Simulation study of the formation of a non-relativistic pair shock
aa r X i v : . [ a s t r o - ph . H E ] J a n Under consideration for publication in J. Plasma Phys. Simulation study of the formation of anon-relativistic pair shock
M. E. Dieckmann † , and A. Bret Department of Science and Technology (ITN), Linkoping University, Campus Norrkoping,60174 Norrkoping, Sweden University of Castilla La Mancha, ETSI Ind, E-13071 Ciudad Real, Spain Instituto de Investigaciones Energéticas y Aplicaciones Industriales, Campus Universitario deCiudad Real, 13071 Ciudad Real, Spain(Received xx; revised xx; accepted xx)
We examine with a particle-in-cell (PIC) simulation the collision of two equally denseclouds of cold pair plasma. The clouds interpenetrate until instabilities set in, which heatup the plasma and trigger the formation of a pair of shocks. The fastest-growing waves atthe collision speed c/5 and low temperature are the electrostatic two-stream mode andthe quasi-electrostatic oblique mode. Both waves grow and saturate via the formationof phase space vortices. The strong electric fields of these nonlinear plasma structuresprovide an efficient means of heating up and compressing the inflowing upstream leptons.The interaction of the hot leptons, which leak back into the upstream region, with theinflowing cool upstream leptons continuously drives electrostatic waves that mediate theshock. These waves heat up the inflowing upstream leptons primarily along the shocknormal, which results in an anisotropic velocity distribution in the post-shock region.This distribution gives rise to the Weibel instability. Our simulation shows that even ifthe shock is mediated by quasi-electrostatic waves, strong magnetowaves will still developin its downstream region.
1. Introduction
Compact objects like neutron stars or black holes that accrete material can emitrelativistic jets. These jets are composed of electrons, positrons and ions. The emissionof relativistic jets by microquasars (Fabian & Rees 1979; Margon 1984), which arestellar-size black holes that gather material from a companion star, and by some ofthe supermassive black holes in the centers of galaxies has been observed directly(Bridle & Perley 1984). The fireball model attributes gamma-ray bursts (GRBs) toultrarelativistic jets, which are emitted during strong supernovae. A direct observationof the ultrarelativistic jets that trigger the GRBs and occur at cosmological distancesis not possible. Their existence can thus not be established unambiguously (Woosley &Bloom 2006). However, observations of a mildly relativistic plasma outflow during thesupernova 1998bw by Kulkarni et al. (1998) lend some support to the fireball model.The efficiency, with which the accreting object can accelerate the jet plasma, isnot constant in time. A variable plasma acceleration efficiency results in a spatiallyvarying velocity profile of the jet plasma. Internal shocks can form at locations with alarge velocity change and these shocks can constitute strong sources of electromagneticradiation (Rees 1978). The prompt emissions of gamma-ray bursts, which are associated † Email address for correspondence: [email protected]
M. E. Dieckmann and A. Bret with internal shocks in ultrarelativistic jets, are visible across cosmological distances andinternal shocks should thus be sources of intense electromagnetic radiation.The relativistic factors of the internal shocks in GRB jets are probably of the orderof a few. A wide range of theoretical and numerical studies have addressed the collisionof lepton clouds at relativistic speeds and the instabilities that sustain the shock andthermalize the plasma that crosses it (Kazimura et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. onrelativistic pair shock et al.
