The duty cycle of radio-mode feedback in complete samples of clusters

The Chandra X-ray Observatory has revealed X-ray bubbles in the intracluster medium (ICM) of many nearby cooling flow clusters. The bubbles trace feedback that is thought to couple the central active galactic nucleus (AGN) to the ICM, helping to stabilize cooling flows and govern the evolution of massive galaxies. However, the prevalence and duty cycle of such AGN outbursts is not well understood. To this end, we study how cooling is balanced by bubble heating for complete samples of clusters (the Brightest 55 clusters of galaxies, hereafter B55, and the HIghest X-ray FLUx Galaxy Cluster Sample, HIFLUGCS). We find that the radio luminosity of the central galaxy only exceeds 2.5 x 10^30 erg s^-1 Hz^-1 in cooling flow clusters. This result implies a connection between the central radio source and the ICM, as expected if AGN feedback is operating. Additionally, we find a duty cycle for radio mode feedback, the fraction of time that a system possesses bubbles inflated by its central radio source, of>69 per cent for B55 and>63 per cent for HIFLUGCS. These duty cycles are lower limits since some bubbles are likely missed in existing images. We used simulations to constrain the bubble power that might be present and remain undetected in the cooling flow systems without detected bubbles. Among theses systems, almost all could have significant bubble power. Therefore, our results imply that the duty cycle of AGN outbursts with the potential to heat the gas significantly in cooling flow clusters is at least 60 per cent and could approach 100 per cent.


INTRODUCTION
A much debated topic in extragalactic astronomy in the last 20 years is the "cooling flow" problem . Since the cooling time of the ICM in the cores of many galaxy clusters is shorter than a few Gyrs (Edge et al. 1992), in the absence of heating the gas is expected to cool down below X-ray temperatures and condense . However, searches for evidence of the high mass depositions rates expected in the standard cooling flow model were unsuccessful (e.g., McNamara & O'Connell 1989). Data for such clusters from the Chandra and XMM-Newton X-ray observatories do not show the signatures of gas cooling below ∼ 2 keV (e.g., Peterson et al. 2001). Therefore, it appears that only a small fraction of the expected cooling is occurring in most cooling flows, suggesting some heating mechanism is operating in these systems.
A possible source of heating has been identified: high resolution images from the Chandra X-ray Observatory have revealed X-ray bubbles in the ICM of many nearby clusters, groups and ellipticals (very few systems were previously known to host bubbles in their atmospheres, e.g, Perseus, Cygnus A). Notable examples include Perseus (Fabian et al. 2000), A2052 (Blanton et al. 2001) and MS 0735.6+7421 (McNamara et al. 2005), among many others that have been discovered. In a study of a large sample of such bubble systems, it was found that a central radio source was present in every system, with the radio plasma often filling the bubbles . The X-ray bubbles are therefore interpreted as regions where the X-ray emitting gas has been displaced by radio plasma produced by energetic out-flows from the central AGN and show that the AGN is injecting significant power into the ICM. The energy that the bubbles inject into the ICM may be important for balancing cooling (McNamara et al. 2000;Fabian et al. 2000;Blanton et al. 2001) and inhibiting star formation in massive galaxies (Croton et al. 2006).
A number of authors Dunn & Fabian 2004;Dunn et al. 2005;Rafferty et al. 2006) have analyzed samples of cooling flow clusters with visible bubbles in their environments. Using a sample of such systems, Rafferty et al. (2006) found that at least 50% of the systems have enough energy in their bubbles to balance cooling, considering only the enthalpy, although the integrated energy is sufficient to offset cooling in all systems ). The enthalpy is only a lower limit for the total energy injected, as Chandra images reveal evidence for the existence of weak shocks (McNamara et al. 2005;Fabian et al. 2006;Forman et al. 2007;Wise et al. 2007) with energies comparable to the bubble enthalpies. Additionally, the energy may still be underestimated due to further adiabatic losses, undetected bubbles (Nusser et al. 2006;Binney et al. 2007) or cosmic ray losses (Mathews & Brighenti 2008).
However, it is not clear whether the bubbles are always present when heating is required. Currently, the bubbles are primarily detected by eye. Some of them have bright rims (e.g., A2052; Blanton et al. 2001) which make them "clear" cases. However, the detectability of a bubble depends on its location, orientation (Enßlin & Heinz 2002;Diehl et al. 2008;Brüggen et al. 2009), its angular size and the depth of the observation. A majority of observed bubbles come from systems where the angle between the jet axis and the line of sight is between 45 • and 90 • (Enßlin & Heinz 2002;Brüggen et al. 2009). As a result, we are likely missing some bubbles in the existing images.
Indeed, many cooling flow systems, which presumably require heating, do not show bubbles in their existing Chandra images (e.g, A1650, A2244; Donahue et al. 2005). It is therefore important to understand the biases and selection effects in the detectability of current X-ray bubble samples. However, images of many of these systems are shallow. In these systems, the question is whether there might exist enough undetected bubble power to balance heating. Also, there might be alternative scenarios for the lack of bubbles in some cooling flow systems, such as the presence of another heating mechanism (e.g., "sloshing"; ZuHone et al. 2010), or perhaps because some systems are in a cooling stage, with high rates of star formation, and might not need heating right now (e.g., A1068; McNamara et al. 2004).
These issues are important in understanding the duty cycle of AGN heating, which we define to be the fraction of time that a system possesses bubbles inflated by its central radio source (and hence shows clear evidence of AGN heating). The duty cycle gives insight into the heating process and is useful in constraining (or as an input to) simulations of galaxy and cluster formation and evolution that include AGN heating.
Previous work to understand the duty cycle of AGN heating in clusters has been done by Dunn et al. (2006), who found an AGN duty cycle of 70% for cooling flows in the B55 sample. However, their selection of cooling flow systems is based on ROSAT data, which are less sensitive to the pres-ence of a cooling flow than higher-resolution data. For ellipticals the results are more controversial. Dong et al. (2010) found a duty cycle for AGN feedback of 50% for a sample of 51 ellipticals. This duty cycle is similar to the Dunn et al. (2010) finding of 9 out of 18 ellipticals having bubbles, but much higher than Nulsen et al. (2009) finding by using 104 ellipticals from Chandra archive, where only 1/4 have bubbles (note however that Nulsen et al. found that the duty cycle varies with the X-ray luminosity, so the above results may be consistent). In this paper we address the question of duty cycle of AGN feedback in complete subsamples of cooling flow clusters using simulations to place limits on AGN heating for the clusters which do not show bubbles in the existing Chandra images. For this purpose we chose the B55 (Edge et al. 1990) and HIFLUGCS (Reiprich & Böhringer 2002) samples, since both samples have already been observed with Chandra. We assume H0 = 70 km s −1 Mpc −1 , ΩΛ = 0.7, and ΩM = 0.3 throughout.

