrium requires that the system's temperature be spatially uniform. We demonstrate that for a collisionless plasma, kinetic equilibrium can be achieved even in the presence of a temperature gradient. Motivated by recent Magnetospheric Multiscale (MMS) magnetopause observations of thin current layers exhibiting electron-scale temperature gradients, here we present an exact solution to the Vlasov–Maxwell system that accounts for the observed electron temperature gradient and reproduces much of the plasma's spatial variation along one dimension. The current layer in the Vlasov–Maxwell model is self-consistently supported by a prominent crescent-shaped electron velocity distribution that naturally arises due to the temperature transition, which agrees with the electron distribution structures measured by MMS during two asymmetric current layers encountered at the magnetopause.

## I. INTRODUCTION

In the context of the thermodynamics and statistical mechanics employed to describe the behavior of ideal (i.e., collisional) gases like air, one often considers an initial setup where two gases of different temperatures are placed next to each other and left to interact. Eventually, after enough time has passed, we intuitively expect the two gases to reach the same temperature—once this has happened, we would say the system has reached “thermodynamic equilibrium.” As we demonstrate in this paper, the situation is more nuanced for collisionless plasmas. Here, we discovered an equilibrium (steady-state) kinetic model for a temperature gradient that (1) is an exact solution to the Vlasov equation governing the evolution of collisionless plasmas, (2) self-consistently solves Maxwell's equations of electromagnetism, and (3) agrees with observations of electron-scale plasma measurements gathered by NASA's Magnetospheric Multiscale mission at the interface between the Sun's solar wind and Earth's magnetic field. Thus, for collisionless plasma systems, counter-intuitively this means that two neighboring regions of a collisionless plasma can have different temperatures and maintain that temperature difference indefinitely.

When a system is said to be in thermal equilibrium, for ideal collisional gases this typically means that the temperature is uniform and the velocity distribution function has relaxed to a Maxwellian (Schroeder, 2000; Gurnett and Bhattacharjee, 2005). For a collisionless plasma governed by the Vlasov–Maxwell system of equations, the concept of thermal equilibrium is more nuanced because temperature differences between constituent plasma species can be sustained for long periods of time in the absence of rapid inter-particle collisions. Consequently, collisionless plasma environments are characterized by the formation and persistence of non-Maxwellian velocity distribution functions (e.g., Ng *et al.*, 2011; 2012; Bessho *et al.*, 2014; 2016; 2017; Wang *et al.*, 2016; 2018; Shuster *et al.*, 2014; 2015).

There have been a great deal of studies making analytic progress and presenting detailed solutions to the Vlasov–Maxwell system for various plasma configurations (e.g., Harris, 1962; Alpers, 1969; Channell, 1976; Mynick *et al.*, 1979; Schindler and Birn, 2002; Camporeale and Lapenta, 2005; and references therein). While solutions for thin current sheets with thicknesses smaller than ion spatial scales have been considered previously (e.g., Sitnov *et al.*, 2006), most equilibrium solutions applicable for modeling magnetospheric current sheet observations do not typically consider scales smaller than a characteristic ion gyroradius. Classes of asymmetric equilibria have been developed that feature ion-scale temperature gradients (e.g., Mottez, 2003; Aunai *et al.*, 2013; Allanson *et al.*, 2017). Force-free equilibria have also been considered (e.g., Neukirch *et al.*, 2009), in some cases yielding spatially oscillatory density and temperature profiles (Wilson *et al.*, 2017). Within the broad class of tangential discontinuities (e.g., Roth *et al.*, 1996), many of these previously reported equilibrium current sheet configurations are applicable to symmetric or force-free current layers exhibiting an ion-scale thickness. Here, we consider the case of an asymmetric, non-force-free, electron-scale current layer suitable for modeling recent observations from NASA's Magnetospheric Multiscale (MMS) mission.

MMS has detected very thin current layers exhibiting electron-scale temperature gradients with spatial scales on the order of a few electron gyroradii in the vicinity of Earth's reconnecting magnetopause (e.g., Shuster *et al.*, 2019; 2021). Such filamentary currents have also been reported in association with reconnection X-line and exhaust regions (Phan *et al.*, 2016), and in the turbulent magnetosheath where a form of small-scale electron-only reconnection has been observed independent of large-scale ion coupling (Phan *et al.*, 2018). Motivated by these discoveries, in this paper we present a model for an electron-scale temperature gradient that exists in exact kinetic equilibrium while self-consistently maintaining electron crescent-like velocity distributions. Our solution to the Vlasov–Maxwell system mimics the observed electron temperature gradient and captures most of the plasma's spatial variations. In this Vlasov–Maxwell model, the current density is supported by a pronounced crescent-resembling electron velocity distribution structure that develops as a result of the transition in temperature across the layer, which is qualitatively consistent with the particle measurements from MMS.

In Sec. II, we present a detailed description of the model and its relevant parameters with an initial comparison to MMS data. Direct comparisons of the plasma moments and velocity-space structure of electron distributions measured by MMS and predicted by our Vlasov–Maxwell solution are discussed in Sec. III. Our summary and conclusions are addressed in Sec. IV.

## II. VLASOV–MAXWELL EQUILIBRIUM FOR AN ELECTRON-SCALE TEMPERATURE GRADIENT

Motivated by recent MMS magnetopause observations (Shuster *et al.*, 2019; 2021), here we seek self-consistent solutions to the Vlasov–Maxwell system suitable for modeling an asymmetric current layer with the following properties: (1) electron-scale electron temperature gradient $Te2>Te1$, (2) crescent-shaped electron velocity distributions *f _{e}*, (3) asymmetric plasma parameters across the layer including $B2<B1$ and $n2>n1$, (4) strong electron bulk velocity

*U*supporting the current density

_{ey}*J*, and (5) hot ($Ti/Te\u224810$), non-drifting (

_{y}*U*= 0) Maxwellian ion distributions

_{i}*f*, where “1” denotes the low density (high magnetic field) magnetosphere (MSP) side of the layer, and “2” denotes the high density (low magnetic field) magnetosheath (MSH) side. We note that for these localized electron-scale current layers observed to be embedded within the larger-scale magnetopause transition itself, the local electron temperature gradient with $Te2>Te1$ differs from the overall trend of the larger-scale magnetopause transition, where the temperature on the MSP side is higher than on the MSH side with

