Issue 
A&A
Volume 579, July 2015



Article Number  A18  
Number of page(s)  5  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/201525710  
Published online  19 June 2015 
The electron distribution function downstream of the solarwind termination shock: Where are the hot electrons?
^{1} Argelander Institute for Astronomy, University of Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
email: hfahr@astro.unibonn.de
^{2} Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
email: jdr@space.mit.edu
^{3} Space Science Center, University of New Hampshire, 8 College Road, Durham, NH 03824, USA
email: daniel.verscharen@unh.edu
Received: 21 January 2015
Accepted: 9 May 2015
In the majority of the literature on plasma shock waves, electrons play the role of “ghost particles”, since their contribution to mass and momentum flows is negligible, and they have been treated as only taking care of the electric plasma neutrality. In some more recent papers, however, electrons play a new important role in the shock dynamics and thermodynamics, especially at the solarwind termination shock. They react on the shock electric field in a very specific way, leading to suprathermal nonequilibrium distributions of the downstream electrons, which can be represented by a kappa distribution function. In this paper, we discuss why this anticipated hot electron population has not been seen by the plasma detectors of the Voyager spacecraft downstream of the solarwind termination shock. We show that hot nonequilibrium electrons induce a strong negative electric chargeup of any spacecraft cruising through this downstream plasma environment. This charge reduces electron fluxes at the spacecraft detectors to nondetectable intensities. Furthermore, we show that the Debye length λ_{D}^{κ} grows to values of about λ_{D}^{κ}/λ_{D} ≃ 10^{6} compared to the classical value λ_{D} in this hotelectron environment. This unusual condition allows for the propagation of a certain type of electrostatic plasma waves that, at very large wavelengths, allow us to determine the effective temperature of the suprathermal electrons directly by means of the phase velocity of these waves. At moderate wavelengths, the electronacoustic dispersion relation leads to nonpropagating oscillations with the ionplasma frequency ω_{p}, instead of the traditional electron plasma frequency.
Key words: plasmas / Sun: heliosphere / solar wind / shock waves / instabilities
© ESO, 2015
1. Introduction
The majority of the plasmaphysics literature on shocks essentially considers general fluxconservation requirements only, leading to the wellknown RankineHugoniot relations (e.g., see Serrin 1959; Hudson 1970; Baumjohann & Treumann 1996; Gombosi 1998; Diver 2001). These relations, however, do not explicitly formulate the internal microphysical processes that generate internal entropy during the conversion from the upstream regime into the downstream plasma. The missing physics in this description evidently leads to the wellknown phenomenon that the set of RankineHugoniot relations is a mathematically unclosed system of equations. Therefore, these relations can only provide unequivocal solutions if additional physical relations are added to the system, such as the assumption of an adiabatic reaction of the plasma ions during their compression into the higherdensity regime on the downstream side (e.g., Erkaev et al. 2000).
The solarwind termination shock is a particular example of a plasma shock for which microphysical effects play an important role. According to recent studies, pickup ions have a crucial influence on overall shock physics at the solarwind termination shock. They are a thermodynamically important additional plasma component, since they extract a significant fraction of the upstream kinetic energy in the form of thermal energy at the termination shock (see Decker et al. 2008). Zank et al. (2010) and Fahr & Siewert (2007, 2010, 2011) have studied kinetic features of this multicomponent shock transition and found relations between upstream and downstream ion distribution functions that are different for solarwind protons and pickup protons. Although these studies discuss the required overadiabatic reaction of pickup ions, a satisfying explanation of all plasma properties observed by Voyager2 (Richardson et al. 2008) is still lacking as demonstrated by Chalov & Fahr (2013). The latter authors show that assuming a significant difference in the behavior of solarwind electrons compared to protons, namely as an independent plasma fluid, leads to an explanation of most of the observed plasma data presented by Richardson et al. (2008) in a satisfying manner.