Complete Samples
In this work we present an analysis of two complete samples: the B55 sample (Edge et al. 1990) and the HIFLUGCS sample (Reiprich & Böhringer 2002). The B55 sample is a 2-10 keV flux limited sample of 55 systems (or 56 since we count both A3391 and A3395s) selected using data from the HEAO-1 and Ariel-V surveys (Edge et al. 1990). HI-FLUGCS is a 0.1-2.4 keV flux limited sample of 64 systems based on ROSAT and ASCA observations at galactic latitude b > 20 • . The two complete samples have 44 systems in common. For the systems that belong only to the B55 sample, the majority of them are low latitude systems (b < 20 • ). These are: AWM7, Perseus, Ophiuchus, A2319, Triungulum Aus, 3C129.1, PKS 0745-191, Cygnus A and A644. M87 is also present only in the B55 sample, since it was excluded from HIFLUGCS based on the fact that the irregular X-ray background and the extended X-ray emission at low latitudes makes the detection of clusters more difficult. A1689 and A3532 are only in the B55 sample due to differences in the X-ray band used to construct the samples. The systems that belong only to the HIFLUGCS sample are in general smaller, cooler ones (e.g., NGC 507, NGC 1399, NGC 1550, and softer-spectrum systems (e.g., Sersic 159/03, A2589, A2657 etc). Sun et al. (2007) showed that there is a class of systems having truncated atmospheres of low entropy gas, with short cooling time, embedded in much more extensive atmospheres of higher entropy gas. Some of these systems are present in our sample and we refer to them as corona systems (they are marked with an asterisk in Table 1, Table 2 and Table 7).
The B55 sample was studied by Dunn et al. (2006) in order to determine the fraction of cooling flow clusters with bubbles. In order to separate the cooling flow clusters from the non-cooling flow clusters, Dunn et al. (2006) used ROSAT data. They defined a cluster to be a cooling flow cluster if t cool < 3×10 9 yr in the core and Touter/Tcenter > 2. They found that 14 out of 20 cooling flow clusters have clear bubbles. Also, Rossetti & Molendi (2010) studied the B55 sample and separated cooling flow from the non-cooling flow clusters using the pseudo-entropy ratio. The HIFLUGCS sample was studied by Sanderson et al. (2006Sanderson et al. ( , 2009, who used a temperature ratio in order to separate the cooling flow clusters; Chen et al. (2007), who used a mass deposition rate ratio; and Hudson et al. (2010), who used the cooling time. However, different definitions of separation give different answers for the cooling flow fraction (for a review see , which might be also affected by the sample bias . In Table 7, Table 7 and Table 7 we list all systems from both complete samples (B55 and HIFLUGCS) in order of right accession. Table 7 gives the X-ray peak and optical coordinates for the brightest cluster galaxies (BCGs), together with the BCG names and the separation between the ICM's X-ray peak and the BCG center. The coordinates of the X-ray peak are well determined (within 1-2 ′′ ) for systems that have a surface brightness peak; for the fainter and more diffuse systems the determination of the separation becomes harder, and we mark these objects with an asterisk in Table 7. The optical coordinates for the BCGs are robustly determined (typically, within 1-2 ′′ ), except in a few cases: A2163, A1689 (the optical image is of poor resolution); A1367 which has 2 BCGs, NGC 3862 and NGC 3842, both of them far from the X-ray center, but neither of them appear to be related to the central X-ray emission; and A2256 which has 2 comparable BCGs, NGC 6331 and UGC 10710, plus a few smaller ones (we choose NGC 6331). However, for some ACIS-I observations, the cluster cores are located in the CCD gaps, which affects the determination of the cluster centre (e.g., A399, A3158, A3562, A2255).

Bubble Systems
There are 22 and 26 bubble systems that are detected in the B55 and HIFLUGCS samples, respectively. There are, in total, 31 systems with bubbles. Among the 31 systems, the bubble identifications for 24 systems are published in the literature: A2204 ), Fornax ), NGC 4636 (Baldi et al. 2009), NGC 5044 , and 20 systems in the Rafferty et al. (2006) sample of clusters with bubbles. For NGC 507, III Zw 54, NGC 1550, A496, A3391, A1060 and A3532 we identified new bubbles by inspecting the Chandra data. For NGC 507, NGC 1550 and A496 more than one Chandra observation is available in the archive. In these cases, we used the ciao tool merge-all in order to add multiple observation for visual inspection of bubbles (see Table 1).
The bubbles were identified and measured using unsharp-mask images made from the exposure-corrected images. For each system, a variety of different smoothing scales was investigated. The scales for which the contrast between the bubbles and the surroundings was the greatest were used for measurement. However, in some systems (e.g., III Zw 54, NGC 1550), the unsharp-mask image might reveal structure that is not related to bubbles. Thus, the unsharp-mask tool cannot be blindly used for bubble detection. Rather, deeper X-ray and radio images are needed to confirm the presence of marginally detected bubbles. As a result, some bubbles (e.g., III Zw 54, A1060, A3391, A3532) need to be further confirmed by deeper X-ray data.
Since the bubbles are presumably created by radio lobes, we investigated the presence of a extended radio emission for these systems. The corona systems A3391 and A3532 have large scale radio lobes (Mauch et al. 2003;Venturi et al. 2001) much larger than the cavities we measured from the unsharp mask images. On the other hand, the corona system A1060 has very small radio lobes ) that fit well inside the cavities that we measured, and for the corona system III Zw 54 there is only a NVSS image available in which any lobe emission is not resolved. For the remaining systems, NGC 507 and NGC 1550 have large-scale radio lobes Dunn et al. 2010), but A496 doesn't have any radio image in the literature besides the NVSS one. As in the case of corona systems A3391 and A3532, for NGC 507 the radio lobes are much larger than the cavity we measure, but for NGC 1550 the radio emission fills very well the cavity that we measure. As we underline above, for some systems there seems to be a discrepancy between the cavities sizes derived from unsharp-mask images and the extent of the radio lobe emission. For the corona class systems, A3391 and A3532, it might be that the X-ray surface brightness contrast is too low and we are not able to measure the full extent of the cavities. In this case we might have underestimated the bubble power (enthalpy). For NGC 507, the unsharp mask shows a clear small depression in the west side of the cluster and this is the bubble that we list in Table 1. Additionally, there appear to be shallow deficits at the location of the radio lobes de Ruiter et al. 1986;Murgia et al. 2011;Giacintucci et al. 2011b), such as at the northern part of the western radio lobe, but deeper X-ray data is needed to confirm this. It might be that such cavities have a flattened "pancake" shape and are filled with entrained material, rather than the perfect void of material and spherical shape that we assumed when we did the calculation, and as a result the contrast between the cavities and the ICM is weakened. This is in agreement with results from Murgia et al. (2011) where it was found that this system has old lobes, and as a result they might have entrained material, expanded and leaked. As a result, if for NGC 507 one uses the radio lobes sizes and spherical geometry assumption, the bubble power might be overestimated by a large factor.
Bubble radii and distances from the cluster centers are listed Table 1. Also, Table 1 lists the buoyancy time, the time required for the bubble to rise buoyantly from the cluster centre to its present location at its terminal velocity, vt (for a description of the analysis see Bîrzan et al. 2004): where R is the projected radial distance from the cluster core to the bubble's center, S is the cross section of the bubble, V is the volume of the bubble and the gravitational acceleration g ≈ 2σ 2 /R (Binney & Tremaine 1987). For the velocity dispersion (σ) the HyperLeda database 1 was used (Paturel et al. 2003); when no measurement of the velocity dispersion was available, σ = 281 km s −1 ) was adopted, which was the average value. a The systems for which we identified possible bubbles. The asterisk denotes the corona class systems (Sun et al. 2007;Sun 2009). b For III Zw 54, A496, A3391, A1060, A3532, the time on source after re-processing the data (the same as in the Table 7). For NGC 507 and NGC 1550 the total time from multiple observations (Obs ID 5801 for NGC 1550). c For the velocity dispersion, the HyperLeda database was used (Paturel et al. 2003). When there were no measurements available, σ = 281 km s −1 was adopted ). d Projected semimajor axis of the bubble. e Projected semiminor axis of the bubble. f Projected radial distance from the cluster core to the bubble's center.

