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Can we trust MHD jump conditions for collisionless shocks?
Antoine Bret
1, 2 ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain Instituto de Investigaciones Energ´eticas y Aplicaciones Industriales,Campus Universitario de Ciudad Real, 13071 Ciudad Real, Spain
Submitted to ApJABSTRACTWhen applied to compute the density jump of a shock, the standard magnetohydrodynamic (MHD)formalism assumes, 1) that all the upstream material passes downstream, together with the momentumand energy it carries, and 2) that pressures are isotropic. In a collisionless shock, shock acceleratedparticles going back and forth around the front can invalid the first assumption. In addition, anexternal magnetic field can sustain stable pressure anisotropies, invaliding the second assumption. Itis therefore unclear whether the density jump of a collisionless shock fulfils the MHD jump or not.Here we try to clarify this issue. A literature review is conducted on 68 articles dealing with Particle-In-Cell simulations of collisionless shocks. We analyze the factors triggering departure from the MHDdensity jump and quantify their influence on ∆ RH , the relative departure from the Rankine-Hugoniotjump. For small departures we propose ∆ RH = + O (10 − − . κ ) t κ − σ O (1) where t is the timescaleof the simulation, σ the magnetization parameter and κ a constant of order unity. The first termstems from the energy leakage into accelerated particle. The second term stems from the downstreamanisotropy triggered by the field (assuming an isotropic upstream). This relation allows to assess towhich extent a collisionless shock fulfils the RH density jump.In the strong field limit and for parallel shocks, the departure caused by the field saturates at a finite,negative, value. For perpendicular shocks, the departure goes to zero at small and high σ ’s so that wefind here a departure window. The results obtained have to be checked against full 3D simulations. Keywords:
Shock waves — MHD INTRODUCTIONSince their discovery during the 19th century(Johnson & Cheret 1998; Salas 2007), shockwaves havebeen the object of innumerable investigations. The fluidequations first used to describe them operate under theassumption that the mean-free-path of the particles ismuch smaller than any other dimension of the systemunder scrutiny. With such a prominent role given to bi-nary collision to randomize the flow at the microscopiclevel, it is reasonable to assume 1) that the pressure isisotropic in both the upstream and the downstream and2) that all the matter upstream goes downstream, to-
Corresponding author: Antoine [email protected] gether with the energy and the momentum it carries .The second assumption allows to apply the conservationlaws between the upstream and the downstream, whilethe first assumption allows to write these laws using fluidmechanics or magnetohydrodynamics (MHD) equations.From there, one derives the jump conditions for the den-sity, pressure, magnetic field, etc. (Fitzpatrick 2014;Goedbloed et al. 2019).Contrary to fluid shockwaves, were dissipation atthe shock front is provided by binary collisions,collisionless shockwaves are mediated by collectiveplasma effects on length scales much shorter thanthe mean-free-path (Sagdeev 1966; Tidman et al. 1971; Radiative shocks (Zel’dovich & Raizer 2002;Mihalas & Weibel-Mihalas 1999) are excluded from the discus-sion.
Bret
Balogh & Treumann 2013). A good example is the earthbow-shock in the solar wind, where the shock front isabout 100 km thick while the mean-free-path at thesame location is of the order of the sun-earth distance(Bale et al. 2003; Schwartz et al. 2011).In the absence of binary collisions to isotropize theflow, to which extent can we assume isotropic pressures?Also, given the mean-free-path is much larger than theshock front, to which extent can we assume all the mat-ter upstream goes downstream, together with the mo-mentum and energy it carries? Indeed, it turns out thatthese two assumptions are far from obvious in a collision-less environment. As a consequence, it is not obviouseither that the fluid or MHD jump conditions derivedfor a collisional fluid, are still valid.Note that we hereafter refer to MHD jump condi-tions derived considering isotropic pressures. Several au-thors adapted them to the case of anisotropic pressures,considering the anisotropy degree as a free parameter(Karimabadi et al. 1995; Erkaev et al. 2000; Vogl et al.2001; Gerbig & Schlickeiser 2011). Yet, the goal of thepresent paper is to compare jump conditions (mainly thedensity jump) of collisionless shocks with the simple andwell known “isotropic MHD” jump conditions, also fre-quently referred to as “Rankine-Hugoniot” (RH) jumpconditions, even though William Rankine and PierreHugoniot derived these relations for a neutral fluid.In the sequel, we shall use interchangeably “isotropicMHD”, “MHD” or “RH”.Two processes have been identified that can trigger anon-RH density jump, • An external magnetic field B can sustain stableanisotropies, breaking the isotropy assumption of MHD.Its strength is characterized by the σ parameter, σ = B / π ( γ − n ( P i m i ) c , (1)where n and γ are respectively the upstream densityand Lorentz factor (measured in the downstream frame).The m i ’s are the masses of the species composing theplasma. • As they accelerate particles, collisionless shocks gen-erate a population which goes back and forth around thefront, breaking the “everything upstream goes down-stream” assumption. As we shall see in Section 3.2, theprocess can be characterized by the parameter, α = F E n v , (2)where F E is the energy fluxes escaping the Rankine-Hugoniot budget and v the upstream velocity. Particle-in-cell (PIC) simulations are undoubtedly thetool par excellence to study non-linear collisionless phe-nomena like collisionless shocks. Because they operatefrom first principles at the microscopic level, they are in-herently kinetic. We therefore present in Section 2 a lit-erature review of PIC simulations of collisionless shocks,magnetized or not, in pair or electron/ion plasmas. Theobservations gathered will then feed Section 3 where de-partures from MHD density jump are modelled.Most of the simulations found in literature use the“reflecting wall” technique to produce a shock. There,a semi-infinite plasma is sent against a reflecting wallwhere it bounces back to interact with itself. Thepresent work focuses on this technique. In this reflect-ing scheme, the simulations are therefore performed inthe downstream frame of the formed shock. By design,such a scheme can only simulate shocks formed by theencounter of 2 identical plasmas.Noteworthily, the less represented “injection method”allows to study shocks produced by the collision of any2 kinds of plasmas (different compositions and/or dif-ferent densities). Shocks arising from the interactionof a jet with a standing plasma can be studied withthis scheme. For example Nishikawa et al. (2009) couldstudy the interaction of a diluted relativistic pair jetwith a unmagnetized pair plasma. While many “re-flecting wall papers” studied shocks in pair plasmas (seeSections 2.1 and 2.2.1), a density ratio different fromunity (Nishikawa et al. (2009) has 0.676) is only achiev-able with the injection method. Still with the injec-tion method, Ardaneh et al. (2016) studied the inter-action of an electron jet with an unmagnetized elec-tron/ion plasma, and commented on the differences be-tween the reflected and injected schemes. Magnetizedsystems have also been explored, with Dieckmann et al.(2019), for example, considering a pair jet colliding withan electron-proton plasma over a guiding magnetic field.As is appears, the injection method truly allows foran extensive exploration of the possible shocks. The re-flected wall scheme restricts the dimension of the param-eters phase space, and to date counts with more studies,which is why we here focus on it. Yet, it would be in-teresting to extend the current analysis to the injectionscheme.Defining now the density ratio between the shock up-stream (subscript “1”) and downstream (subscript “2”)like, r = n n , (3)we shall model ∆ RH , the relative departure from theRH jump r RH , defined by,∆ RH ( σ, α ) ≡ r − r RH r RH . (4) an we trust MHD jump conditions for collisionless shocks? LITERATURE REVIEWWe conducted a literature review of PIC simula-tions of collisionless shocks. We selected 68 articleswhere 1) a shock structure was clearly obtained, witha downstream significantly longer than the overshootregion, if any, right behind the front, and 2) the den-sity jump can be related to its MHD counterpart witha reasonable accuracy, whether explicitly or implicitly.The medium where the shock propagates is homoge-nous (see Tomita et al. (2019) for an inhomogeneouscase). Save a few exceptions like Sironi & Spitkovsky(2009a); Stockem et al. (2012); Plotnikov et al. (2018);Guo et al. (2018), the density jump was not explicitlycompared to its MHD counterpart, for such was not themain goal of the article. It is then possible that a fewpercent discrepancy between the 2 went unnoticed forsome articles. 2.1.
