Shaken Snow Globes: Kinematic Tracers of the Multiphase Condensation Cascade in Massive Galaxies, Groups, and Clusters

The goal of the G12 suite of simulations is to study the evolution and properties of the long-term self-regulated kinetic AGN feedback affecting the X-ray plasma halo. The simulation models a typical cool-core galaxy cluster with central plasma²⁶ entropy K₀ ≃ 15 keV cm² (minimum t_cool/t_ff < 10). The plasma halo is initially perturbed by random fluctuations in density and temperature with 0.3 relative amplitude to model the presence of cosmic weather (G12, Section 2.3). The maximum AMR resolution reaches 300 pc, so that it is possible to evolve the system for several gigayears in a large 1.3³ Mpc³ domain. The static gravitational potential is dominated by the dark matter component with a Navarro–Frenk–White profile; the cluster virial mass is M_vir ≈ 10¹⁵ ${M}_{\odot }$ with gas fraction ≈0.15.

Besides hydrodynamics,²⁷ the two key competing physics are radiative cooling and AGN feedback. The plasma radiative cooling induces the gas to lose thermal energy, and thus pressure support, forming extended warm filaments via nonlinear condensation. The plasma halo emits radiation mainly via bremsstrahlung above 1 keV and line recombination below such soft X-ray regime. The emissivity is ≃n² Λ(T, Z), where Λ is the Sutherland & Dopita (1993) plasma cooling function in collisional ionization equilibrium. The plasma cooling curve incorporates calculations for H, He, C, N, O, Fe, Ne, Na, Si, Mg, Al, Ar, S, Cl, Ca, and Ni, and all stages of ionization. Due to the limited resolution in this run, condensation is halted at the warm phase regime around 10⁴ K. The cooling source term is integrated with an exact solver with a conservative time-step limiter (G12, Section 2.1).

Radiative cooling is counterbalanced by AGN feedback, in the form of massive subrelativistic outflows. The bipolar outflow mass, momentum, and energy are injected through an internal boundary nozzle in the innermost resolved region (G12, Section 2.2)—the locus of the SMBH. The injected velocity is v_out = 5 × 10⁴ $\mathrm{km}\,{{\rm{s}}}^{-1}$, which is typical of observed entrained FRI jets or ionized ultrafast outflows (e.g., Tombesi et al. 2013). The injected kinetic power is self-regulated by the central inflow rate, ${P}_{\mathrm{out}}=(1/2){\dot{M}}_{\mathrm{out}}\,{v}_{\mathrm{out}}^{2}\simeq \varepsilon {\dot{M}}_{\mathrm{in}}\,{c}^{2}$, where ε = 6 × 10⁻³ is the mechanical efficiency able to avoid both overcooling and overheating. The triggering nuclear mass inflow ${\dot{M}}_{\mathrm{in}}$ (r < 500 pc) is dominated by the condensed gas phase (thus linked to the cooling rate of the hot halo) in the form of raining filaments and clouds (a.k.a. CCA—Section 2.2).

The self-regulated, bipolar AGN outflows propagate outwards and gently dissipate the mechanical energy via bubble mixing, turbulence, and weak shocks, thus restoring most of the previously radiated internal energy (i.e., a global quasi-thermal equilibrium), with a duty cycle of the order of the central t_cool (see Gaspari & Sa̧dowski 2017 for a review). This simulation has been shown to be consistent with several spectroscopical X-ray constraints, including the suppression of the cooling flow by over 20 fold and the gigayear survival of the cool core (preserving the positive T gradient; Gaspari et al. 2013a).

While G12 runs model a more realistic AGN feedback injection, the G17 (and previously related) simulations aim to resolve with a maximally feasible resolution (0.8 pc) the top-down multiphase condensation cascade, from the keV plasma phase to the warm gas (10³–10⁵ K) and to the cold (20–200 K) molecular gas. The simulation zooms in, with static mesh refinement, on the central massive galaxy (akin to NGC 5044) in the inner gaseous cool core (52³ kpc³ domain; again, minimum plasma t_cool/t_ff < 10), reaching a 100 Myr evolution. The static potential over such a small domain is dominated by the central galaxy stellar mass M_* ≃ 3.4 × 10¹¹ ${M}_{\odot }$ (with effective radius R_e ≃ 10 kpc). The central SMBH (M_• =3 × 10⁹ ${M}_{\odot }$) is modeled with the pseudo-relativistic Paczyński & Wiita (1980) potential and has a characteristic Bondi radius of 85 pc (G17, Section 2.2).

The plasma radiative emissivity follows the same prescription as described in Section 2.1 ($\simeq {n}^{2}\,{\rm{\Lambda }}$) though now the cooling curve is extended down to the neutral and molecular regime (Figure 1 in G17) following an analytic prescription analogous to reference ISM studies (e.g., Dalgarno & McCray 1972; Kim et al. 2013; Section 2.5 in G17). The included processes are atomic line cooling from hydrogen Lyα, C ii, O i, Fe ii, and Si ii; rovibrational line cooling from H₂ and CO; and atomic and molecular collisions with grains (Koyama & Inutsuka 2000). The typical ionization fraction in the warm/cold phase is of the order of 1% (mimicking the influence of post-AGB stars, AGNs, and cosmic rays).

As we are here interested in the feeding stage and due to the limited integration time, we model only the gentle 4π deposition stage of the mechanical AGN feedback, which prevents the cooling flow catastrophe and the related monolithic collapse of the hot atmosphere (G12, Figure 9). The injected plasma halo heating rate thus balances the average cooling rate in coarse radial shells (G17, Section 2.4), i.e., the hot atmosphere is globally stable but locally unstable. We further include heating of the cold/warm phase (G17, Section 2.5), which is mainly due to the photoelectric effect; however, since massive/early-type galaxies have minor star formation (≲0.01 ${M}_{\odot }$ yr⁻¹), this term is subdominant. We note that we are fully aware of the complexity of the heating and cooling microprocesses in the multiphase ICM/IGrM; these simulations are part of our ongoing numerical campaign to dissect each physics step by step (Section 3.3).

The additional key physics, alongside hydrodynamics, cooling, and heating, is turbulence. As probed naturally in the previous self-regulated AGN outflow runs (and independent studies; Section 1), the hot X-ray halo is continuously stirred by subsonic turbulence (due to the buoyant bubbles, Kelvin–Helmholtz instabilities, and shocks). As these runs are controlled experiments, the diffuse hard X-ray halo is continuously perturbed by subsonic turbulent motions (σ_v ≈ 170 $\mathrm{km}\,{{\rm{s}}}^{-1}$) via a spectral forcing scheme based on an Ornstein–Uhlenbeck random process (G17, Section 2.3), which also reproduces experimental high-order structure functions (Fisher et al. 2008). The gas is only stirred at low-k Fourier modes, allowing the development of a self-consistent and natural turbulence cascade. Such a subsonic, mainly solenoidal turbulence mimics well that produced by AGN feedback during the deposition stage (e.g., Gaspari et al. 2012a). Since we focus in this work on massive galaxies, the driven chaotic gas motions are stronger than the gas rotational velocity (i.e., Ta ≡ v_rot/σ_v < 1).

In this heated and turbulent atmosphere, warm filaments and cold clouds condense out of the hot plasma, some of which rain on the SMBH. In the nuclear region, inelastic collisions allow the angular momentum to mix and cancel via CCA. As shown in our previous series of investigations (cf., Gaspari et al. 2013b, 2015, 2017), CCA displays important properties that can explain diverse observed phenomena, including the rapid flickering of AGNs, their obscuration properties (the broad-/narrow-line region and clumpy torus), the shallow X-ray temperature profiles, and the cospatiality of the inflowing/outflowing multiphase gas in the soft X-ray, optical, and radio bands. In this work, we focus on the multiphase CCA kinematics.

We start from the long-term simulation, which can follow condensation only down to the warm phase but in a long-term AGN outflow feedback evolution (G12; Section 2). Figure 1 addresses a key question: what is the degree of correlation between the velocity dispersions (spectral line width) of the condensed ensemble warm gas and the volume-filling X-ray plasma in cluster cores?

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During the gigayear evolution, the hot halo is continuously perturbed by the cosmic weather and the AGN outflows at large and small radii, respectively. The turbulent motions promote the nonlinear condensation of extended warm ionized filaments (T ≈ 10⁴ K), which are mainly observed in Hα+[N ii] emission. Figure 1 shows that during the top-down multiphase condensation and recurrent AGN feedback cycles (blue points), the ensemble warm phase behaves as a quasi-linear tracer of the turbulent eddy evolution (see also Section 3.2.1). The best-fit linear relation has the following slope and normalization:

$$\begin{eqnarray}&&{\sigma }_{v,\mathrm{los},\mathrm{warm}}\,=\,{0.97}_{-0.02}^{+0.01}\,{\sigma }_{v,\mathrm{los},\mathrm{hot}}+{8.3}_{-5.1}^{+3.5}\,\mathrm{km}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 3 }$$

The linear correlation emerges within a wide ensemble extraction region of size 1–2 condensation radii (for our massive cluster, ∼45 kpc), over which the warm gas forms out of the progenitor X-ray (0.3–8 keV) plasma. At small scales, the single clouds show instead a larger variance driven by the local eddies and cloud-collisional kinematics (see Section 3.2).