To achieve this result in their parameterized study, Chalov & Fahr (2013) had to include preferential heating of the solarwind electrons during the shock passage by a factor of about ten stronger than the proton heating. This type of electron heating at the potential jump of fastmode shocks had been realized earlier by Leroy & Mangeney (1984), Tokar et al. (1986), and Schwartz et al. (1988), and the phenomenon of shockheated electrons also appears in plasmashock simulations when electrons are treated kinetically (see Lembège et al. 2003, 2004). In these cases, the plasma electrons demagnetize due to twostream and viscous interactions and attain downstreamtoupstream temperature ratios of 50 and more.
Leroy et al. (1982) and Goodrich & Scudder (1984) follow a different approach. In their treatments, the plasma electrons carry drifts perpendicular to the shock normal (zdirection) different from the ion motion as a reaction to the shockelectric field. These drifts establish an electric current j_{⊥}, which is responsible for the change of the surface parallel magnetic field B_{∥} in the form 4πj_{⊥}/c = dB_{∥}/ dz. To achieve the same consistency, Fahr et al. (2012) describe the conditions of the upstream and downstream plasma in the bulk frame systems with a frozenin magnetic field. In this framework, the LiouvilleVlasov theorem describes all relevant downstream plasma quantities as an instantaneous kinetic reaction in the velocity distribution function during the transition from upstream to downstream. The excessive electron heating is then the result of the mass and chargespecific reactions to the electric shock ramp, as shown in the semikinetic models of the multifluid termination shock by Fahr et al. (2012) and Fahr & Siewert (2013). According to these studies, electrons enter the downstream side as a strongly heated plasma fluid with negligible mass density that dominates the downstream plasma pressure.
In this paper, we demonstrate why the Voyager1/2 spacecraft did not detect these theoretically suggested hot electrons (see Richardson et al. 2008) when they penetrated into the heliosheath plasma. For the purpose of clarification, we analyze the downstream plasma conditions in more detail under which the detection of preferentially heated electrons would have to take place.
2. Theoretical description of downstream electrons
In the following section, we shall start from a theoretical description of solarwind electrons expected downstream of the termination shock (Fahr & Siewert 2013). We treat them as a separate plasma species, which reacts in a very specific manner to the electricfield structure connected with the shock before adapting to the downstream plasma bulk frame. In the shockatrest system, the shock electric potential ramp decelerates the upstream protons from the upstream bulk velocity U_{1} to the downstream bulk velocity U_{2,p}, which is comparable to the centerofmass flow , where s is the shock compression ratio and m,M denote the masses of electrons and protons, respectively. The downstream magnetic field is frozenin into the centerofmass flow, and all plasma components are eventually comoving with the centerofmass flow (Chashei & Fahr 2013). The electrons, on the other hand, react in a completely different way to this electric potential. First, they attain a strong “overshoot” velocity U_{2,e} which then relaxes rapidly to the centerofmass bulk velocity enforced by the frozenin magnetic field. During this relaxation process, the plasma generates randomized thermal velocity components through the action of the twostream instability or the Buneman instability as well as by pitchangle scattering (see Chashei & Fahr 2013, 2014; Fahr et al. 2014).
Under the assumptions of an instantaneous reaction of the electrons to the electric potential and randomization of the overshoot energy by the Buneman instability and pitchangle scattering to an isotropic distribution in the downstream bulk frame, we obtain (Fahr & Siewert 2013, 2015) the following expression for the electron pressure P_{2,e} on the downstreamside of the shock: (1)The indices p and e denote proton or electronrelevant quantities, and the indices 1 and 2 denote upstream and downstream quantities, respectively. The parameter U denotes the bulk velocity, and s is the shock compression ratio. The parameter c_{1,e} is the average electron thermal velocity on the upstream side, where we assume that solarwind electrons and protons have equal temperatures. We denote the magnetic tilt angle between the shock normal and the upstream magnetic field as α. The functions A(α) and B(α) are given in Fahr & Siewert (2013). In the limit of vanishing thermal pressures and dominating magnetic pressures, the selfconsistent compression ratio s turns out to be s = 1 (see Fahr & Siewert 2013, Eq. (18)). Instead of treating the calculated velocity moment P_{2,e} of the distribution function, we focus on the distribution function f_{2,e} itself to derive the expected electron particle fluxes g_{2,e} as the relevant observable for the Voyager1/2 instrumentation. As motivated in Fahr & Siewert (2013), we assume that the downstream nonequilibrium distribution function f_{2,e} is a general kappadistribution. This convenient choice represents the transition from a thermal core to a suprathermal tail distribution. The kappadistribution is given by (2)where n_{e} is the electron number density, Θ_{e} is the velocity width of the thermal core, and κ_{e} is the specific electron kappa parameter with a range of 3/2 ≤ κ_{2,e,} ≤ ∞. The symbol Γ = Γ(x) denotes the Gamma function of the argument x.
