Fermi acceleration in relativistic collisionless plasma shocks correlates with anisotropic energy gains
FFermi acceleration in relativistic collisionless plasma shocks correlates withanisotropic energy gains
Roopendra Singh Rajawat, Vladimir Khudik, and Gennady Shvets School of Applied and Engineering Physics, Clark Hall, Cornell University, NY 14850,USA Department of Physics and Institute for Fusion studies, The University of Texas at Austin, TX 78712,USA (Dated: 25 February 2021)
Collisionless shocks generated by two colliding relativistic electron-positron plasma shells are studied usingparticle-in-cell (PIC) simulations. Shocks are mediated by the Weibel instability (WI), and the kinetic energyof the fastest accelerated particles is found to be anisotropically modified by WI-induced electric fields.Specifically, we show that all particles interacting with the shock bifurcate into two groups based on theirfinal relativistic Lorentz factor γ : slow ( γ < γ bf ) and fast ( γ > γ bf ), where γ bf is the bifurcation Lorentzfactor that was found to be approximately twice the initial (upstream) Lorentz factor γ . We have found thatthe energies of the slow particles are equally affected by the longitudinal and transverse components of theshock electric field, whereas the fast particles are primarily accelerated by the transverse field component.PACS numbers: 52.27.Ep, 52.65.-y, 52.35.Tc I. INTRODUCTION
Relativistic collisionless shocks are widely viewed asefficient sources of particle acceleration in blazars, super-nova remnants, and gamma-ray burst (GRB) afterglows ,and other high-energy astrophysical objects . Collision-less shocks are ubiquitous in low-density astrophysicalplasmas, where energy is dissipated through effective col-lisions provided by particles’ interactions with turbulentelectromagnetic fields. In the absence of binary col-lisions, such effective collisions enable particle acceler-ation to ultra-relativistic energies. Understanding theemergence of such electromagnetic turbulence via col-lective plasma instabilities is crucial for understandingthe physics of particle acceleration in astrophysical con-texts, where plasma densities are low and binary colli-sions can be mostly neglected. Moreover, the complexityof instability-mediated electromagnetic fields raises ques-tions about the exact mechanism behind the accelerationof fast particles, as well as the factors distinguishing themfrom the majority of particles that never reach high en-ergies.In the specific case of relativistic unmagnetized plasmaflows interacting with each other or with the interstellarmedium (ISM) plasmas, the classic Weibel Instability (WI) is widely viewed as responsible for the spontaneousgeneration of sub-equipartition electromagnetic fields ,mediation of collisionless shocks, and generation of su-perthermal particles in various astrophysical scenar-ios. The WI is a collective electromagnetic instabilitywhich develops in plasmas with anisotropic velocity dis-tributions. Analytical and simulation studies show thatthe WI generates large magnetic fields that can reachthe Alfv`enic limit during its nonlinear stage .The multidimensional particle-in-cell (PIC) simula-tions have long provided the primary tools for studyingthe effects of the WI-generated electromagnetic turbu- lence on the generation of superthermal particles. Forexample, power-law, ( i.e. , non-Maxwellian) particle dis-tributions f ( ε ) ∼ ε − p as a function of the particle energy ε have been predicted based on PIC simulations results,and similar power-law coefficients p have been extractedby several groups, e.g., p (cid:39) . and p (cid:39) . − . ge-ometries. Particle acceleration has been found to be gov-erned by stochastic diffusion, where particles move backand forth across the shock front and gain energy by scat-tering from self-consistent magnetic turbulence throughthe first-order Fermi acceleration mechanism . Consis-tently with phase space diffusion mechanism of particleacceleration, the maximum energy gain of particles inFermi acceleration is observed to scale with accelerationtime ( t acc ) according to (cid:15) max ∝ t / acc . In the aforemen-tioned references, Fermi acceleration is identified by theexistence of non-thermal tail of the distribution functionand kinetic energy carried by the non-thermal particles.Nevertheless, the detailed micro-physics behind parti-cle acceleration is still poorly understood. Because ofthe complex vectorial nature of the electromagnetic tur-bulence inside the collisionless shock itself and in theshocked plasma, systematic tracking of the particles pass-ing through the shock is needed to answer many specificquestions. These include: (i) what is the relative role ofdifferent components of the electric field in particle ac-celeration? (ii) what distinguishes the majority of ther-malized particles in the shocked region from the minorityparticles gaining most of the energy? (iii) what are thetelltale signs of Fermi acceleration that can be extractedfrom such tracking? Note that while magnetic fields aredominant inside the shock, and are primarily responsi-ble for particles thermalizations, they can neither accel-erate nor decelerate charged particles – this is done bythe much weaker electric fields. While particle trackinghas been used in the past to track the accelerated parti- a r X i v : . [ phy s i c s . p l a s m - ph ] F e b cles , it has not been used to investigate the highly-anisotropic nature of particle acceleration, as expressedby relative contributions of the longitudinal (parallel tothe front velocity) and transverse components of the elec-tric field to particles’ energy gains/losses.In this work, we present the details of numericaltracking of representative particles extracted from first-principles 2D PIC simulations of the relativistic, unmag-netized electron-positron (pair) plasma shocks. The 2Dgeometry is sufficient for capturing the basic physics ofparticle acceleration. The rest of the manuscript is or-ganized as follows. In Section II, we describe the ge-ometry and parameters of a first-principles PIC simula-tion. The structure of the shock and the dynamics ofthe bulk plasma (pre-shock and shocked) are discussedin Section III. A detailed balance between work doneon accelerated particles by different electric field com-ponents is discussed in Section IV and the accelerationprocess is described using self-consistent particle trackingin Section IV B. We describe the work-energy bifurcation,which reveals two distinct groups of particles. The par-ticles from the first group gain energy in equal measurefrom the longitudinal and transverse electric fields, whilethose from the second group derive most of their energyfrom the transverse electric field. These distinct groups ofparticles indicate the difference in wave-particle interac-tion between bulk and superthermal plasmas. In SectionV, we discuss the importance of shock reflected particles.The conclusions are presented in Section VI. II. PARTICLE-IN-CELL SIMULATION SETUP
The physical setup of the problem is schematicallyillustrated in Fig. 1(a): two streams of cold electron-positron plasmas counter-propagate along the x -directionand come into the initial contact in the plane marked bya black dashed line. In the rest of the manuscript, we as-sume that the two electrically-neutral streams are mirrorimages of each other, and that their initial Lorentz fac-tors and laboratory frame densities for each of the speciesare γ = 20 and n , respectively. The mirror symmetryenables a standard computationally-efficient approach to modeling colliding plasmas: the stream 2 particles arereflected off a stationary wall placed in the plane of con-tact ( x = 0). Perfectly-conducting boundary conditionsfor the electromagnetic fields are imposed at x = 0.Under this approach, unperturbed streaming plasmais continuously injected through the right boundary. Asthe plasma is reflected by the wall and collides with theincoming plasma, a region of counter-streaming plasmais formed in the overlapping region. The resulting ex-treme anisotropy of the mixed plasma triggers the WIand eventually leads to the formation of a collisionlessshock (red dashed line in Fig. 1(a)) propagating in the+ x -direction. As the incoming cold plasma encountersstrong