Effects of Multi-scale Plasma Waves on Electron Preacceleration at Weak Quasi-perpendicular Intracluster Shocks
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Effects of Multi-scale Plasma Waves on Electron Preacceleration at Weak Quasi-perpendicularIntracluster Shocks
Ji-Hoon Ha, Sunjung Kim, Dongsu Ryu, and Hyesung Kang Department of Physics, School of Natural Sciences UNIST, Ulsan 44919, Korea Department of Earth Sciences, Pusan National University, Busan 46241, Korea (Received; Revised; Accepted)
Submitted to The Astrophysical JournalABSTRACTRadio relics associated with merging galaxy clusters indicate the acceleration of relativistic electronsin merger-driven shocks with low sonic Mach numbers ( M s (cid:46)
3) in the intracluster medium (ICM).Recent studies have suggested that electron injection to diffusive shock acceleration (DSA) could takeplace through the so-called Fermi-like acceleration in the shock foot of β = P gas /P B ≈ −
100 shocksand the stochastic shock drift acceleration (SSDA) in the shock transition of β ≈ − Q ⊥ ) shocksin β ≈ −
100 plasmas by performing particle-in-cell simulations in the two-dimensional domain largeenough to encompass ion-scale waves. We find that in supercritical shocks with M s (cid:38) M ∗ AIC ∼ . M s < . M s ∼ . Keywords: acceleration of particles – cosmic rays – galaxies: clusters: general – methods: numerical –shock waves INTRODUCTIONShocks with low sonic Mach numbers ( M s (cid:46)
5) are ex-pected to form naturally in the hot intracluster medium(ICM) during the large-scale structure formation of theuniverse (e.g., Ryu et al. 2003; Pfrommer et al. 2006;Skillman et al. 2008; Vazza et al. 2009; Hong et al. 2014;Schaal & Springel 2015; Ha et al. 2018a). In particular,weak shocks with M s ∼ . −
3, induced by major merg-ers of galaxy clusters, have been detected in X-ray andradio observations (e.g., Markevitch & Vikhlinin 2007;van Weeren et al. 2010; Br¨uggen et al. 2012; Brunetti& Jones 2014). The so-called radio relics, diffuse elon-
Corresponding author: Hyesung [email protected] gated radio structures detected in the outskirts of merg-ing clusters, are interpreted as synchrotron emissionfrom cosmic ray (CR) electrons accelerated via diffusiveshock acceleration (DSA), also known as the Fermi first-order process, in such merger-driven shocks (see e.g., vanWeeren et al. 2019, for a review).The acceleration of nonthermal particles at collision-less shocks in tenuous astrophysical plasmas involves avery broad spectrum of complex kinetic plasma pro-cesses (e.g., Drury 1983; Balogh & Treumann 2013;Marcowith et al. 2016). It depends on various pa-rameters including the sonic Mach number, M s , theplasma beta, β = P gas /P B (the ratio of thermal tomagnetic pressures), and the obliquity angle, θ Bn , be-tween the upstream background magnetic field directionand the shock normal. For weak shocks in the high- β ICM under consideration in this study, it was shown a r X i v : . [ a s t r o - ph . H E ] F e b Ha et al. through particle-in-cell (PIC) simulations that electronsmight be preaccelerated and injected to DSA mainly inquasi-perpendicular ( Q ⊥ , hereafter) configuration with θ Bn (cid:38) ◦ (e.g., Guo et al. 2014a,b). On the other hand,for strong shocks in β ∼ Q (cid:107) ) configu-ration with θ Bn (cid:46) ◦ owing to the magnetic turbulenceexcited by CR protons streaming upstream (Park et al.2015).Two types of electron preacceleration mechanisms op-erative at weak, high- β , Q ⊥ -shocks have been exploredin the literature so far: (1) the so-called Fermi-like ac-celeration due to the diffusive scattering of electronsbetween the shock ramp and upstream self-generatedwaves in the shock foot (Matsukiyo et al. 2011; Guoet al. 2014a), and (2) the so-called stochastic shockdrift acceleration (SSDA) due to the extended gradient-drift of electrons being confined in the shock transition(Matsukiyo & Matsumoto 2015; Katou & Amano 2019;Niemiec et al. 2019). Figure 1 illustrates the characteris-tic structures of Q ⊥ -shocks with the foot and transitionzone.In the Fermi-like acceleration, electrons are energizedthrough shock drift acceleration (SDA), and reflected bymagnetic mirror at the shock ramp (Guo et al. 2014a).Backstreaming electrons self-generate oblique electron-scale waves in the shock foot via the electron firehoseinstability (EFI) (Guo et al. 2014b; Kang et al. 2019;Kim et al. 2020). This is termed as “Fermi-like” accel-eration, since electrons are scattered between the shockramp and upstream waves in the foot. It differs fromDSA, because electrons do not experience full diffusivetransport back and forth across the entire shock transi-tion.Kang et al. (2019, KRH2019, hereafter) showed that,in high- β plasmas, the Fermi-like acceleration may op-erate only in supercritical Q ⊥ -shocks with M s (cid:38) . λ (cid:46) c/ω pe , where c is the speed of light and ω pe is the electron plasmafrequency), the preacceleration may not proceed all theway to the injection momentum, p inj ∼ p i , th . Here, p i , th = √ m i k B T i2 is the postshock ion thermal momen-tum, where k B is the Boltzmann constant and T i2 is thepostshock ion temperature. Hereafter, the subscripts1 and 2 denote the preshock and postshock quantities,respectively.Considering that the width of the shock transitionzone is several times the gyroradius of postshock thermal ions, i.e., ∆ x shock ∼ r L , i ( T ,i , B ), we argue that electroninjection to DSA must require multi-scale waves, whichcan scatter and confine electrons with the momentum upto ∼ p inj around the shock transition. Such multi-scalefluctuations could be generated in the shock transitionzone by the Alfv´en ion cyclotron (AIC) and ion-mirrorinstabilities due to the ion temperature anisotropy, andby the whistler and electron-mirror instabilities due tothe electron temperature anisotropy (e.g., Matsukiyo &Matsumoto 2015; Guo et al. 2017; Katou & Amano2019). As illustrated in Figure 1, the AIC instabil-ity induces ion-scale waves propagating parallel to thebackground magnetic field, leading to the shock surfacerippling. In the SSDA, electrons undergo the stochas-tic pitch-angle scattering off these multi-scale wavesand stay confined for an extended period in the shocktransition, leading to much greater energy gain via thegradient-drift (Niemiec et al. 2019; Trotta & Burgess2019).In the PIC simulations reported by Guo et al.(2014a,b) and KRH2019, the transverse size of two-dimensional (2D) simulation boxes was L y (cid:46) r L , i , where r L , i is Larmor radius of incoming ions. So the simu-lation domains were not large enough to accommodatethe emergence of ion-scale fluctuations via the AIC andthe ensuing shock surface rippling . On the other hand,the 2D PIC simulations by Niemiec et al. (2019), andthe 2D and 3D hybrid simulations implemented by test-particle electrons by Trotta & Burgess (2019) have thedomains large enough to include ion-scale fluctuationsin the shock transition. So these authors found thatelectrons could be preaccelerated all the way to p inj viathe SSDA. In addition, Trotta & Burgess (2019) foundthat the AIC instability is triggered and the ensuingelectron preacceleration operates only in supercriticalshocks with the Alfv´enic Mach number greater than thecritical Mach number, M A , crit ≈ .
