Thermally Unstable Cooling Stimulated by Uplift: The Spoiler Clusters

Evidence for disturbances in the atmospheres that could be signatures of a bubble or cavity were investigated. Surface brightness profiles of the clusters were extracted from X-ray images, for a series of concentric annuli centered on the brightest pixel. After background subtraction, the resulting surface brightness profile may be fit with an isothermal β-model (Cavaliere & Fusco-Femiano 1976; Branduardi-Raymont et al. 1981):

where I₀ is the central surface brightness, R_c is the core radius of gas distribution, and β is the slope. The single β-model is a poor fit to the surface brightness profiles. We instead fit the surface brightness profile with a double β-model, where the best-fitting double β-model was subtracted from the X-ray images in Figure 1. This difference is then divided by the best-fitting model to produce the residual images shown in Figure 2. This procedure accentuates fluctuations in the surface brightness of the intracluster medium (ICM) revealing substructure. The residual image also reveals structure to the north of RBS 0533 that may be a cavity or bubble. In Section 4.1, we estimate the significance of this region and draw conclusions on whether or not this structure is a bubble. The remaining clusters show evidence of variations in surface brightness, although none of these are likely due to bubbles or AGN feedback.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For instance, sloshing of intracluster gas can create sharp changes in temperature and density that appear as surface brightness edges when projected onto the sky (Markevitch & Vikhlinin 2007). These cold fronts are created by merging halos that displace low-entropy gas from the center of the potential that wraps into a spiral feature owing to sloshing motions. Such spiral features have been observed in many clusters, such as Perseus (Fabian et al. 2006), Virgo (Roediger et al. 2011), Centaurus (Sanders et al. 2016), A496 (Roediger et al. 2012), and A2029 (Paterno-Mahler et al. 2013).

Roediger et al. (2012) performed hydrodynamic simulations to explore the nature and origin of sloshing spiral features in A496. Cold fronts created in this simulation combined to form the observed spiral features, which are not necessarily a result of recent mergers. Ascasibar & Markevitch (2006) have shown that such features can persist for several gigayears. We explored our clusters for evidence of such features in the ICM.

In A2029, a cold front is clearly visible as a sharp change in surface brightness as seen in the top left image of Figure 1. Our residual image of A2029, shown in Figure 2, confirms the continuous spiral feature directly associated with the cold front. Its spiral feature is the largest and most continuous one known, extending outward radially from the center up to approximately 400 kpc (Paterno-Mahler et al. 2013).

Similarly, A2151's residual image also reveals a possible spiral feature. While not as prominent as A2029's sloshing feature, it extends radially outward to at least 81 kpc. We find no clear evidence of sloshing in the remaining objects.

Spectra for each annular region were obtained using the method described in Section 2. Spectra were fit with the absorbed thermal model, phabs(apec). Abundances were determined relative to the values of Anders & Grevesse (1989). Excluding the redshift (frozen to the value of the cluster), all parameters—the column density of neutral hydrogen (N_H), temperature, metallicity, and normalization—were allowed to vary. These values were used to derive the projected electron densities (n_e),

where z is the redshift, N is the model normalization, D_A is the angular diameter distance to the source, and V is the volume of a spherical shell with the inner and outer radii set to the annulus edges. The 1.2 factor is the ratio of the electron density to the hydrogen number density, n_e/n_H (Anders & Grevesse 1989). The cooling time of the ICM was calculated as

where p is the pressure, p = 1.8n_ekT, L_X is the X-ray luminosity within each shell, and Λ(Z, T) is the gas cooling function in terms of abundance and temperature. L_X was determined by first refitting spectra with a phabs × cflux(apec) model. We obtained an estimate for the X-ray bolometric flux by integrating the unabsorbed thermal model between 0.1 and 100 keV. The resulting X-ray flux was then converted to a bolometric X-ray luminosity, L_X.

Finally, we derived the entropy parameter, , and hot gas mass within each spherical shell, M = 1.9 μm_pn_eV, where m_p is the proton mass and μ = 0.62 represents the mean molecular weight of atmospheric plasma. Projected profiles are shown in Figure 3. Of these, A2029 spans a substantially larger temperature (3.0–9.5 keV) than the other clusters. Its density is nearly an order of magnitude higher than the other clusters; however, it still satisfies the threshold for central cooling time in projection.