electromagnetic fields in the shock region, it getsthermalized and forms an isotropic hot plasma in the shocked region. By symmetry, the plasma in this regionbehind the shock (further referred to as the downstreamregion) has a vanishing overall drift velocity. The simula-tion is carried out in the laboratory reference frame of astationary reflective wall, where the downstream plasmais, on average, at rest.The above described problem is numerically solved us-ing a 2D ( y -independent) version of a first-principles PICcode, VLPL . A novel rhombi-in-plane scheme isused for updating the electromagnetic fields, which suc-cessfully suppresses the numerical Cherenkov instability.The non-vanishing electromagnetic field components B y (out-of-plane), E x (longitudinal), and E z (transverse) areassumed to be functions of ( x, z ) and t , and the onlynon-vanishing components of the electron/positron mo-menta are p x and p z . The natural scales for time andspatial length are the inverse values of the plasma fre-quency ω − p and wave number k − p (= c/ω p ), respectively.Here ω p = (cid:0) πn e /γ m e (cid:1) / is the relativistic plasmafrequency, − e and m e are the electric charge and massof an electron. The size of the simulation domain is cho-sen to be L x × L z = 2100 k − p × k − p . Plasma skindepth and plasma frequency are resolved using a gridsize ∆ x = 0 . k − p ; ∆ z = 0 . k − p and a time step∆ t = 0 . ω − p . For all simulations, 16 particles percell per species has been taken. III. REVIEW OF THE SHOCK STRUCTURE
In this section, we review the well-established proper-ties of the shocked (downstream) and pre-shocked (up-stream) plasmas, as well as that of the shock createdby the collision of counter-streaming plasmas .Unless stated otherwise, all figures are plotted at ω pe t (cid:39) n e = n upstream to n s /n = (Γ ad / (Γ ad −
1) + 1 /γ (Γ ad − ∼ . in theshocked region shown in Fig. 1(d) corresponds to the hy-drodynamic shock with an adiabatic constant Γ ad = 3 / v s /c = (Γ ad − γ − / ( γ + 1)) / ∼ . c inthe + x -direction. Effective collisions inside the shock areprovided by turbulent magnetic field plotted in Fig. 1(b),where complex multi-filamentary structures with a typi-cal transverse scale of ∼ k − p can be observed reachingfrom the shock into the upstream region. Magnetic fila-ments are elongated in the direction of the incoming up-stream plasma. While the magnetic field is quasi-staticin the down stream region, it is highly dynamic in the up-stream region. Such time-dependence results in a finitelongitudinal electric field E x .The relative magnitudes of different components ofthe electromagnetic field can be appreciated from therespective plots of their transversely averaged energydensities as shown in Fig. 1(e). The largest energydensity (cid:15) By ( x ) = (cid:104) B y (cid:105) / πγ n m e c is associated with FIG. 1. (a) Two counter-propagating pair plasmas collideand produce two counter-propagating shocks. (b-e) Timesnapshots of the shocked (“downstream”) and pre-shocked(“upstream”) plasmas, and of the collisionless shock at t (cid:39) ω − p . (b) Electron density, (c) magnetic field, (d) trans-versely averaged electron density, and (e) transversely aver-aged electromagnetic field energy density: (cid:15) E x (blue), (cid:15) E z (magenta), and (cid:15) B y (black) lines; inset shows transversely av-eraged energies in the range 720 k − p ≤ x ≤ k − p . the magnetic field (black line), while the smallest one, (cid:15) Ex ( x ) = (cid:104) E x (cid:105) / πγ n m e c belongs to the longitudinalelectric field (blue line). Here (cid:104)(cid:105) defines averaging overthe transverse z coordinate. The intermediate energydensity (cid:15) Ez ( x ) = (cid:104) E z (cid:105) / πγ n m e c is associated withthe transverse electric field (magenta line). Note thatboth (cid:15) By and (cid:15) Ez reach far into the upstream region,forming an important pre-shock region discussed be-low in the context of Fermi acceleration. In our simula-tion, the magnetic field energy density peaks at ∼
20% ofthe equipartition energy in the shock transition region,and decays away from the shock front.