2. Shock formation, the simulation code and the initial conditions
The formation mechanism of a collisionless leptonic shock
We examine the formation of shocks out of the collision of two charge- and current-neutral clouds of electrons and positrons. The plasma we consider is initially unmagne-tized, no ions are present and all lepton species have the same temperature. The absentbinary collisions imply that both lepton clouds will move through each other until plasmainstabilities start to grow. Only three wave modes can develop for our initial conditions.The two-stream modes are purely electrostatic and their wave vector is aligned withthe collision direction. The quasi-electrostatic oblique modes have a wave vector that isoriented obliquely to the collision direction and they belong to the same wave branch asthe two-stream modes. The third mode is the filamentation mode, which is also known asthe beam-Weibel mode (Califano et al. π/ with the collision direction. These three modes grow simultaneously during theshock formation stage. Their growth is eventually halted by nonlinear processes, whichheat up the plasma in the overlap layer and bring it closer to a thermal equilibrium.A leptonic shock can be created in a PIC simulation by the collision of one leptoncloud with a reflecting wall. The reflected leptons move against the inflowing leptonsthat have not yet reached the wall and an overlap layer develops. The instabilities inthis overlap layer let waves grow that heat up the plasma when they saturate. Theexpansion of the heated plasma is limited on one side by the wall and a shock forms onthe other side. The shock evolution is resolved correctly once a downstream region hasformed that is thick enough to decouple the shock from the wall. The formation phaseof the shock may, however, not be resolved correctly by this computationally efficientmethod. The mechanism that triggers the filamentation or beam-Weibel instability isthat particles with oppositely directed current vectors repel each other and particles withparallel current vectors attract each other. The instability saturates by forming currentchannels that collect particles with the same direction of the current vector. Currentchannels that contain particles with oppositely directed current vectors are separatedby magnetic fields. A reflection of a particle by the wall changes its velocity componentalong the wall’s normal direction and, thus, the direction of its current vector. Spatiallyseparated current channels can, however, not form at the wall because the particle is notspatially displaced by the reflection. The suppression of the filamentation instability atthe reflecting wall will affect the spectrum of the unstable waves.This spectrum is resolved correctly if we let two separate lepton clouds collide. If bothclouds differ only in their mean speed, then we have to resolve in the simulation twoidentical shocks that enclose the expanding downstream region. It is computationallyexpensive and unnecessary to track both shocks for a long time. Here we let a long anda short lepton cloud collide. We increase the time interval during which we can observethe shock between the downstream region and the long lepton cloud. The second shockmoves into the opposite direction and it eventually reaches the simulation boundary. Bythat time, it does no longer affect the evolution of the other shock. M. E. Dieckmann and A. Bret
The particle-in-cell (PIC) simulation method
We model the collision of the lepton clouds with a particle-in-cell (PIC) simulation.The PIC simulation code is based on the kinetic plasma model, which approximates eachplasma species i by a phase space density distribution f i ( x , v , t ) . The position vector x and the velocity vector v are treated as independent coordinates, which allows forarbitrary velocity distributions at any given position. The number density of this speciesis the zero’th moment of the distribution n i ( x , t ) = R f i ( x , v , t ) d v and the mean speed ¯ v i ( x , t ) = R v f i ( x , v , t ) d v corresponds to its first moment. The number density andthe mean speed yield the charge density ρ i ( x , t ) = q i n i ( x , t ) and the current density J i ( x , t ) = q i ¯ v i ( x , t ) n i ( x , t ) of the species i . The total charge density ρ ( x , t ) = P i ρ i ( x , t ) and current density J ( x , t ) = P i J i ( x , t ) update the electromagnetic fields via a finitedifference approximation of Ampere’s and Faraday’s laws on a numerical grid. µ ǫ ∂ E ∂t = ∇ × B − µ J ,∂ B ∂t = −∇ × E . The EPOCH code (Arber et al. ∇ · B = 0 and ∇ · E = ρ/ǫ toround-off precision.An ensemble of computational particles (CPs) with the charge q i and mass m i approx-imates the phase space density distribution f i ( x , v , t ) . The relativistic momentum p j ofthe j th CP of species i is updated via a discretized form of the relativistic Lorentz forceequation ∂∂t p j = q i ( E ( x j ) + v j × B ( x j )) and its position is updated via ∂∂t x j = v j .The electric field and the magnetic field are interpolated from the numerical grid to theparticle’s position x j to update its momentum. The current density on the grid, which isused to update the electromagnetic fields, is the sum over all particle currents after theyhave been interpolated from the particle positions to the grid nodes.2.3. The simulation setup
Our two-dimensional simulation box has the length L x along x and L y along y . Thesimulation box is subdivided into the two intervals − . L x < x < and x < . L x . The boundary at . L x is reflecting and that at − . L x is open. The boundaryconditions at y = 0 and y = L y are periodic. We place electrons and positrons with equaldensities n and temperatures T = 10 eV everywhere in the box at t = 0 . The electronsand positrons in the interval with x > have a vanishing mean speed. The electrons andpositrons in the interval x < have the mean speed v = 0 . c along x . No new particlesare introduced while the simulation is running and the simulation is stopped well beforethe end of the inflowing lepton cloud encounters the shock or before the leptons that arereflected by the boundary at x = 0 . L x return to the shock.We normalize the position to the electron skin depth λ s = c/ω p , where ω p =( n e /ǫ m e ) / is the electron plasma frequency of one cloud. Velocities are normalizedto c . Momenta are normalized to cm e and we define p = v m e as the mean momentumof a lepton of the plasma cloud in the half-space x < . The box size L x × L y = 60 × . is resolved by . × grid cells along x and by 760 grid cells along y . Electrons andpositrons are represented by 25 CPs per cell, respectively. The time is normalized to ω − p . The simulation time t sim = 120 , which is subdivided into equal time steps.We normalize the electric field to ω p cm e /e and the magnetic field to ω p m e /e . onrelativistic pair shock Figure 1.
