Mildly relativistic magnetized shocks in electron-ion plasmas -- II. Particle acceleration and heating
Arianna Ligorini, Jacek Niemiec, Oleh Kobzar, Masanori Iwamoto, Artem Bohdan, Martin Pohl, Yosuke Matsumoto, Takanobu Amano, Shuichi Matsukiyo, Masahiro Hoshino
MMNRAS , 1–11 (2020) Preprint 25 January 2021 Compiled using MNRAS L A TEX style file v3.0
Mildly relativistic magnetized shocks in electron–ionplasmas – II. Particle acceleration and heating
Arianna Ligorini, Jacek Niemiec, (cid:63) Oleh Kobzar, Masanori Iwamoto, Artem Bohdan, Martin Pohl, , Yosuke Matsumoto, Takanobu Amano, Shuichi Matsukiyo, and Masahiro Hoshino Institute of Nuclear Physics Polish Academy of Sciences, PL-31342 Krakow, Poland Astronomical Observatory of the Jagiellonian University, PL-30244 Krakow, Poland Faculty of Engineering Sciences, Kyushu University, Kasuga, Fukuoka, 816-8580, Japan DESY, 15738 Zeuthen, Germany Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany Department of Physics, Chiba University, 1-33 Yayoi, Inage-ku, Chiba 263-8522, Japan Department of Earth and Planetary Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Particle acceleration and heating at mildly relativistic magnetized shocks in electron-ion plasma are investigated withunprecedentedly high-resolution two-dimensional particle-in-cell simulations that include ion-scale shock rippling.Electrons are super-adiabatically heated at the shock, and most of the energy transfer from protons to electrons takesplace at or downstream of the shock. We are the first to demonstrate that shock rippling is crucial for the energizationof electrons at the shock. They remain well below equipartition with the protons. The downstream electron spectra areapproximately thermal with a limited supra-thermal power-law component. Our results are discussed in the contextof wakefield acceleration and the modelling of electromagnetic radiation from blazar cores.
Key words: acceleration of particles, instabilities, galaxies:jets, methods:numerical, plasmas, shock waves
High-energy charged particles produce intense nonthermalemission that is observed from many astronomical objectsin the Universe. They are also copiously registered at Earthas cosmic rays (CRs) with energy reaching 10 eV, and pos-sibly even higher at their sources. The acceleration of theseparticles is one of the unsolved questions in astrophysics.Ultra-high-energy cosmic rays (UHECRs) with energies inexcess of ∼ eV are often associated with relativisticshocks formed in jets of active galactic nuclei (AGN) and/orgamma-ray bursts (GRBs). Blazar jets emit strong nonther-mal synchrotron and inverse-Compton radiation across theentire electromagnetic spectrum, which provides evidencefor electron acceleration to ultrarelativistic energies in thesesources. High-energy electrons are most likely also respon-sible for the inverse-Compton component of the GRB jet’safterglow emission at sub-TeV γ -ray energies (Acciari et al.2019a,b; Abdalla et al. 2019). Observations of high-energyneutrinos from the flaring blazar TXS 0506+056 indicate thatrelativistic jets also produce CR hadrons (IceCube Collabo-ration et al. 2018).Extragalactic jets probably harbour shocks whose Lorentz (cid:63) E-mail: [email protected] factor, γ sh , spans the range from near unity to a few hun-dred. Diffusive shock acceleration (DSA) may not work atultra-relativistic ( γ sh (cid:29)
1) magnetized outflows, on accountof inherent superluminal conditions at such shocks, in whichparticle diffusion across the magnetic field is suppressed (e.g.Begelman & Kirk 1990; Niemiec, Ostrowski, & Pohl 2006).Particle-in-cell (PIC) simulation studies confirmed that ultra-relativistic shocks cannot be efficient particle acceleratorsthrough DSA-like processes unless the plasma magnetizationis very small ( σ (cid:46) − , where σ is the ratio of the Poyntingflux to the kinetic energy flux) or the shock is subluminal(Martins et al. 2009; Sironi & Spitkovsky 2009, 2011, see alsorecent review by Pohl, Hoshino, & Niemiec (2020)).As an alternative to the DSA model, collective wave-particle interactions can lead to efficient particle accelera-tion at superluminal magnetized relativistic shocks. Large-amplitude coherent X-mode waves are generated by the syn-chrotron maser instability (SMI) driven by the particles re-flected off the shock-compressed magnetic field. Upstream ofthe shock they form a so-called precursor wave, which hasbeen demonstrated with particle-in-cell (PIC) simulations forultrarelativistic shocks ( γ sh ≥
10) in one-dimensional (1D)(e.g. Langdon, Arons, & Max 1988; Hoshino & Arons 1991;Gallant et al. 1992; Hoshino et al. 1992; Amato & Arons2006; Plotnikov & Sironi 2019) and high-resolution two- © a r X i v : . [ a s t r o - ph . H E ] J a n Ligorini et al. dimensional (2D) studies (Iwamoto et al. 2017, 2018, 2019;Babul & Sironi 2020, see also Plotnikov, Grassi, & Grech(2018)). Strong precursor waves in electron-ion plasma in-duce in their wake large-amplitude longitudinal electrostaticoscillations (Lyubarsky 2006; Hoshino 2008). Nonlinear col-lapse of the Langmuir waves generates nonthermal electronsand ions in a way similar to wakefield acceleration (WFA,Hoshino 2008), known from studies of laboratory plasma (e.g.Tajima & Dawson 1979; Kuramitsu et al. 2008). We recentlydemonstrated that ultrarelativistic shocks with high electronmagnetization in electron-ion plasma can accelerate electronsand ions in the turbulent wakefield to power-law spectra withindex close to 2 (Iwamoto et al. 2019).Efficient heating of electrons to equipartition with the ionswas reported by Sironi & Spitkovsky (2011) to happen beforethe flow reaches the shock. This strong electron-ion couplingwas argued to operate only for shocks with γ sh (cid:38) ( m i /m e ) / ,where m i /m e is the ion-to-electron mass ratio (Lyubarsky2006). PIC simulations of shocks with γ sh (cid:46)
10 that vio-late that condition showed very weak or no wakefield andlittle energy transfer from ions to electrons (Lyubarsky 2006;Sironi & Spitkovsky 2011). However, these simulations wereperformed at low numerical resolution, whereas high resolu-tion is required to properly account for the upstream wavegeneration (Iwamoto et al. 2017).This is the second of two articles in which we studymildly-relativistic strictly perpendicular shocks in electron-ion plasma with unprecedentedly high-resolution and large-scale 2D PIC simulations, that account for ion-scale shockrippling. In Ligorini et al. (2021, hereafter Paper I), we pre-sented the electromagnetic structure of a shock with theLorentz factor γ sh (cid:39) . σ = 0 .
