Electron Injection by Whistler Waves in Non-relativistic Shocks
aa r X i v : . [ a s t r o - ph . H E ] M a r Draft version September 25, 2018
Preprint typeset using L A TEX style emulateapj v. 11/10/09
ELECTRON INJECTION BY WHISTLER WAVES IN NON-RELATIVISTIC SHOCKS
Mario A. Riquelme
Astronomy Department, University of California, Berkeley, CA 94720 andAnatoly Spitkovsky
Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544
Draft version September 25, 2018
ABSTRACTElectron acceleration to non-thermal, ultra-relativistic energies ( ∼ −
100 TeV) is revealed by radioand X-ray observations of shocks in young supernova remnants (SNRs). The diffusive shock accelera-tion (DSA) mechanism is usually invoked to explain this acceleration, but the way in which electronsare initially energized or ‘injected’ into this acceleration process starting from thermal energies is anunresolved problem. In this paper we study the initial acceleration of electrons in non-relativisticshocks from first principles, using two- and three-dimensional particle-in-cell (PIC) plasma simula-tions. We systematically explore the space of shock parameters (the Alfv´enic Mach number, M A , theshock velocity, v sh , the angle between the upstream magnetic field and the shock normal, θ Bn , andthe ion to electron mass ratio, m i /m e ). We find that significant non-thermal acceleration occurs dueto the growth of oblique whistler waves in the foot of quasi-perpendicular shocks. This accelerationstrongly depends on using fairly large numerical mass ratios, m i /m e , which may explain why it hadnot been observed in previous PIC simulations of this problem. The obtained electron energy distri-butions show power law tails with spectral indices up to α ∼ −
4. The maximum energies of theaccelerated particles are consistent with the electron Larmor radii being comparable to that of theions, indicating potential injection into the subsequent DSA process. This injection mechanism, how-ever, requires the shock waves to have fairly low Alf´enic Mach numbers, M A .
20, which is consistentwith the theoretical conditions for the growth of whistler waves in the shock foot ( M A . ( m i /m e ) / ).Thus, if the whistler mechanism is the only robust electron injection process at work in SNR shocks,then SNRs that display non-thermal emission must have significantly amplified upstream magneticfields. Such field amplification is likely achieved by the escaping cosmic rays, so electron and protonacceleration in SNR shocks must be interconnected. Subject headings: acceleration of particles - shock waves - cosmic rays - plasmas INTRODUCTION
Non-thermal electron acceleration is believed to be auniversal feature of non-relativistic collisionless shocksboth in space plasma and astrophysical environments.In the Earth’s bow shock, for example, electrons accel-erated up to few tens of keV are usually observed inquasi-perpendicular regions, i.e, where the angle betweenthe upstream magnetic field and the shock normal is & ◦ (Oka et al. 2006; Gosling et al. 1989). Also, radioand X-ray observations of supernova remnants (SNRs)show synchrotron radiation produced by relativistic, non-thermal electrons accelerated in their forward shocks(e.g. Koyama et al. 1995; Bamba et al. 2003, 2005).Despite its ubiquity, the actual mechanism for shockacceleration of electrons is still a mystery. Thediffusive shock acceleration (DSA) mechanism is themost accepted theory for particle acceleration inshocks (Axford et al. 1977; Krymsky 1977; Bell 1978;Blandford & Ostriker 1978). This theory assumes thatparticles are scattered by MHD turbulence both in theupstream and downstream regions of the shock. Underthese conditions, particles move diffusively in the shockvicinity, gaining energy through many crossings of theshock. However, in order to be able to cross the shock [email protected] many times, the particles need to have Larmor radii thatare larger than the shock width, which is controlled bythe typical ion Larmor radius. This is the still unresolved“injection problem” of the DSA theory, which is partic-ularly stringent for electrons due to their small Larmorradii.Several methods have been applied to studying elec-tron acceleration in non-relativistic shocks. One ap-proach, typically used in the context of low Alfv´enicMach number ( M A ) shocks, uses test particle electronsmoving in electromagnetic fields pre-determined fromhybrid simulations (where ions are modeled as kineticparticles and electrons as a massless fluid). Recently,Guo & Giacalone (2010) showed that electrons can beenergized due to repeated shock crossings as they movealong pre-existing, large-scale magnetic fluctuations inperpendicular shocks. Also, Burgess (2006) used thesame method to show that shock “ripples,” which arefluctuations on the surface of quasi-perpendicular shocks,can provide the scattering necessary for efficient elec-tron acceleration. Although in principle these mecha-nisms may contribute significant electron energization innon-relativistic shocks, they still require a population ofelectrons that are somehow injected upstream with en-ergies significantly above thermal. Understanding theorigin of these particles requires self-consistent kinetic Riquelme & Spitkovskycalculations based on particle-in-cell (PIC) simulations.PIC simulations have been used by several authorsto study the electron acceleration problem in non-relativistic shocks (e.g., Amano & Hoshino 2007, 2009;Umeda et al. 2009). For instance, using two-dimensionalPIC simulations, Amano & Hoshino (2009) showed thatefficient electron acceleration (with spectral index α =2-2.5) can happen in a perpendicular, M A = 14 shockdue to “shock surfing” of electrons on electrostatic wavesfrom Buneman instability excited at the leading edge ofthe shock foot. These simulations were done for rela-tively low mass ratios ( m i /m e = 25) and for a particulargeometry of the magnetic field (strictly out of the planeof the simulation). The dependence of this mechanismon different shock parameters needs to be clarified, andin fact, in this work we find that this mechanism is notvery efficient at realistic mass ratios. Also, using two-dimensional PIC simulations Umeda et al. (2009) showedthat, for M A = 5 shocks, pre-acceleration in Bunemanwaves can be complemented by further energization dueto scattering at the shock ripples, supporting the picturelaid out by Burgess (2006). The energy spectrum of theelectrons in this case, however, does not correspond to apower law tail, but is rather described by two Maxwelliandistributions at different temperatures.One of the main difficulties of PIC studies is that us-ing realistic physical parameters is computationally ex-pensive, so unrealistic approximations have to be made.For instance, simultaneously capturing the dynamics ofelectrons and ions requires resolving time scales as shortas the inverse of the plasma frequency of electrons, ω − p,e , and as long as the inverse of the cyclotron fre-quency of ions, ω − c,i (with ω p,j = (4 πn j e j /m j ) / and ω c,j = e j B/m j c , where c is the speed of light, B isthe magnitude of the magnetic field, and n j , m j , and e j are the density, mass, and electric charge of the j species). Given that ω p,e /ω c,i = ( m i /m e ) / / ( v A /c ),where v A ≡ B/ (4 πn i m i c ) / corresponds to the Alfv´envelocity of the plasma, the computing time in PIC simu-lations is usually reduced by using artificial mass ratios m i /m e ≪ v A /c .Also, these studies are usually made using one- and two-dimensional simulations. Although much of the relevantphysics can still be revealed using these approximations,a complete understanding of the role played by the cho-sen parameters is crucial before any extrapolation to re-alistic setups is made. Thus, in this work we presenta systematic exploration of the space of shock parame-ters, paying special attention to the role of artificial massratios m i /m e on the possible electron energization.The effect of using a reduced ion to electron massratio m i /m e has already been highlighted by previ-ous PIC studies of quasi-perpendicular shock struc-ture (Scholer et al. 2003; Scholer & Matsukiyo 2004;Hellinger et al. 2007). These works have shown that ex-cessively small m i /m e may suppress the appearance ofoblique whistler waves in the foot of low M A shocks,which would lead to periodic shock reformation.In terms of electron energization, the works ofScholer et al. (2003); Scholer & Matsukiyo (2004) alsoshow that whistler waves can lead to significant electronheating in the foot of the shocks, but the investigationof their possible role in non-thermal electron injection remains to be realized .In this work we study the injection of non-thermalelectrons in quasi-perpendicular shocks, using two- andthree-dimensional PIC simulations. We systematicallytest different regimes for the shock velocity, v sh , theplasma magnetization (quantified in M A ), the ion to elec-tron mass ratio m i /m e , and the angle between the shocknormal and the ambient magnetic field, θ Bn . We showthat the presence of oblique whistler waves in the shockfoot does lead to significant electron acceleration withspectral index α ≈ −
4. This acceleration appears toprefer small values of the M A / ( m i /m e ) / ratio, and re-quires θ Bn = 90 ◦ . These two conditions likely explainwhy this effect was not seen by previous PIC studies ofthis problem, where strictly perpendicular shocks withfairly low mass ratios were used. For the realistic value m i /m e = 1836, this acceleration would require low Machnumbers, M A .