A. The potential role of the longitudinal electric field
We further note from Fig. 1(e) that the longitudinalelectric field energy is vanishingly small in the upstreamregion, whereas the transverse electric and magnetic fieldenergies are much stronger and comparable to each other: (cid:15) Ez ≈ (cid:15) By everywhere in the upstream region . Thelatter property is due to the fact that the dominant cur- rent filaments in the upstream (including the pre-shock)region are associated with highly-directional flows of elec-trons and positrons that have not yet undergone any sig-nificant isotropization as can be seen from Fig. 2(b). Thepresence of a small but finite longitudinal electric fieldin the shock transition region has been related to theoblique modes associated with the WI .The role of the transverse component of the electricfield in accelerating superthermal particles has been gen-erally recognized . Its importance is not surprising be-cause of its large amplitude in the pre-shock region. Onthe other hand, the role of the longitudinal electric field inproviding energy Maxwellization to the medium-energyparticles (both downstream and upstream) has not beenpreviously studied. The reason for neglecting the E x component is that it is considerably smaller than E z inthe pre-shock region. On the other hand, the amountsof the mechanical work W ( j ) x ( W ( j ) z ) done by the longi-tudinal (transverse) electric field components on the j thupstream particle interacting with the shock field couldbe comparable with each other. Here we define W ( j ) x,z = q ( j ) (cid:90) + ∞−∞ dtE x,z ( x ( j ) ( t ) , z ( j ) ( t )) v ( j ) x,z ( t ) , (1)where v ( j ) x,z = p ( j ) x,z /m e γ ( j ) are the time-dependent lon-gitudinal (transverse) velocity components of the j thparticle. From here onwards, we assume that the par-ticles are electrons, and q ( j ) = − e . To see why thismight be the case, we note from Figs. 2(a,b) that mostof the counter-streaming ( p x <
0) particles are not yetisotropized, i.e. , | p z | ( j ) (cid:28) | p x | ( j ) . This creates a surpris-ing opportunity for | W x | ( j ) (cid:39) | W z | ( j ) despite | E x | (cid:28) | E z | everywhere in the pre-shock region.Moreover, we find that the two electric field energies, (cid:15) Ez and (cid:15) Ex , are comparable to each other in the down-stream region, see the inset in Fig. 1(e). The relativis-tic pair plasmas incident on the shock region are fullyisotropized behind the shock, as can be observed by com-paring Figs. 2(a) and (b). The out-of-plane magnetic fieldis responsible for effective isotropization of the incidentplasma: magnetic field energy (cid:15) By dominates the down-stream region immediately behind the shock, where itis much larger than the electric field energy. Therefore,just as in the pre-shock region, it is plausible for the twoelectric field components to do comparable mechanicalwork on the incident particles. In the next section weclassify plasma electrons interacting with the shock intotwo categories defined by the relative magnitudes of W x and W z . IV. EMERGENCE OF THE ENERGY BIFURCATION
To quantify the contributions of the longitudinal andtransverse electric fields to individual particles’ kineticenergy increments ∆ ε ( j ) ≡ ( γ ( j ) − γ ) m e c , where γ ( j ) FIG. 2. Phase space densities in the downstream ( x < k − p ), upstream ( x > k − p ), and shock/pre-shock(950 < x < k − p ) regions. (a) Longitudinal (all parti-cles) and (b) transverse (counter-stream particles) momen-tum phase space density (color-coded). (c-e) Decompositionof the kinetic energy gain/loss into the work done by lon-gitudinal (blue line) and transverse (red line) electric fieldsinside several slices: (c) downstream (50 < x < k − p ),(d) inside shock (1050 < x < k − p ), and (e) upstream( x > k − p ). is the final Lorentz factor of the j ’th particle, we breakup all particles located within a given spatial slice L
Based on the bifurcated curves in Fig. 2(c), we identifytwo groups of particles in the downstream region: parti-cles with moderate ( γ < γ bf ) and particles with large( γ > γ bf ) kinetic energies. The first group of parti-cles, which we refer to as the bulk population, is ther-malized to a relativistic Maxwellian distribution. Re-markably, both the longitudinal and transverse electricfields perform equal work on the bulk plasma particles: W x ( γ ) ≈ W z ( γ ) for all γ < γ bf . Note that the bulk popu-lation contains both particles that have been slowed downby the electric fields of the shock ( W x,z ( γ ) < γ < γ )and the ones that have nearly doubled their energy. Toour knowledge, this is the first computational demonstra-tion of the importance of the equal importance of the lon-gitudinal and transverse components of the electric fieldin the Maxwellization of the shocked pair plasma. Whilethe importance of the longitudinal field component E x has been known in electron-ion plasmas , it has notyet been appreciated for collisionless shocks in pair plas-mas .The second group of particles, which we refer to as su-perthermal particles, acquire most of their kinetic energyfrom the transverse electric field, i.e. , W z ( γ ) > W x ( γ ) forall γ < γ bf as shown in Fig. 2(c). The separation pointat γ = γ bf in the work-energy separates the populationof the bulk particles gaining energy in the downstreamregion of the shock from the population of superther-mal particles gaining energy in the course of repetitiveoscillations in the shock/pre-shock region. By carryingout simulations for different periods of time, we have ob-served that while the ratio of the work performed by thetransverse and longitudinal electric field increases withtime for the superthermal population, the value of the bi-furcation Lorentz factor γ bf remains time-invariant. Wehave also carried out simulations with varying initial (up-stream) Lorentz factor ( γ = 2 − γ bf /γ with varying γ . We foundthat for relativistic pair plasma shocks ( γ ≥
5) the ratio γ bf /γ remains constant ( ≈ i.e. , for γ = 2, the ratio γ bf /γ turns outto be much higher ≈ .