The turbulence eddy turnover timescale tied to this coupling region (which is also comparable to the AGN bubble injection scale) is the effective dynamical timescale of the top-down multiphase condensation process (Section 5). For well-resolved objects, it is preferable—but not essential—to cut the very inner region (here 4 kpc) in the presence of jet activity.²⁸ The simulated velocity dispersion distribution has mean σ_v,los ≈ 140 $\mathrm{km}\,{{\rm{s}}}^{-1}$, reaching values up to 250 $\mathrm{km}\,{{\rm{s}}}^{-1}$ during the stronger AGN feedback phases. The logarithmic scatter of the entire warm/hot gas distribution is 0.13 dex. Focusing instead on the deviation from the best-fit line, the rms is 14% (with maximum residuals up to ±40%), which is mainly generated by the AGN duty cycle and the related time hysteresis between driving perturbations and recurrent residual condensation.

In Section 4, we compare the simulated distribution with new ensemble warm gas constraints for 76 clusters, with the results consistent with the simulated range predicted here. For one cluster—Perseus—we can directly probe the correlation here, as direct LOS velocity dispersion detections for both the warm gas and X-ray plasma are available. Specifically, we combine the Fe xxv–xxvi line-width fiducial detection, σ_v,los,hot = 164 ± 10 $\mathrm{km}\,{{\rm{s}}}^{-1}$ (Hitomi Collaboration 2016 ²⁹ ), with a new wide-field SITELLE³⁰ IFTS observation³¹ of the ensemble Hα+[N ii] line width (M. Gendron-Marsolais et al. 2018, in preparation; see Section 4.1 for the analysis method). By fitting the Hα+[N ii] lines of the single spectrum integrated over the same wide extraction region as above, we find σ_v,los,warm = 137 ± 20 $\mathrm{km}\,{{\rm{s}}}^{-1}$. Selecting the hard X-ray band as for the Hitomi iron-lines measurements (≈5–9 keV), we find simulated plasma velocity dispersions that are on average 20% higher than those in the entire X-ray band, since the hard X-ray, less dense gas is more easily accelerated by feedback. Nevertheless, whether or not this correction is applied (orange circle versus triangle in Figure 1), the warm and hot gas velocity dispersions are consistent with having comparable turbulent kinematics within uncertainties, in agreement with the predicted correlation.

We move on from the long-term evolution to the detailed ultra high-resolution kinematics of the top-down multiphase condensation (0.8 pc—ensuring convergence of the main properties), which tracks all the phases down to the molecular regime (G17 and Section 2). While the previous simulation focuses on the realistic feedback process, the current run focuses on the detailed feeding process in a central massive galaxy for a shorter time, 100 Myr, which is still 10 times the central (kiloparsec-scale) cooling time. In this turbulent and heated halo, extended warm filaments (∼10⁴–10⁵ K) condense out of the hot keV plasma atmosphere and produce a condensation rain. The thin outer layer of the filament is ionized and strongly emitting in optical and UV. The spine of the filaments is mostly neutral gas (∼10³ K), containing most of the warm gas mass. The denser peaks further condense into molecular clouds (<100 K) with radii spanning several parsecs to 100 pc for the giant molecular associations. Total molecular masses can reach up to several 10⁷ ${M}_{\odot }$, consistent with recent ALMA data (e.g., David et al. 2014; massive clusters can show even 10⁹ ${M}_{\odot }$; e.g., Vantyghem et al. 2016; Pulido et al. 2017). While temperature radial profiles are fairly flat for all condensed phases, density profiles have logarithmic slope −1, with inner densities up to 10⁻²¹ g cm⁻³ for the molecular phase.

The CCA process is also responsible for efficiently boosting SMBH accretion rates with rapid intermittency of up to two orders of magnitude (e.g., Gaspari 2016 for a brief review). Given that subsonic turbulence is a common state of hot halos (Section 4), CCA is also a recurrent state of observed systems (e.g., McDonald et al. 2011, 2012; Werner et al. 2014; Tremblay et al. 2016; David et al. 2017), although overheated halos can experience a pure hot low-accretion mode, and rotation-dominated halos can be associated with a decoupled thin disk. We refer the interested reader to G17 for the detailed thermodynamic properties of each phase and in-depth discussions; here, we are interested in the statistical kinematic properties related to the CCA rain, in particular confronting the ensemble versus local variance and the mean of the velocity field (i.e., the broadening and shift of the spectral lines) for all gas phases. The limitations of the current simulations and future improvements are discussed in Section 3.3.

3.2.1. Turbulence: Line Broadening

Figure 2 shows the ensemble velocity dispersion in six major temperature bins normalized to the subsonic turbulent velocity, which is stably driven in hot plasma (>5 × 10⁶ K). The different phases correspond to key observational bands, covering the radio, optical/IR, and UV/soft X-ray regimes, as highlighted by different colors in the top panel. Turbulent LOS velocity dispersions are detected through the broadening of the observed spectral lines by measuring the line's FWHM ≃ 2.355 σ_v,los. The lower the temperature, the smaller the contribution of thermal broadening,³² which is ∼1–8 $\mathrm{km}\,{{\rm{s}}}^{-1}$ for molecular and warm gas, respectively. The H₂, CO, H i, [C ii], and Hα+[N ii] lines are all excellent probes of the gas kinematics.

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The top and bottom panels of Figure 2 show the same velocity dispersion diagnostics for an ensemble-beam (excluding the collisional nuclear region) and for a pencil-beam (aperture R ≲ 25 pc) detection, respectively. The blue bars indicate the logarithmic mean and 1–2 standard deviations³³ of the underlying points (not shown), which are tracked every 1 Myr for ∼100 Myr. The ensemble measurement substantially reduces the turbulence intermittency noise and shows again a tight linear correlation throughout the phases, corroborating the result in Section 3.1. The rms deviation from the hot gas turbulent velocity is 13% (brown), which is analogous to the long-term simulation deviation from the best-fit line in Figure 1. The ratio is not unity as condensed structures do not fill the entire halo at every moment in time. This demonstrates that we can use the ensemble warm or cold gas as tracers of the kinematics of the turbulent hot gas, and conversely, we can predict the kinematics of the multiphase CCA cascade from the turbulent plasma halo. The optical/NUV phase near the stable 10⁴ K regime has one of the lowest scatters and better equivalence, making nebular Hα+[N ii] emission an excellent tool to study turbulence (Section 4.1). The FUV phase shows larger mean velocity dispersion, experiencing the most rapid and unstable condensation transition due to the strong line cooling, while tracing the low-mass filament skin (cf., G17 for the multiwavelength synthetic imaging). The molecular clouds, having the lowest volume filling, display the largest scatter. Globally, the condensed gas cannot be treated as ballistic or free-falling objects, as all phases participate in the hydrodynamical layer-within-layer cascade. Note that although some of the condensed gas can be accreted by the SMBH, the phases are continuously replenished by the turbulent condensation rain.

The bottom panel of Figure 2 shows that, in synthetic observations with a pencil beam (small aperture through the center), the velocity dispersion decreases significantly, down to a few 10% of the hot gas value. Therefore, we expect to detect commonly narrow lines with this approach, with FWHM down to a few 10 $\mathrm{km}\,{{\rm{s}}}^{-1}$. At the same time, the scatter increases substantially (the distribution is lognormal with dispersion over all the phases of 0.41 dex), thus a broad component can also be present in pencil-beam measurements (e.g., by using absorption lines against the AGN backlight; Section 4.2). The broad component is typically associated with structures at large radial distance having a small line shift. The narrower component tends instead to track the inner denser clouds (especially for the colder phases), which have experienced inelastic collisions in the nuclear region and are being funneled toward the SMBH. Such clouds can be better probed via major blue-/redshifted lines (Section 3.2.2).

The increased scatter (bottom versus top panel) reflects the chaotic and intermittent nature of turbulence, since single warm/cold structures fill smaller volumes while condensing down the turbulence cascade. The pencil-beam approach indeed tends to sample a few or single clouds (e.g., for the molecular phase, the inner volume filling is 2%, gradually decreasing beyond r > 1 kpc). Because of the turbulence inertial cascade, the velocity dispersion decreases³⁴ as σ_v ∝l^1/3. From characteristic 2 kpc to 20 pc scales, this implies a factor of 0.2 decrease in velocity dispersion, as retrieved for the molecular phase in Figure 2. Warmer phases are less compressed, thus having a lower decrement, as shown by the positive best-fit line slope.

