# AGN Feeding and Feedback in M84: From Kiloparsec Scales to the
Bondi Radius

C. J. Bambic1, H. R. Russell2, C. S. Reynolds3,4,5, A. C. Fabian3, B. R.
McNamara6,7 and P. E. J. Nulsen8,9 1 Department of Astrophysical Sciences,
Peyton Hall, Princeton University, Princeton, NJ 08544, USA
2 School of Physics, Astronomy, University of Nottingham, University Park,
Nottingham NG7 2RD, UK
3 Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK
4 Department of Astronomy, University of Maryland, College Park, MD
20742-2421, USA
5 Joint Space Science Institute (JSI), College Park, MD 20742-2421, USA
6 Department of Physics and Astronomy, University of Waterloo, 200 University
Avenue West, Waterloo, ON N2L 3G1, Canada
7 Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, ON N2L
3G1, Canada
8 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138, USA
9 ICRAR, University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009,
Australia E-mail<EMAIL_ADDRESS>
###### Abstract

We present the deepest Chandra observation to date of the galaxy M84 in the
Virgo Cluster, with over 840 kiloseconds of data provided by legacy
observations and a recent 730 kilosecond campaign. The increased signal-to-
noise allows us to study the origins of the accretion flow feeding the
supermassive black hole in the center of M84 from the kiloparsec scales of the
X-ray halo to the Bondi radius, $R_{\rm B}$. Temperature, metallicity, and
deprojected density profiles are obtained in four sectors about M84’s AGN,
extending into the Bondi radius. Rather than being dictated by the potential
of the black hole, the accretion flow is strongly influenced by the AGN’s
bipolar radio jets. Along the jet axis, the density profile is consistent with
$n_{e}\propto r^{-1}$; however, the profiles flatten perpendicular to the jet.
Radio jets produce a significant asymmetry in the flow, violating a key
assumption of Bondi accretion. Temperature in the inner kiloparsec is
approximately constant, with only a slight increase from 0.6 to 0.7 keV
approaching $R_{\rm B}$, and there is no evidence for a temperature rise
imposed by the black hole. The Bondi accretion rate $\dot{M}_{\rm B}$ exceeds
the rate inferred from AGN luminosity and jet power by over four orders of
magnitude. In sectors perpendicular to the jet, $\dot{M}_{\rm B}$ measurements
agree; however, the accretion rate is $>4\sigma$ lower in the North sector
along the jet, likely due to cavities in the X-ray gas. Our measurements
provide unique insight into the fueling of AGN responsible for radio mode
feedback in galaxy clusters.

###### keywords:

X-rays: galaxies: clusters — galaxies: clusters: M84 — intergalactic medium

##  1 Introduction

Accretion onto active galactic nuclei (AGN) at the centers of massive
elliptical galaxies fuels AGN feedback in clusters of galaxies. The
gravitational potential energy released from plasma flowing onto these
supermassive black holes (SMBHs) powers jets of relativistic particles which
sculpt the surrounding intracluster medium (ICM). In cool-core clusters where
the central cooling time of the hot ($10^{7}-10^{8}$ K) ICM is $\lesssim$ Gyr,
thermalization of the jet kinetic energy provides the heating necessary to
balance radiative cooling, maintaining cluster atmospheres in their observed
quasi-thermal equilibrium and averting a “cooling catastrophe” (Fabian, 1994;
McNamara & Nulsen, 2007; Fabian, 2012).

Deep ($\gtrsim$100 kiloseconds), spatially-resolved Chandra X-ray Space
Telescope observations of nearby galaxy clusters such as Perseus and Virgo
have revealed the signatures of this feedback process: cavities or bubbles
carved out of the ICM by jets (McNamara et al., 2000; Churazov et al., 2001);
weak shocks, ripples, and waves emanating from newly formed bubbles (Sanders &
Fabian, 2007, 2008; Forman et al., 2007); bright filaments formed by gas
cooling around these cavities (Fabian et al., 2003); and turbulent
fluctuations stirred by the buoyant rise of bubbles through clusters (Churazov
et al., 2004; Zhuravleva et al., 2014; Hitomi Collaboration et al., 2016;
Simionescu et al., 2019). Yet, while the X-ray morphology of clusters has
provided insight into how AGN shape their environments on $10\rm{s}-100\rm{s}$
of kiloparsecs (kpc) scales, understanding the connection between AGN and the
sub-kiloparsec scale accretion flows which power them remains a critical
uncertainty in the paradigm of AGN feedback.

Even the basic energetics of large-scale black hole “feeding” is an open
problem (see Abramowicz & Fragile 2013 for a review). In the standard
paradigm, material within the SMBH’s sphere of influence, the Bondi radius
($R_{\rm B}=2GM_{\rm BH}/c_{s}^{2}$ where $M_{\rm BH}$ is the black hole mass
and $c_{s}$ is the speed of sound well beyond $R_{B}$), is destined to either
reach the hole or race away in an outflow. Ionized gas pierces the sphere of
influence at a rate $\dot{M}_{\rm B}$ en route to the hole, where this plasma
is consumed at a rate $\dot{M}$. A fraction $\eta$ of the rest mass power
$\dot{M}c^{2}$ is released by the accretion flow in the form of radiation and
outflows—winds or jets—such that the total power (radiative + outflow) of the
AGN is $L=\eta\dot{M}c^{2}$.