The solution of the linear dispersion relation for two beams, each of which consistsof electrons and positrons with the same number density, mean speed and temperature T = x with the speed modulus . c . The growth rate δ is expressed in units of ω p . The solution of the linear dispersion relation
We have to verify that our box is large enough to resolve the competing unstable modesand we want to determine the wave mode, which grows fastest for the selected initialconditions. We solve for this purpose the linear dispersion relation in order to determinethe spectrum of the growing waves. The solution is computed under the assumptionthat the overlap layer has an infinite size. This condition is approximately fulfilled if thecolliding clouds can interpenetrate for some time before the instabilities grow.The initial velocity spread for T = 10 eV is about v th = 4 . × − c and both cloudsdrift toward each other at v = 0 . c . Thermal effects can be neglected for the ratio v /v th = 44 and the lepton beams are cold. We solve the linear dispersion relation inthe frame of reference in which the total momentum vanishes. The pair clouds move inthis reference frame into opposite x-directions at the speed modulus β ′ ≡ v ′ /c = 1 / .The non-relativistic dispersion equation for a perturbation of the form exp( i k · r − iωt ) and a wave vector k with an arbitrary orientation is (Bret et al. (cid:0) ω ǫ xx − k y c (cid:1) (cid:0) ω ǫ yy − k x c (cid:1) − (cid:0) ω ǫ yx + k x k y c (cid:1) = 0 . (2.1)where δ αβ is the Kronecker symbol and ǫ αβ ( k , ω ) = δ αβ − ω p ω ! + ω p ω X j Z d p p α p β k · (cid:16) ∂f j ∂ p (cid:17) mω − k · p . (2.2)The problem of finding the fastest growing mode has been solved (Bret & Deutsch 2005;Bret et al. f j ( p ) = δ ( p y ) δ ( p x − P j ) .Figure 1 shows the solution of the linear dispersion relation for our plasma parameters.The growth rate peaks at the wave number k x λ s ≈ and its value does not depend on k y λ s for the considered wave number interval. The fastest-growing modes are thus thetwo-stream/oblique modes. Their peak exponential growth rate is δ T S ω p = √ . (2.3) M. E. Dieckmann and A. Bret
The filamentation modes are characterized by a flow aligned component k x = 0 . Califano et al. (1998) estimated their growth rate as δ W ω p = 2 β ′ . (2.4)Figure 1 demonstrates that the growth rate of the filamentation modes with k x = 0 issmaller than that of the two-stream/oblique modes, which confirms the aforementonedapproximations since δ W < δ T S for β ′ = 1 / .We can estimate with the help of Fig. 1 if and how our limited box size will affect thespectrum of growing waves. The simulation employs periodic boundary conditions along y and the box length is L y in this direction. The smallest resolved wave number is thus k c = 2 π/L y or k c λ s = 2 . and waves with k y < k c can not grow. Figure 1 shows that thegrowth rate of the filamentation modes decreases below δ W for k y < k c while that of thetwo-stream/oblique modes remains unchanged. The main effect of the limited box sizealong L y is thus to suppress the wave numbers where the growth of the filamentationinstability is negligible. If our simulation shows that the plasma dynamics is governed bythe two-stream/oblique modes, then we would obtain the same result also for larger L y .
