# Molecular Flows in Contemporary Active Galaxies and the Efficacy of Radio-
Mechanical Feedback

Prathamesh D. Tamhane,1,2 Brian R. McNamara,1,2,3 Helen R. Russell,4 Alastair
C. Edge,5 Andrew C. Fabian,6 Paul E. J. Nulsen,7,8 and Iurii V. Babyk9,10
1Department of Physics and Astronomy, University of Waterloo, Waterloo, ON N2L
3G1, Canada
2Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, ON, N2L
3G1, Canada
3Perimeter Institute for Theoretical Physics, 31 Caroline Street N, Waterloo,
ON N2L 2Y5, Canada
4Department of Physics and Astronomy, University of Nottingham, University
Park, Nottingham NG7 2RD, UK
5Department of Physics, Durham University, Durham DH1 3LE, UK
6Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK
7Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138, USA
8ICRAR, University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009,
Australia
9Department of Physics and Astronomy, University of California, Irvine, 4129
Frederick Reines Hall, Irvine, CA 92697-4575, USA
10Main Astronomical Observatory of the National Academy of Sciences of
Ukraine, 27 Academica Zabolotnoho str., 03143, Kyiv, Ukraine E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Molecular gas flows are analyzed in 14 cluster galaxies (BCGs) centered in
cooling hot atmospheres. The BCGs contain $10^{9}-10^{11}~{}\rm M_{\odot}$ of
molecular gas, much of which is being moved by radio jets and lobes. The
molecular flows and radio jet powers are compared to molecular outflows in 45
active galaxies within $z<0.2$. We seek to understand the relative efficacy of
radio, quasar, and starburst feedback over a range of active galaxy types.
Molecular flows powered by radio feedback in BCGs are $\sim$10–1000 times
larger in extent compared to contemporary galaxies hosting quasar nuclei and
starbursts. Radio feedback yields lower flow velocities but higher momenta
compared to quasar nuclei, as the molecular gas flows in BCGs are usually
$\sim$10–100 times more massive. The product of the molecular gas mass and
lifting altitude divided by the AGN or starburst power — a parameter referred
to as the lifting factor—exceeds starbursts and quasar nuclei by two to three
orders of magnitude, respectively. When active, radio feedback is generally
more effective at lifting gas in galaxies compared to quasars and starburst
winds. The kinetic energy flux of molecular clouds generally lies below and
often substantially below a few percent of the driving power. We find
tentatively that star formation is suppressed in BCGs relative to other active
galaxies, perhaps because these systems rarely form molecular disks that are
more impervious to feedback and are better able to promote star formation.

###### keywords:

galaxies: active, galaxies: clusters: general, galaxies: starburst, galaxies:
Seyfert

††pubyear: 2022††pagerange: Molecular Flows in Contemporary Active Galaxies
and the Efficacy of Radio-Mechanical Feedback–A18

##  1 Introduction

The energy released by active galactic nuclei (AGN) and star formation is able
to lift and perhaps expel interstellar gas (ISM) in galaxies at all epochs.
Gaseous outflows of ionized, atomic, and molecular gas delay and suppress star
formation in galaxies, affecting their evolutionary paths (Alatalo et al.,
2011; Maiolino et al., 2012; Aalto et al., 2012; Cicone et al., 2012; Combes
et al., 2013; Veilleux et al., 2013; Tombesi et al., 2015; Feruglio et al.,
2015; Rupke et al., 2017). For example, energetic feedback from starburst
winds and AGN may imprint the correlation between stellar velocity dispersion
of the host galaxy bulges and the supermassive black holes (SMBHs) masses
(Fabian, 2012; King & Pounds, 2015; Croton & Farrar, 2008; Bower et al., 2008)
and may prevent galaxies from growing to much larger sizes (Scannapieco & Oh,
2004; Beckmann et al., 2017). Burgeoning galaxies at redshift 2 are maintained
on the main sequence of galaxy formation by in- and outflowing gas propelled
by star formation and AGN (Tacconi et al., 2020).

Observations have established that outflows from galaxies that host powerful
AGN are primarily powered by the AGN (Cicone et al., 2014). In the most
massive galaxies, two primary forms or modes of AGN feedback are thought to
operate during a galaxy’s lifetime: the radio and quasar modes (Bower et al.,
2006; Croton et al., 2006). During the quasar mode, nuclear winds drive large-
scale outflows of gas (Tombesi et al., 2015). These winds are thought to eject
interstellar gas from the nascent galaxy which suppresses star formation and
accretion onto the central SMBH (Fabian, 2012; Veilleux et al., 2020). The
mode, quasar or radio, is thought to be governed by the specific accretion
rate onto the black hole (Churazov et al., 2005). For accretion rates lower
than a few percent of the Eddington rate, radio-mechanical feedback ensues
(Russell et al., 2013). Accretion rates approaching the Eddington rate produce
a quasar.

The $M-\sigma$ relation is thought to have been imprinted by quasars accreting
near the Eddington rate following major mergers (Kauffmann & Haehnelt, 2000).
Whether an Eddington flow conserves energy or momentum is consequential to the
form and development of the $M-\sigma$ relation. A hot, adiabatic wind whose
radiative cooling timescale exceeds the outflow timescale conserves energy.
Energy conserving winds lead to $M-\sigma$ relations scaling as $\sigma^{5}$
(Silk & Rees, 1998; Haehnelt et al., 1998). The accretion energy released
during the growth of a nuclear black hole vastly exceeds the potential energy
of the host and thus must couple inefficiently to the surrounding gas (Fabian,
2012; King & Pounds, 2015). A wind accelerated and heated by shock fronts
whose cooling time scale lies below its expansion timescale conserves momentum
(Zubovas & King, 2014). In momentum-conserving outflows, the shocked wind
radiates away its kinetic energy and compresses into a dense gas. Ram pressure
is communicated to the ISM driving outflows to a kiloparsec and perhaps beyond
(Zubovas & King, 2012; King & Pounds, 2015). Momentum conserving winds lead to
$\sigma^{4}$ scaling (Fabian, 1999; King, 2003; Murray et al., 2005) which
lies close to the observed $M-\sigma$ relation (Fabian, 2012).

If the optical depth of dust in the ISM is greater than that due to Thomson
scattering, the coupling between AGN radiation and dust can be significant. In
that case, radiation pressure can power an outflow (Ishibashi et al., 2018).
Depending on the optical depth of the medium around the AGN and the dust
content, momentum ratios $\dot{p}/(L_{\rm AGN}/c)$ can reach values of up to
10 or higher in sources with high AGN luminosity and high dust content, with
large covering fraction leading to higher coupling efficiency. The power ratio
($\dot{E}/L_{\rm AGN}$) can be greater than 0.01.

During radio mode feedback, jets launched from SMBHs inflate bubbles in the
surrounding hot atmosphere, many of which are observed as surface brightness
depressions or cavities in X-ray images. Atmospheric heating by a combination
of shock fronts (Randall et al., 2011), turbulence (Zhuravleva et al., 2014),
and enthalpy released as the cavities rise suppresses and often neutralizes
cooling and star formation (McNamara & Nulsen, 2007, 2012; Fabian, 2012). As
the bubbles rise, they lift low entropy, metal-rich gas that is dispersed in
the inner several 10s of kiloparsecs of BCG’s atmosphere (Nulsen et al., 2002;
Kirkpatrick & McNamara, 2015; Simionescu et al., 2008; Gitti et al., 2011).
Some of this gas may cool and condense in the bubbles’ updrafts into molecular
clouds (Revaz et al., 2008; McNamara et al., 2016; Voit et al., 2017).
Molecular gas is plentiful in brightest galaxies (BCGs) residing at the
centers of cool-core clusters, groups and galaxies (Edge et al., 2002; Salomé
& Combes, 2003; Pulido et al., 2018; Russell et al., 2019; Olivares et al.,
2019).

In Galactic outflows detected in ionized, neutral, and molecular forms,
molecular gas flows usually contain most of the mass. Molecular clouds
directly fuel star formation and nuclear black holes. Therefore, they have the
greatest potential impact on the evolution of galaxies. Several investigations
have explored outflow properties including size, mass, velocity, and their
relationships to galaxy morphology, AGN luminosity, star formation rate, and
stellar mass. Cicone et al. (2014) found a trend between AGN luminosity and
outflow rate. They also found that in starburst galaxies, the ratio of mass
outflow rate to star formation rate, dubbed the loading factor, is typically
$\sim$ 1–4 but increases to upwards of 100 for the most powerful AGN.

Fiore et al. (2017) found that AGN winds are a dominant driving mechanism in
massive galaxies at redshifts of two and below. They concluded tentatively
that winds are capable of sweeping or destroying molecular gas in massive
galaxies. Most studies have focused on relatively small, heterogeneous
samples. Fluetsch et al. (2019) compiled a sample of relatively nearby
galaxies to study these relations with an eye to understanding selection
biases. They found that outflows can be both energy-conserving and radiation
pressure driven, and that less than five percent of the gas escapes the host
galaxies. No correlation was found between molecular gas outflow rates and
radio power. Nevertheless, several instances of jet-powered outflows are known
(Morganti et al., 2015, 2013; Mahony et al., 2016).

Radio mode or radio/mechanical feedback is thought to be the agent maintaining
the inner cooling regions of hot atmospheres thus preventing star formation in
massive galaxies at late times. ALMA observations of central galaxies in
clusters with cooling atmospheres have revealed a far more complex picture
(Russell et al., 2019; Olivares et al., 2019). While stellar populations in
most central galaxies are usually old, those in cooling core clusters are
instead often rich in molecular gas and star formation (Edge, 2001; Salomé &
Combes, 2003; O’Dea et al., 2008; Donahue et al., 2015). For the past two
decades, the Chandra and XMM-Newton observatories have characterized the hot
X-ray emitting atmospheres in and around BCGs with fine detail, revealing
multi-phase, cooling atmospheres. Detailed observations of the molecular gas
content are available for only a dozen systems or so. These studies have
revealed molecular gas filaments trailing behind buoyantly rising X-ray
cavities and interacting with radio lobes (McNamara et al., 2014; Russell et
al., 2014, 2016, 2017b, 2017a; Vantyghem et al., 2016, 2018). These spatial
correlations indicate that radio AGN are disrupting and perhaps expelling
molecular gas from the central galaxies. Molecular gas may also condense in
the updrafts of rising radio bubbles (McNamara et al., 2014; McNamara et al.,
2016; Russell et al., 2019).

Here we analyze molecular flows in central galaxies and compare them to those
in active galaxies from the Fluetsch et al. (2019) (Fluetsch). Fluetsch did
not include outflows driven by radio-mode feedback in central galaxies. In
fact, most studies neglect radio-mode feedback as an effective mechanism for
driving massive outflows. Our goal is to understand the relative impact these
disparate modes of feedback can have on the evolution of galaxies.

