[ Article ]
Journal of the Korean Astronomical Society - Vol. 53, No. 3, pp.59-67
ISSN: 1225-4614 (Print) 2288-890X (Online)
Print publication date 30 Jun 2020
Received 05 Mar 2020 Accepted 14 Apr 2020

# SEMI-ANALYTIC MODELS FOR ELECTRON ACCELERATION IN WEAK ICM SHOCKS

HYESUNG KANG
1Department of Earth Sciences, Pusan National University, Busan 46241 hskang@pusan.ac.kr

Correspondence to: H. Kang

Published under Creative Commons license CC BY-SA 4.0

## Abstract

We propose semi-analytic models for the electron momentum distribution in weak shocks that accounts for both in situ acceleration and re-acceleration through diffusive shock acceleration (DSA). In the former case, a small fraction of incoming electrons is assumed to be reflected at the shock ramp and pre-accelerated to the so-called injection momentum, pinj, above which particles can diffuse across the shock transition and participate in the DSA process. This leads to the DSA power-law distribution extending from the smallest momentum of reflected electrons, pref, all the way to the cutoff momentum, peq, constrained by radiative cooling. In the latter case, fossil electrons, specified by a power-law spectrum with a cutoff, are assumed to be re-accelerated from pref up to peq via DSA. We show that, in the in situ acceleration model, the amplitude of radio synchrotron emission depends strongly on the shock Mach number, whereas it varies rather weakly in the re-acceleration model. Considering the rather turbulent nature of shocks in the intracluster medium, such extreme dependence for the in situ acceleration might not be compatible with the relatively smooth surface brightness of observed radio relics.

## Keywords:

acceleration of particles, cosmic rays, galaxies: clusters: general, shock waves

## 1. INTRODUCTION

Cosmological hydrodynamic simulations predicted that the intracluster medium (ICM) on average encounters shocks several times during the formation of the large scale structures in the Universe (e.g., Ryu et al. 2003; Vazza et al. 2009). As in the case of astrophysical shocks such as the Earth’s bow shock and supernova remnants, ICM shocks are expected to produce cosmic ray protons (CRp) and electrons (CRe) via diffusive shock acceleration (DSA) (e.g. Bell 1978; Drury 1983; Brunetti & Jones 2014). Many merger-driven shocks have been observed and identified as “radio relic shocks” in the outskirts of galaxy clusters through radio synchrotron radiation from shock-accelerated CRe with Lorentz factor γe ∼ 103−104 (e.g. van Weeren et al. 2010, 2019; Kang et al. 2012). Shocks formed in the hot ICM are weak with sonic Mach numbers Ms ≲ 4, which can be inferred from the observed radio spectral index, αν = (${M}_{s}^{2}$ + 3)/2(${M}_{s}^{2}$ − 1), using the test-particle prediction of DSA (e.g. Kang 2011, 2016).

In the following discussion, the shock is specified by the sonic Mach number, Ms, and preshock temperature, T1, where the subscripts, 1 and 2, denote the preshock and postshock states, respectively. The momentum distribution, f(p), scales with the upstream gas density, n1, and so it does not need to be specified. For quantities related to synchrotron emission and cooling, a preshock magnetic field strength B1 = 1 µG, is adopted. The plasma β refers to the ratio of thermal to magnetic pressures, β = Pgas/PB, in the background ICM. Common symbols in physics are used: e.g., me for the electron Corresponding author: H. Kang mass, mp for the proton mass, c for the speed of light, and kB for the Boltzmann constant.

Suprathermal particles above the so-called injection momentum, pinj, have gyroradii large enough to diffuse across the shock transition and may participate in DSA, a.k.a. Fermi 1st-order acceleration, if scattering MHD/plasma waves of sufficient amplitudes are present (e.g., Drury 1983). The pre-acceleration of thermal particles to pinj, i.e., the ‘injection problem’, has been a longstanding key problem in the DSA theory (e.g., Malkov & Drury 2001; Kang et al. 2002; Marcowith et al. 2016). According to plasma simulations of quasi-parallel shocks (Caprioli & Spitkovsky 2014; Caprioli et al. 2015; Ha et al. 2018), some of the incoming protons are specularly reflected by the overshoot in the shock potential and undergo shock drift acceleration (SDA) at the shock front, resulting in the self-excitation of upstream waves via both resonant and non-resonant streaming instabilities. Then the protons are scattered around the shock by those waves, which leads to the formation of the DSA power-law spectrum above pinj ∼ (3.0 − 3.5)pth,p, where pth,p = (2mpkBT2)1/2 is the postshock thermal proton momentum.

