Using Hα Filaments to Probe Active Galactic Nuclei Feedback in Galaxy Clusters

In this section we examine whether the radial extent of the filaments from the cluster center correlates with AGN luminosity. In order to determine the radial extent at a given time, r_fil, we divide the simulated cluster into radial shells of thickness 1.7 kpc, and identify the radius of the outermost shell in which the cold gas⁴ mass exceeds 10³ M_⊙. We have verified that other values between 10³ and 10⁷ M_⊙ give similar results.

The left panel of Figure 1 shows the evolution of r_fil in simulation TI07, along with the kinetic luminosity associated with the AGN jets, averaged over the dynamical timescale of the filaments. This timescale is calculated as Δt_fil = r_fil/v_out, where v_out ≈ 500 km s⁻¹ is the characteristic outflow speed of the filaments in simulations (e.g., Δt_fil ≈ 10⁸ yr for r_fil = 50 kpc). The average kinetic luminosity calculated in this way is similar to the luminosity inferred from the X-ray measurements of cavities inflated in the ICM by jets. Visual inspection reveals that more powerful outbursts of AGN feedback tend to produce more spatially extended filaments.

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The right panel of Figure 1 shows the kinetic luminosity as a function of the spatial extent of the filaments over time. Two distinct trends are visible: one pertains to the early time in the evolution (t < 2 Gyr), when the Hα filaments extend radially out in all directions from the cluster center, as illustrated in Figure 3. After 2 Gyr, the subsequent AGN outbursts result in filaments that are more collimated along the jet axis, and the jet power and the extent of the filaments can be approximately represented by the linear regression fit:

$$\begin{eqnarray}&&\langle {L}_{{\rm{K}}}\rangle =(9.0\pm 0.1)\times {10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}{\left(\displaystyle \frac{{r}_{\mathrm{fil}}}{100\mathrm{kpc}}\right)}^{0.77\pm 0.02}.\end{eqnarray} \tag{ 1 }$$

Note, however, that the $\langle {L}_{{\rm{K}}}\rangle -{r}_{\mathrm{fil}}$ correlation is not strictly linear and is more accurately described by the additive quantile regression (Koenker 2018), illustrated by the solid green line.

This correlation can be understood by considering the freely expanding filaments in a spherically symmetric gravitational potential, defined by the radial acceleration g(r). In the absence of non-gravitational forces, the terminal radius (r_f) of filaments launched with initial velocity v_i from radius r_i is determined by the conservation of energy

$$\begin{eqnarray}&&{\int }_{{r}_{i}}^{{r}_{f}}g(r){dr}=-\displaystyle \frac{1}{2}{v}_{i}^{2}.\end{eqnarray} \tag{ 2 }$$

Assuming that in the spatial region of interest, the acceleration can be represented by a power law, g(r) ∝ −r^α, then the specific kinetic energy at launch is ${k}_{i}=\tfrac{1}{2}{v}_{i}^{2}\propto {r}_{f}^{\alpha +1}$ (if $\alpha \ne -1$). In the scenario where the initial kinetic energy of the filaments is supplied by AGN feedback, this implies ${L}_{{\rm{K}}}\propto {r}_{f}^{\alpha +1}$. In our simulations, α is defined by the cluster potential and varies from −0.6 to −0.2 between 10 and 100 kpc (see Appendix A in Q18). For the Hα filaments that expand freely to ∼100 kpc, this implies ${L}_{{\rm{K}}}\propto {r}_{f}^{0.8}$. Because the ICM is orders of magnitude more dilute than the filaments, the hydrodynamic forces (i.e., ram pressure) acting on the filaments can be neglected. Therefore, a simple ballistic model presented here provides a satisfactory description of their dynamics.

