A MECHANISM FOR STIMULATING AGN FEEDBACK BY LIFTING GAS IN MASSIVE GALAXIES

Several studies have argued that the cooling time and entropy thresholds are a consequence of thermal instabilities that develop in hot atmospheres when the ratio of ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}$ falls below 10 (McCourt et al. 2012; Sharma et al. 2012; Voit et al. 2015). If true, the ratio should more reliably forecast Hα emission than the cooling time or entropy index alone. We find that the ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}\lesssim 10$ criterion as applied in these studies is less reliable. For example, of more than 200 cluster cores studied by Cavagnolo et al. (2008), only five lying below the cooling time/entropy threshold failed to shine with Hα emission, an iconic example being Abell 2029, which we discuss in detail below. In contrast, only 10 of 43 systems in Voit & Donahue (2015) with detectable Hα emission met the ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}\lesssim 10$ criterion.

We illustrate this point by calculating the ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}$ threshold as it applies to three iconic central cluster galaxies for which we plot the cooling time, free-fall time, and their ratio in Figure 1. The center and right panels of Figure 1 show the free-fall time profiles and ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}$ profiles, respectively, for each cluster. The cooling time profiles were derived from deprojected gas density profiles, which removes emission from hot gas at large atmospheric distances seen in projection. The free-fall times were estimated for Abell 2029 and MS 0735+7421 assuming hydrostatic Navarro–Frenk–White (NFW) profile fits to the X-ray data beyond an altitude of about 30 kpc. Those profiles were then grafted to isothermal profiles within 30 kpc whose masses are anchored to the stellar mass, which matches their stellar velocity dispersion profiles. The M87 data were taken from Russell et al. (2015), who adopted the NFW2 profile from Romanowsky & Kochanek (2001). A complete discussion of our methodology and results for a large sample of clusters will be presented in Hogan et al. (2016, in preparation) and Pulido et al. (2016 in preparation).

Zoom In Zoom Out Reset image size

We use these three iconic examples to illustrate a general trend. The central cooling times for all three objects fall near to or below the cooling time threshold of $\sim 5\times {10}^{8}\,{\rm{yr}}$, indicated by a dashed line in the left panel of Figure 1. M87 (Sparks et al. 2004) and MS 0735+7421 (Donahue et al. 1992) shine in nebular emission within $5\mbox{--}30\,{\rm{kpc}}$ of their nuclei, as expected based on their short central cooling times. Hα emission in central galaxies is associated with the presence of molecular clouds (Edge 2001). The third cluster, Abell 2029, has revealed no appreciable Hα emission (McDonald et al. 2010), [O ii] λ3727 emission, or star formation (McNamara & O'Connell 1989) in its central galaxy, despite falling well below the cooling time threshold. All three lie well above ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}\lesssim 10$. By this criterion, shine in Hα emission, yet two do. At the same time, the cooling time threshold predicts that all three should shine with Hα emission, yet Abell 2029 does not. Something is awry. We suggest that new and interesting physics is needed to solve the problem. A more detailed description of these objects is given in Appendices A, B, and C.

Regardless of hosting AGNs spanning 4 decades of radio-mechanical power, the cooling profiles for all three are remarkably similar. Due to their larger distances, the profiles for Abell 2029 (z = 0.077) and MS 0735+7421 (z = 0.216) are unresolved below a few kiloparsecs. Nevertheless, M87's cooling times at 2 and 6 kpc are similar to those of Abell 2029 and MS 0735+7421 at similar altitudes. Based on their similar shapes, it would be tempting to suggest that the cooling time profiles for Abell 2029 and MS 0735+7421 continue to decline into the nucleus despite the vast differences in AGN mechanical power between the three objects.