At the largest scales, a gas inflow with accretion rate $\dot{M}_{\rm B}$ is
formed by gas cooling and gravitational infall under the influence of the
combined galactic and SMBH potential (Quataert & Narayan, 2000). The accretion
rate $\dot{M}$ is then influenced by the structure of this inflow: the angular
momentum (Proga & Begelman, 2003) and effective turbulent viscosity (Narayan &
Fabian, 2011) of the gas, and the relative contributions of hot X-ray emitting
plasma (Matteo et al., 2003) vs. cold atomic and molecular gas (Pizzolato &
Soker, 2005) which may “rain down” through the Bondi radius (Gaspari et al.,
2012; Yang & Reynolds, 2016). Magnetic fields certainly complicate this
picture, with the magnetic flux frozen into the flow (Lubow et al., 1994)
competing with dynamo-generated fields (Brandenburg et al., 1995; Brandenburg
& Subramanian, 2005; Blackman, 2012; Liska et al., 2020) to power relativistic
jets (Blandford & Znajek, 1977; Komissarov, 2001; Tchekhovskoy et al., 2010,
2011) and winds (Blandford & Payne, 1982; Proga, 2000), and thereby influence
the value of $\eta$.

The complexities of this inflow determine the state of the resulting accretion
disk around the black hole and the relative contribution of radiation to the
flow’s structure. For the jetted systems of interest in cluster AGN feedback,
the accretion flow is likely radiatively inefficient, forming a virialized,
geometrically-thick advection-dominated accretion flow (ADAF; Ichimaru, 1977;
Rees et al., 1982; Narayan & Yi, 1994, 1995; Quataert & Narayan, 1999), a
convection-dominated accretion flow (CDAF; Quataert & Gruzinov, 2000), or when
the net magnetic flux reaching the hole is large, a magnetically arrested disk
(MAD; Bisnovatyi-Kogan & Ruzmaikin, 1974; Narayan et al., 2003; Igumenshchev,
2008; McKinney et al., 2012; Avara et al., 2016; Marshall et al., 2018;
Ripperda et al., 2022).

While $\dot{M}_{\rm B}$ is crucial in determining accretion flow structure,
measuring this parameter is a major challenge. Because the true $\dot{M}_{\rm
B}$ cannot be measured directly, large-scale black hole feeding is often
interpreted through a steady, spherically symmetric model of accretion, the
Bondi (1952) solution. Within this framework, $\dot{M}_{\rm B}$ for a given
SMBH mass is specified entirely by the gas density and temperature at $R_{\rm
B}$, quantities which in principle can be measured with deep X-ray
observations.

This choice is one of convenience—there are no strong theoretical reasons to
expect the assumptions of the Bondi solution to hold in real systems. However,
some evidence points to the importance of $\dot{M}_{\rm B}$ in setting
feedback power. Allen et al. (2006), using a small sample of X-ray
observations of nearby elliptical galaxies, found an apparent correlation
between Bondi accretion rate and AGN jet power, as measured from the enthalpy
of jet-blown cavities. This method for inferring jet power is subject to
significant uncertainties, e.g. projection effects and the assumption of
subsonic inflation. Indeed, a follow-up study by Russell et al. (2013) using a
larger sample of elliptical galaxies found a less significant correlation.

A direct correlation between Bondi accretion rate and AGN feedback power has
interesting consequences. The correlation may imply a universality in the
radiatively inefficient accretion flows (RIAFs) which power AGN in early type
galaxies, with $\dot{M}_{\rm B}$ serving as the crucial parameter for
regulating power from radiative ($L_{\rm Rad}$) and jet ($L_{\rm Jet}$)
feedback on $\sim$Gyr timescales. In addition, the correlation could be
leveraged in sub-grid models for galaxy formation, where feedback power from
unresolved AGN must be tuned based on resolvable properties, such as
$\dot{M}_{\rm B}$ (Pillepich et al., 2018). Establishing this correlation
necessitates deep X-ray observations which resolve the density and temperature
at $R_{\rm B}$.

In this paper, we harness the deepest X-ray observations to date of the galaxy
M84 (NGC 4374) to measure the Bondi accretion rate of hot phase ($\gtrsim$ 0.5
keV) gas onto a jetted AGN in an early type galaxy. These measurements are
based on a new Chandra campaign which yielded approximately 730 kiloseconds
(ks) on M84. Combined with legacy data published in Finoguenov & Jones (2001,
2002) and Finoguenov et al. (2008), the observations presented comprise over
840 ks of X-ray data.

M84 is one of only 5 known systems where the Bondi radius can be resolved by
Chandra, despite the observatory’s remarkable sub-arcsecond angular
resolution. The other 4 systems are Sgr A∗ (Baganoff et al., 2003), NGC 3115
(Wong et al., 2014), NGC 1600 (Runge & Walker, 2021), and M87 (Russell et al.,
2015, hereafter HRR15). Even within this small class, M84 stands out. Unlike
Sgr A∗ and NGC 1600, M84 has an X-ray detected AGN. In contrast to NGC 3115, a
Fanaroff & Riley (1974) Type I radio jet is clearly observed in M84. However,
unlike that in M87 which hosts a notably powerful jet, M84’s AGN is not
particularly luminous (more than an order of magnitude dimmer than M87’s AGN)
and our extended campaign caught the SMBH in a relatively quiescent state.
Thus, M84 does not require the same sophisticated treatment of pile-up as was
performed in M87 (HRR15). These factors make M84 an especially useful object
for exploring the interplay of feeding and feedback in elliptical galaxies.