The derived dependence closely corresponds to that shown in Equation (1), measured for the evolution of filaments after 2 Gyr, indicating that in this stage the filaments that form in jet-driven outflows expand freely into the ICM. In contrast, during the first 2 Gyr the expanding filaments collide and are scattered in all directions by the infalling filaments. As a consequence, their spatial extent is suppressed relative to the later stages of evolution, when the "channel" is cleared for the filaments to expand along the jet axis. Thus, the spatial extent of the Hα filaments also provides a direct measure of how far the outflows reach.

The right panel of of Figure 1 also illustrates the properties of several observed clusters that host extended Hα filaments (McDonald et al. 2010, 2015). The kinetic luminosities for these systems are inferred from the size of their X-ray cavities (Rafferty et al. 2006; McDonald et al. 2015). It is interesting to note that in the context of our simulations the Phoenix cluster corresponds to systems in which a large amount of cold gas obstructs the expansion of the outflowing filaments. Our simulations suggest that the Perseus cluster should also be in this category, because its Hα filaments extend in all directions from the cluster center (Conselice et al. 2001). The implication is that about 10⁸ yr ago Perseus had a powerful AGN outburst with a luminosity of ∼10⁴⁵ erg s⁻¹. If so, the kinetic luminosity inferred from the size of the X-ray cavities in Perseus falls short of this value by about one order of magnitude. This apparent discrepancy may be explained if the central AGN has created multiple cavities during this feedback episode, some of which are not identified in observations.

In this section we investigate the degree of correlation between the AGN feedback luminosity and the mass of the Hα filaments, M_fil. The left panel of Figure 2 shows the evolution of the average AGN luminosity (both kinetic and radiative) and M_fil measured from simulation RT02. $\langle {L}_{{\rm{K}}}\rangle$ and $\langle {L}_{{\rm{R}}}\rangle$ are averaged over the dynamical timescale of the filaments, as described in Section 3.1. We measure M_fil as the mass of the H i gas with rotational velocity <300 km s⁻¹. This criterion allows us to distinguish the filaments from a kinematically separate, rotationally supported cold gas disk that forms at the center of the cluster, and is visible in the middle and right panels of Figure 3.

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The resulting relation between $\langle {L}_{{\rm{K}}}\rangle$ and M_fil is shown in the right panel of Figure 2. In this case, a linear regression fit provides a good representation of the data in the first 9 Gyr

$$\begin{eqnarray}&&\langle {L}_{{\rm{K}}}\rangle =(1.22\pm 0.03)\times {10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}{\left(\displaystyle \frac{{M}_{\mathrm{fil}}}{{10}^{10}{M}_{\odot }}\right)}^{0.91\pm 0.01}.\end{eqnarray} \tag{ 3 }$$

In the context of our model, this implies ${M}_{\mathrm{fil}}^{0.9}\propto {L}_{{\rm{K}}}\propto {\dot{M}}_{\mathrm{BH}}$. If AGN feedback in real systems operates in the same way, this suggests that the mass (or equivalently luminosity) of the Hα filaments can be used as a probe of the SMBH accretion rate. Similarly, the relation between $\langle {L}_{{\rm{R}}}\rangle$ and M_fil can be characterized by a power-law index of 1.83 ± 0.03, so that ${L}_{{\rm{R}}}\propto {M}_{\mathrm{fil}}^{1.8}\propto {\dot{M}}_{\mathrm{BH}}^{2}$. This relationship of L_K and L_R with the SMBH mass accretion rate is a consequence of the prescription used in our simulations to describe the allocation of the AGN luminosity between the kinetic and radiative feedback in the radiatively inefficient accretion state (see Section 2). Thus, the correlation with M_fil in Equation (3) applies to radiatively inefficient AGNs and may differ for radiatively efficient AGNs.

This point is illustrated by the high-luminosity outburst at the end of the simulation, represented by the red data points, which do not fall on the correlation. These data points are associated with the radiatively efficient (high-luminosity) state of the SMBH, when it appears as a radio-loud quasar. Large amounts of energy injected into the ICM by jets and radiation partially suppress the formation of the Hα filaments, thus shifting the data points to the left of the relation. It is worth noting that both the AGN luminosity and Hα filament mass are affected by the numerical resolution in RT runs in a similar way (both are higher in lower-resolution runs, see a resolution study in Q18). Consequently, the correlation between the AGN luminosity and filament mass is maintained regardless of resolution.