The scatter related to multiple off-center pencil-beam measurements of the condensed gas line broadening can be used as a new way to quantify the level of small-scale intermittency in the turbulent medium. It is worth noting that the cold molecular phase suffers the largest scatter in line broadening. While the numerous cold clouds trace the condensation out of the peaks of the filamentary warm gas (in turn formed out of the turbulent hot halo), they also experience chaotic collisions and drag with all other phases, in particular at small distances from the SMBH. A fraction of the clouds may turn into young star clusters, decoupling via collisionless dynamics (likely retaining the progenitor cold gas velocity dispersion). However, a significant σ_v in all condensed gas phases implies that turbulent pressure is a key component (dominating over thermal pressure) that prevents most clouds from major collapse (cf., G17), in agreement with the low mean star formation rates and large cloud virial parameters (α ≫ 1) observed in early-type galaxies (e.g., David et al. 2014; Temi et al. 2017).

3.2.2. Bulk/Inflow Motions: Line Shift

In Figure 3, we analyze the bulk motions during the same CCA run via the mean velocity along the line of sight, or analogously, via the spectral line blue-/redshift (as offset from the systemic velocity). We note that our galaxy (stellar) systemic velocity is always null, as the simulation box is centered on the (static) gravitational potential. The ensemble gas detection (top panel) typically shows a small line shift with fairly contained scatter, slightly increasing toward the cooler phase. The logarithmic mean and dispersion of its magnitude over all the phases are $\mathrm{log}| {\bar{v}}_{\mathrm{los}}| /(\mathrm{km}\,{{\rm{s}}}^{-1})=1.59\pm 0.37$. The line centroid would sometimes appear consistent with the galactic dynamics, given the typical measurement uncertainties. The pencil-beam detections (bottom panel) show instead a substantially larger line shift with global logarithmic mean and dispersion, $\mathrm{log}| {\bar{v}}_{\mathrm{los}}| /(\mathrm{km}\,{{\rm{s}}}^{-1})\,=2.07\pm 0.47$. The fastest structures are typically inner clouds that have collided, canceling angular momentum, and are often falling toward the SMBH, within the Bondi capture radius. In both cases, we expect on average a similar fraction of blue- and redshifted lines (at least in emission), as clouds can drift in front of or behind the accretor.³⁵

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Interestingly, the condensed gas kinematics distributions are best fitted by lognormal distributions.³⁶ This is particularly evident when the range increases to several dex as for the velocity shift, since the linear approximation is no longer valid, and the high-end tail of the distribution becomes prominent. The reason is that turbulence continuously drives nonlinear perturbations in all thermodynamic properties with a characteristic lognormal shape (e.g., Gaspari et al. 2014). The turbulence cascade (and related multiphase condensation) is indeed a multiplicative process, with smaller and smaller eddies generated within larger vortices, which can also be seen as a power-law inertial range in Fourier space.

Overall, as shown for the line-broadening kinematics, the adoption of a pencil beam leads to sampling smaller structures, thus tracing a lower velocity dispersion and a larger velocity shift of the infalling multiphase clouds and filaments. This method of probing the inner CCA via observations of narrow and significantly shifted lines is a simple and promising method, which is particularly efficient in absorption against a strong AGN backlight emitting in the band of the targeted gas phase (e.g., GHz radio for CO gas). An excellent case study is A2597 (Tremblay et al. 2016), where ALMA detected three central infalling narrow-line clouds with redshifted velocities of up to 335 $\mathrm{km}\,{{\rm{s}}}^{-1}$ (Section 4.2 for a large data set comparison).

The previous simulations are part of our numerical campaign to dissect the multiphase physics of clusters, groups, and massive galaxies, as we test and disentangle each physical process step by step. We discuss below the main limitations of the current runs. In general, such extra—typically subdominant—physics tends to alter the details of the condensed gas (morphology, ionization layers, etc.), while the statistical thermodynamic and kinematic properties are expected to remain similar, i.e., the gas condenses via the turbulent top-down cascade, feeding the SMBH via the accretion of clouds and filaments that are disordered on large scales.

Magnetic fields and cosmic rays (CRs) can provide non-thermal pressure, in addition to turbulence, and further support the condensation collapse, thus altering the size of the clouds and filaments (e.g., Sharma et al. 2010). The draping provided by the large-scale B-fields around the warm and cold structures is also expected to mitigate the heat and mass exchange between the different phases. On the other hand, the typical strength of magnetic fields is a few micro-Gauss, thus β ≫ 1 in the hot phase, implying that they are dynamically unimportant over most of the volume. Regarding CRs, their transport properties (as diffusion and streaming) out of the Galaxy are highly uncertain. The Fermi telescope has also put severe upper limits on the amount of gamma-ray emission in the ICM, with no significant signal even in stacked analyses (e.g., Huber et al. 2013), implying CR-to-thermal energy ratios of less than a few percent.

Speaking of diffusion processes, anisotropic conduction and viscosity will likely promote the formation of more elongated filamentary structures, with more equilibrated temperature and momentum along the magnetic lines. However, ram-pressure stripping observations (e.g., De Grandi et al. 2016; Eckert et al. 2017b), analytic studies (e.g., Burkert & Lin 2000), and plasma particle-in-cell simulations (e.g., Komarov et al. 2016; Kumar et al. 2017) suggest that the transport Spitzer/Braginskii coefficients are suppressed by at least one to two orders of magnitude due to plasma micro-instabilities (e.g., firehose and mirror) and/or highly tangled magnetic fields below the Coulomb mean free path.

Hot stars and AGN radiation can all contribute to ionize the external layers of warm filaments. While the ionized skin depth varies widely depending on the clump density and location (cf. Valentini & Brighenti 2015), the typical heat input is modest in massive/early-type galaxies (see Loewenstein & Fabian 1990), which have both very low star formation rates and Eddington ratios. Connected to this, stellar feedback is several orders of magnitude lower compared with AGN heating (e.g., Gaspari et al. 2012a). Self-gravity is also negligible as most of the clouds have a large virial parameter because of non-thermal pressure, as found in ALMA data (e.g., David et al. 2014; Temi et al. 2017). Despite its simplicity, our feeding runs showed that radiative emission from line recombination during the condensation cascade down to 10⁴ K, together with mixing with the hot plasma, can reproduce Hα maps similar to those retrieved by SOAR (Gaspari et al. 2015, Section 8). Regardless of the ionization level and microphysics, the turbulence cascade is a top-down process that develops in an analogous way, generating extended filaments and cloud associations that trace the self-similar turbulent eddies.

Overall, although more physics is likely at play in the ICM/IGrM—in addition to the gas dynamics, gravity, radiative multiphase cooling, AGN heating, and turbulence implemented here—the current simulations already provide a robust framework and laboratory to assess the main statistical properties of the multiphase condensation cascade. Passing different multifrequency observational tests (e.g., radial profiles, surface brightness maps, cooling rates, emission measures, AGN variability; cf. G12–G17) gives confidence that the simulations are already capturing the main features of the real condensation process. Our forthcoming works will be important to dissect in depth the above physics and assess any deviation from the current results, thus improving the theoretical understanding of the multiphase gas precipitation.

As shown in Sections 3.1–3.2.1, the primary method that allows us to retrieve the volume-filling turbulent velocity is to analyze the ensemble LOS velocity dispersion. In observations, this can be achieved by extracting a single, integrated spectrum over the maximally feasible radial aperture covering the entire warm (∼10³–10⁵ K) or cold (≲200 K) gas emission region. To show such capability, we applied the ensemble method to the Hamer et al. (2016—H16) sample including 68 well-resolved Hα+[N ii] objects—mostly galaxy cluster cores, plus 12 massive groups. An analogous method is applied to eight other objects not included in the H16 sample and displaying cold/warm gas emission (e.g., McDonald et al. 2012; Werner et al. 2014; and the Perseus cluster, M. Gendron-Marsolais et al. 2018, in preparation). Table 1 lists the retrieved constraints and sample details for all 76 objects.

