Effects of Alfvénic Drift on Diffusive Shock Acceleration at Weak Cluster Shocks

Weak shocks with sonic Mach number typically M_s ≲ a few are expected to form in the intracluster medium (ICM) during the course of hierarchical clustering of the large-scale structure of the universe (e.g., Ryu et al. 2003; Kang et al. 2007). The presence of such shocks has been established by X-ray and radio observations of many merging clusters (e.g., Markevitch & Vikhlinin 2007; Brüggen et al. 2012; Brunetti & Jones 2014). In particular, diffuse radio sources known as radio relics, located mostly in cluster outskirts, could be explained by cosmic-ray (CR) electrons (re-)accelerated via diffusive shock acceleration (DSA) at quasi-perpendicular shocks (e.g., van Weeren et al. 2010; Kang et al. 2012; Kang 2017). Although both CR electrons and protons are known to be accelerated at astrophysical shocks such as Earth's bow shocks and supernova remnant shocks (e.g., Bell 1978; Drury 1983; Blandford & Eichler 1987), the γ-ray emission from galaxy clusters, which would be a unique signature of CR protons, has not been detected with high significance so far (Ackermann et al. 2014, 2016; Brunetti 2017).

In galaxy clusters, diffuse γ-ray emission can arise from inelastic collisions of CR protons with thermal protons, which produce neutral pions, followed by the decay of pions into γ-ray photons (e.g., Miniati et al. 2001; Brunetti & Jones 2014; Brunetti 2017). Using cosmological hydrodynamic simulations, the γ-ray emission has been estimated by modeling the production of CR protons at cluster shocks in several studies (e.g., Ensslin et al. 2007; Pinzke & Pfrommer 2010; Vazza et al. 2016). In particular, Vazza et al. (2016) tested several different prescriptions for DSA efficiency by comparing γ-ray flux from simulated clusters with Fermi-LAT upper limits of observed clusters. They found that non-detection of γ-ray emission could be understood only if the CR proton acceleration efficiency at weak cluster shocks is on average less than 10⁻³ for shocks with M_s = 2–5. On the other hand, recent hybrid plasma simulations demonstrated that about 5%–15% of the shock kinetic energy is expected to be transferred to the CR proton energy at quasi-parallel shocks with a wide range of Alfvén Mach numbers, M_A, (Caprioli & Spitkovsky 2014a). So there seems to exist a tension between the CR proton acceleration efficiency predicted by DSA theory and γ-ray observations of galaxy clusters.

It is well established that CR protons streaming along magnetic field lines upstream of parallel shock resonantly excite Aflvén waves with wavenumber k ∼ 1/r_g via two-stream instability, where r_g is the proton Larmor radius (Wentzel 1974; Bell 1978; Lucek & Bell 2000; Schure et al. 2012). These Aflvén waves are circularly polarized in the same sense as the proton gyromotion, i.e.,  left-handed circularly polarized when they propagate parallel to the background magnetic field. The waves act as scattering centers that can scatter CR particles in pitch-angle both upstream and downstream of the shock, leading to the Fermi first order (Fermi I) acceleration at parallel shocks (Bell 1978).

As CRs are scattered and isotropized in the mean wave frame, the spectral index Γ of the CR energy spectrum, N(E) ∝ E^−Γ, is determined by the convection speed of scattering centers in the shock rest frame, u + u_w, instead of the gas flow speed, u (Bell 1978). Here, u_w is the mean speed of scattering centers in the local fluid frame, or the speed of so-called Alfvénic drift. The direction and amplitude of Alfvénic drift depend on the difference between the intensity of forward waves (moving parallel to the flow) and that of backward waves (moving anti-parallel to the flow), i.e., (δB^f)²–(δB^b)² (Skilling 1975). If forward and back waves have the same intensity or if waves are completely isotropized, i.e., (δB^f)² =(δB^b)², then u_w ≈ 0.

A non-resonant instability due to the electric current associated with CRs escaping upstream is also known to operate on small wavelengths (Bell 2004; Schure et al. 2012). The excited waves are not Alfvén waves and have a circular polarization opposite to the sense of the proton gyromotion, i.e., are right-handed circularly polarized when they propagate parallel to the background magnetic field. This non-resonant instability is more unstable at higher k's (smaller wavelengths), and the ratio of the growth rates of non-resonant to resonant instability is roughly, Γ_nonres/Γ_res ∼ M_A/30 (Caprioli & Spitkovsky 2014b). In cluster outskirts where the magnetic field is observed to have B ∼ 1 μG (e.g., Govoni & Feretti 2004), shocks have M_A ≲ 30 (see below), so resonant instability is expected to be dominant there. As we are interested in cluster shocks here, we focus mainly on Alfvén waves excited by resonant streaming instability.