20, in order to explain the electron in-jection fraction inferred from broadband observations ofSNRs. Thus, if this mechanism happens to be the onlypossible solution for electron injection into the DSA pro-cess in SNR shocks, the M A .
20 condition would implystrong amplification of the upstream magnetic field ofthese shocks.This paper is organized as follows. In § §
3, we describe themechanism that brings electrons to non-thermal energies,by analyzing the evolution of the accelerated particles inone of our simulations. In §
4, we explore different shockparameters, and determine the regimes where electronsacceleration occurs. Our final discussion and conclusionsare presented in § SIMULATION SETUP
We use the electromagnetic PIC code TRISTAN-MP(Buneman 1993; Spitkovsky 2005) in two and three di-mensions . A shock wave is produced by reflecting a coldhigh-speed electron-ion plasma off a conducting wall.The shock forms due to the interaction between the in-coming and reflected beams and propagates away fromthe wall along the x − direction. The incoming plasmacarries a uniform magnetic field, ~B , forming an angle θ Bn with the shock normal (which coincides with the x − axis). The shocked plasma stays at rest with respectto the box, so the simulation is done in the downstreamframe. In two dimensions the simulation box consistsof a rectangle in the xy plane with periodic boundaryconditions in the y − direction.The incoming plasma is injected through a “moving in-jector”, which recedes from the wall in the x − directionat the speed of light. The simulation box is expandedin the + x − direction as the injector approaches the rightboundary. This way the memory and computing time are We note that Levinson (1992, 1994) discussed an analyticaltheory of electron injection due to whistler waves excited by thereturning electrons. These works predicted that electrons shouldbe accelerated in shocks with large Mach numbers, M A &
43. How-ever, both our simulations and the recent work by Kato & Takabe(2010) show neither whistler excitation nor significant electron ac-celeration in such shocks, raising questions about the consistencyof the assumptions that went into the theory. In both two- and three-dimensional simulations the threecomponents of particle velocities and electro-magnetic fields aretracked. lectron Injection in Non-relativistic Shocks 3saved, while following the evolution of all the upstreamregions that are causally connected with the shock. Fur-ther numerical optimization can be achieved by allow-ing the moving injector to periodically jump backward(i.e., in the − x − direction), resetting the fields to its right(see Sironi & Spitkovsky 2009). Since the shock travelsmuch slower than the speed of light, without a jump-ing injector (i.e., with the injector only moving at c ) theupstream region would comprise most of the simulationdomain. However, since we expect the electron accel-eration to occur on scales close to the shock foot (i.e.,at a distance comparable to the typical Larmor radiusof ions, R L,i ), keeping an upstream size of a few shockfoot lengths should be enough to capture the relevant ac-celeration physics. Thus, in our simulations the movinginjector jumps backward every 2000 time steps, with thefirst jump happening after 4000-12000 time steps (de-pending on the length of the shock foot). The injectorjumps backward a distance such that its average veloc-ity is close to the shock speed. This method also helpsto reduce particle heating by the numerical cold beaminstability, which can happen after a long distance prop-agation of the cold plasma over the numerical grid.We ran a series of two- and three-dimensional simula-tions to explore different shock regimes. The run parame-ters are summarized in Tables 1, 2, and 3. The numericalvalue of the speed of light is set to 0 .
45 cells/time stepin all the runs. In the two-dimensional runs presented inthe main part of the paper, ~B is always in the plane ofthe simulation ( xy plane). This condition is changed inAppendix A, where the case of ~B quasi-perpendicularto the simulation plane is considered. THE ACCELERATION MECHANISM
In this section we disentangle the process that gives riseto the electron energization, focusing on one of our shocksimulations where significant acceleration is observed.We use the two-dimensional run 2D-3, with v sh /c = 0 . v A /c = 1 / m i /m e = 400, M A = 7, and θ Bn = 75 ◦ (the other pa-rameters are specified in Table 1). First we will describethe basic features of the shock, and then we will focus onthe process of electron energization. Shock Structure
Quasi-perpendicular shocks are characterized by thepresence of the so-called shock foot. The foot is definedby the existence of a beam of ions reflected by the shock,whose bulk velocity in the upstream frame is close to v sh and has comparable x and z components. Thefoot region can be seen in panels a ) and b ) of Figure 1(between x = 410 and 470 c/ω p,e ), which depicts thephase space distribution functions of the ions. Thisregion covers a distance comparable to the ion Larmorradius, R L,i (calculated with the upstream magneticfield B ), and is located right in front of the shockdensity jump or “ramp”, as shown in panel h ) of thesame Figure. An important feature of the shock foot isthe presence of oblique electromagnetic waves that growon scales of ∼ c/ω p,e , forming an angle of ∼ ◦ withthe shock normal. These waves can be seen in panels e ) − g ) of Figure 1 (between x = 410 and 470 c/ω p,e ),which show the magnetic field along the x − direction, Fig. 1.—
The shock structure for run 2D-3 (Table 1) at tω p,e =10000 ( tω c,i = 10, where ω c,i is defined in the upstream), whosebasic parameters are v sh = 0 . c , M A = 7, m i /m e = 400, and θ Bn = 75 ◦ . The phase space distribution functions of the ions, f i ,on the x − p x and x − p z planes are shown in panels a ) and b ),respectively. The corresponding phase space distribution functionsfor the electrons, f e , are depicted in panels c ) and d ), respectively.The distribution functions are normalized by their maximum value.Panels e ), f ), and g ) show the magnetic field components along the x − and z − axes, and the ion density, respectively. Panel h ) showsthe one-dimensional ion density profile averaged on the y axis. Riquelme & Spitkovsky B x , the magnetic field along the z − direction, B z , andthe ion density, n i , respectively. The oblique modeshave a right-handed circular polarization, and theirphase velocity is comparable to the speed of the shock.All these features allow them to be identified as electronwhistler waves propagating obliquely with respect to thebackground magnetic field.The whistler waves have been studied in the con-text of shock structure evolution, using one-, two-, and three-dimensional PIC simulations, as well ashybrid codes. The two-dimensional studies showthat whistler waves can play an important roleby suppressing the self-reformation of perpendicularand quasi-perpendicular shocks (Hellinger et al. 2007;Lembege et al. 2009; Yuan et al. 2009). This effect,however, would be less important in the fully three-dimensional geometry, as shown by a recent PIC study(Shinohara et al. 2011).The exact generation mechanism of these whistlersis still subject to debate. One candidate mechanism isthe so-called modified two-stream instability (MTSI),driven by the relative cross-field velocity betweenthe electrons and the ions in the foot of the shocks(Wu et al. 1983; Matsukiyo & Scholer 2003, 2006).Since in quasi-perpendicular shocks a significant frac-tion of the ions are reflected into the upstream, theMTSI can be driven by the relative motion betweenelectrons and either the incoming or reflected ions.These two possibilities are usually referred to as MTSI1and MTSI2, respectively. The analytic dispersionrelation calculations show that the MTSI1 and MTSI2will grow if cos( θ ) & − r ) M A / ( m i /m e ) / andcos( θ ) & rM A / ( m i /m e ) / , respectively, where θ is theangle between the magnetic field and the wave vector ofthe waves, and r is the fraction of reflected ions, withtypical values of r ∼ . M A / ( m i /m e ) / the larger is therange of θ where the excitation of whistler waves in thefoot of shocks would be possible.Another possibility for whistler generation is that thesewaves can be an intrinsic component of oblique quasi-perpendicular shocks. Indeed, Krasnoselskikh et al.(2002) proposed an analytical model for the structureof these shocks where the shock ramp is treated asa nonlinear whistler wave. This model shows that if M A . M w ≡ | cos( θ Bn ) | ( p m i /m e ) / /
2, where θ Bn isthe angle between the shock normal and the upstreammagnetic field, ~B , the shock precursor would containa stationary whistler wave train. If M w is exceeded,these whistler waves would become non-linear andwould be rather confined to the shock ramp. But if M A & M nw ≡ | cos( θ Bn ) | ( p m i /m e ) / / / a station-ary solution for the shock structure would no longerbe possible, which would set the condition for shockreformation. In particular for Run 2D-3 ( θ ≈ ◦ and θ Bn = 75 ◦ ) the condition for the whistler generationdue to the MTSI (in any of its two varieties) appearsto be less stringent than for the model proposed byKrasnoselskikh et al. (2002). This is in general the case Fig. 2.—
The energy spectra at different positions for electronsand ions (solid and dashed lines, respectively) are shown for theshock transition region of run 2D-3 at tω c,i = 10. The spectra aremeasured at the x/c/ω p,e = 330 , , h ) of Figure 1, whosecolors match the ones used for the corresponding spectra. for the simulations presented in this paper. But, evenfor runs where cos( θ Bn ) & cos( θ ), the growth of whistlerwaves in the foot of quasi-perpendicular shocks appearsto be favored for small values of the M A / ( m i /m e ) / ratio. Electron Spectrum