This paper is organized as follows. We describe our data analysis in §2
including data reduction, spectral models for the AGN and galactic gas, and
simulations of the detector point spread function (PSF) used for forward
modelling spectral contamination from the AGN. In §3, we present results:
profiles of gas density, temperature, and metallicity approaching and just
within the Bondi radius, and the measured Bondi accretion rates $\dot{M}_{\rm
B}$ and efficiencies $\eta$. We discuss the implications of our measurements
in §4, and conclude in §5.

##  2 Chandra Data Analysis

M84 is a nearby (luminosity distance $D_{L}=$ 16.83 Mpc; redshift $z=$
0.00327) giant elliptical galaxy (type E1) and satellite member of the Virgo
Cluster of galaxies. The galaxy has been the subject of three separate Chandra
ACIS-S campaigns which together yield $\approx$840 ks of data. While earlier
works by Finoguenov & Jones (2001) and Finoguenov et al. (2008) addressed the
detailed structure of M84 and how the X-ray halo is shaped by feedback, our
ultra-deep campaign is concerned primarily with black hole feeding and gas
structure approaching and just within the Bondi radius of the SMBH.

###  2.1 Data Reduction

This work is a follow-up to a similar analysis of M87 by HRR15. Thus, we
follow the same data reduction procedure.

Figure 1: Left: Merged $0.5-2$ keV image, totaling 798.66 ks of cleaned
exposure time with Obs. ID 5908 excluded (see §2.9). Right: 5 GHz radio
contours (white) as measured by the Very Large Array (VLA) overlaid on X-ray
image. Contours correspond to 12 logarithmically-spaced levels in flux from
$2\times 10^{-3}-0.1$ mJy, and colors denote X-ray counts. The
$\mathcal{H}$-shaped morphology is carved out of the halo by radio jets,
forming bright rims about the radio lobes.

Our data reduction was performed using CIAO version 4.11 and the Calibration
Database (CalDB) 4.8.5, updated November 7, 2019 (Fruscione et al., 2006).
This update followed a major revision to the soft energy response brought
about by contaminant build-up over Chandra’s prolific 23 year lifetime (thus
far). Our long campaign was affected by this contamination, and as we show,
the majority of M84’s galactic gas, especially that approaching the Bondi
radius, is cooler than 1 keV and emitting X-rays within the range of degraded
performance. Given the low temperature of the extended emission in M84, the
calibration of the contaminant build up on Chandra’s optical block filters is
particularly important. We therefore verified that temperature, metallicity
and normalization values measured with the new observations are consistent
with the archival observations, which were taken only a few years after
Chandra’s launch and less affected. Using the chandra_repro routine, we
reprocess our data to produce second-level event files, removing bad pixels
based on the analysis reference data library (ardlib), detecting point sources
using wavdetect, and creating light curves to filter out bad time intervals.
To produce merged images, we assume an exposure correction for each Obs. ID’s
exposure map.

###  2.2 X-ray Morphology

Figure 1 displays a merged $0.5-2$ keV image based on all three campaigns.
Similar images can be found in Finoguenov & Jones (2001) and Finoguenov et al.
(2008) from the first two sets of observations.

Using only limited Chandra data, Finoguenov & Jones (2001) were able to
identify the salient features of the galaxy’s X-ray emission. Instead of a
featureless X-ray halo, M84 hosts depressions in emissivity North and South of
the central AGN, coincident with radio lobes produced by Fanaroff & Riley
(1974) Type I jet activity (Laing & Bridle, 1987). These cavities create an
$\mathcal{H}$-shaped structure in the halo gas, which extends $\approx
150^{\prime\prime}$ (12.2 kpc) from the Northernmost edge of the emission to
the faint rim in the Southwest of the image. The crossbar of the $\mathcal{H}$
spans $\approx 46^{\prime\prime}$ (3.7 kpc) and is approximately aligned with
optical dust lanes (Hansen et al., 1985), although the dust lanes are on a
much larger scale, cutting across the X-ray image.

As argued by Finoguenov et al. (2008), these cavities may actually be
comprised of at least two bubbles each, with bright rims (viewed in
projection) demarcating the bubble boundaries. Indeed, our deep observation is
able to clearly detect a tenuous bubble rim extending toward the Southwest in
the image. The Northern bubble is compressed, likely by the ram pressure of
ICM gas as the galaxy moves through the cluster.

While the bubbles are located just to the North and South of the crossbar, the
jet is aligned with the West filament; the galaxy has drifted over time.
Subsequently, ram pressure has swept the Northern bubble back and “bent” the
radio jet—a signature of radio galaxies moving through clusters (Miley et al.,
1972; Owen & Rudnick, 1976; Begelman et al., 1979; Morsony et al., 2013;
McBride & McCourt, 2014). Intriguingly, the Southern bubble has not been swept
in the same direction. There may be a large-scale shear flow across M84, or
the jet may have reoriented itself over the course of the episodes recorded in
the radio lobes, possibly through precession.