_{i}*T*>

_{MSP}*T*due primarily to the contribution of electrons of magnetospheric origin with energies greater than a few keV (e.g., Wang

_{MSH}*et al.*, 2017). Nevertheless, throughout this paper we use the abbreviations “MSP” and “MSH” to indicate “toward the MSP proper” and “toward the MSH proper,” respectively. We restrict our consideration to steady-state systems ($\u2202/\u2202t\u21920$) with spatial variations occurring along only one dimension, which we choose to be the

*x*-direction. Following the approach of Camporeale and Lapenta (2005), we allow the magnetic field $B$ and corresponding magnetic flux function $A$ to have only one component so that $B=Bzz\u0302=(\u2202Ay/\u2202x)z\u0302$. In the present case, we require

*A*to be monotonically increasing so that $Bz>0$. For species

_{y}*s*, we make use of two constants of particle motion within this system: (1) the total energy and (2) the momentum in the

*y*-direction

where $v2=vx2+vy2+vz2$, and $\varphi (x)$ specifies the spatially varying electric potential. For ions (*s* =* i*), $qi=+e$, while for electrons (*s* =* e*), $qe=\u2212e$. Any function of these constants of motion $fs(cs1,cs2)$ will satisfy the steady-state Vlasov equation

Because the ions are non-drifting, after choosing a particular form for *f _{e}*, a self-consistent solution to the Vlasov–Maxwell system is obtained by imposing quasineutrality (

*n*=

_{i}*n*) and solving Ampère's law

_{e}For the ion distribution function *f _{i}*, we choose a Maxwellian population specified by the initial temperature ratio $Ti/Te1$, where the ion temperature

*T*is constant throughout the layer. For the electron distribution function

_{i}*f*, we choose a form that resembles a Maxwellian with a spatially varying temperature controlled by the function $T(ce2)$

_{e}where the $T(ce2)$ function is given by

With $ce2$ given by Eq. (2), the electron distribution satisfies the electron Vlasov equation and takes the following form:

The free parameter *S* (in units of $[eB1rL]\u22121$, or equivalently inverse momentum) influences the thickness of the current layer and the shape of the electron distribution function by determining how sharply the temperature transitions from $Te1$ to $Te2$ in velocity space. On the MSP of the layer, *f _{e}* is a Maxwellian population with temperature $Te1$, while on the MSH side of the layer,

*f*is a hotter Maxwellian with temperature $Te2$. In the center of the current layer,

_{e}*f*exhibits a mixture of these two populations, which we discuss in detail below. All quantities are normalized by reference values taken on the MSP side of the layer; for example, the magnetic field, density, temperature, mass, velocity, and length are normalized by

_{e}*B*

_{1},

*n*

_{1}, $Te1$,

*m*,

_{e}*v*, and

_{th}*r*, respectively, where $vth=Te1/me$ is taken as the electron thermal speed, and $rL=mevth/eB1$ is the thermal electron gyroradius on the MSP side.

_{L}The model coordinate system {*x*, *y*, *z*} corresponds to typical GSE coordinates at the subsolar magnetopause: $+x$ points sunward along the sun-Earth line, $+z$ is directed northward along the magnetic field direction in the magnetosphere, and $+y$ completes a right-handed system and is oriented along the current (opposite to the electron flow). For comparison to the two MMS events presented in Figs. 2–4, the model quantities are plotted with respect to –*x*. Additionally, the MMS data are plotted in field-aligned coordinates where the $\u22a51$ direction points along $(\u2212Ue\xd7B)\xd7B$ (roughly aligned with the $E\xd7B$ direction), and the $\u22a52$ direction points along $\u2212Ue\xd7B$ (roughly along **E** and $\u2207ne$). Since the $\u22a51$ direction is approximately oriented along the model –*y* direction, the figures are adjusted accordingly to facilitate visual comparisons.

### A. Imposing quasineutrality

The electron distribution function *f _{e}* in Eq. (8) depends on both $\varphi $ and

*A*. Because of the chosen form for the $T(ce2)$ function,

_{y}*f*cannot be written in a factorized form that conveniently separates the $\varphi $ and

_{e}*A*dependences. Nevertheless, the quasineutrality requirement enables determination of $\varphi $ as a function of

_{y}*A*. Integrating Eq. (5), the ion density is readily obtained as a function of $\varphi $, which we then equate to the integral expression for the electron density

_{y}We numerically obtain a discretized representation of $\varphi (Ay)$ by first constructing an array with *k* elements of *A _{y}* values, $Ayk$, and then implementing a root-finding algorithm to determine the value $\varphi =\varphi k$ that satisfies Eq. (10) for each $Ayk$. Using this $\varphi k(Ayk)$, we numerically integrate

*f*to acquire the electron moments as functions of

_{e}*A*and $Uey(Ay),\u2009Te(Ay)$. Since

_{y}*A*is assumed to be monotonically increasing, we have $Bz>0$, and so the magnetic field $Bz(Ay)$ may be obtained directly from these particle moments using the pressure balance condition derived from integrating the force balance equation

_{y}where *P*_{1} is the total pressure (a constant) given by $P1=B122\mu 0+n1(Ti+Te1)$. Finally, utilizing the definition of the magnetic vector potential *A _{y}*, we obtain the spatial parameter

*x*explicitly as follows:

from which we find the moments and fields quantities as functions of *x*, such as $Uey(x),\u2009Te(x),\u2009Bz(x)$, etc.

Although Poisson's equation is not strictly satisfied, an electric field *E _{x}* consistent with the potential $\varphi (Ay)$ acquired from quasineutrality is given by $Ex=\u2212\u2202\varphi /\u2202x=\u2212(\u2202\varphi /\u2202Ay)(\u2202Ay/\u2202x)=\u2212(\u2202\varphi /\u2202Ay)Bz$. From a dimensionless Gauss' law, the actual charge difference can be assessed via

where $n\u2032i\u2212n\u2032e=(ni\u2212ne)/n1$ is the normalized charge difference, the electric field $E\u2032x=Ex/(vthB1)$ and spatial coordinate $x\u2032=x/rL$ are dimensionless quantities, $vA,e=B1/\mu 0men1$ is the electron Alfvén speed based on *B*_{1} and *n*_{1}, and $c=1/\mu 0\u03f50$ is the speed of light. For the magnetopause conditions encountered by MMS in the events considered ($B1\u224830$ nT, $n1\u224810$ cm^{−3}, $Te1\u224870$ eV, $vA,e\u2248$ 9000 km/s), the ratio $vA,e/c$ is on the order of 0.03. The peak electric field ranges from about 0.1 to 0.2 $vthB1$, and $\u2202Ex/\u2202x$ is on the order of 0.01 to 0.1 $vthB1/rL$. With these parameters, we estimate the actual charge imbalance in Eq. (13) to be roughly 10^{–4} or 10^{–5}. Thus, quasineutrality is a reasonable approximation for these events.