5. However, note thatNiemiec et al. (2019) considered a shock in a relativelylow- β ( ≈
5) plasma because of severe requirements forcomputational resources, while Trotta & Burgess (2019)focused on shocks in the solar wind with β ≈ M s ≈ − θ Bn = 53 ◦ − ◦ . To perform the PIC simulations withina reasonable time frame, however, we choose β = 50 andthe ion-to-electron mass ratio, m i /m e = 50, for the fidu-cial cases (see Table 1 below). The main goals of thispaper are (1) to find the critical Mach number to trig-ger the AIC instability and the shock surface ripplingin high- β shocks, and (2) to explore how the SSDA can lectron Preacceleration at Weak ICM Shocks Figure 1. (a) Ion number density, (cid:104) n i (cid:105) y, avg /n , averagedover the y -direction, normalized to the upstream ion numberdensity, n , for a supercritical Q ⊥ -shock (M3.0 model). (b)Ion number density, n i ( x, y ) /n , in the x − y plane for thesame model. 2D PIC simulation results are shown in theregion of − . ≤ ( x − x sh ) /r L , i ≤ . ci t ∼
32, where x sh is the shock position. The gyromotion of the reflected ions(green circular arrows) generates the overshoot/undershootstructure in the shock transition, while the backstreamingof the SDA-reflected electrons (magenta cone) induces thetemperature anisotropy and the EFI in the shock foot. Thecolored arrows indicate the regions where DSA (cyan), SSDA(dark green), and Fermi-like acceleration (light green) oper-ate. The labels for the three instabilities, AIC, whistler, andEFI, are placed in the regions where the respective instabil-ities are excited. During a SDA cycle, electrons drift in thenegative z -direction (into the paper here) anti-parallel to theconvection electric field (cid:126)E conv = − (1 /c ) (cid:126)U × (cid:126)B . facilitate the electron preacceleration beyond the pointwhere the Fermi-like acceleration saturates. However, itremains challenging to simulate the true electron injec-tion to DSA beyond p inj in these high- β shocks. The hy-brid approach combined with test-particle electron cal-culations might provide a feasible solution (e.g., Trotta& Burgess 2019), although kinetic processes on electronscales are not properly emulated in such hybrid simula-tions.In a separate paper, Kim et al. (2021) (KHRK2021,hereafter), we examine the properties of the mi-croinstabilities due the ion and electron temperatureanisotropies by carrying out a linear stability analysis for wide ranges of shock parameters. In addition, thepredictions of the linear analysis are compared with 2DPIC simulations with the same set of parameters in aperiodic domain. Below we refer to results from thatwork, when we interpret the properties of plasma wavesin the shock transition zone.This paper is organized as follows. In Section 2, wegive a brief overview of the basic physics of Q ⊥ -shocks.Section 3 describes the numerical details of PIC simula-tions, along with the definitions of various parameters.In Section 4, we present the simulation results, includingthe shock structures, instability analysis, power spectraof self-excited waves, and electron energy spectra. Thedependence of our findings on various model parametersis also discussed in Section 4. A brief summary followsin Section 5. BASIC PHYSICS OF Q ⊥ -SHOCKSThe physics of kinetic plasma processes in collision-less, Q ⊥ -shocks is complex. For comprehensive reviews,readers are referred to Balogh & Treumann (2013) andKrasnoselskikh et al. (2013). A brief overview of somekey problems that are relevant for this study can befound in KRH2019.The structures and ensuing excitation of microinsta-bilities are primarily governed by the dynamics of shock-reflected ions and electrons. Figure 1 illustrates the typ-ical structures of a supercritical Q ⊥ -shock: (1) the shockfoot emerges due to the upstream gyration of reflectedions, and (2) overshoot/undershoot oscillations developdue to the downstream gyration of those reflected ionsin the shock transition zone. The figure also depictsthat the EFI is excited in the shock foot by the SDA-reflected electrons backstreaming along the backgroundmagnetic field (Guo et al. 2014b, KRH2019), whereasthe AIC and whistler instabilities are excited along thefirst and second overshoots, leading to the shock surfacerippling (Niemiec et al. 2019; Trotta & Burgess 2019).2.1. Shock Criticality In Q ⊥ -shocks, incoming ions are reflected mainly bythe electrostatic potential drop at the shock ramp,whereas incoming electrons are reflected by the mag-netic mirror force due to converged magnetic field lines.In the simplest theory, the shock criticality is directlyrelated to ion reflection at the shock ramp when thedownstream flow speed normal to the shock exceeds thedownstream sound speed, which defines the condition forthe so-called fast first critical Mach number , M ∗ f (Ed-miston & Kennel 1984). In addition, there are a fewvarieties of critical Mach numbers, including the secondand third whistler critical Mach numbers , which are re-lated with the emission of dispersive whistler waves and Ha et al. quasi-periodic shock-reformation (Krasnoselskikh et al.2002; Oka et al. 2006). Obviously, these processes de-pend on the shock obliquity angle, θ Bn , and the plasma β , because ion reflection is affected by anomalous resis-tivity and microinstabilities in the shock transition.In Ha et al. (2018b) and KRH2019, examining theshock structure, energy spectra of ions and electrons,and self-excited waves in shock models with M s ≈ − β ( ∼ M ∗ s ∼ . Q (cid:107) and Q ⊥ shocks, while the EFIcritical Mach number for the excitation of the EFI isalso M ∗ ef ∼ . Q ⊥ -shocks. These two critical Machnumbers are closely related, since the oscillations in theshock transition due to ion reflection enhance the mag-netic mirror and electron reflection. The critical machnumber M ∗ s is higher than the fast first critical Machnumber, M ∗ f ∼
1, estimated for β ∼ M A (cid:29) M s ≈ M f in β (cid:29) M s , in-stead of M A , since primarily the sonic Mach numbercontrols both the shock electrostatic potential drop rel-evant for ion reflection and the magnetic field compres-sion relevant for electron reflection.In this work, we explore the shock criticality in termsof the shock surface rippling triggered by the AIC insta-bility, using 2D PIC simulations with a transverse di-mension large enough to include ion-scale fluctuations.2.2. Fermi-like Preacceleration in Shock Foot
As discussed in the introduction, Guo et al. (2014a,b)demonstrated that thermal electrons could be preac-celerated via the Fermi-like acceleration in the foot of M s = 3 shocks in β = 20 ICM plasmas. The key pro-cesses involved in this preacceleration include the fol-lowing: (1) a fraction ( ∼ T e (cid:107) /T e ⊥ > β (cid:38) β plasmasdue to strong magnetization of electrons. Moreover, theEFI is known to be almost independent of the mass ra-tio, m i /m e .Using the PIC simulations for Q ⊥ -shocks with M s =2 − β = 100, KRH2019 found the following forshocks with M s (cid:38) .