Obs. ID | Date | Exposure | $N_{\mathrm{H}}$ | $\Gamma$ | Flux (2-10 keV) | C-Stat/DOF
---|---|---|---|---|---|---
|  | (ks) |  (1022 cm-2) |  |  (10-13 erg cm-2 s-1) |
803 | 19/05/2000 | 28.47 |  0.23${}_{-0.07}^{+0.09}$ |  1.79${}_{-0.19}^{+0.20}$ |  0.99${}_{-0.12}^{+0.14}$ | 142.6/ 184
5908 | 01/05/2005 | 46.08 |  0.16${}_{-0.03}^{+0.03}$ |  2.03${}_{-0.10}^{+0.10}$ |  1.62${}_{-0.11}^{+0.12}$ | 233.7/ 277
6131 | 07/11/2005 | 40.93 |  0.81${}_{-0.31}^{+0.39}$ |  1.68${}_{-0.34}^{+0.36}$ |  0.63${}_{-0.08}^{+0.10}$ | 146.3/ 165
20539 | 05/04/2019 | 39.54 |  0.16${}_{-0.16}^{+0.29}$ |  1.66${}_{-0.31}^{+0.36}$ |  0.50${}_{-0.07}^{+0.09}$ | 121.8/ 141
20540 | 26/02/2019 | 30.17 |  0.08${}_{-0.08}^{+0.28}$ |  1.78${}_{-0.25}^{+0.38}$ |  0.49${}_{-0.08}^{+0.09}$ | 109.5/ 127
20541 | 10/04/2019 | 11.29 |  0.37${}_{-0.37}^{+0.66}$ |  2.11${}_{-0.80}^{+0.97}$ |  0.46${}_{-0.15}^{+0.26}$ | 42.1/ 55
20542 | 18/03/2019 | 34.61 |  0.005${}_{-0.005}^{+0.28}$ |  1.46${}_{-0.18}^{+0.36}$ |  0.48${}_{-0.08}^{+0.08}$ | 120.9/ 122
20543 | 27/04/2019 | 54.34 |  1.57${}_{-0.52}^{+0.63}$ |  2.95${}_{-0.48}^{+0.53}$ |  0.32${}_{-0.04}^{+0.05}$ | 113.5/ 135
21845 | 28/03/2019 | 27.70 |  0.50${}_{-0.32}^{+0.25}$ |  2.05${}_{-0.42}^{+0.34}$ |  0.47${}_{-0.08}^{+0.09}$ | 118.8/ 113
21867 | 13/03/2019 | 23.63 |  0.36${}_{-0.35}^{+0.53}$ |  2.31${}_{-0.52}^{+0.62}$ |  0.34${}_{-0.07}^{+0.08}$ | 80.0/ 104
22126 | 28/02/2019 | 35.10 |  0.22${}_{-0.16}^{+0.20}$ |  1.80${}_{-0.27}^{+0.29}$ |  0.67${}_{-0.09}^{+0.11}$ | 132.3/ 153
22127 | 02/03/2019 | 22.77 |  0.27${}_{-0.21}^{+0.26}$ |  1.75${}_{-0.31}^{+0.32}$ |  0.85${}_{-0.12}^{+0.14}$ | 93.2/ 136
22128 | 03/03/2019 | 23.75 |  0.37${}_{-0.25}^{+0.32}$ |  1.78${}_{-0.36}^{+0.40}$ |  0.70${}_{-0.11}^{+0.14}$ | 94.6/ 124
22142 | 14/03/2019 | 20.77 |  0.74${}_{-0.47}^{+0.64}$ |  2.53${}_{-0.66}^{+0.74}$ |  0.35${}_{-0.08}^{+0.10}$ | 58.5/ 86
22143 | 16/03/2019 | 22.75 |  0.59${}_{-0.34}^{+0.42}$ |  2.05${}_{-0.42}^{+0.45}$ |  0.70${}_{-0.11}^{+0.13}$ | 111.0/ 123
22144 | 15/03/2019 | 31.75 |  0.09${}_{-0.09}^{+0.19}$ |  2.03${}_{-0.26}^{+0.31}$ |  0.46${}_{-0.07}^{+0.08}$ | 128.0/ 142
22153 | 23/03/2019 | 21.08 |  0.89${}_{-0.41}^{+0.51}$ |  2.58${}_{-0.51}^{+0.56}$ |  0.45${}_{-0.08}^{+0.10}$ | 74.2/ 93
22163 | 29/03/2019 | 35.59 |  0.54${}_{-0.27}^{+0.33}$ |  1.86${}_{-0.32}^{+0.34}$ |  0.65${}_{-0.08}^{+0.10}$ | 125.3/ 153
22164 | 31/03/2019 | 32.63 |  0.59${}_{-0.39}^{+0.42}$ |  1.85${}_{-0.41}^{+0.40}$ |  0.60${}_{-0.08}^{+0.10}$ | 134.7/ 142
22166 | 06/04/2019 | 38.56 |  0.41${}_{-0.24}^{+0.31}$ |  2.31${}_{-0.35}^{+0.39}$ |  0.34${}_{-0.05}^{+0.06}$ | 117.2/ 132
22174 | 11/04/2019 | 49.41 |  0.84${}_{-0.30}^{+0.36}$ |  2.20${}_{-0.35}^{+0.38}$ |  0.52${}_{-0.07}^{+0.08}$ | 129.4/ 160
22175 | 12/04/2019 | 27.20 |  0.49${}_{-0.34}^{+0.41}$ |  1.83${}_{-0.40}^{+0.43}$ |  0.54${}_{-0.09}^{+0.11}$ | 93.0/ 114
22176 | 13/04/2019 | 51.39 |  0.57${}_{-0.20}^{+0.23}$ |  2.21${}_{-0.25}^{+0.26}$ |  0.64${}_{-0.06}^{+0.07}$ | 165.3/ 195
22177 | 14/04/2019 | 36.58 |  0.86${}_{-0.32}^{+0.37}$ |  2.59${}_{-0.38}^{+0.40}$ |  0.49${}_{-0.06}^{+0.07}$ | 100.9/ 147
22195 | 28/04/2019 | 38.07 |  0.66${}_{-0.49}^{+0.61}$ |  1.98${}_{-0.50}^{+0.54}$ |  0.41${}_{-0.06}^{+0.08}$ | 133.0/ 122
22196 | 07/05/2019 | 20.58 |  0.97${}_{-0.44}^{+0.56}$ |  2.72${}_{-0.57}^{+0.63}$ |  0.34${}_{-0.07}^{+0.08}$ | 67.1/ 88
New Campaign | 02/2019-05/2019 | 729.26 |  0.49${}_{-0.07}^{+0.07}$ |  2.05${}_{-0.08}^{+0.08}$ |  0.50${}_{-0.02}^{+0.02}$ | 2645.8/ 2973
All Data | 05/2000-05/2019 | 844.74 |  0.44${}_{-0.06}^{+0.06}$ |  2.00${}_{-0.08}^{+0.08}$ |  0.52${}_{-0.02}^{+0.02}$ | 3055.9/ 3328