1. These parameters may be relevant for internal shocksin AGN jets (e.g. Ghisellini, Tavecchio, & Chiaberge 2005).Here we discuss particle acceleration and heating at the shockto estimate the efficiency of WFA and electron-proton cou-pling in the mildly-relativistic regime. Energy transfer be-tween protons and electrons at mildly-relativistic shocks is ofimportance in models of synchrotron and inverse-Comptonemission from blazar jets. Very strong coupling of electronsand ions would have consequences for the radiation spectrathat impose quite strong constraints on the location of theemission sites in the jet and on the pair content (e.g. Sikoraet al. 2013). Weak coupling for mildly relativistic conditionsmay relax some of those constraints.
Results presented in this paper are based on the same PICsimulations as in Paper I. Here we briefly reiterate the sim-ulation setup that is illustrated in Fig. 1. Initially cold, v th , beam = 0, electron-ion plasma is reflected off a conductingwall at the left side of the simulation box. Its interaction withthe incoming particles provides a shock moving in + ˆ x direc-tion. We use a modified version of the relativistic electromag-netic PIC code TRISTAN (Buneman 1993) with MPI-basedparallelization (Niemiec et al. 2008) and the option to traceindividual particles.We perform simulations in 2D3V in the x − y plane, inwhich we track all three components of particle momentaand electromagnetic fields. We probe two configurations of Figure 1.
Illustration of the simulation setup (adapted from Pa-per I). the large-scale perpendicular magnetic field, B , with respectto the simulation plane – the out-of-plane orientation with ϕ B = 90 o ( B = B z ˆ z ) and the in-plane setup with ϕ B =0 o ( B = B y ˆ y ). Together with B , the beam carries themotional electric field E = − v × B , where v is the beamflow velocity.For the beam Lorentz factor γ = 2 .
03, the resultingLorentz factor of the shock is γ sh (cid:39) . σ = 0 .
1, where σ = B / ( µ N i ( m e + m i ) γ c ), with the permeability of freespace, µ , and the upstream ion density, N i (= N e ), in thesimulation (downstream) frame (Hoshino et al. 1992). Themagnetization of electrons, σ e , and ions, σ i , are defined as σ l = B / ( µ N l m l γ c ) , l = e, i , so that 1 /σ = 1 /σ e + 1 /σ i .We assume a reduced ion-to-electron mass ratio of m i /m e =50, for which the electron magnetization is σ e (cid:39) . x -direction, is L x × L y = 160000 × , where ∆ is thegrid cell size. We use the ion skin depth, λ si = c/ω pi , asthe unit of length, and L x × L y (cid:39) × λ . Here, theion plasma frequency is ω pi = (cid:112) e N i /γ (cid:15) m i , where e isthe electron charge, and (cid:15) is the vacuum permittivity. Theunit of time is the relativistic ion cyclotron frequency, Ω ci =( eB ) / ( m i γ ). The simulation time is t max Ω ci = 84 .
3. We alsoran a complementary 1D simulation up to t max Ω ci = 163 . λ se = 80∆, where λ se = (cid:112) m e /m i λ si is the electron skin depth. This unprecedentedlyhigh resolution is necessary to avoid artificial damping of theprecursor waves. The ion skin depth is thus λ si (cid:39) N ppc = 10per species. All parameter values have been selected afterextensive tests that are described in detail in Paper I. We briefly summarize the shock structure that is in detaildiscussed in Paper I. As a point of reference, Fig. 2 showsnormalized maps of the magnetic field oscillations associatedwith the X-mode waves, E x fluctuations, and profiles of trans-versely averaged electric field, (cid:104) E x (cid:105) , at time t = t max for the As discussed in Iwamoto et al. (2017), numerical damping ofhigh-frequency precursor waves may result from the application ofdigital filtering, used to suppress the numerical short-wavelengthCherenkov modes. Precursor waves must therefore be resolved onscales larger than the wavelengths on which the filters operate.A specific resolution depends on the numerical PIC model usedand must be determined through test runs.MNRAS , 1–11 (2020) ildly relativistic shocks in e − p plasmas II Figure 2.
Magnetic field oscillations of the X-mode waves and E x electric field amplitude at time t Ω ci = 84 .
3. Logarithmic scalingis applied, which for the electromagnetic field is sign-preserving (e.g. sgn( B X-mode ) · { | B X-mode | /B , − )] } ), so that fieldamplitudes below 10 − B are not resolved. Panels (a) and (b) refer to the case with ϕ B = 90 o , while panels (d) and (e) refer to ϕ B = 0 o .Panels (c) and (e) show the respective transversely averaged profiles of the electric field, (cid:104) E x (cid:105) , upstream of the shock. simulation runs with ϕ B = 90 o (left) and ϕ B = 0 o (right).For each magnetic-field configuration the SMI operates inagreement with theoretical predictions and produces coher-ent emission of upstream-propagating electromagnetic waves.For out-of-plane magnetic field these precursor waves are en-tirely of the X-mode type, with fluctuating magnetic fieldalong B , whereas for in-plane magnetic field mode conver-sion provides also O-mode waves. The strength of the waves, δB/B (cid:39) .
19 for ϕ B = 90 o and δB/B (cid:39) .