A visualization of various properties of the Vlasov–Maxwell solution described above is featured in Fig. 1, which displays results from twelve different model runs obtained by systematically varying each of the independent parameters *β _{e}*, $Ti/Te1,\u2009Te2/Te1$, and

*S*. Figures 1(a)–1(d) illustrate how the spatial dependence of the magnetic field $Bz(x)$, density

*n*(

*x*), electron bulk velocity $Uey(x)$, and current density $Jy(x)$ changes with three different values (black, blue, and green traces) of the initial electron beta

*β*, ion to electron temperature ratio $Ti/Te1$, electron temperature ratio $Te2/Te1$, and the distribution steepness parameter

_{e}*S*, respectively. Figure 1(e) presents an example of the force balance between $J\xd7B$ (blue) and $\u2207\xb7P$ (green points), where $P=Pi+Pe$, along with decompositions of $\u2207\xb7P$ into $\u2207\xb7Pi$ (red) and $\u2207\xb7Pe$ (black), and $\u2207\xb7Pe$ into $ne\u2207\xb7Te$ (orange) and $Te\xb7\u2207ne$ (aqua) for a model solution with the following parameters:

*S*= 10, $Te2/Te1=2,\u2009Ti/Te1=10$, and $\beta e=0.2$. Scaling relationships with black, blue, and green points corresponding to the twelve model runs in Figs. 1(a)–1(d) (three runs per row) are provided in Figs. 1(f)–1(i) and will be described with more detail in Secs. II B–II E. The parameters displayed in the dashed boxes adjacent to Figs. 1(f)–1(i) were held constant for each set of three runs shown in Figs. 1(a)–1(d). The effect of the free parameter

*S*on the half-thickness of the current layer and the shape of 2D electron distribution slices in

*v*–

_{x}*v*velocity space (taken along

_{y}*v*= 0) is shown in Figs. 1(i) and Fig. 1(j–1), respectively, with 1D cuts along

_{z}*v*= 0 included below each distribution slice. We performed two additional runs for the

_{x}*S*= 2 and

*S*= 5 cases shown in Fig. 1(j) and Fig. 1(k), respectively. The effect that increasing the electron temperature ratio $Te2/Te1$ has on the distribution's structure is presented in Figs. 1(m)–1(o), again with 1D cuts along

*v*= 0 beneath the distribution panels.

_{x}The Vlasov–Maxwell solutions displayed in Fig. 1 are asymmetric ($n2>n1$ and $B2<B1$) and have thicknesses on the order of a few electron thermal gyroradii. The structure of the 2D electron distribution slices significantly depends on the electron temperature ratio $Te2/Te1$ and the steepness parameter *S*. For small values $S\u22722$, there is a relatively smooth transition in velocity-space along *v _{y}* between the hotter electron population with $Te2$ and the colder population with $Te1$ [Fig. 1(j)]. For larger values $S\u227310$, the velocity-space transition is sharp, resulting in pronounced semi-circular (crescent-like) contours of the phase space density [Fig. 1(l)]. For a given

*S*, increasing the electron temperature ratio $Te2/Te1$ exaggerates the transition between the hot and cold populations as seen in Figs. 1(m)–1(o). When the temperature ratio is small $Te2/Te1\u22721.2$, there is only a slight change in the distribution's structure [Fig. 1(m)], whereas increasing the temperature ratio to $Te2/Te1\u22731.5$ visibly amplifies the velocity-space transition [Fig. 1(o)].

### B. Scaling properties across the asymmetric current layer

In this section, we derive several useful scaling relationships between characteristic parameter ratios for the Vlasov–Maxwell solution.

#### 1. Density ratio: $n2/n1$

From quasineutrality in this model, the density ratio $n2/n1$ is set by the ion and electron temperature ratios. On the high-density (MSH) side of the layer as $x\u2192\u221e$, we have $\varphi (x)\u2192\varphi 2$ and $Ay(x)\u2192\u221e$, which means that $T(ce2)\u2192Te2$. In this limit, the electron distribution function becomes

Thus, integrating over velocity space, the MSH electron density *n*_{2} is given by

Equating this expression to the MSH ion density, we solve for $\varphi 2$ as follows:

where the exponent *a* is given by

Substituting Eq. (18) into the expression for the ion density and dividing by *n*_{1}, we obtain the desired density ratio

which is plotted as a function of *T _{i}* in Fig. 1(g) (black trace). Considering the theoretical limits $Ti/Te\u226b1$ and $Ti/Te\u226a1$, which correspond to either infinite or zero ion temperature, the range for

*a*is $0\u2264a\u22643/2$. In the limit where $Ti=Te$, then

*a*= 3/4. For a strong temperature gradient ($Te2/Te1=2$) near the magnetopause ($Ti/Te1=10$), then

*a*= 1/4 and $n2/n1\u22481.2$.

Despite the relatively large ion temperature $Ti/Te1=10$ in Fig. 1(e), the electrons' contribution to the overall force balance is roughly equal to that of the ion $Ti\u2207n$ term (red trace), mainly due to the large $ne\u2207\xb7Te$ term (orange trace) associated with the electron temperature divergence. Even in the limit of infinite ion temperature ($Ti/Te1\u2192\u221e$), the finite $ne\u2207\xb7Te$ term will still contribute significantly to the force balance. This is because increasing *T _{i}* reduces $n2/n1$ as shown in Eq. (20), which in turn decreases the magnitude of $\u2207n$ and thus limits the overall effect of the ion $Ti\u2207n$ term.