3. (1) The fraction of reflectedions increases abruptly and overshoot/undershoot oscil-lations emerge in the shock transition. (2) The instabil-ity parameter for the EFI, I EFI ≡ − T e ⊥ T e (cid:107) − . β . (cid:107) > . (1)increases sharply, where β e (cid:107) = 8 πn e k B T e (cid:107) /B is theelectron β parallel to the magnetic field. Also non-propagating oblique waves appear in the shock foot.(3) The upstream electron energy spectrum develops asuprathermal tail for p (cid:38) p th , e , where p th , e is the post-shock thermal electron momentum. These features allimply that this Fermi-like acceleration could be effectiveonly for supercritical Q ⊥ -shocks with M s (cid:38) M ∗ ef ∼ . M s ∼ . − . λ ∼ − c/ω pe are dominantly excited, andthat the scattering of electrons by those waves reducesthe temperature anisotropy and stabilize the EFI. Thus,we suggested that the preacceleration of electrons bythe Fermi-like acceleration involving multiple cycles ofSDA may not proceed all the way to DSA in weak ICMshocks. This calls for additional mechanisms that couldenergize electrons beyond the point where the Fermi-likeacceleration ceases to operate.2.3. Stochastic Shock Drift Acceleration in ShockTransition
In the transition zone of supercritical Q ⊥ -shocks,three kinds of microinstabilities could be excited (e.g.Guo et al. 2017; Katou & Amano 2019): (1) the AICinstability due to the ion temperature anisotropy ( A i ≡ T i ⊥ /T i (cid:107) >
1) induced by the shock-reflected ions that areadvected downstream; (2) the whistler instability due tothe electron temperature anisotropy ( A e ≡ T e ⊥ /T e (cid:107) >
1) induced by the magnetic field compression at theshock ramp; (3) the mirror instabilities due to A i > A e >
1. The AIC and whistler instabilities ex-cite waves propagating predominantly in the direction lectron Preacceleration at Weak ICM Shocks Table 1.
Model Parameters for PIC Simulations
Model Name M s M A u /c θ Bn β T e = T i [K(keV)] m i /m e L x [ c/ω pe ] L y [ c/ω pe ] ∆ x [ c/ω pe ] t end [ ω − ] t end [Ω − ]M2.0 2.0 12.9 0.038 63 ◦
50 10 (8 .
6) 50 3200 310 0.1 4 . × ◦
50 10 (8 .
6) 50 3200 310 0.1 4 . × ◦
50 10 (8 .
6) 50 3200 310 0.1 4 . × ◦
50 10 (8 .
6) 50 3200 310 0.1 4 . × ◦
50 10 (8 .
6) 50 3200 310 0.1 4 . × ◦
50 10 (8 .
6) 50 3200 310 0.1 4 . × ◦
50 10 (8 .
6) 100 2000 440 0.1 4 . × ◦
50 10 (8 .
6) 100 2000 440 0.1 4 . × ◦
50 10 (8 .
6) 100 2000 440 0.1 4 . × β
20 2.0 8.2 0.038 63 ◦
20 10 (8 .
6) 50 3200 200 0.1 2 . × β
20 2.3 9.4 0.046 63 ◦
20 10 (8 .
6) 50 3200 200 0.1 2 . × β
20 3.0 12.3 0.068 63 ◦
20 10 (8 .
6) 50 3200 200 0.1 2 . × β
100 2.0 18.2 0.038 63 ◦
100 10 (8 .
6) 50 2000 440 0.1 3 . × β
100 2.3 21.0 0.046 63 ◦
100 10 (8 .
6) 50 2000 440 0.1 3 . × β
100 3.0 27.4 0.068 63 ◦
100 10 (8 .
6) 50 2000 440 0.1 3 . × θ
53 2.0 12.9 0.038 53 ◦
50 10 (8 .
6) 50 3200 310 0.1 4 . × θ
73 2.0 12.9 0.038 73 ◦
50 10 (8 .
6) 50 3200 310 0.1 4 . × θ
53 2.3 14.8 0.046 53 ◦
50 10 (8 .
6) 50 3200 310 0.1 4 . × θ
73 2.3 14.8 0.046 73 ◦
50 10 (8 .
6) 50 3200 310 0.1 4 . × θ
53 3.0 19.4 0.068 53 ◦
50 10 (8 .
6) 50 3200 310 0.1 4 . × θ
73 3.0 19.4 0.068 73 ◦
50 10 (8 .