Table 1: Summary of observations. Fits to AGN are obtained using the “M84
Model” as presented in the text, with a VAPEC model for the galactic emission
and an APEC model for the Virgo Cluster “screen.” The photon index $\Gamma$
remains close to 2 as expected for Comptonized emission from an ADAF. We
include local absorption with column density $N_{\rm H}$ to account for
intervening dust lanes or a dusty torus around the AGN.

###  2.3 Spectral Fitting

Unless stated otherwise, we fit spectra simultaneously with the XSPEC spectral
fitting package (Arnaud, 1996) using all Obs. IDs listed in Table 1. Spectra
are extracted using CIAO’s specextract function and grouped such that at least
one count is present in all energy bins over the range of $0.5-7$ keV. Fits
are performed using the modified C-statistic (Cash, 1979) with elemental
abundances taken from Anders & Grevesse (1989) for comparison with past
results. All spectral models are fit with galactic absorption included via a
photoelectric absorption (phabs) model, with a constant galactic column
density of $N_{\rm H,gal}~{}=~{}2.9~{}\times~{}10^{20}$ cm2 as measured by the
HI4Pi Survey (HI4PI Collaboration et al., 2016).

###  2.4 Virgo Cluster Spectral Model

M84 is embedded in the Virgo Cluster, so we must peer through a “screen” of
hard X-ray emission $\gtrsim$1 keV. Because the Virgo Cluster occupies the
entire field of view, we follow HRR15 and choose to use blank sky backgrounds
for all spectral fits. The appropriate blank-sky background dataset was
processed identically to the event file, reprojected to the same sky position,
and normalized so that the count rate matched that of the event file for the
$9.5-12$ keV energy band.

We model the bremsstrahlung emission from the Virgo ICM with a single
temperature APEC plasma emission model. The parameters for this model are
determined by fitting a spectrum extracted from a large region, a
$4.6^{\prime}\times 4.9^{\prime}$ box around M84, excluding point sources (as
detected by wavdetect) and the galaxy, whose X-ray emission is confined within
a $1.9^{\prime}\times 2.6^{\prime}$ box.

This Virgo spectrum is fit with temperature ($T$), metallicity ($Z$), and
normalization free. The fit yields reasonable values, $T$ = 2.32 $\pm$ 0.06
keV and $Z$ = 0.429 $\pm$ 0.04 $Z_{\odot}$, consistent with $T\approx$ 2.3 keV
obtained by Urban et al. (2011) and Ehlert et al. (2013) using the much higher
spectral resolution of XMM-Newton.

Figure 2: Time variability of AGN over all three observational campaigns used
in this work. The observation corresponding to Obs. ID 5908 (red) has a much
higher flux than the others, violating an assumption that the AGN has a
constant luminosity. We remove this observation from our analysis of Bondi
radius scales.

###  2.5 M84 Galactic Gas Spectral Model

The earliest Chandra measurements of M84 by Finoguenov & Jones (2001) showed
an overabundance of metals relative to solar. This overabundance could be
contributed both by iron-peak elements (Fe and Ni) originating from Type Ia
supernovae, or $\alpha$ elements (C, N, O, Al, Si, etc.) produced by Type II
supernovae. Since XMM-Newton lacks the spatial resolution of Chandra,
abundance measurements performed by XMM probe larger length scales than we are
studying; we must constrain $\alpha$ element metallicities ourselves.

We fit the spectrum of the full $4.6^{\prime}\times 4.9^{\prime}$ box with the
galaxy included using a VAPEC model for M84’s galactic gas emission, an APEC
component for the Virgo ICM, and an extra power law component for unresolved
point sources (see §2.6). Since the helium (He) abundance of the VAPEC
component cannot be constrained in the X-ray band, we set the He abundance to
solar. VAPEC iron-peak element abundances $Z_{\rm Fe}$ are tethered together
in the fits, as are all remaining $\alpha$ element metallicities $Z_{\alpha}$.
The APEC temperature and metallicity are fixed based on §2.4, but the
normalization is left free.