3. Simulation results
We discuss the simulation results at selected times and focus on the shock that formsat lower values of x . The first part addresses the wave modes that trigger the formationof shocks. The second part discusses the structure of the shock and the electric fieldsthat mediate it and the final part examines the growth of magnetic fields.3.1. Instability and nonlinear saturation
The two clouds of initially unmagnetized collisionless lepton plasma will move througheach other for some time before plasma instabilities set in. Figure 2(a-c) displays theelectric E x and E y components as well as the magnetic B z component at the time t =7 . . The lepton cloud, which was initially located in the half-space x < , has moved by v t = 1 . towards increasing values of x. Waves have grown in the cloud overlap layer,which spans the interval < x < . at this time. The distribution of E x reveals waveswith a wavelength λ ≈ . . The E y and B z components are closely correlated and bothoscillate rapidly along y.The in-plane electric field components and the out-of-plane magnetic field at the time t = 14 . are displayed in Fig. 2(d-f). Figures 2(a) and (d) show the same distribution of E x except for the larger amplitude. Their spatial confinement demonstrates that thesewaves do not propagate along y . The wave structures belong to electrostatic two-streammodes. The patterns in E y resemble those in B z and their amplitude ratio is comparableto that at the earlier time. The spatial correlation of the field structures in the distributionof E y and B z suggests that they belong to the same waves.We can extract some properties of the waves from a comparison of the amplitude of E y and B z at the times t or t . The ratio of the field energy densities ǫ ( E x + E y ) / and B z / µ is in the given normalization ( E x + E y ) /B z ≈ . The particles of bothclouds move at a speed ≈ v / relative to the waves, which are slow-moving in thereference frame of the overlap layer. The electric force imposed on a charged particle,which moves with v / = 0.1, is 50 times larger than the magnetic force. We concludethat the wave’s magnetic field neither has a significant energy density nor does it affectthe lepton dynamics. The waves are thus quasi-electrostatic and their wavelength alongthe collision direction is ≈ . . The amplitude of the waves has increased by a factor ≈ onrelativistic pair shock Figure 2.
The in-plane electric field and the out-of-plane magnetic field close to the initialcollision boundary at the time t = 7 . (left column) and at t = 14 . (right column): Panels(a, d) show E x , panels (b, e) show E y and panels (c, f) show B z . during the time interval t − t = 7 . . If we assume that the waves grow exponentially,then their growth rate is δ ≈ . in units of ω p , which matches that in Fig. 1.The wave modes that yield the observed electric field can be identified with its spatialpower spectrum. We Fourier-transform the in-plane electric field distribution E p ( x, y ) = E x ( x, y ) + iE y ( x, y ) over the spatial interval . < x < . and over all y and multiply itwith its complex conjugate. Figures 3(a, b) show the power spectra at the times t and t in the quadrant k x > and k y > . The power spectrum at t = 7 . shows wave powerat k x λ s ≈ , which extends up to maximum perpendicular wave number | k y λ s | ≈ .The wavenumber k x λ s = 14 corresponds to a wavelength along x of about . .The flow-aligned wave number k x λ s ≈ of the fastest-growing waves and theextension of wave power to large values of k y agree with the numerical solution of thelinear dispersion relation in Fig. 1. The solution of the linear dispersion relation predictsa peak growth rate that does not depend on the value of k y for the considered wavenumbers. The wave spectrum on Fig. 3(a) does however suggest that waves with a lowvalue of k y grow faster. The growth rate is proportional to the amplitude the wave wouldreach after a given time if its growth would not be limited by nonlinear effects. Theelectric field amplitude, which is necessary to form phase space vortices, decreases withincreasing values of k = | k x + k y | / (O’Neil 1965) and the discrepancy between thespectral distribution in Fig. 3(a) and the solution of the linear dispersion relation in Fig. M. E. Dieckmann and A. Bret y λ S k x λ S −2.5−2−1.5−1−0.500 50 100 150 200050100 (b) k y λ S k x λ S −10123 Figure 3.
The spatial power spectra of the in-plane electric field E p = E x ( x, y ) + iE y ( x, y ) atthe time t = 7 . (a) and t = 14 . (b). The color scale is 10-logarithmic and both spectra arenormalized to the peak value in (a). Figure 4.
The phase space density distribution in the x, p x -plane at the time t = 14 . ofelectrons (a) and positrons (b). The phase space density distribution is averaged over all otherdimensions. The momentum is normalized to p . The color scale is 10-logarithmic. k y .The power spectrum in Fig. 3(b) is still concentrated on the two-stream / oblique modebranch. Its width along k y has diminished, which suggests that thermal damping is atwork; the range of wave numbers k y that are unstable to the oblique mode instability islarge in a cold plasma, while the wave growth is concentrated at low values of k y if theplasma is hot (Silva et al. k x have emerged.The wave amplitudes have thus reached a non-linear regime (Umeda et al. f ( x, p x ) of the electronsand positrons. The overlap layer of both clouds spans the interval − . < x < . Thecounterstreaming clouds have not yet merged along p x . However, the substantial particleacceleration demonstrates that the instability is about to saturate. The density in theoverlap layer is twice that of a single cloud and the density flucutations caused by thewaves are of the order of − (not shown).The filamentation instability starts to grow immediately but two-stream outgrows it. onrelativistic pair shock Figure 5.