If perturbation and turbulence associated with the AGN feedback catalyze the formation of cold gas in the ICM as hypothesized, one would expect that AGN luminosity should also correlate with the velocity dispersion of the Hα filaments. We examine this hypothesis here.

The top panels of Figure 3 illustrate the line-of-sight velocity (v_los) distribution of filaments measured from simulation RT02 at times t = 0.65, 3.61, and 4.94 Gyr. As discussed in Section 3.1, the early stages are characterized by the Hα filaments that extend radially out in all directions. In later stages, however, the filaments are collimated along the jet axis (corresponding to the z-axis), thereby establishing a preferential axis that breaks the spherical symmetry in the velocity dispersion. The filament velocity in these images is weighted by the Hα luminosity, in order to produce a map comparable to those obtained from observations. For this purpose, the Hα luminosity of the gas is assumed to be proportional to the rate of recombination

$$\begin{eqnarray}&&{L}_{{\rm{H}}\alpha }\propto {n}_{p}{n}_{e}{T}_{4}^{-0.942-0.031\,\mathrm{ln}{T}_{4}},\end{eqnarray} \tag{ 4 }$$

where n_p and n_e are the proton and electron number densities, and T₄ ≡ T/10⁴ K (Dong & Draine 2011; Draine 2011). This emission is mainly contributed by the plasma with temperature ∼10⁴ K, and we do not account for the attenuation of the Hα photons due to absorption and scattering.

The bottom panels of Figure 3 show the line-of-sight velocity dispersion (σ_los) of the filaments in the same simulation snapshots. σ_los is calculated in each pixel of the image as a statistical dispersion around the average velocity in each pencil beam (with a cross section of 0.25 kpc × 0.25 kpc) along the line of sight, weighted by the local Hα luminosity. The resulting velocity maps have a lot of similarities with the Hα filaments in Perseus and other clusters (McDonald et al. 2012; Gendron-Marsolais et al. 2018), as well as with other simulations of CCCs (Gaspari et al. 2018; Li et al. 2018).

In order to investigate the relationship with the AGN feedback power, we calculate the velocity dispersion of the entire Hα filamentary network in directions along and perpendicular to the jet axis. We find no statistically significant correlation of the perpendicular components with the feedback power, and measure their time averaged values to be $\langle {\sigma }_{x}\rangle =161\pm 23\,\mathrm{km}\,{{\rm{s}}}^{-1}$ and $\langle {\sigma }_{y}\rangle =162\pm 33\,\mathrm{km}\,{{\rm{s}}}^{-1}$, shown with one standard deviation. Similarly, during the first 2 Gyr, we find no correlation with the value of velocity dispersion along the jet axis, $\langle {\sigma }_{z}\rangle =348\pm 79\,\mathrm{km}\,{{\rm{s}}}^{-1}$. In the later stages (2 Gyr < t < 9 Gyr), however, σ_z correlates positively with the kinetic luminosity. This correlation is illustrated in Figure 4 with an approximate power-law fit

$$\begin{eqnarray}&&\langle {L}_{{\rm{K}}}\rangle =(7.1\pm 0.1)\times {10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}{\left(\displaystyle \frac{{\sigma }_{z}}{300\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{1.67\pm 0.05},\end{eqnarray} \tag{ 5 }$$

and representative additive regression quantiles. As a consequence of the assumptions in our feedback model, a corresponding relation exists between $\langle {L}_{{\rm{R}}}\rangle$ and σ_z. We do not show it here because the kinetic feedback is the primary cause of the correlation. If these measurements are applicable to the real Hα filament systems in observed CCCs, they imply that the observed velocity dispersion is a function of the viewing angle, and may be used as an independent constraint on the spatial orientation of the jets.

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