Bell (1978) noted that resonant instability would produce mostly backward waves in the preshock region, because CR protons streaming upstream excite waves that move parallel to the streaming direction (that is, travel upstream away from the shock in the upstream rest frame), and any forward waves pre-existing in the preshock flow would be damped due to the gradient of the CR distribution in the shock precursor (Wentzel 1974; Skilling 1975; Lucek & Bell 2000). Then, the Alfvénic drift speed in the preshock region may be approximated as u_w1 ≈ +V_A1, where ${V}_{{\rm{A}}}={B}_{0}/\sqrt{4\pi \rho }$ is the local Alfvén speed. See Figure 1 for the velocity configuration in the shock rest frame. Hereafter, the subscripts 1 and 2 refer to the quantities in the preshock and postshock regions, respectively.

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Alfvénic drift in the postshock region was previously considered in studies of CR acceleration at strong supernova remnant (SNR) shocks (e.g., Zirakashvili & Ptuskin 2008, 2012; Caprioli et al. 2009; Lee et al. 2012; Kang 2013). Those studies suggested that owing to the positive gradients of the CR pressure, P_CR, forward waves (moving away from the shock toward the center of supernova explosion) could be dominant in the postshock region, then u_w2 ≈ −V_A2 (see Figure 2).

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The effects of Alfvénic drift should be substantial, only if the Alfvén speed is a significant fraction of the flow speed. In SNR shocks, for instance, the Alfvén Mach number is M_A = u₁/V_A ∼ 20–200, depending on the density of the background medium, yet the Alfvénic drift effects could be appreciable (e.g., Caprioli et al. 2009; Kang 2013). For the ICM in cluster outskirts, the sound and Alfvén speeds are given as ${c}_{{\rm{s}}}\approx 1.14\times {10}^{3}\,\mathrm{km}\,{{\rm{s}}}^{-1}{({k}_{{\rm{B}}}T/5\mathrm{keV})}^{1/2}$ and ${V}_{{\rm{A}}}\,\approx 184\,\mathrm{km}\,{{\rm{s}}}^{-1}(B/1\mu G){({n}_{{\rm{H}}}/{10}^{-4}{\mathrm{cm}}^{-3})}^{-1/2}$, respectively, so

$$\begin{eqnarray}&&\beta \equiv {\left(\displaystyle \frac{{c}_{{\rm{s}}}}{{V}_{{\rm{A}}}}\right)}^{2}\approx 40\left(\displaystyle \frac{{n}_{{\rm{H}}}}{{10}^{-4}\,{\mathrm{cm}}^{-3}}\right)\left(\displaystyle \frac{{k}_{{\rm{B}}}T}{5.2\,\mathrm{keV}}\right){\left(\displaystyle \frac{B}{1\mu G}\right)}^{-2},\end{eqnarray} \tag{ 1 }$$

where k_B is the Boltzmann constant. For M_s ≈ 2–3, the Alfvén Mach number of cluster shocks ranges ${M}_{{\rm{A}}}=\sqrt{\beta }{M}_{{\rm{s}}}\approx 13\mbox{--}19$, which is smaller than that of SNR shocks. Thus, we expect that the Alfvénic drift could have non-negligible effects on DSA at cluster shocks. Note that this definition of β differs from the usual plasma beta by a factor of 1.2 for the gas adiabatic index γ = 5/3; the plasma beta of the ICM has been estimated to be ∼50–100 (e.g., Ryu et al. 2008; Porter et al. 2015).

The transmission and reflection of upstream Alfvén waves at shocks can be calculated by solving conservation equations across the shock transition (e.g., Campeanu & Schlickeiser 1992; Vainio & Schlickeiser 1998, 1999; Caprioli et al. 2009). Vainio & Schlickeiser (1998), for instance, used the conservation of mass flux, transverse momentum, and tangential electric field to calculate them, in the small wave amplitude limit (b ≡ δB/B ≪ 1) in the one-dimensional (1D) plane-parallel geometry. They showed that after purely backward waves cross the shock, forward waves are also generated in the postshock region. Vainio & Schlickeiser (1999, hereafter VS99) extended the work by including the pressure and energy flux of waves across the shock. The transmission and reflection of Alfvén waves and so the ensuing CR spectrum are governed by M_A, β, b, and the properties of upstream waves. For certain shock parameters, the effective compression ratio, r_sc, which is defined as the velocity jump of scattering centers (see Section 3), can be even larger than the gas compression ratio, r, leading to a flatter CR energy spectrum.