Panels (a) and (b) show the electric E x and E y components close to the initial collisionboundary. Panels (c) and (d) show the phase space density distributions in the x, p x -plane ofelectrons and positrons, respectively. The phase space density distributions are averaged overall other dimensions, they are normalized to the same value and displayed on a 10-logarithmicscale. The momentum is normalized to p . The simulation time is t = 58 . . Shock formation
The two-stream instability saturates by forming stable phase space vortices in theelectron and positron distributions (Berk & Roberts 1967) and the same holds duringthe initial saturation stage of the oblique mode instability (Dieckmann et al. b ).Electron phase space vortices are characterized by strong bipolar pulses in the electricfield distribution, which correspond to a localized positive excess charge. Positron phasespace vortices correspond to a localized negative excess charge.The in-plane electric field components at the time t = 58 . are displayed in Fig.5(a,b). The electric E x component shows such bipolar field structures. A large quasi-planar field pulse is located at x ≈ . in the interval starting from y ≈ that goesthrough the periodic boundary at y = 2 . until y ≈ . . The polarity of E x indicates thepresence of a positive excess charge in between both electric field bands. If this quasi-planar bipolar pulse is associated with an electron phase space vortex, then the lattershould be detectable in the electron phase space density distribution even if it has beenintegrated over all values of y .Figures 5(c, d) show the corresponding electron and positron distributions. Figure5(c) confirms the existence of a phase space vortex in the electron distribution at thislocation. The vortex in Fig. 5(c) spans the spatial interval < x < and the momentuminterval − < p x /p < . The mean momenta of the upstream electrons and positronsare modulated by the electrostatic potential of the vortex when they pass it, but they arenot trapped by it. The upstream leptons continue to move to increasing values of x untilthey are thermalized upon entering the downstream region x > , which is characterizedby a dense phase space density distribution between < p x /p < . This thermalizationcan only be accomplished by the field structures seen in the in-plane electric field between x ≈ and x ≈ in Fig. 5(a, b).The distribution of the positrons shows two smaller vortices that surround the largeelectron phase space vortex. The positron vertices are centered at x ≈ . and x ≈ .The zero-crossing of the electric E x component and, thus, the extremal point of theelectrostatic potential at x ≈ . in Fig. 5(a) corresponds to a stable equilibrium point0 M. E. Dieckmann and A. Bret
Figure 6.
Panels (a) and (b) show the electric E x and E y components close to the initial collisionboundary. Panels (c) and (d) show the phase space density distributions in the x, p x -plane ofelectrons and positrons, respectively. The phase space density distributions are averaged overall other dimensions, they are normalized to the same value and displayed on a 10-logarithmicscale. The momentum is normalized to p . The simulation time is t = 120 . for the trapped electrons. Hence it is an unstable equilibrium point for the positrons,explaining why the vortices of positrons and electrons are staggered along x .A small localized cloud of electrons and positrons is centred at x ≈ and p x ≈ . Thecloud is an artifact from our initial conditions. The finite growth time of the electrostaticinstabilities implies that the waves start to grow well behind the front of the plasma cloudthat was initially located in the half-space x > . This charge- and current neutral cloud isstable against electrostatic instabilities, because its extent along x is not sufficiently largeto allow it to interact with the inflowing upstream leptons via a two-stream instability.Figure 6 shows the in-plane electric field distribution and the associated lepton phasespace density distributions at the time t = 120 . We observe strong quasi-planar electricfield structures in the E x -distribution in the interval − < x < . Their amplitude iscomparable to the one that gave rise to phase space vortices in the electron- and positrondistributions at the earlier time. These electrostatic structures in E x have propagatedwell beyond the initial collision boundary x = 0 reaching a position x ≈ − . We findrelatively strong electric field oscillations in E x and E y between < x < . The transitionlayer of this shock thus spans at this time an interval with the width ∆x ≈ .The strong planar waves in the interval − < x < in Fig. 6 are correlated with phasespace vortices in the hot lepton population at low speeds. The vortices of electrons andpositrons are staggered along x . The electrons and positrons that gyrate in these vorticesoriginate from the hot plasma component and they are well-separated along p x from theinflowing upstream leptons. The mean speed of these phase space vortices is less than p x = 0 , which implies that they move towards decreasing values of x . The mean speedof the vortices decreases with an increasing distance from the shock transition layer andthey are thus accelerated away from the downstream plasma. The leptons, which gyratein the vortices, reach a peak momentum ≈ − p .The simultaneous presence in the interval − < x < of the hot leptons that haveleaked from the downstream region and the cooler drifting upstream leptons implies thatthe overall plasma distribution is non-thermal and thus unstable. The electric field ofthe phase space vortices seeds the instability and we observe momentum oscillationsalong p x in the cool inflowing electrons and positrons that increase with x in the interval − < x < − . The oppositely directed oscillations of electrons and positrons result in onrelativistic pair shock −2 −1 0 1 2 3 4 5 611.522.53 (a) X N ( x ) −4 −2 0 2 4 6 8 10 1211.522.53 (b) X N ( x ) Figure 7.