6) 50 4000 310 0.1 6 . × parallel to the background magnetic field, while the mir-ror instabilities induce nonpropagating oblique waves.Katou & Amano (2019) proposed that the electronpreacceleration via SDA could be extended by stochas-tic pitch-angle scattering off these multi-scale waves, be-cause electrons are trapped much longer in the shocktransition zone. They coined the term “stochastic shockdrift acceleration (SSDA)” for such acceleration. Then,Niemiec et al. (2019) performed a 2D PIC simulation for M s = 3 shock with β = 5, θ Bn = 75 ◦ and m i /m e = 100.They observed the emergence of shock surface rippling,accompanied by the plasma waves driven by the threekinds of instabilities in the shock transition, as well asthe EFI-driven obliques waves in the shock foot. Theyalso saw the development of suprathermal tails in boththe upstream and downstream energy spectra of elec-trons that extend slightly beyond p inj by the end of theirsimulations.Furthermore, Trotta & Burgess (2019) performed 2Dand 3D hybrid simulations with test-particle electronsfor Q ⊥ -shocks with M s = 2 . − . β ≈
1, and θ Bn =80 ◦ − ◦ . In typical hybrid simulations, ions are treatedkinetically, while electrons are treated as a charge-neutralizing fluid. So this type of simulations cannotproperly capture electron-driven instabilities. With thatcaveat, they observed that shock surface fluctuations de-velop on ion scales, and that the test-particles electronscould be preaccelerated well beyond p inj at supercriticalshocks with the Alfv´enic Mach number greater than the critical Mach number, M A , crit ≈ .
5. Note that theyconsidered β ≈ M s , crit ≈ M A , crit . NUMERICAL SETUP FOR PIC SIMULATIONSThe numerical code and setup for PIC simulations arethe same as those adopted in KRH2019, except thathere the 2D simulation domain in unit of r L,i is about8 times larger in the transverse direction. TRISTAN-MP code in 2D planar geometry is used (Buneman 1993;Spitkovsky 2005). An ion-electron plasma with Maxwelldistributions moves with the bulk velocity u = − u ˆx toward a reflecting wall at the leftmost boundary ( x =0), so a shock propagates toward the + ˆx direction. Auniform background magnetic field, B , lies in the x - y plane (shock coplanarity plane), and the angle between B and the shock normal is the shock obliquity angle, θ Bn . When B is perpendicular to the simulation plane,the parallel propagating modes such as AIC waves areknown to be suppressed (Burgess & Scholer 2007). Sowe focus on the ‘in-plane’ magnetic field configuration inthis study. In addition, to insure the zero initial electricfield in the flow frame, the initial electric field is setto E = − u / c × B along the + ˆz direction in thesimulation frame.The incoming plasma is specified by the following pa-rameters relevant for the ICM: n i0 = n e0 = 10 − cm − , k B T i0 = k B T e0 = 8 . β = 20 − B = (cid:112) πk B ( n i0 T i0 + n e0 T e0 ) /β . Reduced ion-to-electron mass ratios, m i /m e = 50 − Ha et al.
Figure 2.
Shock structure in the M2.0 and M3.0 modelsin the region of − ≤ ( x − x sh ) ω pe /c ≤
50 at Ω ci t ∼ n i /n . Pan-els (c)-(d) show the magnetic field strength, B/B . Panels(e)-(f) show the ion temperature, (cid:104) T i (cid:107) (cid:105) y, avg /T i0 (red), and (cid:104) T i ⊥ (cid:105) y, avg /T i0 (blue), averaged over the y -direction. to the limitation of available computational resources(where m e c = 0 .
511 MeV). Then, the sound speed ofthe incoming plasma is defined as c s0 = (cid:112) k B T i0 /m i (where Γ = 5 / v A0 ≈ B / √ πn i0 m i in the upstream re-gion.In weakly magnetized plasmas, the sonic Mach num-ber of the shock induced in such a setup can be estimatedas M s ≡ u sh c s0 ≈ u c s0 rr − , (2)where r = (Γ + 1) / (Γ − /M ) is the Rank-ine–Hugoniot compression ratio across the shock. Then,the Alfv´en Mach number of the shock can be calcu-lated as M A ≡ u sh /v A0 ≈ (cid:112) Γ β/ · M s . In addition,the fast Mach number is defined as M f ≡ u sh /v f0 , where v = { ( c + v )+[( c + v ) − c v cos θ Bn ] / } / β (i.e., c s0 (cid:29) v A0 ), M f ≈ M s .These model parameters are given in Table 1. Modelswith different M s are named with the combination ofthe letter “M” and sonic Mach numbers (e.g., the M3.0 model has M s = 3 . β = 50, θ Bn =63 ◦ , and m i /m e = 50. Models with the parametersdifferent from the fiducial values have the names thatare appended by a character for the specific parameterand its value. For example, the M3.0-m100 model has m i /m e = 100, while the M3.0- θ
53 model has θ Bn = 53 ◦ .In PIC simulations, kinetic plasma processes for dif-ferent species are followed on different length scales, theelectron skin depth, c/ω pe , and the ion skin depth, c/ω pi .Here, ω pe = (cid:112) πe n e /m e and ω pi = (cid:112) πe n i /m i arethe electron and ion plasma frequencies, respectively.On the other hand, the shock structure, which is gov-erned by the ion dynamics, evolves in the timescalesof the ion gyration period, Ω − = m i c/eB ∝ m i √ β ,and varies on the length scales of the Larmor ra-dius for incoming ions, r L , i ≡ u / Ω ci ≈ c/ω pe ) · ( M s / (cid:112) β/ (cid:112) ( m i /m e ) / c/ω pe , defined with the incom-ing electron density n e . In addition, ion scales, c/ω pi ,Ω ci and r L , i , defined with n i0 , B and u , are also usedwhen appropriate.Table 1 lists the size of the 2D simulation domain, L x and L y , in the 9 th and 10 th columns, respectively. Inthe M3.0 model, for example, the transverse dimensionis L y = 310 c/ω pe ≈ . r L , i . All the simulations have thespatial resolution of ∆ x = ∆ y = 0 . c/ω pe and include 32particles (16 per species) per cell. The end of simulationtime, t end , is given in the 12 th and 13 th columns. Thetime step is ∆ t = 0 . ω − . RESULTS4.1.