Zoom In Zoom Out Reset image size

Spectra extracted from the inner regions are affected by projected emission from the hotter regions of the atmosphere at higher altitudes. Accurately deriving the profiles of the inner regions of a cluster requires deprojection. This was done using the dsdeproj routine (Russell et al. 2008). Similar to our projected profiles, we fit the deprojected spectra to a single-temperature phabs(apec) model. Again, all parameters except for redshift were allowed to vary. Fitted quantities for temperature, abundance, and model normalization were used to derive the deprojected density, cooling time, entropy, and the hot gas mass profile, as shown in Figure 4. The deprojected profiles indicate that within the central region of each source (≲10 kpc), both the cooling time and the entropy lie below 1 Gyr and 30 keV cm², respectively.

Zoom In Zoom Out Reset image size

These objects span a moderate range of temperatures between 1 and 8 keV, while their radial densities and pressures span at least 2 orders of magnitude in the range of 10⁻¹–10⁻³ cm⁻³ and 10⁻⁹–10⁻¹¹ erg cm⁻³, respectively.

Mass profiles were created following the model presented in Hogan et al. (2017a). The model consists of a Navarro–Frenk–White (NFW) potential and a cored isothermal potential. The former has been shown to be an accurate description of the total gravitating potential of cluster masses on large scales (e.g., Pointecouteau et al. 2005; Vikhlinin et al. 2006) and takes the form of

where ρ is the characteristic gas density and R_s is the scale radius. Although the NFW profile provides a reasonable fit on large scales, its contribution alone underestimates masses inferred from stellar velocity dispersion due to the central galaxy (e.g., Fisher et al. 1995; Lauer et al. 2014; Hogan et al. 2017a). Thus, a cored isothermal potential is needed to account for this, given by

Here σ_* is the stellar velocity dispersion and R_I, which is used solely to prevent the gravitational acceleration from diverging from R → 0, is chosen to be smaller than the scales of interest. The combined NFW and cored isothermal potential, isonfwmass, is implemented as an xspec extension in the package clmass (Nulsen et al. 2010). The model assumes that the cluster atmosphere is spherically symmetric and in hydrostatic equilibrium.

Stable fits were found by following the work of Hogan et al. (2017a). The σ_* parameter was frozen to the inferred stellar velocity dispersions derived from the Two Micron All Sky Survey (2MASS) isophotal K-band magnitude. When unavailable, values were taken from the HyperLEDA database (Makarov et al. 2014), or assumed to be 250 km s⁻¹ when no data were available (Voit & Donahue 2015).

To compute the uncertainties in these quantities, xspec was used to create a Markov Chain Monte Carlo simulation of 5000 iterations. We adopted the 1σ standard deviation as the uncertainties in our mass profiles, as well as the uncertainties of R_s and . The total cluster mass can then be computed at R₂₅₀₀, the radius where the mean density is 2500 times that of the critical density, ρ_c,

where . The best-fitting parameters are shown in Table 2. The ratio of cooling time to freefall time is believed to be related to thermally unstable cooling (Nulsen 1986; Pizzolato & Soker 2005; McCourt et al. 2012), as such freefall time profiles are derived for each cluster. The enclosed mass profiles obtained from fitting are used to calculate the local gravitational acceleration, g = GM/R², which can be used to calculate t_ff:

Table 2.  Mass Fitting Parameters

Cluster	σ*	AISO	Rs	ANFW	R2500	M2500
	(km s−1)	(keV)	(arcmin)	(keV)	(kpc)	(1014 M⊙)
	(1)	(2)	(3)	(4)	(5)	(6)
A2029	336 ± 10a	0.694			693.4
A2151	219 ± 4a	0.295			259.2
A2107	314 ± 25b	0.608			414.6
RBS 0533	306 ± 14b	0.575			228.3
RBS 0540	250 ± 15c	0.384			279.5

Notes. Column (1): equivalent stellar velocity dispersion. Column (2): isothermal potential given by . Column (3): NFW scale radius. Column (4): NFW potential given by . Column (5): R₂₅₀₀. Column (6): M₂₅₀₀.

^aσ_* inferred from 2MASS isophotal K-band magnitude measurements. ^bσ_* measurements from HyperLEDA. ^cAssuming σ_* = 250 km s⁻¹, following Voit & Donahue (2015).