In this paper, we first estimate the effects of Alfvénic drift on the DSA of protons for 1D planar shocks in high beta (β ≥ 1) plasmas, with the transport of Alfvén waves across the shock transition described in VS99. We then consider two other cases, which are physically motivated: (1) postshock waves are isotropized, i.e.,  u_w2 ≈ 0, and (2) forward waves are dominant in the postshock region, i.e., u_w2 ≈ −V_A2. We examine the Alfvénic drift effects in these cases too.

In the next section, the transmission and reflection of upstream Alfvén waves at 1D planar shocks are described. In Section 3, the effects of the drift of Alfvén waves are discussed with the power-law CR proton spectrum in the test-particle limit. A brief summary including implications of our results at weak cluster shocks is given in Section 4.

VS99 derived necessary jump conditions for the transport of Alfvén waves across parallel shocks, whose configuration is illustrated in Figure 1. We here repeat some of them to keep this paper self-contained. The shock moves to the right, so the preshock and postshock flow speeds in the shock rest frame are ${{\boldsymbol{u}}}_{1}=-{u}_{1}\hat{x}$ and ${{\boldsymbol{u}}}_{2}=-{u}_{2}\hat{x}$, respectively. The background magnetic field is given as ${{\boldsymbol{B}}}_{0}=-{B}_{0}\hat{x}$. CR protons streaming upstream along ${{\boldsymbol{B}}}_{0}$ excite backward waves that travel anti-parallel to the background flow in the local fluid frame. The shock amplifies the incoming backward waves and also generates forward waves in the postshock region. The convection speed of backward waves is W_b1,2 = −(u_1,2–V_A1,2) < 0 (to the left) both upstream and downstream of a parallel shock for the high beta plasmas with β ≥ 1 considered here.

We consider nondispersive, circularly polarized Alfvén waves with small amplitudes (b ≡  δB/B ≪ 1), propagating along the mean background magnetic field, ${{\boldsymbol{B}}}_{0}$, at 1D planar shocks. Note that the formulae below do not differentiate the handedness of wave polarization, as the conservation equations do not depend on it.

The relation for the gas compression ratio, r, across the shock jump can be derived from the Rankine–Hugoniot condition including the pressure and energy flux of waves and is given as the following cubic equation,

$$\begin{eqnarray}&&{b}^{2}{M}_{{\rm{A}}}^{2}r\{(\gamma -1){r}^{2}+[{M}_{{\rm{A}}}^{2}(2-\gamma )-(\gamma +1)]r+\gamma {M}_{{\rm{A}}}^{2}\}\\ &&\quad +\,{({M}_{{\rm{A}}}^{2}-r)}^{2}\{2r\beta -{M}_{{\rm{A}}}^{2}[\gamma +1-(\gamma -1)r]\}=0,\end{eqnarray} \tag{ 2 }$$

for a given set of parameters, M_s, β, and b (VS99). Here, γ = 5/3 is used for the ICM gas.

The bottom-left panel of Figure 3 shows the solution of Equation (2), r, for three beta's (β = 1, 10, and 80) and b = 0.1 in the Mach number range of M_s ≲ 5. Because the background magnetic field is parallel to the shock flow (i.e., parallel shocks) and the transverse components of wave fields are small (δB = 0.1B₀), r is almost identical to the gas compression ratio of gas-dynamic shocks, ${r}_{\mathrm{gas}}=(\gamma +1){M}_{s}^{2}/\{(\gamma -1){M}_{s}^{2}\,+\,2\}$, regardless of β. In fact, r would deviate from r_gas, only if b is substantially large or β is small. In the same panel, two such cases with (b = 0.3 and β = 1) and (b = 0.1 and β = 0.5) are shown for comparison, with the green and magenta lines, respectively, to illustrate such dependence.