7. It is expected that by symme-try, positrons and electrons exhibit the same work-energybifurcation.
FIG. 3. The ratio of value of bifurcation Lorentz factor toinitial upstream Lorentz factor ( γ bf /γ ) for different initialLorentz factor ( γ ). Another manifestation of the emergence of the su-perthermal population comes from the energy spectrumof thermalized electrons in the shocked region of theplasma. A typical spectrum plotted in Fig. 4 correspondsto thermalized electrons inside a 100 k − p -wide slice in thedownstream region at t (cid:39) ω − p . We have fitted thenumerically simulated spectrum (black line) to a sum of aMaxwell-J¨uttner (MJ) (red line) and a power-law (blueline) spectra. We choose theoretical distribution functionin the form f (cid:18) γγ (cid:19) = C γγ exp (cid:16) − γ Θ (cid:17) + C (cid:18) γγ (cid:19) − p min (cid:26) , exp (cid:18) − γ − γ cut ∆ γ cut (cid:19)(cid:27) . (2)First term of RHS of the equation (2) corresponds to2D MJ distribution, and the second term shows the powerlaw with an exponential cutoff. Θ = k B T /m e c is the dimensionless temperature, k B is the Boltzmannconstant, C and C are normalizing constants such that C = 0 for γ < γ min . γ cut and ∆ γ cut show the begin-ning of high energy cutoff and high energy spread, respec-tively. In Fig. 4(a), theoretical MJ distribution is plot-ted for k B T = 9 . m e c , which is in excellent agreementwith typical temperature predicted by Rankine-Hugoniotcondition k B T = 0 . γ − m e c = 9 . m e c forcomplete thermalization in the downstream region. Thepower-law is plotted for p = 2 . γ min = 3 γ , γ cut = 10 γ and ∆ γ cut = 6 γ . Deviation from the MJ spectrum isclearly observed for γ > γ . Understanding the ori-gins of the two groups of particles (bulk (group I) and superthermal (group II)) requires that we examine indi-vidual particle trajectories in detail: their entrance intothe shock, subsequent interaction with the shock, andtransition into the downstream region. FIG. 4. (a) Electron energy spectra inside the 50 k − p < x < k − p downstream slice: simulated (black line) and its fitto the sum of a Maxwell-J¨uttner (red line) and a power-law γ − . (blue line) spectra. (b) Electron energy spectra of theupstream reflected particles inside 1150 k − p < x < k − p range; inset shows their transversely averaged density at t =2040 ω − p . B. Particle Tracking Results
To understand field-particle interactions with differentregions of the shock, we tracked electrons based on theirfinal energy ε fin = γm e c and position x fin with respectto the shock. Specifically, four classes of particles wereconsidered: (I) two classes of the bulk particles with γ <γ bf that ended up downstream of the shock’s position x sh ( t ) = v sh t (top row of Fig. 5), and (II) two classes ofsuperthermal particles with γ (cid:29) γ bf (bottom row of Fig.5). Group I electrons comprise those that gained energy,as exemplified by a representative particle in Fig. 5(a),and those that lost energy, as exemplified by Fig. 5(b).Group II electrons that gained a significant amount ofenergy from the shock comprise those crossed the shockinto the downstream region, as shown in Fig. 5(c), andthose that reflected from the shock into the upstreamregion, as shown in Fig. 5(d).The black lines in Fig. 5 indicate electrons’ trajectories x ( t ). Shock’s trajectory separates the blue (upstream) re-gion from the gray (downstream) region. The dotted redline indicates the edge of the pre-shock region x p − sh ( t )defined in such a way that the magnetic energy declinesby exp − from its peak: (cid:15) By ( x p − sh ) /(cid:15) By ( x sh ) = 1 /e (seeFig. 1(e) for a representative profile of (cid:15) By as a functionof x ). The blue lines in Figs. 5(a-d) indicate electrons’Lorentz factors γ ( t ). By comparing particle trajecto-ries and energy changes, it is easy to deduce when thosechanges have occurred. FIG. 5. Time-dependent trajectory and kinetic energy of fourrepresentative electrons. Color-coded: transversely-averagedelectron density separated by the shock line x sh ( t ). Dotted redline: the boundary of the pre-shock x p − sh ( t ). Black lines (leftscale): horizontal trajectories x ( t ), blue lines (right scale):energies γ ( t ). (a,b) Typical bulk electrons gaining (a) andlosing (b) energy. (c,d) Superthermal electrons moving intodownstream (c) and upstream (d) regions. Particles of the first and second classes gain or losemoderate amounts of energy that are comparable to theirinitial energies ε = γ m e c . Those particles cross theshock once, become thermalized, and become a part ofthe bulk plasma in the downstream region. The down-stream plasma primarily consists of these two classes ofparticles, as they form the Maxwellian portion of thespectrum shown in Fig. 4. As the bifurcation in Fig. 2(c)indicates, these two classes of particles, on average, gain(for W x,z >