Shock Criticality and Surface Rippling
To understand the shock structures, we first look atthe 2D spatial distributions of the ion number densityand magnetic field strength for the M2.0 (subcritical)and M3.0 (supercritical) models in Figure 2(a)-(d). Inthe M3.0 model, overshoot/undershoot oscillations de-velop and ripples appear in the shock transition zonealong the shock surface, while the overall shock struc-ture is relatively smooth in the M2.0 model. Panels(e) and (f) of Figure 2 show that the ion temperatureanisotropy, A i >
1, is generated in the shock transi-tion due to the shock-reflected ions in the M3.0 model,while A i ≈ lectron Preacceleration at Weak ICM Shocks ( b )( a ) (cid:1) AIC M s b = 1 0 0 b = 5 0 b = 2 0
0] and yω pe /c = [0 , β = 50 are shown by the black circles con-nected with the black line, while the models with β = 100and 20 are presented by the blue triangles and red squares,respectively. (b) Ion number density, (cid:104) n i (cid:105) y, avg ( x ), averagedfor yω pe /c = [0 , (cid:104) n i (cid:105) x, avg ( y ),averaged for ( x − x sh ) ω pe /c = [ − , ci t ∼
32 are used. stability. In particular, the characteristic length of theripples is λ ripple ∼ c/ω pe ∼ c/ω pi for the fiducialM3.0 model, which is consistent with the wavelength ofthe AIC-driven waves with the maximum growth rate(i.e., λ AIC ∼ c/ω pi ). In addition, the rippling wavespropagate along the shock surface with the Alfv´en speedin the shock overshoot, v A , os = B os / √ πn os m i ∼ . c ,where B os and n os are the magnetic field strength andthe ion number density of the shock overshoot, respec-tively. Hence, we regard that the rippling waves havethe characteristics of the waves driven by the AIC in-stability (Lowe & Burgess 2003).Using both linear theory and hybrid simulations, Garyet al. (1997) presented the instability condition for the AIC instability: I AIC = T i ⊥ T i (cid:107) − − S p β α p i (cid:107) > , (3)where β i (cid:107) = 8 πn i k B T i (cid:107) /B is the ion β parallel to themagnetic field. The fitting parameters are α p ≈ . S p ≈ . β i (cid:107) ≈ −
50 (see their Figure 8). Thiscondition signifies that the AIC instability tends to bestabilized at lower β i (cid:107) due to the stronger magnetizationof ions. For a given value of β i (cid:107) , the AIC growth rateincreases with increasing A i , which in turn depends onthe fraction of reflected ions. Sine the ion temperatureanisotropy is higher at stronger shocks, the transitionzone is expected to be more unstable against the AICinstability in shocks with higher M s .Here, using the simulation results for the anisotropy A i , we calculate the instability parameter, I AIC , whichis shown in Figure 3(a). For M s = 2, I AIC (cid:46)
0, so theAIC instability is stable, which is consistent with thesmooth shock structure shown in Figure 2. The insta-bility parameter increases steeply around M s ∼ . − . M ∗ AIC ≈ . β shocks. We note that this is similar to the crit-ical Mach number for ion reflection, M ∗ s , reported inKRH2019, because the AIC instability is triggered bythe shock-reflected ions.As can be inferred from Equation (3), Figure 3(a)shows that with similar anisotropy A i ’s, I AIC decreasesas β decreases from 100 (blue triangles) to 20 (redsquares) owing to stronger magnetization at lower β .Thus, we expect that the AIC critical Mach numberwould be somewhat higher at lower- β shocks. For exam-ple, Hellinger & Mangeney (1997) estimated M ∗ AIC ∼ β ≈
1, using hybrid simulations. Trotta & Burgess(2019) obtained a similar value, M ∗ AIC ∼ . β ≈
1, as mentioned before.Figure 3(b) shows the y -averaged ion number densityprofile, (cid:104) n i (cid:105) y, avg ( x ), for the models with M s = 2 −
3. Itdemonstrates that the shock becomes supercritical for M s (cid:38) .
3, developing substantial overshoot/undershootoscillations in the shock transition. Figure 3(c), onthe other hand, shows the ion number density profile, (cid:104) n i (cid:105) x, avg ( y ), averaged over the shock transition zone inthe x direction, along the y axis (parallel to the shocksurface) for the same four models. The mean wave-lengths of the shock surface ripples are λ ≈ c/ω pe ∼ r L , i for the M3.0 model (black), while λ ≈ c/ω pe ∼ r L , i for for the M2.5 model (green). Note that here the Lar-mor radius of incoming ions, r L , i ∝ u ∝ M s , scalesapproximately with the shock Mach number. The vari- Ha et al. - 2 - 1 - 4 - 3 - 2 - 1 - 2 - 1 - 2 - 1 - 4 - 3 - 2 - 1 - 2 5 0 - 2 0 0 - 1 5 0 - 1 0 0 - 5 0 005 01 0 01 5 02 0 02 5 03 0 0 x - x s h [ c / w p e ] y[c/ w pe] - 2 5 0 - 2 0 0 - 1 5 0 - 1 0 0 - 5 0 0 x - x s h [ c / w p e ] - 2 5 0 - 2 0 0 - 1 5 0 - 1 0 0 - 5 0 0 5 0 05 01 0 01 5 02 0 02 5 03 0 0 M 3 . 0M 2 . 3 ( a ) ( b ) ( f )( e )( d ) ( c ) x - x s h [ c / w p e ] - 1 . 5- 1 . 0- 0 . 50 . 00 . 51 . 01 . 5 B z / B M 2 . 0
PBz(ky) w h i s t l e r - 2 w h i s t l e rA I C s h o c k t r a n s i t i o n : ( x - x s h ) / r
L , i = [ - 1 , 0 ] f a r d o w n s t r e a m : ( x - x s h ) / r
L , i = [ - 2 . 8 , - 1 . 8 ] k y / 2 p [ w p e , 2 / c ]k y / 2 p [ w p e , 2 / c ]k y / 2 p [ w p e , 2 / c ] A I C w h i s t l e r B B B Figure 4. (a)-(c) Magnetic field fluctuations, B z /B , in the region of ( x − x sh ) ω pe /c = [ − , +50] at Ω ci t ∼
32 for the fiducialmodels with M s = 2, 2.3, and 3. The black arrows indicate the direction of the upstream magnetic field, B . (d)-(f) Magneticpower spectrum, P B z ( k y ) ∝ ( k y / π )( δB z ( k y ) /B ), is calculated for the shock transition region of ( x − x sh ) /r L , i ≈ [ − . , . x − x sh ) /r L , i ≈ [ − . , − .