Table 1.  Detected Line Broadening and Shift for the Ensemble Warm and Cold Phases in Observed Massive Galaxies Within Cluster/Group Cores

Object	σv,los ($\mathrm{km}\,{{\rm{s}}}^{-1}$)	${\bar{v}}_{\mathrm{los}}$ ($\mathrm{km}\,{{\rm{s}}}^{-1}$)	Rex (kpc)	Object	σv,los ($\mathrm{km}\,{{\rm{s}}}^{-1}$)	${\bar{v}}_{\mathrm{los}}$ ($\mathrm{km}\,{{\rm{s}}}^{-1}$)	Rex (kpc)
A1348 (a)	Hα+[N ii]: 281 ± 3	Hα+[N ii]: −76 ± 8	18.0	A1663 (a)	Hα+[N ii]: 241 ± 3	Hα+[N ii]: 34 ± 9	13.7
A1060 (a)	Hα+[N ii]: 90 ± 2	Hα+[N ii]: −54 ± 14	1.2	A133 (a)	Hα+[N ii]: 122 ± 2	Hα+[N ii]: 0 ± 14	7.2
A1668 (a)	Hα+[N ii]: 166 ± 3	Hα+[N ii]: −72 ± 10	10.9	A2052 (a)	Hα+[N ii]: 162 ± 4	Hα+[N ii]: −31 ± 11	6.4
A2415 (a)	Hα+[N ii]: 170 ± 3	Hα+[N ii]: −85 ± 12	11.2	A2495 (a)	Hα+[N ii]: 128 ± 4	Hα+[N ii]: 0 ± 11	8.9
A2566 (a)	Hα+[N ii]: 149 ± 3	Hα+[N ii]: 12 ± 10	10.1	A2580 (a)	Hα+[N ii]: 85 ± 5	Hα+[N ii]: −44 ± 14	9.4
A2734 (a)	Hα+[N ii]: 229 ± 3	Hα+[N ii]: −42 ± 11	6.1	A3112 (a)	Hα+[N ii]: 258 ± 3	Hα+[N ii]: 0 ± 11	8.9
A1084 (a)	Hα+[N ii]: 135 ± 3	Hα+[N ii]: 59 ± 12	11.2	A3581 (a)	Hα+[N ii]: 188 ± 3	Hα+[N ii]: −19 ± 11	5.4
A3605 (a)	Hα+[N ii]: 236 ± 3	Hα+[N ii]: 36 ± 9	7.4	A3638 (a)	Hα+[N ii]: 171 ± 3	Hα+[N ii]: −34 ± 12	8.6
A3806 (a)	Hα+[N ii]: 85 ± 5	Hα+[N ii]: −95 ± 13	5.9	A3880 (a)	Hα+[N ii]: 255 ± 3	Hα+[N ii]: −35 ± 11	13.2
A3998 (a)	Hα+[N ii]: 123 ± 4	Hα+[N ii]: −30 ± 11	17.8	A4059 (a)	Hα+[N ii]: 203 ± 3	Hα+[N ii]: −65 ± 10	7.9
A478 (a)	Hα+[N ii]: 129 ± 4	Hα+[N ii]: −126 ± 15	17.0	A496 (a)	Hα+[N ii]: 125 ± 4	Hα+[N ii]: 0 ± 12	8.1
A85 (a)	Hα+[N ii]: 149 ± 4	Hα+[N ii]: −172 ± 12	8.3	Hydra-A (a)	Hα+[N ii]: 211 ± 3	Hα+[N ii]: −102 ± 10	7.9
N4325 (a)	Hα+[N ii]: 110 ± 4	Hα+[N ii]: −40 ± 13	6.7	Rc 0120 (a)	Hα+[N ii]: 182 ± 3	Hα+[N ii]: −38 ± 10	4.7
Rc 1524 (a)	Hα+[N ii]: 192 ± 3	Hα+[N ii]: 0 ± 9	15.6	Rc 1539 (a)	Hα+[N ii]: 187 ± 3	Hα+[N ii]: 80 ± 14	13.7
Rc 1558 (a)	Hα+[N ii]: 138 ± 3	Hα+[N ii]: −66 ± 33	13.8	Rc 2101 (a)	Hα+[N ii]: 88 ± 5	Hα+[N ii]: 0 ± 24	7.1
R0000 (a)	Hα+[N ii]: 144 ± 4	Hα+[N ii]: −17 ± 11	3.5	R0338 (a)	Hα+[N ii]: 190 ± 3	Hα+[N ii]: 15 ± 10	8.6
R0352 (a)	Hα+[N ii]: 190 ± 3	Hα+[N ii]: −23 ± 14	17.8	R0747 (a)	Hα+[N ii]: 191 ± 3	Hα+[N ii]: 39 ± 4	24.8
R0821 (a)	Hα+[N ii]: 122 ± 4	Hα+[N ii]: 20 ± 13	20.4	S555 (a)	Hα+[N ii]: 232 ± 3	Hα+[N ii]: 61 ± 10	11.5
Rc 1436 (a)	Hα+[N ii]: 122 ± 4	Hα+[N ii]: 68 ± 11	14.6	A1111 (a)	Hα+[N ii]: 113 ± 3	Hα+[N ii]: −277 ± 12	21.9
A1204 (a)	Hα+[N ii]: 221 ± 3	Hα+[N ii]: 109 ± 15	26.3	A2390 (a)	Hα+[N ii]: 231 ± 2	Hα+[N ii]: −91 ± 35	26.0
A3378 (a)	Hα+[N ii]: 80 ± 4	Hα+[N ii]: −54 ± 12	20.6	A3639 (a)	Hα+[N ii]: 157 ± 3	Hα+[N ii]: −56 ± 40	18.8
A383 (a)	Hα+[N ii]: 244 ± 2	Hα+[N ii]: −178 ± 46	24.2	Rc 0132 (a)	Hα+[N ii]: 205 ± 3	Hα+[N ii]: −82 ± 8	24.8
Rc 0331 (a)	Hα+[N ii]: 183 ± 3	Hα+[N ii]: −128 ± 13	14.5	Rc 0944 (a)	Hα+[N ii]: 185 ± 4	Hα+[N ii]: −304 ± 11	25.5
Rc 2014 (a)	Hα+[N ii]: 206 ± 3	Hα+[N ii]: 53 ± 32	18.6	Rc 2129 (a)	Hα+[N ii]: 114 ± 3	Hα+[N ii]: 158 ± 11	30.5
R1651 (a)	Hα+[N ii]: 177 ± 3	Hα+[N ii]: −12 ± 10	17.5	S780 (a)	Hα+[N ii]: 188 ± 2	Hα+[N ii]: 258 ± 10	33.0
Z3179 (a)	Hα+[N ii]: 80 ± 8	Hα+[N ii]: −66 ± 17	14.8	A3444 (a)	Hα+[N ii]: 133 ± 3	Hα+[N ii]: −80 ± 16	35.7
R0439 (a)	Hα+[N ii]: 184 ± 3	Hα+[N ii]: 76 ± 11	26.1	A795 (a)	Hα+[N ii]: 309 ± 2	Hα+[N ii]: −146 ± 7	11.1
A3017 (a)	Hα+[N ii]: 189 ± 3	Hα+[N ii]: −333 ± 9	25.0	A1795 (b)	Hα: 205 ± 1	Hα: 60 ± 11	50
A1991 (a)	Hα+[N ii]: 109 ± 4	Hα+[N ii]: −107 ± 14	16.4	N1275 (e)	Hα+[N ii]: 137 ± 20	Hα+[N ii]: −43 ± 32	45
A2597 (b)	Hα: 241 ± 1	Hα: 76 ± 10	17	Se 15903 (b)	Hα: 160 ± 2	Hα: −121 ± 12	23
M87 (f)	[C ii]: 153 ± 11	[C ii]: −62 ± 11	3.8	Rc 1257 (a)	Hα+[N ii]: 128 ± 2	Hα+[N ii]: 0 ± 3	6.8
N5044 (a), (d)	Hα+[N ii]:190 ± 2	Hα+[N ii]: −77 ± 4	5.6	Rc 1511 (a)	Hα+[N ii]: 208 ± 2	Hα+[N ii]: −7 ± 2	5.0
	CO: 177 ± 10	CO: −169 ± 8	4	Rc 1304 (a)	Hα+[N ii]: 176 ± 2	Hα+[N ii]: −39 ± 3	3.3
A3574 (a)	Hα+[N ii]: 106 ± 2	Hα+[N ii]: −27 ± 4	1.5	A194 (a)	Hα+[N ii]: 90 ± 1	Hα+[N ii]: 19 ± 3	1.5
S805 (a)	Hα+[N ii]: 121 ± 2	Hα+[N ii]: 35 ± 4	2.7	S851 (a)	Hα+[N ii]: 209 ± 4	Hα+[N ii]: −23 ± 8	3.6
N4636 (c)	[C ii]: 153 ± 3	[C ii]: 22 ± 3	1.8	N533 (a)	Hα+[N ii]: 138 ± 3	Hα+[N ii]: −37 ± 5	2.8
	[O i]: 99 ± 12	[O i]: 75 ± 12	1	N5846 (a), (c)	Hα+[N ii]: 118 ± 2	Hα+[N ii]: 30 ± 4	1.3
H62 (a)	Hα+[N ii]: 103 ± 2	Hα+[N ii]: −27 ± 4	3.3		[C ii]: 202 ± 4	[C ii]: −25 ± 4	2.5
N5813 (a), (c)	Hα+[N ii]: 133 ± 6	Hα+[N ii]: 48 ± 15	1.5	N6868 (c)	[C ii]: 216 ± 3	[C ii]: 125 ± 3	2.8
	[C ii]: 178 ± 4	[C ii]: 96 ± 4	3.7		[O i]: 215 ± 13	[O i]: 137 ± 13	1
	[O i]: 116 ± 15	[O i]: 30 ± 15	1	N7049 (c)	[C ii]: 168 ± 3	[C ii]: 78 ± 3	3.2