Un-magnetized shocks
All the articles examined but Keshet et al. (2009);Stockem et al. (2012) pertaining to the un-magnetizedregime, relativistic or not, display a shock in agreementwith the MHD requirements.
For shocks in pair plasmas (Spitkovsky 2005; Kato2007; Chang et al. 2008; Spitkovsky 2008a; Keshet et al.2009; Sironi & Spitkovsky 2009b; Bret et al. 2013,2014; Dieckmann et al. 2016; Dieckmann & Bret 2017;Li et al. 2017; Dieckmann & Bret 2018; Pelletier et al.2019; Lemoine et al. 2019c,b,a; Vanthieghem et al.2020), the longest simulation (Keshet et al. 2009) wasran up to 11925 ω − p , where ω p is the electronic plasmafrequency.In Stockem et al. (2012) the authors carefully mea-sured the departure from MHD, as the very goal ofthe paper was to “assess the impact of non-thermallyshock-accelerated particles on the MHD jump conditionsof relativistic shocks”. Pushing the simulation up to2395 ω − p , a +7% departure for the density jump wasfound.Although Keshet et al. (2009) did not precisely mea-sured the density jump, Figure 1 of their article shows aMHD jump at 2250 ω − p and a slight departure (+3 . ω − p due to energy leakage in accelerated parti-cles (see Section 3.2). For shocks in electron/ion plasmas (Spitkovsky2008b; Kato & Takabe 2008; Martins et al. 2009;Dieckmann et al. 2010b; Niemiec et al. 2012; (cid:1) (cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1) (cid:1) (cid:1) [ ] (cid:1) (cid:1) (cid:1)(cid:1) (cid:1) (cid:1) [ ] (cid:1) (cid:1) (cid:1)(cid:1) (cid:1) (cid:1) [ ] [ ][ ] - - (cid:2) n ( degrees ) (cid:3) [ ] [ ] [ ] [ ] Figure 1.
Values of ∆ RH defined by Eq. (4) for the 6articles dealing with magnetized shocks in pair plasmas, interms of θ n and σ . Red means mean negative departure∆ RH < − positive departure. Green meansthe shock fits the MHD density jump to within 4%. Thelabels correspond to the references: [1] = Spitkovsky (2005),[2] = Sironi & Spitkovsky (2009a), [3] = Bret et al. (2017),[4] = Gallant et al. (1992), [5] = Iwamoto et al. (2017), [6]= Plotnikov et al. (2018). Parametric studies are indicatedby vertical lines. [4,5,6] are all for θ n = 90 ◦ . Fiuza et al. 2012; Stockem et al. 2014a,b; Ruyer et al.2015; Stockem Novo et al. 2015; Ruyer et al. 2017;Naseri et al. 2018; Moreno et al. 2020), the longest sim-ulation time was 4111 ω − pi (Niemiec et al. 2012). Nosignificant departure from the MHD jump was detectedin any article.The case of pair plasmas suggests that departure fromMHD due to accelerated particles requires running thesimulation for several thousands of electronic plasma fre-quencies to be perceptible. In electron/ion plasmas, thistranslate to running the simulation for several thousandsof ionic plasma frequencies. The longest run examinedin this respect was 4111 ω − pi (Niemiec et al. 2012), where ω pi is the ionic plasma frequency. Yet the density jumpis not measured accurately enough . Pushing simula-tions beyond this time scale for non-MHD effects to be-come clear, requires so far Hybrid simulations discussedin Section 2.3.An important feature observed is related to acceler-ated particles. Their effect on the shock is not steady.As specified in Keshet et al. (2009), “simulations donot reach a steady state; rather, an increasing frac-tion of shock energy is transferred to energetic parti-cles and magnetic fields throughout the simulation timedomain”. We shall comment further on this point inSection 3.2. See footnote 13 for more on particle acceleration in Niemiec et al.(2012).
Bret
Table 1.
