Electrons under the dominant action of shockelectric fields
^{1} Argelander Institute for Astronomy, University of Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
email: hfahr@astro.unibonn.de
^{2} Space Science Center, University of New Hampshire, 8 College Road, Durham, NH 03824, USA
Received: 4 December 2015
Accepted: 14 January 2016
We consider a fast magnetosonic multifluid shock as a representation of the solarwind termination shock. We assume the action of the transition happens in a threestep process: In the first step, the upstream supersonic solarwind plasma is subject to a strong electric field that flashes up on a small distance scale Δz ≃ U_{1}/ Ω_{e} (first part of the transition layer), where Ω_{e} is the electron gyrofrequency and U_{1} is the upstream speed. This electric field both decelerates the supersonic ion flow and accelerates the electrons up to high velocities. In this part of the transition region, the electric forces connected with the deceleration of the ion flow strongly dominate over the Lorentz forces. We, therefore, call this part the demagnetization region. In the second phase, Lorentz forces due to convected magnetic fields compete with the electric field, and the highly anisotropic and energetic electron distribution function is converted into a shell distribution with energetic shell electrons storing about 3/4 of the upstream ion kinetic energy. In the third phase, the plasma particles thermalize due to the relaxation of free energy by plasma instabilities. The first part of the transition region opens up a new thermodynamic degree of freedom never before taken into account for the electrons, since the electrons are usually considered to be enslaved to follow the behavior of the protons in all velocity moments like density, bulk velocity, and temperature. We show that electrons may be the downstream plasma fluid that dominates the downstream plasma pressure.
Key words: plasmas / Sun: heliosphere / solar wind
© ESO, 2016
1. Introduction
Recent literature on astrophysical shocks concludes with increasing conviction that a shock, such as the solarwind termination shock, is a phenomenon in which multifluid effects play an important role. This means that, for the understanding of global shock physics, effects arising from different plasma species must be taken into account while all species have to fulfill the RankineHugoniot conservation laws concertedly. For instance, previous studies indicate that pickup ions substantially modify the structure of the solarwind termination shock. In addition, pickup ions are generally considered an important fluid component since they transport a major fraction of the entropized upstream kinetic energy in the form of thermal energy into the downstream regime of the shock (see Decker et al. 2008). Zank et al. (2010) and Fahr & Siewert (2007, 2010, 2011) found relations between the upstream and downstream ion distribution functions that are different for solarwind protons and for pickup protons, respectively. However, these refinements did not lead to a fully satisfying representation of some properties of the shocked plasma observed by Voyager 2 (Richardson et al. 2008). In fact, Chalov & Fahr (2013) demonstrate that it is necessary to allow for solarwind electrons to behave like a third, independent plasma fluid to achieve an agreement between the model results and the plasma data presented by Richardson et al. (2008). Their parameter study shows that the bestfit results for the observed proton temperature are then achieved if the electrons are assumed to be heated to a temperature that is higher than the downstream proton temperature by a factor ≳10. We refer to the works of Leroy & Mangeney (1984), Tokar et al. (1986), and Schwartz et al. (1988) on electron heating at fastmode shocks. Also, plasma shock simulations, in which electrons are treated kinetically, show preferential shock heating of the plasma electrons (see Lembège et al. 2003, 2004). Twostream and viscous interactions lead to a demagnetization of the electrons; i.e., they are not Lorentzwound by the magnetic fields and attain an increase in their temperature by a factor of 50 or more on the downstream side of the shock. In a different approach, Leroy et al. (1982) and Goodrich & Scudder (1984) predict that electrons carry out perpendicular electric drifts that are different from those of the ions due to the shockelectric field, thereby establishing an electric current that modifies the surfaceparallel magnetic field.