Note. These constraints can be used by other studies as proxies of the turbulent velocities and/or bulk motions of the diffuse ICM/IGrM. The ensemble FWHM/2.355 = σ_v,los and velocity offset (from the systemic velocity) ${\bar{v}}_{\mathrm{los}}$ are derived from the single integrated spectrum extracted within the detectable emission region R_ex (major axis length; Section 4.1); for clusters/groups, the median is R_ex ≃ 14/3 kpc. All objects are galaxy clusters, except for the last 15 groups. The rest-frame wavelengths of the tabulated lines are Hα 6562.8 Å, [N ii] 6548/6583 Å, [C ii] 157.7 μm, [O i] 63.2 μm, and CO(2–1) 1.3 mm. Where the line shift is consistent with no offset from the systemic velocity, a null shift is reported. The object prefixes A, S, H, N, R/Rc, Se, and Z are abbreviations for the Abell, Abell Supplementary, HCG, NGC, RXJ/RXCJ, Sérsic, and Zwicky catalogs, respectively.

References. (a) Newly computed from Hamer et al. (2016) VIMOS IFU (VLT) data. (b) Newly computed from McDonald et al. (2012) Magellan and Keck data. (c) Herschel data from Werner et al. (2014). (d) Newly computed from ALMA data (Temi et al. 2017; velocity offsets dispersion method). (e) Newly computed from SITELLE (CFHT) data (M. Gendron-Marsolais et al. 2018, in preparation); R < 4 kpc excised. (f) Herschel data from Werner et al. (2013).

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The observational analysis to retrieve the gas FWHM (σ_v,los) and line offset from the systemic velocity (shift) was carried out as follows, by taking the H16 sample as reference. Initially, the total spectra of the continuum and line-emitting regions are extracted from the data cube (e.g., VIMOS IFU for the H16 sample). For the line emission, this is done by taking the Hα flux maps (emission detected at S/N > 7). A masked cube is then created by discarding all spaxels in each wavelength channel with no emission. The remaining spaxels are summed to give a total flux value for that channel, producing a total spectrum for the line-emitting regions. Table 1 lists the extraction radius,³⁷ with a median R_ex ≃ 14 kpc for clusters and 3 kpc for groups.

The total continuum spectrum is determined in the same way, using a collapsed white-light image in place of the Hα map and discarding spaxels where the continuum flux is less than one-tenth of the peak flux from the galaxy center.³⁸ The Hα masking is then inverted (discarding only spaxels with Hα detection) and the standard deviation of the remaining spaxels in each channel is calculated to provide an estimate of the error related to the two total spectra. The kinematics of the ionized gas and stellar components of the galaxy are then determined by fitting Gaussian profiles to the relevant total spectrum (a triplet fit to the Hα+[N ii] complex for the line-emitting gas and a negative doublet fit to the NaD absorption³⁹ for the stellar component) using a χ² minimization procedure (see H16 for more details). Finally, the FWHM of the line-emitting gas is extracted directly from the fit.⁴⁰ The velocity difference between the Hα emission and the NaD absorption gives the velocity offset. It is worth noting that the total uncertainty is dominated by the systematic error (spectral sampling and instrumental line spread function), while random noise is drastically reduced due to the aggregation of several hundred spectra for each source (${\rm{S}}/{\rm{N}}\propto N/\sqrt{N}$, with N the number of spectra).

As a diagnostic tool to understand the kinematic properties with different methods, we propose a novel diagram confronting the line broadening versus the line shift magnitude. Figure 4 shows the observed data points for warm and cold gas, which are compared with the simulation predictions (shaded contours). The shaded contours show the simulation bivariate distributions with mean and 1–3 standard deviations found in the previous run (Section 3.1), which are best fitted by lognormal distributions. As the Section 3.1 run probes a large dynamical range and varying cluster regimes, we use as the reference line-broadening normalization the mean of its entire hot gas distribution, σ_v,los,hot ≃ 140 $\mathrm{km}\,{{\rm{s}}}^{-1}$ (which is comparable to that of the warm gas; Figure 1), and show the logarithmic standard deviation from such a mean. The simulated global velocity offsets are discussed in Section 3.2.2.

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The top panel in Figure 4 shows that the ensemble detection—both in simulations and observations—cover a specific section of the $\mathrm{log}{\sigma }_{v,\mathrm{los}}-\mathrm{log}| {\bar{v}}_{\mathrm{los}}|$ diagram, namely the top-left region, which is the locus of substantial line broadening and relatively low velocity offsets. The scatter in velocity dispersion is mild due to the ensemble/single-spectrum approach, which decreases the statistical noise of single clouds and filaments. The log mean of the two distributions differs by 3% and 6% along the broadening and shift axis, respectively. The overlap within 2σ is evident to the naked eye, with essentially a null correlation angle and analogous symmetry. Along the broadening/shift direction, the observed data have a mildly larger standard deviation (5%/6%). Given the limitations of the models and observational biases, we deem the simulations and observations to be in good agreement. For example, the observed velocity offsets can be very sensitive to the redshift measurement of the host galaxy, with systematics not always captured. Further, coherent warm gas structures dominating the field of view have the tendency to increase the LW velocity offsets. Running a 2D Kolmogorov–Smirnov (KS_2d)⁴¹ two-sample test results in a p-value of 0.02, implying that the null hypothesis cannot be rejected at the customary confidence levels of 99% and 99.9%. This corroborates the visual inspection that the two data sets do not deviate by a large amount but are perhaps not drawn from an identical distribution. Nevertheless, perfect equivalence shall not be expected considering the detailed differences between the synthetic and real value measurements.

The above analysis benefiting from a large (76) sample size shows that turbulence in galaxy cluster/group cores is contained within a relatively narrow window, σ_v,los ≈100–250 $\mathrm{km}\,{{\rm{s}}}^{-1}$, which implies subsonic Mach numbers Ma_1d ∼ 0.1–0.3. This corroborates indirect X-ray hot gas estimates (Section 1); e.g., Hofmann et al. (2016) and Zhuravleva et al. (2017) find a similar range via plasma density fluctuations in the core, with σ_v,los within 50 $\mathrm{km}\,{{\rm{s}}}^{-1}$ from our values (interesting cases are A2052, A85, and A1795), although their uncertainties remain substantial because of the density contamination tied to substructures. The upper and lower limits set via XMM-RGS (e.g., Ogorzalek et al. 2017) and cosmological simulations (e.g., Lau et al. 2009, 2017; Nagai et al. 2013) further support such range of subsonic ICM/IGrM turbulence. Last but not least, it is remarkable that the SITELLE warm gas constraint matches the direct high-resolution σ_v,los observed by Hitomi in the archetypal galaxy cluster Perseus (Section 3.1). While waiting for the next-generation of X-ray instruments (e.g., XARM—the successor of Hitomi—and Athena; Ettori et al. 2013) to give us high-precision turbulence detections, this analysis allows us to use warm and cold gas velocity dispersions as robust proxies for the hot gas turbulent velocities in large samples (note that future X-ray instruments will still require several days of exposure for just one object).

For five objects (Table 1), we report literature detections in more than one band (besides Perseus, which was tackled in Section 3.1). Except for NGC 5846, the four other galaxies have ensemble broadening comparable between the cold and warm gas within ∼50 $\mathrm{km}\,{{\rm{s}}}^{-1}$, with NGC 5044 and NGC 6868 two exemplary cases. The velocity offsets are also broadly aligned. On the other hand, the currently available extraction regions are often dissimilar; indeed, we observe larger values, all associated with larger extraction regions. Upcoming systematic multiwavelength investigations—which our team is currently undertaking with Chandra, XMM, ALMA, VLT, SOFIA, HST, Magellan, SOAR, SITELLE, and IRAM—will be key to calibrate the multiphase kinematics over the same extraction region and with similar depth. At present, the ensemble Hα+[N ii] nebulae appear to be the best channel to test the volume-filling turbulence and related velocity dispersion.