The phase space distribution functions for electrons isdepicted in panels c ) and d ) of Figure 1. Although theelectrons are mainly heated at the shock, significant elec-tron energization also occurs in the foot region. This isseen from the electron spectra shown in Figure 2, whichare measured at several positions in the shock region.The positions are marked by the vertical lines in Figure1h, with colors matching the ones of the correspondingspectra in Figure 2. We see that the two energy dis-tributions measured in the downstream (red and blacklines) show a high-energy power law tail with spectralindex α ≃ . e ≃ e ≈ v sh ( m i /m e ) /c . Also,two downstream ion spectra, measured at the locationsof the vertical black and red lines in panel h ) of Fig-ure 1, are depicted by dashed lines. These spectra showan incomplete ion thermalization, and essentially no ionacceleration, with only a small, exponentially decreas-ing bump at energies a few times above thermal. Noticethat the downstream electron temperature is close to themean kinetic energy of the ions (for m i /m e = 100), im-plying energy equilibration between both species. Fi-nally, the spectra depicted in Figure 2 are fully evolved,lectron Injection in Non-relativistic Shocks 5in the sense that they do not show significant variationsin time. This will be the case in all the simulations shownin this paper after a time of ∼ ω − c,i . The Physics of Acceleration
In this section we illustrate the physics of the electronacceleration process by focusing on the evolution of atypical non-thermal electron. In the three panels of Fig-ure 3 we plot the energy of the electron, represented bythe solid, black line. In order to identify the source of theenergy gain, we also plot the accumulated energy gain, ǫ j , due to the work done by the electric field along the j axis (red, green, and blue lines represent j = x , y , and z , respectively).This energy gain is calculated as ǫ j ≡ R v e,j E j e/m e dt ,where v e,j and E j correspond to the electron velocity andelectric field along the j axis, respectively. Since thesecumulative energies are shown on the logarithmic scale,we plot their absolute values, and use dotted lines whenthey correspond to negative quantities. In each panelwe also highlight a small time interval, whose startingand final points are marked by yellow and red circles,respectively. The particle trajectories corresponding toeach of these intervals are shown on top of Figures 5 and6 by a solid, black line. The black circle on top of thehighlighted intervals marks the time of each field snap-shot. The initial increase in the electron energy happensin the first highlighted interval, shown in Figure 3 a , withthe particle location depicted in Figures 5 a and 6 a . Wecan see that this initial energization happens in the shockfoot as the particle moves through the region of ampli-fied whistler waves. The energy gain is initially due toincrease in ǫ y , with ǫ x contributing most of the energygain after tω p,e ≈ ǫ y ) ǫ x has alreadyincreased significantly, its contribution to the total en-ergy of the electron is almost exactly compensated by adecrease in ǫ z (see rising dotted blue and solid red linesbetween tω p,e = 1737 and 1920 in Figure 3 a ). Indeed,the particle motion until this point was dominated bythe ~E × ~B drift, which does not allow any net work to bedone by the electric field. However, as soon as electricfield fluctuations on scales comparable to the electronLarmor radius appear, this almost perfect cancellationstops, and a net energy gain becomes possible. Since the y − axis is quasi-parallel to the initial field ~B , the initialenergization due to E y already suggests an important fea-ture of the whistlers waves: their electric field has a non-negligible component along the magnetic field direction.This parallel electric field component can be explained bythe obliquity of the waves with respect to the initial mag-netic field, which makes the projection of whistler electricfield on ~B have a magnitude ∼ sin( θ ) | ~E | . The energygain due to the parallel electric field can be verified fromFigure 4, where we have plotted in red and green the en-ergy gain due to the electric field perpendicular and par-allel to ~B ( ǫ ⊥ and ǫ || , respectively). Indeed, we can seethat ǫ || is what dominates the initial energy gain of theelectron, confirming that the presence of the electric fieldparallel to the magnetic field is the essential componentin the initial electron energization. This feature makesthis process fundamentally different from fast Fermi ac-celeration where the shock acts as a magnetic mirror, en- ergizing the electrons due to the electric field componentperpendicular to the magnetic field. Finally, we note thatthis initial energization is described in the frame wherethe downstream plasma is at rest (usually called normalincidence frame). The picture, however, should be essen-tially the same if described from any other frame like, forinstance, the de Hoffman-Teller frame. In particular, theinitial energization will continue to be dominated by theelectric field component parallel to the magnetic field.This is due to the relativistic invariance of ~E · ~B , whichwould make ~E · ~B/ | ~B | a nearly invariant quantity underany non-relativistic frame transformation.After the initial energization driven by the whistlers,the acceleration due to E z (the convective electricfield) starts to play a more relevant role, as can beseen from Figure 3 b . We see that in the highlightedinterval (marked by the two vertical, dashed lines) theenergy increase is driven by a jump both in ǫ z and ǫ y . In Figures 5 b and 6 b we can see that during theseenergy increases the electron moves mainly along the+ y − direction, and stays close to the shock ramp. Therelevance of the energy gain due to E z is more obvious atlater times, when the electron Larmor radius has becomecomparable to a sizable fraction of the foot length. Asshown in Figure 3 c , most of the energy is gained due toan increase in ǫ z . The particle trajectory correspondingto the marked interval (between the two vertical, dashedlines) is plotted in Figures 5 c and 6 c . We can seethat, as it moves (mainly along the + y − direction), theparticle jumps a couple of times from the upstream tothe downstream and vice versa, gaining energy in a waysimilar to shock drift acceleration.Apart from dominating the initial electron energiza-tion, the energy gain along the magnetic field, ǫ || , also in-creases the particle’s ability to move along the magneticfield lines. This is important because if an electron veloc-ity along ~B becomes larger than v sh / cos( θ Bn ), it will bepossible for the particle to move along the + x − directionfaster than the shock. When this happens, the electronwill be able to escape upstream, increasing its chances tobe further scattered by the whistlers waves. This quali-tative picture already shows that having θ Bn = 90 ◦ is animportant component in the acceleration process. Thisangle dependence of the acceleration is confirmed in thefollowing section where the importance of the shock pa-rameters: θ Bn , m i /m e , v sh , and M A for electron accel-eration is studied. Finally, it is important to point outthat the evolution of thermal electrons is fundamentallydifferent to the one of the non-thermal particles, which iswhat we described in this section. Thermal electrons gainsome energy due to a random combination of the paral-lel and perpendicular electric field mainly in the rampand shock transition region, and then are rapidly incor-porated into the downstream medium. ELECTRON ACCELERATION REGIME
In this section, we seek to determine the physicalconditions under which the electron acceleration due towhistler waves happens. To do this, we analyze the ac-celeration behavior by varying different shock parametersone by one. Riquelme & Spitkovsky
Fig. 3.—
The time evolution of the energy of an accelerated electron, Γ e − e is the electron Lorentz factor), is depicted bythe solid, black line in panels a ), b ), and c ), which show different time intervals. Besides the electron energy, the accumulated energygain, ǫ j , due to the work done by the electric field along the j axis is also plotted (red, green, and blue lines represent j = x , y , and z ,respectively). Since these cumulative energies are shown on the logarithmic scale, only their magnitudes are plotted. Thus, in order tokeep the sign information, the negative values of ǫ j < a ), b ), and c )mark the intervals corresponding to the electron trajectories tracked in panels a ), b ), and c ) of Figures 5 and 6. Fig. 4.—
The time evolution of the energy Γ e − e isthe electron Lorentz factor) of the accelerated electron analyzed in § a ) of Figure 3 is plottedin black solid line. Besides the total energy, the cumulative energygain due to the work done by the electric field perpendicular andparallel to ~B ( ǫ ⊥ and ǫ || , respectively) is also plotted in red andgreen lines, respectively. Angle Dependence
We illustrate the importance of the angle θ Bn for elec-tron acceleration using Figure 7. The downstream spec-trum of the electrons at tω p,e = 10000 ( tω c,i = 10) isplotted for simulations that only differ in their angle θ Bn .The rest of the parameters are the same as for simula-tion 2D-3 in Table 1. We can see that the hardest spec-trum is reached for θ Bn = 70 ◦ (red line), with α ≈ . θ Bn = 90 ◦ case (purple line), which corresponds to a Maxwelliandistribution with a small increase in the number of par-ticles at energies a few times above thermal. This re-sult confirms the picture suggested in § y − axis. This is in fact what wenoticed when tracking the trajectory of the acceleratedelectron analyzed in § θ Bn < ◦ the softer spectra can be explained by theless efficient confinement of the particles to the shockvicinity. A quasi-perpendicular configuration keeps theelectrons close to the shock, not allowing them to easilyescape upstream. From that perspective, it is reasonableto think that the most efficient acceleration will happenwhen cos( θ Bn ) is just large enough to allow the electronsto move at v sh in the + x − direction, with decreasingefficiency for larger values of cos( θ Bn ). Mass Ratio Dependence
We now explore the ion to electron mass ratio m i /m e dependence of the electron acceleration. To do this, weuse three runs in two dimensions with the same param-eters M A = 7, θ Bn = 75 ◦ , and v sh = 0 . c , but with m i /m e = 25, 100 and 400 (run 2D-7, 2D-8, and 2D-3 of Table 1, respectively). Their downstream spectraare shown by the solid lines in Figures 8 a -8 c . We canclearly see the role played by m i /m e in the electron ac-celeration. While for the mass ratios m i /m e = 25 and100 cases, the spectral index α ≈ . ≈ .