15 for ϕ B = 0 o ,is much smaller than at high- γ sh shocks, where δB/B (cid:29) ϕ B = 90 o and the Alfv´en Ion Cyclotron (AIC) temperature-anisotropyinstability for ϕ B = 0 o . The wavelength of the ripples is λ rippl (cid:39) . λ si and λ rippl (cid:39) λ si , respectively. For out-of-plane magnetic field the precursor waves are on average emit-ted obliquely to the shock normal, which reflects wave emis-sion in a direction normal to the local tangent to the rip-plings’ arcs and the effects of retardation and aberration, asthe ripples move with v rippl ≈ . c along the shock surface.The AIC-generated shock front corrugations in the in-planecase are of slightly lower amplitude, and the waves are mostlypropagating along the shock normal with only a weak obliquecomponent.In all simulations we observe density filaments upstream ofthe shock that are generated by the parametric filamentationinstability (Kaw, Schmidt, & Wilcox 1973; Drake et al. 1974).Likewise, in each case electrostatic wakefield of moderate am-plitude is excited in the upstream plasma. Using the so-calledstrength parameter, a = eδE/m e cω , where δE is the electric-field amplitude and ω is the wave frequency (Kuramitsu etal. 2008), we find with significant fluctuations a (cid:39) . − . ϕ B = 90 o and a (cid:39) . ϕ B = 0 o . ϕ B = 90 o Fig. 3 shows electron and ion phase-space distributions acrossthe shock, as well as the mean particle kinetic energy, (cid:104) γ − (cid:105) m l c . Electrons streaming toward the shock are ac-celerated toward it and gradually heated, reaching γ ≈ λ si aheadof the shock the mean electron energy commences a steadybut slow growth that results from interactions with the wake-field. We know from Fig. 2 that further away the amplitudeof the wakefield does not exceed (cid:104) E x (cid:105) /B c ≈ × − , andcloser than 50 λ si to the shock it is only marginally larger thanthat. Incoming electrons interacting with a Langmuir wave ofsuch amplitude should show an acceleration-deceleration pat-tern in the mean electron energy that we can indeed see inFig. 3(e). There is little or no net gain in energy, but the co-herent oscillations in velocity can be effectively regarded asheating to the maximum energy (Hoshino 2008): (cid:15) max γ m e c ≈ e (cid:104) E x (cid:105) L (cid:39) ξa (cid:112) ξa (1 + β ) (cid:39) . , (1)where L ≈ /k L is the scale-length of the wakefield and ξ =1 / a = 0 . γ ≈ MNRAS000
15 for ϕ B = 0 o ,is much smaller than at high- γ sh shocks, where δB/B (cid:29) ϕ B = 90 o and the Alfv´en Ion Cyclotron (AIC) temperature-anisotropyinstability for ϕ B = 0 o . The wavelength of the ripples is λ rippl (cid:39) . λ si and λ rippl (cid:39) λ si , respectively. For out-of-plane magnetic field the precursor waves are on average emit-ted obliquely to the shock normal, which reflects wave emis-sion in a direction normal to the local tangent to the rip-plings’ arcs and the effects of retardation and aberration, asthe ripples move with v rippl ≈ . c along the shock surface.The AIC-generated shock front corrugations in the in-planecase are of slightly lower amplitude, and the waves are mostlypropagating along the shock normal with only a weak obliquecomponent.In all simulations we observe density filaments upstream ofthe shock that are generated by the parametric filamentationinstability (Kaw, Schmidt, & Wilcox 1973; Drake et al. 1974).Likewise, in each case electrostatic wakefield of moderate am-plitude is excited in the upstream plasma. Using the so-calledstrength parameter, a = eδE/m e cω , where δE is the electric-field amplitude and ω is the wave frequency (Kuramitsu etal. 2008), we find with significant fluctuations a (cid:39) . − . ϕ B = 90 o and a (cid:39) . ϕ B = 0 o . ϕ B = 90 o Fig. 3 shows electron and ion phase-space distributions acrossthe shock, as well as the mean particle kinetic energy, (cid:104) γ − (cid:105) m l c . Electrons streaming toward the shock are ac-celerated toward it and gradually heated, reaching γ ≈ λ si aheadof the shock the mean electron energy commences a steadybut slow growth that results from interactions with the wake-field. We know from Fig. 2 that further away the amplitudeof the wakefield does not exceed (cid:104) E x (cid:105) /B c ≈ × − , andcloser than 50 λ si to the shock it is only marginally larger thanthat. Incoming electrons interacting with a Langmuir wave ofsuch amplitude should show an acceleration-deceleration pat-tern in the mean electron energy that we can indeed see inFig. 3(e). There is little or no net gain in energy, but the co-herent oscillations in velocity can be effectively regarded asheating to the maximum energy (Hoshino 2008): (cid:15) max γ m e c ≈ e (cid:104) E x (cid:105) L (cid:39) ξa (cid:112) ξa (1 + β ) (cid:39) . , (1)where L ≈ /k L is the scale-length of the wakefield and ξ =1 / a = 0 . γ ≈ MNRAS000 , 1–11 (2020)
Ligorini et al.
Figure 3.
Phase-space distribution in log( p/mc ) of electrons andions across the shock at time t Ω ci = 84 .
3, averaged over the spa-tial coordinate y . From top to bottom, shown are the x - and y -components of electron momentum and the x -component of ionmomentum. Panel (d) displays in blue for ions and in red for elec-trons the mean kinetic energy in units of that of far-upstream ions.Inserted is a rescaled kinetic-energy profile of electrons (e). as a pressure gradient. Frequency and wavenumber match-ing are consequences of energy and momentum conservation.In our case, nonlinear stimulated Raman scattering turns alarge-amplitude electromagnetic (pump) wave into a Lang-muir wave and a scattered electromagnetic (light) wave. Ifthe pump wave frequency is much larger than the plasma fre-quency, we have Forward Raman Scattering (FRS), and thescattered electromagnetic wave and the Langmuir wave prop-agate in the same direction as the pump wave. In subsequentstages the scattered electromagnetic waves can further decayinto other electromagnetic waves and Langmuir waves, andeventually broadband wave spectra can be generated. In theupstream plasma frame, the waves propagate outward, but inthe simulation frame part of the electromagnetic and Lang-muir waves can propagate in downstream direction (Hoshino2008). Fig. 2(e) indicates a much higher wakefield amplitudeclose to the shock than in the far-upstream region, which re-flects amplification of precursor-wave emission through shockrippling. We demonstrated in Section 3.2.5 of Paper I, thatin the region extending as far as ( x − x sh ) /λ si ≈
70 fromthe shock the wakefield moves toward the shock, which weinterpret as a signature of nonlinear FRS, during which thewakefield can collapse and transfer energy to particle heatand bulk acceleration.As at ultrarelativistic shocks, the downstream-propagatingLangmuir waves can resonantly interact with the electronsthrough the so-called phase-slippage effect, boost them to-ward the shock, and create an asymmetric wing in the p x phase-space distribution. The Langmuir waves also facili-tate shock-surfing acceleration (SSA) that would accelerate Figure 4.