On the one hand, electrons are known to be injected and accelerated preferentially at quasi-perpendicular shocks, which involves kinetic processes on electron kinetic scales much smaller than ion scales (e.g., Balogh & Truemann 2013). Earlier studies on the electron preacceleration via self-generated waves focused mainly on high Mach number shocks in β ∼ 1 plasma, which are relevant for supernova blast waves (e.g., Levinson 1992, 1996; Amano & Hoshino 2009; Riquelme & Spitkovsky 2011). Guo et al. (2014) showed, through particle-in-cell (PIC) simulations, that in weak quasi-perpendicular shocks in high beta ICM plasma, a small fraction of incoming electrons are reflected due to the magnetic mirror and energized via SDA, while the backstreaming electrons excite oblique waves via the electron firehose instability (EFI). Due to scattering of electrons between the shock ramp and EFI-induced waves in the shock foot, the pre-accelerated electrons seem to form a DSA power-law spectrum through a Fermi I-like acceleration. However, Kang et al. (2019) showed that such pre-acceleration is effective only in supercritical shocks with Ms ≳ 2.3. Moreover, they suggested that suprathermal electrons may not be energized all the way to pinj ∼ 150pth,e (where pth,e = (2mekBT2)1/2), because the growth of longer waves via the EFI is saturated. On the other hand, Trotta & Burgess (2019) and Kobzar et al. (2019) have demonstrated through hybrid simulations with test-particle electrons (β ≈ 1) and PIC simulations (β ≈ 5), respectively, that at supercritical quasi-perpendicular shocks the rippling of shock surface excited by Alfvén Ion Cyclotron (AIC) instability could induce multi-scale fluctuations, leading to the pre-acceleration of electrons beyond pinj. Whereas the critical Mach number above which the shock rippling becomes active was estimated to be MA,crit ≈ 3.5 for β ≈ 1 shocks (Trotta & Burgess 2019), this problem needs to be investigated for higher β shocks.

Although the DSA model seems to provide a simple and natural explanation for some observed properties of radio relics, such as thin elongated shapes, postshock spectral steepening due to aging electron population, and polarization vectors indicating perpendicular magnetic field directions, there remain some unresolved problems that need further investigation. First of all, the pre-acceleration of thermal electrons to suprathermal energies and the subsequent injection into the DSA process still remains rather uncertain, especially at subcritical shocks with Ms ≲ 2.3 (Kang et al. 2019). Secondly, the fraction of observed merging clusters with detected radio relics is only ∼ 10 % (Feretti et al. 2012), while numerous quasi-perpendicular shocks are expected to form in the ICM (Wittor et al. 2017; Roh et al. 2019). Thirdly, in a few cases, the sonic Mach number inferred from X-ray observations is smaller than that estimated from radio spectral index of radio relics, i.e., MX < Mradio (Akamatsu & Kawahara 2013; Kang 2016). Thus reacceleration of fossil CRe, pre-existing in the ICM, has been suggested as a possible resolution for these puzzles that the DSA model with ‘in situ injection only’ leaves unanswered (e.g., Kang et al. 2012, 2017; Kang 2016).

Based on what we have learned from the previous studies, here we propose semi-analytic models for the momentum distribution function of CRe, f(p), in the two scenarios of DSA at weak quasi-perpendicular shocks in the test-particle regime: (1) an in situ acceleration model in which electrons are injected directly from the background thermal pool at the shock, and (2) a reacceleration model in which pre-existing fossil CRe are accelerated. Although it remains largely unknown if and how CRe are accelerated at subcritical shocks, in this paper we take a heuristic approach and assume that DSA operates at shocks of all Mach numbers.

In the next section we describe in details the semianalytic models for f(p) along with an in-depth discussion of the underlying physical justification. In Section 3 we demonstrate how our model can be applied to weak shocks in the ICM and discuss observational implications. A brief summary will be given in Section 4.

## 2. SEMI-ANALYTIC DSA MODEL

The physics of collisionless shocks depends on various shock parameters including the sonic Mach number, Ms, the plasma β, and the obliquity angle, θBn, between the upstream background magnetic field direction and the shock normal (e.g., Balogh & Truemann 2013). For instance, collisionless shocks can be classified as quasi-parallel (Qll, hereafter) shocks with θBn ≲ 45º and quasi-perpendicular (Q, hereafter) shocks with θBn & 45º ≳ CRp are known to be accelerated efficiently at Qll-shocks, while CRe are accelerated preferentially at Q-shocks (Gosling et al. 1989; Burgess 2007; Caprioli & Spitkovsky 2014; Guo et al. 2014).

In this study, we focus on the electron acceleration at Q-shocks with Ms ≲ 4 that are expected to form in the ICM. Most of the kinetic problems involved in the electron acceleration, including the shock criticality, excitation of waves via microinstabilites, and wave-particle interactions, have been investigated previously for shocks in β ∼ 1 plasma such as the solar wind and the interstellar medium (see Balogh & Truemann 2013; Marcowith et al. 2016). Although a few studies, using kinetic PIC simulations, have recently considered weak shocks in the high β ICM environment (Guo et al. 2014; Matsukiyo & Matsumoto 2015; Kang et al. 2019; Kobzar et al. 2019), full understanding of the electron injection and acceleration into the regime of genuinely diffusive scattering has yet to come.

The main difficulty in reaching such a goal is the severe computational requirements to perform PIC simulations for high β shocks; the ratio of the proton Larmor radius to the electron skip depth increases with β 1/2 . Moreover, to properly study these problems, PIC simulations in at least two-dimensional domains extending up to several proton Larmor radii are required, because kinetic instabilities induced by both protons and electrons may excite waves on multiple scales that propagate in the direction oblique to the background magnetic fields.