###  2.6 Unresolved Point Sources

M84 is known to host a substantial number of X-ray binary (XRB) point sources
(Finoguenov & Jones, 2002). While many of these XRBs can be masked out,
unresolved XRBs and AB/CV stars represent a source of hard emission which can
affect temperature and abundance measurements. We follow the common practice
of modeling these unknown populations using a simple power law model with
fixed photon index $\Gamma_{\rm XRB}=1.6$ (Goulding et al., 2016). The
normalization of this power law is left free in the fits to the VAPEC+APEC
model, which are designed to provide adequate statistics for constraining the
$\alpha$ element metallicity, $Z_{\alpha}$.

Unfortunately, the contaminant build-up which has degraded Chandra’s soft
energy response prevents us from constraining the $\alpha$ element metallicity
from the new extended campaign, even with ample source counts available. Thus,
the only Obs. IDs used for determining $Z_{\alpha}$ come from legacy
observations: Obs. IDs 803, 5908, and 6131. We find a reasonable constraint on
$Z_{\alpha}$, approximately 0.45 times solar abundance. This value changes
only slightly to 0.51 solar when the power law component is neglected. For all
remaining spectral fits in this paper, we fix $Z_{\alpha}$ to 0.45 and fit
only the temperature, normalization, and the iron-peak metallicities in the
VAPEC model.

###  2.7 Accounting for AGN Contamination

While the AGN is a point source, Chandra’s point spread function (PSF)
distributes AGN photons across a number of pixels, including those containing
photons produced by gas at Bondi radius scales. Therefore, we must account for
AGN contamination when fitting spectra extracted from these sub-kpc scales.

The AGN and galactic gas are spectrally distinct; however, they are not
spectrally separable with Chandra’s spectral resolution. We find that when
fitting the AGN and galactic gas together, we are unable to adequately
constrain the parameters for both models. In response to this limitation, we
follow the example of HRR15 and forward-model the AGN contamination by
simulating Chandra’s energy-dependent PSF’s effect on the measured AGN
spectrum. We first fit the spectrum of the AGN based on fixing the parameters
of the galactic gas spectral model (§2.5). Using the parameters obtained from
this fit, we produce an AGN spectrum free of contributions from the galactic
gas or the Virgo screen. This spectrum is fed into the Chandra Ray Tracer tool
(ChaRT; Carter et al., 2003) which simulates the detector response. The MARX
software package (Davis et al., 2012) is used to produce a second-level event
file of the simulated AGN.

Figure 3: Left: Sectors ($10^{\prime\prime}$ in radius) overlaid on merged
$0.5-2$ keV image of M84 central region. The East (magenta) and West (green)
regions are aligned perpedicular with the AGN jet which is approximately
aligned with the Western filament (Figure 1). Right: Background-subtracted
$\mathbf{0.5-7}$ keV surface brightness profiles in sectors. The upper panel
shows profiles of the AGN (triangles) as well as the broadband data, while the
lower panel shows the AGN-subtracted surface brightness profile, used to
compute the deprojected density profiles.

###  2.8 AGN Spectral Model

Previous estimates based on the most recent SMBH mass measurements by Walsh et
al. (2010) place the angular scale of the SMBH Bondi radius at approximately
$1^{\prime\prime}$ (Russell et al., 2013); however, measurements in the
literature differ by as much as a factor of $\approx$4 (Bower et al., 1998;
Maciejewski & Binney, 2001). Thus, we choose to define the region of AGN
contamination as a circle with a radius of $1^{\prime\prime}$ centered on the
peak of the AGN surface brightness distribution. When we refer to the “AGN
spectrum” in this paper, we are referring to the spectrum extracted from this
region which includes the AGN, galactic gas emission, Virgo Cluster emission,
and unresolved XRBs and AB/CV stars.

The Comptonized emission from the AGN is modeled as a redshifted power law
(Russell et al., 2010; Siemiginowska et al., 2010). Due to the presence of the
intervening dust lanes as well as a possible “dusty torus” surrounding the
AGN, we allow for local photoelectric absorption through a zphabs model. Thus,
the spectrum of the $1^{\prime\prime}$ AGN region is fit using the “M84
model,” phabs(zphabs(zpowerlw)+VAPEC+APEC+powerlaw). While this model seems
complicated, the only free parameters are the local column density and the
photon index $\Gamma$ and normalization of the zpowerlw component.

Appendix A2 of HRR15 presents the method for obtaining the parameters for the
remaining components. Spectra are extracted from a
$2^{\prime\prime}-4^{\prime\prime}$ annulus circumscribing the AGN and fit
with a phabs(VAPEC+APEC+powerlaw) model for the galactic gas, Virgo ICM, and
unresolved point source emission. The APEC component is fixed based on fits in
§2.5 and the $T$ and $Z$ from §2.4 (note that all normalizations are scaled to
the appropriate region areas). We then fit the annulus spectrum with a free
VAPEC temperature, iron-peak metallicity, and normalization as well as a free
powerlaw normalization with fixed photon index $\Gamma_{\rm XRB}=1.6$ (see
§2.5 and §2.6).

The VAPEC normalization used in the AGN fits is determined by boosting the
normalization from the annulus fit centered at $3^{\prime\prime}$ to the
$1^{\prime\prime}$ AGN circle based on a power law extrapolation of the
$1^{\prime\prime}-40^{\prime\prime}$ surface brightness profile. For the
powerlaw component, we assume the normalization is constant from
$1^{\prime\prime}-3^{\prime\prime}$.