Download table as:  ASCIITypeset image

The enclosed cluster mass and freefall time profiles are shown in the top left and top right panels of Figure 5, respectively. The latter was used to create t_cool/t_ff profiles shown in the bottom panel. The minimum t_cool/t_ff values lie between 20 and 50, with A2029 residing at the lower end of this range and A2107 being at the higher end. Our min(t_cool/t_ff) profiles differ from previous results but agree within a 1σ error for values calculated for A2029 (McNamara et al. 2016; Hogan et al. 2017a), A2151 (Pulido et al. 2018), and A2107 (Hogan et al. 2017a). These differences are likely due to contrasts in the size and number of spatial bins used for spectral extraction.

Zoom In Zoom Out Reset image size

X-ray cavities are typically identified by surface brightness deficits of about 20%–30% relative to the surrounding medium (McNamara & Nulsen 2007, 2012). Panagoulia et al. (2014) showed that ∼20,000 counts within 20 kpc of the core are often required to clearly detect a cavity. Only A2029, RBS 0533, and RBS 0540 satisfy this criterion. Calzadilla et al. (2019) studied the surface brightness fluctuations in A1664 to determine whether two depressions surrounding the BCG are cavities or due to noise. Through significance tests, they determined that the regions were significant fluctuations and thus cavities.

We approached our analysis of surface brightness fluctuations by comparing the counts in the images of Figure 1, N_I, to the counts in the best-fitting β-model image, N_M. The model represents the undisturbed cluster atmosphere. We used the residual images in Figure 2 as a reference point for the location of potential bubbles in the ICM. We put circular regions of radius r over these depressions at distance R from the center and calculate the signal-to-noise ratio (S/N) in these regions to determine their significance. The signal within each generated region is calculated as

and the S/N within a region is estimated by

We calculate N_M by fitting a double β-model and single β-model to the surface brightness profiles with both elliptical and circular annuli centered on the brightest pixel. The residual images reveal one cluster, RBS 0533, that possesses bubble-like structure to the north of its center. Since this bubble has no rims, making size estimates difficult, the size of the bubble was determined by calculating the S/N where the structure fades into the background (S/N < 5). This was done by overlaying box regions with fixed length and width, corresponding to 5.7 and 1.7 kpc, respectively. Regions are placed in succession of one another moving radially outward, beginning at roughly 2.5 kpc, where the depression is visible (see Figure 6). We find that at approximately R = 20 kpc (the edge of the yellow rectangle) the S/N falls below 5. Since the shape of the bubble is also unknown, we estimate the bubble as being spherically symmetric, which encloses the boxed regions (magenta circle).

Zoom In Zoom Out Reset image size

The S/Ns for the deficit in this circular region and for the other circular regions of interest are marked in Figure 6. The best-fit single and double β-models for elliptical and circular annuli are used to determine the deficits in these regions, and the S/N is calculated within each region using Equation (9). The details of this are given in Table 3.

The surface brightness deficit of this candidate bubble reaches 31% with an S/N of ∼28. These values are consistent with both elliptical and circular β-models, indicating that the structure is resilient to model parameters. Thus, it is likely real and is roughly consistent with the emissivity expected for an evacuated cavity relative to its surroundings (McNamara & Nulsen 2007).

Colored regions to the south of the center (cyan and yellow circles) were found to be insignificant and correspond to an excess of roughly 6% at most for the single circular β-model and a deficit close to 2% relative to the double elliptical β-model. The S/N is relatively low for all models in these regions, which suggests that the structure we see within these regions is likely not real, or rather, it may be an artifact from the model.

The red circle to the west of the center is a region of excess, which reaches levels of about 39% and 47% relative to the model corresponding to S/Ns of 13 and 15 for an elliptical and circular double β-model, respectively. This indicates that this structure is likely a real feature.

Extending this analysis to the other clusters, we find that no other clusters have significant structure associated with feedback, although the sloshing feature in A2029 has an S/N > 20, making it a conclusive detection, whereas the sloshing feature in A2151 has an S/N ≲ 5, making its detection uncertain.