7, re-spectively, the m i /m e = 400 case shows the decreaseof the spectral index to α ≈ . . In order to showthe correlation between this acceleration and the growthof the whistler modes, in Figure 9 we have plotted theelectric field along ˆ x , E x , in the shock transition re-gion of these three simulations (panels a ), b ), and c ) for m i /m e =25, 100, and 400, respectively). These plotsshow how the shock foot becomes dominated by largeamplitude whistler waves as the mass ratio increases from m i /m e = 25 to 400. Also this Figure confirms the length Since the typical time scale for quasi-perpendicular shock evo-lution is given by the ion cyclotron period, ω − c,i , the comparisonbetween simulations of different mass ratio is performed at a fixedtime tω c,i = 10. We use this criterion because, for all the massratios in our study, tω c,i = 10 corresponds to the typical time atwhich the downstream spectrum becomes homogeneous, with nosubstantial variations as a function of position. lectron Injection in Non-relativistic Shocks 7 Fig. 5.—
The electric field along ˆ x , E x , for the two-dimensionalrun 2D-3 ( M A = 7, v sh = 0 . c , θ Bn = 75 ◦ , and m i /m e = 400).Panels a ), b ), and c ) correspond to times tω p,e = 1950, 3370, and4840. The field is normalized in terms of E ≡ B v in /c , where v in is the speed at which the upstream plasma is injected, as seenfrom the downstream medium (for run 2D-3, v in = 0 . c ). On topof each panel, the trajectory of an accelerated electron is trackedby the black line, with the yellow and red circles marking the initialand final time of each trajectory. The black circle, marks the timeof each field snapshot. scale of the whistlers waves ( ∼ c/ω p,e ) determined in §
3, which is consistent with previous dispersion relationcalculations (Wu et al. 1983). In this study of mass ra-tio dependence we also want to include the possibility ofelectron energization by “shock surfing” of electrons, dueto electrostatic waves produced by the Buneman instabil-ity in the foot of the shocks (Amano & Hoshino 2009).These waves appear in the leading edge of the shock footon scales of ∼ c/ω p,e , and are due to the relative velocitybetween the upstream electrons and the ions reflected bythe shock (Buneman 1958). As electrons get scatteredby these waves, they gain energy due to the work per-formed by the fluctuating electric field, combined withthe convective field of the upstream plasma. Since the Fig. 6.—
Same as in Figure 5 but for the ion density, n i . Thedensity is normalized in terms of the upstream ion density, n i, .The arrows represent the orientation of the magnetic field on the xy plane. wave vector of the fastest growing Buneman mode, ~k bun ,is parallel to the ion beam, the growth of the waves andtheir effect on the electron acceleration are better re-solved when the ion motion is parallel to the plane ofthe simulation ( xy plane). This is achieved if the mag-netic field, ~B , is quasi-perpendicular to the xy plane,implying that the Buneman acceleration will be sup-pressed in our two-dimensional runs with the magneticfield in the simulation plane (Amano & Hoshino 2009).We used this behavior to distinguish the contribution ofthe “shock surfing” mechanism to electron accelerationby comparing the results of the two-dimensional simula-tions with analogous three-dimensional runs, where the ~k bun is resolved by adding a third dimension of a few c/ω p,e . The corresponding downstream spectra of thethree-dimensional runs with m i /m e = 25, 100, and 400(runs 3D-1, 3D-2, and 3D-3 of Table 2, respectively) aredepicted using dashed lines in Figures 8 a , 8 b , and 8 c .We can see significant differences between the two- and Riquelme & Spitkovsky Fig. 7.—
The downstream spectrum of the electrons at tω p,e =10000 ( tω c,i = 10) is plotted for simulations like run 2D-3 ( M A = 7, v sh = 0 . c , θ Bn = 75 ◦ , and m i /m e = 400), but using differentangle θ Bn . The purple, orange, red, blue, and black lines representthe cases θ Bn = 90 (run 2D-1), 80 (run 2D-2), 70 (run 2D-4),60 (run 2D-5), and 45 (run 2D-6). For comparison a Maxwelliandistribution is shown in black dashed line. three-dimensional spectra for m i /m e = 25, implying animportant contribution to acceleration due to the shocksurfing mechanism. Indeed, when analyzing the trajec-tories of the energetic particles of the three-dimensionalrun with m i /m e = 25, we find that most of their energy isgained as the electrons get scattered by Buneman wavesin the leading edge of the foot. The difference betweentwo- and three- dimensional runs, however, is substan-tially reduced for the case m i /m e = 100, and almostdisappears for m i /m e = 400, showing that the relativeimportance of Buneman acceleration, compared to theenergization due to whistler waves, decreases for morerealistic mass ratios. Further details on this mass ratiodependence of the shock surfing mechanism are given inAppendix A, where the acceleration of electrons is stud-ied for two-dimensional runs with ~B quasi-perpendicularto the plane of the simulation. The results presentedin Appendix A also demonstrate that the accelerationdue to whistler waves disappears if the direction along ~B is not resolved by the simulation, which makes thetwo-dimensional configuration used in the main part ofthis paper (i.e., with ~B in plane), the most suitable forthe study the electron injection due to whistlers. Also,the convergence between the two- and three-dimensionalruns depicted in Figure 8 c shows that whistler accelera-tion is well modeled by our two-dimensional simulations,with no appreciable changes due to the additional thirddimension of a few c/ω p,e . This result, however, doesnot rule out significant three dimensional effects when ascale comparable to a few whistler scales (i.e., a few 10 c/ω p,e ) is resolved along the third dimension. We leavethis possibility as a subject of future investigation. Shock Velocity Dependence
It is interesting to see how this electron accelerationmay change if a different shock velocity is used. We testthe shock velocity dependence by running simulationswith the same parameters as for the two-dimensional sim-ulations shown in Figure 8 but using v sh = 0 . c , whichcorresponds to v in = 0 . c (simulations 2D-18, 2D-19, Fig. 8.—
The downstream electron spectra for simulations with M A = 7, θ Bn = 75 ◦ , and v sh = 0 . c , but different ion to elec-tron mass ratios, m i /m e = 25, 100, and 400, are depicted inplots a ), b ), and c ), respectively. The solid lines represent two-dimensional runs, while the dashed lines correspond to their three-dimensional counterparts. The parameters of the two-dimensionalsimulations are described in Table 1 (runs 2D-7, 2D-8, and 2D-3for m i /m e = 25, 100, and 400, respectively) and the ones of thethree-dimensional runs are specified in Table 2 (runs 3D-1, 3D-2,and 3D-3 for m i /m e = 25, 100, and 400, respectively). A con-vergence between the two- and three-dimensional results can beseen as m i /m e increases to more realistic values, implying a de-crease in the contribution of the “shock surfing” mechanism to theacceleration of electrons. and 2D-20 in Table 1). Notice that in order to keep thesame M A = 7 the magnetic field also has to be reduced bya factor of 3.3. The resultant downstream electron spec-tra are shown in Figure 10 for m i /m e = 25, 100, and 400(red, black, and green lines). These spectra essentially re-produce the ones corresponding to two-dimensional runswith v sh = 0 . c , shown in Figure 8. The same spectralindex α dependence on m i /m e is obtained, with a de-crease in α when passing from m i /m e = 100 to 400. Theoverall normalization of the power law tails, however,seems to drop by a factor of ∼ v sh = 0 . c cases. This result suggests that the mecha-nism for electron acceleration due to growth of whistlerwaves has a rather weak dependence on the shock veloc-ity, as long as M A is kept constant. It is important tonotice, however, that these measured spectral indices arefor the same inclination angles that produce the lowest α in the case of v sh = 0 . c ( θ Bn = 75 o ). Thus, in thiscase we are assuming that the angle θ Bn at which α isminimized does not depend on the shock velocity. Thisassumption needs to be confirmed by further explorationof the angle dependence of acceleration for different shockvelocities. A thorough study aimed to clarify this pointlectron Injection in Non-relativistic Shocks 9 Fig. 9.—