The key point is, despite short central cooling times, the ratios of cooling time to free-fall time all lie well above 10. Therefore, all should be thermally stable and devoid of Hα emission, yet two are not. McCourt et al. (2012) suggested that an upward departure from the ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}\lesssim 10$ criterion in systems with bright Hα emission, such as M87 and MS 0735+7421, may be a consequence of a temporary decrease in central gas density in response to AGN heating. This explanation cannot be excluded out of hand given the scatter in the central cooling time profiles of clusters (Panagoulia et al. 2014). However, that the cooling profiles in Figure 1 are so similar, despite ongoing AGN activity in both M87 and MS 0735+7421, indicates that a dramatic increase in AGN power need not lead to a dramatic upward response in atmospheric cooling time, the radial run of ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}$, or central gas density.

This point is further illustrated in Figure 2, where we plot the radial run of gas density in the hot atmospheres of the three objects discussed here. In addition, we have included Abell 1835, which hosts one of the largest reservoirs of molecular gas ($5\times {10}^{10}\,{M}_{\odot }$) and one of the highest star formation rates ($\sim 200\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$) known (McNamara et al. 2006, 2014). Figures 1 and 2 illustrate the surprisingly small variation in both gas density and cooling time of the hot atmospheres, despite an enormous range of AGN power. The objects shown in Figure 2 span 5 decades in AGN energy and nearly 4 decades in molecular gas mass. Yet the variation in their gas densities at 10 kpc, which is where most studies find a minimum value in ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}$, lie in the narrow range of $(2\mbox{--}10)\times {10}^{-2}\,{\mathrm{cm}}^{-3}$. Furthermore, their cooling times at 10 kpc are nearly identical. Despite vast differences in their star formation rates, their molecular gas masses, and, most importantly, their AGN power, their central X-ray gas densities and cooling times are remarkably steady, showing little evidence of cycling indicated in precipitation models. Evidently AGNs contribute little to the scatter in central cooling times found by Panagoulia et al. (2014), but instead probably reflect variations in their halo masses. The resilience of the central gas density and central cooling time to powerful AGN outbursts is, in fact, a key feature of hot atmospheres stabilized by continual and gentle AGN feedback (McNamara & Nulsen 2012).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

This is not to say that all clusters with multiphase gas fail the ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}\lesssim 10$ instability criterion. But we find that systems that obey the criterion do so because their central cooling times are short and not because free-fall times are long. Among the several studies that have examined this criterion (Sharma et al. 2012; Voit & Donahue 2015; Voit et al. 2015; Loubser et al. 2016), all did so by calculating the ratio of the average cooling time in radial bins and dividing by an estimate of the free-fall time from a given altitude using a similar approach to ours. These studies assumed that thermal instability ensues if and where the minimum value of the ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}$ profile falls below 10. Minima are usually found at a radius of about 10 kpc, consistent with Figure 1. Their measured locations depend on several physical and nonphysical effects, including the relative slopes of the gas and free-fall time profiles and instrumental resolution. The largest uncertainty concerns the value of the free-fall time, as the acceleration is difficult to measure using standard techniques: stellar velocity dispersions are in short supply, and hydrostatic mass measurements are difficult to measure using standard techniques. The limited number of available velocity dispersions led Voit et al. (2015), understandably, to calculate the free-fall time adopting a velocity dispersion floor of $\sigma =250\,\mathrm{km}\,{{\rm{s}}}^{-1}$. While reasonable on average, this assumption biases ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}$ artificially low as stellar velocity dispersions in central cluster galaxies often lie well above $300\,\mathrm{km}\,{{\rm{s}}}^{-1}$. For example, M87 and Abell 2029, with velocity dispersions of $\sigma \sim 340\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (Gebhardt et al. 2011) and $\sigma \sim 400\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (Fisher et al. 1995), respectively, drive ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}$ well above 10 (Figure 1).

