Cool-Core Cycles and Phoenix
Deovrat Prasad Prateek Sharma Arif Babul G. Mark Voit Brian W. O'Shea


firstpage–lastpage
On extrema of the objective functional for short-time generation of single qubit quantum gates
    Alexander Pechen^1[Corresponding author. E-mail: apechen@gmail.comapechen@gmail.com; Webpage: http://www.mathnet.ru/eng/person17991mathnet.ru/eng/person17991]   and Nikolay Il'in^2
    
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Recent observations show that the star formation rate (SFR) in the Phoenix cluster's central galaxy is ∼ 500 M_⊙ yr^-1. Even though Phoenix is a massive cluster (M_200≈ 2.0× 10^15 M_⊙; z≈ 0.6) such a high central SFR is not expected in a scenario in which feedback from an active galactic nucleus (AGN) maintains the intracluster medium (ICM) in a state of rough thermal balance. It has been argued that either AGN feedback saturates in very massive clusters or the central supermassive black hole (SMBH) is too small to produce enough kinetic feedback and hence is unable to quench the catastrophic cooling. In this work, we present an alternate scenario wherein intense short-lived cooling and star formation phases followed by strong AGN outbursts are part of the AGN feedback loop. Using results from a 3D hydrodynamic simulation of a standard cool-core cluster (M_200∼ 7×10^14 M_⊙; z=0), scaled to account for differences in mass and redshift,
we argue that Phoenix is at the end of a cooling phase in which an AGN outburst has begun but has not yet arrested core cooling. This state of high cooling rate and star formation is expected to last for ≲ 100 Myr in Phoenix.     


Galaxy Clusters, Cooling flow, Black Hole, AGN feedback, Intra-cluster medium 



§ INTRODUCTION

Recent deep multi-wavelength observations of the Phoenix cluster by <cit.> have detected a star formation rate of ∼ 500 M_⊙ yr^-1 (z≈0.6) in the central galaxy (r<50 kpc). Such a high star formation rate is not expected in the standard AGN feedback scenario, even in a massive galaxy cluster like Phoenix (M_200≈ 1.8×10^15 M_⊙).  However, it is consistent with the condensation rate of a pure cooling flow in the intracluster medium (ICM). Phoenix is therefore unlike CL09104, another massive galaxy cluster in which the high star formation rate (70-200 M_⊙ yr^-1) is due to an ongoing merger (). Given that the cooling time, t_ cool≡ 3nk_BT/[2 n_e n_iΛ], in the central r<20 kpc of Phoenix is an order of magnitude shorter than in any other observed galaxy cluster, <cit.> have speculated that Phoenix has experienced an evolutionary pathway different from other cool core clusters. 

The intracluster medium (ICM) is expected to become multiphase whenever min(t_ cool/t_ ff) ≲ 10 (with t_ ff≡√(2r/g)), leading to star formation and AGN outbursts (). Most cool core clusters are observed to be in a state of rough thermal balance, with min(t_ cool/t_ ff) between 10-20 ().  Few have  < 10, but Phoenix has  ∼ 1 at r≈3 kpc ().  None with  < 5 were previously known, making Phoenix a significant outlier.

Phoenix has a large X-ray luminosity (≈ 4.7×10^45 erg s^-1; ) and a kinetic feedback power ∼ 10^46 erg s^-1 inferred from the cavities the AGN has produced, assuming the AGN has shut off and the cavities are rising buoyantly ().  The cavity power is therefore comparable to the X-ray luminosity, but it has been argued that the mechanical jet power may be saturated and unable to quench the cooling flow, perhaps because of a central black hole mass that is small compared to the halo mass ().  