In the next step, we determine the adequate value of κ_{2,e} for electrons downstream of the shock associated with a pressure given by Eq. (1). For this purpose, we determine the associated pressure P_{2,e,κ} (i.e., the pressure resulting as the second velocity moment of the above distribution, Eq. (2), see Heerikhuisen et al. 2008) that is equal to the pressure given by Eq. (1) (i.e., the electron pressure found in the multifluid approach by Fahr & Siewert 2013). We obtain the following relation for P_{2,e,κ}: (3)As shown in Fahr & Siewert (2013), we can define the factor Π by (4)where P_{1,p} is the upstream solarwind proton pressure. The factor Π describes the change of thermal core velocities from upstream to downstream. We quantify this factor later in Sect. 5. Using an upstream proton temperature of T_{1,p} = 2 × 10^{4} K and Π ≈ 1, we obtain an average energy for the thermal core electrons of .
From the above relation Eq. (3), we first obtain: (5)which further simplifies to (6)in the region behind the shock, where U_{2,e} ≈ U_{2,p}, using the short notation Λ(α) ≡ A(α)sin^{2}α + B(α)cos^{2}α.
Relating the thermal velocity c_{1,e} of the upstream electrons to the thermal velocity c_{1,p} of the upstream protons through T_{1,p} = T_{1,e}, we obtain (7)We assume that the upstream solarwind Mach number of the protons (i.e., μ_{1,p} ≡ U_{1}/c_{1,p}) is of order 8, which then leads to (8)The terminationshock compression ratio observed by Voyager2 is s ≃ 2.5 (Richardson et al. 2008), which leads to (9)In the case of a perpendicular shock (i.e., α ≃ 90°), Λ(π/ 2) = A(π/ 2) = s leading to (10)Assuming a value of Π = 1, we obtain the result κ_{2,e} = 1.517. This kappa index κ_{2,e} for the shocked downstream solarwind electrons characterizes a highly suprathermal electron spectrum with a powerlaw nearly falling off as v^{5} as shown in Eq. (2). Consequently, we can write for the resulting distribution function of the downstream electrons, (11)with κ_{2,e} ≃ 1.517.
The distribution function in Eq. (11) is easily transformed into the spectral electron flux g_{2,e}(v) = 4πv^{3}f_{2,e}(v). We normalize velocities as x = v/ Θ_{2,e} and find (12)In Fig. 1, we show these spectral electron fluxes for different indices κ_{2,e}.
Fig. 1
Normalized spectral electron fluxes downstream of the solarwind termination shock according to Eq. (12). We show the kappadistributed fluxes for different values of κ_{2,e}. The special case κ_{2,e} = 1.517 is our result for the perpendicular solarwind termination shock according to Eq. (10). 
3. Electric equilibrium potential
In the following, we calculate the electric equilibrium potential Φ up to which a spacecraft charges up when entering the heliosheath under the assumption that both electrons and protons have kappadistributions and that emission processes are negligible in the plasma. Fahr & Siewert (2013) show that all ions (solarwind as well as pickup ions) treated as one fluid can be characterized as one joint kappafunction with a joint kappaindex κ_{2,i} ≃ 2, depending on the pickup ion abundance downstream of the shock (see Fig. 2 of Fahr & Siewert 2013). Therefore, we describe electrons with Eq. (11) and protons with the following distribution downstream of the shock: (13)We assume that quasineutrality prevails outside of perturbed Debye regions, i.e., n_{2,e} = n_{2,i}.