#### 2. Magnetic field ratio: $B2/B1$

Pressure balance across the current layer requires

which is readily obtained by integrating the force balance equation $J\xd7B=\u2207\xb7P\u2192\u2212Bz(\u2202Bz/\u2202x)/\mu 0=\u2202Pxx/\u2202x$. Dividing both sides of Eq. (21) by $n1Te1$, we have

where $\beta e=2\mu 0n1Te1/B12$. Using Eq. (20), we can solve for $B2/B1$ in terms of the ion and electron temperatures

where $\beta e*$ is given by

and where *β _{e}* is chosen such that $0<\beta e<\beta e*$. Equation (23) is plotted as a function of

*β*in Fig. 1(f) (black trace). For $Ti>Te1$ and $Te2>Te1$, the expression in brackets under the square root in Eq. (23) is negative. This implies that for a given set of temperature parameters, the ratio $B2/B1$ in Eq. (23) would become imaginary if $\beta e>\beta e*$. As $\beta e\u2192\beta e*$, we have $B2/B1\u21920$, and the structure of the Vlasov–Maxwell solution undergoes a bifurcation: for $\beta e>\beta e*$, the current layer becomes symmetric with a

_{e}*B*reversal and a corresponding

_{z}*A*profile that is no longer a monotonically increasing function of

_{y}*x*. Since the focus of this paper is on asymmetric equilibria, we consider only cases for which $\beta e<\beta e*$.

#### 3. Approximation for the maximum $Uey/vth$

From Ampère's law in Eq. (4), we can obtain an approximation for the maximum $Uey/vth$ as a function of the previously derived parameter ratios. Near the maximum velocity $Uey*$ at the center of the current layer, we estimate $dB/dx\u2248(B2\u2212B1)/L\u2009rL$, where the dimensionless quantity $L=\Delta x/rL$ is a number specifying the width of the layer in units of *r _{L}*, and we can approximate the density as $ne\u2248(n2+n1)/2$. Inputting these estimations into Ampère's law, we have

Solving for $Uey*/vth$, we find

For the runs presented in Fig. 1(c), choosing a thickness in the range *L* = 2–3 provides a reasonable estimate of how the peak *U _{ey}* scales with density, temperature, and magnetic field strength [see the magenta and purple traces in Fig. 1(h)]. Section II C describes a more sophisticated approach for obtaining an explicit representation of

*U*and hence a more accurate approximation for $Uey*$.

_{ey}### C. Piecewise Maxwellian limit for *f*_{e}

_{e}

In the limit as $S\u2192\u221e$, the $T(ce2)$ function in Eq. (7) approaches a step function in velocity-space along the *v _{y}* direction. Thus, in this limit

*f*approaches a piecewise Maxwellian

_{e}where $fe1=fe0\u2009exp\u2009[\u2212ce1/Te1],\u2009fe2=fe0\u2009exp\u2009[\u2212ce1/Te2]$, and $vy*=eAy(x)/me$ is the spatially dependent velocity marking the velocity-space transition between $fe1$ and $fe2$. This conveniently allows the moments of *f _{e}* to be computed analytically in a piecewise fashion.

The exact expression for the electron density *n _{e}* as a function of $\varphi $ and

*A*in this limit is given by

_{y}where $erf(x)$ is the error function, and $\Phi (x)$ is the cumulative distribution function for the standard normal distribution. As $Ay\u2192\u2212\u221e$, we have that $ne\u2192n1$, while for $Ay\u2192+\u221e$, Eq. (29) reduces to Eq. (15). To compute the *v _{y}* integrals, we make use of the symmetry of the Gaussian functions to conveniently adjust the limits of integration, and we utilize the relationship between the cumulative distribution function $\Phi (x)$ and the error function $erf(x)$ (see the Appendix for full derivations).

For the electron bulk velocity *U _{ey}*, we find

The black trace in Fig. 1(h) is a plot of Eq. (30) evaluated at *A _{y}* = 0, which corresponds to a spatial location close to where the maximum

*U*occurs, and hence represents a more accurate scaling relationship for the ratio $Uey*/vth$. This can be understood by considering how the electron distribution in Eq. (8) depends on

_{ey}*A*. Starting on the MSP side with $Ay=\u2212\u221e$, the electron distribution $fe=fe1$ is composed entirely of colder electrons with temperature $Te1$. As

_{y}*A*increases, electrons from the hotter $Te2$ population begin to be added to the $\u2212vy$ side of the distribution. The contribution to

_{y}*U*from these hotter $Te2$ electrons on the $\u2212vy$ side surpasses the colder $Te1$ electrons' contribution on the $+vy$ side, resulting in the net

_{ey}*U*< 0 signature throughout the layer. As more $Te2$ electrons are added to the distribution, the magnitude of

_{ey}*U*increases up until roughly

_{ey}*A*= 0, where the electron nongyrotropy is large and the distribution takes the form of Eq. (27) with $vy*=0$. Increasing

_{y}*A*beyond

_{y}*A*= 0 adds additional hot $Te2$ electrons to the $+vy$ side of the distribution, which makes the electron distribution more gyrotropic and decreases $|Uey|$ up until the MSH side where $Ay=+\u221e$ and $fe=fe2$. Thus, the location of maximum

_{y}*U*occurs close to where the most $Te2$ electrons appear on the $\u2212vy$ side of the distribution, which is near

_{ey}*A*= 0 ($vy*=0$).

_{y}### D. Phase-space structures and electron orbits

Figures 2(a)–2(c) show how the phase space structure of the modeled current layer in the *x*-*v _{y}* plane depends on various choices of the initial electron beta

*β*with a comparison to the phase space structure of an electron-scale layer observed by MMS on December 23, 2016, in Fig. 2(e) (more details about this event are presented in Sec. III). Figure 2(d) shows a characteristic phase space trajectory in three-dimensional

_{e}*x*-

*v*-

_{x}*v*space for an energetic electron with initial parameters {

_{y}*x*

_{0}, $vx0,\u2009vy0$} = {$12\u2009rL,\u20093\u2009vth$, 0} taken from the solution with $\beta e=0.468$, which is close to the maximum $\beta e*$ given by Eq. (24) for the initial parameters and ratios used in Figs. 2(a)–2(d):

*S*= 10, $Ti/Te1=10$, and $Te2/Te1=1.7$.