###  2.9 AGN Variability

We perform the fit described in §2.8 for all Obs. IDs individually in addition
to a simultaneous fit. The measured parameters for the AGN model are shown in
Table 1 and plotted in Figure 2.

Note that the AGN, unlike the galactic gas emission, is highly variable, with
nearly an order of magnitude variation in flux over the three campaigns.
Serendipitously, the AGN was relatively quiescent during 2019 (observations
following the break in Figure 2), implying that the vast majority of data is
subject to minimal AGN contamination. Unfortunately, one observation, Obs. ID
5908, caught a state of outburst. Because we are modeling the AGN based on the
statistically powerful simultaneous fit to all usable observations, we
necessarily make the assumption that the AGN flux is constant with time. Obs.
ID 5908 (shown in red in Figure 2) breaks this assumption and is thus omitted
from the remaining analysis of Bondi radius scales.

###  2.10 Simulating the AGN Spectrum

The fit to the AGN spectrum is consistent with a photon index of
$\Gamma\approx 2$, in accord with other similar ADAF spectra (Gu & Cao, 2009;
Younes et al., 2011; Yang et al., 2015; Younes et al., 2019). With this
result, we simulate how the AGN spectrum free of contributions from the
galactic gas, Virgo, and XRB/AB/CVs would appear to Chandra’s ACIS-S detector.

Appendix A3 of HRR15 describes this process in detail. We boost the
normalization of the spectrum input to ChaRT and use the aspect solution file
for Obs. ID 20543 (the longest exposure observation). Pile-up is negligible in
M84 and not included in our modeling.

The MARX tool is used to produce a second level event file and exposure map
for the simulation, and we reproject the simulation to the coordinate system
of the observation. We have accounted for galactic emission in the AGN
spectrum by measuring the galactic background from
$2^{\prime\prime}-4^{\prime\prime}$ and modeling the surface brightness (SB)
profile to account for the increase in background into $1^{\prime\prime}$.
However, our choice of model introduces a systematic error which may cause us
to inadvertently over or under-subtract the AGN by a small amount—enough to
impact the measured temperature.

Subtraction of the AGN can be tested by comparing the hard band ($4-7$ keV) SB
profile of the simulation with that of the data (see Appendix A). Bright,
lumpy, soft emission from galactic gas, which may vary on scales of the PSF,
is entirely subdominant in the $4-7$ keV band. Instead, AGN, Virgo ICM, and
unresolved point source emission dominates. If the AGN is under-subtracted, a
hard band excess will manifest itself at the scales of the PSF. However, if
the AGN is over-subtracted, there should be a drop in hard emission at small
scales. Because the Virgo screen is uniform over these small scales, any
discontinuities in the AGN-subtracted SB profile is evidence of spatial
variation in the unresolved point source flux.

Our AGN simulation leaves a modest hard band excess of $7\%$ at PSF scales. We
compensate for this excess by boosting the AGN simulation 7% such that the
AGN-subtracted $4-7$ keV band SB profile flattens from
$1^{\prime\prime}-2^{\prime\prime}$ (see Figure 9). Continuity of the AGN-
subtracted SB at these small scales indicates that the unresolved point source
emission is relatively constant with radius, and our assumption that the
XRB/AB/CV normalization is constant with radius obtains.

We extract spectra for both the observations and AGN simulation in
$1^{\prime\prime}$ radial bins extending from
$1^{\prime\prime}-10^{\prime\prime}$. Simulated spectra are fit using an
absorbed, redshifted power law model, and the fit parameters are fixed for the
AGN components in the combined “M84 model” used in §2.8. We compute the error
bars on temperature and metallicity for the two points in the innermost
$2^{\prime\prime}$ of each sector by boosting or diminishing the AGN
normalization by 5%, marginalizing over uncertainties in the AGN flux. The AGN
normalization is set to 0 beyond the innermost three annuli in each sector
since the AGN’s PSF is insignificant beyond $3^{\prime\prime}$.

Sector | $R_{\rm B}$ | $T(R_{\rm B})$ | $Z(R_{\rm B})$ | $n_{e}(R_{\rm B})$ | Index | $\dot{M}_{\rm B}$ | $\eta$ | $L_{\rm B}/L_{\rm Edd}$
---|---|---|---|---|---|---|---|---
| pc | keV |  Z⊙ |  cm-3 | $\alpha$ | $10^{-3}\>\mathrm{M}_{\odot}\mathrm{yr}^{-1}$ | $\times 10^{-6}$ | $\times 10^{-4}$
North | $49.0_{-5.3}^{+6.6}$ | $0.71_{-0.05}^{+0.04}$ | $0.14_{-0.03}^{+0.04}$ | $0.11\pm 0.09$ | $-1.15\pm 0.19$ | $1.57_{-1.01}^{+1.03}$ | $17.62_{-7.03}^{+31.64}$ | $0.84_{-0.54}^{+0.56}$
East | $48.4_{-5.4}^{+5.7}$ | $0.72_{-0.04}^{+0.05}$ | $0.09_{-0.02}^{+0.03}$ | $0.45\pm 0.04$ | $-0.80\pm 0.12$ | $6.35_{-1.35}^{+1.58}$ | $4.37_{-0.93}^{+1.28}$ | $3.39_{-0.78}^{+0.93}$
West | $42.8_{-5.9}^{+4.6}$ | $0.82_{-0.04}^{+0.09}$ | $0.12_{-0.02}^{+0.02}$ | $0.47\pm 0.03$ | $-0.93\pm 0.09$ | $5.52_{-1.31}^{+1.21}$ | $5.03_{-0.98}^{+1.68}$ | $2.94_{-0.75}^{+0.72}$
South | $58.7_{-6.9}^{+8.2}$ | $0.60_{-0.05}^{+0.05}$ | $0.29_{-0.07}^{+0.09}$ | $0.26\pm 0.02$ | $-1.21\pm 0.33$ | $\leq 4.88$ | $\geq 5.68$ | $\leq 2.60$
All | $48.1_{-4.7}^{+5.3}$ | $0.73_{-0.02}^{+0.02}$ | $0.15_{-0.02}^{+0.02}$ | $0.27\pm 0.04$ | $-0.87\pm 0.09$ | $3.74_{-0.89}^{+1.05}$ | $7.42_{-1.72}^{+2.46}$ | $1.99_{-0.50}^{+0.61}$