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Following VS99, the cross-helicity is defined as

$$\begin{eqnarray}&&{H}_{{\rm{c}}}=\displaystyle \frac{{(\delta {B}^{{\rm{f}}})}^{2}-{(\delta {B}^{{\rm{b}}})}^{2}}{{(\delta {B}^{{\rm{f}}})}^{2}+{(\delta {B}^{{\rm{b}}})}^{2}},\end{eqnarray} \tag{ 3 }$$

where δB^b and δB^f are the magnetic fields of backward and forward waves, respectively. In the preshock region, backward waves are expected to be dominant for CR-mediated shocks (see Introduction), so we assume H_c1 ≈ −1.

For power-law energy spectra of waves with slope q, I(k) ∝ k^−q, the transmission and reflection coefficients for backward and forward waves, respectively, in the postshock region are derived from the equations for transverse momentum and tangential electric field, as follows:

$$\begin{eqnarray}&&T\equiv \displaystyle \frac{\delta {B}_{2}^{{\rm{b}}}}{\delta {B}_{1}}=\displaystyle \frac{{r}^{1/2}+1}{2{r}^{1/2}}{\left(r\displaystyle \frac{{M}_{{\rm{A}}}+{H}_{{\rm{c}}1}}{{M}_{{\rm{A}}}+{r}^{1/2}{H}_{{\rm{c}}1}}\right)}^{(q+1)/2},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&R\equiv \displaystyle \frac{\delta {B}_{2}^{{\rm{f}}}}{\delta {B}_{1}}=\displaystyle \frac{{r}^{1/2}-1}{2{r}^{1/2}}{\left(r\displaystyle \frac{{M}_{{\rm{A}}}+{H}_{{\rm{c}}1}}{{M}_{{\rm{A}}}-{r}^{1/2}{H}_{{\rm{c}}1}}\right)}^{(q+1)/2}\end{eqnarray} \tag{ 5 }$$

(Vainio & Schlickeiser 1998). Note that these coefficients are independent of the wavenumber. According to hybrid simulations of collisionless shocks by Caprioli & Spitkovsky (2014b), for shocks with M_A ≲ 30 where resonant streaming instability dominantly operates, the spectrum of excited magnetic turbulence in the precursor is consistent with I(k) ∝ k⁻¹. So we adopt q = 1. With these coefficients, the downstream cross-helicity can be estimated as

$$\begin{eqnarray}&&{H}_{{\rm{c}}2}={H}_{{\rm{c}}1}\cdot \displaystyle \frac{{T}^{2}-{R}^{2}}{{T}^{2}+{R}^{2}}.\end{eqnarray} \tag{ 6 }$$

The top panels of Figure 3 show T, R, and H_c2, calculated with b = 0.1, q = 1, and H_c1 = −1. One can see that incident backward waves are amplified across the shock with T > 1, while forward waves are generated with 0 < R < 1 (greater R for higher β) in the postshock region. The ensuing downstream cross-helicity ranges −1 < H_c2 ≲ −0.85 for the shocks considered here. We note that the quasi-linear treatment adopted here should break down for nonlinear waves, which are expected to develop via streaming instabilities at strong shocks.

3.1. Scattering Center Compression Ratio and CR Spectral Index

The CR transport at shocks can be described by the diffusion-convection equation,

$$\begin{eqnarray} & & \displaystyle \frac{\partial f}{\partial t}+(u+{u}_{{\rm{w}}})\displaystyle \frac{\partial f}{\partial x}=\displaystyle \frac{1}{3}\displaystyle \frac{\partial (u+{u}_{{\rm{w}}})}{\partial x}\cdot p\displaystyle \frac{\partial f}{\partial p}\\ & & +\displaystyle \frac{\partial }{\partial x}\left[\kappa (x,p)\displaystyle \frac{\partial f}{\partial x}\right]+\displaystyle \frac{1}{{p}^{2}}\displaystyle \frac{\partial }{\partial p}\left({p}^{2}{D}_{{pp}}\displaystyle \frac{\partial f}{\partial p}\right),\end{eqnarray} \tag{ 7 }$$

where f(x, p, t) is the pitch-angle-averaged phase space distribution function for CRs, u is the flow speed, u_w is the local speed of scattering centers, κ(x, p) is the spatial diffusion coefficient, and D_pp is the momentum diffusion coefficient (Skilling 1975; Bell 1978; Schlickeiser 1989). The effects of Alfvénic drift enter through u_w, which is given here as u_w1 = H_c1V_A1 and u_w2 = H_c2V_A2 in the preshock and postshock regions, respectively.