Radio Properties
Since the bubbles are thought to be the result of the interaction between the central radio source and the ICM, it is of interest to determine whether a central radio source is present. Mittal et al. (2009) performed an radio analysis for the HIFLUGCS sample using literature information (e.g., NVSS, VLSS, FIRST etc) plus archived VLA radio data. As a result, for the HIFLUGCS sample the presence of a central radio source is based on the Mittal et al. (2009) analysis except for A754, A1060, A1650 and A1736. These sources show the presence of a central radio source coincident with the BCG location (see Table 7 for references). Mittal et al. (2009) described a radio source as central if there is a radio source within 50h −1 71 kpc of the X-ray peak, except A3562, A2142, A4038 and A3376 where a cut of 12h −1 71 kpc was used. For the systems that are in B55 but not in the HIFLUGCS sample we established the presence of a central radio source based on information from the literature. We define a radio source as central if its location is coincident with the BCG location. Table 7 lists the presence of a radio source for our sample and the systems marked with an asterisk are the ones with central radio lobes.
Also, Table 7 lists the 1.4 GHz monochromatic radio luminosities for the central radio sources, calculated as follows: where α is the spectral index assuming Sν ∼ ν −α and S1.4GHz is the flux density at 1.4 GHz. Mittal et al. (2009) found that there is a good correlation between the total radio luminosity and the monochromatic radio luminosity (except for 2A 0335+096, A3376, MKW 3S and A4038). When available, the radio luminosities are from Mittal et al. (2009). Otherwise, they are computed using information from the literature (e.g. FIRST, TGSS, NVSS surveys, Bîrzan et al. 2004;Lindblad et al. 1985;Govoni et al. 2009;Baum & O'Dea 1991;Venturi et al. 2001). In cases where a spectral index was not available, a value of -1 was adopted. For the systems with no detected central radio source, Table 7 lists the upper limits from the NVSS images (the numbers without errors in the above table) or from Mittal et al. (2009) for the HIFLUGCS sample.

Merging Activity
Our systems range from giant ellipticals and poor clusters to massive clusters, and they often reside in active environments with various degrees of merging activity. Since merging activity is part of cluster formation and evolution and might be different for cooling flow versus non-cooling flow systems, we searched the literature for such information for all our systems. This information is listed in Table 7. However, we do not make any distinction between major mergers and minor ones.
Signatures of merging activity can be seen in optical, Xray, and radio data. In optical data, merging may be inferred from the detection of substructures in the cluster, such as local peaks in the spatial distribution of galaxies or localized velocity substructure. This is the case of A2142 ), A1367 (Cortese et al. 2004), A1689 ), A2063 and MKW3S (Krempec-Krygier & Krygier 1999, A3395s , among many others (see Table 7).
Furthermore, the X-ray data can reveal shocks that might be related to merger activity (e.g., A754, Henry et al. 2004), the presence of a significant velocity gradient (e.g., A576, Dupke et al. 2007a), detection of a low entropy gas (e.g., A3266, Finoguenov et al. 2006), and detection of cold fronts Ghizzardi et al. 2010) and sloshing activity (e.g., A1795, . Cold fronts were first detected in Chandra data Vikhlinin et al. 2001a) as contact discontinuities at the boundary of a cool, dense gas that is moving subsonically through a hotter, more diffuse surrounding gas (for a review, see Markevitch & Vikhlinin 2007). Using the B55 sample, Ghizzardi et al. (2010) found that a large fraction of systems have cold fronts.
Sloshing was introduced by Markevitch & Vikhlinin (2001) to explain cold fronts in the relaxed cluster A1795. Other examples of sloshing are found in A1644  and Ophiuchus (Million et al. 2010), among many others (Markevitch & Vikhlinin 2007). Sloshing activity is thought to be a gravitational phenomenon, due to oscillations in the dark matter potential which might be set off by minor mergers (Ascasibar & Markevitch 2006), or hydrodynamical in nature Fujita et al. 2004). Furthermore, ZuHone et al. (2010) performed simulations of sloshing activity, and they show that such activity might be responsible for quenching the cooling in some cooling flow clusters.
Lastly, radio haloes and relics are indicators of merging activity. Haloes and relics are thought to be formed in major mergers (van Weeren et al. 2011), and radio mini-haloes may be indicators of sloshing (ZuHone et al. 2011a). In Table 7, systems which have a radio mini-halo, radio halo or a radio relic are indicated.

General ICM Properties
All systems were observed with the Chandra ACIS detector in imaging mode, and the data were obtained from the Chandra Data Archive. Details of the observations are given in Table 7.
The Chandra data were reprocessed with CIAO 4.2 using CALDB 4.2.1 and were corrected for known timedependent gain and charge transfer inefficiency. Blank-sky background files, normalized to the count rate of the source image in the 10 − 12 keV band, were used for background subtraction. 2 The analysis of the X-ray data closely followed that of Rafferty et al. (2008). Briefly, X-ray spectra were extracted in circular annuli with ∼ 3000 counts centered on the centroid of the cluster emission. For the fainter systems we used fewer counts (2000) for the inner region to allow derivation of the properties as close to the core as possible (e.g., Triangulum Aus, A2657, ZwCl 1215, III Zw 54, A2147, A1736, EXO 0423.4-0840, MKW 8). Also, in the case of the corona systems, due to the very small cores, we used for the inner region fewer than 3000 counts (2000 counts for A3558, 1000 counts for A3376, and 500 counts for A3395s, 3C129.1, Coma). In some systems, due to the diffuse nature of the cluster emission or to substructure, the centroid was difficult to identify precisely. These systems are noted in Table 7. Spectra and their associated weighted responses were made for each annulus using CIAO and were fit in XSPEC version 12.5.1 (Arnaud 1996).
Gas temperatures and densities were found by deprojecting the spectra with a single-temperature plasma model (MEKAL) with a foreground absorption model (WABS) using the PROJCT mixing model. We derived the cooling times using the deprojected densities and temperatures found above and the cooling curves from APEC spectral model Smith et al. (2001). The pressure in each annulus was calculated as p = nkT, where we have assumed an ideal gas and n ≈ 2ne. Lastly, the entropy is defined as S = kT n −2/3 e and has units of keV cm 2 .
To investigate whether or not the bubbles have enough energy to balance cooling, ones needs the luminosity of the cooling gas inside the cooling radius. Within the cooling radius, radiative energy losses must be replaced to prevent the cooling of large quantities of gas. We define the cooling radius as the radius within which the gas has a cooling time less than 7.7 × 10 9 yr, the look-back time to z = 1 for our adopted cosmology. To find the total luminosity inside the cooling radius, we performed the deprojection using a single-temperature model. Table 7 gives values of t cool for all clusters, and r cool and Lx(< r cool ) for only our subsample of clusters that require heating (see Section 5.1).
To derive densities as close to the core as possible, we additionally use the onion peeling deprojection method described in Rafferty et al. (2008) inside the innermost annulus used for spectral deprojection, assuming that the temperature and abundance of the gas are constant in the inner region used in spectral deprojection and equal to the derived emission-weighted values. For a number of objects, constraints on the density profile inside the innermost annulus were not improved by this method (see Table 7). For these cases, we report only the emission-weighted values from the innermost annulus.