Electrons with large $\u2212vy$ (+$v\u22a51$) penetrate deepest into the MSP side of the layer (up to several *r _{L}* in the –

*x*direction), which produces the visible protrusion of phase space density toward the low-density (MSP) side. Prior to this region of enhanced phase space density in $\u2212vy$, there is also a depletion of electrons with $+vy$ ($\u2212v\u22a51$) that, depending on the initial

*β*, can extend up to tens of

_{e}*r*toward the higher density (MSH) side of the layer. The overall shift of phase space density toward $\u2212vy$ corresponds to the bulk $E\xd7B$ = $\u2212ExBzy\u0302$ drift in this region. Just as $vy*$ denotes the velocity-space transition between $fe1$ and $fe2$ [see Eq. (27)], here the curve defining the edge in phase space is given by $vy*(x)=eAy(x)/me$. This curve is found by setting the argument of the hyperbolic tangent function equal to 0 in Eq. (8). Similarly, any electron orbit's projection in the

_{L}*x*-

*v*plane [magenta traces in Figs. 2(a)–2(d)] is governed by $Ay(x)$ from the expression for $ce2$ in Eq. (2): $vy(x)=(ce2+eAy(x))/me$. The MMS observations in Fig. 2(e) have $\beta e=0.33$ and are qualitatively consistent with these Vlasov–Maxwell model solutions. Figure 2(e) shows that MMS detected a slanted enhancement of phase space density in $+v\u22a51$ preceded by a tilted depletion of phase space density in $\u2212v\u22a51$ that roughly aligns with the $A\u22a51(x\u22a52)$ curve obtained from integrating the magnetic field over the interval.

_{y}At a given spatial location *x*, velocity distributions in *v _{x}*-

*v*($v\u22a51$-$v\u22a52$) space are vertical slices of the phase space density structures shown in Figs. 2(a)–(c) and 2(e). Thus, velocity space distributions in

_{y}*v*-

_{x}*v*represent a mixture of both the low-temperature electron population and the energetic penetrating electrons. The projection of the particle's orbit in the

_{y}*v*-

_{x}*v*plane is approximately circular [rightmost projection in Fig. 2(d)]. The direction of an electron's net drift in the

_{y}*y*-direction depends on both the electron's initial energy and its proximity to the current layer, which determine whether the $\u2207B$ drift (pointing in the $+y$ direction for electrons) overpowers the $E\xd7B$ drift (directed along –

*y*). Since there are no forces acting parallel to the magnetic field in the

*z*-direction, the particle's initial velocity $vz0$ remains unchanged.

The visible elongation of the electron orbit in $+x$ toward the high density (low magnetic field) side in Fig. 2(d) is responsible for the skewed spatial structure of the layer with its tail extending toward the $+x$ direction (e.g., Fig. 1). The orbit's shape is understood by considering the combined effects of $Ex(x)$ and $Bz(x)$ on the gyrating electron. The particle's orbit in the *x*-*v _{x}*-

*v*phase space [Fig. 2(d)] is closed as a result of the solution's symmetries. As the electron gyrates toward the center of the current sheet (toward –

_{y}*x*), the increasing electric force $\u2212eEx$ accelerates the electron faster toward the central layer. The electron's $|vx|$ increases up until it encounters the maximum

*E*near $x\u224810\u2009rL$. At this stage, the electron's gyrofrequency is higher due to the larger magnetic field strength, so

_{x}*B*works to quickly reverse the electron's velocity from $\u2212vx$ to $+vx$. The rapidly gyrating electron with large acquired $+vx$ promptly exits the central current layer region and travels far (up to tens of

_{z}*r*depending on

_{L}*β*) into the high temperature side of the layer back to its original

_{e}*x*location. During this stage, the electric force $\u2212eEx$ acts to decelerate the electron. On the high temperature (low magnetic field) side of the layer (toward +

*x*), the electron's gyrofrequency is smaller, but eventually

*B*rotates the electron's velocity back toward $\u2212vx$, and the cyclic orbit repeats. The acceleration by

_{z}*E*in this model is analogous to the acceleration from the normal electric field reported by Bessho

_{x}*et al.*(2016) in the context of asymmetric reconnection simulations, except here electrons do not perform meandering, crescent-shaped orbits since there is no magnetic field reversal in our model.

### E. Thickness of the current layer

Figures 1 and 2 show that the thickness of the current layer, *L*, visibly increases in any of the following limits: $\beta e\u2192\beta e*$, $Te2/Te1\u2192\u221e$, and $S\u21920$. When we let $S\u2192\u221e$ (discussed in Sec. II C above), the velocity-space boundary between the $Te1$ and $Te2$ electron populations becomes infinitely sharp, yet *L* cannot become smaller than the scale of a thermal electron gyroradius, *r _{L}*. In the $S\u2192\u221e$ limit, the velocity space gradient term $|(F/me)\xb7\u2207vfe|\u2248|(eBz/me)vx\u2202fe/\u2202vy|\u2192\u221e$, which must balance with a corresponding structure in the spatial gradient term $v\xb7\u2207fe=vx\u2202fe/\u2202x\u2192\u221e$ along the curve $vy*(x)$.

In the limit as $\beta e\u21920$, the magnetic field ratio $B2/B1\u21921$. In this case, the magnetic pressure dominates the plasma pressure so that the particles' dynamics do not appreciably alter the magnetic field profile. In this situation, in order for *β _{e}* to vanish, either $B1=B2=B0\u2192\u221e$, or $Pe1=n1Te1\u21920$. Alternatively, in the limit as $\beta e\u2192\beta e*$ given in Eq. (24), we have $B2/B1\u21920$ from Eq. (23), which is shown by the scaling relationship between $B2/B1$ and

*β*in Fig. 1(f). In this limit, there are two possibilities: either (1) $B1\u2192\u221e$ (corresponding to an infinitely sharp magnetic wall), or (2) $B2\u21920$. If $B1\u2192\u221e$, then for a finite electron temperature we have $rL\u21920$, and the layer becomes arbitrarily small to sustain the infinite magnetic gradient. If $B2\u21920$ with a finite

_{e}*B*

_{1}[similar to Fig. 1(a) or Fig. 2(c)], then the slope of $Ay(x)$ on the MSH side of the layer approaches 0. Thus, as $Ay(x)$ flattens, the skew of the layer toward the MSH side extends infinitely since the vanishing

*B*

_{2}is unable to rotate the electron's velocity back toward the MSP side of the current layer.