CR particles then experience the velocity change from u₁ + H_c1 V_A1 to u₂ + H_c2 V_A2 across the shock, as they are isopropized in the local wave frame. Then the compression ratio of scattering centers, defined as the velocity jump of scattering centers, is given as

$$\begin{eqnarray}&&{r}_{\mathrm{sc}}\equiv \displaystyle \frac{{u}_{1}+{H}_{{\rm{c}}1}{V}_{{\rm{A}}1}}{{u}_{2}+{H}_{{\rm{c}}2}{V}_{{\rm{A}}2}}=r\displaystyle \frac{{M}_{{\rm{A}}}+{H}_{{\rm{c}}1}}{{M}_{{\rm{A}}}+{r}^{1/2}{H}_{{\rm{c}}2}}.\end{eqnarray} \tag{ 8 }$$

Thus, r_sc can be different from the gas compression ratio, r, from Equation (2), depending on the cross-helicity. The bottom-middle panel of Figure 3 shows that r_sc, calculated for 1D planar shocks (VS99), depends on β and can be greater than r_gas. But for β ≫ 1, r_sc ≈ r_gas, as M_A ≫ 1.

At weak cluster shocks, the CR pressure is dynamically insignificant, that is, shocks are in the test-particle regime, in which the CR energy spectrum, N(E), is represented by a power-law form. Then, its power-law index, Γ, is determined by r_sc as

$$\begin{eqnarray}&&{\rm{\Gamma }}=\displaystyle \frac{{r}_{\mathrm{sc}}+2}{{r}_{\mathrm{sc}}-1}\end{eqnarray} \tag{ 9 }$$

(Bell 1978). The bottom-right panel of Figure 3 shows Γ calculated for 1D planar shocks. Flattening of N(E) due to Alfvénic drift could be substantial for β ≈ 1 (red solid lines), for which even Γ < 2 is predicted. It can be seen that for β ≫ 1 (see blue dot-dashed lines), Γ ≈ Γ_gas as r_sc ≈ r_gas.

3.2. CR Acceleration Efficiency

In the test-particle regime, the amplitude of the CR proton spectrum can be fixed by setting it at the injection momentum, p_inj, and then the momentum distribution function at the shock position, x_s, is given as

$$\begin{eqnarray}&&f({x}_{s},p)={f}_{N}{\left(\displaystyle \frac{p}{{p}_{\mathrm{inj}}}\right)}^{-({\rm{\Gamma }}+2)}\exp \left[-{\left(\displaystyle \frac{p}{{p}_{\mathrm{cut}}}\right)}^{2}\right],\end{eqnarray} \tag{ 10 }$$

where f_N is the normalization factor (Kang & Ryu 2010). The cut-off momentum, p_cut, represents the maximum momentum of CR protons that can be accelerated within the shock age, t_age, and is given as ${p}_{\mathrm{cut}}\propto {u}_{1}^{2}{B}_{0}{t}_{\mathrm{age}}$. As long as p_cut ≫ m_pc, the CR energy density does not depend on its exact value if Γ > 2.

Here, we define p_inj as the minimum momentum above which protons can cross the shock transition and participate in the Fermi I acceleration process, and we describe it using the injection parameter, Q_i, as

$$\begin{eqnarray}&&{p}_{\mathrm{inj}}\equiv {Q}_{{\rm{i}}}\cdot {p}_{\mathrm{th}},\end{eqnarray} \tag{ 11 }$$

where ${p}_{\mathrm{th}}=\sqrt{2{m}_{p}{k}_{{\rm{B}}}{T}_{2}}$ is the proton thermal peak momentum of the postshock gas with temperature T₂ (Kang & Ryu 2010). Using hybrid simulations, Caprioli et al. (2015) demonstrated that the injection momentum increases with the shock obliquity angle, Θ_Bn, and Q_i ≈ 3.3–4.6 for quasi-parallel shocks (Θ_Bn ≲ 45°) with M_A = 5–50 and β ≈ 1. The injection parameter should be affected by the strength of self-generated MHD turbulence, which in turn depends on M_A and β, in addition to Θ_Bn. It is also expected to increase in time as the particle spectrum extends to higher energies for strong shocks with p⁻⁴ momentum distribution, considering that the CR conversion efficiency cannot be greater than 100%. More accurate estimation of Q_i for weak cluster shocks in high β ICM plasmas, however, could only be made through kinetic plasma simulations; however, its value has not yet been precisely defined (see, e.g., Caprioli & Spitkovsky 2014a; Caprioli et al. 2015).