0) or lose (for W x,z <
0) approximately equalamounts of energy from both components of the electricfield. This is related to the fact that the downstreamregion of the plasma contains almost equal amounts ofelectromagnetic energies (cid:15) Ex and (cid:15) Ez associated with thelongitudinal and transverse electric field components, re-spectively (see inset in Fig. 1(e)). Additional mixing be-tween longitudinal and transverse momenta p x and p z isprovided by the magnetic field B y which is much largerthan either E x or E z components of the electric field.A small number of particles which are either reflectedby the shock, or diffuse from the downstream to upstream region, do not immediately cross the high-field region be-tween the shock and the pre-shock boundary shown by adashed line in Figs. 5(c) and (d). Such particles can stayin the pre-shock region for a long time, gaining signifi-cant energy from the strong transverse electric field. Anexample of a particle belonging to the third class of bulkelectrons that gain considerable energy while eventuallymoving through the shock is shown in Fig. 5(c). Thisspecific particle (which we label as j = 3) stays in thepre-shock region for almost (∆ t ) (3)p − sh ≈ ω − p , gains∆ ε (3) ≈ ε by experiencing numerous rapid energychanges that can be characterized as first-order Fermiacceleration, and eventually crosses the shock transitioninto the downstream region.In agreement with the energy bifurcation curve, W (3) z ≈ W (3) x , i.e. , superthermal electrons crossing intothe downstream region gain more than an order of magni-tude from the transverse component of the electric fieldthan from the longitudinal one. The reason for this isthat superthermal electrons spend a long period of time(∆ t ) (3)p − sh in the pre-shock region, where they are sub-jected to E z (cid:29) E x . In combination with isotropizationprovided by a strong magnetic field in the pre-shock re-gion, this results in W (3) z (cid:29) W (3) x . V. PARTICLES’ REFLECTION BY THE SHOCK INTOTHE UPSTREAM REGION
At the same time, a minority of electrons that interactwith the pre-shock region for a long time eventually getreflected and move into the upstream region. The num-ber of reflected electrons and positrons is much smallerthan of those propagating past the shock into the down-stream region. Qualitatively, this is related to the factthat the combination of the transverse magnetic and elec-tric fields in the pre-shock region creates a stronger de-flecting force for the particles traveling in the positive x -direction than for their counterparts with v x <
0. There-fore, the pre-shock creates an effective one-way barrierthat makes it easier for the thermalized particles to dif-fuse downstream from the shock than to reflect back intothe upstream region.The energy spectrum of the reflected electrons popu-lation is shown in Fig. 4(b). It peaks at a much higherLorentz factor γ (up)(peak) ≈ γ than the γ (down)(peak) ≈ γ peakof the energy spectrum of the downstream electron pop-ulation. Therefore, based on the plots of the averaged W x ( γ ) and W z ( γ ) in Fig. 2(e), we conclude that mostof the reflected pairs gain most of their energy from thetransverse electric field component than from the longi-tudinal one. A typical trajectory and energy gain plotsfor a representative class-four particle are shown in Fig.5(d). The particle spends roughly the same time inter-acting with the pre-shock as the one shown in Fig. 5(c),gains approximately the same energy, and eventually be-comes a counter-streaming particle penetrating deep intothe upstream region.Next, we discuss the importance of the counter-streaming particles for seeding the WI. The counter-streaming population propagating ahead of the shock,plotted in the inset of Fig. 4(b) and also observed inFig. 