Finally, we can obtain an estimate for the total energy required to inflate the bubble in RBS 0533, which is given by its enthalpy,

where p is the pressure within the cavity assuming that the cavity is in pressure balance with its surroundings, V is the volume of the cavity, and γ is the ratio of specific heat capacities. Here, γ is 4/3 for a relativistic gas and 5/3 for a nonrelativistic monatomic gas. Throughout our analysis, we assume that cavities are filled by a relativistic ideal gas, so E = 4pV. The age of a cavity is best represented by the buoyancy timescale, t_buoy, which is given by (Bîrzan et al. 2004; Vantyghem et al. 2014)

where S is the bubble’s cross section, V is the volume of the bubble, g is the local gravitational acceleration, and C = 0.75 is the drag coefficient (Churazov et al. 2001). The gravitational acceleration is estimated as g = GM/R², where M is the total enclosed mass found in Section 3.4. The buoyancy time of the bubble is ∼1.3 × 10⁷ yr, which can be used to estimate the jet power, or the mechanical power of the AGN that would be required to inflate a spherical bubble of this size, given by

For a bubble of size r = 7.5 kpc at distance R = 10 kpc, the jet power is (3.5 ± 0.2) × 10⁴³ erg s⁻¹.

Among the factors that govern the detection of cavities, high central surface brightness favors their detection in cool cores. Bîrzan et al. (2012) used a simulation to define the properties of bubbles that remain undetected. They concluded that most bubbles are undetected when the angle between the bubble-to-core axis and line of sight is small, or when they lie at large distances from the core.

Apart from RBS 0533, cavities may exist in these systems but remain undetected. Several studies have investigated the statistical properties of cluster cavities that are drawn from the Chandra archive. One such survey found a cavity detection frequency of 41% for a sample size of 75 clusters (Bîrzan et al. 2012), while a sample of 133 systems biased toward cool core clusters found a detection rate of 52% (Shin et al. 2016).

In the brightest 55-cluster sample (B55), it was shown by Dunn & Fabian (2006) that 20 of these clusters require heating to offset cooling. At least 14 of the 20 clusters have clear bubbles, and only one of these does not harbor a central radio source.

Finally, Dunn & Fabian (2008) studied the 42 clusters from the B55 and brightest cluster samples with Chandra data. Of those, 23 have a central radio source. Defining cooling flow clusters as those with a significant central temperature drop and a short central cooling time, they found that 14 of the 42 clusters met these criteria and 6 of those harbor bubbles.

The distribution of radio powers for this sample is consistent with expectations for cool core clusters (Hogan et al. 2015). Although only one of the five objects in our sample has a significant surface brightness depression that may be an X-ray bubble, apart from A2107, four of five possess a central radio source shown in Figure 7. It is noteworthy that the central radio source in RBS 0533 does not coincide with the surface brightness depression associated with the putative X-ray bubble. While this does not exclude the possibility that it is a “ghost cavity” whose radio emission has faded (Bîrzan et al. 2004), it is not a resounding confirmation that the surface brightness depression is indeed a radio/X-ray bubble. Our thermal instability analysis below assumes, conservatively, that it is indeed a bubble capable of lifting low-entropy gas outward.

Zoom In Zoom Out Reset image size

X-ray cavities not only displace hot gas but also may draw metal-enriched plasma out from the centers of clusters at rates of tens to hundreds of solar masses per year (Kirkpatrick et al. 2011; McNamara et al. 2014). The maximum radius that metals can be uplifted to, referred to as the iron radius, R_Fe, is correlated with P_jet according to the relation (Kirkpatrick & McNamara 2015)

Here P_jet is in units of 10⁴⁴ erg s⁻¹ and R_Fe is defined as the radial bin farthest from the cluster’s center where the 1σ error bars for the metallicity profiles (along the jet and orthogonal to it) do not overlap (Kirkpatrick et al. 2011). The scatter in the relation is large, approximately 0.87 dex. In clusters with known cavities, metal-enriched gas preferentially, but not exclusively, lies along the bubbles and thus the radio jets.

As a first step in a search for uplifted gas, we created metallicity maps using the contour binning algorithm of Sanders (2006). The metallicity maps are intended to provide a visual representation of the radial abundance distributions with respect to the radio sources. However, they are not used in our analytical evaluation of the relationship between the off-nuclear atmospheric gas abundance and the radio sources. Clusters were binned using a minimum S/N of 70 per bin to maximize the number of bins generated while retaining a high enough count per bin that uncertainties in metallicity do not dominate. No attempts were made to create maps for A2107, as its exposure time is too short, and thus its low number of counts would not allow us to generate enough bins with the required S/N for any meaningful analysis.