8] (red) at Ω ci t ∼
32. For these models, r L , i ≡ u / Ω ci ≈ c/ω pe ) · ( M s / (cid:112) β/ (cid:112) ( m i /m e ) /
50. Note that here the wavenumber k y is normalized with the downstream electron skindepth, ω pe , ≈ √ rω pe . The blue and green vertical lines denote the wavenumbers with the maximum growth rate for the AICand whistler instabilities, respectively. ation of (cid:104) n i (cid:105) x, avg along the shock surface is insignificantfor M s (cid:46) . Plasma Waves in Shock Transition
As discussed in Section 2.3, in the shock transi-tion region of supercritical shocks, the ion tempera-ture anisotropy ( A i ) can trigger the AIC instability andion-mirror instability, while the electron temperatureanisotropy ( A e ) can induce the whistler instability andelectron-mirror instability. Due to the large mass ra-tio, typically electron-driven waves grow much faster onmuch smaller scales, compared to ion-driven waves.According to the linear analysis in KHRK2021, for themodel parameters relevant for the fiducial M3.0 model,i.e., A i ≈ . A e ≈ . β = 50, and m i /m e = 50, themaximum growth rates for the whistler, electron-mirror,AIC and ion-mirror instabilities are γ WI ≈ . ce , γ EM ≈ . ce , γ AIC ≈ . ci , and γ IM ≈ . ci ,respectively. So the growth rates of the four instabili- ties have the following order: γ WI (cid:29) γ EM (cid:29) γ AIC > γ IM . (4)The growth time scales of the instabilities, τ inst ≡ /γ inst , expressed in units of Ω − , are τ WI ≈ . τ EM ≈ . τ AIC ≈ .
3, and τ IM ≈ . A i ≈ . A e ≈ . β = 50, and m i /m e = 50, by contrast, the linear anal-ysis estimates that γ WI ≈ . ce , γ EM ≈ . ce ,and γ IM ≈ . ci , while the AIC instability is stable.So electron-scale waves could be excited mainly by thewhistler instability, regardless of M s or the shock crit-icality. Also note that ion-scale fluctuations could be lectron Preacceleration at Weak ICM Shocks B z /B , forthree shock models are shown in Figure 4(a)-(c). Inthe M3.0 model, we observe multi-scale waves with thewavelengths ranging from electron to ion scales in theshock transition region, ( x − x sh ) /r L , i ≈ [ − , B . In theM2.0 model shown in Figure 4(a), by contrast, primarilyelectron-scale waves with small amplitudes are observedin the shock transition, while weak oblique waves prob-ably due to the ion-mirror instability appear in the fardownstream region.The black lines in Figure 4(d)-(f) show the magneticpower spectra, P B z , in the shock transition region shownin the upper panels (a)-(c). In the M3.0 model, P B z indi-cates the presence of multi-scale waves in the wide rangeof wavenumbers, k y / π ∼ [0 . − . ω pe , /c , corre-sponding to λ ∼ [20 − c/ω pe , ∼ [11 . − c/ω pe1 . In particular, the wavelength of the ion-scale wavesdriven by the AIC instability, λ ∼ c/ω pe , ∼ c/ω pe , is similar to the size of shock surface surfaceripples, λ ripple , as shown in Figure 2. P B z shows sub-stantial powers also on the electron-scale waves with k y / π ∼ . ω pe , /c ( λ ∼ c/ω pe , ∼ . c/ω pe )driven by the whistler instability. In the M2.3 model,the electron-scale waves are relatively more dominantthan the ion-scale waves. In contrast to those supercrit-ical cases, the ion-scale waves are almost not inducedin the subcritical M2.0 model,. These results are con-sistent with the fact that multi-scale plasma waves canbe triggered by the AIC instability only in supercriticalshocks with M s (cid:38) . P B z (red lines) in the far down-stream region of ( x − x sh ) ω pe /c = [ − , − A e in the far Here, the wavenumber k y is given in terms of the downstreamelectron skin depth, ω pe , ≈ √ rω pe , since the properties of theexcited waves are determined by the local parameters in the shocktransition zone. - 5 0 0 5 0 1 0 01 . 01 . 52 . 02 . 5- 2 . 5 - 2 . 0 - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 50 . 00 . 51 . 01 . 52 . 02 . 5 2 5 2 6 2 7 2 81 . 01 . 52 . 02 . 5 g x - x s h [ c / w p e ] W c i t p ^ /mec p l l / m e c g sim, g drift W c i t s i m u l a t i o n d r i f t ( c )( b )( a ) Figure 5. (a) Trajectory of a selected electron with respectto the shock location during Ω ci t ∼ − .
5, taken from theM3.0 model simulation. (b) Corresponding trajectory of thesame electron in the p (cid:107) − p ⊥ momentum space. (c) The blueline shows the evolution of the Lorentz factor, γ sim , in thesimulation frame during Ω ci t ∼ −
28, while the red lineshows the energy gain, γ drift = − ( e/m e c ) (cid:82) E z dz , estimatedusing the motional electric field in the shock transition zone. downstream region. So the magnetic power of electron-scale waves is significantly reduced there as well. (2) Insupercritical shocks, ion-scale waves remain relativelysubstantial in the far downstream, even after experi-encing nonlinear evolution. (3) In subcritical shocks, asmall amount of ion temperature anisotropy could ex-cite nonpropagating, oblique waves via the ion-mirrorinstability in the far downstream region, which can beseen in panels (a) and (d) of Figure 4.4.3. Electron Preacceleration via SSDA
To understand the preacceleration mechanism in oursimulations, we examine how electrons gain energy inthe M3.0 model shock. Figure 5 shows the trajectory ofone selected electron that gains energy via the SSDA.During Ω ci t ≈ −
28, from blue to green in panel (a),it is confined in the region of ( x − x sh ) ω pe /c = [ − , Ha et al. - 5 - 4 - 3 - 2 - 1 - 5 - 4 - 3 - 2 - 1 - 2 - 1 - 5 - 4 - 3 - 2 - 1 - 2 - 1 - 5 - 4 - 3 - 2 - 1 ( g - 1)dN/d g M 2 . 0 ( u p s t r e a m ) K R H 2 0 1 9 ( M 2 . 0 ) ( a ) M 2 . 0
M 2 . 3 ( u p s t r e a m ) K R H 2 0 1 9 ( M 2 . 3 ) ( b ) M 2 . 3( c ) M 2 . 5 g - 1 ( g - 1)dN/d g M 2 . 5 ( u p s t r e a m ) K R H 2 0 1 9 ( M 2 . 5 ) ( d ) M 3 . 0 g - 1 M 3 . 0 ( u p s t r e a m ) K R H 2 0 1 9 ( M 3 . 0 )
Figure 6.