The electric field along ˆ x , E x , in the shock transition re-gion of simulations 2D-7, 2D-8, and 2D-3, which only differ in their m i /m e parameter given by m i /m e =25, 100, and 400, respectively.The field is shown at tω c,i = 10 ( tω p,e = 2500, 5000, and 10000,respectively), and is normalized in terms of E , which correspondsto the convective electric field given by B v in /c (where v i = 0 . c is the injection velocity of the upstream plasma as seen from thedownstream frame). will be presented elsewhere. Alfv´enic Mach Number Dependence
We test the effect of varying M A by running two-dimensional simulations that are analogous to the onesshown in Figure 8, but also using the Mach numbers M A = 3 . M A = 14. We do this by changingthe magnitude of the magnetic field while keeping thesame shock velocity v sh = 0 . c . We use the mass ratios m i /m e = 25, 100, 400, and 1600, in order to simulta-neously take into account the mass ratio dependence foreach of the studied Mach numbers. The parameters ofthe simulations are summarized in Table 1 (where for m i /m e = 25, 100, 400, and 1600 the M A = 3 . M A = 7 simulations are called 2D-7, 2D-8,2D-3, and 2D-9, and the M A = 14 simulations are called2D-14, 2D-15, 2D-16, and 2D-17).The corresponding downstream electron spectra for M A = 3 .
5, 7, and 14 are shown in panel a), b),and c) of Figure 11 (with the red, black, green, andblue lines showing the spectra for m i /m e = 25, 100, Fig. 10.—
The downstream electron spectra for the 2D-18, 2D-19,and 2D-20 simulations, with v sh = 0 . M A = 7, θ Bn = 75 ◦ , and m i /m e =25, 100, and 400 (depicted by red, black, and green lines,respectively). A dependence of the spectral index α on m i /m e similar to the one found for the v sh = 0 .
14 case is observed. α also drops as theMach number goes from 14 to 3.5. This tendency isconsistent with our finding that the electron accelera-tion is driven by the growth of whistler waves in thefoot of quasi-perpendicular shocks. Indeed, our analy-sis presented in § M A / ( m i /m e ) / . M A = 7 and m i /m e = 25, shownby the red line in plot b ) we obtain a rather soft spec-trum, with α ∼
6. The acceleration efficiencies, in termsof the fraction of non-thermal particles, follow a similartrend, with the number of non-thermal particles beinglarger for smaller values of M A / ( m i /m e ) / . The typicalfraction of non-thermal particles in our runs ranges be-tween ∼ ∼
10 to 30% .For each Alfv´enic Mach number, we explored the angledependence of the electron acceleration by using severalvalues for θ Bn . Thus, the spectra presented in Figure 11correspond to the hardest spectra for each M A , whose an-gles are: θ Bn = 75 o for M A = 3 . θ Bn = 75 o for M A = 7,and θ Bn = 60 for M A = 14. However, these angles weredetermined using a single mass ratio m i /m e = 400. Thusour results assume that the angle of the hardest spectralindex depends only weakly on the used mass ratio. Thisassumption needs to be confirmed by further explorationof the angle dependence of this acceleration mechanismfor different values of m i /m e . Despite this uncertainty,our results show, in general, a hardening of the electronspectra as the M A / ( m i /m e ) / ratio decreases. This canalso be seen from Figure 12, which shows a compilation ofthe spectral indices for the cases depicted in Figure 11 asa function of the mass ratio and Mach number. The yel-low dashed line, which corresponds to M A = ( m i /m e ) / ,clearly separates the regions of small α from regions of These percentages are estimated by considering the electronswith energies in the range where the power-law component of thespectrum is larger than the thermal component.
Fig. 11.—
The downstream electron spectra are shown for simulations with M A = 3 .
5, 7, and 14, in panels a ), b ), and c ), respectively.The variations in M A are obtained by only changing the magnitude of the magnetic field, so in all the cases the shock velocity has thevalue v sh = 0 . c . The realistic mass ratio m i /m e = 1600 was also included, so the red, black, green, and blue lines represent cases with m i /m e = 25, 100, 400, and 1600, respectively. We can see from these spectra the overall trend to have smaller spectral index α either whenthe Mach number is reduced or when the mass ratio is increased. The parameters of each of these simulations are compiled in Table 1(where for m i /m e = 25, 100, 400, and 1600 the M A = 3 . M A = 7simulations are called 2D-7, 2D-8, 2D-3, and 2D-9, and the M A = 14 simulations are called 2D-14, 2D-15, 2D-16, and 2D-17). Fig. 12.—
A compilation of the spectral indices for the 12 simu-lations depicted in Figure 11 shown as a function M A and m i /m e .The yellow dots correspond to the actual grid point locations,while the rest of the diagram is colored using linear interpola-tion of the measured values of α . The yellow dashed line cor-responds to M A = ( m i /m e ) / . We see that regions of hardand soft spectra tend to be separated by this line, which marksthe theoretical estimate for the limit between growth and suppres-sion of whistler waves (Wu et al. 1983; Matsukiyo & Scholer 2003;Krasnoselskikh et al. 2002). rather soft spectra. In the cases with realistic mass ratio( m i /m e = 1600) the spectral index goes from α = 2 . M A changes from 3.5 to 14. We will discuss the con-sequences that this result has for electron acceleration inSNR shocks below.Finally, we emphasize that this acceleration mecha-nism does not show a significant dependence on the cho-sen value of β e . This is verified by directly comparing run2D-3 and 2D-21, which only differ in their β e ( β e =0.5 and0.05, respectively) and show essentially no difference intheir final electron spectra. SUMMARY AND CONCLUSIONS
In this paper we studied the problem of electron accel-eration in non-relativistic electron-ion shocks, using two- and three-dimensional PIC simulations. By systemati-cally exploring the space of shock parameters, and usingrealistic ion to electron mass ratios, we identify a newacceleration mechanism, that is able to accelerate elec-trons starting from fairly low temperatures. Thus, itconstitutes a possible candidate to solve the well knownelectron “injection problem” of the diffusive shock accel-eration (DSA) theory. This mechanism confirms the ideathat non-thermal electron acceleration can be caused byelectron scattering due to foot waves, which are producedby the electron-ion counter-streaming (see, for example,Cargill & Papadopoulos 1988). We have described thephysics of this mechanism, and identified the physicalregime where this acceleration is most efficient. Theacceleration occurs preferentially in quasi-perpendicularshocks, and is driven by oblique whistler waves excitedin the shock foot.We found that, for simulations that only differ in theirion to electron mass ratio m i /m e , the spectral indextends to harden as the m i /m e grows from 25 to 1600.This trend is confirmed for M A = 3 .