Simulation results for Refs. [1,4,5,6] of Figure 1.∆
RH,max is the maximum relative deviation from the RHdensity jump (see Eq. 4). This deviation is reached for σ = σ m . γ is the upstream Lorentz factor measured in thedownstream frame.Ref. [1] [4] [5] [6]∆ RH,max −
50% +27% − − σ m > . . . . − γ
15 40 & 10
40 10
Magnetized shocks
In a magnetized media, we have one more source ofdeparture from MHD. There, collisionless shocks canstill accelerate particles which will break the “every-thing upstream goes downstream” MHD assumption.But now in addition, the field can sustain stable pres-sure anisotropies and prompt departures from isotropicMHD. Considering the shock behaviour strongly de-pends on the external field strength and on its orien-tation, the physics of magnetized collisionless shocks isextremely rich.The upstream field strength is measured by the σ parameter defined by Eq. (1). The field orientationis measured by the angle θ n it makes with the shockfront normal. In the non-relativistic regime, this angleis Lorentz invariant in the direction of the shock prop-agation up to order ( v/c ) , where v is the speed of theframe to which the field is transformed. In the relativis-tic regime, a perpendicular or a parallel field remain soin any frame. The only articles mentioned here wherean oblique field is considered in a relativistic setting areSironi & Spitkovsky (2009a, 2011). There, the angle ofthe upstream field with the shock normal is given in thesimulation frame, that is, the downstream frame.2.2.1. Magnetized shocks in pair plasmas
Figure 1 summarizes the results for the 6 articlesfalling into the present category in terms of θ n and σ (see references in Figure 1 caption).In [2] (Sironi & Spitkovsky 2009a) the departure isabout −
3% for θ n ≤ ◦ and goes down to −
13% for45 ◦ .In [3] (Bret et al. 2017), the field is parallel and in-troduces a downstream anisotropy responsible for thedeparture from the MHD jump. For σ = 3, the jumpwas reduced by − .The departure in [1] (Spitkovsky 2005) is directly re-lated to the perpendicular field, and the density jump issaid to saturate at 2 instead of 4 for larger σ ’s.The perpendicular shocks of Refs. [4,5,6] are intrigu-ing. The σ -ranges of MHD departure of [4,5] and [6] donot overlap. [4] (Gallant et al. 1992) finds an increase of the density jump reaching a maximum of +27% for σ = 0 .
1. [5] (Iwamoto et al. 2017) finds a decrease ofthe density jump reaching −
9% for σ = 0 .
3. Finally,[6] (Plotnikov et al. 2018) also finds a decrease reaching −
4% for σ = 2 . − .Table 1 summarises the main features of these works.We shall comment further on these results in Section3.1. 2.2.2. Magnetized shocks in electron/ion plasmas
Here 22 articles were analyzed, from parallel tonormal orientations and σ ’s ranging from 6 . − (Kato & Takabe 2010) to 0.25 (Dieckmann et al.2010a).Figure 2 pictures the results in the ( θ n , σ ) phase space.Besides some PIC simulations performed in Guo et al.(2018), all the simulations fulfilled the RH density jump.As specified earlier, a comparison with RH was not thepoint of some works so that a few percent discrepancymay have escaped the analysis.Guo et al. (2018) did perform a detailed comparisonwith the RH jump for 16 simulations . Discrepan-cies with RH range from -0.6% (run “Ms5beta8”) to-7% (run “Ms3beta8”). Only discrepancies < − σ stemsfrom different values of the upstream parameters β p =16 πn k B T /B (see Section 3.1).In summary all the RH-departure in the examinedarticles come from the field and decrease the densityjump . Jump increase stemming from accelerated par-ticles seem to demand a few 10 Ω − ci (see Section 2.3) tobe observed while the longest simulation in the presentsection was ran up to 559Ω − ci (Fang et al. 2019) whereΩ ci is the ionic cyclotron frequency. See Lichnerowicz (1976) or Kulsrud (2005), Chapter 6, Eq. (36)with B y = 0. Guo et al. (2018) does not measure the field in terms of σ but interms of β p = 16 πn k B T /B . For the purpose of the presentstudy, we compute the σ used in Guo et al. (2018) from the for-mula for the Alfv´enic Mach number M A given below Eq. (4) ofGuo et al. (2018), M A = M s p Γ β p /
2. We then take σ = 1 /M A . See Section 3.1 and Plotnikov et al. (2018) for a discussion of thejump increase in Gallant et al. (1992) an we trust MHD jump conditions for collisionless shocks? (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1)(cid:1)(cid:1)(cid:1) [ ][ ] [ ][ ][ ] [ ][ ] [ (cid:15)(cid:28) ][ ] [ ][ ] (cid:1) [ ] [ ][ ][ ][ ] (cid:62)(cid:21)(cid:64) (cid:1) [ ][ ][ ][ ][ ][ ] (cid:62) ] (cid:1) (cid:1) [ ][ ] (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) [ ] - - (cid:2) ( degrees ) (cid:3) [ (cid:20)(cid:19) ][ (cid:22) ][ (cid:21)(cid:21) ] (cid:1) (cid:62)(cid:20)(cid:26) ] [ (cid:24) ] n Figure 2.
Values of ∆ RH defined by Eq. (4) for the 22 articles dealing with magnetized shocks in electron/ion plasmas, in termsof θ n and σ . The green circle means the shock fits MHD jump condition. The red circle means it does not (departure < − Hybrid results
Hybrid codes treat part of the medium as a fluid,and the rest through the PIC method. In some, thefluid part is the plasma while the PIC part is devotedto accelerated particles (Bai et al. 2015; Casse et al.2018; van Marle et al. 2018). In others, the elec-trons are the fluid part while the ions are dealt withwith PIC (Sugiyama 2011; Gargat´e & Spitkovsky 2012;Guo & Giacalone 2013; Caprioli & Spitkovsky 2014a;Haggerty & Caprioli 2019; Caprioli & Haggerty 2019).The advantage of the method is clearly that it allowsto run the simulations longer at a similar computationalcost. Such a feature is necessary to render the back-reaction of accelerated particles on the shock itself. In-deed, among the articles examined, the only ones whoran the simulations longer than 10 ion cyclotron pe-riods Ω − ci are Hybrid, with the 2 longest simulations,Bai et al. (2015) and Haggerty & Caprioli (2019), push- ing the computation up to 10800Ω − ci and 6000Ω − ci re-spectively.Haggerty & Caprioli (2019) and Caprioli & Spitkovsky(2014a) indicate a downstream Maxwellian reachingonly 80% of the expected MHD temperature after 6000and 2500Ω − ci respectively, due to “energy leakage” intoaccelerated particles. Regarding the density jump, Fig-ure 3 shows its increase in terms of the simulated timelengths for various works. A significant difference (from+0 to +75%) is noticeable between Haggerty & Caprioli(2019) and Caprioli & Haggerty (2019), due to the waythe fluid electrons are modeled. Such is a challenge ofHybrid simulations: giving up a first principles (PIC) The density jump in Haggerty & Caprioli (2019) can be inferredapproximately from its Figure 3 and seems to fit RH. However,it must be somewhat higher, as the downstream temperature islower than its RH value. Due to this uncertainty on the measureddensity jump, this reference is not listed on the present Figure 3.
Bret [ ] [ ] [ ] [ ] [ ] [ ] [ ]
100 500 1000 5000 10 ( (cid:1) ci - ) (cid:2) RH Figure 3.