In recent years, the Atacama Large Millimeter Array (ALMA) has observed
molecular gas in central galaxies with unprecedented spatial and velocity
resolution. This combined with sensitive X-ray imaging and nebular line
spectroscopy have provided a vivid picture of radio-mode feedback. Such work
has revealed some of the most massive molecular gas flows known in the
universe. Here we examine ALMA observations of BCGs to decompose the flow and
non-flow components. We investigate the relative efficiency and driving power
of different feedback modes and we examine their roles in suppressing star
formation.

Throughout this paper, we adopt $H_{0}=$ 70 km s-1 Mpc-1, $\Omega_{M}=$ 0.3
and $\Omega_{\lambda}=$ 0.7.

##  2 Sample selection

Archival ALMA observations of 12 BCGs were analyzed. All targets in our sample
lie below a redshift of 0.6 and span the range in the molecular gas mass of
107 – 1011 M⊙. Star formation rate ranges between a few to $\sim$ 600 M⊙ yr-1
and radio mechanical AGN power between 1042 – 1046 erg s-1. Their stellar
masses are greater than 1011 M⊙. These properties are summarized in table 1.
Most ALMA archival targets were observed based on their large molecular gas
reservoirs (Edge et al., 2002; Pulido et al., 2018). Therefore, our sample is
biased, incomplete and does not represent all central cluster galaxies. We
consider this bias throughout our analysis. We have included NGC 1275, the
central galaxy in the Perseus cluster, observed with the IRAM telescope
(Salomé et al., 2011).

The Fluetsch et al. (2019) sample of 45 galaxies with gaseous outflows lies
within redshifts of $z<0.2$, providing a good reference sample to compare to
central galaxies. They include four ULIRGs for which the molecular properties
were determined from far-infrared OH transitions. They classify galaxies as
Seyfert, low-ionization nuclear emission-line region (composite) and starburst
(star-forming) galaxies based on the BPT-[SII] diagram. They include galaxies
with flows in neutral and ionized phases. We refer to these categories as AGN,
composite, and starburst galaxies, respectively.

##  3 Data Analysis

We used automated science pipeline scripts provided by ALMA with the data sets
to reduce and calibrate the data. For Cycle 0 Early Science data (Abell 1835
and Abell 1664), scripts were modified according to instructions provided on
the ALMA
website111https://casaguides.nrao.edu/index.php/Updating_a_script_to_work_with_CASA_4.2
to re-calibrate the data with casa 4.2. All other data were processed in casa
version 4.7.2. Additional flagging was done for some BCGs such as phase center
correction (Abell 1835), total flux calibration (Phoenix) and self-calibration
was performed on some datasets to improve the signal to noise.

casa tasks uvcontsub and clean were used to subtract the continuum and
generate image data cubes, respectively. We used Briggs weighting with
parameter 2 to improve sensitivity to faint filaments. The rms noise in the
data cubes was found to be close to the theoretical rms.

##  4 Comparison of Molecular Flow Sizes, Speeds, Masses and Power

The outflows studied by Fluetsch are generally unresolved spatially due to
their small angular sizes. Their masses and velocities were measured by
fitting at least two gaussians to the molecular gas velocity profiles. The
area and width of the broader gaussian component provided mass and velocity
estimates. In some cases, the flows are resolved and visibly surround the
galaxy. However, their morphologies are usually indiscernible due to their
small angular sizes.

In contrast, the flows in BCGs are well resolved with a variety of
morphologies and velocity patterns. Molecular gas in BCGs is often
filamentary, extending out from the center with multiple spectral components
(Russell et al., 2019; Olivares et al., 2019). In most BCGs, the molecular gas
lies outside of the nucleus in an unsettled state. Molecular gas is rarely
observed in ordered motion about the nucleus such as disks. We consider this
off-nucleus gas as a flow in BCGs. Some of this gas may be flowing towards the
BCG as opposed to a pure outflow. A simple, comprehensive characterization of
their flows that can be compared to the Fluetsch et al. (2019) systems would
be desirable.

Although some filaments have smooth velocity gradients, they are much more
extended compared to the sizes of molecular gas disks observed in galaxies
(Rose et al., 2019a; Boizelle et al., 2017). Their position-velocity (PV)
diagrams do not show the characteristic ‘S’-shaped curve that represents
rotation (see Fig. A15). Some of the molecular gas in A1664 may be forming a
disk of molecular gas in the centre (Russell et al., 2014). Similarly, the
circumnuclear gas reservoir in phoenix has a smooth velocity gradient from
$-$200 to 200 km s-1 suggestive of a disk (Russell et al., 2017b). But the
extended filaments that we consider as flows have distinct velocity structures
from the circumnuclear gas. Molecular gas disks are detected in very few BCGs
such as HydraA (Rose et al., 2019a) and Abell 262 (Russell et al., 2019). We
have excluded those BCGs from our sample. In the following subsections, we
describe the process of estimating flow properties in BCGs and compare them to
flows in the Fluetsch sample.

###  4.1 Flow velocities

Spectra from data cubes were extracted in beam-sized regions centered on each
pixel in the entire emission region. They were fitted with a model consisting
of one Gaussian component using the lmfit Python
package222https://lmfit.github.io/lmfit-py/. A significance of greater than
$3\sigma$ was imposed for the detection of line emission, evaluated by
performing 1000 Monte Carlo iterations. Integrated CO images and velocity
centroid maps of BCGs analyzed here are presented in Appendix 1.

Fluetsch et al. (2019) identified outflows spectroscopically, based primarily
on the detection of two velocity components in the spectrum of the entire
emission region. They calculated flow speed using FWHM${}_{\rm
broad}/2+|v_{\rm broad}-v_{\rm narrow}|$, where FWHMbroad is the full width at
half maximum of the broad component in the extracted spectrum of the source
and $v_{\rm broad}$ and $v_{\rm narrow}$ are the mean speeds of broad and
narrow components. This method can be applied only when narrow and broad
components are identified. In the absence of multiple components, Fluetsch
concluded non-detection of an outflow. To allow for a comparison to their
method, we extracted spectra in the entire emission region including all
emission structures and fit them with one or two Gaussian components. The
prescription described above was applied to estimate outflow velocities in
BCGs according to their method.

The outflows in many Fluetsch galaxies are barely resolved spectrally and few
are resolved spatially. Therefore, the outflow speeds and gas masses must be
large enough for the spectrum to deviate from gaussianity. Applying this
method to Abell 2597, Abell 1835, AS1101 and NGC4696, the summed spectra
deviate only marginally from gaussianity. Their flows would therefore have
been missed were they too distant to be spatially resolved. Nevertheless, some
of these systems harbor some of the most massive molecular gas flows known.

We identify gas flows in BCGs and determine their velocities and sizes
individually from spatially extended emission components. Gas speeds were
estimated as flux weighted average speed along the spatially extended
filaments or clumps of gas from their velocity maps within regions we
considered as flows. The regions and the corresponding velocity maps are shown
in the Appendix. If more than one filament or clump is present, the average
speeds of all extended components are considered as the flow speed. Figure 1
compares Fluetsch’s methodology to ours in systems where both methods can be
applied. The flow speeds estimated using the two methods are correlated with a
Pearson correlation coefficient of 0.77 and a p-value of 0.001. Speeds
estimated using the Fluetsch method are two times higher compared to our
method, except for RXCJ0821. The molecular gas in RXCJ0821 lies in two clumps
north of the BCG’s nucleus. Their mean velocities are close to the BCG’s
systemic velocity. However, their FWHMs are significantly larger than their
mean relative velocities, yielding much higher flow velocities using the
Fleutsch method. The Fluetsch method captures a small fraction of the gas that
is flowing at high speeds and overestimates the flow velocity. In what
follows, we adopt speeds estimated using our method. However, were we to
adjust Fluetsch velocities to velocities obtained using our method by dividing
them by a factor of two, it does not change our results qualitatively (see
Appendix B).

Figure 1: Comparison of flow speeds estimated using the Fluetsch method
against our method. The black dashed line is the 1:1 relation.

Figure 2 shows the distribution of flow speeds for all systems. BCG flow
speeds lie in the range 50–300 km s-1 with a mean of 173 km s-1. Abell 1664
with a speed of 579 km s-1 is a moderate outlier. By comparison, the flow
speeds of starburst and AGN galaxies lie in the range 50–600 km s-1 and
100–800 km s-1, respectively. The mean outflow velocity for AGN and starburst
galaxies is 447 km s-1 and 243 km s-1, respectively. Thus BCG flow speeds on
average lie $\sim$60% below AGN hosting galaxies but are similar to starburst
galaxies.

Figure 2: The distribution of flow speeds in BCG, AGN hosting, composite and
starburst galaxies. The flows in BCGs have relatively lower speeds compared to
AGN hosting galaxies.

###  4.2 Flow sizes

Flow projected sizes are determined from the projected lengths of the
filaments in ALMA CO images. The velocity profiles of the gas in filaments in
most BCGs are distinct from the gas in the central regions of BCGs.
Filamentary gas in cluster central galaxies generally has narrow velocity
widths with full width at half maximum velocities $\lesssim$100km s-1 and mean
recessional speeds of several hundred km s-1. These filaments are considered
off-nuclear flows. To examine the filaments we imaged the data cube over the
velocity ranges observed in the filament. We then measured the filament sizes
from the BCG nucleus to the most distant part of the filament. When multiple
filaments are present their sizes are averaged. For example, PKS0745 has three
long filaments in the SE, NW and SW directions. We take the length of each
filament from the BCG center marked as x in Figure 14 to the average outermost
edge of the filament as the size of the flow.

In Figure 3 a histogram of flow size in BCGs, AGN hosting, composites and
starburst galaxies are presented. BCG flow sizes generally range between 2 and
15 kpc. Phoenix, RXCJ1539 and NGC1275 are exceptions with flows extending out
to $\sim$21, 24 and 33 kpc, respectively. The average molecular flow size is
$\sim$12 kpc in BCGs. AGNs span a broad range of flow sizes from 30 pc to 3
kpc. They are generally much smaller and more compact than those in BCGs with
an average flow size of only $\sim$1 kpc. Molecular flow sizes in starburst
and composite galaxies range between 100 pc and 1 kpc. Thus, the molecular
flows in active galaxies studied by Fluetsch are generally confined to the
nuclear regions. In contrast, the molecular flows in BCGs tend to be 10 to
1000 times larger extending well into the bulge and beyond.

Figure 3: Histogram of flow radii in BCGs, AGN, composites and starburst
galaxies.