With a similar level of consistency, Fahr et al. (2012) describe the conditions of the upstream and downstream plasma in bulkframe systems with frozenin magnetic fields. They treat the transition from the upstream side to the downstream side of the shock as an instantaneous kinetic reaction in the velocity distribution function via the LiouvilleVlasov theorem, predicting all of the relevant downstream plasma quantities. In this model, a semikinetic representation of the multifluid termination shock with mass and chargespecific reactions of protons and electrons to the electric shock ramp leads to excessive electron heating as described by Fahr et al. (2012) or Fahr & Siewert (2013). Based on these studies, we assume that electrons enter the downstream side as a strongly heated, massless plasma fluid that dominates the downstream plasma pressure. A recently published multifluid termination shock reconstruction by Zieger et al. (2015), which describes the electrons as a separate, independently reacting fluid, supports this notion. This study finds that electrons are preferentially heated with respect to ions, converting overshoot velocities into thermal energy supporting the claims made by Chalov & Fahr (2013), Fahr & Siewert (2015), and Fahr et al. (2015). Furthermore, the study finds that the partial pressure of the heated electrons plays a dominant role in the downstream plasma flow.
In the following study, we investigate more carefully for what reason and in which way the plasma electrons behave as a preferentially heated fluid at the shock crossing.
2. The shock electric field and charge separation
Strongly spacedependent electric and magnetic fields are typical for the transition region of astrophysical shocks, and none of the conventional testparticle descriptions, such as the expansion of the particle motion into different forms of electromagnetic drifts, apply under these conditions. This is certainly the case when electric forces locally flash up and become strongly dominant over Lorentz forces so that the latter forces can be safely neglected for the firstorder motion analysis. In this section, we look into the details of this special situation that arises at shocks such as the solar windtermination shock.
We assume that the shock is characterized by a threephase transition in space. In the first phase (the demagnetization region), electric forces dominate the deceleration of the plasma. In the second phase, Lorentz forces dominate the deceleration over the electric forces. While in the first two phases the transition is isothermal for protons, the third phase on the downstream side represents the region in which the particle thermalization due to streaming plasma instabilities takes place. We focus on the first phase of the shock transition (the demagnetization region). We illustrate the threephase shock in Fig. 1.
Fig. 1 Illustration of the threephase shock. Phase I: electric forces dominate the deceleration of the plasma (demagnetization region); Phase II: magnetic Lorentz forces dominate; Phase III: quasiequilibrium relaxation of the plasma and thermalization through plasma instabilities. 

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We first estimate the typical field conditions at a shock of this nature. In a sufficiently small surface element of the termination shock, the curvature of the shock surface is negligible. We approximate this surface element under consideration as a planar surface and, consequently, assume a onedimensional geometry of the system. In this approximation, the shock parameters only depend on the coordinate z parallel to the shock normal. We assume a representation of the ion bulkflow velocity U as a function of z of the form (1)where ΔU = U_{1}−U_{2}. Indexes 1 and 2 characterize the asymptotic upstream and downstream conditions of the flow, respectively. The quantity α determines the transition length at the shock. We choose the coordinate system in which B lies in the xzplane. In addition, we assume that the Bfield is frozen into the main momentum flow (i.e,. the ion flow). Therefore, the frozenin condition under steadystate conditions, ∂B/∂t = ∇ × [U × B] = 0, yields (2)in our onedimensional coordinate system, where B_{z} = B_{z1} = const. and B_{x} = B_{x1}U_{1}/U(z).
The zcomponent of the ion momentum equation is written as (3)where m_{p} is the proton mass e is the elementary charge, E is the electric field that is parallel to the flow direction , P_{p} is the proton pressure, and n_{p} is the proton density (see also Verscharen & Fahr 2008; Fahr & Siewert 2013). In our isothermal approximation (T_{p} = const.) for the first phase of the shock transition, we find for the pressuregradient term in the momentum equation by applying the fluxconservation requirement Φ_{e} = Φ_{p} = n_{p}U = const. that (4)where k_{B} is the Boltzmann constant. This leads to the following expression for the electric field (5)where is the ion thermal speed. The correction term on the righthand side is small compared to 1 for highMachnumber shocks in the first phase of the shock transition, which justifies neglecting the pressure gradient. With Eq. (1), the electric field as a function of z is given by (6)The electric field E is selfconsistently generated by the charge separation between protons and electrons according to Gauss’ law: (7)
3. Dominance of electric forces over Lorentz forces
We can assume that, as long as the electric force is much stronger than the Lorentz force on the electrons, the motion of a single electron with electron velocity v_{ez} corresponds to a permanent linear acceleration according to (8)Integrating leads to (9)We assume that the initial energy of an electron with a velocity equal to the bulkflow velocity on the upstream side, , is negligible compared to the resulting downstream energy. We note that protons and electrons have equal upstream bulk velocities U_{1} as well as equal upstream densities n_{1}. With this assumption and Eq. (5) in the limit , we find for the zdependent kinetic energy of the electron that (10)For the downstream electron kinetic energy, we obtain with Eq. (10) that (11)with the observed compression ratio s = U_{1}/U_{2} ≈ 2.5 (see Richardson et al. 2008). Therefore, the shock electrons may consume about 84% of the upstream proton kinetic energy, which corresponds to about 0.7 keV or v_{ez2} ≈ 10^{9} cm/s, respectively.