### 2.1. Particle Injection to DSA

In this section, we review the current understanding of the injection problem that has been obtained previously through plasma hybrid and PIC simulations. Suprathermal particles, both protons and electrons, with p ≳ 3pth,p could diffuse across the shock both upstream and downstream, and participate in the DSA process, because the shock thickness is of the order of the gyroradius of postshock thermal protons. Thus the injection momentum is often parameterized as

 (1)

where the injection parameter is estimated to be in the range Qi,p ∼ 3.0 − 3.5, according to the hybrid simulations of Qll shocks in β ∼ 1 plasma (Caprioli & Spitkovsky 2014; Caprioli et al. 2015; Ha et al. 2018). On the other hand, Ryu et al. (2019) showed that the DSA power-law with Qi,p ≈ 3.8 gives a postshock CRp energy density less than 10 % of the shock kinetic energy density for Ms ≲ 4, i.e., ECRp < 0.1Esh (where Esh = ${\rho }_{1}{u}_{s}^{2}/2$).

The electron injection at Q-shocks involves somewhat different processes, which can be summarized as follows: (1) the reflection of some of the incoming electrons at the shock ramp due to magnetic deflection, leading to a beam of backstreaming electrons, (2) the energy gain from the motional electric field in the upstream region through shock drift acceleration (SDA), (3) the trapping of electrons near the shock due to the scattering by the upstream waves, which are excited by backstreaming electrons via the EFI, and (4) the formation of a suprathermal tail for ppref with a power-law spectrum, which seems consistent with the test-particle DSA prediction (Guo et al. 2014; Matsukiyo & Matsumoto 2015). Here, pref represents the lowest momentum of the reflected electrons above which the suprathermal power-law tail develops. This is again parameterized as

 (2)

with the injection parameter, which is assumed to be in the range Qi,e ∼ 3.5 − 3.8 as in the case of pinj (e.g., Guo et al. 2014; Kang et al. 2019).

Recently, Kang et al. (2019) showed that the electron pre-acceleration through the combination of reflection, SDA, and EFI may operate only in supercritical Q-shocks with Ms ≳ 2.3 in β ∼ 100 plasma. In addition, they argued that the EFI alone may not energize the electrons all the way to pinj, unless there are preexisting turbulent waves with wavelengths longer than those of the EFI-driven waves. As mentioned earlier, on the one hand, Trotta & Burgess (2019) and Kobzar et al. (2019) showed through 2D simulations that the suprathermal tail may extend to beyond pinj in the presence of multi-scale turbulence excited by the shock rippling instability. But Trotta & Burgess (2019) suggested that the critical Mach number, at which the shock surface rippling starts to develop, is Ms ≈ 3.5 in β ≈ 1 plasma. Hence, we still need to answer the following questions in future studies: (1) if and how the electron injection occurs at subcritical shocks with Ms ≲ 2.3, and (2) how the critical Mach number for the shock surface rippling varies with shock parameters such as β and θBn.

On the other hand, X-ray and radio observations of several radio relics indicate the efficient electron acceleration even at subcritical shocks with 1.5 ≲ Ms ≲ 2.3 (van Weeren et al. 2019). Hence, in the discussion below, we heuristically assume that the DSA power-law spectrum of the accelerated electrons, fe,injp−q (where q = 4${M}_{s}^{2}$ /(${M}_{s}^{2}$ −1)), develops from ∼ pref all the way to the cutoff momentum peq (see below) at Q-shocks of all Mach numbers. This hypothesis needs to be examined in future studies for high β shocks.

Nevertheless, we refer to Figure 4 of Park et al. (2015), in which the acceleration of both protons and electrons at strong Qll-shocks (Ms = 40) were investigated through 1D PIC simulations. There, electrons form a DSA power-law for ppref, because local fields become quasi-perpendicular at some parts of the shock surface due to turbulent magnetic field amplification driven by the strong non-resonant Bell instability.

### 2.2. Test-Particle Solutions for the Injection-only Case

Here we adopt the test-particle solutions of DSA, because dynamical feedbacks of CRp and CRe are expected to be insignificant at weak ICM shocks (e.g., Ryu et al. 2019). Then the isotropic part of the momentum distribution function at the shock position can be approximated by a power-law spectrum with a superexponential cutoff. For the CRp spectrum,

 ${f}_{\mathrm{p}.\mathrm{i}\mathrm{n}\mathrm{j}}\left(p\right)\mathrm{\approx }{f}_{\mathrm{i}\mathrm{n}\mathrm{j}}\cdot {\left(\frac{p}{{p}_{\mathrm{i}\mathrm{n}\mathrm{j}}}\right)}^{-q}\mathrm{exp}\left(-\frac{{p}^{2}}{{p}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{2}}\right).$ (3)

The normalization factor at pinj is given by

 ${f}_{\mathrm{i}\mathrm{n}\mathrm{j}}=\frac{{n}_{\mathrm{p},2}}{{\pi }^{1.5}}{p}_{\mathrm{t}\mathrm{h},\mathrm{p}}^{-3}\mathrm{exp}\left(-{Q}_{\mathrm{i},\mathrm{p}}^{2}\right),$ (4)

where np,2 is the postshock proton number density. The maximum momentum of CRp at a shock age t can be estimated as

 $\frac{{p}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{m}_{\mathrm{p}}c}\mathrm{\approx }\frac{\sigma -1}{6\sigma }\frac{{u}_{\mathrm{s}}^{2}}{{\kappa }^{\mathrm{*}}}t,$ (5)

where σ = n2/n1 is the shock compression ratio and κ∗ is the diffusion coefficient at p = mpc (Kang & Ryu 2011). For ICM shocks, pmax/mpc ≫ 1, so the exponential cutoff at pmax is not important for weak shocks.

