Studying the Effect of Shock Obliquity on the $\gamma$-ray and Diffuse Radio Emission in Galaxy Clusters

Observations of diffuse radio emission in galaxy clusters indicate that cosmic-ray electrons are accelerated on $\sim$Mpc scales. However, protons appear to be accelerated less efficiently since their associated hadronic $\gamma$-ray emission has not yet been detected. Inspired by recent particle-in-cell simulations, we study the cosmic-ray production and its signatures under the hypothesis that the efficiency of shock acceleration depends on the Mach number and on the shock obliquity. For this purpose, we combine ENZO cosmological magneto-hydrodynamical simulations with a Lagrangian tracer code to follow the properties of the cosmic rays. Our simulations suggest that the distribution of obliquities in galaxy clusters is random to first order. Quasi-perpendicular shocks are able to accelerate cosmic-ray electrons to the energies needed to produce observable radio emission. However, the $\gamma$-ray emission is lowered by a factor of a few, $\sim$3, if cosmic-ray protons are only accelerated by quasi-parallel shocks, reducing (yet not entirely solving) the tension with the non-detection of hadronic $\gamma$-ray emission by the \textit{Fermi}-satellite.


Introduction
The existence of peripheral, elongated and often polarised radio emission in galaxy clusters, so-called radio relics, gives evidence of cosmic-ray electrons being accelerated by shock waves in the intracluster medium (see [1] and references therein). Cosmic-ray protons should undergo the same acceleration mechanism, but no evidence of their presence has been found yet. The Large Area Telescope on-board of the Fermi-satellite [2] is searching for signatures of the cosmic-ray protons, which should produce γ-ray emission through collisions with the thermal gas. Yet no detection of these γ-rays has been confirmed and for a variety of clusters the upper flux limits have been estimated to be in the range of 0.5 − 2.2 × 10 −10 ph/s/cm 2 above 500 MeV [3]. Extended searches for the γ-ray emission in the Coma cluster [4] and the Virgo cluster [5] have set the flux limits above 100 MeV to 5.2 × 10 −9 ph/s/cm 2 for the former and to 1.2 × 10 −8 ph/s/cm 2 for the latter.
Recent results from particle-in-cell simulations [6][7][8] suggest that the efficiency of shock acceleration does not only depend on the shock strength but also on the shock obliquity, e.g., the angle between the shock normal and the underlying upstream magnetic field. Cosmic-ray protons should be accelerated more efficiently by diffusive shock acceleration (DSA) in parallel shocks [6]. In contrast, cosmic-ray electrons should prefer a perpendicular configuration as they are first accelerated by shock drift acceleration before they are injected into the DSA cycle [7,8].
In our recent work, we have tested if, in galaxy clusters, the additional dependence on the shock obliquity can explain the missing γ-ray emission and still produce detectable radio relics [9]. In this contribution we present the most relevant results from that work and include new results.

Cosmological MHD Simulations
The cosmological magneto-hydrodynamical (MHD) simulation presented in this work has been carried out with the ENZO-code [10]. In our simulation, we solve the MHD equations (see Section 2.1 in [10]) using a piecewise linear method [11] in combination with hyperbolic Dedner cleaning [12]. We re-simulate a single galaxy cluster with a final mass of M 200 (z ≈ 0) ≈ 9.74 × 10 14 M . The cluster shows a major merger at z ≈ 0.27, which is strong enough to produce detectable radio relics (see [9] for further information). We simulate a 250 3 ∼ Mpc 3 comoving volume from z ≈ 30 to z ≈ 0, starting from a root grid of 256 3 cells and 256 3 dark matter particles. Furthermore, using five levels of Adaptive Mesh Refinement (AMR), we refine 2 5 times a ≈ 25 3 Mpc 3 region centred around our massive cluster, resulting in a final resolution of 31.7 kpc for a large portion of the cluster volume. For the seeding of the large scale-magnetic fields, we use a primordial magnetic seed field with a comoving value of B 0 = 10 −10 G along each direction.