5, 7, and 14, andfor shock velocities v sh = 0 . c and 0 . c . On theother hand, simulations that only differ in their Alfv´enicMach numbers also show a progressive hardening of theirspectra as M A is reduced from 14 to 3.5. This massratio/Alfv´enic Mach number dependence of the acceler-ation is consistent with theoretical arguments suggest-ing that the growth of whistler waves in the foot ofquasi-perpendicular shocks would be favored when M A . ( m i /m e ) / . Although the physics of this whistler exci-tation is still a subject of debate, this condition appearsto hold independently of whether the whistlers are gener-ated by the MTSI (Wu et al. 1983; Matsukiyo & Scholer2003, 2006), or if they are just explained as an intrinsiccomponent of the structure of quasi-perpendicular shocks(Krasnoselskikh et al. 2002). We also found a strong de-pendence of the acceleration on the angle between themagnetic field and the shock normal, θ Bn , which needsto satisfy θ Bn = 90 ◦ . On the other hand, we found thatthe shape of the spectra do not depend significantly onthe shock velocity, although a factor of ∼ TABLE 1Parameters for the two-dimensional simulations
Run c/ω p,e L y / ( c/ω p,e ) β e = β i v sh /c M A θ Bn m i /m e N ppc ◦
400 1002D-2 10 102 0.5 0.14 7 80 ◦
400 1002D-3 10 102 0.5 0.14 7 75 ◦
400 1002D-4 10 102 0.5 0.14 7 70 ◦
400 1002D-5 10 102 0.5 0.14 7 60 ◦
400 1002D-6 10 102 0.5 0.14 7 45 ◦
400 1002D-7 15 34 0.5 0.14 7 75 ◦
25 1002D-8 10 77 0.5 0.14 7 75 ◦
100 1002D-9 10 102 0.05 0.14 7 75 ◦ ◦
25 102D-11 10 102 0.01 0.14 3.5 75 ◦
100 102D-12 10 102 0.01 0.14 3.5 75 ◦
400 102D-13 10 102 0.01 0.14 3.5 75 ◦ ◦
25 102D-15 10 102 0.2 0.14 14 60 ◦
100 102D-16 10 102 0.2 0.14 14 60 ◦
400 102D-17 10 102 0.2 0.14 14 60 ◦ ◦
25 1002D-19 15 51 0.005 0.042 7 75 ◦
100 322D-20 10 77 0.005 0.042 7 75 ◦
400 652D-21 10 102 0.05 0.14 7 75 ◦
400 100
Note . — We list the electron skin depth c/ω p,e in terms of number of grid cells,the transverse size of the simulation box in terms of c/ω p,e , the beta parameter ofthe different plasma particles β j ( ≡ p j /B / π , where p j is the pressure of particle“ j ”), the upstream medium shock velocity, v sh , the Alfv´enic Mach number, M A , theangle between the upstream magnetic field and the shock normal, θ Bn , the ion toelectron mass ratio, m i /m e , and the total number of particles per cell, N ppc , in thesimulation. TABLE 2Parameters for the three-dimensional simulations
Run c/ω p,e L y × L z / ( c/ω p,e ) β e = β i v sh /c M A θ Bn m i /m e N ppc × ◦
25 1003D-2 5 77 × ◦
100 253D-3 5 51 × ◦
400 50
Note . — We list the electron skin depth c/ω p,e in terms of number of grid cells, thetransverse size of the simulation box in terms of c/ω p,e , the beta parameter of the differentplasma particles β j ( ≡ p j /B / π , where p j is the pressure of particle “ j ”), the upstreammedium shock velocity, v sh , the Alfv´enic Mach number, M A , the angle between the upstreammagnetic field and the shock normal, θ Bn , the ion to electron mass ratio, m i /m e , and thetotal number of particles per cell, N ppc , in the simulation. bution was observed when the shock velocity was reducedfrom v sh = 0 . c to v sh = 0 . c .This dependence of the acceleration on the shock pa-rameters can be explained in terms of two requirements:1) the shock needs to be able to excite whistler wavesin the foot, which translates into the condition M A . ( m i /m e ) / , and 2) the accelerated electrons need tostay in the shock foot for a time long enough to increasetheir chances of being scattered by the whistler waves.The second requirement explains why the efficiency ofthe acceleration drops substantially when θ Bn = 90 ◦ .In that case, energized electrons cannot move along theshock normal, ˆ n , since they are tied to the magneticfield. Thus, they are rapidly advected into the down-stream medium of the shock. This does not happen inan oblique shock, where accelerated particles can developa significant velocity along ˆ n . Given that the growthof whistler waves does not depend on the shock veloc-ity, v sh , the weak dependence of the acceleration on v sh implies that the ability of the accelerated electrons tostay in the shock foot is also independent of v sh . We can verify this point as follows. The initial electron en-ergization is driven by the electric field of the whistlerwaves (in particular, its component parallel to the mag-netic field). This electric field has a typical length scaleof ∼ c/ω p,e and a maximum amplitude of ∼ E (where E ≡ B v sh /c ). Thus, the energy gain in eachscattering will be given by ∼ eE c/ω p,e . There-fore, an electron that moves along ~B with that en-ergy, will have x − velocity larger than the shock veloc-ity if 100 eE c/ω p,e cos( θ Bn ) & m e v sh , which implies100 cos( θ Bn ) & M A / ( m i /m e ) / . We can see that thiscondition is independent of the shock velocity, and con-firms that θ Bn and the M A / ( m i /m e ) / ratio are thecrucial parameters for the acceleration mechanism.The maximum electron energy that we measure is con-sistent with the electron Larmor radii R L,e being com-parable to that of the ions R L,i , as can be verified fromFigure 13, where we plot the downstream spectrum ofelectrons for run 2D-3 ( M A = 7, v sh = 0 . c , θ Bn = 75 ◦ ,and m i /m e = 400). The electron spectrum is shown asblack line, and the thermal and power law tail fits are2 Riquelme & Spitkovsky Fig. 13.—
The downstream energy spectra for electrons and ions(black and green lines, respectively) are shown for the shock tran-sition region of run 2D-3 ( M A = 7, v sh = 0 . c , θ Bn = 75 ◦ , and m i /m e = 400). The thermal and power law tail fits are shownin red dashed lines. The vertical dotted line marks the electronLorentz factor corresponding to equal electron and ion Larmor radii(Γ e = ( m i /m e )( v sh /c )). depicted in red dashed lines ( α ≈ . R L,e = R L,i , which coincides with the maximum en-ergy of the electrons. However, this maximum energyis statistically limited by the number of macroparticlesused in the simulations. Thus, higher energy electronsmay in principle be possible. In Figure 13 we also de-pict the corresponding downstream ion spectrum, whichshows incomplete thermalization and the absence of apower law tail.We also explored the contribution to acceleration givenby the previously proposed “shock surfing” mechanism,which is driven by the excitation of Buneman waves inthe leading edge of the foot (Amano & Hoshino 2009).We find that its contribution is strongly dependent onthe artificial mass ratio m i /m e , becoming negligible forrealistic values of m i /m e .From the observational viewpoint, our results are con-sistent with in-situ electron spectrum measurements inthe Earth’s bow shock, in which typically M A ≈ − α ≈ −
4, with the power-law part of the spectrumextending smoothly out of the thermal part of the dis-tribution, which is essentially what we observe in oursimulations. Similar results were more recently reportedby Oka et al. (2006), using data from the Geotail space-craft. We emphasize, however, that the application of ourresults to the Earth’s bow shock measurements is basedon simulations with v sh = 0 . c and 0 . c , which are afactor of at least ∼
100 larger than the shock velocitiestypically found in the solar system. This extrapolation ismade due to the finding that the crucial parameters forthe acceleration due to whistlers are M A and θ Bn , withthe shock velocity dependence being much weaker. Thisextrapolation remains to be confirmed by shock simula- tions with orders of magnitude smaller v sh (or v A ).In the context of electron acceleration in SNR shocks,we can confirm the potential importance of this injec-tion mechanism by estimating the Alfv´enic Mach num-ber necessary to explain the typical fraction of parti-cles injected into the DSA process, η inj . According tothe modeling of broadband observations of SNRs, η inj ranges between ∼ − , for magnetic fields amplifiedonly by shock compression ( B ∼ µG ), and ∼ − ,for significant magnetic amplification consistent with re-cent X-ray observations ( B ∼ µG ) (see, for example,Zirakashvili & Aharonian 2010). Also, in order to beinjected, the particles need to have Larmor radii closeto the size of the shock transition region, whose char-acteristic scale corresponds to the Larmor radii of theions. Thus, we can estimate the maximum slope of thenon-thermal part of the spectrum necessary to satisfythese requirements. If we assume that the normalizationof the non-thermal spectrum is such that the power lawtail dominates for energies very close to the peak of thethermal distribution (which is approximately what wesee in our simulations), we get that the necessary spec-tral index should satisfy α ≈ − log( η inj ) / log( c/v sh ) . (1)Thus, for a typical shock velocity of v sh = 3000 km/s,and η inj = 10 − , we get that α ≈
4. From the resultssummarized in Figure 12, we see that, for realistic massratios ( m i /m e = 1600), the maximum Mach number thatwould give α . M A = 14. This estimateshows that the injection mechanism presented in this pa-per is a viable solution to the injection of electrons intothe DSA mechanism in SNR shocks only if the Alfv´enicMach number is smaller than ∼
20. This result impliesthat significant magnetic field amplification must occurin the upstream region of SNR shocks. Indeed, if the up-stream field is not amplified, the typical Alfv´enic Machnumber for SNR shocks would be ∼ v sh = 3000 km/s propagating in a n i = 1 cm − plasmawith a 3 µ G ISM field. This is interesting considering thatsignificant magnetic amplification in SNR shocks basedon X-ray observations has been recently reported (Ballet2006; Uchiyama et al. 2007). These observations showthe existence of thin, non-thermal rims, which are inter-preted as synchrotron emission by TeV electrons acceler-ated at the shocks. The rapid variability and thinness ofthe rims (which depend on the synchrotron cooling timeof the electrons) have allowed to estimate the strengthof the field, suggesting downstream amplitudes ∼ ∼
25, considering that the field compres-sion at the shock itself would contribute an extra factorof ∼ M A .