Energy spectra for electrons (red) and ions (blue) down-stream of the shock ( x/λ si = 113 − t Ω ci = 84 .
3. Theenergy axis is scaled with the particle mass, m = m e , m i . Verticaldashed and dash-dotted lines mark the initial kinetic energies ofthe electrons and ions, respectively. Also shown with thin red dot-ted lines are two relativistic 2D Maxwellians that together (thickdotted line) is an eyeball fit to the electron spectrum. Inset (b)highlights the high-energy part of the electron spectrum, in whichthe double Maxwellian (red), a power-law ∝ ( γ − − . (yellow),and their sum (green) are shown with dotted lines. trapped electrons in the y -direction and cause the anisotropyin the p ye distribution toward positive momenta that is ev-ident in Fig. 3(b). The trajectory analysis in Section 4.1.3demonstrates energy gains and losses correlated with motionin y -direction. Note that also the ions show similar distur-bances in their phase-space distribution.Despite the presence of coherent precursor waves, the bulkenergy of electrons increases by only 5% before they hit theshock (see Fig. 3e). Most of the energy transfer from ions toelectrons takes place at the shock and in the near-downstreamregion, as we discuss next. Fig. 4 shows kinetic-energy spectra of electrons and ions attime t Ω ci = 84 .
3, taken in the near-downstream region at x/λ si = 113 − .
5% and 10 .
6% of the initial ion kinetic energy,respectively. In the simulation frame the mean kinetic energyin the upstream region should be similar to that downstream,and so about 8% of the energy are available for transfer toelectromagnetic waves and turbulence.The ion spectrum tends towards a 2D relativisticMaxwellian distribution, dNdγ ∝ γ exp (cid:18) − mc kT γ (cid:19) . (2) MNRAS , 1–11 (2020) ildly relativistic shocks in e − p plasmas II Figure 5.
Electron trajectories ending at the circles on a map of normalized magnetic-field strength, | B z | /B . The map is linearly scaledand refers to the end times t a , t b , and t c , that are marked in panels (d) and (e). The trajectories span 0 . − . In each panel a selectedelectron trajectory is plotted in red. Panel (d) shows the evolution of the average kinetic energy (red line) as well as the the x - (greenline) and y - (blue line) components of the average electric work (cf. Eq. 3). Panel (e) shows the averaged magnetic moment (red line) andthe magnetic field, B z (blue line), along the trajectories. The high-energy particles with Lorentz factors up to γ i ≈ γ e − m e /m i (cid:39) .
2, and theother one is about 6 times colder. The electron spectrum alsoshows a weak supra-thermal component at ( γ − /m i (cid:38) p ≈
2, shownwith the yellow dotted line in Fig. 4(b).
Although coherent precursor waves and wakefield exist up-stream, they cause only little energy transfer from ions toelectrons. Nevertheless, superadiabatic particle heating isobserved at the shock, and some supra-thermal electronsare produced. On account of weak electron-ion coupling inthe upstream region, the ions enter the shock with muchlarger energy than do electrons, reach deeper in the shock- compressed magnetic field, and cause charge separation pro-ducing electrostatic field in upstream direction. With out-of-plane field, B = B z , particle motion is constrained in the x − y plane, and so the cross-shock electric potential cannotprovide electron heating parallel to B . In this section we an-alyze particle trajectories to study electron energization atthe shock.Fig. 5 illustrates the main stages of electron energizationas they cross the shock. The particles are selected in the far-upstream region at the same x -location, and they approachthe shock ramp at the same time. Some trajectories are shownwith thin grey lines in panels (a)-(c), and a single electronin each panel is highlighted in red. The temporal evolutionof the kinetic energy averaged over all traced particles, notonly those visible in panels (a) to (c), is shown in red in panel(d), where we also plot the normalized work done by the E x (green line) and E y (blue line) electric-field components,( γ − E i = − em e c (cid:68) (cid:90) tt dt (cid:48) v i ( t (cid:48) ) E i ( x ( t (cid:48) )) (cid:69) , (3)where i = x, y . The local electric field is measured at the MNRAS000
Although coherent precursor waves and wakefield exist up-stream, they cause only little energy transfer from ions toelectrons. Nevertheless, superadiabatic particle heating isobserved at the shock, and some supra-thermal electronsare produced. On account of weak electron-ion coupling inthe upstream region, the ions enter the shock with muchlarger energy than do electrons, reach deeper in the shock- compressed magnetic field, and cause charge separation pro-ducing electrostatic field in upstream direction. With out-of-plane field, B = B z , particle motion is constrained in the x − y plane, and so the cross-shock electric potential cannotprovide electron heating parallel to B . In this section we an-alyze particle trajectories to study electron energization atthe shock.Fig. 5 illustrates the main stages of electron energizationas they cross the shock. The particles are selected in the far-upstream region at the same x -location, and they approachthe shock ramp at the same time. Some trajectories are shownwith thin grey lines in panels (a)-(c), and a single electronin each panel is highlighted in red. The temporal evolutionof the kinetic energy averaged over all traced particles, notonly those visible in panels (a) to (c), is shown in red in panel(d), where we also plot the normalized work done by the E x (green line) and E y (blue line) electric-field components,( γ − E i = − em e c (cid:68) (cid:90) tt dt (cid:48) v i ( t (cid:48) ) E i ( x ( t (cid:48) )) (cid:69) , (3)where i = x, y . The local electric field is measured at the MNRAS000 , 1–11 (2020)
Ligorini et al.
Figure 6.
Trajectory of a highly energetic electron overlaid on alinear map of B z magnetic field (a). The map corresponds to time t Ω ci = 23 .