Spectra were extracted within each bin and fitted with a phabs(apec) model. Temperature, metallicity, and normalization were allowed to vary. Column densities were frozen at the value obtained from the LAB Survey (Kalberla et al. 2005). The resulting metallicity maps are shown in Figure 7.

Higher-metallicity gas aligned parallel and antiparallel to the jet axis is a strong indicator of metal-enriched gas being uplifted. The slightly asymmetric map near the center of RBS 0533 hints at this mechanism without being prominent enough to be deemed significant.

To further constrain the lack of evidence for uplifted metal-enriched gas in these four clusters, spectra were extracted from the profiles of annular sector bins with openings lying between 50° and 90°. One of these profiles is along where the extended radio emission from the jet is located (“along-jet”). Extracted spectra in these regions are assumed to reflect the impact of the AGN on the gas. Other profiles are extracted along a direction orthogonal to or offset from the jet axis (“off-jet”) and represent the undisturbed atmosphere. Spectra from these profiles are assumed to be representative of the average prior AGN outbursts. Extracted spectra were fit in the same manner as those used to make Figure 7. The results of this analysis are shown in Figure 8. A2029 demonstrates slight evidence of a trend in higher-metallicity gas along the jet axis, but only in bins between approximately 20 and 60 kpc outward from its center.

Zoom In Zoom Out Reset image size

To determine whether the along-jet and off-jet profiles differ significantly, and thus the likelihood that an abundance excess along the jet axis is significant, we perform two statistical tests for each of our clusters. The results of both statistical tests are shown in Table 4.

In our first test we compare the means for two profiles, using them to calculate the likelihood that they are drawn from the same distribution using a two-tailed t-test. We choose to perform our tests at the 95% significance level (α = 0.05) with the null hypothesis that differences between mean profile metallicities are insignificant. In each of the clusters, the null hypothesis cannot be rejected at the chosen significance level.

Table 5.  Cold Gas Measurements for the Spoiler Clusters: A Comparison of Hα Measurements from the ACCEPT Database versus Other Sources

				Corrected
Cluster	BCG	SFRUV	LACCEPT,Hα	LHα	SFRHα
		(M⊙ yr−1)	(1040 erg s−1)	(1040 erg s−1)	(M⊙)	(108 M⊙)
(1)	(2)	(3)	(4)	(5)	(6)	(7)
A2029	IC 1101	<1.72	<0.643	<0.44 (1)	<0.035	<17 (2)
A2151	NGC 6041	<0.38	<0.141	0.032 (3)	0.003	<3.1 (2)
A2107	UGC 09958	<0.57	<0.179	⋯	⋯	⋯
RBS 0533	NGC 1550	<0.14	<0.016	⋯	⋯	<0.47 (4)
RBS 0540	MCG-01-12-005	0.4 ± 0.09	<0.011	<0.014 (5)	<0.001	⋯

Note. Column (1): cluster. Column (2): BCG. Column (3): ultraviolet SFR from Hoffer et al. (2012). Column (4): ACCEPT database Hα luminosity (Cavagnolo et al. 2008). Column (5): Hα luminosity corrected for our chosen cosmology (see end of Section 1). Column (6): SFR from calculated Hα luminosity using SFR_Hα = 7.9 × 10⁻⁴² L_Hα (Kennicutt 1998). Column (7): molecular gas measurement from CO observations. References to L_Hα and measurements: (1) McDonald et al. 2010; (2) Salomé & Combes 2003; (3) McDonald et al. 2011; (4) O’Sullivan et al. 2018; (5) Cavagnolo et al. 2009.

Download table as:  ASCIITypeset image

In the second statistical test, a two-sample Kolmogorov–Smirnov (K-S) test is performed to determine whether both the along-jet and off-jet metallicity profiles are independent of one another, or come from the same distribution. Again, a significance level of α = 0.05 is chosen with a null hypothesis, that both distributions are drawn from the same parent sample. The null hypothesis is rejected at this significance level if the calculated K-S statistic, D_K-S, satisfies the condition given by , where n and m are the sizes of the along-jet and off-jet profiles, respectively. Similarly to the previous test, the null hypothesis cannot be rejected in any of the clusters.