## III. SPATIAL EVOLUTION OF THE MMS OBSERVATIONS AND THE VLASOV–MAXWELL MODEL

In this section, we compare the Vlasov–Maxwell equilibrium solution directly to MMS observations of two thin current sheets exhibiting electron-scale temperature gradients recently found in the vicinity of Earth's magnetopause (Shuster *et al.*, 2019; 2021). The Fast Plasma Investigation (FPI) onboard MMS measures a full 3D distribution function for ions and electrons at unprecedented time cadences of 150 and 30 ms, respectively (Pollock *et al.*, 2016), which is fast enough to resolve electron-scale current layers with thicknesses on the order of a few electron gyroradii. Table I lists relevant plasma parameter ratios observed by MMS for the two events featured in Figs. 3 and 4 alongside ratios from the Vlasov–Maxwell (VM) model runs chosen for comparison.

Event . | November 28, 2016 . | December 23, 2016 . | ||
---|---|---|---|---|

Ratio . | MMS . | VM . | MMS . | VM . |

$n2n1$ | 1.15 | 1.03 | 1.27 | 1.06 |

$B2B1$ | 0.48 | 0.48 | 0.38 | 0.38 |

$TiTe1$ | 11 | 11 | 13 | 13 |

$Te2Te1$ | 1.15 | 1.20 | 1.18 | 1.50 |

$(Te2Te1)\u22a5$ | 1.19 | 1.20 | 1.48 | 1.50 |

β _{e} | 0.37 | 1.44 | 0.33 | 0.59 |

$\beta e*$ | 0.48 | 1.86 | 0.39 | 0.69 |

$\beta e\beta e*$ | 0.77 | 0.77 | 0.85 | 0.85 |

$|Ueyvth|max$ | 0.22 | 0.16 | 0.39 | 0.41 |

Event . | November 28, 2016 . | December 23, 2016 . | ||
---|---|---|---|---|

Ratio . | MMS . | VM . | MMS . | VM . |

$n2n1$ | 1.15 | 1.03 | 1.27 | 1.06 |

$B2B1$ | 0.48 | 0.48 | 0.38 | 0.38 |

$TiTe1$ | 11 | 11 | 13 | 13 |

$Te2Te1$ | 1.15 | 1.20 | 1.18 | 1.50 |

$(Te2Te1)\u22a5$ | 1.19 | 1.20 | 1.48 | 1.50 |

β _{e} | 0.37 | 1.44 | 0.33 | 0.59 |

$\beta e*$ | 0.48 | 1.86 | 0.39 | 0.69 |

$\beta e\beta e*$ | 0.77 | 0.77 | 0.85 | 0.85 |

$|Ueyvth|max$ | 0.22 | 0.16 | 0.39 | 0.41 |

Not all of the parameter ratios can be exactly matched due to the relative simplicity of the model, which assumes the distribution function is isotropic on either side of the current layer and varies only along one spatial dimension, the implications of which will be discussed in more detail below. For the model, the isotropy assumption requires that ratios of the total, perpendicular, and parallel temperatures across the layer are the same: $Te2/Te1=(Te2/Te1)\u22a5=(Te2/Te1)\u2225$, whereas for the MMS observations this may not necessarily be the case. For instance, in the second MMS event we consider, there is a mechanism producing an electron temperature anisotropy $Te\u2225/Te\u22a5\u22481.3$ on the MSP side [Fig. 4(d)] that is not included in the model. Thus, we choose these model run parameters by first matching the electron perpendicular temperature ratio $(Te2/Te1)\u22a5$, which is the parameter that most influences whether or not an electron crescent distribution structure will be evident in the $v\u22a51$-$v\u22a52$ velocity space [see Figs. 1(m)–1(o)]. Additionally, we matched the ion to electron temperature ratio $Ti/Te1$, and we adjusted *β _{e}* so that the magnetic field ratio $B2/B1$ in the model would match the MMS observations. As a consequence of these parameter choices, the model density ratios $n2/n1$ are smaller than what was observed by MMS. Although the chosen

*β*parameters do not match exactly with the MMS observations, they are still on the order of about unity (

_{e}*β*for the model runs and MMS observations all fall between 0.5 and 1.5), and we note that matching $B2/B1$ results in the modeled ratio $\beta e/\beta e*$ matching exactly with the MMS observations due to the relationship derived in Eq. (23). For these run parameters, the modeled maximum electron velocities of the current layers are close to the observed MMS values, which were $(Uey)max$ $\u2248$ 0.2

_{e}*v*for the first event, and $(Uey)max$ $\u2248$ 0.4

_{th}*v*for the second.

_{th}We chose the velocity-space steepness parameter *S* so that the modeled *v _{x}*-

*v*electron distribution structures would best resemble the observed $v\u22a51$-$v\u22a52$ distribution signatures observed by MMS. As we showed in Figs. 1(j)–1(l), increasing

_{y}*S*sharpens the velocity space boundary of the crescent-shaped population. Although the perpendicular electron temperature ratio has the largest effect on the distribution's structure, for sufficiently low values of this steepness parameter $S\u22721$ corresponding to broader current layers, the crescent structure would not be discernible even if a large temperature gradient was present. Since the electron-scale layers passed by MMS in only a few tenths of a second, it is difficult to precisely determine the current sheet thicknesses, and hence the exact value of

*S*, to use for the corresponding model runs. Using multi-spacecraft timing analysis, we estimated the structure velocity to be

*V*$\u2248$ 30 km/s for the event on November 28, 2016 (Fig. 3), and

*V*$\u2248$ 60 km/s for December 23, 2016 (Fig. 4). Thus, we estimate the thickness

*L*of these layers to be roughly 5 or 6 km, with corresponding half-thicknesses $L/2$ of about 2.5 to 3

*r*(since

_{L}*r*$\u2248$ 1 km for these events), which is consistent with the model scaling results [Fig. 1(i)]. This range of thicknesses corresponds to values of

_{L}*S*between roughly 1 and 10.