6, when the particle first interacts with the shock front.Panel (b) shows the evolution of the total kinetic energy and theelectric work (cf. Eq. 3) as in Fig. 5, but here for a single electron.The black vertical line at time t (cid:39) . particle position and t Ω ci = 20. Finally, in panel (e) weplot in red the average of the magnetic moment, µ , and the B z profile (blue line) along the particle trajectories. Here, µ = p ⊥ / (2 m e | B p | ), where p ⊥ is the transverse momentum ofthe particles in the local magnetic field, B p , and µ = µ ( t ).As electrons approaching the shock interact with the wake-field, far upstream most of the particles only oscillate inthe electrostatic field but do not significantly gain energy.Closer to the shock the wakefield is stronger and the elec-trons may become decoupled from the bulk flow (compareSironi & Spitkovsky 2011). SSA kicks in, and the decoupledelectrons are accelerated in the y -direction, which should bevisible as anisotropy in the p ye − x phase-space distribution.To be noted from Fig. 5(a) is that only a small fraction ofparticles are decoupled from the bulk flow and show small-amplitude oscillations in their trajectories.A significant impact of the upstream waves can be observedjust in front of the shock. Fig. 5(a) shows strong waves emit-ted by a shock ripple that form an arc-like feature in the B z distribution and corresponding E x and E y components (notshown). The E x wave field is strong enough to effectively stopan electron and decouple it from the bulk flow. This causeswiggles in the trajectory, well visible for the highlighted elec-tron at ( x/λ si , y/λ si ) ≈ (33 . , .
2) and the particles below itat y/λ si ≈ . − y/λ si ≈ . − .
4, af-fected by the upper ripple. Note, that at time t a the affectedelectrons have already passed through the waves. After de-coupling, the electrons gain significant energy by the E x fieldand to a lesser degree by the motional E y field. The strongelectrostatic E x field is a combination of the strong wakefieldclose to the ripple and the standard cross-shock field. Thejump in the average magnetic moment (panel (e)) indicatesthat energization at this stage is nonadiabatic.This scenario of electron acceleration close to the shock is acrucial and necessary step for their subsequent energization. It fully relies on strong rippling of our mildly relativistic shockand is, to our knowledge, described here for the first time.Deeper in the shock, electrons are adiabatically heatedwhile they undergo E × B drift in the − E x B z ˆ y directionwith the same speed as the shock ripples. The combination ofpenetration and drifting of ions causes the charge-separationelectric field in the ripples whose components have ampli-tudes E x ≈ E y (cid:29) E . In the simulation frame the electronsare dragged together with the ripples and are accelerated bythe E y field and decelerated by the E x field. Hence the com-ponents of work ( γ − E x and ( γ − E y diverge at t (cid:38) t a .The drifting of electrons is evident in Fig. 5(b).Obviously, the behavior of individual particles may differfrom that of the average particle. Electrons that decoupledfrom the bulk flow in the upstream region and started gy-rating there may hit the shock with unfavorably low energy.After adiabatic heating at the shock they will be advecteddownstream and form the thermal pool together with elec-trons that passed through the shock without interacting withthe ripple structures.Having non-adiabatically gained energy around time t a ,the average electron resides at the overshoot, and its furtherenergy evolution is predominantly adiabatic. Fig. 5 illustratesthat around time t b the magnetic moment is conserved whilethe magnetic field becomes weaker, which results in a grad-ual energy loss due to decompression. Later, around t ≈ t c ,the mean electron energy again increases, associated with asecond jump of the magnetic moment. The particle trajec-tories in panel (c) show that the gyroradii of some electronsare comparable to the scale of the turbulent field, allowingresonant scattering off these waves. Essentially all electronswith Lorentz factor γ e (cid:38)
10 can gain energy in this way andpopulate the high-energy wing of the spectrum downstream.At times t > t c the average electron energy saturates, prob-ably for want of large-scale turbulence with which they couldinteract. The turbulence exists only in a narrow zone down-stream of the overshoot, and its scale is commensurate withthe ripple wavelength.The features discussed here for the average electron can ex-plain the downstream electron spectrum as shaped by a singlepopulation of particles that was energized in the shock. Theparticles, that decoupled from the bulk flow in the upstreamregion and were energized through ripple-mediated processes,can experience further acceleration in resonant interactionswith turbulence. A consequence is the smooth transition be-tween the low-energy and high-energy component in the elec-tron spectrum. Some energization at the shock can be de-scribed as a stochastic second-order Fermi process. An exam-ple is given by the particle shown in Fig. 6. After decouplingfrom the flow this electron experienced multiple instances ofinelastic scattering in the shock transition, gaining energy onaverage and finally reaching γ e ≈
80. Such particles form thesupra-thermal power-law portion of the electron spectrum.Most of the scattering is less successful, and the electronsare in the bulk of the population. In our mildly relativisticmagnetized shock, the layer of strong electromagnetic turbu-lence is relatively narrow, and stochastic scattering does notprovide significant energy gain to the average electron.Double-Maxwellian downstream electron distributionswere reported before for ultrarelativistic superluminal shocks(Sironi & Spitkovsky 2011). However, their high-energy spec-tral component results from energization through wakefield
MNRAS , 1–11 (2020) ildly relativistic shocks in e − p plasmas II Figure 7.
Phase-space distributions of electrons and ions at time t Ω ci = 84 . in the shock upstream, that is not efficient in our mildly rel-ativistic shock. Here, the formation of the hotter Maxwellianand its supra-thermal component entirely results from elec-tron interactions with turbulence at the shock and down-stream, that is related to the shock ripples. ϕ B = 0 o For the in-plane case the phase-space distributions of elec-trons and ions and the mean kinetic-energy profiles presentedin Fig. 7 are very similar to those for the simulation with ϕ B = 90 o . The amplitudes of the precursor waves and thewakefield are compatible in both runs, and so essentially thesame physical mechanisms of wave-particle interaction op-erate. In particular, electrons can be accelerated by strongwakefield propagating toward the shock, reach energies inexcess of γ e ∼
20, and develop anisotropy in the p xe distri-bution. SSA works as well, although the corresponding p ze anisotropy along the motional electric field cannot be ob-served in our 2D simulation. The bulk energy gain of elec-trons before they reach the shock is again about 5% of theirinitial kinetic energy, and the electrons and ions are far fromenergy equipartition.Fig. 8 shows downstream particle spectra. As with out-of-plane magnetic field, the ions are in the process of ther-malization, and the reflected particles undergo SDA at theshock. The electron distribution is close to a combination oftwo 3D Maxwellians with moderately different temperature, Figure 8.
Spectra of electrons in red and ions in blue downstreamof the shock ( x/λ si = 87 −
97 at x sh /λ si ≈ t Ω ci = 84 . Figure 9.