That central cooling time is driving the ratio is evident in Loubser et al. (2015), who studied star formation in a sample of 18 central galaxies, four of which are forming stars. The central cooling times of all four star formers lie below 0.5 Gyr, and their ratios of cooling time to free-fall time are claimed to lie below 10, i.e., they apparently obey both the ${t}_{{\rm{c}}}\lesssim 5\times {10}^{8}\,{\rm{yr}}$ and ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}\lesssim 10$ criteria. The central cooling times and ratios of cooling time to free-fall time for the remaining red central galaxies, which are largely devoid of star formation, exceed 1 Gyr and 10, respectively. Thus, these objects are consistent with both criteria. However, the free-fall times for the short and long cooling time systems are consistent with a single value with small dispersion. The average free-fall time for the star-forming systems and dormant systems are $\langle {t}_{\mathrm{ff}}\rangle \,=\,0.064\pm 0.016\,{\rm{Gyr}}$ and $\langle {t}_{\mathrm{ff}}\rangle \,=\,0.055\pm 0.014$, respectively. Therefore, ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}$ in this study, as in others, is insensitive to the free-fall time and is governed entirely by the cooling time, t_c. We performed an analysis for Voit & Donahue (2015) and found, similarly to Loubser et al. (2015), that dividing by the free-fall time only increases the scatter in the fundamental relationships between central cooling time, entropy, Hα emission (Cavagnolo et al. 2008), and star formation (Rafferty et al. 2008). Therefore, the ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}\lesssim 10$ criterion, as it has been applied, does not indicate the onset of cooling instabilities.

Three-dimensional, high-resolution simulations of the effects of AGN feedback on cooling atmospheres have offered new insights into AGN feedback. For example, Li & Bryan (2014) and Li et al. (2015) modeled the response of X-ray atmospheres to AGN feedback over a period of several gigayears using an adaptive mesh refinement code, while Prasad et al. (2015) used two- and three-dimensional hydrodynamic simulations. Despite significant differences in approach, their model predictions are broadly similar to each other. In these simulations, hot atmospheres experience large swings in gas density, cooling time, molecular gas mass, and star formation rate in response to AGN power output over a span of several gigayears. The AGN power variations and star formation rates roughly correlate with rising and falling levels of molecular gas cooling from hot atmospheres. For example, Li et al. (2015) found swings in the minimum value of the cooling time and minimum value of ${t}_{{\rm{c}}}/{t}_{{\rm{ff}}}$ that vary by two orders of magnitude and factors of 25, respectively, as the atmosphere breaths in response to variations in AGN power. Prasad et al. (2015) found similarly large amplitude swings in minimum ${t}_{{\rm{c}}}/{t}_{{\rm{ff}}}$ and jet power. The molecular gas mass in these models likewise varies by three to four orders of magnitude as it is consumed by star formation. In the Li et al. (2015) model, the molecular gas levels peak when the black hole, and presumably the radio AGN, are at maximum accretion and power, respectively. The star formation rate, jet power, and molecular gas mass all move roughly together, albeit with a lag in time, over the several-gigayear simulations. As the cold gas is consumed by star formation, the jet power diminishes as its fuel supply subsides. Declining jet power causes the atmosphere to contract; the atmospheric gas density rises, causing ${t}_{{\rm{c}}}$ to drop as ${t}_{{\rm{c}}}/{t}_{{\rm{ff}}}$ approaches a minimum near unity; and the cooling cycle begins anew.

While these models capture the quasi-periodic nature of AGN feedback, we are unable to match the observed minimum values of ${t}_{{\rm{c}}}/{t}_{{\rm{ff}}}\simeq 20$ for MS 0735, M87, and Abell 2029 to the predicted minimum cooling time, jet power, molecular gas mass, and star formation rates at any time during the several gigayears spanning the models. For their minimum values of ${t}_{{\rm{c}}}/{t}_{{\rm{ff}}}$, the models generally predict star formation rates, molecular gas, and cooling times at levels exceeding those observed. Only Abell 1835, for which we find a minimum ${t}_{{\rm{c}}}/{t}_{{\rm{ff}}}\simeq 10$, may correspond to an acceptable solution at $\sim 4\mbox{--}5$ Gyr in the models of Li et al. (2015) and Prasad et al. (2015), as molecular gas builds up at late times.