For the pencil-beam (small aperture, below a few arcsec) detections in massive galaxies, we report the published value from the literature and a few new detections. A larger sample requires new observational programs (e.g., one recently approved in ALMA Cycle 5—PI: A. Edge). The most used pencil-beam approach is to detect absorption lines (e.g., due to H i and CO clouds having significant optical depth) against the (radio) AGN, which acts as a backlight source. The BCG stellar light can also be used as backlight, in combination with NaD absorption. The analysis procedure is then analogous to that above, extracting the systemic velocity offset and FWHM from Gaussian fitting of the absorption lines in the continuum-subtracted spectrum. Hogan (2014) discusses in detail the observational reduction with different instruments such as VLA and WRST (see Tremblay et al. 2016 for ALMA data). Although more challenging due to the low S/N in massive galaxies, small-aperture observations can be used to track small-scale clouds via emission features such as CO(2—1). Table 2 lists the 47 fiducial detections for the 19 available objects, with the addition of the mean properties of the Maccagni et al. (2017—M17) radio galaxy sample and of the Perseus IRAM regions.

Table 2.  Detected Line Broadening and Shift for the Pencil-beam Cold/Warm Phases in Observed Massive Galaxies Within Cluster/Group Cores

Object	σv,los ($\mathrm{km}\,{{\rm{s}}}^{-1}$)	${\bar{v}}_{\mathrm{los}}$ ($\mathrm{km}\,{{\rm{s}}}^{-1}$)
A2390 absor. broad (a)	H i: 191 ± 13	H i: −110 ± 35
absor. narrow (a)	H i: 42 ± 8	H i: 145 ± 35
Z8276 absorption (a)	H i: 27 ± 5	H i: 219 ± 35
A1795 absorption (a)	H i: 255 ± 21	H i: 0 ± 30
Z8193 absorption (a)	H i: 92 ± 8	H i: 0 ± 30
N6338 absorption (a)	H i: 109 ± 8	H i: −200 ± 30
R1832 absorption (a)	H i: 85 ± 6	H i: −773 ± 80
R1558 absor. broad (a)	H i: 56 ± 3	H i: 43 ± 35
absor. narrow (a)	H i: 18 ± 2	H i: 160 ± 35
R1350 absorption (a)	H i: 119 ± 8	H i: 0 ± 30
A2597 absor. broad (a)	H i: 175 ± 15	H i: 0 ± 35
absor. narrow (a)	H i: 94 ± 10	H i: 250 ± 35
absor. narrow (b)	CO: 6 ± 2	CO: 240 ± 60
absor. narrow (b)	CO: 6 ± 2	CO: 275 ± 60
absor. narrow (b)	CO: 6 ± 2	CO: 335 ± 60
N1275 absor. broad (a)	H i: 203 ± 15	H i: 0 ± 35
absor. narrow (a)	H i: 28 ± 6	H i: 50 ± 25
PKS 1353 absorption (a)	H i: 66 ± 10	H i: −53 ± 35
Cygnus-A absorption (a)	H i: 115 ± 8	H i: 0 ± 35
Hydra-A absorption (a)	H i: 33 ± 4	H i: 0 ± 35
4C55.16 absorption (c)	H i: 88 ± 6	H i: −399 ± 35
R1603 absor. broad (a)	H i: 136 ± 25	H i: −30 ± 25
absor. narrow (a)	H i: 57 ± 19	H i: −155 ± 30
N4636 emission (d)	CO: 26 ± 8	CO: 140 ± 8
emission (d)	CO: 26 ± 4	CO: 210 ± 4
N5846 emission (d)	CO: 23 ± 2	CO: −231 ± 2
emission (d)	CO: 15 ± 3	CO: −155 ± 3
emission (d)	CO: 20 ± 2	CO: 111 ± 2
N5044 absorption (e)	CO: 5 ± 1	CO: 260 ± 20
emission (d)	CO: 126 ± 25	CO: 0 ± 26
emission (d)	CO: 76 ± 7	CO: −149 ± 8
emission (d)	CO: 73 ± 13	CO: 29 ± 13
emission (d)	CO: 67 ± 11	CO: −557 ± 12
emission (d)	CO: 47 ± 8	CO: 42 ± 9
emission (d)	CO: 43 ± 8	CO: −81 ± 7
emission (d)	CO: 41 ± 5	CO: −96 ± 6
emission (d)	CO: 39 ± 8	CO: −47 ± 11
emission (d)	CO: 37 ± 2	CO: 28 ± 2
emission (d)	CO: 37 ± 9	CO: −313 ± 9
emission (d)	CO: 31 ± 6	CO: −207 ± 7
emission (d)	CO: 30 ± 8	CO: −193 ± 7
emission (d)	CO: 28 ± 4	CO: −274 ± 4
emission (d)	CO: 20 ± 4	CO: −227 ± 4
emission (d)	CO: 18 ± 4	CO: −133 ± 4
emission (d)	CO: 11 ± 3	CO: −108 ± 3
emission (d)	CO: 10 ± 2	CO: −574 ± 3
A3716 absorption (f)	NaD: 34 ± 25	NaD: 100 ± 10
Perseus emission (g) [log]	$\langle \mathrm{CO}\rangle :1.9\pm 0.3$	$\langle \mathrm{CO}\rangle :1.9\pm 0.4$
66 radio galaxies (h) [log]	$\langle {\rm{H}}\,{\rm{I}}\rangle :1.7\pm 0.3$	$\langle {\rm{H}}\,{\rm{I}}\rangle :1.7\pm 0.6$

Note. These constraints can be used in other studies requiring the kinematics of small-scale filaments or accreting clouds. Analogue of Table 1. The rest-frame wavelengths are H i 21 cm, CO(2-1) 1.3 mm, and NaD 5890/5896 Å. Where the shift is consistent with no offset from the systemic velocity, a null shift is reported. Except for A3716, all absorptions are against the radio AGN.

References. (a) VLA, WRST, and ATCA data from Hogan (2014 and references within). (b) ALMA data from Tremblay et al. (2016). (c) WRST data from Vermeulen et al. (2003). (d) Newly computed from ALMA CO(2–1) center and off-center emission (Temi et al. 2017): N4636 and N5846 are new ALMA Cycle 3 observations, while N5044 is newly reduced from Cycle 0 data (with S/N ≥ 6). (e) ALMA data from David et al. (2014). (f) MUSE data from Smith & Edge (2017): NaD absorption against the stellar light of the A3716 BCG (central E sector). (g) IRAM low-resolution data (159 regions; log mean and rms) from Salomé et al. (2006, 2008). (h) Log mean and rms of H i absorbers for the WRST Maccagni et al. (2017) sample (non-central radio galaxies).

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Figure 4 (bottom panel) shows the observational detections as red points (identical non-circle symbols denote the same host galaxy), compared with the simulation results for all of the condensed gas with 1σ–3σ confidence intervals. For the pencil-beam approach, the Section 3.2 run covers a meaningful range to statistically test the broadening distribution; the reference mean σ_v,los,hot is the same as in the previous section, but now the condensed gas has a ratio lower than 1:1 due to the pencil-beam sampling (Figure 2, bottom panel). The simulated velocity offsets over all of the condensed gas are discussed in Section 3.2.2.

As anticipated by the numerical analysis, it is clear that this method substantially boosts the logarithmic scatter in both the broadening and shift dimensions up to nearly 0.5 dex. The mean line broadening (≃40 $\mathrm{km}\,{{\rm{s}}}^{-1}$) is significantly lower than the ensemble measurement, as predicted by the simulations. The mean velocity offset has instead larger values ≳100 $\mathrm{km}\,{{\rm{s}}}^{-1}$. This is because the pencil beam is sampling smaller and typically fewer condensed elements. About one-third⁴² of the central galaxies observed in absorption display a dual component: either with a large line broadening and small shift or with a large shift and fairly contained broadening. A mild anticorrelation appears to be present, with the faster (and denser) clouds associated with the nuclear inflow toward the SMBH sink region. It is interesting to observe that the broad component in absorption is often a reasonable proxy for the ensemble velocity dispersion, as it tends to sample multiple clouds along the LOS. For instance, the pencil-beam broad components of A2390, A1795, A2597, and N1275 all reside within ≲50 $\mathrm{km}\,{{\rm{s}}}^{-1}$ from the actual ensemble value, although there are exceptions, such as Hydra-A (which has an abnormal 5 kpc disk).