Comparison of downstream electron spectra in 2D (blueline for ϕ B = 90 o , red line for ϕ B = 0 o ) and 1D simulations(orange line). The spectra are calculated in a region 5 λ si − λ si downstream of the shock front. each described by dNdγ ∝ γ (cid:112) γ − (cid:18) − m e c kT e γ (cid:19) . (4)A single 3D Maxwellian provides a slightly worse fit. In-set (b) of Fig. 8 features a weak supra-thermal componentwith spectral index p ≈
2, similar to that in the out-of-planesimulation.Downstream of the shock electrons and ions are far fromenergy equipartition, as in the setup with ϕ B = 90 o . To facil-itate a direct comparison, in Fig. 9 we once again show theelectron spectra for both 2D runs and also for a 1D simu-lation. With in-plane field, the downstream electrons carry12 .
6% of the initial ion kinetic energy, the ions account for77 . MNRAS000
6% of the initial ion kinetic energy, the ions account for77 . MNRAS000 , 1–11 (2020)
Ligorini et al.
Figure 10.
Trajectories of electrons (a-c), the evolution of the average kinetic energy and electric work (d), and the magnetic moment andthe magnetic-field profile (e) in the simulation with ϕ B = 0 o . The format is the same as in Fig. 5, except that the normalized magnitudeof the magnetic field, | B | /B , is shown in panels (a-c) and (e), and the acceleration rates are split into components parallel (blue line)and perpendicular (green line) to the local magnetic field. For ( γ − E par we also show its y -component in orange. The inset in panel(b) shows zoom-in of the fields configuration in a region marked with a square box, where red arrows present the in-plane electric field,black lines the contours of the A z -component of the vector potential (displaying the in-plane magnetic field lines), and the green thickline shows the region with strong electric field parallel to the magnetic field. case, but comparable in both 2D runs. In 1D simulations thecoupling is slightly stronger, the electrons gain 16 ,
8% of theenergy, which is still well below equipartition.
As for the out-of-plane run in Section 4.1.3, we describe elec-tron heating and acceleration by analyzing the behaviour oftraced particles. Fig. 10 illustrates the main electron ener-gization phases in a format similar to that in Fig. 5. We splitthe work done by the electric field into components paralleland perpendicular to the local magnetic field (see caption ofFig. 10).Qualitatively, with in-plane B the electron energizationproceeds as described for ϕ B = 90 o . Generally, particles thatlater gain significant energy are decoupled from the bulk flowwhen interacting with large-amplitude waves emitted by theripples. An initial energy gain at time t = t a is providedby the perpendicular electric field components, E x and the motional electric field, E z . The magnetic moment increases.Note, that magnetic gyration proceeds in the x − z plane, andthe E × B drift motion due to the E x field is normal to thesimulation plane and hence not visible.At the shock the particles are adiabatically heated, and themagnetic moment remains roughly constant until the elec-trons come close to the overshoot at time t b (Fig. 10b). Thenthe magnetic moment starts to increase, and particles signifi-cantly gain energy as work of the y -component of the parallelelectric field. The inset in panel (b) shows that magnetic-field-aligned electric fields exist everywhere along the mag-netic overshoot. Their structure in the larger context of theovershoot is shown in Fig. 11.At the overshoot the density of ions is much larger thanthat of electrons, on account of ion deceleration duringreflection in the shock-compressed magnetic field. Charge-separation causes a strong electrostatic field associated withthe cross-shock potential that reflects the ions and pulls theelectrons toward downstream. In a laminar shock the cross- MNRAS , 1–11 (2020) ildly relativistic shocks in e − p plasmas II Figure 11.
Normalized charge density (left panel) and the electric-field component aligned with the magnetic field (right panel) attime t Ω ce = 24 . E par has a substantial amplitude. shock electric field is aligned with the shock normal, i.e.,lies in x -direction in our setup and flips sign at the over-shoot. The red arrows in Fig. 11 in principle conform withthis picture, but it is also evident that the electric field fol-lows a complex set of thin and highly warped filaments. Thecharge-separation field thus acquires a component along themagnetic field, that is largely in the y -direction. The red tra-jectory in Fig. 10(b) gives an example of an electron thatmoves freely along the magnetic field and is efficiently accel-erated by the aligned electric field.Electrostatic field perpendicular to the shock normal isquite common in nonrelativistic systems. Electrons that areaccelerated by the cross-shock potential can excite the two-stream instabilities (Thomsen et al. 1983; Goodrich & Scud-der 1984) and may do so here. Fig. 11 demonstrates modu-lation of the overshoot structure by the shock ripples. Thepresence of modulations at scales of the ion skin depth andsmaller suggests that other instabilities may also operate inthe overshoot. Disentangling the coupling between these un-stable modes is difficult and beyond the scope of this paper.We do note that the role of the E (cid:107) field is observable onlywith the in-plane magnetic-field configuration. The strengthof this electric field component is much larger than that typi-cally observed at nonrelativistic shocks and may be specific tomildly relativistic and magnetized shocks. After crossing theovershoot the average kinetic energy of the particles and themagnetic moment keep increasing, despite the magnetic-fielddecompression. As with out-of-plane B , this nonadiabaticacceleration is due to gyroresonant or stochastic scatteringof electrons off downstream turbulence created at shock frontripples (Fig. 10c). Note, that magnetic-field aligned electricfield persists in an extended region past the overshoot. This work is the second of two articles in which we usePIC simulations to investigate mildly-relativistic superlumi-nal magnetized shocks in electron-ion plasma. Paper I de-scribed the electromagnetic shock structure, plasma instabil- ities, and waves. Here we discussed particle acceleration andheating.It has been suggested that wakefield acceleration at ultra-relativistic shocks may account for the production of highlyenergetic particles through nonlinear collapse of the waves(Lyubarsky 2006; Hoshino 2008; Iwamoto et al. 2017, 2018,2019). At mildly relativistic shocks, shock ripples at timesgenerate strong precursor waves that lead to nonlinear wake-field amplitudes. Then nonlinear FRS is triggered, produc-ing downstream-propagating wakefield that accelerates elec-trons through the phase-slippage effect combined with SSA(Hoshino 2008). However, the WFA is less efficient than inthe ultrarelativistic regime, and only few particles reach thebulk energy of upstream ions, much less exceed it. Conse-quently, and in contrast to ultrarelativistic shocks, electronsand ions do not reach equipartition by the time they arriveat the shock.The ion-to-electron energy transfer is comparable for in-plane and out-of-plane magnetic field. In both cases it isfar below equipartition, with electrons on average carrying11% −