Figure 3 features an electron-scale current layer observed on November 28, 2016 (Shuster *et al.*, 2019), with electron distribution slices that exhibit a gradual temperature transition with $(Te2/Te1)\u22a5=1.19$, and hence, they lack a pronounced crescent-shaped distribution. Nevertheless, there is still a strong bulk electron flow signature with $Ue\u22a51\u2248$ 700 km/s supported by a visible shift in the electron distribution function in the $v\u22a51$-direction (see Fig. 3, times *t*_{4} and *t*_{5}). For the corresponding Vlasov–Maxwell solution, a lower distribution steepness parameter *S* = 3 is chosen to model the gradual velocity-space transition. Figure 4 presents a second electron-scale current layer observed on December 23, 2016 (Shuster *et al.*, 2019; 2021), which features an electron bulk flow of $Ue\u22a51\u2248$ 1300 km/s (almost twice that of the first event) and a pronounced crescent-shaped electron distribution function corresponding to the substantially larger perpendicular electron temperature gradient with $(Te2/Te1)\u22a5=1.48$. A higher steepness parameter *S* = 10 is chosen for the Vlasov–Maxwell model to capture the sharper velocity-space transition, which results in a crescent-like distribution structure comparable to the MMS measurements for this event. In particular, comparing the 1D cuts of the 2D MMS distributions *t*_{4}-*t*_{6} to the modeled distribution locations *x*_{4}-*x*_{6} in Fig. 4, there is a notable bump in phase space density occurring within 4000 to 8000 km/s in $v\u22a51$ (1 to 2 *v _{th}* in $\u2212vy$ for the model distributions). This signature corresponds to the more pronounced crescent-shaped distribution that emerges due to the combined effects of the higher perpendicular temperature ratio and larger

*S*parameter.

The layouts of Figs. 3 and 4 are the same: panels (a)–(g) show an overview of the MMS plasma moments and fields data, and panels (*t*_{1}-*t*_{7}) show a 2D slice of the electron distribution structure in the $v\u22a51$-$v\u22a52$ plane observed by MMS for seven times indicated by the vertical gray bars in panels (a-g). Below each 2D slice, 1D cuts of the distribution taken along $v\u22a52=0$ are included with uncertainties for the electron phase space density measurements shown as gray error bars with a light yellow background (e.g., Gershman *et al.*, 2015). The same moment and field quantities from the Vlasov–Maxwell model are shown in panels (h)–(n), with panels (*x*_{1}-*x*_{7}) showing 2D slices in the *v _{x}*-

*v*plane of the modeled electron distribution structure at seven spatial locations indicated by the vertical gray bars in panels (h)–(n), with 1D cuts taken along

_{y}*v*= 0 included below. Figure 3(a) and Fig. 4(a) show MMS 1 magnetic field data from the fluxgate magnetometer (FGM) (Russell

_{x}*et al.*, 2016), and Figs. 3(f) and 4(f) include electric field data from the electric field double probes (EDP) (Lindqvist

*et al.*, 2016).

Going from the MSP to MSH side of the layer [right to left in Figs. 3(a)–3(g) and 4(a)–4(g)], both of the MMS electron-scale temperature gradient events maintain approximate pressure balance and exhibit qualitatively similar plasma properties, each of which are captured by the Vlasov–Maxwell solutions [Figs. 3(h)–3(n) and 4(h)–4(n)]: (1) the magnetic field strength decreases by about a factor of 2 or 3, (2) the density increases on the order of 10%, (3) the ion temperature is about an order of magnitude larger than the electron temperature, (4) there is a strong electron bulk velocity signature on the order of 1000 km/s supported by enhanced electron phase space density in the $v\u22a51$ ($\u2212vy$) direction, (5) the electron pressure divergence $\u2207\xb7Pe$ is dominated by the $ne\u2207\xb7Te$ term, and (6) there is an enhanced DC electric field $E$ that deviates from $Ue\xd7B$ due to the nonzero $\u2207\xb7Pe$. The structure of each current layer is also skewed slightly toward the MSP (low density) side of the layer, which is evident for instance in the $Ue\u22a51$ and $E\u22a52$ profiles. The gradient structure of the layers observed by MMS also agrees with the Vlasov–Maxwell models based on comparing the density gradient (aqua trace) and bulk velocity gradient (magenta trace) terms in Figs. 3(b), 3(c), 4(b), and 4(c) to Figs. 3(i), 3(j), 4(i), and 4(j). Since these layers are approximately steady-state structures moving past the spacecraft, single spacecraft MMS gradients can be inferred via $\u2202/\u2202x\u2248V\u22121\u2202/\u2202t$ [solid aqua and magenta traces in Figs. 3(b), 3(c), 4(b), and 4(c)], which are more consistent with the model results [in Figs. 3(i), 3(j), 4(i), 4(j)].

There are also aspects of the current layers observed by MMS that are not captured by the simplified 1D kinetic model. The electron distribution function *f _{e}* in Eq. (8) is based on a varying temperature function $T(ce2)$ that only depends on

*v*($v\u22a51$); thus, the modeled

_{y}*f*is symmetric about the other two velocity-space directions

_{e}*v*and

_{x}*v*($v\u22a52$ and $v\u2225$). However, the MMS distributions do not reflect such symmetry. Because the model does not account for this structure of the electron distribution in the $v\u2225$ (

_{z}*v*) direction, the model incorrectly represents the parallel electron temperature $Te\u2225$ profile [blue traces in Figs. 3(k) and 4(k)]. Consequently, despite the matched perpendicular electron temperature ratio, the modeled total electron temperature ratio $Te2/Te1$ is larger than what was observed by MMS, which leads to an artificially large deviation between the

_{z}*E*and $UeyBz$ terms, and an exaggerated variation in the modeled

_{x}*T*-component of the electron temperature tensor [green trace in Figs. 3(k) and 4(k)] when compared to the MMS observations [green trace for the $Te,\u22a51,\u22a51$-component in Figs. 3(d) and 4(d)]. These limitations are consequences of the relative simplicity and assumed symmetry of the 1D model. Nevertheless, the model successfully reproduces many qualitative and quantitative properties of the bulk plasma parameters and produces electric and magnetic fields that are self-consistent with the electron distribution's crescent-like structure observed by MMS.

_{eyy}## IV DISCUSSION AND CONCLUSIONS

We formulated an exact Vlasov–Maxwell equilibrium solution suitable for modeling thin (electron gyroradius scale) asymmetric current layers observed by MMS at the magnetopause. Our solution self-consistently produces a spatially evolving, nongyrotropic electron crescent-type distribution structure in velocity space similar to the kinetic electron structures detected by MMS. Nongyrotropic, crescent-shaped electron velocity distribution functions can be indicative of non-ideal energy conversion processes associated with the electron diffusion region of magnetic reconnection (Burch *et al.*, 2016; Bessho *et al.*, 2017; Torbert *et al.*, 2018; Webster *et al.*, 2018). While electron crescents may be a necessary consequence of asymmetric reconnection dynamics, they alone do not constitute a sufficient condition for reconnection to occur. Electron crescent distributions appear more generally as the manifestation of diamagnetic drifts and finite Larmor radius effects occurring within well-magnetized plasmas (Egedal *et al.*, 2016; Rager *et al.*, 2018), which is consistent with the crescent-like electron distributions produced in our 1D solution where the associated particle orbits remain closed in phase space throughout the current layer.