Values of ∆ RH defined by Eq. (4) in terms of thesimulated time lengths for various Hybrid references. Thelabels correspond to the references: [1] = Guo & Giacalone(2013), [2] = Caprioli & Spitkovsky (2014a), [3] =Gargat´e & Spitkovsky (2012), [4] = Casse et al. (2018), [5]= Caprioli & Haggerty (2019), [6] = Sugiyama (2011), [7] =Bai et al. (2015). description of the electrons is the price to pay for longersimulation times. Much depends then of the fluid clo-sure implemented, as evidenced by these two references. MODELLING OF THE DENSITY JUMPDeviations from the RH density jump observed so farare small (except maybe Caprioli & Haggerty (2019),see Section 2.3). We can therefore devise a first ordermodelling of ∆ RH ( σ, α ), the departure from RH givenby Eq. (4), writing∆ RH ( σ, α ) ∼ α ∂ ∆ RH ∂α + σ ∂ ∆ RH ∂σ , (5)where all derivatives are considered in ( σ, α ) = (0 , α parameter determines the departure from RHdue to accelerated particles. From Eq. (2), we see it isproportional to F E , the escaping energy flux. In turn,it is known that the ability of shocks to accelerate par-ticles depends on their magnetization, the angle of thefield with the shock normal or the sonic Mach num-ber M . In addition, F E is also an increasing functionof time. Therefore, strictly speaking, we should write α = α ( σ, θ n , M , t ). A theory of the density jump ac-counting for all these parameters is out of the scope ofthis work. Yet, among them the time variable is promi-nent since, α ( σ, θ n , M , t = 0) = 0 , ∀ ( σ, θ n , M ) . (6)She shall therefore focus on the time dependence of α ,thus deriving its order of magnitude instead of its pre-cise value. We will elaborate further on the effects ofaccelerate particles in Section 3.2. 3.1. Field effect on the density jump
To which extent can we assume isotropic distributionfunctions in a collisionless plasma? Here it seems rel-evant to single out the magnetized and un-magnetizedcases.
In a un-magnetized plasma , an anisotropic dis-tribution function is Weibel unstable (Weibel 1959;Kalman et al. 1968). Although Weibel’s result wasonly obtained for Maxwellian distribution functions withanisotropic temperatures, it seems reasonable to conjec-ture that any anisotropic distribution function is un-stable (see Silva et al. (2019); Silva (2020) for an efforttoward a mathematical proof). We could also refer tothe ample literature on collisionless “Maxwellianization”(see Bret (2015) and references therein), starting withthe “Langmuir paradox” (Langmuir 1925).Indeed, observations of the solar wind show that inthe small field limit (high β k ), the protons temperaturebecomes isotropic (Bale et al. 2009; Schlickeiser et al.2011; Maruca et al. 2011). It seems therefore than pastthe front turbulence, the downstream anisotropy of acollisionless shock should relax to isotropy on a timescale related to the instability growth rate. The magnetized case is different because a magnetizedVlasov plasma can sustain stable anisotropies (Gary1993). A strong enough field B can therefore maintainan anisotropic upstream and/or an anisotropic down-stream. An isotropic upstream can turn anisotropic asit goes to the downstream depending on the magnetiza-tion parameter σ .How should a residual downstream anisotropy modifythe density jump? A hint can be given by the fact thatthe field tends to reduce the degrees of freedom D of theplasma. It therefore increases its macroscopic adiabaticindex Γ = 1 + 2 /D . And since the RH density jump is adecreasing function of Γ, the presence of the field shouldlower it. This has been seen in the literature review.A quantitative assessment of this reduction impliesdetermining the downstream anisotropy in terms of theupstream properties. As noted earlier, the effect is espe-cially interesting for the parallel case because the MHDjump of a parallel shock is σ -independent. Making anansatz on the kinetic evolution of the plasma throughthe front, Bret & Narayan (2018) derived for a paral-lel shock in a non-relativistic pair plasma (strong shocklimit, upstream Γ = 5 / , r = 12 (cid:16) − σ + p ( σ − σ −
1) + 5 (cid:17) See r + of Eq. (3.5) of Bret & Narayan (2018), with χ = ∞ . an we trust MHD jump conditions for collisionless shocks? (cid:1) - (cid:2) RH Figure 4.
Values of − ∆ RH defined by Eq. (4) in terms of σ , for Guo et al. (2018) (red) and Bret et al. (2017) (blue).Dashed lines stem from Eqs. (8,11). Plain lines = best fitsof the form ∆ RH = aσ , with a ∈ R . = 4 − σ + O ( σ ) . (7)After some algebra we get in Eq. (5) ∂ ∆ RH ∂σ = − . (8)The perpendicular case is quite different as the MHDjump is already reduced by the field like , r = 4 − σ + O ( σ ) . (9)Applying the same method than for the parallel case,Bret & Narayan (2019) derived for the perpendicularone (see Appendix A), r = 4 − σ + O ( σ ) , (10)where 86 / ∼ . ∂ ∆ RH ∂σ = − . (11)At the present stage it is premature to accurately con-trast the model with the simulations, since a full blowntheory should account for the composition of the plas-mas (pair or e/i) and relativistic effects. Yet, the ordersof magnitude can be checked at least with Guo et al.(2018) and Bret et al. (2017). The latter is relativistwhile the former is not.The values of ∆ RH obtained in Guo et al. (2018) fora perpendicular shock can be fitted by ∆ RH ∼ − . σ . Taylor expansion of Eq. (7) of Bret & Narayan (2019), with σ = M − A . Computed from the data recorded in Table 4 of Guo et al. (2018). (cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1) (cid:2) p = (cid:2) p = (cid:2) p = (cid:2) p = (cid:3) - (cid:4) RH Figure 5.