At the 95% level, then, there is no significant difference between metallicity profiles in any clusters, and thus there is no evidence of uplifted metal-enriched gas. Indeed, extracting spectra along random directions not aligned with the jet axis generally yields profiles that show no evidence of trends in our sample. Subsequent statistical analysis of our profiles only further confirms the results.

Central galaxies located at the bases of hot atmospheres are often associated with molecular clouds, star formation, and nebular emission. Chandra observations have shown that they are prevalent when systems lie below the central cooling time and entropy thresholds (1 Gyr and 30 keV cm², respectively), while those above are usually devoid of cool gas and star formation (Cavagnolo et al. 2008; Rafferty et al. 2008).

On a more fundamental level, hot atmospheres should be susceptible to thermally unstable cooling when the ratio, t_cool/t_ff, falls below unity (Nulsen 1986; Pizzolato & Soker 2005; McCourt et al. 2012). In this context, the cooling time and entropy thresholds would be necessary but insufficient criteria. However, the local value of t_cool/t_ff almost never lies below 10, even in systems experiencing vigorous star formation (Hogan et al. 2017b). Others have suggested that thermally unstable cooling occurs when this ratio lies well above unity, in the range of 10–30 (Gaspari et al. 2012; McCourt et al. 2012; Sharma et al. 2012).

Inspection of Figures 4 and 5 indicates that both the cooling time threshold and criterion are satisfied and that the min(t_cool/t_ff) ∼ 20–50. The central galaxies in this sample should be thermally unstable and should be forming stars and shining by nebular emission. They are not. This failure to respond to both criteria calls for a third possible criterion, possibly uplift, which we have investigated in detail here.

We find surface brightness depressions consistent with an X-ray bubble only in RBS 0533. However, it lacks bright rims composed of low-entropy gas lifted from the inner region of the atmosphere. Furthermore, RBS 0533 and the remaining clusters show no other evidence of substantial uplifted atmospheric gas that would trigger thermally unstable cooling once the gas reaches an altitude where t_cool/t_ff falls below unity. Therefore, the observations are consistent with the hypothesis that uplift may be a significant factor in driving thermally unstable cooling. This investigation does not constitute proof but may indicate that we are on the right track.

Another factor that may trigger thermally unstable cooling is mild atmospheric turbulence (Gaspari et al. 2018; Voit 2018). Turbulence may be induced by the peculiar motion of the central galaxy and mergers. However, in this context the driving mechanism would most likely be the central AGN. The absence of X-ray bubbles would imply the absence of a mechanism to drive the turbulence imparted on the lifted gas. The sloshing spiral in A2029 seen in Figure 2 is evidence of a merger and may indicate that it produced insufficient levels of turbulence required to trigger instabilities.

On the other hand, modest atmospheric turbulence may be a factor leading to the thermal stability of these systems through turbulent heating (Hitomi Collaboration et al. 2016; Zhuravleva et al. 2018). The situation is far from clear but will be further explored with the X-ray Imaging and Spectroscopy Mission and future X-ray observatories equipped with microcalorimeter spectrometers.

These objects were originally selected for observation on the basis of having upper limits on Hα luminosity as listed in the ACCEPT database (Cavagnolo et al. 2008). We have since performed an exhaustive literature search for more recent nebular and molecular mass measurements for each of the clusters. The results from this search are summarized in Table 5.

ACCEPT lists an Hα luminosity for RBS 0533 as L_Hα < 0.016 × 10⁴⁰ erg s⁻¹. Two other studies that probed its central galaxy’s (NGC 1550) CO emission are in tension. O’Sullivan et al. (2018) detected no CO (2−1) or CO (1−0) emission, arriving at an upper limit for molecular hydrogen of M_H2 < 0.47 × 10⁸ M_⊙. However, Nakanishi et al. (2007) claimed a detection of CO (3−2) deriving a molecular gas mass of M_H2 = 4.3 × 10⁸ M_⊙. Clearly these measurements are inconsistent with each other. The apparent CO (3−2) line is broad, spanning a significant fraction of the receiver’s ∼445 km s⁻¹ bandwidth, leaving little room for the baseline continuum to be evaluated. Taking all into account, we adopt the O’Sullivan et al. (2018) upper limit. RBS 0533's central cooling time and entropy both lie below their respective thresholds (10⁹ yr and 30 keV cm²), and thus it is expected to shine in nebular emission.

