defined in this study. These indicate that although energetically sub-dominant, equatorial shocks could have possibly observable imprints.
From Figure 5(d), one can see that⟨vs⟩increases during the peak period ofFφ andFCR. The adiabatic blast wave solution for a point explosion requires the density gradient steeper thanρ−3for accelerating shock fronts [79]. Apparently, the blast wave assumption does not hold for merger-driven shocks, since the kinetic and gravitational energies of merging clumps are continuously dissipated through shocks and additional energies are supplied by secondary infall and multiple minor mergers. In addition,⟨vs⟩ includes the contributions not only from the shock propagation, but also from turbulent flow motions ahead of shocks. As a matter of fact, large fluctuations in⟨vs⟩as well as in⟨Ms⟩reflect complicated flow dynamics of clusters.
81]. For instance, [80] identified the temperature jump corresponding toMs∼3 in the pre-merger stage (A1758N & A1758S). In addition, [81] also found the proof of equatorial shocks on the equatorial plane of an pre-merger cluster pair. Such results are in agreement with the prediction of equatorial shocks discussed in this Section.
Finally, we note that the properties of merger shocks, as well as their positions relative to X-ray peak and DM clumps, should depend on a number of merger parameters. For example, according to simple calculation in the system satisfying energy conservation, Mach number of merger-driven shocks sensi- tively depends on the clump mass ratio [82] . More comprehensive investigation of such dependence requires a very large sample of simulated merging clusters, and we will leave it as future works.
III Proton Acceleration at Weak Quasi-parallel Shocks in Intracluster Medium
In the observed merging galaxy clusters, the radio synchrotron emission indicates that the ICM shocks accelerate CRs via DSA, like most astrophysical shocks. While the previous studies using hydrodynamic simulations have been focused on the properties and dynamical evolution of ICM shocks as shown in the Section II, it is necessary to understand particle acceleration process in such shocks. In this Section, we examine the detailed proton acceleration mechanism at the weak ICM shocks including the kinetic plasma processes. Note that all results shown here are originally presented in the paper (Ha, J.-H., Ryu, D., Kang, H., & van Marle, A. J. 2018, The Astrophysical Journal, 864, 105; [83]).
Just like Earth’s bow shocks and supernova remnant shocks, ICM shocks are thought to accelerate cosmic ray (CR) protons and electrons via diffusive shock acceleration (DSA,a.k.a. Fermi I accelera- tion) [20–22]. Although the acceleration of relativistic electrons by putative shocks has been observed as the so-called giant radio relics such as the Sausage relic in the merging cluster CIZA J2242.8+5301 [39], the presence of CR protons produced by ICM shocks has yet to be confirmed [84–86]. Inelastic colli- sions of CR protons with thermal protons followed by the decay of neutral pions can produce diffuse γ−ray emission, which has not been detected so far by the Fermi-LATγ−ray telescope [87]. According to theoretical estimations using cosmological hydrodynamics simulations that adopt prescriptions for theCR proton acceleration efficiencyat shocks, η(Ms), non-detection ofγ−ray emission from galaxy clusters constrainsη(M)to be less than 10−3for 2≤Ms≤5 [31].
The key element in estimating the DSA efficiency is the so-called ‘injection process’, which ener- gizes thermal protons to the suprathermal energies sufficient to diffuse across the shock. In order to understand CR injection and early acceleration at collisionless shocks, kinetic plasma processes have been studied through particle-in-cell (PIC) and hybrid plasma simulations [23–29]. In PIC simulations, both ions and electrons are treated kinetically, therefore wave-excitation and wave-particle interactions can be followed from first principles. In hybrid simulations, on the other hand, only ions are treated kinetically, while electrons are treated as charge-neutralizing fluid with zero-mass. They are suitable for studying ion acceleration well into full Fermi-I acceleration regime on many ion gyration periods, since they are computationally much less expensive than PIC simulations. However, collisioness shock formation and particle acceleration involve kinetic processes due to both electrons and protons, so PIC simulations should be more appropriate in the investigation of the ion injection problem.
Comprehensive studies by Caprioli and collaborators using hybrid simulations showed that CR ions are accelerated efficiently with η∼0.05−0.15 at strong quasi-parallel shocks withMs≳5 and θBn≲45◦, whereθBn is the obliquity angle between the shock normal and the background magnetic field direction [26–28]. The magnetic field obliquity angleθBnis one of the key parameters that govern the characteristics of shocks: quasi-parallel shocks with θBn≲45◦ versus quasi-perpendicular shocks withθBn≳45◦. In particular, [28] showed that at quasi-parallel shocks a substantial fraction of ions im- pinging on the shock potential barrier can be specularly reflected, when the quasi-periodically reforming
shock potential is in a high state (i.e., e∆φ >12miv2x). The reflected ions escaping upstream along the parallel magnetic field generate low-frequency waves and amplify the transverse magnetic fields via CR ion-driven instabilities, transforming upstream quasi-parallel fields to locally quasi-perpendicular fields in the shock transition layer. Then ions arriving subsequently at the shock are reflected at locally per- pendicular portions of turbulent magnetic fields [88]. Due to perpendicular components of the magnetic field the reflected ions gain sufficient energies via multiple cycles of shock drift acceleration (SDA), and then start participating to the Fermi I cycle of shock acceleration. Thus reflection of ions at the overshoot of the shock potential, self-excitation of turbulent waves, and SDA are integral parts of ion injection at quasi-parallel shocks.