Values of − ∆ RH defined by Eq. (4) in terms of σ for the runs listed in Table 4 of Guo et al. (2018). Thedashed line stems from Eq. (11). For a given β p , an MHD-departure window is clear. The values of ∆ RH obtained in Bret et al. (2017) fora parallel shock can be fitted by ∆ RH ∼ − . σ . Theorders of magnitude fit well Eqs. (8) for the parallel caseand (11) for the perpendicular one.These results are displayed on Figure 4. The disper-sion around the fit for Guo et al. (2018) (red) is im-portant due to the various values of β p explored. Thedispersion around the fit for Bret et al. (2017) is far lessimportant because all the runs dealt with plasmas ini-tially cold.We may finally comment on the “departure windows”observed on Figure 1 for pair perpendicular shocks,and on the +27% increase of the jump observed inGallant et al. (1992) (ref [4] of the present Figure 1).This increase was attributed to the emission of electro-magnetic waves at the shock front. As commented inPlotnikov et al. (2018) (ref [6] of the present Figure 1),this should be a dimension effect as Gallant et al. (1992)is the only 1D simulation of the 4 references.The “departure windows” in [5,6] (Iwamoto et al.2017; Plotnikov et al. 2018) were left unexplained. Be-sides their weak amplitude (-9 and -4% respectively for the maximum departure) they may simply arise fromthe following process: at small σ the jump is in agree-ment with RH, as evidenced in all the simulations anddiscussed in Section 3.1. Then the field generates ananisotropy which triggers a negative departure from theisotropic MHD jump. Yet, for even higher σ ’s, the grow- Computed from Figure 4a of Bret et al. (2017). The “window‘” in Iwamoto et al. (2017) in visible on Figure 15of Iwamoto et al. (2017). Determined from Figure 2 of Iwamoto et al. (2017) and from Fig-ure 2 of Plotnikov et al. (2018).
Bret ing anisotropy drives the jump to 2, and MHD does thevery same (see Figure 6 in Gallant et al. (1992) and Fig-ure 2 in Plotnikov et al. (2018)). We may therefore ex-pect departure windows for perpendicular shocks sinceMHD and PIC simulations have the same limits in boththe weak and the strong field limits.This is confirmed on Figure 5 where we plotted againFigure 4 for Guo et al. (2018), simply joining the pointssharing the same β p . An MHD-departure window isevidenced for each curves, centered around values of σ similar to those observed on Figure 1 for pair plasmas.Such a feature is absent in parallel shocks since MHDis insensitive to the field. In parallel geometry, we canjust expect a departure threshold beyond which kineticeffects progressively drive the collisionless density jumpaway from its MHD counterpart.3.2. Accelerated particles effect on the density jump
In a fluid shock particles constantly share their en-ergy with each others through binary collisions. In col-lisionless shocks, it has been known for long that par-ticles can be accelerated (Krymskii 1977; Axford et al.1977; Blandford & Ostriker 1978; Bell 1978a,b). By go-ing back and forth around the front and/or escapingupstream, these particles escape the Rankine-Hugoniotbudget. Bret & Pe’er (2018a,b) derived some require-ments for particle accelerations, that are all fulfilled inthe present cases.A simple calculation derived from Berezhko & Ellison(1999) allows to conclude that accelerated particlesshould increase the density jump. We outline ithere for completeness. Considering the non-relativisticregime for simplicity, we start writing the conserva-tion equations between the upstream and the down-stream with subscripts “1” and “2” respectively. La-beling n i , v i , P i , Γ the density, velocity pressure and adi-abatic index of the fluid we have, n v = n v − F m , (12) n v + P = n v + P − F p , (13)12 n v + ΓΓ − P v = 12 n v + ΓΓ − P v − F E , (14)where F m , F p , F E are the mass, momentum and energyfluxes escaping the Rankine-Hugoniot budget becauseof accelerated particles. It turns out that n v ≫ F m , n v ≫ F p while F E in Eq. (14) is not negligible withrespect to n v (see Berezhko & Ellison (1999) and ref-erences therein). We can therefore neglect F m , F p inEqs. (12,13) and solve the system for n . From Eq.(12) we derive v = ( n /n ) v . Using this expression toeliminate v from Eqs. (13,14) allows to derive two dif-ferent expressions for P . Equaling them yields a second degree polynomial in n that can be solved exactly. Theshock solution reads, r = 1 + Γ M + p M ( α (Γ −
1) + 1) − M + 1(1 − α )(Γ − M + 2 , (15)where α is defined by Eq. (2) and, M = n v Γ P , (16)is the upstream sonic Mach number. Clearly the den-sity jump (15) can be arbitrarily high as α →
1. Sucha feature can be elaborated further by modeling α (Berezhko & Ellison 1999; Vink et al. 2010).In the strong shock limit M → ∞ it reduces to, r ∞ = p α (Γ −
1) + 1 + Γ(1 − α )(Γ − , = Γ + 1Γ − α Γ + 2Γ + 12(Γ −
1) + O ( α )= 4 + 163 α + O ( α ) for Γ = 5 / . (17)After soma algebra we find in Eq. (5), ∂ ∆ RH ∂α = + 43 , (18)so that the relative deviation stemming from acceler-ated particles should read + α . Now, α is not con-stant in time because the energy F E poured into cosmicrays is not. For example, the maximum energy of ac-celerated particles grows like t / for relativistic shocks(Sironi et al. 2013a; Plotnikov et al. 2018) and t for non-relativistic ones (Caprioli & Spitkovsky 2014b).These results allow to phenomenologically assess the α coefficient in (17). Density jump departures stemmingfrom accelerated particles were notified in Keshet et al.(2009) and Stockem et al. (2012) (see Section 2.1) .For magnetized shocks, such departures were notifiedin the references featured in Figure 3, among whichCaprioli & Haggerty (2019) stands out as the only onewhere the density jump is clearly evaluated at varioustimes. These results suggest altogether that the depar-ture reaches a few percents for run times of the orderof 5 . time units. For magnetized shocks, this “time Niemiec et al. (2012) ran their (un-magnetized) simulation upto 4111 ω − pi and saw no sign of particle acceleration. Yet, theauthors themselves found it odd as they wrote in the conclu-sion: “In PIC simulations one uses few computational particlesto represent very many real electrons or ions and thus introducesartificial collisionality. Would that impact, and possibly prevent,particle pre-acceleration in our simulations?”. an we trust MHD jump conditions for collisionless shocks? σ, θ n , M ) in pairsand electron/ion. It turns out that the accelerationefficiency has been studied extensively in magnetizedpair (Sironi & Spitkovsky 2009a) or electron/ion plas-mas (Sironi & Spitkovsky 2011; Caprioli & Spitkovsky2014a; Guo et al. 2014a, 2018), and it was found thatthe aforementioned order of magnitude is representativeof the full range of possibilities (Sironi 2020).Assuming that the departure grows like βt κ with κ >