Similarly, the test-particle spectrum of CRe can be expressed as

 ${f}_{\mathrm{e},\mathrm{i}\mathrm{n}\mathrm{j}}\left(p\right)\approx {f}_{\mathrm{r}\mathrm{e}\mathrm{f}}\cdot {\left(\frac{p}{{p}_{\mathrm{r}\mathrm{e}\mathrm{f}}}\right)}^{-q}\mathrm{exp}\left(-\frac{{p}^{2}}{{p}_{\mathrm{e}\mathrm{q}}^{2}}\right).$ (6)

The normalization factor at pref is given by

 ${f}_{\mathrm{r}\mathrm{e}\mathrm{f}}=\frac{{n}_{\mathrm{e},2}}{{\pi }^{1.5}}{p}_{\mathrm{t}\mathrm{h},\mathrm{e}}^{-3}\mathrm{exp}\left(-{Q}_{\mathrm{i},\mathrm{e}}^{2}\right),$ (7)

where ne,2 is the postshock electron number density. The cutoff momentum, peq, can be derived from the equilibrium condition that the DSA momentum gains per cycle are equal to the synchrotron/iC losses per cycle (Kang 2011):

 ${p}_{\mathrm{e}\mathrm{q}}=\frac{{m}_{e}^{2}{c}^{2}{u}_{s}}{\sqrt{4{e}^{3}q/27}}{\left(\frac{{B}_{1}}{{B}_{\mathrm{e},1}^{2}+{B}_{\mathrm{e},2}^{2}}\right)}^{1/2},$ (8)

where the ‘effective’ magnetic field strength ${B}_{\mathrm{e}}^{2}$ = B2 + ${B}_{\mathrm{r}\mathrm{a}\mathrm{d}}^{2}$ takes account for radiative losses due to both synchrotron and iC processes, where Brad = 3.24 µG(1+z)2 corresponds to the cosmic background radiation at redshift z. Here, we assume the Bohm diffusion for DSA, and set z = 0.2 as a reference epoch and so Brad = 4.7 µG. For typical ICM shock parameters, it becomes

 (9)

where the magnetic field strength is expressed in units of µG. Again, peq/mec ≫ 104 , so the exponential cutoff is not important for weak shocks.

With the DSA model spectra given in Equations (3) and (6), if Qi,p = Qi,e as assumed here, then the ratio of fp,inj to fe,inj at p = pinj can be estimated as

 ${K}_{p/e}\equiv \frac{{f}_{\mathrm{p},\mathrm{i}\mathrm{n}\mathrm{j}}\left({p}_{\mathrm{i}\mathrm{n}\mathrm{j}}\right)}{{f}_{\mathrm{e},\mathrm{i}\mathrm{n}\mathrm{j}}\left({p}_{\mathrm{i}\mathrm{n}\mathrm{j}}\right)}={\left(\frac{{p}_{\mathrm{t}\mathrm{h},\mathrm{p}}}{{p}_{\mathrm{t}\mathrm{h},\mathrm{e}}}\right)}^{q-3}={\left(\frac{{m}_{\mathrm{p}}}{{m}_{\mathrm{e}}}\right)}^{\left(q-3\right)/2},$ (10)

where Kp/e is equivalent to the CRp-to-CRe number ratio. For example, in the case of a Ms = 3.0 shock with q = 4.5, Kp/e = 280, but with the caveat that protons (electrons) are accelerated at Qll (Q) shocks.

In Figure 1, we illustrate the thermal Maxwellian distribution and the test-particle power-law spectrum, fp,inj(p), for protons, which are demarcated by the magenta line of pinj. The shock parameters adopted here are Qi,p = 3.8, Ms = 3 and T1 = 5.8×107 K, and q = 4.5. Also, the thermal Maxwellian distribution and fe,inj(p) for electrons are demarcated by the green line given by pref with Qi,e = 3.8. This clearly demonstrates that, in order to get injected to DSA, the reflected electrons need to be energized by a factor of pinj/pref$\sqrt{{m}_{\mathrm{p}}/{m}_{\mathrm{e}}}$.

Semi-analytic functions for the momentum distribution, f(p)p4 , in a Ms = 3.0 shock, based on the test-particle DSA model. The red line shows the injected population, fe,inj(p), in Equation (6). The black dashed line shows a power-law spectrum of pre-existing fossil electrons, fe,foss(p), with a slope s = 4.7 and a cutoff momentum pc/mec = 103 . The blue dot-dashed line shows the spectrum of re-accelerated electrons, fe,reacc(p) in Equation (12). The green vertical line denotes pref = Qi,epth,e with Qi,e = 3.8, above which suprathermal electrons are reflected at the shock ramp and accelerated by Fermi-I acceleration. Note that the amplitude of fe,reacc(p) scales with the adopted normalization factor, fo, and so the relative importance between fe,reacc and fe,inj depends on it. The proton spectrum, including both the postshock Maxwellian and injected DSA power-law components, is shown by the black solid line for comparison. The magenta vertical line demarcates the injection momentum, pinj = Qi,ppth,p with Qi,p = 3.8, above which particles can undergo the full DSA process across the shock transition.