Observational trends with molecular gas masses (Edge 2001) may offer additional insight. For example, no clear trend between total molecular gas mass and AGN power is found in central cluster galaxies (McNamara et al. 2011). The relationship between molecular gas mass and jet power reveals a 3-decade scatter in both variables, superposed, perhaps, on a weak trend. Our main point is that high molecular gas masses do not necessarily lead to powerful AGN activity, at least when considering only central cluster galaxies (this may not be true for lower jet power elliptical galaxies). This point is clearly illustrated by the enigmatic MS 0735+7421. At $\sim {10}^{46}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ over the past few hundred megayears, MS 0735+7421 is the most energetic AGN outburst known. More importantly, its central atmospheric gas density at 10 kpc is similar to others with vastly lower AGN power, and its molecular gas mass lies well below much weaker AGNs. Prasad et al. (2015) have argued, using their model, that the large scatter between molecular gas mass and jet power arises because the molecular gas is locked up in kiloparsec-scale disks (e.g., Gaspari et al. 2012) that are unable to fuel the central black hole. Whether this is generally true is not clear. ALMA observations, which are in short supply, would be required to resolve the molecular gas and to test the molecular disk hypothesis. The main difficulty comparing observation to the models is that we are unable to identify a reliable observational marker of the time-evolutionary state of these systems (${t}_{{\rm{c}}}/{t}_{{\rm{ff}}}$ is unsuited). Once we do, ALMA will in principle test whether objects with the highest molecular gas masses are evolutionarily advanced.

The crux of the problem, in our view, is that we do not observe the large-amplitude swings in atmospheric gas density (Figure 2) or cooling time (Figure 1) that correspond to atmospheric "overheating" implied by simulation (Gaspari et al. 2012; Li & Bryan 2014; Li et al. 2015). Nor do we find that ${t}_{{\rm{c}}}/{t}_{{\rm{ff}}}\lesssim 10$ corresponds to the onset of cooling instability. However, Li & Bryan (2014) and Li et al. (2015) pointed out that cooling is enhanced in their simulation by turbulence and when AGNs lifted hot gas to higher altitudes (McNamara et al. 2014; Voit & Donahue 2015). Both processes tend to increase the infall time of the cooling gas driving the local value ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}$ toward unity. This is an important result that we believe is key to understanding the thermal instability of hot atmospheres.

Studies of the Perseus Cluster have revealed an association between molecular clouds, Hα filaments, and buoyantly rising X-ray bubbles (Salomé et al. 2011). Tendrils of molecular gas have apparently been lifted tens of kiloparsecs in altitude from NGC 1275 into the Perseus Cluster, with velocity fields consistent with inflow and/or outflow. ALMA observations of several other central cluster galaxies (including Abell 1835 and PKS 0745-191) have since revealed similar molecular cloud complexes lying beneath buoyantly rising X-ray cavities. Clouds are seen in thin filaments, with velocity fields and locations that further suggest a close relationship between X-ray bubbles and molecular gas (McNamara et al. 2014; Russell et al. 2016).

The Cycle 0 observations of Abell 1835 (McNamara et al. 2014) revealed two velocity components: a fast component of ${10}^{10}\,{M}_{\odot }$ of molecular clouds traveling with projected speeds of $200\mbox{--}500\,\mathrm{km}\,{{\rm{s}}}^{-1}$, and a slowly moving, $4\times {10}^{10}\,{M}_{\odot }$ component of molecular gas. The fast component lies beneath buoyantly rising X-ray cavities at altitudes of $5\mbox{--}10\,{\rm{kpc}}$, while the slow component lies at the center of the galaxy associated with $\sim 200\,{M}_{\odot }{\mathrm{yr}}^{-1}$ of star formation. The fast clouds are likely an outflow propelled by the rising X-ray bubbles. However, it is unclear how the molecular clouds are accelerated. Are molecular clouds themselves lifted and accelerated by the bubbles and jets, or is molecular gas condensing from hot, keV gas lifted in the bubbles' wakes? The answer may be "both." Acceleration is an issue not just in clusters but in active galaxies in general (Cicone et al. 2014; Morganti et al. 2015), where the molecular outflows are thought to be driven by jets (e.g., Nesvadba et al. 2008; Wagner & Bicknell 2011) and winds.