13% of the initial ion kinetic energy or six times theirown initial energy. Going beyond adiabatic compression atthe shock, most of the energy transfer takes place at theshock and immediately downstream. The downstream elec-tron spectra are close to thermal distributions with smallsupra-thermal components. Shock rippling is crucial for elec-tron energization. Strong precursor waves emitted by theshock ripples excite wakefield that decouples particles fromthe bulk flow. They are then further accelerated in the elec-trostatic field at the shock.With the out-of-plane magnetic field, electrons crossingthe shock ramp experience adiabatic heating in the over-shoot. Particles that were significantly accelerated at therippled shock front now have gyro-radii comparable to scaleof the downstream turbulence, and they can gain more en-ergy through resonant wave-particle interactions. Stochasticsecond-order Fermi-like scattering also provides some parti-cles with supra-thermal energies. Similar processes shape thedownstream energy spectra for the in-plane magnetic fieldscenario. In addition, significant energization happens at theovershoot, where charge-separation associated with the cross-shock potential and modified by the shock ripples reorganizesthe electric field into a complex structure of warped filaments.Electric and magnetic fields partially align, leading to efficientelectron acceleration.We argued in Paper I that SMI-generated precursor wavesshould persist in realistic 3D systems and generate wakefield.We expected that the precursor waves have similar strengthin 3D and in 2D, and that shock rippling operates to amplifythem. This rippling operates on small scales and is not to beconfused with MHD-scale rippling that results from fluctua-tions in the upstream medium (Lemoine, Ramos, & Gremil-let 2016). Similar conclusions should apply to the particle-energization efficiency. For both magnetic-field configurationswe observe similar processes that heat and accelerate elec-trons and thermalize the ions, although for the in-plane con-figuration in 2D cross-field diffusion is suppressed (Jokipii,Kota, & Giacalone 1993). The coherence of the precursorwaves is likely not higher in 3D than in lower-dimensionalsimulations. We conclude that the particle-acceleration effi-ciency is expected to be at most the same in 3D as it isin 2D simulations, and we do not anticipate a stronger ion-
MNRAS000
MNRAS000 , 1–11 (2020) Ligorini et al. electron coupling in three-dimensional studies of ion-electronshocks. At ultrarelativistic superluminal shocks, in which theenergy transfer occurs in the turbulent precursor, the level ofcoupling increases with the shock obliquity as the strengthof the precursor wave is larger for obliquities approaching astrictly perpendicular configuration (Lyubarsky 2006; Sironi& Spitkovsky 2011). Future studies may explore whethershock rippling can enhance the coupling at not strictly per-pendicular mildly relativistic shocks.The absence of energy equipartition between ions and elec-trons has important astrophysical implications. The presenceof ions in blazar jets appears to be energetically required(Celotti & Ghisellini 2008). It is evident that high-energyemission from blazars requires a particle energy much in ex-cess of that seen in our simulation. A second-stage accelera-tion to TeV-scale energies is required, whatever the type ofradiating particle. Diffusive shock acceleration at mildly rel-ativistic shocks is a possible process, because it can extracta large fraction of the jet energy flux and the post-shockplasma can be reasonably well confined (Marcowith et al.2016; Pelletier et al. 2017). In blazar jets internal shocks aremagnetized and likely superluminal though, which stronglylimits the efficiency of shock acceleration and suggests thatother processes may be at play. Our simulations suggest thatWFA is not one of them, at least for mildly relativistic sys-tems. Weakly magnetized ultrarelativistic shocks can also notaccelerate to very high energies (Reville & Bell 2014). It ispossible that shocks in blazar jets do not produce the high-energy particles that produce their X-ray and gamma-rayemission, and instead turbulence provides stochastic acceler-ation in a larger volume (e.g. Chen, Pohl, & B¨ottcher 2015;Chen et al. 2016).The pre-acceleration at the shock provides the initial con-dition for the subsequent acceleration to the TeV scale. It alsoshapes the electron spectra below the GeV scale, whose obser-vational consequences can be used to make inferences aboutelectron-ion coupling in blazar jets. If there were equiparti-tion between ions and electrons, one would find few electronswith Lorentz factor γ e (cid:46) . Indeed, internal-shock modelsfor blazar jets tend to require electron-ion equipartition toreproduce blazars SEDs (see, e.g., Spada et al. 2001). It isalso implicitely called for in the radiation modelling, whereit appears as minimum Lorentz factor of electrons (Dermeret al. 2009; Archambault et al. 2016), otherwise the modelspredict too much flux in X-rays and soft gamma rays. Thereis explicit evidence for a low-energy cut-off at γ ≈ in theelectron spectrum in the lobes of a radio galaxy (Blundell etal. 2006).Leptonic blazar models are based on either external Comp-ton scattering or the internal synchrotron-self-Compton pro-cess, or a combination of the two. External Compton scat-tering with strong electron-ion coupling, i.e. γ (cid:38) , wouldgive X-ray spectra that are much harder than observed (see,e.g., Sikora et al. 2013). Better agreement is reached, if theelectrons would have cooled to low energies, resulting in alow-energy spectral tail ∝ γ − , that would produce consis-tent X-ray spectra by synchrotron-self-Compton scattering.The energy loss time of γ = 100 electrons in blazar jets is onthe order of years, and so cooling tails would not be expectedduring flares, but rather for baseline conditions. Our simula-tions suggest that there is a large population of low-energyelectrons on account of limited electron-ion coupling, from which particles are accelerated to very high energies. Thenthe low-energy part of the electron spectrum would not needcooling to form, and the observed X-ray spectra would benaturally explained.Recently, a high-energy neutrino event was observed inassociation with the flaring blazar TXS 0506+056 (Ansoldiet al. 2018). This result requires the presence of ultrarela-tivistic protons in these sources and seems to exclude purepair plasma in the jets, at least for this particular source.In mixed plasma composed of ions, electrons, and positronsan additional process may operate that accelerates electronsand positrons. Reflected ions gyrating at the shock emit left-handed elliptically polarized magnetosonic waves with fre-quencies that are harmonics of the ion cyclotron frequency(the ion SMI). If the spectrum of the ion-emitted waves hasenough power at high harmonics at and above the cyclotronfrequencies of positrons and electrons, then positrons travers-ing the shock can be efficiently accelerated to non-thermalenergies through resonant relativistic synchrotron absorptionof the magnetosonic waves (Hoshino & Arons 1991; Hoshinoet al. 1992). Non-thermal electrons are also created, but notas efficiently on account of the left-handed polarization of theion waves (Amato & Arons 2006; Stockem et al. 2012). Newhigh-resolution and large-scale PIC simulations are necessaryto investigate in 2D this scenario for mildly relativistic shocksin ion-pair plasmas. ACKNOWLEDGEMENTS
J.N. acknowledges inspiring discussions with Marek Sikora.This work has been supported by Narodowe Centrum Naukithrough research projects DEC-2013/10/E/ST9/00662(A.L., J.N., O.K.), UMO-2016/22/E/ST9/00061 (O.K.) and2019/33/B/ST9/02569 (J.N.). This research was supportedby PLGrid Infrastructure. Numerical experiments wereconducted on the Prometheus system at ACC CyfronetAGH. This work was supported by JSPS-PAN BilateralJoint Research Project Grant Number 180500000671. Partof the numerical work was conducted on resources providedby the North-German Supercomputing Alliance (HLRN)under projects bbp00003, bbp00014, and bbp00033.