The total lepton density N ( x ) in units of the initial total density n at the time t = 58 . (a) and t = 120 (b). a strong current, which induces an electric field. The electric field oscillates in spaceand its oscillation amplitude decreases in unison with the net current in the direction ofdecreasing values of x . We can describe this oscillation in terms of a product between asinusoidally oscillating electric field and an envelope function.A spatially varying envelope function gives rise to a ponderomotive force (Kono et al. < x < in Fig. 6(c,d) are composed of a hot dilutecomponent and the cool dense upstream leptons. Both populations gradually mix andthey merge to a single one at x ≈ . We observe electric fields in this interval in Fig.6. These fields show some piecewise planar structures, which correspond to phase spacevortices with a limited extent along y . The two strongest localized structures at x ≈ are separated by a perpendicular E y field at y ≈ . . These localized structures are likelyto be the result of an instability of initially planar phase space vortices or phase spacetubes. Indeed, two-dimensional PIC simulations (Oppenheim et al. x, p x plane. Strong small-scale electric fields are observed up to x ≈ . Theabsence of phase space vortices with x > demonstrates that the lepton distribution inthis interval is no longer unstable to electrostatic instabilities.The phase space density distribution of the leptons was uniform in the interval Figure 8. The total lepton distribution f t ( v x , v y ) averaged over − . < x < − . is shown inpanel (a), that averaged over . < x < . in panel (b) and that averaged over . < x < . is shown in panel (c). The color scale is 10-logaritmic and normalized to the peak value in (a). electrons and positrons into opposite directions. The density rises from about N ( x ) = 1 to N ( x ) ≈ over a few electron skin depths. The plasma compression factor of about 3is the one expected for a strong nonrelativistic shock (Zel’Dovich et al. Secondary instabilities and magnetic field generation The electric fields associated with the phase space vortices heat up the leptons viaLandau damping (Landau 1946; O’Neil 1965) and their collapse scatters them in phasespace. The effects of this heating on the lepton distribution is visualized by Fig. 8, whichshows the phase space density as a function of p x and p y at three positions along x . Thedistribution has been integrated over y and over an interval along x of width 0.3.The distribution in Fig. 8(a) has been sampled far upstream of the shock. The upstreamleptons constitute the cold dense beam that is located at p x ≈ p . The leaked leptonsform a hot and dilute beam that moves at p x ≈ − p . The mean speed of the hot leptonbeam exceeds that expected from a specular reflection, since the shock is moving toincreasing values of x . Figure 8(b) reveals that the inflowing upstream leptons have beenheated up by the time they reach the position x ≈ . They are distributed over a widervelocity range and their peak value of the phase space density has thus decreased. Thetemperature is of the order of 100 eV. These leptons are immersed in a hot dilute leptoncomponent. Its thermal momentum spread is of the order of p and the temperature isthus about one keV. The inflowing upstream leptons form a hot beam at p x ≈ p in 8(c)that is only about twice as dense as the leptons in the hot population.The waves observed close to x = 2 in Fig. 6(a,b) suggest that the velocity distributionin Fig. 8(b) is still unstable to an electrostatic instability. It can not be the two-streaminstability because that one requires two beams that are well-separated along p x . Thisdistribution can, however, still be unstable to the electron acoustic instability. Alike thewell-known ion acoustic instability, which is driven by a drift between cold ions andhot electrons, the electron acoustic instability can develop if cold electrons drift relativeto a hot electron species. Waves grow if the drift speed between the hot and the coldelectron species exceeds several times the thermal spread of the cold electron species(Gary 1987). This condition is fullfilled in Fig. 8(b). We note in this context that althoughthe phase space density of the hot leptons is two orders of magnitude less than that of onrelativistic pair shock Figure 9. Panels (a) and (b) show the out-of-plane component B z of the magnetic field at thetimes t = 58 . and t = 120 , respectively. The color scale is the same for both panels. Forcomparison: The