###  4.3 Flow masses

Molecular gas masses in BCGs are calculated using the integrated intensity of
the CO line emission. Flow masses were estimated from the CO intensity of
spectra extracted from flow regions. The total molecular gas mass was
estimated using the spectrum extracted from a region enclosing all detected CO
emission. The molecular gas mass is inferred using an empirical expression
calibrated for the CO(1-0) line. The integrated line flux ratios of $I_{\rm
CO(2-1)}/I_{\rm CO(1-0)}$ $\approx$ 3.2 and $I_{\rm CO(3-2)}/I_{\rm CO(1-0)}$
$\approx$ 7.2 are used to convert integrated CO(2-1) and CO(3-2) flux
densities to CO(1-0), respectively. The corresponding brightness temperature
ratios are CO(2-1)/CO(1-0) = 0.8 and CO(3-2)/CO(1-0) = 0.8. These ratios are
based on observed ratios at CO(3-2), CO(1-0) or CO(2-1) and CO(1-0) in BCGs
(Russell et al., 2016; Vantyghem et al., 2016, 2017, 2018). They correspond to
an excitation temperature of $\sim$20–25 K and high densities $\sim$ 104 cm-3.
Integrated CO(1-0) flux is converted into molecular gas mass using the
equation (Solomon & Vanden Bout, 2005; Bolatto et al., 2013)

$\begin{split}M_{\rm mol}=&1.05\times 10^{4}\Bigg{(}\frac{X_{\rm CO}}{X_{\rm
CO,Gal}}\Bigg{)}\bigg{(}\frac{1}{1+z}\bigg{)}\\\ &\bigg{(}\frac{S_{\rm
CO}\Delta v}{\rm Jy\,km\,s^{-1}}\bigg{)}\bigg{(}\frac{D_{\rm L}}{\rm
Mpc}\bigg{)}^{2}\,{\rm M_{\odot}},\end{split}$ (1)

where $S_{\rm CO}\Delta v$ is the integrated flux density of the CO(1-0) line,
$D_{\rm L}$ is the luminosity distance, $z$ is the redshift of the galaxy and
$X_{\rm CO}$ is the CO-to-H2 conversion factor, with $X_{\rm CO,Gal}$ =
2$\times$1020 cm-2 (K km s-1)-1.

The adopted value of $X_{\rm CO}$ lends to a factor of two or more uncertainty
in the gas mass estimates. Its value depends on temperature, density and
particularly, the metallicity of the molecular gas (Bolatto et al., 2013).
Estimates of $X_{\rm CO}$ are available for Milky Way ($X_{\rm CO,Gal}$) and
nearby galaxies. No direct estimate of $X_{\rm CO}$ is available for BCGs.
However, studies have shown that the hot atmospheres surrounding BCGs, from
which the molecular clouds have likely condensed, have metallicities lying
between $\sim$0.6–0.8 $Z_{\odot}$, close to if not below solar metallicity.
Furthermore, the line widths of individual molecular gas clouds are comparable
to those in the Milky Way (Rose et al., 2019b; Tremblay et al., 2016; David et
al., 2014; Heyer & Dame, 2015). CO line ratios are consistent with optically
thick molecular gas (Russell et al., 2019). Vantyghem et al. (2017) found that
the optically thin 13CO to optically thick 12CO line ratio in one BCG
indicated abundances lying within a factor of two of the Galactic value.
Therefore, we have adopted $X_{\rm CO,Gal}$ for BCGs in our calculations apart
from Phoenix.

The BCG in the Phoenix cluster is undergoing star formation at a rate of
$\sim$500 – 800 M⊙ yr-1 (McDonald et al., 2012) similar to LIRG/ULIRGs. It has
been shown that the use of $X_{\rm CO,Gal}$ in ULIRGs may overestimate the
amount of molecular gas by a factor of five (Downes et al., 1993; Solomon et
al., 1997; Downes & Solomon, 1998). Therefore, we have adopted $X_{\rm CO}$ =
0.4$\times$1020 cm-2 (K km s-1)-1 for Phoenix following Russell et al.
(2017b). Fluetsch et al. (2019) use $X_{\rm CO,Gal}$ in their calculations for
all galaxies except for LIRGs and ULIRGs for which they use $X_{\rm CO}$ of
0.4$\times$1020 cm-2 (K km s-1)-1 and 0.6$\times$1020 cm-2 (K km s-1)-1,
respectively, which is in agreement with the factor we assume.

Radio interferometers such as ALMA can underestimate the total flux if the
angular size of the emission region is larger than the maximum recoverable
scale given by $\sim$0.6$\lambda$/Dmin, where $\lambda$ is the wavelength of
the observation and Dmin is the shortest baseline. Single dish telescopes
generally do not suffer from this problem due to their large maximum
recoverable scales. Therefore, we compared our molecular gas mass measurements
with molecular gas masses obtained from single-dish observations (Edge, 2001)
using the same CO-to-H2 conversion factor as ours to quantify the missing flux
in our observations. The ALMA molecular gas masses are within a factor of the
single-dish molecular gas mass measurements. ALMA detected 60 percent more
molecular gas in RXCJ0821, while in A1664, A1835 and NGC5044 the ALMA masses
lie 25 to 40 percent below the single dish mass measurements. In the latter
systems, ALMA may have resolved away some extended molecular gas. These
discrepancies do not qualitatively affect our results.

Figure 4 shows the molecular gas mass associated with the out(in)flows on a
logarithmic scale. BCG molecular flow masses lie between 0.3–12 $\times
10^{9}$ M⊙ apart from the much smaller masses in NGC 5044 and NGC4696 of 107.7
M⊙ and 107.5 M⊙, respectively. BCG flow masses differ markedly from those in
Fluetsch et al. (2019). BCGs possess on average $\sim$100 times more molecular
gas in flows compared to composites (107.5–108.5 M⊙) and starburst galaxies
(106–108 M⊙). On the other hand, a broad mass range is found in AGN galaxies,
lying between $10^{6}$–$10^{9}$ M⊙. Nevertheless, the most massive flows in
BCGs exceed those in AGN by more than a factor of 10.

However, the total molecular gas masses in BCGs are comparable to those in
Fluetsch et al. (2019) (Fig 5). By comparison, the average total molecular gas
mass in BCGs is $\sim$1010 M⊙, while it is 5$\times 10^{9}$ M⊙ in AGN and
3$\times 10^{9}$ M⊙ in composite and starburst galaxies.

About half of the total molecular gas mass in BCGs is found in filaments. Some
BCGs like Perseus are filament-dominated, in which almost all of the molecular
gas is in filaments. In AGN hosting galaxies the amount of molecular gas in
the outflow is only $\sim$10 percent. It is even lower in composite and
starburst galaxies at $\sim$3 percent.

On a cautionary note, many BCGs analyzed here were selected for observation
based on their high molecular gas masses during the early ALMA cycles.
Therefore they do not represent the molecular gas masses of average BCGs. We
have kept this in mind as we draw scientific inferences from the data.

Figure 4: Histogram of the mass of molecular gas in outflows of BCGs, AGNs,
composites and starburst galaxies. Figure 5: Histogram of total molecular gas
mass in BCGs, AGNs, composites and starburst galaxies.

###  4.4 Measurement uncertainties

The true sizes of molecular flows and their velocities are systematically
underestimated due to projection. Furthermore, when observed in emission we do
not know what side of the nucleus the filaments are located. Unless absorption
against the continuum is observed, it is very difficult to determine whether
the gas is moving towards or away from the AGN (Rose et al., 2019b).
Therefore, whether the filaments in an individual object are flowing in or out
is unknown. Both inflows and outflows can be present at the same time.
Observed velocities will be lower if both inflowing and outflowing gas is
present along the line of sight. For example, in PKS0745, flux weighted
average velocity is only 50 km s-1, however, there is a component at $\sim$200
km s-1 in the extended filament. The low-velocity gas could have detached from
the flow and slowed down or infalling onto the central galaxy. These effects
would introduce a factor of a few uncertainty in velocity in most objects.
Although these effects are difficult to quantify, we adopt 0.3 dex uncertainty
in velocity.

The primary mass uncertainty comes from the application of $X_{\rm CO}$, which
contributes 0.3 dex (Bolatto et al., 2013). Statistical uncertainties due to
the fitting the total intensities add about $\lesssim$20 percent. Thus the
average error on the mass of the outflow may be as large as 0.31 dex.
Orientation and projection introduce scatter in both velocity and radius
measurements. Including this scatter, the overall average error on flow
properties is $\sim$0.5 dex.

BCG | $z$ |  log $M_{\rm flow}$ |  log $M_{\rm tot}$ | $R_{\rm flow}$ | $v_{\rm flow}$ | $\dot{M}_{\rm flow}$ |  log $P_{\rm Mech}$ |  log $L_{\rm Nuc}$ | SFR |  log ($M_{\star}$) | Ref.
---|---|---|---|---|---|---|---|---|---|---|---
|  |  ($M_{\odot}$) |  ($M_{\odot}$) | (kpc) |  (km s-1) |  ($M_{\odot}\,{\rm yr^{-1}}$) |  (erg s-1) |  (erg s-1) |  ($M_{\odot}\,{\rm yr^{-1}}$) |  ($M_{\odot}$) |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12)
2A0335+096 | 0.0346 | 8.31 | 8.59 | 2 | 153 | 17 | 43.949 | $<40.80$ | 0.47 | 11.73 | [a]
Abell 1664 | 0.128 | 9.56 | 9.87 | 6.3 | 579 | 341 | 44.04 | $<41.50$ | 13.18 | 11.1 | [b]
Abell 1795 | 0.063 | 9.36 | 9.52 | 8.3 | 191 | 54 | 43.133 | $<41.24$ | 3.47 | 11.84 | [c]
Abell 1835 | 0.252 | 10 | 10.6 | 12 | 61 | 52 | 45 | $<42.78$ | 117.49 | 10.57 | [d]
Abell 2597 | 0.0821 | 10 | 10.3 | 15 | 61 | 42 | 44.278 | $<41.73$ | 3.98 | 11.51 | [e]
PKS0745-191 | 0.1028 | 9.7 | 9.89 | 4.1 | 50 | 63 | 45.7 | 42.11 | 13.49 | 11.71 | [f]
Phoenix | 0.596 | 10.5 | 11.1 | 21.4 | 332 | 501 | 45.85 | 44.37 | 616.6 | 12.48 | [g]
RXCJ1504 | 0.216902 | 9.7 | 10.4 | 18 | 108 | 30 | 44.96 | 42.66 | 85.11 | 11.7 | [h]
RXCJ0821 | 0.109 | 10.21 | 10.32 | 4 | 28 | 72 | 44.4 | $<41.14$ | 37 | 11.06 | [i]
NGC5044 | 0.00928 | 7.4 | 7.56 | 2.8 | 167 | 2 | 42.78 | 39.70 | 0.22 | 11.38 | [j]
NGC4696 | 0.00987 | 7.52 | 7.95 | 4 | 222 | 2 | 43.11 | $<39.67$ | 0.16 | 11.79 | [k]
RXCJ1539 | 0.075766 | 9.51 | 10.11 | 23 | 203 | 29 | 42.76 |  | 1.86 | 11.36 | [k]
Abell S1101 | 0.05639 | 9.03 | 9.06 | 8.3 | 160 | 21 | 44.89 | $<40.99$ | 1 |  | [k]
NGC1275 | 0.01756 | 9.02 | 9.68 | 33.3 | 117 | 17 | 43.9 | 42.71 | 70.79 | 11.38 | [l,m]
Table 1: Properties of BCGs in the sample. Columns: (1) BCG name, (2)
redshift, (3) mass of the flow, (4) total molecular gas mass, (5) radius of
the flow, (6) speed of the flow, (7) flow rate, (8) mechanical power of the
AGN, (9) nuclear 2-10 keV luminosity, (10) star formation rate, (11) stellar
masses estimated from 2MASS K-band mangitudes following Main et al. (2017),
(12) references: [a] Vantyghem et al. (2016); [b] Russell et al. (2014); [c]
Russell et al. (2017a); [d] McNamara et al. (2014); [e] Tremblay et al.
(2016); [f] Russell et al. (2016); [g] Russell et al. (2017b); [h] Vantyghem
et al. (2018); [i] Vantyghem et al. (2017); [j] David et al. (2014); [k]
Olivares et al. (2019); [l] Salomé et al. (2011); [m] Lim et al. (2008).