20 in these shocks. Also, thereare theoretical reasons to believe that this field growthcould happen in the upstream medium of the shocks, dueto streaming instabilities driven by cosmic rays (CRs) asthey get accelerated in these environments (Bell 2004,2005; Riquelme & Spitkovsky 2009, 2010).The M A .
20 condition for electron injection is alsosupported by a recent PIC study of a two-dimensional,perpendicular shock with in-plane magnetic field and pa-lectron Injection in Non-relativistic Shocks 13rameters: M A ≈ v in = 0 . c , and m i /m e = 30(Kato & Takabe 2010). This study showed the absenceof electron acceleration and whistler waves in the foot ofthe shock. Indeed, the shock foot is dominated by theWeibel instability, which plays a fundamental role in theformation of the shock itself. Although the strict per-pendicularity of the shock ( θ Bn = 90 ◦ ) may contributeto reducing the electron acceleration, we believe that thesuppression of the whistler waves in this low magnetiza-tion shock would make electron injection unlikely for anyvalue of θ Bn .Thus, this result reinforces the idea of very large mag-netic amplification associated with the synchrotron rimsseen by the X-ray observations of SNRs. It is importantto emphasize that in this paper we have concentratedonly on the injection part of the acceleration process. Af-ter the injected electrons reach energies such that theycan move diffusively in the shock vicinity, they would befurther accelerated via the DSA up until the ∼ − R L,cr . Therefore, given that the electron injec-tion mechanism presented here operates on length scalescomparable to a few c/ω p,i (which is much smaller than R L,cr ), it is plausible that electron acceleration can hap-pen locally, in regions where ~B is quasi-perpendicular tothe shock normal.The likely global picture of the shock acceleration pro-cess then unfolds as follows. On large scales upstreamof the efficiently accelerating shock, the magnetic fieldis likely quasi-parallel to the shock normal. This con-figuration is generally conducive to ion (cosmic ray) ac-celeration and subsequent escape. Escaping cosmic raysamplify and reorient the upstream magnetic field closerto the shock via current-driven instabilities. The re-sulting magnetic turbulence advected to the shock willbe roughly isotropic, with regions of quasi-perpendicularmagnetic field intermittently crossing the shock. This lo- cally transverse field will be efficient in injection of elec-trons via the whistler mechanism. Once pre-accelerated,the electrons will join the ions in DSA on scales largerthan the shock foot in the turbulence that is driven bythe cosmic rays. This speculative picture underscoresthe interdependence of electron and ion acceleration, aswithout cosmic rays the amplification and reorientationof magnetic field needed for electron injection would behard to achieve. Observationally, this scenario is con-sistent with the large scale magnetic field in SN1006pointing along the axis that connects the “polar caps”of bright non-thermal emission (Ballet 2006). Also, thedegree of linear polarization should be smaller in SNRrims with strong synchrotron emission, as we expect thefield direction near the shock to be randomized by theamplified turbulence (Stroman & Pohl 2009). On largerscales, the radial magnetic fields inferred from radio po-larization measurements of synchrotron-emitting regionsof SNR shocks are also consistent with our model (Dickelet al. 1991, DeLaney et al. 2002).The results presented in this paper correspond to onlya partial exploration of the space of shock parameters.Studying this problem using realistic mass ratios is com-putationally expensive, so testing all the possible com-binations of parameters of interest is complicated andwill require further work. For instance, one of our basicassumptions is that the angle θ Bn at which the acceler-ation is maximized has a weak dependence on the ionto electron mass ratio. We believe that this assumptionneeds to be investigated by performing a more completeexploration of the acceleration efficiency dependence on θ Bn and m i /m e . In a similar way, the dependence ofthe acceleration efficiency on the shock velocity v sh re-quires further study. We found that, when passing from v sh = 0 .
14 to 0.42, the slope of the power law part of theelectron spectra for the mass ratios m i /m e = 25, 100,and 400 does not change significantly. However, this re-sult was obtained only in the case of θ Bn = 75 o and M A = 7. We think that it would be interesting to con-firm this weak velocity dependence by performing a thor-ough exploration of the shock parameter space at lowervelocities. Finally, comparing two-dimensional simula-tions with analogous three-dimensional runs where thetypical whistler length scale ( ∼ c/ω p,e ) is resolved inthe third dimension would also be useful to confirm ourresults and to enquire about possible additional three-dimensional effects. Although we do not expect the qual-itative picture presented here to change significantly, webelieve that these studies would give us more accuracyin our estimates of the electron injection efficiencies as afunction of the shock parameters.With these caveats in mind, in this paper we haveidentified a new possible mechanism for electron injec-tion into the DSA process in non-relativistic shocks.The obtained spectra are consistent with in-situ mea-surements of electron energy distributions at the Earth’sbow shock. Also, our results would explain the observedelectron acceleration in SNR shocks, implying very largeupstream magnetic field amplification in these environ-ments. Thus, if this mechanism proves to be the onlypossible solution for electron injection in non-relativisticblast waves, it would reinforce the inferred strong con-nection between particle acceleration and magnetic fieldamplification in SNR shocks.4 Riquelme & Spitkovsky Fig. 14.—
The downstream electron spectra at tω c,i = 10 for the two-dimensional simulations 2Db-1 (red line), 2Db-2 (black line), and2Db-3 (green line), described in Table 3. The simulations are characterized by having a magnetic field quasi-perpendicular to the simulationplane, and only differ in their mass ratios, which have values m i /m e = 25, 100, and 400, respectively. M. A. R. thanks the support from the Department ofAstrophysical Sciences of Princeton University. This re-search was supported by NSF grant AST-0807381. Thesimulations presented in this article were performed oncomputational resources supported by the PICSciE-OIT High Performance Computing Center and VisualizationLaboratory of Princeton University. This research alsoused resources of the National Energy Research ScientificComputing Center, which is supported by the Office ofScience of the U.S. Department of Energy under Con-tract No. DE-AC02-05CH11231.