The difficulty in lifting ${10}^{10}\,{M}_{\odot }$ of molecular clouds by jets and bubbles has been discussed in detail by Russell et al. (2016), McNamara et al. (2014), and David et al. (2014), who found in all cases that AGNs release enough energy to account for the kinetic energy in molecular gas flows. However, whether low-density jets have sufficient momentum to accelerate molecular clouds, whose densities exceed jet densities by four or five orders of magnitude, is unclear.

The total momentum flux (force) available from a kinetic-energy-dominated jet of power P_j is $({{\rm{\Gamma }}}_{j}+1){P}_{j}/({{\rm{\Gamma }}}_{j}{v}_{j}$), where v_j is the jet speed and ${{\rm{\Gamma }}}_{j}$ is the corresponding Lorentz factor. The buoyant force due to a bubble is ${F}_{B}={\rho }_{e}{gV}$, where $g={v}_{K}^{2}/R$ is the acceleration due to gravity. Estimating the volume as $V=H/(4p)={P}_{j}{t}_{j}/(4p)$ gives ${F}_{B}=({P}_{j}/{v}_{B})[{v}_{K}^{2}/(4p/{\rho }_{e})]$, where the mean speed of the bubble as it formed is ${v}_{B}=R/{t}_{j}$, and H is the bubble's enthalpy. The factor in square brackets is of order unity, and the mean speed of the bubble is comparable to the sound speed. Therefore, the buoyant force exceeds the jet ram pressure unless the flow speed of the jet is transonic relative to the atmosphere or slower. The small cross sections of jets make them even less effective at lifting.

Bubble buoyancy is generally more effective at lifting than the ram pressure of the jet that inflated it. The lifting ability of radio bubbles in general is limited by Archimedes's principle, which prohibits them from lifting more weight than they displace. The displaced mass in Abell 1835 is uncomfortably close to the $\sim {10}^{10}\,{M}_{\odot }$ of molecular gas flowing behind the bubbles (McNamara et al. 2014), while in PKS 0745–191 the bubbles displace roughly 10 times the molecular gas mass (Russell et al. 2016). However, David et al. (2014) found that the molecular gas mass substantially exceeds the displaced mass in NGC 5044. Therefore, bubbles may be able to lift the molecular gas in Abell 1835 and PKS 0745–191, while NGC 5044's bubbles would be unable to do so.

Additional clues may be found from the radio sources themselves. Observations of the Abell 2597 and Abell 1795 central galaxies have shown that their radio jets bend by roughly 90° at the locations of molecular clouds and knots of star formation (McNamara et al. 1996; Salomé & Combes 2004, Tremblay et al. 2016, in preparation). The sharp bending suggests that a collision between the ensemble of molecular clouds and jets halted the jets' forward momenta (McNamara et al. 1996). Therefore, the jets are unlikely to be accelerating the molecular clouds appreciably, at least over the time the jets have been in contact with the molecular clouds. In fact, the molecular clouds in Abell 2597 are moving below the circular speed at their radius (Tremblay et al. 2016, in preparation) and should then be falling in rather than being driven out.

In summary, observation indicates that some molecular gas is lifted directly by radio jets and bubbles. But the often large molecular gas masses relative to the hot gas mass displaced by X-ray bubbles suggests that some or most is lifted in the hot phase. Lifting hot, volume-filling gas behind the bubbles is easier, and the coolest gas may be able to condense into molecular clouds on the same timescales in which bubbles rise to their observed locations (McNamara et al. 2014; Russell et al. 2016).