##  5 Energy, Momentum, AGN radiation

AGN release mechanical energy and radiation capable of displacing the gas
around them. In ULIRGs and quasars, AGN power is correlated with the kinetic
power of the flows (Cicone et al., 2014), consistent with the AGN’s energetic
output driving the flows. The output is in the form of fast nuclear winds and
radiation, and radiation pressure from young stars. In this section, we
compare molecular flow energy and momentum fluxes in BCGs to the radiative
energy and momentum fluxes of their AGN. We then compare those relations with
AGN, composite and starburst galaxies to study the difference between these
systems.

###  5.1 Flow momentum flux vs $L_{\rm AGN}/c$

Apart from a few known quasars hosted in BCGs with radio bubbles, the
mechanical AGN power dominates the power budget usually by an order of
magnitude or more (Russell et al., 2013). Nevertheless, some systems with
powerful radio activity also emit nuclear X-rays. For the sake of
completeness, we compare the nuclear radiation emerging from BCGs to that of
the active galaxies in Fleutsch.

Here, $L_{\rm AGN}$ denotes the total isotropic radiated power emerging from
the BCG’s nucleus. Russell et al. (2013) estimated the nuclear X-ray
luminosities ($L_{\rm nuc}$) in the $2-10$ keV energy range using photometric
and spectroscopic methods. For those systems, we adopted their photometric
value for $L_{\rm nuc}$. For the remaining sources (Phoenix and RXCJ0821), we
used archival $Chandra$ observations to estimate nuclear luminosity/upper
limits using Russell’s photometric method. The presence or absence of a
nuclear point source was verified by generating an image in the 3–7 keV band
and visually inspecting for a central point source. When absent, upper limits
for nuclear luminosities are presented. $L_{\rm nuc}$ was converted to total
bolometric AGN radiative luminosity ($L_{\rm AGN}$) by multiplying it by a
bolometric correction. For Compton thick AGN, the bolometric correction is
$\kappa_{bol}\sim 30$, with an intrinsic uncertainty of 0.2 dex (Brightman et
al., 2017). Our targets are likely Compton thin, so we expect the bolometric
correction to be smaller than the one for Compton thick AGN. Low luminosity
type-2 AGN have X-ray bolometric correction factors of $\sim$10 (Lusso et al.,
2012). Therefore, we adopt $\kappa=10$ for our calculations. AGN X-ray
luminosity is sensitive to short-term AGN variability and can be
underestimated.

The flow momentum flux is calculated as $\dot{M}v$, where $\dot{M}$ is the
mass flow rate in gm s-1 and $v$ is the flow speed in cm s-1. $\dot{M}$ is
estimated by dividing the mass of the flow by the time it takes the flow to
reach its projected size ($R/v$). The rate of momentum output in radiation
from the AGN is $L_{\rm AGN}/c$.

In figure 6, the flow momentum flux is plotted against radiative momentum flux
$L_{\rm AGN}/c$. The diagram is intended to probe the ability of radiation
emitted by the nucleus to drive a flow. Indicated in the figure are flow to
AGN radiation momentum flux ratios $\dot{M}v/(L_{\rm AGN}/c)=1,5,20$.
According to nuclear wind-driven models, systems in which the ratio lies
between 1 and 5 are able to be driven by radiation. Higher values of momentum
flux ratios can be obtained by radiation driven flows when the gas in the
central regions has high IR optical depth, as in highly obscured AGNs and
ULIRGs. However, at kpc scales, the optical depth of the medium is generally
too low for outflows to attain momentum flux ratios greater than 5. The energy
conserving flows generally have momentum flux ratios above 5. All points lie
above the one-to-one line, indicating that radiation is generally unable to
drive the flows in these systems. These quantities are correlated in AGN and
composite galaxies with a Pearson-r correlation coefficient of 0.73 at greater
than 99 percent significance, including systems with either $L_{\rm AGN}$ or
$\dot{M}$ upper limits. This indicates that even if AGN radiation pressure is
unable to power a flow on its own, it may contribute significantly in high
power AGN (Fluetsch et al., 2019).

As expected, BCGs show no correlation between the flow momentum flux and the
AGN radiative momentum flux (Pearson r-value of 0.24, the 50 percent
confidence level). This lack of correlation suggests that radiation pressure
has little influence on flows in BCGs.

The momentum ratios for AGN and composite galaxies lie in the range predicted
by nuclear wind-driven outflow models discussed in section 7. Still, the large
scatter in the relationship makes it difficult to determine whether flows in
these systems are energy conserving, momentum conserving, or a combination of
both. In BCGs, the rate of change of flow momentum exceeds the radiative
momentum input by $\geq 20$, more than the maximum theoretical prediction for
energy conserving flows. The only exception is Phoenix, which has a ratio of
$\sim$0.2. Phoenix is the most powerful and highly star-forming BCG. It has an
active AGN depositing a large amount of energy into its surrounding medium in
both mechanical and radiative form in nearly equal amounts. That is reflected
in the plot above, where the nuclear luminosity of Phoenix is $\sim 10^{46}$
erg s-1, similar to quasars. Phoenix also has a high star formation rate
(McDonald et al., 2019). Therefore, the conditions in the Phoenix BCG are
similar to those in ULIRGs and quasars. Therefore, Phoenix is expected to be
like ULIRGs, with a radiation pressure driven, momentum conserving flow.
However, the molecular gas in Phoenix is closely tied to its radio bubbles,
rather than star formation or the quasar. The bubbles are apparently doing
most if not all of the work.

Figure 6: Relationship between the flow momentum rate and the AGN radiation
momentum rate. The three dashed lines represent nuclear wind-driven model
predictions for energy conserving (20:1), momentum conserving (1:1), and
radiation pressure-driven (5:1) flows, respectively. Red circles are denoted
by BCGs, AGNs by blue circles, composites by light blue triangles and star-
forming galaxies by blue stars. Symbols with white circles in the middle are
galaxies with flows detected in OH and symbols with black marker edges are
fossil flows from the Fluetsch sample, respectively. The black point with
error bars in the bottom right corner represents the average error bar on each
point.

###  5.2 Momentum

Momentum may be conserved in some gas flows. After the hot shocked gas
radiates away most of the kinetic energy, its momentum is left behind and
drives the flow. As discussed in sections 4.3 and 4.1, flow speeds in BCGs on
average are similar to flow speeds in star-forming galaxies and lower than
AGNs. However, higher molecular gas masses are often found in BCGs. Therefore,
their flow momenta are higher on average compared to AGN, composite or
starburst galaxies (see figure 7). A broad range of flow momenta are found in
AGNs, lying between 1046–1050 g cm s-1. The flow momenta in BCGs lies between
1049–1050 g cm s-1 with an average of 2.3$\times$1050 g cm s-1. This figure is
five to ten times larger than in AGN, composite and starburst galaxies whose
average flow momenta are 6.6$\times$1049, 1$\times$1049 and 1.7$\times$1049 g
cm s-1, respectively. A small fraction of AGN galaxies also have a high flow
momentum.

Figure 7: The histogram of flow momenta ($M_{\rm flow}\left\langle v_{\rm
flow}\right\rangle$) in BCGs, AGN, composite and starburst galaxies.

##  6 Gas Flows and Mechanical Power

BCGs often host powerful radio/mechanical AGN. The mechanical power output of
the AGN is estimated as the power required to inflate the buoyantly rising
bubbles of relativistic plasma that are fed through jets from the vicinity of
the central SMBH (McNamara & Nulsen, 2007). These bubbles are observed as
cavities in X-ray images. X-ray cavities are detected in all sources in our
sample. The energy required to inflate a bubble is given by $E=4PV$, where $P$
and $V$ are pressure and volume of the bubble, respectively, where the bubble
is approximately in pressure equilibrium with the surrounding ICM (Churazov et
al., 2000). The mechanical power of the AGN ($L_{\rm mech}$) can be obtained
by dividing the energy by the buoyancy time of the bubble. Cavity power
measurements were taken from the literature (Bîrzan et al., 2012; Hlavacek-
Larrondo et al., 2015; McDonald et al., 2015; Vantyghem et al., 2018;
Calzadilla et al., 2019).

Their mechanical powers lie between $\sim 10^{43}$–1046 erg s-1, one to two
orders of magnitude larger than their nuclear radiation powers. Their
mechanical powers are comparable to the radiative powers of AGN and some
composite galaxies. Using the mechanical powers as $P_{\rm drive}$ for BCGs in
the relationship between $P_{\rm drive}$ and $\dot{M}$ using linear regression
gives a slope and intercept of 0.42$\pm$0.25 and 0.97$\pm$0.40 $M_{\odot}$,
respectively. It is shown in Fig. 8 by a dotted line. The Pearson correlation
coefficient 0.61 with a p-value of 0.02 indicates, again, a weak correlation.
Thus, despite being comparable to the radiative powers of AGN and composite
galaxies, mechanical powers are poorly correlated with the molecular gas flow
rate.

This poor correlation is true for all systems including those in Fluetsch.
While trends are seen and AGN have ample power to drive the flows, the process
is complex and inefficient. Perhaps most AGN power is either radiated away or
it is being deposited in other forms.

The ample mechanical powers and close association of molecular gas filaments
with X-ray cavities in several systems suggests molecular clouds are lifted by
the rising bubbles or are condensing in their updrafts. Whether this is true
in all systems is not clear. Evidence suggests that at least some molecular
gas condenses from cooling hot gas lifted behind the X-ray cavities. Filaments
may grow due to interpenetration of hot and cold gas (Liu et al., 2019).
Extended molecular filaments are detected towards multiple generations of
cavities in the Perseus cluster (Salomé et al., 2006). However, in some
instances, the mass of molecular gas is close to and may exceed the displaced
atmospheric gas mass, which would be difficult to explain by uplift unless the
molecular gas was created by multiple AGN cycles (Russell et al., 2019). The
scatter in the trend between the molecular filament mass and the mechanical
power indicates a complex relationship between molecular gas and the AGN.