Qualitatively similar semi-circular, crescent-resembling electron distributions were reported by (Camporeale and Lapenta, 2005) [see their Fig. 3, cases (a) and (b)] in their study of symmetric, bifurcated current layers applicable to magnetotail geometries. The similar structures develop primarily due to the dependence on the constant of motion associated with the electron's *y*-momentum [$ce2$ in our Eq. (2)], but there are also several distinctions to note. In their study, the distribution function's dependences on $\varphi $ and *A _{y}* could be separated for the forms of

*f*that were chosen based on the separable forms considered by Schindler and Birn (2002). Additionally, the temperature parameter

_{e}*T*is kept constant in their solutions, and thus, the relative magnitude between the two electron populations on either side of the cutoff velocity $vy*$ is different than in our solution. Several forms for the ion distribution

_{e}*f*were also considered in their study, whereas here for simplicity we model the ions with a non-drifting Maxwellian population in the electron-scale layer (i.e., we view the current layer from the frame of reference moving with the ion bulk flow, where the ion diamagnetic drift balances the $E\xd7B$ drift so that

_{i}**U**

_{i}= 0). We note that for particular choices of $\beta e*$ [see Eq. (24)], our solution also permits symmetric, bifurcated-type current layers, although our focus in this paper is on the asymmetric case applicable to the MMS magnetopause observations.

The two parameters in our model which most influence the crescent-shaped distribution structure are (1) the perpendicular electron temperature ratio across the layer $(Te2/Te1)\u22a5$, and (2) the velocity space steepness parameter *S,* which affects the thickness and strength of the current layer by controlling the sharpness of the velocity-space transition between the two electron populations of differing temperatures. These parameters may be adjusted for facilitating comparison to MMS encounters of electron-scale current layers with varying temperature ratios and thicknesses, some of which may support a strong current density without exhibiting a visibly apparent crescent-shaped distribution signature (Shuster *et al.*, 2019). An electron crescent distribution in $v\u22a51$-$v\u22a52$ is likely to be detected in sufficiently thin ($S\u22733$) electron-scale current layers that exhibit a strong perpendicular electron temperature gradient $(Te2/Te1)\u22a5\u22731.5$. However, for broader current layers ($S\u22721$) with a weaker electron temperature gradient $(Te2/Te1)\u22a5\u22721.2$, a crescent-shaped distribution is less likely to be observed. In the limit as $S\u21920$, the gradient vanishes as the electron distribution reduces to a uniform Maxwellian. On the other hand, in the limit as $S\u2192\u221e$, the electron distribution function becomes piecewise Maxwellian in velocity-space, which conveniently permits analytic solutions to the self-consistent plasma moments and fields.

Having an exact Vlasov–Maxwell solution for the case of an asymmetric gradient in the electron density, bulk velocity, and temperature is relevant to recent MMS magnetopause observations of terms in the Vlasov equation $\u2207fe$ and $v\xb7\u2207fe$ (Shuster *et al.*, 2021). The form of the electron distribution we chose takes two electron temperatures $Te1$ and $Te2$ as initial parameters to specify the size of the overall temperature gradient across the layer. The self-consistent solution also yields gradients in the electron density and bulk velocity, consistent with the MMS events presented here, and relevant to recent observations of similar electron-scale gradients found in association with magnetopause current layers. Our solution may be used to further validate MMS observations of the spatial gradient term $v\xb7\u2207fe$ and future measurements of the velocity-space gradient term $(F/me)\xb7\u2207vfe$. For the spatial gradient term, one could fly several virtual spacecraft probes through our 1D layer at various $\Delta x$ spacings to simulate how the $\u2207fe$ gradient distribution structures seen by MMS vary in velocity-space, and assess the accuracy of the spatial gradient computation for a given MMS spacecraft separation (e.g., $\Delta x\u2248$ 10 km).

## ACKNOWLEDGMENTS

We especially thank the MMS instrument teams for their dedication and commitment to providing unprecedented, high-quality data sets. This research was supported in part by NASA grants to the Fast Plasma Investigation, FIELDS team, and Theory and Modeling program of the MMS mission. J.R.S. was supported by NASA under Grant Nos. 80NSSC19K1092 and 80NSSC21K0732. S.W. was supported by DOE under Grant No. DE-SC0020058. J.N. was supported by NASA under Grant No. 80NSSC21K1046.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

All MMS data are made publicly available through the Science Data Center (SDC) at the Laboratory for Atmospheric and Space Physics (LASP) via at https://lasp.colorado.edu/mms/sdc/public/, LASP (2015). The codes used to solve the one-dimensional, steady-state Vlasov-Maxwell system of equations and generate the results shown in the figures will be made available upon reasonable request.

### APPENDIX: DERIVATION OF ELECTRON MOMENTS IN THE PIECEWISE MAXWELLIAN LIMIT

Here, we compute electron moments in the limit $S\u2192\u221e$ where *f _{e}* becomes the piecewise Maxwellian distribution defined in Eq. (27) as discussed in Sec. II C.

The electron density *n _{e}* is derived as follows:

where $vth,2=Te2/me$. As $Ay\u2192\u2212\u221e$, we have that $ne\u2192n1$, while for $Ay\u2192+\u221e$, Eq. (28) reduces to Eq. (15). To compute the *v _{y}* integrals, we make use of the symmetry of the Gaussian functions to conveniently adjust the limits of integration, and we utilize the relationship between the cumulative distribution function for the standard normal distribution $\Phi (x)$ and the error function $erf(x)$

where the error function is defined as

Invoking quasineutrality, the expression for *n _{e}* in Eq. (28) can be equated to $ni=n1\u2009exp\u2009[\u2212e\varphi /Ti]$, and the resulting equation can be numerically solved for $\varphi (Ay)$ via

For the electron bulk velocity *U _{ey}*, we have

To arrive at equation (A), we made use of the result

For *P _{exx}*, we have

where we have made use of the following Gaussian integral:

For *P _{eyy}*, we have

Evaluating the integral *I* in Eq. (A22), we find

Therefore,

where *n _{e}* and

*U*are derived above, and we used the result that

_{ey}