Assuming that f(x_s, p) is anchored to the postshock Maxwellian distribution at p_inj, the normalization factor is given as

$$\begin{eqnarray}&&{f}_{N}=\displaystyle \frac{{n}_{{\rm{H}}2}}{{\pi }^{1.5}}{p}_{\mathrm{th}}^{-3}\exp (-{Q}_{{\rm{i}}}^{2}),\end{eqnarray} \tag{ 12 }$$

where n_H2 is the postshock hydrogen number density (Kang & Ryu 2010).

Figure 4 illustrates how the test-particle spectrum in Equation (10) depends on the sonic Mach number, M_s. We adopt the relevant parameters for cluster shocks, k_BT₁ = 5.2 keV, ${n}_{{\rm{H}}1}={10}^{-4}\,{\mathrm{cm}}^{-3}$, and B₀ = 1 μG , resulting in β ≈ 40. We set Q_i = 3.5 as a representative value, as we here model mostly parallel shocks with small obliquity angles. (Below, we also consider Q_i = 3.8 as a comparison case.) The CR spectra shown have the power-law indices, Γ's, from Equations (8) and (9), which are calculated with H_c2 estimated according to VS99 for 1D planar shocks (also with H_c2 = 0, see Section 3.3). The cut-off momentum for t_age = 10⁸ years is drawn for an illustrative purpose.

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With the spectrum in Equation (10) and Γ > 2 for weak shocks, the CR injection fraction can be estimated as

$$\begin{eqnarray}\xi & \equiv & \displaystyle \frac{1}{{n}_{{\rm{H}}2}}{\displaystyle \int }_{{p}_{\min }}^{{p}_{\mathrm{cut}}}4\pi f({r}_{\mathrm{sc}},p){p}^{2}{dp}\\ & \approx & \displaystyle \frac{4}{\sqrt{\pi }({\rm{\Gamma }}-1)}{Q}_{{\rm{i}}}^{3}\exp (-{Q}_{{\rm{i}}}^{2}),\end{eqnarray} \tag{ 13 }$$

if we take p_min = p_inj as the lower boundary of the CR momentum distribution (Kang & Ryu 2010). According to this definition, the CR injection fraction depends mainly on Γ and Q_i, as normally p_cut ≫ m_pc.

We also define the CR acceleration efficiency as the ratio of the downstream CR energy flux to the shock kinetic energy flux, as follows,

$$\begin{eqnarray}&&\eta \equiv \displaystyle \frac{{f}_{\mathrm{CR}}}{{f}_{\mathrm{kin}}}=\displaystyle \frac{{u}_{2}{E}_{\mathrm{CR}}}{(1/2){\rho }_{1}{u}_{{1}^{3}}}\end{eqnarray} \tag{ 14 }$$

(Kang & Ryu 2013). Here, the postshock CR energy density is given as

$$\begin{eqnarray}&&{E}_{\mathrm{CR}}=4\pi {m}_{p}{c}^{2}{\int }_{{p}_{\min }}^{{p}_{\mathrm{cut}}}(\sqrt{{p}^{2}+1}-1)f({x}_{s},p){p}^{2}{dp},\end{eqnarray} \tag{ 15 }$$

where the particle momentum p is expressed in units of m_pc. Again, we take p_min = p_inj in the calculation of E_CR below. Note that in general, the CR injection fraction and the DSA efficiency sensitively depend on how one specifies p_min, because the CR number is dominated by nonrelativistic particles with p ∼ p_inj.

The left panel of Figure 5 shows the power-law slope, Γ, estimated with H_c2, which is calculated according to VS99. Here, β varies in the ranges relevant to cluster shocks, β = 10–80, as does the background magnetic field as B₀ = 1 μG (β/40)^−1/2. One can see that at weak cluster shocks, H_c2 based on VS99 could flatten the CR spectrum slightly, compared to gas-dynamic shocks without Alfénic drift (green line). But for β ≫ 1, the dependence of Γ on β is rather weak.