### 2.3. Re-acceleration of Fossil CR Electrons

For the preshock population of fossil CRe, we adopt a power-law spectrum with slope s and a cutoff momentum pc for p > pref:

 ${f}_{\mathrm{e},\mathrm{f}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right)={f}_{\mathrm{o}}\cdot {\left(\frac{p}{{p}_{\mathrm{r}\mathrm{e}\mathrm{f}}}\right)}^{-s}\mathrm{exp}\left(-\frac{{p}^{2}}{{p}_{\mathrm{c}}^{2}}\right),$ (11)

where pc/mec = 103−105 is considered in this discussion, and the normalization factor, fo, determines the amount of fossil CRe. Then, the re-accelerated population at the shock can be calculated semi-analytically by the following integration:

 ${f}_{\mathrm{e},\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{e}}\left(p\right)=q\cdot {p}^{-q}{\int }_{{p}_{\mathrm{r}\mathrm{e}\mathrm{f}}}^{p}{p}^{\text{'}q-1}{f}_{\mathrm{e},\mathrm{f}\mathrm{o}\mathrm{s}\mathrm{s}\left({p}^{\text{'}}\right)d{p}^{\text{'}}}$ (12)

(Drury 1983; Kang & Ryu 2011). In Figure 1, we show an example of fe,foss(p) with s = 4.7, and pc/me = 103 marked by the black dashed line, while its re-accelerated spectrum, fe,reacc(p), at a Ms = 3 shock is marked by the blue dot-dashed line. Note that the normalization factor is set to fo = fref in Equation (7) with Qi,e = 3.8 for the purpose of illustration only.

For a power-law fossil population without a cutoff, fe,fossps , the re-accelerated spectrum can be obtained by direct integration (Kang & Ryu 2011): for ppref

 (13)

Although we do not explicitly show it here, the reaccelerated spectrum of pre-existing protons, fp,reacc(p), can be described by the same integration as Equation (12), except that the lower bound should be replaced with pinj and fe,foss(p) should be replaced with a preexisting proton population, fp,pre(p), with appropriate parameters, s, pc, and fo.

## 3. APPLICATION TO RADIO RELICS

### 3.1. DSA Model Spectrum

We use the DSA models given in Equations (6) and (12) to calculate the energy spectrum of accelerated electrons at weak shocks propagating into the preshock gas with T1 = 5.8 × 107 K. Panel (a) of Figure 2 shows the injection spectrum, fe,inj(p), for shocks with Ms = 1.6 − 4.0 (with increment ∆Ms = 0.2). Considering that the synchrotron emission from mono-energetic electrons with the Lorentz factor, γe, peaks around the characteristic frequency, νpeak ≈ 130MHz(γe/104)2 (B/1 µG), we compare the amplitude of fe,inj(prad), where prad = 104mec, at the green vertical dashed line. The magenta line in Figure 3(a) illustrates how fe,inj(prad) depends on the shock Mach number. In the case of the in situ acceleration model, fe,inj(prad) increases by a factor of 4.2 × 103 for the range of Ms = 2.0 − 3.0. This strong dependence is even stronger at lower Mach number, so that fe,inj(prad) decreases almost by a factor of 90, when Ms decreases from 2.0 to 1.8. This implies that the radio surface brightness could vary extremely sensitively with Ms, when a radio relic consists of multiple shocks with slightly different Mach numbers (Roh et al. 2019).

Semi-analytic DSA model for f(p)p4 for shocks with Ms = 1.6 − 4.0 and T1 = 5.8 × 107 K: (a) injected spectrum, fe,inj(p), with Qi,e = 3.8, (b) re-accelerated spectrum, fe,reacc(p), for the power-law spectrum of fossil electrons with s = 4.5 and pc/mec = 103 , (c) the same as (b) except pc/mec = 104 . The blue dashed lines correspond to the models with Ms = 3.0 and q = s = 4.5. Note that the re-accelerated spectrum scales with the normalization factor of fe,foss, i.e., fe,reacc ∝ fo. The green vertical lines denote prad/mec = 104 .

(a) Amplitude of f(prad) at prad = 104mec for the re-acceleration case with slope s = 4.5 and cutoff momentum p/mec = 103 (black circles), 104 (red circles), and 104 (blue circles). The magenta line with asterisks shows f(prad) for the injection-only case. The semi-analytic DSA spectra (Qi,e = 3.8) shown in Figure 2 are used. (b) CRe pressure, ECRe, in units of Esh = ρ1us2/2. The preshock ECRe,1 due to fossil CRe with s = 4.5 and pc/mec = 103 is indicated by open circles, while the postshock ECRe,2 due to re-accelerated CRe is indicated by closed circles. The magenta line with asterisks shows ECRe,2 for the injection-only case. Again, the semi-analytic DSA spectra in Figure 2 are used.