REFERENCESAmano, T., & Hoshino, M. 2007, ApJ, 661, 190Amano, T., & Hoshino, M. 2009, ApJ, 690, 244Axford, W. I., Leer, E., & Skadron, G. 1977, 15th Int. CosmicRay Conf., 11, 132Ballet, J. 2006, Adv. in Space Res., 37, 1902Bamba, A., Yamazaki, R., Ueno, M., & Koyama, K. 2003, ApJ,589, 827Bamba, A., Yamazaki, R., Yoshida, T., Terasawa, T., & Koyama,K. 2005, ApJ, 621, 793Bell, A. R. 1978, MNRAS, 182, 147Bell, A. R. 2004, MNRAS, 353, 550Bell, A. R. 2005, MNRAS, 358, 181Blandford, R. D., & Ostriker, J. P. 1978, ApJ, 221, L29Buneman, O. 1958, Phys. Rev. Lett., 1, 8Buneman, O. 1993, “Computer Space Plasma Physics”, TerraScientific, Tokyo, 67Burgess, D. 2006, ApJ, 653, 316Cargill, P. J., & Papadopoulos, K. 1988, ApJ, 329, 29DeLaney, T., Koralesky, B., Rudnick, L., & Dickel, J. R. 2002,ApJ, 580, 914Dickel, J. R., van Breugel, W. J. M., & Strom, R. G. 1991, AJ,101, 2151Gosling, J. T., Thomsen, M. F., and Bame, S. J., 1989, J.Geophys. Res., 94, 10011Guo, F., & Giacalone, J. 2010, ApJ, 715, 406Hellinger, P., Tr´avn´ıˇcek, P., Lemb`ege, B., & Savoini, P. 2007,Geophysical Research Letters, 34, L14109Kato,T. N., & Takabe , H. 2010, ApJ, 721, 828Koyama, K., Petre, R., Gotthelf, E. V., Hwang, U., Matsuura,M., Ozaki, M., & Holt, S. S. 1995, Nature, 378, 255Krasnoselskikh, V. V., Lemb`ege, B., Savoini, P., & Lobzin, V. V.2002, Physics of Plasma, 9, 1192Krymsky, G. F. 1977, Sov. Phys. Dokl., 23, 327 Kulsrud, R., & Pearce, W. P. 1969, ApJ, 156, 445Lemb`ege, B., Savoini, P., Hellinger, P., & Tr´avn´ıˇcek, M. 2009,Journal of Geophysical Research, 114, A03217Levinson, A. 1992, ApJ, 401, 73Levinson, A. 1994, ApJ, 426, 327Matsukiyo, S., & Manfred Scholer. M. 2003, Journal ofGeophysical Research, 108, 1459Matsukiyo, S., & Manfred Scholer. M. 2006, Journal ofGeophysical Research, 111, 6104Oka, M., Terasawa, T., Seki, Y., Fujimoto, M., Kasaba, Y.,Kojima, H., Shinohara, I., Matsui, H., Matsumoto, H., Saito,Y., and Mukai, T. 2006, Geophysical Research Letter, 33,L24104Riquelme, M. A., & Spitkovsky, A. 2009, ApJ, 694, 626Riquelme, M. A., & Spitkovsky, A. 2010, ApJ, 717, 1054Scholer, M., Shinohara, I., & Matsukiyo, S. 2003, Journal ofGeophysical Research., 108, 1014Scholer, M. & Matsukiyo, S. 2004, Annales Geophysicae, 22, 2345Shinohara, I., Fujimoto, M., Takaki, R., & Inari, T. 2011, in prep.Sironi, L., & Spitkovsky, A. 2009, ApJ, 698,1523Spitkovsky, A. 2005, AIP Conf. Proc, 801, 345, astro-ph/0603211Stroman, W. & Pohl, M. 2009, ApJ, 696, 1864Uchiyama, Y., Aharonian, F. A., Tanaka, T., Takahashi, T., &Maeda, T. 2007, Nature, 449Umeda, T., Yamao, M., & Yamazaki, R. 2009, ApJ, 695, 574Wu, C. S., Zhou, Y. M., Tsai S-. T., Guo, S. C., Winske, D., &Papadopoulos, K. 1983, Physics of Fluids, 26, 5Yuan, X., Cairns, I. H., Trichtchenko. L., Rankin, R., & Danskin,D. W. 2009, Geophysical Research Letters, 36, L05103Zirakashvili, V. N., & Aharonian, F. A. 2010, ApJ, 708, 965APPENDIX
MASS RATIO DEPENDENCE OF THE “SHOCK SURFING” ACCELERATION
In this appendix we explore the mass ratio dependence of the electron energization due to shock surfing acceleration(Amano & Hoshino 2009). This acceleration is driven by the presence of electrostatic waves, produced by the Bunemaninstability (Buneman 1958), in the foot of quasi-perpendicular shocks. The Buneman waves grow in the leading edgeof the foot due to the relative velocity between the upstream electrons and the shock-reflected ions, and have a typicalscale comparable to ∼ c/ω p,e . The small length scale of these waves, which can be comparable or even smallerlectron Injection in Non-relativistic Shocks 15 Fig. 15.—
The electric field along ˆ x , E x , in the shock transition region of simulations 2Db-1 (panel a ), 2Db-2 (panel b ), and 2Db-3 (panel c ), which are described in Table 3. These simulations are characterized by having a magnetic field quasi-perpendicular to the simulationplane, and only differ in their m i /m e parameter given by m i /m e =25, 100, and 400, respectively. The field is shown at tω c,i = 10, and isnormalized in terms of E , which corresponds to the convective electric field given by B v in /c (where v i = 0 . c is the injection velocity ofthe upstream plasma as seen from the downstream frame). TABLE 3Parameters for the two-dimensional simulations with magnetic fieldquasi-perpendicular to simulation plane
Run c/ω p,e L y /c/ω p,e β e = β i v sh /c M A θ Bn m i /m e N ppc ◦
25 1002D-2b 15 34 0.5 0.14 7 75 ◦
100 1002D-3b 10 51 0.5 0.14 7 75 ◦
400 1002D-4b 10 51 4.5 0.14 21 75 ◦
25 1002D-5b 10 26 4.5 0.14 21 75 ◦
100 1002D-6b 15 34 0.045 0.042 7 75 ◦
25 1002D-7b 10 26 0.045 0.042 7 75 ◦
100 100
Note . — We list the electron skin depth c/ω p,e in terms of number of grid cells,the transverse size of the simulation box in terms of c/ω p,e , the beta parameter ofthe different plasma particles β j ( ≡ p j /B / π , where p j is the pressure of particle“ j ”), the upstream medium shock velocity, v sh , the Alfv´enic Mach number, M A ,the angle between the upstream magnetic field and the shock normal, θ Bn , the ionto electron mass ratio, m i /m e , and the total number of particles per cell, N ppc , inthe simulation. than the electron Larmor radii, makes the Buneman waves a potentially important means for electron scattering andenergization in the foot of quasi-perpendicular shocks. As the upstream electrons encounter these waves, they gainenergy due to the work performed by the fluctuating electric field, combined with the convective field of the upstreamplasma. The wave vector of the fastest growing mode is parallel to the velocity of the beam of returning ions, so thesewaves are best studied either by three-dimensional simulations or by two-dimensional runs where the magnetic field isquasi-perpendicular to the simulation plane, so that the gyrational motion of the ions is resolved on the plane Using a two-dimensional simulation with shock parameters v in = 0 . c ( v sh ≈ . c ), M A = 14, θ Bn = 90 o (with ~B perpendicular to the simulation plane), and m i /m e = 25, Amano & Hoshino (2009) showed that an electron spectrumwith index α ≈ v in = 0 . c ( v sh ≈ . c ), M A = 7, θ Bn = 75 o (and the angle between ~B and ˆ z equal to15 ◦ ), but different mass ratios: m i /m e = 25 , tω c,i = 10 measured in the downstream medium. Although in all three cases a non-thermalcomponent can be seen, the maximum energy reached by the electrons is about the same (Γ e − ≈ m i /m e = 400 case, the closeness between thepeak of the thermal part of the distribution and the maximum energy of the accelerated particles reduces significantlythe relative fraction of non-thermal particles compared with m i /m e = 25 and 100 cases. Also, notice that in these As in the main part of the paper, in this appendix the two-dimensional simulations will be in the xy plane. x − axis is depicted for the same simulations shown in Figure14. The Buneman waves correspond to the fluctuations on ∼ c/ω p,e scale that appear in the leading edge of the shock(at x/c/ω p,e = 115 − a ), b ), and c ), respectively). The black dashed line in thethree panels marks the position of the shock density peak (overshoot) in the three simulations. When compared tothe convective electric field ( E = B v in /c ), we see that the maximum electric field of the Buneman waves is reducedby a factor of ∼ m i /m e passes from 25 to 400.The same maximum energy for the accelerated electrons is observed in similar simulations but using M A = 21(simulations 2Db-4 and 2Db-5). Also, when trying cases similar to the ones shown in Figures 14 and 15 but using v sh = 0 .
042 (simulations 2Db-6 and 2Db-7), a decrease in the maximum energy to Γ e − ≈ . m i /m e .Thus, in general, the maximum energy attainable by the shock surfing mechanism appears to depend mainly on theshock velocity, and shows no changes for different values of m i /m e . Therefore, as the mass ratio m i /m e is increased,the peak of the thermal part of the distribution gets closer to this maximum energy. This implies that the fractionof non-thermal electrons is reduced when m i /m e approaches more realistic values. We conclude that this mechanismwould not contribute significantly to electron acceleration in the fully realistic case of m i /m ee