In this work, we argue that Phoenix is not fundamentally different from other cool-core clusters and does not necessarily represent a failure of kinetic SMBH feedback. We suspect instead that Phoenix is near the end of a short-lived cooling state, with AGN feedback just turning back on after a ≪ 1 Gyr starburst[Note added during pre-publication: The age  of the cavities in Phoenix from  the latest observations is estimated to be < 10 Myr ()]. Given that the fraction of young stellar population is small, the current starburst cannot be long lived.  Similar events occur in the fiducial simulation of <cit.>,[The 3D hydrodynamic NFW+BCG run without stellar depletion.] in which a strong AGN outburst follows each phase of core cooling. This transitional state lasts for only ∼ 50-100 Myr, after which the AGN heats and disperses the core to a state more typical of cool core clusters.  In simulations, such cooling phases are seen across a range of halo masses but are most frequent in the most massive halos, given a fixed efficiency for converting mass accretion into kinetic feedback power ().  Cooling phases are more frequent in very massive halos because even strong feedback is unable to push low-entropy gas far from the center.  



§ COOLING AND INITIAL AGN HEATING PHASE


In the simulation discussed here, the ICM was initialised in hydrostatic equilibrium in a gravitational potential given by the sum of a Navarro-Frenk-White (NFW) potential for the dark matter halo and a singular isothermal sphere (SIS) potential for the central brightest cluster galaxy (BCG).  The fiducial run initially has min(t_ cool/t_ ff) ≈ 7 at r≈10 kpc and a minimum cooling time ≈ 200 Myr. An accretion rate (Ṁ_ acc) calculated at 0.5 kpc determines the power ϵṀ_ acc c^2 of bipolar jets of kinetic feedback injected at small radii (for details see ).
Accretion and jet injection are minimal at first because the galaxy cluster is initialized in hydrostatic equilibrium. After a core cooling time (≈ 200 Myr) the cooling-flow accretion rate rises, resulting in enhanced AGN jet power.  AGN outbursts then heat the core, push gas around, and raise t_ cool/t_ ff, keeping the average Ṁ_ acc well below the cooling-flow value. Throughout its lifetime, the cool core undergoes multiple radiative cooling and AGN heating cycles. 


Figure <ref> shows density (left panels) and temperature (right panels) slices along the jet axis during a cooling phase at t=2.62 Gyr (top panels) and the subsequent AGN outburst at t =2.64 Gyr (bottom panels). Those epochs correspond to a strong starburst before AGN feedback couples to the ICM and quenches star formation (see the right panel in Fig. 4 of ). 

Figure <ref> shows projected surface brightness maps of the X-ray gas (0.5-10 keV) at the same two epochs as Figure <ref>. During the cooling phase the X-ray surface brightness is relatively smooth within r ∼ 10 kpc (upper panel). But as AGN jets begin to make their way out of the core, pronounced surface brightness features appear in the form of bright rims around the jet-driven cavities (lower panel).

Morphologically, Phoenix appears to be in a state similar to the t=2.64 Gyr snapshots (compare our Figs. <ref> & <ref> with Figs. 1, 2, and 5 in ). Current star formation, as traced by UV and [OII], appears enhanced at the periphery of the X-ray cavity created by the AGN outburst. The bipolar X-ray cavities observed in Phoenix are attached to the center and have a major axis of ∼ 25  kpc, rather similar to (but somewhat bigger than) those in our simulations at 2.64 Gyr.

Cooling phases followed by AGN outbursts are expected to be more frequent in more massive halos for a fixed accretion efficiency because of the deeper potential well (for details, see Section 3.2.2 in ). This relationship suggests that the likelihood of observing a massive halo in a cooling state may be much greater than for a lower mass halo, possibly explaining why the first detection of a galaxy cluster at the end of a cooling phase happens to be a very massive cluster. 




§ THERMODYNAMIC PROFILES


Phoenix exhibits profiles of entropy, cooling time, and t_ cool/t_ ff in its core, r<0.02× r_200 (∼ 50 kpc), that are unique compared to any other observed cluster. However,a comparison of the Phoenix cluster's scaled[We assume WMAP9 () cosmology to compute the characteristic parameters used to rescale the profiles.] thermodynamic profiles with those from our simulation at epochs corresponding to
the transitions between cooling and AGN outburst phases in our fiducial run reveals close similarities.