Panels (b) and (c) of Figure 2 show the reaccelerated spectra, fe,reacc(p), of the fossil electron spectrum, fe,foss(p), with s = 4.5 and pc/mec = 103 and 104 , respectively. Again, we set fo = fref with Qi,e = 3.8 as in Figure 1. For stronger shocks with Ms ≥ 3.0, the re-accelerated spectrum is flatter than the fossil spectrum, i.e., , qs. Hence fossil CRe serve only as seed particles, and so fe,reacc(p) does not depend on the cutoff momentum, pc. For weaker shocks with Ms < 3.0, however, q > s. Hence, fe,reacc(p) depends on pc for p > pc, as can be seen in Figure 2. Thus, in the case of weaker shocks with q > s, the cutoff pc should be high enough to obtain the simple power-law, fe,reacc(p) ∝ ps for p > prad. Again, panel (a) of Figure 3 shows how fe,reacc(prad) for the re-acceleration model depends on Ms for the three models of pc/mec = 103 (black), 104 (red), and 105 (blue). It increases only by a factor of 55 for the range of Ms = 2.0 − 3.0 for the models with pc/mec = 103 . In the case of pc/mec = 105 , it increases by an even smaller factor, 9.6, for Ms = 2.0 − 3.0. Thus, the amplitude of fe,reacc(prad) of radio-emitting electrons for the re-acceleration model has a much weaker dependence on Ms, compared to fe,inj(prad) for the in situ acceleration model. Moreover, this dependence becomes weaker if the cutoff momentum is higher than pc/mec ≳ 105 .

Panel (b) of Figure 3 shows the ratio ECRe/Esh due to the following three CRe spectra: (1) fe,reacc(p) in the postshock region, (2) fe,foss(p) with pc/mec = 103 in the preshock region, and (3) fe,inj(p) in the postshock region. With the adopted value of Qi,e = 3.8, the in situ acceleration model (magenta asterisks) predicts that ECRe,2/Esh ≈ 10−3.5 − 10−3 for 1.6 ≲ Ms ≲ 4.4. In the case of the re-acceleration model with s = 4.5, pc/mec = 103 , and fo = fref, the postshock ratio (closed circles) varies rather slowly, ECRe,2/Esh ≈ 10−2.3 − 10−2.1. Of course, this ratio is arbitrary here and scales with the adopted normalization factor, fo, of the fossil CRe population. Note that the preshock ratio (open circles), ECRe,1/Esh, decreases with increasing Ms, because the denominator increases with ${M}_{s}^{2}$ .

We now calculate the synchrotron volume emissivity, jν(ν), due to fe,reacc(p) shown in Figure 2, with the postshock magnetic field strength, B2 = B1$\sqrt{1/3+2{\sigma }^{2}/3}$ (where B1 = µG), in order to illustrate how the radio spectrum changes with the shock Mach number. We do not show explicitly the emissivity spectrum for the in situ acceleration case, since both fe,inj(p) and jν(ν) are simple power-laws with an exponential cutoff.

The top panels of Figure 4 shows νjν, while the bottom panel shows its spectral index, αν = −d ln jν/d ln ν. Note that jν scales with the adopted normalization factor, fo and is plotted in arbitrary units here. For stronger shocks with q < s, jν(ν) is a power-law with αq = (q − 3)/2 = (${M}_{s}^{2}$ + 3)/2(${M}_{s}^{2}$ − 1) with a cutoff. For weaker shocks with q > s, however, the radio spectrum depends on the cutoff momentum of the fossil CRe spectrum, pc, as well as Ms, as expected from fe,reacc(p) in Figure 2. At these weaker shocks, the slope, αν, gradually increases from (s − 3)/2 to (q − 3)/2, as the frequency increases. Hence, the fossil CRe power-law should extend to well above pc/mec ≳ 105 , in order for the spectral index to be determined by the slope of fossil CRe, i.e., αs = (s − 3)/2, for ν ≲ 10 GHz (see panel (f) of Figure 4). For example, if the power-law spectrum of fossil CRe extends only up to pc/mec ≲ 103 , the radio spectral index due to postshock CRe becomes (q − 3)/2 for ν ≳ 153 MHz (see panel (d) of Figure 4).

Top panels: Synchrotron spectrum, νjν, due to fe,reacc(p) re-accelerated at shock with Ms = 1.6 − 4.0 and T1 = 5.8 × 107 K in the presence of fossil CRe with the slope, s = 4.5, and the cutoff momentum, a) p/mec = 103 , (b) p/mec = 104 , (c) p/mec = 105 . The semi-analytic DSA spectra (Qi,e = 3.8) shown in Figure 2 are used. The emissivity νjν is plotted in arbitrary units. The blue dashed lines correspond to the models with Ms = 3.0 and q = s = 4.5. The green vertical lines denote ν = 153 MHz. Bottom panels: Synchrotron spectral index, αν = −d ln jν/d ln ν, for the radiation spectra shown in the top panels. Note that the results for the in situ acceleration model are not shown because the corresponding synchrotron spectrum is a simple power-law with a cutoff.