0, we then set,43 βt κ = O (10 − ) for t = O (5 . ) , (19)so that, α = O (5 − κ − − κ ) t κ , = O (10 − − . κ ) t κ , (20)where t is measured in the dominant unit of the simu-lation. Note that the scaling of the maximum energy ofthe accelerated particles does not have to translate tothe energy flux F E (see Eq. 2) leaking into acceleratedparticles, since the most energetic ones are but a few.In all the papers examined, the only one from which itwas possible to extract a time dependent variation of F E is Caprioli & Haggerty (2019). Its Figure 2 suggests κ = O (1). Further works would be welcome to narrowdown the value of κ and the time scale of 5 . set byEq. (19). 3.3. Summary
Gathering the results obtained in Section 3.1 for thefield effects, and in Section 3.2 for accelerated particles,we can complete Eq. (5) and write,∆ RH ( σ, α ) ∼ + O (10 − − . κ ) t − σ ( / θ n = 0 , / θ n = π/ . (21)Since we are interested in the order of magnitude ofthe field correction, we can aggregate the results for θ n =0 and π/ RH ( σ, α ) ∼ + O (10 − − . κ ) t κ − σ O (1) , (22) where κ is of order unity. As commented above, thefirst term, + O (10 − − . κ ) t κ , should be representative, in order of magnitude , of the full spectrum of shockspopulating the ( σ, θ n , M ) phase space.To which extent can we make the same claim for thecorrection term ∝ σ , due to the field driven anisotropy?The field correction obviously vanishes for σ = 0, ∀ ( θ n , M ). Regarding the θ n variation, Figures 1 and 2present the results of 9 simulations at various angles inpair plasmas, and 31 in electron/ion plasmas. No dif-ference of order of magnitude has been detected withrespect to Eq. (22) for the coefficient of σ .As for the incidence of M , all the pair shocks fea-turing Figure 1 are strong, that is, M ≫
1. As forthose in electron/ion presented on Figure 2, they spansonic Mach numbers ranging from 2 (Guo et al. 2018;Ha et al. 2018; Kang et al. 2019) to M ≫
1. Again nodeviation, order of magnitude wise, has been detectedfrom the coefficient of the σ correction reported in Eq.(22).The only part of the phase space parameter which hasnot been tested in the literature is the deviation from RHin weak shocks in pair plasmas. If the model developedin Bret & Narayan (2018, 2019); Bret & Narayan (2020)is further confirmed by PIC simulation, then this gap ofweak shock in pairs will be filled. SUMMARY AND CONCLUSIONThe present work represents an attempt to determinewhen the RH density jump can be applied to collisionlessshocks. A tentative answer is given by Eq. (22) whichis valid as long as ∆ RH , the relative departure from theRH density jump, is small.The departure is the sum of 2 terms. One, positive,arises from the accelerated particles escaping the RHbudget, and grows with time. The second is negativeand stems from the pressure anisotropy sustained bythe field.The literature review clearly evidences positive de-partures arising from particles acceleration, and nega-tive ones from field driven anisotropies. We didn’t findstudies considering both effects together, contemplatingfor example the possibility that they compensate eachother.What about the long times and/or strong field regime,beyond the validity of Eq. (22)?Can accelerated particles drive a density jump arbi-trary high on the long run? Some theoretical modelssuggests so (Berezhko & Ellison 1999; Vink et al. 2010)together with some simulations (Caprioli & Haggerty2019). Yet, since the maximum energy of these particlessaturates with time (Sironi et al. 2013b), the energy flux0 Bret F E escaping the Rankine-Hugoniot budget in Eq. (14)could also saturate.As for the large σ regime, we have to distinguish par-allel shocks from perpendicular ones.For parallel shocks, the MHD jump is insensitive tothe field. The jump departure is therefore going toincrease until it saturates since a large σ will eventu-ally trigger an anisotropy yielding an asymptotic den-sity jump. For example, for a strong shock in 3D withΓ = 5 /
3, the MHD jump remains 4 while the collisionlessone tends to 2, resulting in the largest possible departure∆ RH = − σ jumps are identi-cal for MHD and collisionless shocks. The negative jumpdeparture should therefore reach a maximum for inter-mediate values of σ and vanish before and after. Sucha “departure window” has been retrieved for relativisticpair shocks (see Iwamoto et al. (2017); Plotnikov et al.(2018) and Figure 1 of the present work) and non-relativistic electron/ion shocks (Guo et al. 2018). In thelater case, Figure 5 shows that the location and mag-nitude of the window depend on the upstream protontemperature parameter β p . At any rate, ∆ RH nevergoes below −
10% in any of the aforementioned studies.The dimensionality of the works involved in thepresent study may be its main limitation. Eventhough the formalism of Bret & Narayan (2018, 2019);Bret & Narayan (2020) is 3D, simulations underscrutiny feature at best 3 velocity dimensions but only2 spatial dimensions (2D3V). To with extent can theconclusions be generalized to 3D space? The reducednumber of spatial dimensions can have important effectson particle acceleration and/or the external field effect.Comparisons between 2D and 3D results found for ex-ample that some particles trapping occur in 2D and not in 3D (Cruz et al. 2017; Trotta & Burgess 2018). Also,considering only 2 spatial dimensions necessary excludeswaves and instabilities with a wave vector k orientedalong the excluded spatial dimension. Indeed, theoreti-cal explorations of the full unstable k -spectrum of somebeam-plasma (or Weibel-like) instabilities, found it istruly 3D (Kalman et al. 1968; Dieckmann et al. 2008;Bret 2014; Novo et al. 2016).Finally, Matsumoto et al. (2017) (injection scheme,Mach number = 22 .