We have shown in Section 4.1 that its atmosphere harbors a possible X-ray bubble. Therefore, the absence of spatially extended nebular emission is intriguing but not necessarily inconsistent with our hypothesis that thermally unstable cooling is stimulated by uplift. McNamara et al. (2016) suggested that bubbles must lift cool, atmospheric gas to an altitude where t_cool/t_ff < 1. The cooling time of the atmospheric gas at the center of RBS 0533 is ∼10⁸ yr. Based on Figure 5, this gas must be lifted to an altitude of nearly 40 kpc to meet this cooling criterion. However, the observed bubble, at least in projection, extends to roughly half this distance. Therefore, it is plausible that the bubble has not lifted enough atmospheric gas to stimulate thermally unstable cooling at an observable level.

A2151's Hα luminosity is listed as L_Hα < 0.141 × 10⁴⁰ erg s⁻¹ in ACCEPT. McDonald et al. (2011) recently ejected Hα emission in its BCG, NGC 6041, with a luminosity of L_Hα ∼ 3 × 10³⁸ erg s⁻¹. Therefore, NGC 6041 has a detectable level, albeit a modest level of molecular gas. However, emission at this level lies well below the luminosity where the cooling time threshold seen in Cavagnolo et al. (2008) becomes prominent at, L_Hα ∼ 10⁴⁰ erg s⁻¹. An Hα detection of this magnitude is expected as the accumulation of gas from stellar winds, supernovae, and external accretion that may be unrelated to uplift and thermally unstable cooling. In A2107, we find no measurements for cold gas, and so the ACCEPT upper limit is adopted.

RBS 0540's Hα luminosity, <0.014 × 10⁴⁰ erg s⁻¹ (Cavagnolo et al. 2009), agrees reasonably well with the value quoted in ACCEPT, <0.011 × 10⁴⁰ erg s⁻¹. This is also the only cluster in our sample with a detection for SFR_UV of 0.4 ± 0.09 M_⊙ yr⁻¹ (Hoffer et al. 2012).

A2029's Hα luminosity is listed in the ACCEPT database as L_Hα < 0.643 × 10⁴⁰ erg s⁻¹. However, McDonald et al. (2010) found a more restrictive upper limit of <4.41 × 10³⁹ erg s⁻¹. This upper limit is surprising, as its central cooling time and bright, cuspy X-ray emission are similar to clusters with nebular emission luminosities exceeding this limit by more than three orders of magnitude. Why A2029's central galaxy lies dormant while others with similar or less extreme atmospheric properties burgeon with star formation has been a mystery for decades.

Further complicating the matter, A2029's central galaxy hosts a large and relatively powerful radio source P_1.4 ∼ 10⁴¹ erg s⁻¹. Therefore, it should in principle be able to lift hot gas outward, making it susceptible to thermally unstable cooling. However, no evidence for large cavities was found in A2029 by Paterno-Mahler et al. (2013) or in our analysis. A2029's radio source is long and thin (Figure 7). It lacks jets feeding high-volume lobes seen in other powerful sources such as Hydra A and MS 0735+096, which are lifting vast quantities of atmospheric gas (Kirkpatrick & McNamara 2015). It is unclear why this is so. It may be a consequence of atmospheric sloshing that may be sweeping the radio source back into a wide-angle tail morphology (Paterno-Mahler et al. 2013).

In the context of this discussion, the absence of prominent X-ray cavities or radio lobes may indicate that its radio source is incapable of lifting an appreciable amount of atmospheric gas. Its mechanical power may be too small despite its powerful synchrotron emission. Therefore, its atmosphere remains thermally stable, at least for the time being. It is unclear why this would be. Croston et al. (2018) have pointed out that Fanaroff and Riley (FR) type I and II radio galaxies have different particle contents, with FR I galaxies having higher jet (mechanical) power for their synchrotron luminosities than FR II galaxies. Perhaps A2029's radio source is composed of light particles akin to an FR II radio source rather than an FR I, which is commonly found at the centers of clusters.