The top left panel in Figure <ref> shows the azimuthally-averaged, emissivity-weighted scaled entropy profile
() of X-ray gas (0.5-10 keV) for two different Phoenix-like cooling cycles, the t=1.85-2.0 Gyr (red lines) and the t=2.55-2.7 Gyr cycles (blue lines). The solid lines show the entropy profiles during the cooling phase and the dot-dashed lines show the profiles at the start of the corresponding AGN outburst phases. Entropy in the central r≲ 0.005r_200 (∼ 10 kpc) falls below 10 keV cm^2 during the cooling phase and in the case of the second cycle, remains below this value even 20 Myr after the onset of AGN feedback, as the jets drill their way out of the core. The scaled entropy profile of Phoenix (solid black line with circles) shows strikingly similar behaviour.  In the discussion below, we focus on the second cycle, which as discussed in 2, more closely resembles Phoenix  from a morphological perspective.

During the intense cooling phase, the cooling time (bottom left panel) in the central r≲ 0.005 r_200 (≲ 10 kpc) region of our simulated cluster falls to ∼ 10^-4t_ cool (r_200) (∼ 10s of Myr), and remains below 100 Myr in the core (r<0.005 r_200 kpc) during the early stages of the AGN outburst phase. While the magnitude of Phoenix cluster's
cooling time profile, t_ cool(r), is a factor of 2 lower at all radii, the scaled t_ cool curve is remarkably similar to our simulation results from r∼ 0.1 r_200 (∼ 200 kpc)  down to 0.001 r_200 (∼ 2-3 kpc). 

The scaled emissivity-weighted temperature plot (top right panel, Figure <ref>) shows that the temperature profile of Phoenix over the radial range r < 0.1r_200 is likewise consistent with the cooling phase of our simulated cool core cluster during the second cycle.
The individual wiggles in the temperature profile are due the presence of AGN cavities. 

The bottom right panel in Figure <ref> shows the scaled t_ cool/t_ ff profiles from our simulation.
Before the AGN turns on, t_ cool/t_ ff reaches its minimum value of 0.01× t_ cool,200/t_ ff,200 (∼ 2-3) at r∼ 0.002 r_200 (∼ 5) kpc and rises approximately linearly toward larger radii.  Phoenix similarly exhibits an approximately linear rise in from 0.001 r_200 (∼ 2-3 kpc) out to large radii. The simulation and Phoenix scaled profiles are in excellent agreement.

Figure <ref> shows how the (unscaled) t_ cool/t_ ff profile from our simulation
varies during a 150 Myr period that includes the intense cooling and the subsequent AGN outburst phases at t=2.55-2.7 Gyr. Line colours represent time as shown in the colour bar. During the initial cooling phase, t_ cool/t_ ff declines in the central r<30 kpc, and the plasma becomes unstable to multiphase condensation, producing cold gas clumps that raise the core density and fuel a strong AGN outburst. These profiles show the first signs of AGN heating at 2.64 Gyr within r<3 kpc. At 2.65 Gyr the effect of AGN heating can be seen extending to r<10 kpc, although min(t_ cool/t_ ff) is still ≈ 5. By 2.68 Gyr, the AGN outburst has driven the core to min(t_ cool/t_ ff)≈ 10, and has driven it to ≈ 20 by 2.7 Gyr.       
The figure shows that min(t_ cool/t_ ff) ≲ 5 is maintained for the first ≲20 Myr of the active AGN phase. Similar behaviour is seen towards the end of an earlier cooling cycle at
t=1.85-2.0 Gyr (see Fig. <ref>, as well as the right panel of Fig. 4 in ). 