8) explicitly compared electronacceleration in 2D and 3D simulations for quasi-perpendicular electron/ion shocks. They found the ac-celeration to be more efficient in 3D than in 2D, be itwith an out-of-plane or an in-plane field .A very similar system was studied in Guo et al.(2014a) (reflecting wall, Mach number = 3). There,2D simulations were also compared to 3D ones. Yet,in contrast with Matsumoto et al. (2017), it was foundthat the 2D in-plane field configuration “is a good choiceto capture the acceleration physics of the full 3D prob-lem”. Perhaps the discrepancy with Matsumoto et al.(2017) is due to the different Mach numbers or the dif-ferent methods. Therefore, it seems that as concluded inBohdan et al. (2017), “true 3D simulations are urgentlyneeded to resolve this issue”.ACKNOWLEDGMENTSThis work has been achieved under projects ENE2016-75703-R from the Spanish Ministerio de Econom´ıa yCompetitividad and SBPLY/17/180501/000264 fromthe Junta de Comunidades de Castilla-La Mancha.Thanks are due to Colby Haggerty, Thales Silva, ArnoVanthieghem, Laurent Gremillet, Damiano Caprioli andLorenzo Sironi for valuable inputs.APPENDIX A. PROOF OF EQ. (10)We start from Eq. (25) of Bret & Narayan (2019) where the parameter χ is proportional to the sonic Mach number.This equation already has the adiabatic index Γ = 5 /
3. The equation for the density jump r in the strong shock limit χ → ∞ is given by the coefficient of χ of Eq. (25) in Bret & Narayan (2019). The result is the 3rd degree polynomial, P ( r ) ≡ (2 r ( r + 2) − rσ + 2( r − r + 8 = 0 , (A1) Figure 4d of Matsumoto et al. (2017) suggests that the totalamount of energy in accelerated electrons could be closer to the3D case for 2D in-plane, than for 2D out-of-plane simulations. an we trust MHD jump conditions for collisionless shocks? σ = 0, the 2 roots are r = 1 and 4. We then set r = 4 + kσ with k ∈ R , and perform aTaylor expansion of P (4 − kσ ) up to first order in σ . The zeroth order vanishes, and the first order also if k = − / Ardaneh, K., Cai, D., & Nishikawa, K.-I. 2016, TheAstrophysical Journal, 827, 124,doi: 10.3847/0004-637x/827/2/124Axford, W., Leer, E., & Skadron, G. 1977, Proc. 15thInternational Cosmic Ray Conference, 11, 132Bai, X.-N., Caprioli, D., Sironi, L., & Spitkovsky, A. 2015,ApJ, 809, 55, doi: 10.1088/0004-637X/809/1/55Bale, S. D., Kasper, J. C., Howes, G. G., et al. 2009, Phys.Rev. Lett., 103, 211101,doi: 10.1103/PhysRevLett.103.211101Bale, S. D., Mozer, F. S., & Horbury, T. S. 2003, Phys.Rev. Lett., 91, 265004Balogh, A., & Treumann, R. 2013, Physics of CollisionlessShocks: Space Plasma Shock Waves, ISSI ScientificReport Series (Springer New York).https://books.google.es/books?id=mR4 AAAAQBAJBell, A. R. 1978a, Mon. Not. R. Astron. Soc, 182, 147—. 1978b, Mon. Not. R. Astron. Soc, 182, 443Berezhko, E. G., & Ellison, D. C. 1999, ApJ, 526, 385,doi: 10.1086/307993Blandford, R., & Ostriker, J. 1978, Astrophysical Journal,221, L29Bohdan, A., Niemiec, J., Kobzar, O., & Pohl, M. 2017,ApJ, 847, 71, doi: 10.3847/1538-4357/aa872aBret, A. 2014, Physics of Plasmas, 21, 022106Bret, A. 2015, Journal of Plasma Physics, 81, 455810202Bret, A., & Narayan, R. 2018, Journal of Plasma Physics,84, 905840604, doi: 10.1017/S0022377818001125—. 2019, Physics of Plasmas, 26, 062108,doi: 10.1063/1.5099000Bret, A., & Narayan, R. 2020, Laser and Particle Beams,17, doi: 10.1017/S0263034620000117Bret, A., & Pe’er, A. 2018a, Journal of Plasma Physics, 84,905840311, doi: 10.1017/S0022377818000636—. 2018b, Laser and Particle Beams, 36, 458,doi: 10.1017/S0263034618000472Bret, A., Pe’er, A., Sironi, L., S¸adowski, A., & Narayan, R.2017, Journal of Plasma Physics, 83, 715830201,doi: 10.1017/S0022377817000290Bret, A., Stockem, A., Fiuza, F., et al. 2013, Physics ofPlasmas, 20, 042102, doi: 10.1063/1.4798541Bret, A., Stockem, A., Narayan, R., & Silva, L. O. 2014,Physics of Plasmas, 21, 072301, doi: 10.1063/1.4886121 Caprioli, D., & Haggerty, C. 2019, in International CosmicRay Conference, Vol. 36, 36th International Cosmic RayConference (ICRC2019), 209.https://arxiv.org/abs/1909.06288Caprioli, D., & Spitkovsky, A. 2014a, ApJ, 783, 91,doi: 10.1088/0004-637X/783/2/91—. 2014b, ApJ, 794, 47, doi: 10.1088/0004-637X/794/1/47Casse, F., van Marle, A. J., & Marcowith, A. 2018, PlasmaPhysics and Controlled Fusion, 60, 014017,doi: 10.1088/1361-6587/aa8482Chang, P., Spitkovsky, A., & Arons, J. 2008, ApJ, 674, 378,doi: 10.1086/524764Cruz, F., Alves, E. P., Bamford, R. A., et al. 2017, Physicsof Plasmas, 24, 022901, doi: 10.1063/1.4975310Dieckmann, M. E., & Bret, A. 2017, Journal of PlasmaPhysics, 83, 905830104, doi: 10.1017/S0022377816001288—. 2018, MNRAS, 473, 198, doi: 10.1093/mnras/stx2387Dieckmann, M. E., Bret, A., & Shukla, P. K. 2008, NewJournal of Physics, 10, 013029,doi: 10.1088/1367-2630/10/1/013029Dieckmann, M. E., Folini, D., Hotz, I., et al. 2019, A&A,621, A142, doi: 10.1051/0004-6361/201834393Dieckmann, M. E., Murphy, G. C., Meli, A., & Drury,L. O. C. 2010a, A&A, 509, A89,doi: 10.1051/0004-6361/200912643Dieckmann, M. E., Sarri, G., Doria, D., Ynnerman, A., &Borghesi, M. 2016, Physics of Plasmas, 23, 062111,doi: 10.1063/1.4953568Dieckmann, M. E., Sarri, G., Romagnani, L., Kourakis, I.,& Borghesi, M. 2010b, Plasma Physics and ControlledFusion, 52, 025001, doi: 10.1088/0741-3335/52/2/025001Erkaev, N. V., Vogl, D. F., & Biernat, H. K. 2000, Journalof Plasma Physics, 64, 561,doi: 10.1017/S002237780000893XFang, J., Lu, C.-Y., Yan, J.-W., & Yu, H. 2019, Research inAstronomy and Astrophysics, 19, 182,doi: 10.1088/1674-4527/19/12/182Fitzpatrick, R. 2014, Plasma Physics: An Introduction(Taylor & Francis).https://books.google.es/books?id=0RwbBAAAQBAJFiuza, F., Fonseca, R. A., Tonge, J., Mori, W. B., & Silva,L. O. 2012, PhRvL, 108, 235004,doi: 10.1103/PhysRevLett.108.235004 Bret