Upstream electron energy spectra (red solid lines)at Ω ci t ∼
32 in the fiducial models with M s = 2 −
3. Thespectra are taken from the region of ( x − x sh ) /r L , i = [0 , +1]and the black dashed lines show the Maxwellian distributionsin the upstream. The blue solid lines show the upstreamelectron energy spectra at Ω ci t ∼
30, taken from KRH2019,in which the simulation domain, L y /r L,i , is about 8 timessmaller than that of the simulations in this study. while undergoing stochastic pitch-angle scattering as canbe seen in panel (b). The duration of this confinementis longer than the acceleration time of a single SDA cy-cle ( ∼ Ω − ). Panel (c) compares the variation of theLorentz factor in the simulation (blue line) with the en-ergy gain of γ drift = − ( e/m e c ) (cid:82) E z dz (red line), whichis expected to accumulate from the drift along the mo-tional electric field in the shock transition zone. We con-firm that the preacceleration realized in the simulatedshock is consistent with the SSDA mechanism proposedby previous studies (Katou & Amano 2019; Niemiecet al. 2019). Although electrons can be energized byboth the Fermi-like acceleration and SSDA, the mostenergetic electrons are produced mainly by the SSDA.In KRH2019, the 2D simulation domain was too smallin the transverse direction to include the emergence ofthe shock surface rippling via the AIC instability. Asa result, the SDA-reflected electrons gain energy onlythrough the Fermi-like acceleration in that study. Fig-ure 6 compares the upstream electron energy spectrain the fiducial models of the current study (red) withthe corresponding spectra reported in KRH2019 (blue).The figure clearly demonstrates that in the case of su-percritical shocks, through the SSDA, electrons can beaccelerated further to higher energies in the new simula-tions than in the simulations of KRH2019. In subcriti-cal shocks, on the other hand, the AIC instability is nottriggered and the ensuing SSDA does not occur even inthe new simulations with a larger simulation domain. y [rL,i]y [rL,i] y [rL,i]y [rL,i] x - x s h [ r L , i ] x - x s h [ r
L , i ] x - x s h [ r
L , i ] - 2 . 5 - 2 . 0 - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 - 2 . 5 - 2 . 0 - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 0 . 00 . 51 . 01 . 52 . 02 . 53 . 0 ( f ) M 3 . 0 - (cid:2) (cid:1) (cid:2) (cid:1) b b n i / n ( a ) M 3 . 0 Figure 7.
Ion number density, n i ( x, y ) /n , in the region of( x − x sh ) /r L , i = [ − . , .
6] and y/r L , i = [0 , .
4] at Ω ci t ∼ β , m i /m e , and θ Bn .The fiducial model M3.0 has M s = 3, β = 50, m i /m e = 50,and θ Bn = 63 ◦ . See Table 1 for the parameters of othermodels. The PIC simulation of a β ≈ β ≈ γ inj ∼ β shocks are quite demand-ing. In fact, simulating electron energization all the wayto injection to DSA in β ≈
100 shocks would requiremuch larger simulation domains and much longer simu-lation times.4.4.
Dependence on the Model Parameters
In this section, we examine how our findings dependon the simulation parameters such as β , m i /m e and θ Bn . According to our linear analysis, for shocks with M s = 3, the wavelength with the maximum growth rateranges λ AIC ∼ − c/ω pi for β = 20 − λ AIC is almost independent of the mass ratio, m i /m e (see KHRK2021). First, we compare the ion densitydistributions in the six M s = 3 models with differentparameters in Figure 7. Several points are noted: (1)For all the models, the length scale of induced ripplesranges λ ∼ − . r L , i ( ∼ − c/ω pi ), which is in agood agreement with the linear analysis. (2) Comparingthe three models with different β in the upper panels, wesee some traces of ion-scale ripples linger further down-stream in the β = 20 case, whereas they are mostly lectron Preacceleration at Weak ICM Shocks - 4 - 3 - 2 - 1 - 4 - 3 - 2 - 1 - 4 - 3 - 2 - 1 - 4 - 3 - 2 - 1 PBz(ky)
M 3 . 0 M 3 . 0 - b b ( a ) ( b ) ( c )( d ) M 3 . 0 M 3 . 0 - m 1 0 0 M 3 . 0 M 3 . 0 - (cid:1) (cid:1)
PBz(ky)
M 2 . 0 M 2 . 0 - b b b - d e p e n d e n c e m i / m e - d e p e n d e n c e (cid:1) B n - d e p e n d e n c e
M 2 . 0 M 2 . 0 - m 1 0 0 ( e ) k y / 2 p [ r L , i - 1 ] M 2 . 0 M 2 . 0 - (cid:1) (cid:1) ( f ) k y / 2 p [ r L , i - 1 ]k y / 2 p [ r L , i - 1 ] Figure 8.
Magnetic power spectra, P B z ( k y ) ∝ ( k y / π )( δB z ( k y ) /B ), for the M3.0 models (upper panels) and the M2.0 models(lower panels) with various simulation parameters in the region of ( x − x sh ) /r L , i = [ − . , .