Lagrangian Analysis
We track the evolution of cosmic rays using Lagrangian tracer particles (see [9] for more details). The tracer particles follow, both, the advection of the baryonic matter and the enrichment of shock-injected cosmic rays in time. In post-processing, the tracers are advected in a sub-box consisting of 256 3 cells of the finest grid of the simulation. The sub-box is centred around the mass centre of our galaxy cluster at z ≈ 0. The tracers are first injected into the box at z ≈ 1 following the mass distribution of the gas. During run-time, additional tracers are injected from the boundaries following the mass distribution of the entering gas. In total, we generate N p ≈ 1.33 × 10 7 tracers with a final mass resolution of m tracer ≈ 10 8 M .
The tracers are advected linearly in time using the velocities at their location: v =ṽ + δv. Here, v is the interpolated velocity between the neighbouring cells and δv (see Equation (1) in [9]) is a correction term to cure for a possible underestimate due to mixing in complex flows (see [13] for more details).
The local gas values are assigned to every tracer and other properties are computed on the fly. Subsequently, we apply a shock-finding method based on the temperature jump between the positions of a tracer at two consecutive timesteps, similar to the method described in [14]. Every time a shock is recorded, the Mach number and the corresponding shock obliquity are computed. The latter is calculated using the velocity jump ∆v = v post − v pre between the pre-and post-shock velocity of the tracer: In the equation above, the index i refers to either the pre-or post-shock values. Across each shock, we compute the kinetic energy flux as F Ψ = 0.5 · ρ pre v 3 sh , where ρ pre is the pre-shock density and v sh is the shock velocity. The thermal energy flux, F th = δ(M)F Ψ , and cosmic-ray energy flux, F CR = η(M)F Ψ , are computed using the acceleration efficiencies δ (M) and η (M) given in [15]. The efficiency, η(M), is taken from Kang and Ryu [15] and it includes the effects of magnetic field amplification at the shocks and thermal leakage of suprathermal particles. We include (as in [9,16]) the effect of re-acceleration by computing an effective η eff (M) that is interpolated from the acceleration efficiencies of acceleration and re-acceleration given in Kang and Ryu [15].
We use the formula given in [17] to compute the radio emission from the shocked tracers: ( The quantities that are taken from the grid are: A the surface area of a tracer, n e the number density of electrons, T d the downstream temperature, B the magnetic field strength and the acceleration efficiency η (M) depending on the Mach number M taken from [15]. We notice that the application of η(M) to Equation (2) is limited to spectra flatter than s ≈ −3, because Equation (2) has been derived in energy space while η(M) has been derived in momentum space. However, our modelling is accurate enough for the radio frequency, produced by electrons with Lorentz factor of γ > 10 3 , that we are investigating here [17]. The other quantities are the observed frequency band, ν obs = 1.4 GHz, the equivalent magnetic field of the cosmic microwave background, B CMB = 3.2 · (1 + z) 2 µG and electron-to-proton ratio ξ e = 0.01. Following Hoeft and Brüggen [17] we assume that the minimum electron energy is 10 times the thermal gas energy, while the minimum proton energy is fixed to 780 MeV. We use the same approach as in [18][19][20] to compute the γ-ray emission. We refer to Appendix C of our previous publication [9] for a summary of the method.

Results
In [9] we studied how linking the shock acceleration efficiency to the shock obliquity can affect the acceleration of cosmic rays by predicting the amount of radio and γ-ray emission produced by either quasi-perpendicular or quasi-parallel shocks. Following Figure 3 of Caprioli and Spitkovsky [6] we define quasi-perpendicular shocks as θ ∈ [50 • , 130 • ] and quasi-parallel shocks as θ ∈ [0 • , 50 • ] & [130 • , 180 • ]. However, a more detailed analysis in Wittor et al. [9] showed that the effects on the acceleration of cosmic rays are not very sensitive to the selection of θ. We found that the distribution of shock obliquities in a galaxy cluster roughly follows the distribution of random angles in three-dimensional space, ∝ sin(α). Just based on this, one can expect to observe more quasi-perpendicular shocks than quasi-parallel shocks. Hence, the acceleration of cosmic-ray electrons should be more favoured than the acceleration of cosmic-ray protons. For the results on how this affects the radio and γ-ray emission, we point to our previous publication [9] as they are similar to the ones presented below.
In this contribution we present a closer analysis of the same cluster at the epoch of the peak of the total radio emission. We show the projection of the gas density overlayed with the radio contours in Figure 1. The cluster is still in a very active phase after it experienced a major merger at z ≈ 0.27, and several smaller gas clumps are still falling onto the cluster.
First, we measure the distribution of pre-and post-shock obliquities at z ≈ 0.2. The left panel of Figure 2 shows the measured distributions consistent with isotropy. The right panel of Figure 2 shows the distribution of pre-shock obliquities for different selections in the shock Mach numbers. While the obliquity distributions of M < 3 shocks roughly follow the distribution of random angles, stronger (and rarer) shocks are found to cluster at specific obliquity values, related to single large-scale magnetic structures in the cluster volume.
In the following we perform a similar analysis as in our previous work [9] to investigate how coupling the shock acceleration efficiencies to the shock obliquity affects the γ-ray emission 3.1 and the radio emission 3.2. Projected gas density (colours) and radio contours at z ≈ 0.2. Two radio relics can be seen on the right (P radio ≈ 3.42 × 10 31 erg/s/Hz) and left (P radio ≈ 2.26 × 10 32 erg/s/Hz) side of the cluster centre.