We choose an electron with v_{ez1} = U_{1} and follow its trajectory through the shock. This electron fulfills v_{ez}(z) = U_{e}(z) throughout its transit. With this condition and flux conservation, n_{e}U_{e} = const., we directly determine the upstreamtodownstream density ratio for the electrons as (12)which translates to U_{e2} = 40U_{1}.
The electric acceleration of the electron, however, only dominates compared to the acceleration due to Lorentz forces as long as (13)This condition limits the resulting velocity to (14)which we write in normalized units as (15)where Ω_{px1} = eB_{x1}/m_{p}c. We define the normalized threshold velocity v_{thr}/U_{1} as the righthand side of Eq. (15). Beyond this limit, Lorentz forces begin to compete with electric forces and need to be taken into account. We show this condition in Fig. 2.
Fig. 2 Electron bulk velocity U_{e} from Eq. (10), threshold velocity v_{thr} from Eq. (15), and U_{e} + c_{e1}, where is the upstream electron thermal speed, as functions of the spatial coordinate z. We use the following parameters: s = 2.5, U_{1} = 4 × 10^{7} cm/s, B_{x1} = 2 × 10^{7} G, α = 10^{8} cm, and T_{e1} = 2 × 10^{4} K. 

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The black solid line shows the threshold velocity v_{thr} in units of the upstream bulk velocity U_{1}. Considering again an electron with v_{ez}(z) ≈ U_{e}(z), we determine the profile of U_{e} as a function of z from Eq. (10). We show this profile as the red dashed line in Fig 2. The bulk velocity fulfills the condition in Eq. (15) in the range between z ≈ −1.5α and z ≈ −0.1α. We also plot the sum of the electron bulk velocity and the upstream electron thermal speed as the green dashed line. This line indicates the region in which the majority of the solarwind electrons fulfill Eq. (15). This condition is fulfilled between z ≈ −1.2α and z ≈ −0.1α. As these considerations show, the demagnetization region is restricted to the region z< 0.
In the limit of a quasiparallel shock, Ω_{px1} → 0 which leads to v_{thr} → ∞. Therefore, the quasiparallel shock is solely a result of the electrostatic interaction while Lorentz forces are negligible. On the other hand, a quasiperpendicular shock still has a finite v_{thr}, which can be >U_{e} depending on the finite shock parameters U_{1}, Ω_{px1}, and α.
4. Motion under firstorder Lorentz forces
We assume that, within a finite region, the electric forces are strongly dominant according to Eq. (13). The zerothorder motion of an electron is then determined by (16)while the firstorder Lorentz force is given by (17)Under the assumption that the electric field is purely oriented along the zdirection, we obtain (18)We furthermore assume that the upstream electron velocity is negligible. Now we allow for a firstorder velocity component in the ydirection (i.e., the direction, where ) due to the firstorder Lorentz force : (19)where is given by (20)With Eq. (19), we find (21)Using , we obtain the solution (22)where Ω_{ex}(z) denotes the local electron gyrofrequency based on B_{x}, and ⟨Ω_{ex}⟩_{z} is the average gyrofrequency in the integration domain.