Figure 8: The power of the driving mechanism is plotted against the molecular
flow rate. The red symbols indicate BCGs. For starburst galaxies, their star
formation power is used as $P_{\rm drive}$. Dark blue circles, light blue
triangles and murky blue stars represent AGN, composite and starburst galaxies
from Fluetsch et al. (2019), respectively. The white filled symbols are flows
detected in OH, and fossil flows are denoted by symbols with a black border.
The dashed and dotted lines are the best fit lines for BCGs and Fluetsch
galaxies, respectively.

##  7 Lifting mechanism

Figure 9: The figure shows the relationship between the kinetic power of flows
and the mechanical powers in BCGs. For the Fluetsch galaxies, the $x$-axis
represents the bolometric AGN luminosity. The black lines represent flow
powers 5 percent (dashed), 0.5 percent (dashed-dotted) and 0.05 percent
(dotted) that of bolometric AGN luminosity. Symbols are as in figure 8.

The relationship between the kinetic energy flux of the gas flow and the
lifting force provides a measure of the coupling between the power source and
the gas. It is an easily-determined quantity in real and model systems and
thus places interesting constraints. We define the molecular flow kinetic
energy flux as 0.5$\dot{M}v^{2}$, where $\dot{M}$ is the mass flow rate and
$v$ is the average speed of the flow. The power for the Phoenix cluster BCG is
indicated with the sum of its nuclear radiation and mechanical powers.

Figure 9 shows that the kinetic energy fluxes of molecular gas in most systems
lie near or below a few percent of the AGN or starburst power. The
radio/mechanically-driven BCGs are indistinguishable from the others in most
BCGs, indicating that all mechanisms couple inefficiently to the molecular
clouds and at similar levels. Some BCGs such as A1835, A2597 and PKS0745 lie
much below others as a result of low flow speeds. The gas in these systems may
have decoupled from the flow and slowing down. While a trend is observed in
other systems, the scatter in kinetic energy flux at a given power spans 2-3
decades. Some scatter may be attributable to AGN variability. But the radio
AGN show similar scatter to nuclear AGN and starbursts. Mechanical power is
averaged over $10^{7}-10^{8}~{}\rm yr$, which is comparable to the timescales
for accelerating the molecular clouds. Nuclear AGN vary on much shorter
timescales (Schawinski et al., 2015). So the scatter probably indicates both
the weak coupling and complex nature of feedback.

The three lines in Fig 9 were chosen to reference the fractions of AGN
radiation power expected in energy conserving (5%), momentum conserving
(0.5%), and radiation pressure driven (0.05%) flows. Most Seyfert and
composite galaxies lie in the theoretically expected ranges. Thus, the
coupling between AGN power and molecular flow kinetic energy in BCGs and other
systems are broadly similar, despite very different acceleration mechanisms.

Figure 10: Histogram of flow distance times the flow mass divided by the power
of the driving mechanism for BCGs, AGN, composite and starburst galaxies. This
quantity is referred to as the lifting factor in the text.

###  7.1 Lifting Factor

Molecular gas flows in BCGs generally have higher masses and extend to larger
distances on average than the systems in Fluetsch et al. (2019). These
properties indicate that radio AGN are generally more capable than nuclear
winds, radiation, and starbursts at propelling molecular clouds out of the
centers of galaxies, at least at the present epoch.

This phenomenon is explored further in Figure 10, where we compare the product
of the mass and flow size per unit power. We refer to this as the lifting
factor. The radio-mechanical power is adopted for BCGs. For AGN in the
Fluetsch sample, we adopted the bolometric AGN luminosity. For starburst
galaxies, we adopted the star formation power estimated from star formation
rates using the relation $P_{\rm SFR}$ (erg s-1) $=2.5\times 10^{41}$ SFR
($M_{\odot}$) from Veilleux et al. (2005). This conversion factor is
consistent with the factor measured independently in the central galaxy Abell
1835 (McNamara et al., 2006) which is included in our sample. This single
example offers reassurance that the conversion relation of Veilleux et al.
(2005) applies to BCGs.

Figure 10 shows BCGs have higher lifting factors on average than other outflow
systems, indicating that radio-mechanical feedback is more effective per unit
power at lifting large masses to greater distances than nuclear AGN and
starbursts. The lifting factor for starbursts (H2 galaxies) is considerably
larger than for AGNs. The starbursts overlap the lifting factors of the
weakest BCG flows, but BCGs outperform all by more than an order of magnitude
using this figure of merit.

Figure 10 can be interpreted in the context of Figure 9. In Figure 9 we show
that the molecular kinetic energy fluxes against driving power are of similar
magnitude between the BCGs and galaxies in the Fluetsch sample. At first
blush, Figure 9 and Figure 10 appear to be inconsistent with each other. This
apparent inconsistency is attributable to the large differences in timescale
and size of the flows. The AGN in Fluetsch are driving smaller masses of
molecular gas over shorter distances but at higher speeds. The much larger
masses, distances and timescales in BCGs compensate such that the kinetic
energy fluxes per unit power are similar. However, the total mass displaced
over time is much larger in BCGs for a given mean power.

The large range of lifting factors may be due to variations in flow
characteristics including the volume affected by radio, nuclear, and star-
formation activity and the ability of the working surfaces to couple to the
ambient gas. This coupling is always weak but varies greatly (Fig. 9).

Radiation pressure wind energy is released on smaller scales close to the AGN,
where radiation intensity is strong, and the particle number density is high.
At larger distances the medium becomes tenuous, and the radiation intensity
drops rapidly. The outflow then transitions into a momentum conserving flow
and rapidly slows down (Veilleux et al., 2020). Therefore, radiation driven
flows are less efficient at lifting a large amount of mass to vast distances.

The large lifting factor of radio-mechanical feedback in BCGs is attributable
to the large volume of impact and the relatively long timescales radio bubbles
are able to lift gas in the surrounding interstellar medium and atmosphere.
Radio bubbles couple to the gas in the inner kpc (Mukherjee et al., 2018; Guo
et al., 2018) lifting the low entropy gas in their wakes to high altitudes
through drift and entrainment (Pope et al., 2010). This is observed in real
systems as high metallicity atmospheric gas located in the wakes of rising
X-ray bubbles extending in some instances to altitudes of tens of kpc
(Kirkpatrick & McNamara, 2015).

Radio bubbles encompass a large range of sizes and volumes, with diameters of
a few kpc to over 200 kpc. A typical bubble appearing as an X-ray cavity is
elliptical in projection with an average semi-major and semi-minor axis of
11.4 kpc and 7.7 kpc, respectively (Rafferty et al., 2006; Calzadilla et al.,
2019; McDonald et al., 2015; Hlavacek-Larrondo et al., 2015; Vantyghem et al.,
2017). Assuming ellipsoidal 3D shape, the average volume of a bubble is 1068
cm3. In contrast, the average volume of the AGN wind-driven outflows is $\sim
10^{65}$ cm3, assuming cylindrical geometry with a radius and height of
$\sim$1 kpc, respectively. Starbursts typically occur within the central 1
kpc. The energy injection region is typically up to $\sim$2 kpc in extreme
situations, after which the flow starts slowing down (Schneider et al., 2020).
Thus, the effected volume is approximately 1665–1066 cm-3. The volume would be
smaller considering a more realistic bi-conical geometry. Therefore the
working volume of radio-mechanical feedback is much larger and affects a much
larger mass for a given ambient gas density.

Another key factor is the long lifting timescales of radio/X-ray bubbles.
Their typical observed ages lie between 10–20 million years (Bîrzan et al.,
2004). But many survive for $\sim$ 100 million years or longer (Brienza et
al., 2021). While the jet launching phase typically lasts for $\sim$10 million
years, comparable to a typical quasar lifetime (Martini & Schneider, 2003;
Bird et al., 2008; Schawinski et al., 2015), radio bubbles propelled outward
by buoyancy continue to draw in and lift gas in their updrafts. Therefore, a
single radio event can continue to drive gas outward long after a quasar of
similar power shuts down. Once it shuts down the gas slows down in the
galaxy’s gravitational potential and by ram pressure and drag forces.

We have considered here only the displaced molecular gas in BCGs. The effects
shown in Figure 10 are likely larger. Rising radio bubbles also lift and
displace hot gas from the central atmospheres. $Chandra$ has observed metal
enriched gas preferentially along the radio jet axes (Kirkpatrick et al.,
2011; Simionescu et al., 2008; Gitti et al., 2011). Similar features are
reproduced in hydrodynamic simulations indicating 109–1010 M⊙ of metal rich
gas lifted from central regions of galaxy clusters (Duan & Guo, 2018; Qiu et
al., 2020; Li et al., 2015).

The atmospheric mass displaced by cavities is comparable to or larger than the
molecular gas masses (Russell et al., 2019). Fluetsch et al. (2019) found the
molecular masses in their sample exceeded the atomic and neutral phases by
roughly 40 times. Therefore, in order to compare our systems with Fluetsch, we
have not included the hot gas masses. But were we to do so the flow rates and
masses would increase by a factor of two or more.

Figure 11: Histogram of log of mass loading factor in BCGs, AGNs, composite
and starburst galaxies.

##  8 Star Formation

Here we explore relationships between feedback, outflows, and star formation.
We adopt BCG star formation rates (SFR) from McDonald et al. (2018) who used
several methods to estimate the SFRs. For Pheonix, they decomposed the
spectrum to separate the AGN component from stellar radiation. For the
remaining objects, they adopted the logarithmic mean of multiple SFRs for each
system found from the literature. The average logarithmic scatter in
measurements of SFRs for BCGs is determined to be 0.28 dex. The SFR
measurement for RXCJ0821 is taken from O’Dea et al. (2008).

###  8.1 Mass loading factor

The mass loading factor $\eta$ is defined as the ratio of the molecular gas
flow rate to the star formation rate ($\eta=\dot{M}/SFR$). A high value of
$\eta$ indicates that the AGN can sweep the gas from the galaxy as quickly as
star formation consumes it. Low $\eta$ indicates that star formation will
consume a significant amount of gas before it can be removed from the inner
regions of the galaxy. The histogram in figure 11 shows the distribution of
$\eta$ for different galaxy types, including the fossil galaxies in the
Fluetsch sample. The majority of the star-forming galaxies have mass loading
factors $\leq$1\. This is expected from feedback models for star-forming
galaxies in which supernova explosions are the primary outflow driving
mechanism. The galaxies hosting AGN from the Fluetsch sample and BCGs have a
broad range of $\eta$ from 0.1 to $\sim$100\. However, the majority of those
galaxies have $\eta>1$.

High values of $\eta$ found preferentially in BCGs and AGN galaxies indicate
that much gas is displaced from these systems before it can form stars.
Therefore, star formation may be suppressed or quenched assuming the gas
leaves the galaxy and does not return. In the next section we show that very
little molecular gas leaves these systems.