Cooling Time, Temperature Drop and the Thermal Stability Criterion
Our goal is to understand the biases and selection effects in the detectability of current X-ray bubble samples and to place limits on the fraction of systems that can have enough bubble power to balance cooling. As a result, from these complete samples we need a subsample of systems in which feedback is expected to operate (i.e., cooling flow systems). The cooling time in the core is a basic selection criterion for cooling flows: if the cooling time of a system is smaller than its age, which is typically a significant fraction of the Hubble time, then one expects that this system needs heating to prevent cooling. In order to calculate the central cooling times we used archival Chandra data. The minimum radius at which we could derive reliable cooling times depends on the integration time and proximity of the source (see Table 7 for the radius of the inner annulus used in the deprojection). Since we want to compare the cooling times for all the systems at a single physical radius, as close as possible to the nucleus, we computed the cooling time at 1 kpc using the surface brightness profiles to obtain densities (see Section 3.1). For some systems the extrapolation did not work well, as the surface brightness profile is too noisy or drops towards the centre or there is significant substructure that is inconsistent with the deprojection method. This was the case for: A401, A1367, A1736, A2256, A3667, A2319, A119, ZwCl 1215, A2657, A3562, A644 and A2255. For some other systems, deprojection did not work because the surface brightness profile drops or flattens towards the centre or because they have X-ray-bright AGNs (such as Perseus) or large bubbles. This is the case for A133, A478, A2597, Perseus, A2199, A4059 and Cygnus A. In these cases, the cooling time at 1 kpc was extrapolated from the cooling time profile from the single-temperature model deprojection. This extrapolation was generally small, since for all these systems the inner bin has a radius below 10 kpc. For some systems there is a large difference between the cooling time from the innermost annulus (used in spectral deprojection) and the cooling time at a radius of 1 kpc. In some cases this is due to a very peaked surface brightness profile. This is the case for MKW 8, A2634, A400, III Zw 54, A576, A1060, A3391, A2142, A3532, Coma, A3376, A3395s, and A3558.
Additionally, we calculated the central temperature drop, since for a cooling flow cluster the temperature is expected to drop towards the center. We calculated the drop as the ratio between the highest temperature in the profile and the temperature of the innermost annulus. Table 7 lists the temperature drop values for all systems. Some systems have no entry since the temperature profile increases towards the center, or the profile was too noisy and the temperature drop value was insignificant within errors. The calculated temperature drop depends on the size of the innermost annulus. However, the temperature typically varies slowly with radius, so it will not affect our estimates. Based on the temperature drop criterion only 2 systems with short cooling times at 1 kpc (less than 1 ×10 9 yr) do not show a temperature drop (A1651 and A2063). However, the temperature declines are probably related to the local virial temperature, so the cooler clusters, where the virial temperature of the central galaxy is closer to that of the cluster, have smaller central temperature declines. The temperature decline is not used in this paper as a criterion to separate the cooling flow systems from non-cooling flow systems, but just as an additional test after the systems where already separated by using a different criterion, such as cooling time.
An alternative way to select cooling flows is based on the thermal stability of the gas Sharma et al. 2012). Using the sample of clusters, Voit et al. (2008) found that star formation and H-α emission (and hence cooling) seem to occur only if, at some location in the cluster, the following condition is met: where Λ(T ) is the cooling function calculated using APEC spectral model (Smith et al. 2001), and fc is the factor by which the magnetic field suppresses the conductivity below the Spitzer value. Assuming that the effective thermal conductivity can be expressed as a multiple, fc, of the Spitzer value, this parameter provides a measure of the stability of the gas to local cooling. For large values of ηmin, thermal conduction overwhelms radiative cooling, preventing local cooling throughout the ICM. For small values of this parameter, local cooling can run away, so that parts of the ICM may cool to low temperatures, resulting in the deposition of the cooled gas. If AGN are powered by the accretion of hot gas, then ηmin should have little significance for the feedback process. However, if the AGN are fueled by cooled ICM and the effective conductivity is comparable to the Spitzer value, ηmin determines the systems where fuel may be available. Voit et al. (2008) found that values of ηmin 5 correspond approximately to an inner cooling time of 5 × 10 8 yr. This agrees well with our findings (see Table 7 for the ηmin values), the exeptions are A2065, A3391 and A400 which have ηmin < 5, but central cooling times of 5 × 10 8 yr. However the last 2 systems are corona systems (with small cool cores, e.g., Sun et al. 2007) with shallow Chandra observations and we were not able to get closer to the mini-core.

RESULTS FROM X-RAY AND OTHER OBSERVATIONS
In this section, we examine how the central cooling time and ηmin relate to other properties of the cluster, such as the separation between the X-ray centroid and the optical centre (an indication of how relaxed a system is) and the central radio source luminosity. There is a separation between clusters, such that objects with radio luminosities above 2.5 × 10 30 erg s −1 Hz −1 generally have cooling times at 1 kpc less than 5 × 10 8 yr and have bubbles. There are no high radio luminosity objects with longer cooling times. A similar plot is showed in , but they were no able to probe the CF/NCF separation threshold since they used NVSS radio fluxes, and were not able to distinguish between central radio source emission and halo emission. Sun (2009) found a similar cutoff in radio luminosity for their strong cool-core systems (with cooling times below 1 Gyr). Rafferty et al. (2008) studied star formation in a sample of 39 systems in different cooling stages and found a sharp threshold for the onset of star formation that occurs when the central cooling time falls below ∼ 5 × 10 8 yr (see also ). This number agrees well with our results shown in Figure 1 left. The only objects in our sample with a central cooling time 5×10 8 yr and with a high radio luminosity ( 2.5× 10 30 erg s −1 Hz −1 ) are A3391, A400 and A1689. The first two are corona systems for which we were not able to probe the cooling time in the tiny cores. For A3391 we see bubbles in the existing Chandra image. The fact that there are no objects with high radio luminosity (> 2.5× 10 30 erg s −1 Hz −1 ) and long cooling times (> 5×10 8 yr) indicates that the central AGN radio-mode heating in clusters is sensitive to the presence of gas with short cooling times. This finding fits well with the AGN feedback scenario.

Cooling Time and Radio Luminosity
Also in Figure 1 we plotted in red the systems that have evidence in the literature for merger activity. We do not differentiate between a minor or major merger or sloshing (see Section 2.4 for details). Figure 1 shows that mergers are present in both cooling-flow and non-cooling flow systems. We note, however, that the typical strength of the merger likely differs between the two classes. Voit et al. (2008) argued that a central cooling time of ∼ 5 × 10 8 yr is equivalent to a value of 5 for ηmin, which agrees well with our results, shown in Figure 1 right, which plots ηmin versus the radio luminosity. A3391, A400 and A2065 have central cooling times higher than ∼ 5 × 10 8 yr and ηmin smaller than 5. However the first two systems, as mentioned in the discussion of Figure 1 left, are corona systems with shallow Chandra observations and we were not able to get closer to the mini-core. On the other hand, A1689 has a central cooling time of ∼ 5 × 10 8 yr, but ηmin much larger than 5. Figure 2 shows a strong relationship between central cooling time and the separation between X-ray centroid and the optical centre (see Table 7). No object with central cooling time less than ∼ 10 9 yr has a separation above 12 kpc, and almost no objects with a separation above 12 kpc (exceptions A401, A119) have a cooling time below ∼ 10 9 yr. For the systems with large separation between the optical centre and the X-ray centroid indicates that these systems are not dynamically settled that they are still going through significant merger activity. This trend is similar to the one in Figure 1 left, but much tighter. The only system with a cooling time below ∼ 10 9 yr that has a separation higher than 12 kpc is A4038. This system might have experienced major merger activity, which might have been created the radio relic present on the outskirts of the cluster (Slee & Roy 1998). Furthermore, Rafferty et al. (2008) shows that the systems with larger separation do not have star formation. This is another indicator that these systems do not need heating.