It is noteworthy that the electron motion in the ydirection does not depend on the electricfield configuration in the transition region as long as the Lorentz force can be considered to be of higher order than the electric force. Therefore, the curling of the particle trajectory around the frozenin field only happens after a time τ_{Le} when the action of the Lorentz force changed the velocity by about its magnitude, (e/c)v_{ez}B_{x}τ_{Le} ≈ m_{e}v_{ez}, yielding τ_{Le} ≈ m_{e}c/eB_{x} = 1/Ω_{ex}. During this period, the electron moves in the zdirection by the amount Δz = τ_{Le}v_{ez} = v_{ez}/ Ω_{ex}. With our results for v_{ez} from Sect. 3, we find that the permitted extent of the overall transition region can roughly be estimated as Δz ≈ v_{ez}/ Ω_{ex} ≈ 10^{8} cm.
5. Discussion and conclusions
In this paper, we study the action of shockelectric fields on electrons and ions entering from the upstream regime of the shock into the shock transition region. We split the latter into three consecutive regions: the first is demagnetized and operates like an electric double layer where the action of the upflashing shockelectric field strongly dominates over Lorentz forces. In the second region, Lorentz forces due to the piledup magnetic fields compete with or even dominate compared to electric forces. In the third phase, the overshooting energy is transferred into proton and electron heating by the action of kinetic plasma instabilities.
In Sect. 3, we determine the electron velocity profile. The downstream electron velocity before entering phase three of the shock is about 40 times the upstream velocity, while the downstream electron density is about 0.025 the upstream density. The shocked plasma opens up a new thermodynamic degree of freedom allowing that a substantial fraction (about 84 percent) of the upstream ion kinetic energy gets stored in the overshoot velocities of the electrons which, in the second region of the shock transition, is deposited into a shell in velocity space centered around the downstream ion bulk velocity. In that case, a major fraction of the upstream ion bulkflow energy is available to be transferred into electron thermal energy. We determine the threshold velocity v_{thr}, which defines the separation between phases one and two of the shock transition. The electron motion is dominated by electric forces for all electrons with a velocity <v_{thr}. For typical terminationshock parameters, we find that the region in which this condition is fulfilled for electrons with the electron bulk velocity extends from z = −1.5α to z = −0.1α.
In Sect. 4, we treat Lorentzforces as a firstorder correction to the plasma deceleration. We show that the firstorder shockperpendicular component of the electron velocity is determined by the average electron gyrofrequency in the shock layer. With this result, we determine the size of the demagnetization region (phase one) as Δz ≈ 10^{8} cm for typical solarwind parameters.
In earlier work (see Chalov & Fahr 2013; Fahr & Siewert 2013, 2015), we emphasized the fact that the creation of energetic electrons at the plasma passage over the shock is essential to fulfill the thermodynamic entropy requirements (Fahr & Siewert 2015) and to arrive at downstream plasma properties that nicely fit the Voyager2 measurements of Richardson et al. (2008). Zieger et al. (2015) recently published a multifluid study of the solarwind termination shock, which strongly supports our claim for the occurrence of energetic downstream electrons by showing that the Voyager2 measurements allow for a reasonably good theoretical fit with the model results only if the appearance of energetic downstream electrons is taken into account.
The remaining question as to why these predicted energetic electrons were not detected by the Voyager plasma analyzers, can most probably be answered along the argumentation developed in a recent paper by Fahr et al. (2015). These authors show that the energetic electrons create a strongly increased electric chargeup of the spacecraft detectors and thereby, owing to electric screening, impede the detectors in measuring countable fluxes of these electrons.
We intend to continue the studies presented in this Letter by kinetically analyzing the physics relevant for phases II and III of the threephase shock.
Acknowledgments
We appreciate helpful discussions with Marty Lee. This work was supported in part by NSF/SHINE grant AGS1460190.
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All Figures
Fig. 1 Illustration of the threephase shock. Phase I: electric forces dominate the deceleration of the plasma (demagnetization region); Phase II: magnetic Lorentz forces dominate; Phase III: quasiequilibrium relaxation of the plasma and thermalization through plasma instabilities. 

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In the text 
Fig. 2 Electron bulk velocity U_{e} from Eq. (10), threshold velocity v_{thr} from Eq. (15), and U_{e} + c_{e1}, where is the upstream electron thermal speed, as functions of the spatial coordinate z. We use the following parameters: s = 2.5, U_{1} = 4 × 10^{7} cm/s, B_{x1} = 2 × 10^{7} G, α = 10^{8} cm, and T_{e1} = 2 × 10^{4} K. 

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In the text 