###  8.2 Does Radio-mode Feedback Suppress Star Formation?

Figure 12: The figure shows the total molecular gas mass plotted against the
SFR of host galaxies. The dashed line and orange region show the best-fit line
and 1$\sigma$ confidence interval for BCGs, respectively. The dashed-dotted
line and the green shaded region show the same for all Fluetsch galaxies. All
symbols are as in figure 8.

Figure 12 shows the relationship between the total molecular gas mass and the
star formation rate for BCGs and the Fluetsch galaxies. Two clear trends are
seen in Figure 12. Linear regressions of the form log SFR ($M_{\odot}$ yr-1) =
$a$ log $\frac{M_{\rm tot}}{10^{9}M_{\odot}}$ $+$ $b$ were fitted separately
to the data for the Fluetsch galaxies and BCGs. The values of parameters ($a$,
$b$, $\Delta a$, $\Delta b$) for BCGs and Fluetsch galaxies are (1.13, 0.23,
0.26, 0.24) and (1.66, 0.67, 0.43, 0.23), respectively. $\Delta a$ and $\Delta
b$ are the 1$\sigma$ errors. The best fit-lines for BCGs and Fluetsch galaxies
are shown by the dashed and dashed-dotted lines, respectively.

The values of the parameter $a$ indicate the two populations have different
slopes. BCGs have a lower total star formation rate per unit molecular gas
mass compared to galaxies from the Fluetsch sample. The segregation becomes
strikingly apparent for total molecular gas masses greater than $\sim$109
$M_{\odot}$. To test the statistical significance of the difference observed
between the two best-fit lines, we conducted a two-sample t-test with a null
hypothesis that the coefficients of the two linear regressions are equal. The
t-statistic for the slope is 5.6 with a p-value smaller than 0.005. Similarly,
the t-statistic for the intercept is 5.5 with a p-value smaller than 0.005.
Therefore, we can confidently reject the null hypothesis. The observed
difference between BCGs and Fluetsch galaxies in the relationship between
their SFR and $M_{\rm tot}$ in log space is statistically significant. Given
the small sample size, we cannot determine whether the separate trends are due
to selection bias.

To further investigate these potentially interesting trends, we compared the
star formation in BCGs and galaxies from Fluetsch with the Kennicutt-Schmidt
(KS) relation. KS relates the star formation rate surface density
($\Sigma_{\rm SFR}$) to the total (H I+H2) cold gas surface density
(Kennicutt, 1998). The KS relation is well-characterized, fits a broad range
of galaxy classes, and scales non-linearly as $\Sigma_{\rm SFR}$ $\propto$
$\Sigma_{\rm HI+H_{2}}^{1.4}$.

We compared the SF law in BCGs and Fluetsch galaxies with the KS relation. We
adopted star formation rates in BCGs listed in Table 1, and the remaining star
formation rates were taken from Fluetsch et al. (2019). In most Fluetsch
galaxies, star formation occurs in the circumnuclear region within $\sim$1
kpc. Star formation rates in galactic disks typically lie between 1–20
$M_{\odot}$ yr-1 (Kennicutt, 1998). Therefore, we have adopted the area of a 1
kpc radius region around the nucleus to estimate gas and star formation
surface densities for Fluetsch galaxies with star formation rates greater than
20 $M_{\odot}$ yr-1. For the remaining galaxies from the Fluetsch sample, we
obtained their angular diameters from NED333https://ned.ipac.caltech.edu/. We
used blue band diameters from the RC3 - Third Reference Catalog of Bright
Galaxies (Corwin et al., 1994) and estimated areas assuming elliptical shape.
In eight BCGs, the areas of the star formation regions were estimated using
UV-band images from the Hubble Telescope. The star formation regions in these
BCGs are confined within roughly 3 kpc to 10 kpc of the center. Star formation
in Phoenix and Perseus extend up to $\sim$25 kpc in filamentary structures.
For the remaining BCGs, we adopted a conservative area of the inner 3 kpc
region centered on the nucleus. In some BCGs (e.g., Abell S1101, RXCJ1539.5,
PKS0745), molecular gas is more extended than the star forming region.
Therefore, the molecular gas surface densities may be overestimated by a
factor of $\sim$2 in these systems.

Figure 13 shows the star formation law in BCGs and Fluetsch galaxies when only
molecular gas surface density is used for all galaxies. The best-fit relation
for all galaxies is given by:

$\Sigma_{\rm SFR}=10^{-3.09\pm 0.49}\Sigma_{\rm H_{2}}^{1.34\pm 0.17},$ (2)

where, $\Sigma_{\rm SFR}$ is in $M_{\odot}$ yr-1 kpc-2 and $\Sigma_{\rm
H_{2}}$ is molecular gas surface density in $M_{\odot}$ pc-2. The overall
distribution appears consistent with the KS-relationship. The dispersion
around the KS relation is larger than the observational uncertainties alone,
implying real variance from galaxy to galaxy. Some of this variance may be
attributed to variations in the $X_{\rm CO}$ factor used to convert CO surface
brightness to molecular gas mass (see Kennicutt & Evans, 2012, for review, and
references therein). Most BCGs fall into the intermediate density star
formation regime occupied by normal spiral disks galaxies and the inner
regions of the Milky Way. The star formation surface densities of most
Fluetsch galaxies are higher than the BCGs and are typical of those observed
in circumnuclear starbursts and ULIRGs. This is to be expected as the Fluetsch
sample contains several ULIRG and starburst systems.

Figure 13: The figure expresses the relationship between the molecular gas
surface density and the SFR surface density for BCGs and FL galaxies. The grey
line shows the global Kennicutt-Schmidt relation (Kennicutt, 1998). Symbols
are as in figure 8.

The Fluetsch galaxies lie systematically above the KS relation, such that they
have higher than normal star-formation-rate surface densities compared to
their molecular gas mass surface densities. Whether this departure is real or
due to measurement bias is unclear. The KS relation is properly characterized
as the sum of the atomic and molecular gas surface densities. We consider here
only the molecular gas mass because H I emission is rarely observed in BCGs,
whose cold gas reservoirs are likely dominated by molecular gas with H2/H I
ratio unity or above (Babyk et al., 2021).

Why BCGs have lower star formation rates per molecular gas mass compared to
the Fluetsch galaxies is unclear. The location of the Fluetsch galaxies above
the KS law while the BCGs follow the law suggests the Fluetsch galaxies have
higher than normal star formation rate densities. The BCGs are outwardly
normal. This difference may be related to the fact that half of the total
molecular gas in BCGs is found in off-nuclear filaments and is rarely seen in
circumnuclear disks (Russell et al., 2019; Olivares et al., 2019). The
filaments, being dynamically disordered and perhaps younger than disk gas, may
be less prone to gravitational instabilities leading to star formation. The
magnetic field may further reduce fragmentation. Mixing with hot surrounding
gas and magnetic reconnection may reheat the cold filaments before star
formation can ensue (Fabian et al., 2011; Churazov et al., 2013). The gas in
the nuclear regions is also dynamically disturbed in many BCGs. The lack of
ordered structure in the nucleus suggests that the gas is dynamically young or
is continually destroyed and reformed within the central galaxy, preventing it
from being consumed by star formation on short timescales. Perhaps one more of
these factors reduces the efficiency of star formation per unit local free-
fall time in BCGs compared to AGN, composite and starburst galaxies. These
suggestions are speculative and merit rigorous treatment in the future.

##  9 Fate of the Outflowing Gas

The final disposition of the outflowing gas depends on many factors including,
the loading factor, the speed of the flow relative to the escape speed, and
the rates of formation and destruction of molecular clouds. Figure 11
indicates broad similarity across all feedback modes in nearby AGN, albeit
with a large scatter. Loading factors near and above unity indicate that winds
are as effective or more effective than star formation at depleting the gas
supply.

The degree to which star formation is delayed or suppressed depends largely on
whether the molecular gas escapes the galaxy. Fluetsch found that in systems
with the highest outflow speeds, only a few percent of the molecular gas
escapes entirely. In most instances escape is negligible. If the currently
outflowing gas returns to the galaxy in molecular form, it would be available
for future star formation. This scenario is almost certainly true in BCGs,
where molecular gas velocities are low compared to both the free-fall speeds
and escape speeds (Russell et al., 2017b, a, 2019). This is compounded by ram
pressure forces on the gas by the surrounding hot atmosphere. The emerging
picture is a fountain where hot and possibly cold molecular gas is lifted
behind the bubbles. The gas eventually cools. Most will likely return to the
galaxy, but some molecular gas may be destroyed. The details are unclear.

To compare the molecular gas speeds to the escape speeds, we adopt the
Hernquist profile to model a galaxy’s gravitational potential. The estimated
escape velocities are found using the following formula:

$v_{\rm esc}=\sqrt{\frac{2GM}{r+a}}$ (3)

where $a$ is the scale radius, which is related to the effective radius as
$R_{e}\sim 1.8153a$, and $M$ is the stellar mass of the galaxy taken from
table 1. To obtain an approximation to the escape velocity for the Phoenix
cluster BCG, we adopt the value 17 kpc from McDonald et al. (2012). For the
remaining galaxies, we use an empirical relation between effective radius and
stellar mass for local elliptical galaxies from McIntosh et al. (2005) to
determine $R_{e}$. This method overestimates the effective radius for Phoenix
by $\sim$65 percent. Therefore, we adjusted effective radii to take this
factor into account. The escaped gas was defined as the molecular gas with a
velocity greater than the escape velocity of the galaxy. To estimate the
escaped gas fraction, we integrated the spectrum above the positive and below
negative escape velocity and compared it to the total integrated intensity of
the flow.

We found that for all BCGs the escaped gas fraction is negligible. The high-
velocity gas in Abell 1664 has sufficient velocity to escape the central
region of the BCG. But it lies well below the escape velocity of the central
cluster halo and will likely return.

As the gas cannot leave the galaxy, it will likely return. Thus some of the
molecular gas in BCGs is almost certainly flowing inward. Discussed in section
4.2, it is extremely difficult to distinguish an outflow from an inflow with
gas seen in emission. However, molecular gas is seen in absorption against the
nuclear continuum of some systems has revealed both in- and out-bound
molecular gas with velocities between $-45~{}\rm km~{}s^{-1}$ and $283~{}\rm
km~{}s^{-1}$ (David et al., 2014; Tremblay et al., 2016; Rose et al., 2019b).
The absorption velocities are broadly consistent with the molecular fountain
model.

##  10 Concluding Remarks

We have shown that mechanical energy released by radio jets can have a far
more dramatic impact on galaxies compared to nuclear radiation (QSOs) and
winds in contemporary active galaxies. This is also true in ancient massive
galaxies lying at high redshift. The relative impact of radio-mechanical
feedback, characterized here as the lifting factor, is qualitatively similar
to the situation in high redshift radio galaxies experiencing both quasar and
radio-mechanical feedback. In a study of 24 radio galaxies lying at $z\sim 2$,
Nesvadba et al. (2017) examined the energy and momentum imparted on the
surrounding ionized gas by star formation, radio jets, and nuclear AGN. They
found that while starburst winds play a minor role, radio jets are generally
more effective than quasars at powering gas motions.