Gallant, Y. A., Hoshino, M., Langdon, A. B., Arons, J., &Max, C. E. 1992, ApJ, 391, 73, doi: 10.1086/171326Gargat´e, L., & Spitkovsky, A. 2012, ApJ, 744, 67,doi: 10.1088/0004-637X/744/1/67Gary, S. 1993, Theory of Space Plasma Microinstabilities,Cambridge Atmospheric and Space Science Series(Cambridge University Press)Gerbig, D., & Schlickeiser, R. 2011, The AstrophysicalJournal, 733, 32.http://stacks.iop.org/0004-637X/733/i=1/a=32Goedbloed, H., Keppens, R., & Poedts, S. 2019,Magnetohydrodynamics of Laboratory and AstrophysicalPlasmas (Cambridge University Press),doi: 10.1017/9781316403679Guo, F., & Giacalone, J. 2013, ApJ, 773, 158,doi: 10.1088/0004-637X/773/2/158Guo, X., Sironi, L., & Narayan, R. 2014a, ApJ, 794, 153,doi: 10.1088/0004-637X/794/2/153—. 2014b, ApJ, 797, 47, doi: 10.1088/0004-637X/797/1/47—. 2017, ApJ, 851, 134, doi: 10.3847/1538-4357/aa9b82—. 2018, ApJ, 858, 95, doi: 10.3847/1538-4357/aab6adHa, J.-H., Ryu, D., Kang, H., & van Marle, A. J. 2018,ApJ, 864, 105, doi: 10.3847/1538-4357/aad634Haggerty, C. C., & Caprioli, D. 2019, ApJ, 887, 165,doi: 10.3847/1538-4357/ab58c8Iwamoto, M., Amano, T., Hoshino, M., & Matsumoto, Y.2017, ApJ, 840, 52, doi: 10.3847/1538-4357/aa6d6fJohnson, J., & Cheret, R. 1998, Classic Papers in ShockCompression Science: Edition en anglais, High pressureshock compression of condensed matter (Springer).https://books.google.es/books?id=256ws-XBjzwCKalman, B. G., Montes, C., & Quemada, D. 1968, Phys.Fluids, 11, 1797Kang, H., Ryu, D., & Ha, J.-H. 2019, ApJ, 876, 79,doi: 10.3847/1538-4357/ab16d1Karimabadi, H., Krauss-Varban, D., & Omidi, N. 1995,Geophysical Research Letters, 22, 2689,doi: 10.1029/95GL02788Kato, T. N. 2007, ApJ, 668, 974, doi: 10.1086/521297Kato, T. N., & Takabe, H. 2008, ApJL, 681, L93,doi: 10.1086/590387—. 2010, ApJ, 721, 828, doi: 10.1088/0004-637X/721/1/828Keshet, U., Katz, B., Spitkovsky, A., & Waxman, E. 2009,ApJL, 693, L127, doi: 10.1088/0004-637X/693/2/L127Krymskii, G. 1977, Doklady Akademii Nauk SSSR, 234,1306Kulsrud, R. M. 2005, Plasma Physics for Astrophysics(Princeton, NJ: Princeton Univ. Press)Langmuir, I. 1925, Phys. Rev., 26, 585 Lemoine, M., Gremillet, L., Pelletier, G., & Vanthieghem,A. 2019a, PhRvL, 123, 035101,doi: 10.1103/PhysRevLett.123.035101Lemoine, M., Pelletier, G., Vanthieghem, A., & Gremillet,L. 2019b, PhRvE, 100, 033210,doi: 10.1103/PhysRevE.100.033210Lemoine, M., Vanthieghem, A., Pelletier, G., & Gremillet,L. 2019c, PhRvE, 100, 033209,doi: 10.1103/PhysRevE.100.033209Lezhnin, K. V., Fox, W., Schaeffer, D. B., et al. 2020, arXive-prints, arXiv:2001.04945.https://arxiv.org/abs/2001.04945Li, R., Zhou, C. T., Huang, T. W., Qiao, B., & He, X. T.2017, Physics of Plasmas, 24, 042113,doi: 10.1063/1.4980832Lichnerowicz, A. 1976, Journal of Mathematical Physics,17, 2135, doi: 10.1063/1.522857Lyubarsky, Y. 2006, ApJ, 652, 1297, doi: 10.1086/508606Martins, S. F., Fonseca, R. A., Silva, L. O., & Mori, W. B.2009, ApJL, 695, L189,doi: 10.1088/0004-637X/695/2/L189Maruca, B. A., Kasper, J. C., & Bale, S. D. 2011, Phys.Rev. Lett., 107, 201101,doi: 10.1103/PhysRevLett.107.201101Matsumoto, Y., Amano, T., Kato, T. N., & Hoshino, M.2017, Phys. Rev. Lett., 119, 105101,doi: 10.1103/PhysRevLett.119.105101Mihalas, D., & Weibel-Mihalas, B. 1999, Foundations ofRadiation Hydrodynamics, Dover Books on Physics(Dover Publications).https://books.google.es/books?id=C3h3DQAAQBAJMoreno, Q., Dieckmann, M. E., Folini, D., et al. 2020,Plasma Physics and Controlled Fusion, 62, 025022,doi: 10.1088/1361-6587/ab5bfbNakanotani, M., Matsukiyo, S., Hada, T., & Mazelle, C. X.2017, ApJ, 846, 113, doi: 10.3847/1538-4357/aa8363Naseri, N., Bochkarev, S. G., Ruan, P., et al. 2018, Physicsof Plasmas, 25, 012118, doi: 10.1063/1.5008278Niemiec, J., Pohl, M., Bret, A., & Wieland , V. 2012, ApJ,759, 73, doi: 10.1088/0004-637X/759/1/73Nishikawa, K. I., Niemiec, J., Hardee, P. E., et al. 2009,ApJL, 698, L10, doi: 10.1088/0004-637X/698/1/L10Nishimura, K., Matsumoto, H., Kojima, H., & Gary, S. P.2003, Journal of Geophysical Research (Space Physics),108, 1182, doi: 10.1029/2002JA009671Novo, A. S., Bret, A., & Sinha, U. 2016, New Journal ofPhysics, 18, 105002Otsuka, F., Matsukiyo, S., & Hada, T. 2019, High EnergyDensity Physics, 33, 100709,doi: 10.1016/j.hedp.2019.100709 an we trust MHD jump conditions for collisionless shocks?13