Chandra has revealed columns of high-metallicity gas along and behind X-ray bubbles in clusters. Gas with high metallicity approaching and sometimes exceeding the solar value, enriched by stellar evolution, accumulates around central galaxies. The metal-rich columns of gas extending tens to hundreds of kiloparsecs in elevation are thought to trace hot flows lifted outward by radio bubbles (Simionescu et al. 2008; Kirkpatrick et al. 2009, 2011; Werner et al. 2010; Werner et al. (2011)). This phenomenon is also seen in hydrodynamic simulations where metal-enriched gas in the central galaxy is propelled outward in the updraft of rising bubbles (Pope et al. 2010; Gaspari et al. 2011). Estimated flow rates of several tens of solar masses per year would be sufficient to account for the observed molecular gas masses. The altitudes achieved are roughly proportional to the square root of the jet power (Morsony et al. 2010; Kirkpatrick & McNamara 2015) and are on the order of tens of kiloparsecs for typical cluster AGNs, and several hundred kiloparsecs in the most powerful systems such as MS 0735+7421 and Hydra A (Simionescu et al. 2008; Kirkpatrick et al. 2011). Therefore, the lifting of hot gas by radio bubbles would be at least a plausible source of fuel for molecular cloud condensations at high altitudes.

Among the puzzling results from early ALMA observations of clusters are the surprisingly low radial velocities and velocity widths of the molecular gas. In response to the slow inner cloud velocities in Abell 1835, we suggested the mundane and unlikely possibility that the slow (radially) moving nuclear clouds are rotationally supported in a disk viewed in the plane of the sky (McNamara et al. 2014). While this interpretation may apply to Abell 1835 itself, cloud velocities lying well below the stellar velocity dispersion or circular speeds have now been seen in several systems (David et al. 2014; Russell et al. 2015, 2016, Tremblay et al. 2016, in preparation), indicating a phenomenon unrelated to rotational support or orientation effects. Traveling well below their expected gravitational speeds, either the clouds have had no time to relax in their gravitational potential wells and thus are surprisingly young, or they are pinned by magnetic fields or dynamic pressure to the hot gas from which they cooled, or some combination (Russell et al. 2016). Regardless of the cause, if molecular cloud velocities are representative of the speeds of condensing clouds, their velocities are several times lower than their hosts' stellar velocity dispersions and below their expected free-fall speeds.

Following the arguments of Nulsen (1986), low-entropy ($\lesssim 1\,\mathrm{keV}$) gas should condense into molecular clouds when lifted to an altitude where the ratio of its cooling time to infall time, ${t}_{{\rm{c}}}/{t}_{{\rm{I}}}$, approaches unity. Here, the infall time can be longer than the free-fall time, as indicated by ALMA observations. We postulate that systems with short central cooling times ${t}_{{\rm{c}}}\lesssim 5\times {10}^{8}\,\mathrm{yr}$, yet lacking X-ray cavities powerful enough to lift low-entropy gas to altitudes where ${t}_{{\rm{c}}}/{t}_{{\rm{I}}}\sim 1$, remain thermally stable and thus do not shine with nebular emission.