During the t=1.85-2.0 Gyr cycle, the cooling rate in the central 10 kpc is ∼ 200 M_⊙ yr^-1 (about a third of the steady cooling flow rate), leading to accumulation of ≳ 5×10^9 M_⊙ of cold gas (T<10^4 K) by 1.95 Gyr.
The cooling rate dies down after this as the jets heat up the core, partly evaporating the cold gas leading to a slight decrease in the total cold gas.  During the second cooling cycle at t=2.55-2.7 Gyr, we observe a similar behaviour as well as enhanced cooling mostly along the periphery of the cavities at 2.62-2.66 Gyr as the jet cavities inflated.  The latter leads to an 
accumulation of about 2×10^9 M_⊙ of cold gas (T<10^4 K) and a dropout of > 10^9 M_⊙ of intermediate temperature (10^4 K < T < 5×10^6) gas.  
This is an example of positive feedback ).
Cold gas mass starts to decline after 2.66 Gyr as the shocks and mixing gradually heat up the core.

For comparison, Figure <ref> also shows the scaled profiles (magenta solid lines)
for a steady cooling flow applied to the mass model of Phoenix (M_200=2.5 × 10^15 M_⊙, c_200=10, V_c=250 km s^-1) and appropriate boundary conditions at 500 kpc (n_e=10^-3 cm^-3, K=1057 keV cm^2). 
The steady cooling flow model is similar to the observed profiles (; see also ) but so are the profiles from our simulations (once the mass and redshift differences are taken into account) with feedback in the cooling phase. Thus, similarity to a cooling flow solution does not necessarily imply an absence of feedback heating. 



§ OBSERVATIONAL AND THEORETICAL LIMITATIONS


While the similarities between Phoenix and the cooling phases of our simulated cluster are encouraging, observations of other cool core clusters show that few have 1 ≲min(t_ cool/t_ ff) ≲ 10.  However, the temporal distribution of min(t_ cool/t_ ff)
in our simulation (see Figure 11 of ) suggests that systems with min(t_ cool/t_ ff) ≲ 5 should be present. 
 
One possibility is that Phoenix-like cooling episodes are more common in our idealized non-cosmological simulations than in real clusters.  For example, our idealized simulations do not include cosmological structure formation, which can produce dynamical perturbations capable of promoting multiphase condensation at higher levels of  (), preventing lower levels from being reached.  Consequently, the temporal distribution of min (t_ cool/t_ ff) for a single, idealized, non-cosmological simulated cluster might not be representative of a heterogenous cluster population with very different
accretion histories.  Additionally, our
simulations () does not include magnetic fields or anisotropic thermal conduction, which may also lower the incidence of very small .  All of the above effects will result in less Phoenix-like states than in our simulations.

Another possibility is that low values of min(t_ cool/t_ ff) like the one observed in Phoenix can be detected only in deep observations that provide a large enough photon count at r < 10 kpc. In <cit.>, the large photon count  allowed accurate removal of the AGN X-ray point source, revealing sharply peaked diffuse emission indicative of unusually low entropy gas. Earlier observations without accurate point-source removal estimated the central temperature of Phoenix to be ≈ 7 keV and the central entropy to be >10 keV cm^2 (see Phoenix [SPT-CLJ2344-4243] in Figs. 3, 5 of ), significantly greater than the values obtained by . 


A greater photon count also enables more finely grained radial binning, which improves the spatial resolution achievable with deprojection. Figure <ref> tries to quantify the effects of spatial resolution on observational estimates of min(t_ cool/t_ ff). It shows how min(t_ cool/t_ ff) varies with time in our fiducial simulation for different inner cut-off radii. The dot-dashed magenta line shows min(t_ cool/t_ ff) for r>10 kpc, the dashed black line for r>5 kpc and the solid orange line for r>0.5 kpc. At high resolution, the dips in min(t_ cool/t_ ff) are more pronounced while the larger values remain unchanged because the region with t_ cool/t_ ff≲ 5 is relatively small. In contrast, the minimum t_ cool/t_ ff achieved in a heating phase is at r≳10 kpc (see Fig. <ref>). Observations with high angular resolution (enabled by a high photon count) are thus needed to accurately extract min(t_ cool/t_ ff) for clusters in a cooling phase.  Some of the deep observations from  do find min() ≈ 5 in elliptical galaxies. However, it remains possible that the lowest observed values of min(t_ cool/t_ ff) are biased low because of noise. Similar interpretations have been made for the low min(t_ cool/t_ ff) clusters in , in which nearly half of the clusters with confirmed CO detection lie between 7-12 with a distribution peaking at 10-12 (see Figure 11, ). Making more definitive comparisons with observations will require deprojection of mock observations of simulated clusters, so that the values of min(t_ cool/t_ ff) derived from the simulations can more accurately reflect what observers actually measure.  However, such an analysis is beyond the scope of this short paper.