On the other hand, at quasi-perpendicular shocks the reflected ions are advected downstream along with the background magnetic fields typically after one gyromotion, so they may undergo only a few cycles of SDA and do not reach energies sufficient to be injected to Fermi I acceleration [26]. Since the gyrostream of the reflected ions penetrates upstream less than one ion gyroradius from the shock ramp, turbulent waves are not excited efficiently in the precursor of quasi-perpendicular shocks [27].
Consequently, Fermi-I acceleration of protons is efficient only at quasi-parallel shocks, which will be the main focus of this study.
We further comment about particle reflection at the shocks. Firstly, ion reflection is governed by both the shock potential and the magnetic mirror at the shock, however, the fraction of ion reflection satisfying the reflection condition in terms of shock potential dominates over that satisfying the condition for magnetic mirror conditions (i.e., the particles are not in the loss cone). Secondly, while upstream ions are mainly reflected by the shock potential, the reflection of upstream electrons are suppressed by the shock potential because electrons have opposite charge. And thus, electron reflection is mainly governed by the magnetic mirror at the shock surface. In Section VII, we provide the detailed physics regarding electron reflection at the shock surface.
Unlike collisional shocks, physics of collisionless shocks include complex kinetic plasma processes such as particle reflection, self-excitation of waves, and wave-particle interaction, well beyond MHD Rankine-Hugoniot jump conditions [29]. From studies of collisionless shocks such as Earth’s bow shocks and interplanetary shocks, the concept ’shock criticality’ has been adopted. ’Subcritical shock’
denotes that the shock kinetic energy can be fully dissipated through resistivity so the shock structure is smooth without complex plasma waves. ’Supercritical’ shock, on the other hand, the shock kinetic energy cannot be fully dissipated through resistivity alone, so a substantial fraction of incoming ions must be reflected upstream to satisfy the Rankine-Hunoniot jump conditions [89]. Such shock criticality can be described through the certain critical Mach number. For instance, the fast mode Mach number of shock satisfies the condition,Mf>Mf∗whereMf∗ stands for thefast first critical Mach number, then such shock is ’supercritical’ shock. We also provide different critical Mach numbers regarding different criteria in Section VII.
In typical astrophysical environment, the plasma betaβ =Pgas/PB∼1, so the Alfvén Mach number MAis often used to characterize a shock and the critical Alfvén Mach numberMA∗ ≈2.76 is commonly quoted for quasi-perpendicular shocks [29]. As for the papers by Caprioli and collaborators cited above,
CR acceleration at supercritical shocks has been explored through plasma simulations mainly forβ≲1 cases (i.e.,MA∼Ms). In the typical hot ICM plasma, however,β∼100, so shocks have relatively high Alfvén Mach number,MA≈10Ms≈20−30, but low sonic Mach number,Ms≈2−3. Note that for these ICM shocks,Mf≈Ms. To our knowledge, the supercriticality of weak (Ms≈2−4) quasi-parallel shocks in such highβ environment has not yet been studied by PIC or hybrid simulations.
PIC simulations ofMs=3 shocks in highβ ICM plasmas were considered by [23, 24], focusing mainly on electron acceleration at quasi-perpendicular shocks. They showed that for the shock model withβ =20,θBn=63◦, andMs=3, about 20 % of incoming ions are reflected at the shock and gain a small amount of energy via a few cycles of SDA. Those energized ions overcome the potential barrier and advect downstream along with the magnetic fields. Note that their simulations were not intended to study ion acceleration in the DSA regime in sufficiently long ion gyro-time scales.
Excitation of magnetic turbulence by protons streaming upstream of quasi-parallel shocks is an in- tegral part of injection and acceleration of CR particles. There are two dominant modes: (1) resonant streaming instability which excites left-handed circularly polarized waves [20], and (2) nonresonant current-driven instability which excites right-handed circularly polarized wave [90]. Using hybrid sim- ulations of quasi-parallel shocks inβ ∼1 plasma, [27] showed that the resonant streaming instability is dominant in the precursor of shocks withMA≲30, while the nonresonant current-driven instability operates faster at stronger shocks withMA≳30. The magnetic field amplification factor increases with increasing Alfvén Mach number as⟨B/B0⟩2∝MA. Both instabilities amplify primarily transverse com- ponents of the magnetic field, so they generate locally perpendicular fields in the shock foreshock and downstream region, which in turn facilitate SDA of the reflected ions and reflect subsequently arriving ions. Eventually, excited turbulent waves act as scattering centers both upstream and downstream of the shock, which are required for the Fermi I acceleration.
In this Section, we examine the physics of ‘shock criticality’ in weak ICM shocks by using PIC simulations. To identify ion injection and early stage acceleration, we study shock structures and ion energy spectra. In order to understand the nature of CR ion-driven instabilities and turbulent magnetic field amplification, we perform Fourier analysis of upstream self-excited magnetic field components.
We then discuss dependence of ion injection and CR ion-driven instabilities on the pre-shock conditions such asMs,β, andθBn.