For example, Abell 2029 is apparently thermally stable, i.e., ${t}_{{\rm{c}}}/{t}_{{\rm{I}}}\gt 1$ throughout its hot atmosphere. Figure 1 shows that the mean atmospheric cooling time within 10 kpc is ${t}_{{\rm{c}}}=(3\mbox{--}5)\times {10}^{8}\,\mathrm{yr}$. Over this volume, ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}=20\mbox{--}30$ and its atmosphere remains thermally stable, despite its short central cooling time. The atmosphere will become unstable where ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}\lesssim 1$. Its AGN must then lift hot gas from the inner 10 kpc to altitudes between ∼280 and 400 kpc, where the free-fall time is approximately equal to the cooling time of the low-entropy gas lifted from within 10 kpc. Relaxing the thermal instability criterion to ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}\lesssim 10$ implies lifting altitudes between 15 and 30 kpc. Upon lifting the cooler, denser central gas into the lower-pressure atmosphere at higher altitudes, the lifted gas will expand and cool. Being denser than its surroundings, it should detach from the outflowing gas behind the bubble and fall back to the galaxy as it condenses into molecular clouds. If the cooling gas remains tethered to the surrounding atmosphere by magnetic fields or dynamical pressure, as ALMA observations suggest, its infall time, being governed by the terminal speed, would exceed the free-fall time. This would reduce the lifting altitude required to initiate cooling, such that ${t}_{{\rm{c}}}/{t}_{{\rm{I}}}$ approaches unity. In this picture, Abell 2029 fails to shine in Hα emission because its radio emission, despite being fairly powerful, lies within 25 kpc of the nucleus (Paterno-Mahler et al. 2013). Abell 2029 has apparently not developed cavities capable of lifting low-entropy gas to the altitudes required to destabilize it (see Figure 3).

A similar analysis for MS 0735+7421 gives similar figures. However, unlike Abell 2029, its powerful X-ray bubbles are lifting hot gas to altitudes upward of 300 kpc (Kirkpatrick et al. 2011), well beyond the elevations required to stimulate cooling. As expected, its Hα emission extends $20\mbox{--}40\,{\rm{kpc}}$ in the central galaxy with a luminosity among the highest known in a cluster (Donahue et al. 1992). The third example, M87, lying in a cooler atmosphere, will destabilize at lower altitudes and thus requires a lower-power AGN to stimulate cooling. Its average cooling time within a kiloparsec or so lies below $\sim {10}^{8}\,\mathrm{yr}$. Therefore, following on the previous examples, its bubbles must lift this gas to altitudes of only $5\mbox{--}20\,{\rm{kpc}}$ or so to initiate thermal instability. Shock fronts, bubbles (Forman et al. 2007), and metal-enriched gas columns (Simionescu et al. 2008) are observed in M87 to elevations exceeding 10 kpc, and nebular emission is observed within a similar volume (Sparks et al. 2004), which again is consistent with our model.

Perhaps the best example is the Perseus Cluster, which contains $\sim {10}^{10}\,{M}_{\odot }$ of molecular gas centered on NGC 1275 and in filaments extending to altitudes of $30\mbox{--}50\,{\rm{kpc}}$ (Fabian et al. 2003; Salomé et al. 2011). The cooling time of its ambient gas exceeds the free-fall time by more than an order of magnitude and should be thermally stable as ${t}_{{\rm{c}}}/{t}_{\mathrm{ff}}\gt 10$ at these elevations. However, the association between molecular gas, Hα filaments, and its system of X-ray bubbles is consistent with low-entropy gas lifted to altitudes where it can cool and return to fuel star formation in NGC 1275.

Finally, we have examined archival Chandra images of the five "spoilers" in Cavagnolo et al. (2008), those systems whose central atmospheric cooling times fall below the cooling time threshold yet lack detectable Hα emission. Like Abell 2029, and shown here in Figure 4, only one, RBS 0533, has revealed a possible cavity located approximately 10 kpc to the southeast of its centroid, but otherwise no prominent cavity systems are seen (Abell 2029 is among the five). The archival images shown in Figure 4 have not been exposed deeply enough to exclude possible faint cavities at large radii. Nevertheless, until deeper exposures are obtained, the data in hand are consistent with our phenomenological model. Stimulated feedback is thus a viable and testable alternative to the conceptually important thermal instability (McCourt et al. 2012) and precipitation (Voit et al. 2015) models, which are inconsistent with observation.

Zoom In Zoom Out Reset image size