DATA AVAILABILITY
The data underlying this article will be shared on reasonablerequest to the corresponding author.
REFERENCES
Abdalla H. et al., 2019, Nature, 575, 464Acciari V. A. et al., 2019, Nature, 575, 455Acciari V. A. et al., 2019, Nature, 575, 459Amato E., Arons J., 2006, ApJ, 653, 325Ansoldi S. et al., 2018, ApJ, 863, L10Archambault S. et al., 2016, MNRAS, 461, 202Babul, A.-N. & Sironi, L. 2020, MNRAS, 499, 2884Begelman M. C., Kirk J. G., 1990, ApJ, 353, 66Blundell K. M., Fabian A. C., Crawford C. S., Erlund M. C.,Celotti A., 2006, ApJ, 644, L13MNRAS , 1–11 (2020) ildly relativistic shocks in e − p plasmas II Buneman, O., 1993, in Computer Space Plasma Physics: Simula-tion Techniques and Software, pag.67-84 Terra Scientific Pub-lishing Company (TERRAPUB), TokyoBurgess D., Scholer M., 2007, Phys. Plasmas, 14, 012108Celotti A., Ghisellini G., 2008, MNRAS, 385, 283Chen X., Pohl M., B¨ottcher M., 2015, MNRAS, 447, 530Chen X., Pohl M., B¨ottcher M., Gao S., 2016, MNRAS, 458, 3260Dermer C. D., Finke J. D., Krug H., B¨ottcher M., 2009, ApJ, 692,32Drake J. F., Kaw P. K., Lee Y. C., Schmid G., Liu C. S., Rosen-bluth M. N., 1974, Phys. Fluids, 17, 778Gallant Y. A., Hoshino M., Langdon A. B., Arons J., Max C. E.,1992, ApJ, 391, 73Ghisellini G., Tavecchio F., Chiaberge M., 2005, A&A, 432, 401Goodrich C. C., Scudder J. D., 1984, J. Geophys. Res., 89, 6654Hoshino M., Arons J., 1991, Phys. Fluids B, 3, 818Hoshino M., 2008, ApJ, 672, 940Hoshino M., Arons J., Gallant Y. A., Langdon A. B., 1992, ApJ,390, 454Aartsen M. G. et al., 2018, Science, 361, eaat1378Iwamoto M., Amano T., Hoshino M., Matsumoto Y., 2017, ApJ,840, 52Iwamoto M., Amano T., Hoshino M., Matsumoto Y., 2018, ApJ,858, 93Iwamoto M., Amano T., Hoshino M., Matsumoto Y., Niemiec J.,Ligorini A., Kobzar O., Pohl, M., 2019, ApJ, 883, L35Jokipii J. R., Kota J., Giacalone J., 1993, Geophys. Res. Lett., 20,1759Kaw P., Schmidt G., Wilcox T., 1973, Phys. Fluids, 16, 1522Kruer W. L., 1988, The physics of laser plasma interactions, Read-ing, MA, Addison-Wesley Publishing Co.Kuramitsu Y., Sakawa Y., Kato T., Takabe H., Hoshino M., 2008,ApJ, 682, L113Langdon A. B., Arons J., Max C. E., 1988, Phys. Rev. Lett., 61,779Lemoine M., Ramos O., Gremillet L., 2016, ApJ, 827, 44Ligorini A., Niemiec J., Kobzar O., Iwamoto M., Bohdan A., PohlM., Matsumoto Y., et al., 2021, MNRAS, 501, 4837Lyubarsky Y., 2006, ApJ, 652, 1297Marcowith A., Bret A., Bykov A., Dieckman M. E., O’C Drury L.,Lemb`ege B., Lemoine M., et al., 2016, Rep. Prog. Phys., 79,046901Martins S. F., Fonseca R. A., Silva L. O., Mori W. B., 2009, ApJ,695, L189Niemiec J., Ostrowski M., Pohl M., 2006, ApJ, 650, 1020Niemiec J., Pohl M., Stroman T., Nishikawa K.-I., 2008, ApJ, 684,1174Pelletier G., Bykov A., Ellison D., Lemoine M., 2017, Space Sci.Rev., 207, 319Plotnikov I., Grassi A., Grech M., 2018, MNRAS, 477, 5238Plotnikov I., Sironi L., 2019, MNRAS, 485, 3816Pohl M., Hoshino M., Niemiec J., 2020, Prog. Part. Nucl. Phys.,111, 103751Reville B., Bell A. R., 2014, MNRAS, 439, 2050Sikora M., Janiak M., Nalewajko K., Madejski G. M., ModerskiR., 2013, ApJ, 779, 68Sironi L., Spitkovsky A., 2009, ApJ, 698, 1523Sironi L., Spitkovsky A., 2011, ApJ, 726, 75Spada M., Ghisellini G., Lazzati D., Celotti A., 2001, MNRAS,325, 1559Stockem A., Fi´uza F., Fonseca R. A., Silva L. O., 2012, ApJ, 755,68Tajima T., Dawson J. M., 1979, Phys. Rev. Lett., 43, 267Thomsen M. F., Barr H. C., Gary S. P., Feldman W. C., ColeT. E., 1983, J. Geophys. Res., 88, 3035 This paper has been typeset from a TEX/L A TEX file prepared bythe author. MNRAS000