Any metallic body embodied in the heliosheath plasma downstream of the shock charges up to an electric equilibrium potential Φ_{2}, which guarantees equal fluxes of ions and electrons reaching the metallic surface of this body per unit time (geometry taken to be planar). This behavior leads to the following requirement (after dropping downstream indices “2” for simplification): (14)where β_{e}(Φ) and β_{i}(Φ) denote the Boltzmann screening factors for electrons and ions, respectively. These factors describe the fraction of particles that can reach the wall against the electric potential Φ. The assumption of isotropic distribution functions leads to (15)We expect that the resulting equilibrium potential Φ only affects the lowestenergy part of the distribution functions. Therefore, the Gaussian core of the kappadistributed particles is the only screened population, leading to (16)which can be rewritten as (17)We solve the remaining integrals in the above expression with Θ_{2,i}/ Θ_{2,e} ≈ m/M, leading to (18)We find (with ) (19)where μ_{1,e} and denote the upstream solarwind electron and pickup ion Mach numbers. These numbers are given by values of the order μ_{1,e} ≃ 10 and . Therefore, (20)which leads to the following potential (21)As a consistency check, we note that this expression leads to the classical plasmaphysics formula for Φ = Φ_{c} in the limit of Maxwellian distributions (i.e., κ_{e} = κ_{i} → ∞) with identical temperatures K.
In Fig. 2, we show the resulting electric potential Φ as a function of the prevailing kappaindex κ_{2,e} of the shockheated downstream electrons. This profile shows that the expected equilibrium potential drops to values of Φ ≤ −30 V in the range of expected indices 1.5 ≤ κ_{e} ≤ 2. This potential does not allow electrons with energies below 30 eV to reach the detector.
Fig. 2
Equilibrium potential of a metallic body in the presence of kappadistributed ions and electrons according to Eq. (21). We show the potential as a function of κ_{2,e} for different values of κ_{2,i}. We assume U_{1} = 400 km s^{1}. 
4. Degenerated Debye length
The degeneration of the Debye length is a direct consequence of highly nonthermal kappatype distribution functions. The electric screening by plasmaelectron distributions with a largely extended powerlaw tail is significantly less efficient compared to the screening by Maxwellian thermal electrons with temperature T_{e}. Maxwellian thermal electrons lead to the classical Debyescreening length of . The effect of degenerating Debye lengths has been recognized and emphasized by Treumann et al. (2004), finding that the resulting Debye length may easily increase by factors of for low electron kappaindices of κ_{e} ≃ 1.5. With a more relaxed approach, yet along the lines of these authors’ discussion, we obtain the following very similar conclusions.
For the description of the effective screening of kappa electrons, we simply replace the Maxwellian temperature T_{e} by the corresponding electron kappa temperature , which is given by(22)Consequently, we find (23)Assuming that the core of the kappa distribution is identical to the Maxwellian core (i.e., ), we obtain the following result for the effective Debye length: (24)This modified Debye length has an interesting effect on the propagation of plasma waves. The general dispersion relation for electronacoustic plasma waves (e.g., see Chen 1974, Eqs. (4)–(48)) is given by (25)In the general case of , and for very small wavevector values of , this dispersion relation allows for a branch of electronacoustic waves that propagate with a phase/group velocity of (26)This branch is a special kappamode propagating with a typical phase or group velocity that directly depends on the electron kappa index κ_{e}. Testing plasma acoustic waves in this range of large wavelengths should hence directly reveal the prevailing kappa index κ_{e} and, therefore, the character of the suprathermal downstream electrons.
On the other hand, the limit of larger wavevectors with allows for a branch of nonpropagating waves (i.e., standing oscillations) with plasma eigenfrequencies given by (27)With Eqs. (22) and (24), we can write (28)for these oscillations. Under these conditions, the plasma, surprisingly enough, does not oscillate with the electron plasma frequency ω_{e}, but it does oscillate with the ion plasma frequency ω_{p}, which is a phenomenon that only quite rarely occurs in nature. The plasma oscillations recently registered by Voyager1 (Gurnett et al. 2013), which were interpreted as electron plasma oscillations, may perhaps be reinterpreted as this type of ion plasma oscillations. As such, they would allow us to infer environmental plasma densities of the order n ≃ (m/M)0.1 cm^{3} ≤ 10^{4} cm^{3}.