0] at Ω ci t ∼
20. In the fiducial M2.0and M3.0 models, M s = 3, β = 50, m i /m e = 50, and θ Bn = 63 ◦ . See Table 1 for the parameters of other models. Note that herethe wavenumber k y is normalized with the Larmor radius for incoming ions, r L , i ≈ c/ω pe ) · ( M s / (cid:112) β/ (cid:112) ( m i /m e ) / erased by the thermal motions in the β = 100 case. (3)Comparing the three models with different θ Bn , we findthat the shock surface in the model with θ Bn = 73 ◦ isrelatively more unstable than that of the model with θ Bn = 53 ◦ . This is mainly because the motional electricfield is stronger for higher θ Bn , and so the SDA-reflectedions are more energetic. (4) The AIC instability is in-sensitive to m i /m e , so the two models with m i /m e = 50and 100 produce similar results.Figure 8 compares the magnetic field power spec-tra for the M3.0 and M2.0 models with different val-ues of β , m i /m e and θ Bn . In all the models with M s = 3 (upper panels), multi-scale waves in the range of k y r L , i / π ∼ [0 . −
10] ( λ ∼ [0 . − . r L , i ) are induced forthe considered ranges of the simulation parameters. Inall the models with M s = 2 (lower panels), by contrast,mainly electron-scale waves are excited, as expected.In Figure 9, we examine the upstream electron energyspectra for the same set of models shown in Figure 8.The figure shows that the preacceleration depends onlyweakly on β and m i /m e , while it is more efficient withlarger θ Bn due to the stronger motional electric field. Inthe M3.0- θ
73 model, in which the simulation was car-ried out for a longer time, Ω ci t ∼
50 (magenta line inFigure 9(c)), some of the most energetic electrons wereaccelerated to p inj ∼ p th , i ( γ inj ∼ β supercritical shocks, as previously shownin Niemiec et al. (2019) and Trotta & Burgess (2019). In all the models with M s = 2, however, the energy spec-tra seem consistent with the single SDA cycle (e.g., Guoet al. 2014b), and neither the Fermi-like acceleration northe SSDA are effective.Based on the results described in this section, we con-clude that the preacceleration of electrons and the shockcriticality are independent of m i /m e , but depend some-what weakly on β ( ≈ − θ Bn , as long as theshock parameters satisfy the subluminal condition, i.e., θ Bn ≤ arccos( u sh /c ) (see KRH2019). SUMMARYIn supercritical Q ⊥ -shocks, a substantial fraction ofincoming ions and electrons are reflected at the shockramp (e.g. Krasnoselskikh et al. 2013). The gyromo-tion of the reflected ions in the immediate upstream anddownstream of the shock ramp generates the foot andthe transition structures, respectively (see Figure 1).The reflected electrons backstreaming along the back-ground magnetic field can experience the Fermi-like ac-celeration in the shock foot (Guo et al. 2014a,b; Kanget al. 2019), whereas the downstream advected electronsmay undergo the SSDA in the shock transition (Katou &Amano 2019; Niemiec et al. 2019). In both the acceler-ation mechanisms, the primary energy source is the gra-dient drift along the motional electric field in the shocktransition. In the Fermi-like acceleration, electrons arescattered back and forth between the shock ramp and2 Ha et al. - 5 - 4 - 3 - 2 - 1 - 5 - 4 - 3 - 2 - 1 - 2 - 1 - 5 - 4 - 3 - 2 - 1 - 2 - 1 - 2 - 1 - 5 - 4 - 3 - 2 - 1 (cid:1) B n - d e p e n d e n c em i / m e - d e p e n d e n c e b - d e p e n d e n c e ( g - 1)dN/d g M 3 . 0 M 3 . 0 - b b ( a ) ( b ) M 3 . 0 M 3 . 0 - m 1 0 0 ( f )( e )( d ) ( c )
M 3 . 0 M 3 . 0 - (cid:1) (cid:1) (cid:1) W c i t = 5 0 ] ( g - 1)dN/d g g - 1 M 2 . 0 M 2 . 0 - b b g - 1 g - 1 M 2 . 0 M 2 . 0 - m 1 0 0 M 2 . 0 M 2 . 0 - (cid:1) (cid:1)
Figure 9.
Upstream electron energy spectra at Ω ci t ∼
20 for the M3.0 models (upper panels) and the M2.0 models (lowerpanels) with various simulation parameters. The spectra are taken from the region of ( x − x sh ) /r L , i = [0 , +1] and the blackdashed lines show the Maxwellian distributions in the upstream. In the fiducial M2.0 and M3.0 models, β = 50, m i /m e = 50,and θ Bn = 63 ◦ . See Table 1 for the parameters of other models. In panel (c) the magenta line shows the energy spectrum atΩ ci t ∼
50 for the M3.0- θ
73 model. the upstream waves self-generated via the EFI in theshock foot. In the SSDA, electrons undergo stochasticpitch-angle scattering off the multi-scale waves, whichare excited by both the ion and electron temperatureanisotropies, T i ⊥ /T i (cid:107) and T e ⊥ /T e (cid:107) , in the shock transi-tion.In KRH2019, we performed 2D PIC simulations tostudy electron preacceleration in Q ⊥ ICM shocks. Wefound that the Fermi-like acceleration involving multi-ple SDA cycles can operate only in supercritical shockswith the sonic Mach number greater than the criticalMach number M ∗ ef ≈ .
3. In this work, with the specificaim to examine the SSDA due to ion-driven instabilities,we have extended the work of KRH2019 by consideringthe 2D simulation domain large enough to encompassion-scale waves in the transverse direction. As a result,the new set of simulations can include the excitation ofmulti-scale waves from the electron skin depth to the ionLarmor radius, c/ω ep (cid:46) λ (cid:46) r L , i .The main results can be summarized as follows:1. Adopting the numerical values for T i ⊥ , T i (cid:107) , and β i (cid:107) in the shock transition zone of the simulated modelswith M s = 2 −
3, we estimated the instability param-eter, I AIC , defined in Equation (3). Considering boththe behavior of I AIC and the PIC simulation results,we suggest that the critical Mach number above whichthe AIC mode becomes unstable is M ∗ AIC ≈ . β ≈ − M s in supercriticalshocks with M s (cid:38) M ∗ AIC , while the shock structuresseem relatively smooth for subcritical shocks.3. In the transition zone of supercritical shocks, ion-scale waves can be generated by the AIC and ion-mirrorinstabilities due to the ion temperature anisotropy( T i ⊥ /T i (cid:107) > T e ⊥ /T e (cid:107) > β plasma under consideration. In the case of subcriticalshocks with small anisotropies, on the other hand, pri-marily electron-scale waves are induced by the whistlerinstability, while ion-scale waves with small amplitudescould be excited by the ion-mirror instability.4. In β ≈ −
100 supercritical shocks, electrons are con-fined within the shock transition for an extended periodand gain energy by the SSDA, as suggested by previousstudies for β ∼ − lectron Preacceleration at Weak ICM Shocks Q ⊥ ,shocks in the ICM.5. The shock criticality in terms of triggering the AICinstability (or shock surface rippling) depends ratherweakly on the simulation parameters such as m i /m e and θ Bn for the ranges of values considered here. How-ever, the critical Mach number, M ∗ AIC , tends to besomewhat higher at lower β ( ∼