2(a), is essential for maintaining the shock. For ex-ample, the density of counter-streaming particles deter-mines the growth rate and saturation of the secondaryWI manifested magnetic field filaments in the upstreamregion, as shown in the Fig. 1(d). Note that the densityof the counter-streaming particles decreases as they moveaway from the shock transition region. To illustrate thiseffect more, we plot in Fig. 6 the transversely-averageddensity of electrons with Lorentz factors within the fol-lowing ranges: (1) γ < γ < γ bf (blue line), (2) γ bf <γ < γ (up)(peak) (orange line), and (3) γ (up)(peak) < γ < γ (up)(peak) (yellow line).The general trend for all three ranges is the same:higher density in the downstream rather than in the up-stream region. It confirms that the particles more easilyescape into the downstream than into the upstream be-cause of the deep penetration of the transverse electricand magnetic fields into the upstream region shown inFig. 1(e). Another reason behind the lower density ofaccelerated particles in the upstream is that the reflectedparticles seed the secondary Weibel instability in the up-stream region, thereby losing energy in the process . In-terestingly, while the density of high-energy populations(ranges 2 and 3) close to the shock is much lower thanthat of the thermal population of the downstream region(range 1), the former extend much further upstream thanthe latter. FIG. 6. Transversely averaged electron densities for kineticenergy ranges γ < γ < γ bf (blue line), γ bf < γ < γ (up)(peak) (orange line), and γ (up)(peak) < γ < γ (up)(peak) (yellow line). However, the counter-streaming particle density in-creases with time as more particles are continuously ac-celerated by the shock , and may be the reason why cur- rent PIC simulations do not reach a steady-state .Not surprisingly, some of the most energetic electronscan be found reflected upstream of the shock. The tra-jectory of one such simulated particle shown in Fig.7(a)(blue line, left scale) demonstrates that the most ener-getic class-four particles “surf” around the shock andgain energy continuously (blue line, right scale). Thetemporal evolution of the decompositions of the kineticenergy change into W (4) x ( t ) and W (4) z ( t ) are plotted inFig. 7(b). The jumps in W (4) z ( t ) clearly coincide withmultiple scatterings of the particle around the shock re-gion. This scattering of particle in the shock transitionregion can be identified as a beginning of the first orderFermi acceleration. FIG. 7. Time evolution of horizontal trajectory and kineticenergy of a fast electron reflected by the shock. (a) Trans-versely averaged density is color-coded. Black line: particletrajectory x ( t ), blue line: Lorentz factor γ ( t ). (b) Mechani-cal work performed by longitudinal (black line) and transverse(orange line) electric field components, and the Lorentz factor(blue line) of the particle. VI. CONCLUSIONS
In conclusion, we have studied particle acceleration viathe unmagnetized relativistic collisionless shock in pair(electron-positron) plasmas by means of a first-principles2D particle-in-cell simulation code. A collisionless shockis formed in the overlapping region of counterstreamingcolliding plasmas and excites the classic WI. The par-ticles are accelerated by WI-generated electromagneticfields by a novel mechanism called “First order Fermi ac-celeration”. The WI-generated electromagnetic fields arehighly anisotropic in the shock/pre-shock region, how-ever, both electric fields have an almost equal magnitudein the downstream region. We have demonstrated thatin the downstream region, particles are divided into twogroups based on the final energy: slow ( γ < γ bf ) andfast ( γ > γ bf ) group of particles. For relativistic shocks,The value of the Lorentz factor which separates bothgroups of the particles turn out to be γ bf ≈ γ . Theslow group of particles surprisingly take equal amount ofwork from both longitudinal and transverse electric fieldcomponents while fast particles take their energy mostlyfrom transverse field component.Nonetheless, it would be interesting to observe bifur-cation in electron-ion shocks, where due to slow mobility,ions might develop bifurcation later than electrons. ACKNOWLEDGMENTS
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