In Figure 4 (bottom panel), we also plot the mean and deviation of the H i absorption detections in the M17 sample of 66 radio galaxies (yellow bars). Although comprising mostly non-central radio/elliptical galaxies and requiring deeper follow-up observations for each target, the H i absorption broadening and shift are consistent with the above central galaxies and simulation properties, with a slightly lower average line shift. An interesting case study is the early-type galaxy PKS 1718, where F. M. Maccagni et al. (2018, in preparation) find redshifted clouds in H i and H₂, and CO clouds infalling within the Bondi radius with velocities up to 345 ± 20 $\mathrm{km}\,{{\rm{s}}}^{-1}$, as found very similarly in the A2597 central galaxy (Tremblay et al. 2016). Other interesting cases are PKS 1740 (Allison et al. 2015), PKS 1657 (Moss et al. 2017), and NGC 3998 (Devereux 2018). This suggests that the top-down condensation and CCA are also likely common phenomena in low-mass/non-central galaxies, given that plasma atmospheres are expected to be ubiquitous (e.g., Anderson et al. 2015) and subsonically perturbed by any AGN type (radio, quasar, etc.) and/or cosmic flow.

The pencil-beam method can be further used in emission and off center. For instance, taking advantage of ALMA high angular resolution and CO sensitivity, we probed several giant molecular associations in the massive galaxies NGC 4636, NGC 5846, and NGC 5044 (Temi et al. 2017). Remarkably, the emission features show a similar mean and scatter to the above H i absorption features. Commonly, small-scale clouds with low broadening (<35 $\mathrm{km}\,{{\rm{s}}}^{-1}$) are associated with large velocity offsets above 100 $\mathrm{km}\,{{\rm{s}}}^{-1}$. At the same time, the inspected masses, radii, and cospatiality of the giant molecular associations are consistent with the G17 simulation, corroborating the incidence of in situ cooling via the multiphase cascade. Line absorption against the stellar light of a background galaxy is another promising way to retrieve the kinematics of gas at varying clustercentric distances (e.g., Smith & Edge 2017). Notice that the beam must be small to achieve a proper pencil-beam detection (below the kiloparsec or a few arcsec scale), otherwise the measurement will tend toward the ensemble approach; the IRAM data of Perseus (orange) is an example of an intermediate regime with a beam of 4 kpc (12 arcsec).

Focusing on the comparison between the observed and simulated distributions in the $\mathrm{log}{\sigma }_{v,\mathrm{los}}-\mathrm{log}| {\bar{v}}_{\mathrm{los}}|$ diagram, the mean differs by 2% along both axes. The fairly good overlap within 2σ is again evident to the naked eye. The observed data have mildly larger/lower standard deviation (4%/10%) along the broadening/shift direction, respectively. Running a KS_2d two-sample test (keeping in mind the limitations discussed in Section 4.1) results in a p-value of 0.14 (0.18 including the M17 and IRAM points). The absence of low p-values implies that the null hypothesis cannot be rejected, even at the less significant 95% level. The point to take away is that the two distributions are similar, though not necessarily identical. In particular, the simulation shows a milder anticorrelation (0.54 rad difference in angle rotation), although the scarcity of observed points in the bottom-left section of the diagram may be attributed to the difficulty in detecting both small shift and narrow lines in the spectra.

Overall, granted that the sample requires larger statistics, both simulations and data agree well on the same picture: the pencil-beam method is useful to track the inner infalling clouds (selecting the narrow features with a large velocity shift) or to have a preliminary estimate of the large-scale chaotic motions (selecting the broad features). The wide range of velocity shifts is key to allowing a clean separation of such components in the energy spectrum (e.g., in the radio band). The ensemble measurement instead provides a robust and direct constraint on the volume-filling turbulent motions, as the ensemble condensed elements actively participate in large-scale kinematics via the top-down multiphase condensation cascade. The combination of the two approaches provides a powerful complementary diagnostic of the global and local gas kinematics.

A majorly debated topic in recent literature concerns the dominant criterion governing the formation of the condensed structures in the ICM/IGrM. Previous studies proposed that the ratio of the plasma cooling time to free-fall time⁴³ falling below a few tens is the triggering threshold of thermal instability (a.k.a. the TI ratio). The mean value of the TI ratio is highly uncertain. McCourt et al. (2012) and Sharma et al. (2012) propose a value of 10. Gaspari et al. (2012b) simulations show a TI ratio between 8 and 25 (see their Figure 3, bottom-right panel), which is similarly adopted by Voit et al. (2015a). Valentini & Brighenti (2015) find a TI-ratio threshold that can reach a value of 70 in early-type galaxies. Taken at face value and over different studies, the uncertainty on the TI ratio can be an order of magnitude. It is important to note that this theoretical uncertainty accumulates on top of the intrinsic scatter due to system-to-system variations mainly tied to the cooling time (more in Section 5.2).

Four key issues arise with the TI ratio. First, the unity problem: why is the condensation occurring at such an elevated threshold, well above 1, which should be instead the physically sound transition⁴⁴ ? Second, the large 1 dex theoretical uncertainty in the actual threshold—even for similar systems—hints at the fact that the free-fall time is not the primary timescale of the condensation process. Third, observed condensed clouds do not follow ballistic orbits but are drifting with subvirial velocities (e.g., McNamara et al. 2014; Russell et al. 2016). Last but not least, there is the major observational hurdle of retrieving the free-fall times, i.e., the total masses, including the galactic and cluster baryonic and dark matter masses, which are notoriously challenging to constrain.

Another suggested criterion (Hogan et al. 2017b) is to consider the cooling time below a fixed threshold, e.g., 1 Gyr. Although empirically effective and less noisy, this has the limitation of being applicable only to some classes of objects, e.g., massive clusters, but not groups or galaxies (e.g., O'Sullivan et al. 2017). Finally, the uplift action of the AGN cavity may significantly increase the gas total inflow time (t_in) well above the free-fall timescale, and bring it to altitudes where t_cool < t_in (e.g., McNamara et al. 2016). However, it appears too restrictive to consider just one specific directional (bipolar) and short stage of the feedback cycle, namely the trailing phase. The full process includes bubble inflation, uplift, and cocoon shocks, with isotropic turbulence being generated as a common by-product (this is true even if the driver is a merger or sloshing event). The turbulence time described below can be considered as a more universal concept of the reduced inflow time (with a clear quantitative observable), i.e., gas drifts in the halo regardless of the radial direction (inflow or outflow) and location (behind or around cavities).

Given the results in Section 3, namely the cold and warm gas following the progenitor chaotic kinematics, we show in Figure 5 that a robust and physically motivated condensation criterion is the ratio of the cooling time to the turbulence eddy turnover time t_cool/t_eddy ≈ 1, which marks the multiphase state of the cluster/group core. The condensation criterion can be alternatively written and interpreted as the velocity ratio C = σ_v/v_cool, similar to a new dimensionless "Mach" number, where v_cool ≡ r/t_cool is the cooling velocity.

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Figure 5 shows the two characteristic timescales and the locus of extended Hα+[N ii] filaments, for clusters (left) and groups (right). In the left panel, the blue curves represent the plasma cooling time of observed non-cool-core (NCC) clusters (Cavagnolo et al. 2008) and cool-core (CC) clusters (following the recently updated constraints by Panagoulia et al. 2014 and Hogan et al. 2017b). The CC profiles are best fitted by a broken power law: at large radii following the cosmic adiabatic baseline (Tozzi & Norman 2001), K ∝ r^1.1 (for a typical 5 keV cluster, R₅₀₀ ≈ 1.1 Mpc and K₅₀₀ ≈ 1600 keV cm²; Sun et al. 2009), while at small radii, $K\simeq 95.4{(r/100\mathrm{kpc})}^{2/3}\ \mathrm{keV}\,{\mathrm{cm}}^{2}$. The (X-ray) plasma cooling time is

$$\begin{eqnarray}&&{t}_{\mathrm{cool}}=\displaystyle \frac{3{k}_{{\rm{b}}}T}{{n}_{{\rm{e}}}{\rm{\Lambda }}},\end{eqnarray} \tag{ 4 }$$

where ${n}_{{\rm{e}}}={({k}_{{\rm{b}}}T/K)}^{3/2}$ and Λ(T, Z) is the plasma cooling function (Sutherland & Dopita 1993; Z = 0.3 Z_⊙ for clusters). The observed profiles are typically contained within 0.25 dex (blue bands; ≈90% confidence level). The 0.5–1 Gyr dotted black band crudely separates the observed clusters hosting (below) or not hosting (above) extended warm filaments traced via Hα+[N ii] emission (e.g., Hogan et al. 2017b).