Likewise, Kellermann et al. (2016) found that 20% of contemporary quasars
lying between $z=0.2-0.3$ are radio-loud, double-lobed systems, akin to those
at higher redshift. Furthermore, Jarvis et al. (2019) noted that compact radio
jets with relatively modest radio luminosities hosted by some quasars lying
within the Kellermann et al. (2016) redshift range are interacting with and
driving outflows of nebular gas. Assuming radio jets are operational during
20% of the life of a quasar, as the Kellermann et al. (2016) results may
imply, their large lifting factors indicate that radio-mechanical feedback
would have a significant effect on the evolution of their host galaxies.

A difference between the Nesvadba study and ours apart from epoch is the
ability to examine the three forms of feedback occurring simultaneously in the
same galaxies. While this approach is rarely possible in the contemporary
Universe, some central cluster galaxies are simultaneously experiencing
powerful nuclear AGN, starbursts, and radio-mechanical feedback. For example,
IRAS 09104, is a radio galaxy with large X-ray cavities surrounding a quasar
host lying at $z=0.44$. Its quasar power exceeds its radio-mechanical cavity
power by more than an order of magnitude. Yet its cavities are driving $\sim
4.5\times 10^{10}~{}\rm M_{\odot}$ of molecular gas out of the central galaxy
with little help from the quasar (see O’Sullivan et al., 2021). The same is
true for the iconic Phoenix cluster central galaxy (see Russell et al., 2017b)
where radio-mechanical feedback is the primary source of energy and momentum
input to its $\sim 10^{10}~{}\rm M_{\odot}$ well of molecular gas. We now
understand that this feedback mechanism has been operating effectively through
much of the history of central cluster galaxies (Ma et al., 2013; Hlavacek-
Larrondo et al., 2015) and a broader spectrum of massive galaxies (Nesvadba et
al., 2017).

Powerful radio AGN are found commonly in massive giant ellipticals (Best et
al., 2003, 2005; Heckman & Best, 2014) often hosting dense, high pressure
atmospheres. Radio AGN are also found in giant ellipticals in less dense
atmospheres capable of driving both HI and molecular flows (Morganti et al.,
2005). In these systems the radio energy may be a significant fraction of the
thermal energy in their atmospheres, indicating that the radio jets are
heating the atmospheres and driving away hot gas from their elliptical galaxy
hosts (Webster et al., 2021b; Morganti et al., 2021).

How far down the radio luminosity function radio-mode feedback is affecting
galaxy evolution is unknown. Radio surveys such as NVSS and First detect
primarily low radio luminosity galaxies (Tadhunter, 2016). But even low power
radio galaxies trace much higher mechanical powers (Bîrzan et al., 2008;
Croston et al., 2018) and presumably higher lifting factors. Recent
observations suggest that the mechanical feedback could be significant in 10%
of Seyfert galaxies (for example, Webster et al., 2021a) whose hosts are
spiral rather than elliptical galaxies. Even dwarf galaxies experience AGN
feedback (Manzano-King et al., 2019), some driven by mechanically-powerful
radio jets (Mezcua et al., 2019; Davis et al., 2022).

The upshot here is that when active, radio jets can strongly influence the
evolution of galaxies from their nascency to mature contemporary galaxies. The
systems studied here do not represent the general population of galaxies.
Instead, they are snapshots of an active period through which most massive
galaxies transit. It would be premature to draw broad conclusions about the
influence of radio-mode feedback on the general population of galaxies and
their nuclear black holes. This must await further investigations of large
samples of galaxies observed across the electromagnetic spectrum from radio to
X-rays.

##  11 Summary

Molecular gas properties are examined in 14 active galaxies (BCGs) centered in
clusters with cooling atmospheres. The molecular gas properties and power
output from AGN and star formation were compared to 45 local active galaxies
compiled by Fluetsch et al. (2019). Our results are summarized as follows:

$\bullet$ BCGs centered in cooling atmospheres contain $\sim 10^{8}$
$M_{\odot}$ to upwards of 1010 $M_{\odot}$ of molecular gas. Thirty to seventy
percent of the gas lies outside of the nucleus in extended, filamentary
structures that appear to be moving relative to the BCG. Gravitationally
stable structures, such as large-scale disks, are rare (Olivares et al., 2019;
Russell et al., 2019). In contrast, only a few to ten percent of the total
molecular gas mass in contemporary AGN and starburst galaxies examined by
Fluetsch et al. (2019) is flowing inward or outward. The remainder presumably
lies in disks or other stable structures.

$\bullet$ Molecular clouds surround or lie beneath X-ray cavities inflated by
radio jets in many systems. The clear association of molecular clouds with
X-ray cavities in systems such as the Perseus cluster (NGC 1275), the Phoenix
cluster, Abell 1835 and others indicate that molecular clouds are being lifted
outward, clearing the nucleus of gas. Molecular clouds may also be condensing
from atmospheric gas lifted in the updrafts of the rising radio bubbles
(cavities). The molecular mass lifted by radio bubbles can exceed the masses
of gas flows studied by Fluetsch et al. (2019) by factors of 10–100.

$\bullet$ Radio bubbles are able to lift molecular material to altitudes of
$\sim 10$ kpc and beyond, with flow sizes on average $\sim$10 times larger
compared to active galaxies compiled in Fluetsch et al. (2019). Cluster
outflows tend toward lower velocities than those in the comparison sample.
However, their momentum fluxes are, on average, an order of magnitude larger.

$\bullet$ We introduce the lifting factor, a parameter that is the product of
the mass and size of the molecular flow divided by the driving power. The
driving power is characterized by the measured AGN power or the starburst
power. This parameter indicates that radio-mode feedback is vastly more
capable of driving large gas masses to higher altitudes per unit driving power
than other active systems. This result is consistent with similar measurements
for powerful radio galaxies at redshifts beyond 2 (Section 11).

$\bullet$ Loading factors vary broadly in BCGs from $\sim 0.1-100$. This range
is similar to the active galaxies compiled by Fluetsch et al. (2019). Only a
few percent at most of the AGN mechanical power in BCGs is transmitted to the
molecular gas. This fraction is also similar to the systems in Fluetsch et al.
(2019).

$\bullet$ The star formation rate per gram of molecular gas in BCGs is five to
ten times lower than the most massive systems with AGNs and starbursts studied
by Fluetsch et al. (2019). As molecular gas is closely associated with star
formation, this deficit indicates, tentatively, that star formation in BCGs is
suppressed compared to other systems.

$\bullet$ Molecular cloud velocities in BCGs lie well below escape speeds.
Similar to the systems in Fluetsch et al. (2019), little or no molecular gas
is able to escape the galaxy. Radio-mechanical feedback is likely driving a
fountain of hot and cold gas that eventually returns to the central galaxy.
This process may delay or suppresses star formation relative to other active
galaxies.

Radio-mechanical feedback (radio-mode) is more complex than is commonly
assumed. Radio sources not only heat their surrounding atmospheres but also
drive molecular clouds out of their host galaxies. In some instances, radio-
mode feedback may promote the condensation of molecular clouds from their hot
atmospheres that would sustain long-term feedback needed to prevent the
outsized growth of galaxies. Radio lobes are more capable of driving large
masses of molecular gas to higher altitudes than contemporary starbursts and
quasars. These effects may be most prominent in dense cluster atmospheres
where the cooling gas supply is plentiful and buoyancy forces are large.
Radio/X-ray bubbles encompass large volumes of gas that vastly exceed other
AGN, where buoyancy effectively drives gas outward long after the AGN has
ceased to power the radio lobes.

Understanding when molecular clouds are inbound or outbound is fraught with
uncertainty. That most molecular gas lies off the nucleus shows much of it is
outbound. But because their speeds lie well below the escape speed, some of
the gas must be inbound. Our results will thus be biased to some degree. But
quantities such as the lifted mass will be affected by no more than a factor
of two and will not qualitatively affect our conclusions. Lacking a complete
sample, we are subject to selection bias, which likely will affect properties
of lower molecular gas mass systems that are undersampled. Selection biases
will be addressed as sample sizes increase and unbiased samples become
available.

## Acknowledgements

BRM acknowledges support from the Natural Sciences and Engineering Council of
Canada. ACE acknowledges support from STFC grant ST/P00541/1. HRR acknowledges
support from an STFC Ernest Rutherford Fellowship and an Anne McLaren
Fellowship. We thank the referee for their comments which significantly
improved the manuscript. This paper makes use of the following ALMA data:
ADS/JAO.ALMA#2012.1.00837.S, ADS/JAO.ALMA#2011.0.00374.S,
ADS/JAO.ALMA#2015.1.00623.S, ADS/JAO.ALMA#2012.1.00988.S,
ADS/JAO.ALMA#2012.1.00837.S, ADS/JAO.ALMA#2013.1.01302.S,
ADS/JAO.ALMA#2016.1.01269.S, ADS/JAO.ALMA#2011.0.00735.S,
ADS/JAO.ALMA#2015.1.01198.S. The National Radio Astronomy Observatory is a
facility of the National Science Foundation operated under cooperative
agreement by Associated Universities, Inc. The scientific results reported in
this article are based in part on data obtained from the Chandra Data Archive.

This research made use of python (Van Rossum & Drake, 2009), Astropy (Astropy
Collaboration et al., 2013, 2018), matplotlib (Hunter, 2007), numpy (van der
Walt et al., 2011; Harris et al., 2020), and scipy (Virtanen et al., 2020). We
thank their developers for maintaining them and making them freely available.

## Data Availability

The ALMA data used in this article is publicly available at
https://almascience.nrao.edu/aq/.

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##  Appendix A Description of targets

The molecular gas distribution in BCGs is often clumpy and asymmetric.
Filaments exist in most systems with narrow velocity widths and smooth
gradients. Filaments are often found behind rising X-ray cavities or encasing
the cavities. The fraction of molecular gas in filaments ranges from greater
than 70 per cent to greater than 30 per cent. Most systems exhibit multiple
velocity components in the molecular gas spectra. The spatial extent of the
filaments determines the size of the flow. In the following subsections, we
briefly describe molecular gas morphology and kinematics in each target in our
sample.

Figure 14: ALMA CO integrated intensity map overlaid on HST images (left
column), velocity centroid maps (middle column), and FWHM maps (right column)
for all BCGs in our sample are shown. For RXCJ1539, DSS red band image is
shown. The beam size is indicated as a grey ellipse. The crosses indicate the
location of the AGN in each BCG. The emission region enclosed by black
ellipses or polygons in the right panels of each row indicates the region we
consider as a flow in each BCG. Figure A14 (Continued):  Figure A14
(Continued):  Figure A14 (Continued):

###  A.1 2A0335+096

The molecular gas distribution in 2A0335+096 has a filamentary structure with
the filament extending SE to NW of the BCG. There is a large reservoir of
molecular gas south of the BCG at a velocity of $-240$ km s-1. The north
filament appears discontinuous and extends $\sim$ 3 arcseconds (2 kpc) and has
an average velocity of 200 km s-1. The PV diagram along the BCG does not show
a sign of a rotating disk (Vantyghem et al., 2016, see Fig. 11). We consider
both the N filament and the southern molecular gas as a flow in this BCG.