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The middle and right panels of Figure 5 show the injection fraction, ξ, and the CR acceleration efficiency, η, respectively, calculated with the test-particle spectrum in Equation (10) with the slope Γ shown in the left panel. Here, the adopted values of k_BT₁ and n_H1 are the same as in Figure 4. Both ξ and η strongly depend on Q_i through the normalization factor f_N, due to the exponential nature of the tail in the Maxwellian distribution. While $\xi \propto {Q}_{{\rm{i}}}^{3}\exp (-{Q}_{{\rm{i}}}^{2})$ from Equation (13), the CR acceleration efficiency can be approximated as $\eta \propto {Q}_{{\rm{i}}}^{5}\exp (-{Q}_{{\rm{i}}}^{2})$ for weak shocks with power-law spectra much steeper than p⁻⁴ (dominated by nonrelativisitc particles). So ξ decreases by a factor of 7 as Q_i increases from 3.5 to 3.8, while η decreases roughly by a factor of 6 or so.

For cluster shocks with M_s ≲ 3, ξ ≲ 3.2 × 10⁻⁴ and η ≲ 2.2 × 10⁻² for Q_i = 3.5, while ξ ≲ 4.6 × 10⁻⁵ and η ≲ 3.6 × 10⁻³ for Q_i = 3.8. This indicates that the estimated CR injection fraction and acceleration efficiency could easily differ by an order of magnitude, depending on the adopted Q_i. For parallel shocks with small obliquity angles (i.e., Θ_Bn ≲ 15°), however, we expect that Q_i is unlikely to be much larger than 3.8 (Caprioli et al. 2015).

3.3. Cases with H_c2 ≈ 0 and H_c2 ≈ + 1

The overall morphology of cluster shocks, induced mainly by merger-driven activities in turbulent ICMs, is expected to be quite complex and different from simple 1D planar shocks (see, e.g., Ha et al. 2018; Vazza et al. 2017). Rather, it can be characterized by portions of spherically expanding shells, composed of multiple shocks with different properties. In addition, vorticity is generated behind curved shock surfaces, leading to a turbulent cascade over a wide range of length scales and turbulent amplification of magnetic fields in the postshock flow (see, e.g., Ryu et al. 2008; Vazza et al. 2017). Then, downstream waves could be isotropized through various MHD and plasma processes in the postshock region, resulting in zero cross-helicity, H_c2 ≈ 0 (equal strengths of T and R). Note that Fermi II acceleration should be operative in this case, but it is expected to be much less efficient than Fermi I acceleration.

In addition, as mentioned in the Introduction, the CR particle distribution peaks at the shock (i.e., decreases downstream) in spherical shocks or even in evolving planar shocks in which the CR pressure at the shock is increasing with time. In that case, the gradient of P_CR is expected to damp backward waves, leaving dominantly forward waves with H_c2 ≈ +1 in the postshock region (Bell 1978; Zirakashvili & Ptuskin 2008; Caprioli et al. 2009). Hence, we here quantitatively examine the effects of Alfvénic drift in these physically motivated cases with H_c2 = 0 and H_c2 = +1, as phenomenological models.

In the panels for r_sc and Γ of Figure 3, the magenta and cyan lines show H_c2 = 0 and H_c2 = +1 cases, respectively. In fact, the scattering center compression ratio is minimized for H_c2 = +1 (see Equation (8)). So this represents the case with the greatest impact of Alfvénic drift (the largest Γ). Moreover, Figure 4 compares the models with H_c2 estimated according to VS99 and the models with H_c2 = 0 (isotropic waves), demonstrating how the Alfvénic drift may affect the CR spectrum.

Figure 6 shows Γ, ξ, and η for the cases with H_c2 = 0 (magenta lines) and H_c2 = +1 (cyan lines), for the model parameters relevant to cluster shocks and Q_i = 3.5 and 3.8. The case for 1D planar shocks with β = 40, calculated by following the VS99 approach (black lines), where −1 ≲ H_c2 ≲ −0.85, is also plotted for comparison. Again, the green lines show the results for gas-dynamic shocks without Alfvénic drift. The scattering center compression ratio, r_sc, is smaller for larger H_c2, resulting in steeper Γ, hence, smaller ξ and η. For weak cluster shocks with 2 ≲ M_s ≲ 3 and isotropic downstream waves with H_c2 = 0, the CR acceleration efficiency is 10⁻³ ≲ η ≲ 10⁻² for Q_i = 3.5 and 2 × 10⁻⁴ ≲ η ≲ 1.5 × 10⁻³ for Q_i = 3.8. For the case of dominantly forward waves with H_c2 = +1, on the other hand, $7\times {10}^{-4}\,\lesssim \eta \,\lesssim \,4\times {10}^{-3}$ for Q_i = 3.5 and 10⁻⁴ ≲ η ≲ 7 × 10⁻⁴ for Q_i = 3.8. Our results indicate that η could be reduced by "a factor of up to ∼5" due to Alfvénic drift alone. Thus, to quantify the CR acceleration efficiency, it could be crucial not only to constrain the injection parameter Q_i through plasma simulations, but also to account for Alfvénic drift effects.