Subsample of Clusters that Require Heating
Our goal is to understand the biases and selection effects in the detection of X-ray bubbles, and to place limits on the fraction of systems that can have enough bubble power to balance cooling (these are the cooling-flow systems, CFs). However, in systems in which no cooling is expected to occur (i.e., non-cooling flows, NCFs), one also expects no heating. Therefore, these systems should not be considered in our analysis. As a result, a subsample in which feedback is expected to operate must be defined from these complete samples.
In Figure 1 (left panel) we found a clear separation between CF and NCF systems. Furthermore, we found that the criteria using the thermal stability parameter and cooling time to separate CF and NCF systems give similar results. Based on these results, we now define a subsample requiring heating based on the thermal stability parameter (ηmin less than 5), since this criterium is more physically motivated. Additionally, based on the clear separation between clusters seen in Figure 2 with a cut-off value of inner cooling time (at 1 kpc) of 1 × 10 9 yr, and a cut-off value of 12 kpc for the projected separation between the X-ray peak and optical centroids, we further limited the subsample to systems that have such a separation of less than 12 kpc. In total, there are 49 systems (32 for B55 and 41 for HI-FLUGCS) that meet these criteria (ηmin < 5 and separation < 12 kpc), 31 of which have detected bubbles (22 for B55, 26 for HIFLUGCS). Table 2 lists the systems that meet these criteria and also lack detected bubbles. Finally, based on the central cooling time (at 1 kpc) and the separation (see Figure 2), we also consider a somewhat larger subsample: those systems with central cooling times below ∼ 10 9 yr and separations smaller than 12 kpc. This subsample includes the previous main heating subsample, plus 7 extra systems (see Table 2). There are a total of 56 systems (39 for B55, 47 for HIFLUGCS) in this extended subsample. Therefore, there are 16 systems in B55 and 16 systems in HIFLUGCS (19 systems in total) that are excluded from the samples of systems that need heating.

Simulations
For the systems without bubbles from our extended subsample (which have cooling times at 1 kpc smaller than ∼ 10 9 yr), we performed simulations in order to find the locations and the sizes of bubbles that can be present in the cluster and remain undetected. These systems are listed in Table 2.
No attempt was made to simulate the effects of bright rims around the bubbles, since that would complicate the model greatly. When present, these can enhance bubble contrast, making them more evident. As a result, if undetected bubbles had bright rims, we may be overestimating their possible energy content. Regardless of this, the modeling used here provides good limits on the energy that may be contained in undetected bubbles.
We simulated a 3 dimensional cluster using a spherically symmetric β profile model for the emissivity (de-  Table 7 for details and references). The filled symbols denote the 31 systems with bubbles (see Section 2.2). The red symbols denote the systems that appear to be undergoing some merger activity such as major mergers, minor mergers, or sloshing. This information is taken from the literature (see Table 7 for references). Over-plotted are the lines at 5 × 10 8 yr, 1 × 10 9 yr, 3 × 10 9 yr and 2.5 × 10 30 erg s −1 Hz −1 (see text for details). right: Thermal stability parameter (see Section 3.2) versus the monochromatic 1.4 GHz radio luminosity. The symbols are the same as on the left panel.
rived from the existing archive observations, see Table 2). The parameters of the β model were fixed to those derived from the azimuthally averaged surface brightness profiles. When a double-β model was preferred over a single-β model, the double-β model was used in the simulation. A cube of 500×500×500 elements was populated with the corresponding emissivities.
For simplicity, spherical bubbles were simulated in this cube by setting the emissivities inside the bubbles to zero. However, it is important to notice that observationally it is unclear whether the X-ray cavities are devoid of X-ray emitting gas or instead are filled with hot, underdense gas. Then, by integrating the emissivity along lines of sight through the cube, we obtained the 2 dimensional surface brightness map. This map was used as input for the MARX simulator 3 in order to obtain a simulated Chandra image. For MARX we used a spectrum extracted from the data, with the integration time and position set as in the data. As in the case of the systems from Section 2.2, we used the ciao tool merge all to add all the available observations for each system. This total integration time is used in the simulations (see Table  2 for details).
We assume that the bubbles expand adiabatically into a β-model atmosphere. Therefore, the radius of the bubble at at the distance r from the centre of the cluster is given by: is the bubble size at the cluster center, rc is the core radius and β is the parameter of the beta model. To calculate the ages we assume that the bubbles rise buoyantly from the centre of the cluster to their current radius (giving t buoy , see section 2.2 for details). In order to calculate the initial size, R b (0), with which the bubbles are injected, we assumed: 4pV ∼ Pcavt ∼ LXt, where p is the central pressure of the cluster, LX is the bolometric X-ray luminosity inside r cool (see Section 3) and t is the time between the outbursts. We adopted a time between outbursts of 10 8 yr, motivated by the observations of clusters with multiple generations of bubbles, such as Perseus (Fabian et al. 2000). We ran multiple simulations for a cluster, with the locations of the bubbles and the angle in the plane of sky (φ) randomized, and with an angle from the line of sight (θ) of 90 • . The resulting images were inspected to determine if the bubbles remains visible for the whole of the 10 8 yr outburst cycle. If a bubble become undetectable when its buoyancy age was less than 10 8 yr, such a bubble may be present, but undetectable, in the real cluster and the heating rate could be sufficient to balance cooling. In cases where the bubble would have remained visible for the whole cycle, the simulation was repeated using smaller values of Pcav, until the bubble becomes undetectable in less than 10 8 yr. Under our assumptions, this value of Pcav determines an upper limit on the possible AGN power in such systems. These systems are A3112, A1650, A1689, RXCJ 1504.1-0248, A2244, and Ophiuchus. Therefore, these systems have upper limits less than the value required to balance cooling under our assumptions (see Table 2). a Systems without bubbles in the existing Chandra images, but with a separation < 12 kpc and η min < 5 or a central cooling time (at 1kpc) < 1 Gyr. The asterisks denote the corona class systems (Sun et al. 2007;Sun 2009 . Otherwise, we listed the time on source after re-processing (the same as the ones in Table 7). c Velocity dispersions were taken from the HyperLeda database (Paturel et al. 2003). When none were available, σ = 281 km s −1 was adopted . d The normalisation of the isothermal surface brightness profile or β-profile (Cavaliere & Fusco-Femiano 1976) in units of 10 −9 ph s −1 cm −2 pix −2 . e The background contribution for the isothermal surface brightness profile in units of 10 −9 ph s −1 cm −2 pix −2 . f This is the bubble injection radius for the first run (Lx, see Table 7); the injection radius for the actual run is 5.8 kpc for RXCJ 1504.1-0248 (Lx/10), 5.8 kpc for A1650 (Lx/5), 4.7 for A3112 (Lx/5), 7.1 for A1689 (Lx/5), 5.4 for Ophiuchus (Lx/5), 5.64 kpc for A2244 (Lx/2).