§ DISCUSSION


We find that many of the unique features of the Phoenix cluster core are consistent with a numerical simulation of kinetic AGN feedback fueled by cold gas accretion. They do not necessarily signal a different evolutionary pattern for Phoenix, but rather motivate higher sensitivity observations of the most centrally peaked systems to assess how compatible the observed distribution of min(t_ cool/t_ ff) is with the simulations. 

Phoenix has a high current SFR but its fraction of young stars indicates that it has not been forming stars at that rate for very long. The observed amount of molecular gas (M_ cold>10^10 M_⊙) most likely has originated in gas cooling from the surrounding hot atmosphere over 50-120 Myr (). Pure cooling for a few hundred Myr (Figure <ref>) is enough to make the cluster look like a pure cooling flow, consistent with our interpretation of a short cooling cycle. Moreover, pure cooling in Phoenix for ∼ 1 Gyr would create a cold gas reservoir of 10^12 M_⊙, more than an order of magnitude larger than what is observed ().

We cannot rule out that Phoenix has a black hole with an unusually low mass for its halo mass and therefore cannot produce enough kinetic feedback, as suggested by <cit.>.  However, if the black hole is indeed ineffectual, the feedback failure must be a very recent phenomenon otherwise, for the reasons noted above, 
we would expect a higher fraction of young stars and a much higher reservoir of cold gas than observed. 
 used the scaling relations between spheroid stellar mass and central black hole mass () to estimate a mass ∼ 1.8×10^10 M_⊙ for the SMBH, which would make it one of the most massive in the universe. Owing to its large Eddington accretion rate, such a massive SMBH can provide sufficient kinetic feedback through accretion at 10^-2 times the Eddington rate.  The cavity power inferred in Phoenix may therefore be an underestimate, given our hypothesis that the AGN is in the initial stages of an outburst and may still be rapidly inflating its cavities.



§ CONCLUSIONS


The principal conclusions of this work are as follows:

    
  * Surface brightness snapshots of our cluster simulations, along with the corresponding radial profiles of density, entropy, temperature, and  during transitions from a cooling phase to an AGN outburst, are similar to observations of Phoenix (although the halo mass of the simulated cluster is a factor of few smaller).
    
  * Phoenix is not necessarily fundamentally different from other cool-core clusters. We may have  caught it at the end of a cooling state, just as AGN feedback is turning back on. The thermodynamic and various timescale profiles from our simulation is in very good agreement with Phoenix profiles once the profiles are scaled to account for mass and redshift differences.
    
  * Phoenix does not necessarily signal the failure of kinetic SMBH feedback. Similar phases may also be seen in lower mass halos.  However, the cooling phases in simulated systems are more frequent in more massive halos, given a fixed efficiency parameter. 
    
  * Deeper observations with a higher photon count within r<10 kpc of centrally peaked galaxy clusters may be required to measure min(t_ cool/t_ ff) accurately, since t_ cool/t_ ff during a cooling phase is close to the minimum value only over a small range in radius (r ≪ 10 kpc).




§ ACKNOWLEDGMENTS

DP is supported by Chandra theory grant no. TM8-19006X (G. M. Voit as PI) and NSF grant no. AST-1517908 (B.W.O'Shea as PI). BWO acknowledges further support from NASA ATP grant NNX15AP39G.
A.B. acknowledges support from NSERC through the Discovery Grant program.
P.S. acknowledges a Swarnajayanti Fellowship from the Department of Science and Technology, India (DST/SJF/PSA-03/2016-17) and thanks the Humboldt Foundation for support that enabled his sabbatical at MPA Garching. We thank Michael McDonald for useful discussions and providing Phoenix mass profile.

mnras