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We study the effects of Alfvénic drift on the DSA of CR protons at weak shocks in high beta ICM plasmas. We assume that upstream Alfvén waves are self-excited by CR protons via resonant streaming instability at parallel shocks (Lucek & Bell 2000; Schure et al. 2012). Such waves are mostly backward, moving anti-parallel to the background flow (Bell 1978), so they can be characterized by the cross-helicity of H_c1 ≈ −1 (see Equation (3) for the definition of H_c). As CR protons are scattered and isotropized in the local wave frame, the scattering center compression ratio, r_sc, in Equation (8), which accounts for the mean drift of Alvén waves, determines the spectral index, Γ, of the CR spectrum in the test-particle limit.

We first consider 1D planar shocks where the transport of Alfvén waves across the shock transition is described in the small wave amplitude limit (b ≡ δB/B ≪ 1) (Vainio & Schlickeiser 1998, and V99). In this limit, as noted by VS99, Alfvénic drift may increase or decrease r_sc, depending on the shock parameters. This results in the CR spectra either flatter or steeper, compared to that for gas-dynamic shocks without Alfvénic drift. For shocks with M_s ≲ 3 and β ≡ (M_A/M_s)² ∼ 40–80, a mixture of backward and forward waves are present in the postshock region with the postshock cross-helicity estimated to −1 ≲ H_c2 ≲ −0.85, leading to only a slight decrease of Γ (see Figure 3). That is, for weak cluster shocks, r_sc ≈ r_gas and Γ ≈ Γ_gas, and so the effects of Alfvénic drift on the DSA efficiency are only marginal (see Figure 5).

We then consider two additional, physically motivated cases: (1) downstream waves are isotropic with H_c2 ≈ 0, and (2) they are dominantly forward with H_c2 ≈ +1. The former could be realistic, if waves are isotropized via a variety of MHD and plasma processes including turbulence while they cross the shock transition. The latter may be relevant, if the CR pressure distribution peaks at the shock as in spherical SNR shocks or evolving planar shocks. In these two cases, Alfvénic drift causes the CR spectrum to be steeper, which results in significant reductions of the CR injection fraction, ξ, and the CR acceleration efficiency, η (see Figure 6). In the case of H_c2 ≈+1, for example, the CR proton acceleration efficiency for shocks with M_s ≲ 3 and β ≈ 40 could be reduced by "a factor of up to five," compared to that for gas-dynamic shocks. So, we conclude that the Alfvénic drift effects on the DSA efficiency could be substantial at weak cluster shocks.

We note that the CR acceleration efficiency is most sensitive to the injection momentum, or, the injection parameter, Q_i, defined in Equation (11). Increasing Q_i from 3.5 to 3.8 (about 10%), for instance, reduces η by a factor of five to seven. For parallel shocks with small obliquity angles, we expect that Q_i = 3.5–3.8 would be a reasonable range. Thus, in order to reliably estimate the CR proton acceleration efficiency at weak cluster shocks, it is important to understand the kinetic plasma processes that govern the particle injection to Fermi I acceleration at collisionless shocks at high beta plasmas.

We suggest η could vary in a wide range of 10⁻⁴−10⁻² for weak cluster shocks with M_s ≈ 2–3, depending on H_c2, Θ_Bn, and β. Such an estimate could be smaller by up to an order of magnitude than those adopted in previous studies, such as that of Vazza et al. (2016). Thus, this study implies that there remains room for the DSA prediction for CR proton acceleration at cluster shocks to be compatible with non-detection of γ-ray emission from galaxy clusters (Ackermann et al. 2014, 2016). Yet, we emphasize that eventually detailed quantitative studies of DSA at weak cluster shocks using kinetic plasma simulations should be crucial for solving this problem.

We thank the anonymous referee for constructive comments. H.K. was supported by the Basic Science Research Program of the NRF of Korea through grant 2017R1D1A1A09000567. D.R. was supported by the NRF of Korea through grants 2016R1A5A1013277 and 2017R1A2A1A05071429. The authors also thank R. Schlickeiser for helpful comments during the initial stage of this work.