Biases in Bubble Samples
Samples of observed bubbles could suffer from biases due to detectability and projection effects. To assess the degree to which such biases might be present, we simulated an A4059like cluster with two bubbles with a variety of sizes and positions. We then attempted to detect the bubbles by eye (without knowing their locations or sizes a priori). The measured properties can then be compared to the intrinsic ones to understand possible biases.
In total, 89 simulations were performed. Bubble radii were varied between 10 kpc and 60 kpc, and the distance from the core to the bubble was varied between 20 kpc and 80 kpc. Bubbles with radii greater than the bubble-to-core distance were not simulated. Figure 3 left shows a comparison between the measured bubble radii and the true ones. Approximately 44% of the simulated bubbles were undetected. Of these, the majority had an axis between the bubble and core that was oriented near to the line of sight (θ 45 • ) or were small bubbles at large distances from the core. Of those that were detected, the measured bubble radius is on average close to the true one. The greatest deviations from equality are generally for bubbles with smaller angles to the line of sight.
A similar situation also applies to the comparison between the measured distance to the core and the true one, shown in Figure 3 right. It is clear that most of the undetected bubbles lie at large distances from the core or, again, Detection fraction Figure 4. Detection fraction as a function of the true bubble radius and the true (unprojected) distance from the bubble centre to the cluster core. The fraction has been summed over all angles between the bubble-to-core axis and the line of sight.
have small values of θ. For detected bubbles, the average distance is close to the true one in most cases. These results imply that when bubbles are detected, their properties are probably not grossly in error, particularly in an average sense. Lastly, we can estimate the detection rate for bubbles as a function of bubble radius and distance to the core by summing over all angles of the bubble-to-core axis to the line of sight. This detection rate is shown in Figure 4. As expected, the detection fraction depends strongly on the bubble size and on the distance from the core. However, it is currently unclear as to which fraction of real bubbles are expected to fall into this parameter space. Figure 5 shows the bubble power (the heating rate) versus the bolometric X-ray luminosity (the cooling rate) within the cooling radius (at which t cool < 7.7 × 10 9 yr) for our sample of cooling flows. The fraction of systems with detected bubbles for the main subsample is ≈ 0.69 (22/32) for the B55 and ≈ 0.63 (26/41) for HIFLUGCS. We note that this sample is based on two criteria: ηmin < 5 and X-ray-tooptical core separation < 12 kpc. If, however, the AGN are not fueled by cooled gas, then the ηmin criterion is not relevant to this analysis. For the larger samples based on cooling time cut-off the fraction is ≈ 0.56 (22/39) for the B55 and ≈ 0.55 (26/47) for HIFLUGCS. This implies a duty cycle in cooling flows of at least 55% to 69%. This duty cycle agrees well with previous findings (Dunn et al. 2006;Nulsen et al. 2009;Dong et al. 2010) and with the radio-loud fraction of BCGs in cooling flows . It also agrees with the radio-loud fraction of ∼ 30% found for massive galaxies by Best et al. (2005), when adjusted for the fraction of such systems that are in cooling flows (∼ 50%, see Section 5.1).

The Duty Cycle
From Figure 5 we can conclude that most of the systems that lack detected bubbles could have enough bubble power to balance cooling and remain undetected in existing images. A few systems have upper limits on the bubble power below the 4pV line, implying that they cannot have sufficient bubble power to balance cooling under our assumptions. However, one of our primary assumptions is that the bubbles are in the plane of the sky. As noted in Section 1, the orientation of the bubble axis relative to the line of sight has a strong effect on the detectability of the bubbles. If the angle from the bubble axis to the line of sight is 30 • , it is difficult to detect even large bubbles. As such an orientation is expected to occur in roughly 15% of all systems (assuming the jet axes are randomly oriented), it is possible that ∼ 2-3 of the systems without detected bubbles in our sample have significant bubble power that is hidden. Unfortunate orientation might therefore account for all of the systems for which the upper limit on the bubble power is insufficient to balance cooling. Therefore, with existing data, one cannot exclude the scenario that almost all cooling flow clusters have bubbles with enough power to balance cooling, implying a duty cycle for such activity of up to 100%.
However, of the 7 intermediate cooling flow systems (with a central cooling time between ∼ 5 -10 × 10 8 yr), only A2063 shows extended radio emission. Thus, if the remaining intermediate systems do have bubbles, they must be "ghost", like those of A2597 . It is more probable that these systems have previously had a cooling core that was heated by a merger, such as A2142, A3571, A1651, A2244, and corona system A576 (see Section 6.4). Therefore, these systems might not need bubbles in order to balance the internal heating.
Also, it is important to mention that some of the systems without bubbles are corona-class systems (denoted by the blue symbols in Figure 5), which often have large radio lobes (all such systems in our sample except A576, MKW 8, A2589 and A3558 have lobes) embedded in lower density gas surrounding the bright X-ray core, which reduces the contrast of any bubbles and makes them harder to detect. Such systems may well have undetected bubbles. They include A400, A3376, A2634, A3395s, Coma, and 3C129.1.
However, it is not fully understood what makes these systems different from larger cooling cores or whether the radio mode feedback cycle applies to them. Since the radio lobes are inflated beyond the cooling region and the fraction of the outburst energy coupled to the corona is small, their feedback process may be very inefficient, and heating required to balance the internal cooling might be supplied by SNe or by heat conduction (Sun et al. 2007). Alternatively, radio heating through weak shocks may be important in these systems (for a discussion on the importance of weak shocks for AGN heating, see the review of McNamara & Nulsen 2012). When the corona class systems are excluded, we find a duty cycle of 76% (= (22 − 3)/(32 − 7)) for the B55 main subsample and 77% (= (26 − 3)/(41 − 11)) for the HIFLUGCS main subsample.
Lastly, we find that the fraction of systems with detected bubbles for both complete samples is very similar. The two subsamples overlap for 44 systems (see Section 2.1), however the HIFLUGCS sample extends to lower-mass systems. In general, our results are most relevant for clusters, since there are only 10 ellipticals and poor clusters in our sample (3 for B55 and 9 for HIFLUGCS), and 8 of them have bubbles.

Radio Luminosity Cut-Off
Figure 1 left shows that there is a radio luminosity cut-off for massive clusters, such that systems with a central radio luminosity above 2.5 × 10 30 erg s −1 Hz −1 are found only in cooling flows (see Section 4.1). This supports the idea that the radio-mode outbursts are self-regulating, with increased radio luminosity (and hence feedback power) seen in sources where cooling is likely occurring.
We also note that the luminosity of central radio sources does not appear to be lower on average when there are no detected bubbles. In such systems, the radio source lacks bright lobes and is core dominated but is generally as bright as the systems with radio lobes and bubbles. This is the case of A1689, A1644, RXCJ 1504.1-0248, which represent ∼ 10% of the high-luminosity objects (see Section 6.3). A1689 is an interesting case, as it is not part of the main subsample since it has a thermal stability parameter much higher than 5, and it has a radio halo . This system may have been heated up by a major merger (Andersson & Madejski 2004) that did not destroy the cool core.

Duty Cycle of Radio Mode Feedback
As discussed in Section 5.4, our simulations imply an AGN heating duty cycle of at least ∼ 60% and up to 100%. This result implies that cooling flow systems do not undergo long periods of time in which no significant cavities are present in their hot atmospheres. Therefore, it appears from our results that the atmospheres of cooling flows are undergoing almost non-stop energy injection through bubbles inflated by the central AGN. The exact nature of this heating remains unclear (for a review, see ), but our results clearly point to a connection between the presence of gas that is unstable to cooling (traced by low values of ηmin) and the bubbles. This connection supports the idea that cold gas accretion, which might be induced by mergers or cooling of the ICM, fuels the outbursts that create the bubbles instead of hot accretion (i.e., Bondi accretion).
The heating duty cycle that we have derived is distinct from the jet duty cycle, which is the fraction of time that the radio jet is active. Our results do not address the jet duty cycle, which, when traced only by the bubble population, could range from small values if the jets operate in short, intermittent bursts (which inflate the cavities and then stop) to values approaching unity if instead the jets are continuous (and the bubbles detach from them as they rise). However, we found that almost every source with cooling time below ∼ 5 × 10 8 yr has radio lobes. Our results should be useful as constraints on (or inputs to) simulations that include radiomode AGN heating.