Figure 5 shows the relative values of jν and αν at three typical observation frequencies, νobs = 153 MHz, 608 MHz, and 1.38 GHz (e.g., van Weeren et al. 2010). Similarly to the case of f(prad), both jν(νobs) and αν(νobs) vary strongly with Ms for the in situ acceleration model (magenta asterisks). For example, j153MHz increases by a factor of 3.7 × 103 , j608MHz by a factor of 6.5 × 103 , and j1.38GHz by a factor of 9.2 × 103 , as Ms increases from 2.0 to 3.0; the Mach number dependence is a bit stronger at higher observational frequencies. Again, the re-acceleration models (filled circles) exhibit much weaker dependence on Ms. For the model with pc/mec = 103 (black filled circles), j153MHz increases by a factor of 48, and j1.38GHz by a factor of 120 for Ms = 2.0 − 3.0. For the model with pc/mec = 105 (blue filled circles), j153MHz increases only by a factor of 14, and j1.38GHz by a factor of 16 for Ms = 2.0 − 3.0.

Top panels: Amplitudes of jν at 153 MHz, 608 MHz, and 1.38 GHz for the synchrotron spectra shown in Figure 4. Three models with fossil CRe with slope s = 4.5, and cutoff momentum p/mec = 103 (black circles), 104 (red circles), and 105 (blue circles) are shown. The magenta lines with asterisks show the same quantities for the injection-only case. Note that jν scales with the adopted normalization factor, fo and is plotted in arbitrary units here. Bottom panels: Synchrotron spectral index, αν = −d ln jν/d ln ν, for the same models shown in the upper panels. Note that the black line (re-acceleration case with p/mec = 103 ) coincides with the magenta line (injection-only case), which also corresponds to αq(Ms) = (q − 3)/2. The green horizontal lines denote αs = (s − 3)/2 = 0.75.

For stronger shocks with qs, even the reacceleration models have a spectral index that follows the injection index, αq (magenta asterisks); thus, in the bottom panels of Figure 5, all symbols (black, red, blue, and magenta) overlap for Ms ≥ 3.0. For weaker models with q > s, the spectral index depends again on pc. For the models with pc/mec = 105 (blue filled circles), αναs for Ms < 3. In the case of the models pc/mec = 103 (black filled circles), the fossil CRe serve as only seed particles, so the spectral indices at the three observational frequencies become the same as the injection index, αq(Ms).

In conclusion, if a radio relic is composed of multiple shocks with slightly different Mach numbers (Roh et al. 2019), the surface brightness fluctuations could be much larger in the in situ acceleration model, compared to the re-acceleration model. The variations in the spectral index profile should be much smaller, however. Relatively smooth profiles of radio flux along the edge of some observed radio relics, such as the Sausage relic (Hoang et al. 2017) and the Toothbrush relic (van Weeren et al. 2016), probably indicate that re-acceleration plays a significant role there.

## 4. SUMMARY

Based on recent studies using plasma kinetic simulations (e.g., Guo et al. 2014; Matsukiyo & Matsumoto 2015; Park et al. 2015; Kang et al. 2019; Trotta & Burgess 2019; Kobzar et al. 2019), we suggest semi-analytic DSA models for the electron (re-)acceleration at weak Q-shocks in the test-particle regime. They rely on the following working assumptions: (1) at Q-shocks of all Mach numbers in the test-particle regime (i.e., Ms ≲ 4), electrons can be pre-accelerated from the thermal pool by both electron and ion kinetic instabilities and injected to the DAS process, and (2) the momentum distribution function of (re)-accelerated electrons follows the prediction of the DSA theory for ppref = Qi,epth,e with Qi,e ≈ 3.5 − 3.8. However, it remains uncertain if and how subcritical shocks with Ms ≲ 2.3 could inject electrons to the DSA process (Kang et al. 2019) or reaccelerate pre-existing fossil CR electrons through DSA. We include the electron (re-)acceleration in subcritical shocks here, because in some of observed radio relics the shock Mach number is estimated to be less than 2.3 (e.g., van Weeren et al. 2016).

Then, the momentum distribution of accelerated electrons can be represented by the simple power-law with a cutoff given in Equation (6) for the in situ acceleration model. For the re-acceleration model with fossil CRe, which is specified by three parameters, the slope, s, the cutoff, pc, and the normalization factor, fo, the reaccelerated spectrum can be integrated semi-analytically as in Equation (12).

We explore how our model spectrum of CRe varies with the parameters such as Ms, s, and pc in the case of weak shocks with Ms = 1.6 − 4.4 for the two types of DSA models: the in situ acceleration model and the re-acceleration model. The main findings can be summarized as follows:

• 1. For stronger shocks with q = 4${M}_{s}^{2}$ /(${M}_{s}^{2}$ − 1) ≤ s, the re-accelerated spectrum becomes a powerlaw, fe,reacc(p) ∝ p−q , and does not depend on pc. The radio synchrotron spectrum likewise becomes a power-law with αναq = (q − 3)/2 = (${M}_{s}^{2}$ + 3)/2(${M}_{s}^{2}$ − 1) and an appropriate cutoff.
• 2. For weaker shocks with q > s, on the other hand, fe,reacc(p) depends on the cutoff pc. Only for pc/mec ≳ 105 , the radio synchrotron spectrum has a spectral index, αναs = (s−3)/2 for observation frequencies in the range νobs ≈ 100 MHz − 10 GHz.
• 3. If pc/mec ≲ 103 , the fossil CRe provide only seed particles to DSA, and hence the spectral index is similar to the injection index, αναq.
• 4. In the in situ acceleration model, the radio synchrotron emissivity, jν, depends strongly on Ms, and it increases by a factor of 103 − 104 as Ms increases from 2.0 to 3.0. But it varies by a factor of only about 15 in the re-acceleration model with pc/mec = 105 for the same range of Ms. In the case of a lower cutoff at pc/mec = 103 , j153MHZ increases by a factor of 48, and j1.38GHZ by a factor of 120 for the same range of Ms.