γ-ray Emission
In Figure 3, we show the total integrated γ-ray emission and radial γ-ray emission profiles produced in the cluster at this epoch. The total γ-ray emission received from inside r 200 is ∼1.03×10 45 ph/s, which is above the corresponding Fermi-limits (see [9] for the exact computations) of the Coma (0.035 × 10 45 ph/s) cluster and just below the limits of A2256 (1.075 × 10 45 ph/s). The γ-ray emission resulting from cosmic-ray protons that have been accelerated only by quasi-parallel shocks is ∼0.31×10 45 ph/s. This is still above the lowest upper limit of the Coma cluster. The observed drop in γ-ray emission is consistent with the fact that at low Mach numbers only ∼one-third of all shocks are quasi-parallel (see Figure 2). Consistent with our findings from [9], we conclude that the missing γ-ray emission cannot be entirely reproduced by limiting the acceleration of cosmic-ray protons to quasi-parallel shocks.

Radio Emission
We observe two radio relics on the left (hereafter relic one) and right (hereafter relic two) side of the cluster core (see Figure 1). Both relics are in the range detectability by modern radio observations. Figure 4 shows the complex geometry of the magnetic field lines in the relic regions. We observe that the morphologies of the relics do not change significantly, if only either quasi-perpendicular (middle panel) or quasi-parallel (right panel) shocks are able to accelerate cosmic-ray electrons.
In the first relic, only ∼ 46% of the radio emission is produced by cosmic-ray electrons that have been accelerated by quasi-perpendicular shocks. This is consistent with the distribution of obliquities (see Figure 2), as relic one is mostly produced by a higher Mach number shocks with M ∼ 3 − 5 (see Figure 5), that do not follow the distribution of random angles in a three-dimensional space (see Figure 2). On the other hand, ∼59% of the shocks producing relic two are quasi-perpendicular, as it is produced by weaker shocks with M ∼ 2 − 3 (see Figure 5).
However, consistent with [9], we find that both simulated relics remain visible if the acceleration of electrons is limited to quasi-perpendicular shocks.

Discussion
Combining MHD-simulations and Lagrangian tracers, we continued our study on how restricting the shock acceleration efficiencies to the obliquity affects cosmic rays in galaxy clusters. At the epoch of the highest radio emission, we examined how cosmic rays, that have been accelerated by either quasi-parallel or quasi-perpendicular shocks, contribute to the resulting γ-ray and radio emission. We chose this epoch for our investigation as the two radio relics are the most prominent.
Our findings agree with our results from [9]: The distribution of shock obliquities follows the distribution of random angles in a three-dimensional space. Furthermore, we discovered that this only holds for low Mach numbers M ≤ 3. The distribution of shock obliquities for the few high Mach numbers M ≥ 3 does not show this trend, as they tend to cluster around single magnetic field structures.
Consistent with our findings from [9], the γ-ray emission drops by a factor of ∼3 if only quasi-parallel shocks are able to accelerate the cosmic rays. Yet, this drop is not large enough the explain the low upper limits set by the Fermi-satellite [3], especially in the case of the Coma cluster [4].
On the other hand, the radio emission remains observable if only quasi-perpendicular shocks are able to accelerate cosmic rays. This also holds if the majority of the radio emission is produced by a strong quasi-parallel shock. This supports our conclusion from [9] that it is possible that the cosmic-ray electrons in observed radio relics have only been accelerated by quasi-perpendicular shocks.
We mention that we do not include any other additional mechanisms, such as cosmic-ray re-acceleration by cluster weather or turbulence (e.g., [21]) which would produce further cosmic-ray protons. On the other hand, we do not allow any spatial diffusion of the cosmic-rays, that would reduce the γ-ray flux through proton accumulation in the cluster outskirts (e.g., [22,23]).