Cooling Flow Systems without Bubbles
Figure 1 left shows that the systems with bubbles have short cooling times ( 5 × 10 8 yr) and the majority of them have high radio luminosities ( 2.5× 10 30 erg s −1 Hz −1 ). However, as discussed in the previous section, some systems with short cooling times do not appear to have bubbles (e.g., Rafferty et al. 2008). Here we further examine these systems to understand whether they are likely to have bubbles that are simply missed in observations. In particular, we examine their radio properties, star formation properties and merger activity.
When the radio luminosity is considered, we find that many of the objects with high radio luminosity ( 2.5× 10 30 erg s −1 Hz −1 ) that do not show bubbles in the existing Chandra images are "coronae" (Sun et al. 2007;Sun 2009). In this category are A400, A3376, A2634, A3395s and Coma. The only coronae from this category for which we found bubbles are A3391 and A3532. However, all these coronae have large radio lobes Bagchi et al. 2006;Bock et al. 1999;Becker et al. 1995;Venturi et al. 2001). The interaction between these radio lobes and the ICM is expected to create bubbles, as in A3391 and A3532, but deeper X-ray images are needed to detect any bubbles embedded in the low density gas of these systems. Another object with high radio luminosity, short cooling time and no visible bubbles in the existing Chandra image is EXO 0423.4-0840, which has a radio source at the centre of the cluster with very small lobes . Again, the interaction between these lobes and the ICM might create bubbles, but we are not able to see the  Table 2 (see Section 5.1 for details). Different symbols denote different subsamples: circles for the main subsample (η min < 5 and separation < 12 kpc) and triangles for the extended sample (cooling time at 1 kpc between 0.5-1 Gyr and separation < 12 kpc). The filled symbols denote the systems with bubbles, and the blue symbols denote the corona-class systems, which are marked with an asterisk in Table 2. The diagonal lines indicate Pcav = L X assuming pV , 4pV or 16pV as the energy deposited. The limits on the 4pV line are not strictly upper limits, but simply mark the systems where the limit is at least this large.
bubbles because the Chandra point spread function is too large to resolve them. The remaining category of objects that have high radio luminosity, but lack bubbles, are massive clusters. This may be the case for A1644, A3112 and RXCJ 1504.1-0248. Except possibly A3112 , these systems lack radio lobes. Therefore, these systems may well lack substantial bubbles, with the possible exception of A3112, which has the shortest cooling time and some hints of radio lobes.
These systems may be in a cooling stage, between AGN outbursts. Binney & Tabor (1995) postulated that there may be short bursts of nuclear activity alternating with periods of quiescent cooling. While A3112 shows very little star formation (Hicks et al. 2010), RXCJ 1504.1-0248 has a very high star formation rate (Ogrean et al. 2010;Rafferty et al. 2008) and a mini-halo ) most probably related to the cooling flow activity, as in Perseus (Gitti et al. 2002). A1644 also has high rates of star formation (Crawford et al. 1999;O'Dea et al. 2010). Furthermore, sloshing might provide enough heat to balance the inner cooling of the gas (ZuHone et al. 2010(ZuHone et al. , 2011b. This may be the case for A1644 which is thought to be undergoing a merger between the main cluster and a subcluster Johnson et al. 2010).
The cooling flow systems with a radio luminosity below 2.5 × 10 30 erg s −1 Hz −1 are mostly ellipticals, groups or poor clusters (e.g., NGC 1550, NGC 4636, NGC 507, NGC 5044, Fornax, MKW 4, AWM 7, A262, A133) and most of them show bubbles in the existing Chandra images, except AWM 7, which is probably undergoing a merger , and MKW 4. Both AWM 7 and MKW 4 do not have any lobe radio emission, but they have star formation activity (McNamara & O'Connell 1989), and thus they might be in a cooling stage, as proposed above for the more massive clusters like RXCJ 1504.1-0248.
Another category of cooling flow objects with low radio luminosity are the corona class systems: A1060, III ZW 54, 3C 129.1, A3558, A2589 and MKW 8. For A1060 and III Zw 54 we measured bubbles (see Section 2.2), and 3C 129.1 has extended radio emission, and thus might have bubbles that are not visible in the existing shallow Chandra image. On the other hand, MKW 8, A2589 and A3558 do not have extended radio lobes (see Table 7).
There are also massive clusters with low radio luminosity: Ophiuchus, A1650 and A2065. The central galaxies in these clusters only show unresolved nuclear radio emission and no star formation, but A2065 and Ophiuchus might be in the midst of mergers (see Table 7). Also Ophiuchus has a radio mini-halo Murgia et al. 2010a), and shows evidence of sloshing in its core (Million et al. 2010), which might be related (ZuHone et al. 2011a). As in the case of RXCJ 1504.1-0248, the mini-halo in Ophiuchus might be related to the previous cooling flow activity, and the sloshing has revived this old radio emission. Furthermore, Eckert et al. (2008) and Nevalainen et al. (2009) show evidence of possible hard X-ray emission, which might be of non-thermal origin, caused by Compton scattering of cosmic microwave background radiation by the same population of relativistic electrons responsable for the mini-halo emission. As a result, Ophiuchus, A1650 and A2065 might be similar to the intermediate class systems, those with a cooling time between ∼ 5 − 10 × 10 8 yr, which presently are not expected to have bubbles since they are heated by major mergers, but may previously have had cooling flow activity and bubbles (see Section 6.4).
Lastly, we note that small bubbles may be present in these systems but missed in current observations. Most of the B55 and HIFLUGCS systems are at moderate redshifts (around z = 0.05), with a few systems above a redshift of 0.1 (A1689 and RXC J1504.1-0248). Therefore, bubbles such as those seen in M87 (e.g., Forman et al. 2005) would be too small to image. For example, if we extrapolate one of the inner cavities from M87, with a radius of 12 arcsec , to a higher redshift of 0.05 we will obtain a radius of 1 arcsec. As a result, this kind of bubble would be completely missed in current X-ray observations.

Cooling Flow Versus Non-Cooling Flow Systems
The CF/NCF dichotomy has been studied a great deal recently using observations (e.g., Sanderson et al. 2006Sanderson et al. , 2009Chen et al. 2007;Cavagnolo et al. 2009;Pratt et al. 2009;Hudson et al. 2010) and simulations (e.g., Poole et al. 2008;Burns et al. 2008;McCarthy et al. 2008;Guo & Oh 2009). Using a statistically selected sample, Sanderson et al. (2009) found that the dichotomy seems to be real and not an archival bias. There are two main underlying origins for the CF/NCF dichotomy. The first assumes that the separation occurs early due to pre-heating  through mergers (Poole et al. 2008;Burns et al. 2008) or from TeV gamma-rays from blazars (Pfrommer et al. 2012).
The second interpretation assumes that the separation occurs late, such that a CF cluster can be destroyed due to merger Rossetti et al. 2011) or powerful AGN outburst (Guo & Oh 2009). However, Pfrommer et al. (2012) argue that the AGN heating is likely insufficient to turn a CF into a NCF cluster, but the impact of AGN induced turbulence could result into a NCF atmosphere on time scale larger than 1 Gyr (Parrish et al. 2010;Ruszkowski & Oh 2010). From Figures 1 (left panel) and 2, we found that clusters are separated into two different groups based on the central cooling time, central source radio luminosity and the separation between the X-ray peak and the optical centroid. Clusters with low central cooling times ( 5 × 10 8 yr) and smaller separation ( 12 kpc) have a cooling flow activity and many of them show bubbles in their atmospheres. Clusters with longer cooling times ( 10 9 yr), low central radio luminosities ( 2.5× 10 30 erg s −1 Hz −1 ) and larger separations (above 12 kpc) often show large haloes and relics likely due to merger shocks.
For the cooling flow systems, most have visible bubbles, or have extended radio emission from the central source (e.g., A3112), or appear to be in a cooling stage (e.g., A1644 and RXCJ 1504.1-0248). The systems with a cooling time between ∼ 5 × 10 8 yr and 10 9 yr (A576, A2244, A3571, A2142, A2065, A2063, A1651, A4038) do not have visible bubbles. This result agrees with the findings of Rafferty et al. (2008), since they are above the threshold for the onset of star formation (and hence significant cooling is probably not occurring). For the central radio source, none of them have more than point-like central radio emission (see Table 7), except A2063 which has small radio lobes and possible cavities (Kanov et al. 2006). Due to the fact that most of them are experiencing some sort of merger activity, diffuse emission might be also present in some of them, such as A2142, A3571, and A4038 (see Table 7 for references and M. Rossetti private communication for the detection of a radio halo in A2142).
Among the 8 clusters with intermediate cooling times, only A2063 and A1651 lack the central temperature decrease typical of cooling flow clusters (see Table 7 and Section 3.2). Therefore, these systems share properties of both cooling flow and non cooling flow systems, and thus may represent an intermediate class of systems which are presently heated by a major merger but previously might have had a cooling flow activity and bubbles. However, one cannot say whether they will evolve toward non-cooling flow as Rossetti et al. (2011) have suggested, or if they might restart the cycle of cooling and heating again at some point.
Lastly, for the objects with central cooling times above ∼ 10 9 yr, none show any extended emission for the central radio source, except A2147 which shows small lobes (see Table 7), and they do not have visible bubbles A2 147 might have been a corona system which was destroyed by the merger (Sun et al. 2007). However, this process has low probability even for corona systems (Sun et al. 2007). All these systems with a central cooling time above ∼ 10 9 yr might be experiencing majors mergers due to the continuing growth of large-scale structure. Therefore, they are classified as non-cooling flow systems for which one does not expect the presence of bubbles.
We can conclude that cooling flow systems (including the intermediate systems) and non-cooling flow systems are well separated, expecially when we look at the central cooling time versus the separation between the X-ray peak and the optical core (see Figure 2). Furthermore, the intermediate systems separate from the cooling flow systems, with a central cooling time below ∼ 5 × 10 8 yr, if we look at the central radio luminosity (see Figure 1 left) and the presence of an extended central radio source (see Table 7). The intermediate systems have a radio luminosity below 2.5 × 10 30 erg s −1 Hz −1 and they do not have extended central radio sources. However, from observations there is no conclusive way to know if a cluster like A2256 will become a cooling flow system (or if an object like A2065 will become a non cooling flow one).
The merging NCF systems and the CF systems are quite separate in Figure 1. Thus any transitions between the two types must be rare and/or fast. However, a transition from CF to NCF caused by a major merger or a very powerful AGN outburst was found to be improbable (Poole et al. 2008;Pfrommer et al. 2012). Transitions in the opposite direction could be the result of gas cooling or the injection of low entropy gas during a merger. However, the speed of the former process is determined by the cooling time of the central gas, while it remains unclear whether the latter process is possible since the turbulence at the cluster centre is sustained for several gigayears after a cluster merger . Further simulations are required to establish which, if any, of these transitions are physically plausible and could be consistent with the data.