###  A.2 Abell 1664

In Abell 1664, the molecular gas is distributed in a high velocity filament E
of the BCG center, a blob to the north of the BCG and another blob coincident
with the AGN. The E filament extends up to $\sim$6.37 kpc from the BCG center,
whereas the north blob is $\sim$5.8 kpc away from the BCG in projection. The
high velocity filament has a velocity of 590 km s-1, whereas the blob co-
incident with the AGN could be part of a molecular gas disk. We consider the
high velocity filament as the flow as shown by elliptical region in Fig 14.
Russell et al. (2014, Fig. 4 and 5) show PV diagram along the flow filament
and across the BCG. There is a possibility of molecular gas disk across the
BCG, the flow filament has a different velocity structure than the central gas
reservoir and is not part of the potential molecular gas disk.

###  A.3 Abell 1795

The molecular gas in A1795 is distributed in two filaments extending to the
north and southwest of the BCG centre, and in a central region around the
BCG’s nucleus. The north filament is curled up around a radio jet (see Russell
et al. (2017a)) with a projected length of 4.2 arcsec (5.1 kpc) and the SW
filament is more extended at a projected length of 6 arcsec (7.3 kpc). Both
filaments have a distinct velocity structure compared to the central clump,
which has a velocity centroid of $\sim$1 km s-1 with respect to systemic
velocity. The SW filament has a shallower velocity gradient with velocity
centroid lying between $-$80 to $-$180 km s-1 from the tail-end to the base,
respectively. In the N filament, the velocity gradient increases from $-$270
km s-1 near the edge to 0 km s-1 at the base. The two filaments are considered
as flows in our analysis. The PV diagrams presented in Russell et al. (2017a)
show no evidence of a rotating disk. We consider the N and the S filament as
flows in our analysis.

###  A.4 Abell 1835

The molecular gas in Abell 1835 is mostly located within 3 kpc of the BCG
centre and peaks within a few tens of km s-1 of the galaxy’s systemic
velocity. However, there are filaments extending $~{}\sim$12 kpc towards N and
SE of the nucleus behind X-ray cavities. The N filament has a narrow component
with a velocity centre close to the systemic velocity and a broad component
with a velocity centre at $-$230 km s-1. The SE filament has an average
velocity shift of 40 km s-1 and another faint $\sim$200 km s-1 component. The
PV along the BCG does not show a clear sign of a molecular gas disk (see Fig.
A15), although most of the molecular gas in the centre is unresolved. The
filaments appear to be part of a bipolar outflow. These regions we consider as
flows are shown in Fig. 14.

###  A.5 Abell 2597

In Abell 2597, most of the molecular gas is distributed in bright filaments
draped around radio bubbles close to the nucleus and in a fainter, elongated
filament extending $\sim$15 kpc south of the BCG. The brighter filaments are
$\sim$6.3 kpc long and contain $\sim$50 per cent of molecular gas. Abell 2597
has a complex velocity structure. Filaments have a shallow constant velocity
gradient between velocities 60 to 40 km s-1. In the innermost 2 kpc, the
gradient changes from $-$120 km s-1 to 120 km s-1. The system also has
absorption features at 240, 275, and 335 km s-1 (Tremblay et al., 2016), which
indicates that the gas is moving at fast speeds close to the nucleus. The
regions considered as flows are shown in Fig. 14. See Tremblay et al. (2018)
for PV diagrams of the filaments and the central gas structure.

###  A.6 PKS0745+091

The molecular gas in PKS0745+091 is distributed in three filaments extending
north, southwest (SW), and southeast (SE) directions, respectively. The
northern filament is most extended at 5.73 kpc followed by SW and SE filaments
at projected lengths of 3.6 and 2.7 kpc, respectively. The north and SW
filaments appear to be trailing behind an X-ray cavity. The PV diagrams reveal
a a velocity gradient from $-$150 to 50 km s-1 (see Fig. 8, Russell et al.,
2016). We consider these filaments as flows in this BCG and are shown in Fig.
14.

###  A.7 Phoenix

Phoenix is one of the most luminous galaxy clusters. The BCG in the Phoenix
cluster is massive and has an ongoing star formation at a rate of 610 M⊙
year-1. It contains a large amount of molecular gas. The molecular gas in
Phoenix is distributed around the centre and in three filaments extending
20-24 kpc to N, SE, and S of the BCG centre. Filaments show a smooth velocity
gradient with narrow FWHM ($<$250 km s-1), indicating ordered flow motion
(also see Fig. 5, Russell et al., 2017b). The velocity centroids vary in the
range between 280 km s-1 to 0 km s-1 in NW filament and $\sim$250 km s-1 to
$\sim$570 km s-1 in the SE filament and 250 to 50 km s-1 in the S filament.
These filaments shown in Fig. 14 are considered flows in this BCG.

###  A.8 RXCJ1504

RXCJ 1504 has a central clump and a disturbed, clumpy filament extending
radially outward to the W the BCG centre. The projected size of the filament
is 18 kpc from the BCG centre, and it contains roughly 30 per cent of the
total molecular gas in the system. It has a smooth velocity gradient with
velocities between 90 km s-1 near the tail to $-$210 km s-1 at the base with
an FWHM less than 100 km s-1 throughout most of the filament. We consider this
filament as a flow in this BCG which is shown as a region enclosed by a
polygon in Fig. 14.

###  A.9 RXCJ0821+0752

The molecular gas in RXCJ0821+0752 is distributed in two major clumps. The
first clump, which is $\sim$ 3 kpc north of the BCG centre, contains the
majority ($\sim$60%) of molecular gas. It has a velocity centroid of 24 km s-1
compared to the systemic velocity. The secondary clump is located 3 kpc west
of the first clump and has a velocity centroid of $\sim$20 km s-1. The entire
molecular gas distribution is offset from the BCG. The position velocity
diagram doesn’t show any sign of rotation (see Fig. A15). The region
considered as a flow is shown in Fig. 14.

###  A.10 NGC 5044

Most of the molecular gas in NGC 5044 is distributed around the central
galaxy, and some of it is detected as individual clouds of molecular gas NE,
NW and E of the BCG. There is a very high-velocity cloud blueshifted to
velocities of $\sim-500$ km s-1 close to the nucleus. The PV diagram across
the BCG doesn’t show any sign of a rotating disk. We consider the blueshifted
components and the clouds of molecular gas as parts of flow in this galaxy.
They are shown in Fig. 14.

###  A.11 NGC 4696

The molecular gas in NGC 4696 is distributed in a large clump roughly
cospatial with the BCG nucleus and in a $\sim$4 kpc long curved filament S of
the BCG. The central gas reservoir has velocities within 50 km s-1 of the BCGs
systemic velocity, whereas the filament has a velocity gradient from 120 km
s-1 in the outer parts to $\sim$350 km s-1 in the inner part. The PV diagrams
do not show any sign of a rotating disk. This extended filament is considered
a flow in NGC 4696.

###  A.12 RXCJ1539

RXCJ1539 has two large filaments to the E and W of the BCG and a central clump
of molecular gas co-spatial with the BCG. The E and W filaments extend out to
24 and 27 kpc, respectively. Another small spur of molecular gas is detected
to the N of the BCG extending out to 18 kpc. The W filament has a roughly
constant velocity of $\sim$50 km s-1. The E filament is mostly redshifted to
velocities of up to 200 km s-1, but also contains a blueshifted component at a
velocity of $-$210 km s-1. The N filament, however, is entirely blueshifted to
velocities of up to $-$285 km s-1. The inner 7 kpc radius region around the
nucleus has velocities lying between $\sim$0 to 50 km s-1. It is no
unambiguous evidence of smooth gradient or ordered motion indicating a
rotating disk in the PV diagrams shown in Fig. A15, as the central reservoir
is unresolved. The three regions we consider as a flow in this BCG are shown
in Fig. 14.

###  A.13 AS1101

The molecular gas in AS1101 appears as a single extended filament to the N of
the BCG. The filament has a smooth velocity gradient from 140 km s${-1}$ close
to the BCG to 40 km s-1 in the outer parts. It is extended out up to 8.23 kpc
N of the BCG. Some blueshifted emission is detected to the W of the BCG at
velocities of $-$130 km s-1. The PV diagrams show a smooth velocity gradient
across the filament. The filament is indistinguishable from the circumnuclear
gas in this data. The region we consider as a flow is shown in Fig. 14.

Figure A15: The position-velocity diagrams of BCGs in our sample. The left
panel shows an integrated intensity image where the magenta lines indicate the
axis used to make PV diagrams. The right panels show the PV diagrams
corresponding to the labels shown in the left panel when PV diagrams along
multiple axes are shown. The dashed vertical lines show the position of the
BCG centre. In all images, the east is to the left and the north is up. The
position velocity diagrams do not show any sign of symmetric ordered motion
about the nucleus in filaments in any of the BCGs. Figure A15 (Continued):

##  Appendix B Flow properties with adjusted velocities for Fluetsch galaxies

In Fig. 1, we showed that applying the Fluetsch method to BCGs overestimates
flow velocities by a factor of two. Assuming that a similar systematic
difference in velocities applies to galaxies in the Fluetsch sample, we divide
their outflow velocities by two to estimate new velocities. It results in a
reduction of molecular outflow rates by half, kinetic energies of outflows by
a factor of 8 and momenta by a factor of 4. Figure A18 shows all the relevant
figures where outflow properties for galaxies in the Fluetsch sample are
calculated using new velocities. It does not affect correlation coefficients
for Fluetsch outflows. The average flow velocities of BCGs lie only 30% below
average flow velocity in AGN hosting galaxies and $~{}\sim 20$% faster than
flow velocities in starburst galaxies. The flow momenta for AGN, composite and
starburst galaxies are 3.3$\times$1049, 5.2$\times$1048 and 8.6$\times$1048 g
cm s-1, respectively. They are 10–15 times smaller than flow momenta in BCGs.
The loading factors do not change significantly. More galaxies lie within
theoretical maximum limits for energy and momentum conserving flows. At the
same time, the ratio of kinetic power of the flow to the power of the driving
mechanism falls below 0.05% for some galaxies, indicating a weak coupling. The
lifting factor remains unaffected, as it does not depend on the flow velocity.
Thus, qualitatively, these results are consistent with the results presented
in the sections above.

Figure A18: We show figures 9, 6, 8, 2, 7, 11 from top left to bottom right
panels, respectively, with the molecular flow rate, momentum, kinetic energy
of the flow calculated with adjusted flow velocities for fluetsch galaxies as
described in Appendix B.