5. Calculation of Π in view of the downstream electron instabilities
In Sect. 2, we introduced the quantity Π (see Eq. (4)), which denotes the ratio of the thermal core widths, . Until this point, we did not determine a reasonable value for Π. Previous expressions based on an upstreamdownstream transformation of thermalcore velocities (Fahr & Siewert 2013) appear to be irrelevant since their calculation relies on the assumption that upstream core electrons are independently transformed simply into downstream core electrons according to the LiouvilleVlasov theorem. In reality, however, all upstream electrons overshoot to the downstream side where the action of instabilities, such as the twostream instability or the Buneman instability, redistribute and isotropize them. In the case of the twostream instability (e.g., see Chen 1974), electrons can excite ion oscillations as long as their velocities are greater than the thermal core velocities of the protons. This fast relaxation of the electron distribution function toward the downstream ion distribution function then leads to a quasiequilibrium distribution.
Consistent forms of such quasiequilibria between particle distribution functions and turbulence power spectra have been investigated in Sect. 2.5 in Fahr & Fichtner (2011) and, for steady state conditions, by Yoon (2011, 2012) and Zaheer & Yoon (2013). In all of these cases, the asymptotic state results in kappa distributions. Also, in our case, as a result of a shockinduced electron injection with velocityspace diffusion and relaxation described by a phasespace transport equation of the type (29)we expect solutions in form of kappa distributions. In fact, as shown by Treumann et al. (2004), this kind of transport equations leads to a quasiequilibrium distribution in the form of a kappa distribution with a thermal core given by the downstream ion velocities: . We finally find (using Eq. (4) in Fahr & Siewert 2013) (30)with and B(s,α) = s^{2}/A^{2}(s,α).
The above expression for a perpendicular shock with s = 2.5 (Richardson et al. 2008) leads to Π(α = π/ 2) = 1.33. This finally shows that the quantity Π is, in fact, of order unity, verifying a posteriori all of our above results that we calculated for Π = 1. Using this more precise value for Π, Eq. (10) leads to a marginally different value for the kappaindex of κ_{2,e} = 1.522.
6. Summary and conclusions
Previous studies suggest that stronglyheated solarwind electrons should appear in measurements as accelerated suprathermal particles downstream of the termination shock. However, these electrons were not observed by Voyager in the heliosheath. We investigate this apparent contradiction and find that heliosheath electrons are distributed according to a kappatype distribution function with an extended suprathermal tail (see Fig. 1). These highly suprathermal kappadistributed electrons lead to a
strong negative charging of all metallic bodies exposed to this plasma environment, consequently also charging up the Voyager spacecraft.
A spacecraft potential of the order −30 V, as calculated in Fig. 2, has a significant effect on the Voyager electron measurements in the heliosheath. Under these conditions, it repels thermal electrons in the energy range below 30 eV, leading to an increase in the previously determined upper limit of 3 eV (Richardson et al. 2008) for the electron temperature. The ions, on the other hand, are accelerated into the Faraday cups. The difference in the derived bulk speeds, however, is negligible: corrected for a −30 V potential, the radial downstream proton bulk velocity increases from 130 km s^{1} to about 132 km s^{1}.
In addition, this suprathermal distribution of downstream electrons also results in an unusually enlarged Debye length. As a consequence of this effect, the phase velocity v_{φ} = ω/k of electrostatic plasma waves depends on the effective kappa temperature of the electrons in the heliosheath plasma environment. The detection of these plasma waves allows us to infer the effective kappa electron temperature as an observable quantity. These distributions also permit a type of nonpropagating standing waves with the ion plasma frequency ω_{p} as their eigen frequency.
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All Figures
Fig. 1
Normalized spectral electron fluxes downstream of the solarwind termination shock according to Eq. (12). We show the kappadistributed fluxes for different values of κ_{2,e}. The special case κ_{2,e} = 1.517 is our result for the perpendicular solarwind termination shock according to Eq. (10). 

In the text 
Fig. 2
Equilibrium potential of a metallic body in the presence of kappadistributed ions and electrons according to Eq. (21). We show the potential as a function of κ_{2,e} for different values of κ_{2,i}. We assume U_{1} = 400 km s^{1}. 

In the text 
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