The eddy turnover timescale, i.e., the time a turbulent vortex requires to gyrate—thus producing density fluctuations in a stratified halo—is

$$\begin{eqnarray}&&{t}_{\mathrm{eddy}}=2\pi \,\displaystyle \frac{{r}^{2/3}{L}^{1/3}}{{\sigma }_{v,L}},\end{eqnarray} \tag{ 5 }$$

where σ_v,L is the velocity dispersion at the injection scale and using the Kolmogorov cascade σ_v ∝ l^1/3 appropriate for subsonic turbulence (Gaspari & Churazov 2013; we use here the approximation that smaller radii contain eddies of smaller size, l ∼ r).⁴⁵ Turbulence is mainly injected via AGN feedback in the core. While it was previously difficult to assess the plasma kinematics, the ensemble measurement makes it now trivial to apply the turbulence velocity dispersion derived from one of the condensed phases. We use the mean three-dimensional (${\sigma }_{v}=\sqrt{3}\,{\sigma }_{v,1{\rm{d}}}$) ensemble velocity dispersion, σ_v,L ≃ 242 $\mathrm{km}\,{{\rm{s}}}^{-1}$, found in the AGN feedback simulation and similarly in the observational sample (Figure 4). We remark that the ensemble dispersion is comparable in all phases, cold, warm, or hot. The injection scale L can be traced by the diameter of the bubble inflated by the AGN outflows/jets, which start to decelerate as they deposit kinetic energy (or, alternatively, from the size of the warm gas nebula; more below).⁴⁶ From the large high-quality sample of Shin et al. (2016), observed 5 keV cluster bubbles have a typical diameter L ≈ 25 kpc. Given the weak dependence on the injection scale (∝L^1/3), the eddy time scatter for a given system is driven by σ_v, which has a 90% confidence level of 0.2 dex (Figure 4).

The key result from Figure 5 is that, despite its simplicity, the crossing of t_cool and t_eddy traces the condensation region. For our analyzed galaxy clusters (Table 1), the logarithmic median and dispersion of the extent radius (≈R_ex) are $\mathrm{log}R/\mathrm{kpc}=1.14\pm 0.29$, i.e., a 2σ range of 3.5–52.5 kpc (which is close to that found by McDonald et al. 2010 in a smaller sample of CC clusters with detected warm filaments; dashed ellipse). This range is tracked by our proposed dimensionless cooling number C ≈ 1 (see the overlapping blue and red bands). Around such a threshold, turbulence in a stratified halo drives significant density/entropy perturbations in the hot phase, $\delta \rho /\rho \sim {\sigma }_{v,1{\rm{d}}}/{c}_{{\rm{s}}}$ (cf. Gaspari et al. 2014); the overdense hot gas rapidly cools down, promoting the multiphase cascade down to warm filaments and molecular clouds. The cospatiality between the different phases, in particular the soft X-ray and radial extent of Hα gas, corroborates this scenario (e.g., Hogan et al. 2017a). We remark that this process differs from linear thermal instability models, which assume tiny perturbations growing exponentially against the restoring buoyancy force⁴⁷ in a heated halo (indeed, CCA can develop even in a non-heated atmosphere, given significant turbulent fluctuations).

An important result is that within the condensation region, both timescales do not deviate drastically from each other (C ∼ 1), except in the very inner region, implying that both generation of fluctuations and cooling act at fairly concurrent times, inducing the drop-out of warm filaments with multiple scales within the extent radius (a fraction of which triggers the central AGN). In other words, C ∼ 1 naturally traces what other studies refer to as the "precipitation"–feedback balance threshold. If the plasma cooling timescale were too short with perturbations that are weak and/or injected at very large scales (C ≪ 1), a monolithic overcooling of the X-ray halo would develop without extended multiphase structures. This is expected to be infrequent (e.g., A2029 and A2107) because of the AGN feedback self-regulation which elevates t_cool back while decreasing t_eddy via the injection of turbulence. Conversely, when the cooling time is too long⁴⁸ (C ≫ 1), even significant perturbations (e.g., driven by mergers) cannot induce multiphase condensation, thus preserving the NCC structure (an exemplary case is the Coma cluster).

The right panel in Figure 5 shows the same timescales as above but for 1 keV groups/massive galaxies. Typical massive groups have Z = 1 Z_⊙ and K₅₀₀ ≃ 356 keV cm² at R₅₀₀ ≃ 470 kpc (Sun et al. 2009). Cluster-like NCC groups with a flat and elevated entropy core seem nonexistent (although a more complete sample is required; see also O'Sullivan et al. 2017). Following Sun et al. (2009) constraints, we plot as "NCC" the upper envelope of their observed groups. At large radii, the profiles follow the adiabatic baseline, while within R₅₀₀ the CC groups trace a similar scaling to that found by Panagoulia et al. (2014). Given the non-flattening entropy profiles of groups, the Hα filament threshold is best taken within 5–15 kpc and not as a straight line, where multiphase groups show significantly lower entropy corresponding to t_cool ∼ 100 Myr (Werner et al. 2014). Regarding the eddy time, following the observational constraints by Shin et al. (2016), the driven AGN bubbles in groups show smaller average injection scales with diameters L ≈ 5 kpc. Because of the self-regulated, thus diminished, AGN feedback power (in part counterbalanced by the smaller deposition volume), the average turbulent velocity dispersions are lower, too. For our Hα+[N ii] emitting groups, the median is ≈20% lower, so we use a reference 3D velocity dispersion σ_v,L ≃ 190 $\mathrm{km}\,{{\rm{s}}}^{-1}$ (see also Ogorzalek et al. 2017).

The same main result for clusters applies to groups, but with the condensation region (C ≈ 1) now more compressed between radii of 0.6–15 kpc. For our group sample, we find a logarithmic median and dispersion for the extent radius of $\mathrm{log}R/\mathrm{kpc}=0.44\pm 0.24$, i.e., a 2σ range of 0.9–8.3 kpc. McDonald et al. (2011) find a similar median around a few kiloparsec, with a slightly wider range, 0.5–18 kpc (dashed ellipse). Unlike clusters, some massive groups without extended multiphase filaments (e.g., NGC 4472 and NGC 1399) and residing at the CC–NCC transition can still have low cooling times (below 50 Myr) within the inner r < 500 pc, crossing below the eddy time. This is likely related to the ubiquitous presence of central kiloparsec-scale warm "coronae" in groups (Sun 2009).

It is interesting to point out that the condensation radius is often comparable to the AGN bubble radius, and thus to the injection scale (again a sign of tight self-regulated feedback), for both clusters and groups. Therefore, if an observation is lacking any evident cavity detection and if the goal is to quickly estimate the C ratio, then L ≈ 2R_ex can be used as an alternative estimate for the injection scale. This keeps the t_eddy calculation solely based on the condensed gas properties. We remark that the C ratio can be purely retrieved from observational data (e.g., Hα+[N ii]) without requiring velocity dispersions or injection scales from simulations (though the two have been proven to be consistent; Figure 4).

Addressing the key issues introduced in Section 5.1, the C ≈ 1 criterion is able to solve the unity problem, which should be the natural transition of physical processes. It also does not show the one order of magnitude theoretical uncertainty of the TI ratio (which has possible threshold values t_cool/t_ff ≈ 8–70), though still retaining the ±0.25 dex intrinsic scatter due to system-to-system variations mainly tied to the cooling time. Perhaps more important, the eddy time is observationally much easier and robust to measure compared with the challenging total masses, as the large-scale velocity dispersion can be computed in several low-T phases via the ensemble method (Section 3.2). Indirect hydrostatic masses are required to compute t_ff; however, solely considering turbulence and AGN feedback,⁴⁹ hydrostatic masses in the core can be off by a factor of several (e.g., Gaspari et al. 2011b), inducing a major systematic uncertainty in the TI ratio. On the other hand, the C criterion takes advantage of the substantial noise reduction provided by the integrated spectrum measure, which is directly accessible from multiple frequency bands.

We note t_ff is a lower limit to the eddy time, as the condensed structures do not escape the cluster or group central potential. Moreover, AGN feedback self-regulates based on the halo X-ray luminosity (thus mass), so we still expect some degree of secondary correlation between these two dynamical timescales. Finally, while the hot gas cooling time shows the largest variation between CC and NCC systems, it is important to note that the physical C crossing threshold is still set by t_eddy. The crossing gradually decreases from massive clusters to groups and small galaxies, from 0.5 Gyr to ≲100 Myr, as the eddy time has a total mass trend of roughly ∝M^0.2.

In conclusion, we expect groups to display condensed structures more frequently than in clusters, but with a much more concentrated topology and with lower warm and cold gas masses. An excellent case study for the presence of soft X-ray turbulence/perturbations, cospatial warm filaments, and cold molecular gas is NGC 5044 (e.g., Gastaldello et al. 2009; David et al. 2017). Over different environments, the C ≃ 0.6–1.8 range (90% interval) probes well the condensation region of observed systems and thus the multiphase state of the core. Future studies should extend the sample of clusters, groups, and especially low-mass galaxies, taking advantage of the linked kinematic properties between the hot, warm, and cold gas phases, and thus complementing the thermodynamical constraints set via high-resolution CCD imaging.