CONCLUSIONS
The duty cycle of radio mode feedback was determined for two overlapping, complete samples of cooling flow clusters: B55 and HIFLUGCS. All members of both samples have been observed with Chandra. As a result, these observations allow us with certainty to put limits on the non-detection of bubbles in some systems and to detect more bubbles in others. We found bubbles in 7 systems that were not previously reported (NGC 507, III Zw 54, NGC 1550, A496, A3391, A1060 and A3532).
We identified a total of 49 cooling flow systems (32 for B55 and 41 for HIFLUGCS) using two criteria: ηmin < 5 and X-ray to optical core separation < 12 kpc. However, we caution that ηmin is relevant to AGN heating only if the AGN is fueled by cooled gas (as opposed to accretion of hot gas, i.e., Bondi accretion). Our results imply that ∼ 60% of all clusters are cooling flows: ∼ 57% for B55 and ∼ 64% for HIFLUGCS. These numbers agree well with the result of Hudson et al. (2010) for HIFLUGCS of 72%. If we exclude the corona systems (13 in total, 7 for B55, and 11 for HIFLUGCS), the percentage of cooling flow systems is ∼ 45% (∼ 45% for B55 and ∼ 47% for HIFLUGCS). Out of 49 cooling flow systems, 31 have bubbles (22 for B55, and 26 for HIFLUGCS). There are 7 intermediate cooling flow systems (7 for B55, and 6 for HIFLUGCS), and none of them have bubbles. Therefore, the duty cycle of radio-mode feedback is at least ∼ 69% for B55 and ∼ 63% for HIFLUGCS. If we exclude the corona class systems, we found a duty cycle of ∼ 76% for B55 and ∼ 77% for HIFLUGCS.
We found that the more luminous central radio sources (those with L1400 2.5 × 10 30 erg s −1 Hz −1 ) are only found in cooling flows (as are the X-ray bubbles). This result supports the AGN feedback scenario, in which cooling from the hot atmosphere results in AGN outbursts (traced by their luminous radio emission) that in turn heat the gas and prevent periods of intense, long-duration cooling.
However, some bubbles could be missed in existing observations. For the systems without detected bubbles, we used simulations to limit the energy that could be present in undetected bubbles, finding that the power being injected by the central AGN into most of these systems could balance cooling, although the bubbles remain undetected. Therefore, with the existing data, one can not exclude the possibility that all cooling flow clusters have bubbles with enough power to balance cooling, implying a duty cycle of up to ∼ 100%. This result needs to be investigated further, especially using radio observations, to see if all cooling flow clusters have radio lobes that might create bubbles.

ACKNOWLEDGMENTS
LB's work at Pennsylvania State University was supported by Chandra X-ray grant AR9-0018x. PEJN's work was supported in part by NASA grant NAS8-03060. LB thanks George Pavlov and Niel Brandt for their support at Pennsylvania State University and J. Nevalainen and D. Eckert for useful discussions about the Ophiuchus Cluster.  a The X-ray core positions from this work; the asterisks mark the uncertain core positions (due to the lack of the cooling core or to the presence of a confusing structure). b Notes on some systems (noted with an asterisk). NGC 507 -this system has a smaller satellite nearby. A400 -there are two cD galaxies in this system: A400-42, to the north, and A400-43 to the south. There is super-massive binary black hole at the centre of A400 , we located the X-ray centre and the CDG centre on the northern AGN. NGC 1399this system has a satellite galaxy. Zw III 54 -a posible satellite galaxy is nearby. A3158 -this system has another cD galaxy in the centre part of the cluster, PGC 013652, and the location of the optical centre (PGC 013641) lies in the CCD gap. EXO 0422-086 -this system has a couple of satellite galaxies. A3266 and A3391 -the central cD galaxies in these systems appear to be interacting and have an extended corona. A3376 -this is not the central cD galaxy of the cluster, but the one that corresponds to the centroid of the X-ray emission. A576 -there is a another, similarly sized cD galaxy in the central part of the cluster, PGC020784. A1060another similarly sized cD galaxy is nearby, NGC 3309. A1367 -there is another D galaxy, NGC 3842 located north of the X-ray center, which has large scale radio lobes (Becker et al. 1995;Owen & Ledlow 1997), but neither NGC 3842 or NGC 3862 correspond to the X-ray peak. NGC 3862 hosts also a radio galaxy, 3C 264 (Lara et al. 1999;Condon et al. 1998). A3532 -this system has a satellite nearby. A1689 -there is another equally bright galaxy nearby; the optical image is of poor resolution. MKW 8 -there is another equally bright galaxy nearby. A2163the identification of the BCG is uncertain because the optical image is of poor resolution. A2256 -the BCG identification is uncertain since there is another similarly sized cD galaxy in the field (UGC 10726), plus several smaller ones, such as CGCG 355-026 and MCG+13-12-014. Between the four cDs, only MCG+13-12-014 have an associated radio source (in NVSS). A2255 -there is another similarly sized cD galaxy nearby, PGC 059831. A2634 -this is a galaxy pair system; the pair is NGC 7720 NED01; A4038 -there is another smaller cD nearby, PGC 072437. c The CDG core positions are taken from the literature. d The separation between the optical centre and the X-ray center.  Table 5. Central Radio Sources, Bubbles, and Mergers