downstream region at t = 58 . is enclosed by shocks at x ≈ and x ≈ ,while the correctly resolved shock at t = 120 is located at x ≈ . the inflowing upstream leptons their number density, which we obtain by integrating thephase space density along p x , is of the same order. The interaction of counterstreaminglepton beams with a similar density results in rapidly growing instabilities. To the bestof our knowledge the acoustic instability in pair plasma has not yet been explored. Herewe can not unambigously show that it exists in pair plasma, because the electric fieldmay also be the residual field of a phase space hole that formed previously.The distribution in Fig. 8(c) appears to be stable against electrostatic instabilities sincewe do not observe significant electric field oscillations in the region x > . The velocitydistribution of the leptons in this region is, however, not a Maxwellian. Therefore theplasma contains free energy that can be released by a collisionless instability. The thermalanisotropy contained in the total lepton velocity distribution f t ( v x , v y ) and measured inthe rest frame of the downstream plasma can be estimated as A = R f t ( v x , v y )( v x − p / m e ) dv x dv y R f t ( v x , v y ) v y dv x dv y − . (3.1)A value A = 0 would say that the thermal energy in the x direction equals that in the y direction, which would imply that there is no thermal anisotropy. We obtain the value A ≈ from the data shown in Fig. 8(c). Such a large anisotropy value results in theWeibel instability in its original form (Weibel 1959; Morse & Nielson 1969).Figure 9 confirms that a magnetic field has grown in the downstream region. Strongmagnetic fields with approximately the same peak amplitude are present at both times.The amplitude of the magnetic field exceeds that observed in Fig. 2(c, f) by a factor 3 andit equals that of the electric field in the given normalization in Fig. 6(a,b). Nevertheless,the magnetic force, which is acting on a lepton that is moving at the speed of . relativeto these field patches, will still be an order of magnitude weaker than the electric forcein the shock transition layer.The strong downstream magnetic fields are not correlated with electric field structures(See Figs. 5(a, b) and 6(a,b)) and they are thus not driven by an oblique mode instability.The Weibel instability drives magnetowaves with a negligible electric field and withthe same magnetic field direction as the one observed here and this instability is thuscompatible with the simulation data. The Weibel instability yields magnetic fields withan energy density that can reach up to 10% of the thermal energy for strong temperature4 M. E. Dieckmann and A. Bret anisotropies (Morse & Nielson 1971; Stockem et al. K. The magneticpressure of a field B z = 0 . is thus a few percent of the cumulative thermal pressure ofelectrons and positrons and within the range that is accessible to the Weibel instability.We can compare the maximum size of the magnetic patches in Fig. 9 to the gyroradiusof an electron in that field. The magnetic field ˜ B z and the collision speed ˜ v in physicalunits can be calculated from the normalized ones via B z = e ˜ B z /m e ω p and ˜ v = v c .The gyroradius of an electron that moves at the speed v / relative to the stationaryperpendicular magnetic field, which is normalized to λ s = c/ω p , is then r g /λ s = v / B z .Taking v = 0 . and B z = 0 . gives r g ≈ λ s , which is about three times the coherencescale of the largest magnetic field patches. The magnetic field patches are also notstationary on the time scale needed to perform a gyro-orbit. This time equals for amaximum amplitude B max = 0 . in our normalization π/B max ≈ , which exceedsthe simulation time. The leptons can thus not complete a full gyro-orbit. The magneticfield will instead deflect leptons by a small angle that depends on where the lepton enteredthe patch and on how long it stayed inside the patch. The magnetic field will thus scatterthe leptons of the directed beam in Fig. 8(c). The repeated scattering of the leptons willeventually thermalize their distribution. 4. Summary We have examined the formation and the initial evolution of a non-relativistic leptonicshock. The shock was created by letting two spatially uniform clouds of equally denseelectrons and positrons collide at a relative speed of 0.2c. The absence of binary collisionsimplied that both clouds initially interpenetrated and formed an overlap layer. 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