Considering that the spatial profiles of radio flux and spectral index vary rather smoothly along the edge of some observed radio relics (e.g., van Weeren et al. 2016; Hoang et al. 2017), and that giant radio relics on Mpc scales are likely to consist of multiple shocks with different Ms (e.g., Roh et al. 2019), our results imply that the re-acceleration of fossil CRe is important in understanding the origin of radio relics.

## Acknowledgments

This work was supported by a 2-Year Research Grant of Pusan National University. The author thanks D. Ryu for stimulating discussions at the initial stage of this work.

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### Figure 1.

Semi-analytic functions for the momentum distribution, f(p)p4 , in a Ms = 3.0 shock, based on the test-particle DSA model. The red line shows the injected population, fe,inj(p), in Equation (6). The black dashed line shows a power-law spectrum of pre-existing fossil electrons, fe,foss(p), with a slope s = 4.7 and a cutoff momentum pc/mec = 103 . The blue dot-dashed line shows the spectrum of re-accelerated electrons, fe,reacc(p) in Equation (12). The green vertical line denotes pref = Qi,epth,e with Qi,e = 3.8, above which suprathermal electrons are reflected at the shock ramp and accelerated by Fermi-I acceleration. Note that the amplitude of fe,reacc(p) scales with the adopted normalization factor, fo, and so the relative importance between fe,reacc and fe,inj depends on it. The proton spectrum, including both the postshock Maxwellian and injected DSA power-law components, is shown by the black solid line for comparison. The magenta vertical line demarcates the injection momentum, pinj = Qi,ppth,p with Qi,p = 3.8, above which particles can undergo the full DSA process across the shock transition.

### Figure 2.

Semi-analytic DSA model for f(p)p4 for shocks with Ms = 1.6 − 4.0 and T1 = 5.8 × 107 K: (a) injected spectrum, fe,inj(p), with Qi,e = 3.8, (b) re-accelerated spectrum, fe,reacc(p), for the power-law spectrum of fossil electrons with s = 4.5 and pc/mec = 103 , (c) the same as (b) except pc/mec = 104 . The blue dashed lines correspond to the models with Ms = 3.0 and q = s = 4.5. Note that the re-accelerated spectrum scales with the normalization factor of fe,foss, i.e., fe,reacc ∝ fo. The green vertical lines denote prad/mec = 104 .

### Figure 3.

(a) Amplitude of f(prad) at prad = 104mec for the re-acceleration case with slope s = 4.5 and cutoff momentum p/mec = 103 (black circles), 104 (red circles), and 104 (blue circles). The magenta line with asterisks shows f(prad) for the injection-only case. The semi-analytic DSA spectra (Qi,e = 3.8) shown in Figure 2 are used. (b) CRe pressure, ECRe, in units of Esh = ρ1us2/2. The preshock ECRe,1 due to fossil CRe with s = 4.5 and pc/mec = 103 is indicated by open circles, while the postshock ECRe,2 due to re-accelerated CRe is indicated by closed circles. The magenta line with asterisks shows ECRe,2 for the injection-only case. Again, the semi-analytic DSA spectra in Figure 2 are used.

### Figure 4.

Top panels: Synchrotron spectrum, νjν, due to fe,reacc(p) re-accelerated at shock with Ms = 1.6 − 4.0 and T1 = 5.8 × 107 K in the presence of fossil CRe with the slope, s = 4.5, and the cutoff momentum, a) p/mec = 103 , (b) p/mec = 104 , (c) p/mec = 105 . The semi-analytic DSA spectra (Qi,e = 3.8) shown in Figure 2 are used. The emissivity νjν is plotted in arbitrary units. The blue dashed lines correspond to the models with Ms = 3.0 and q = s = 4.5. The green vertical lines denote ν = 153 MHz. Bottom panels: Synchrotron spectral index, αν = −d ln jν/d ln ν, for the radiation spectra shown in the top panels. Note that the results for the in situ acceleration model are not shown because the corresponding synchrotron spectrum is a simple power-law with a cutoff.

### Figure 5.

Top panels: Amplitudes of jν at 153 MHz, 608 MHz, and 1.38 GHz for the synchrotron spectra shown in Figure 4. Three models with fossil CRe with slope s = 4.5, and cutoff momentum p/mec = 103 (black circles), 104 (red circles), and 105 (blue circles) are shown. The magenta lines with asterisks show the same quantities for the injection-only case. Note that jν scales with the adopted normalization factor, fo and is plotted in arbitrary units here. Bottom panels: Synchrotron spectral index, αν = −d ln jν/d ln ν, for the same models shown in the upper panels. Note that the black line (re-acceleration case with p/mec = 103 ) coincides with the magenta line (injection-only case), which also corresponds to αq(Ms) = (q − 3)/2. The green horizontal lines denote αs = (s − 3)/2 = 0.75.