Table 2: Summary of measurements at $1^{\prime\prime}$, density profile index
$\alpha$, accretion rates $\dot{M}_{\rm B}$, efficiencies $\eta$, and
Eddington ratios $L_{\rm B}/L_{\rm Edd}$ for Bondi accretion in each sector.
Because of small-scale cavities evident in the AGN-subtracted SB profiles in
the South sector (Figure 3), we are only able to obtain limits on the density
$n_{e}$ and quantities derived from density in this region.

###  2.11 Profile Deprojection

To obtain density profiles, we follow HRR15 and first compute background-
subtracted surface brightness (SB) profiles of the inner
$1^{\prime\prime}-10^{\prime\prime}$ ($\sim 0.8$ kpc) in sectors, referred to
as North, East, West, and South respectively. The North and South sectors are
aligned with the radio jet, while the East and West sectors are anti-aligned.
We subtract off the Virgo and X-ray background from the SB profiles based on a
measurement of SB taken far away from the galaxy and free of point sources.
The sectors and SB profiles are shown in Figure 3.

SB is a projection of a 3D distribution of X-ray emission onto a 2D plane. By
assuming spherical symmetry, we can deproject each sector’s SB profile,
“peeling back” shells of X-ray emission to determine a volumetric emissivity
at each profile radius. Spherical symmetry is a poor assumption for M84’s
highly-structured $\mathcal{H}$-shape; however, the assumption may obtain more
readily around the quasi-spherical halo in the inner $10^{\prime\prime}$.

We apply the deprojection method of Kriss et al. (1983) to the AGN-subtracted
SB profiles, panel (c) in Figure 3. This method only strictly applies when the
SB is monotonically increasing inward. Cavities, evident in the significant
drop in the AGN-subtracted surface brightness profiles for the North and South
sectors at a radius of $1^{\prime\prime}$, violate this assumption and prevent
us from obtaining anything more than an upper limit on $n_{e}$ in the South
sector.

Even though the AGN simulation is always sub-dominant to the observed SB
(panel (b) in Figure 3), the simulation is brightest in the South where the
observed profiles show a depression in SB. We note that in the jet-aligned
sectors, the average SB of the innermost radial bin is $\sim$20% less than
that in the off jet-axis sectors. However, the AGN simulation tends to favor
more photons in the jet-aligned sectors, with $\sim$26% more photons in the
North and South compared to the East and West. Chandra’s PSF at sub-arcsecond
scales is subject to a hook feature which is captured in the ChaRT-MARX
simulation. Because the simulation PSF is asymmetric, a simple re-alignement
of the simulation is insufficient to eliminate the cavities from the AGN-
subtracted SB.

Our emissivity profiles, temperatures, and metallicities are all measured in
the same radial bins/ sectors. Thus, we are able to use the temperatures and
metallicities to determine the number density of X-ray emitting electrons
$n_{e}$ from the emissivity profiles. In this way, we obtain profiles of
$n_{e}$ in each sector separately.

###  2.12 Contour Binning

The large signal-to-noise afforded by our deep observations allows us to
produce maps tracing the large-scale temperature and metallicity structure in
M84. For this task, we use the contour binning method presented in Sanders
(2006) and made possible through the contbin software package (Sanders, 2016).

Contour binning groups adjacent pixels of similar surface brightness to
achieve a requested signal-to-noise ratio. The method groups gas expected to
be spectrally similar, allowing us to extract spectra with high signal-to-
noise. Thus, contour binning produces accurate temperature maps of spatially-
resolved extended sources with non-smooth surface brightness distributions.

We use a signal-to-noise ratio of 32 and set the smooth signal-to-noise
parameter to 20. Because of M84’s $\mathcal{H}$-shaped emissivity
distribution, contours tend to be elongated along the filaments, connecting
regions which are too spatially separated to be causally connected. We thus
constrain the shape of the contours using contbin’s constrainval parameter,
which we set to 1.2. The spectra extracted from the regions defined by contbin
are fit using the spectral model defined in §2.5; however, we do not include
an XRB/AV/CV component in our fits as this component tends to be negligible on
the kiloparsec (kpc) scales relevant for the maps.

##  3 Results

In this section, we present profiles of temperature, metallicity, and
deprojected density measured in four separate sectors—two aligned with the jet
axis and two anti-aligned (Figure 3a). We use these measurements to calculate
the accretion rates $\dot{M}_{\rm B}$ and efficiencies $\eta$ for Bondi
accretion in each of the sectors, where $L_{\rm Jet}+L_{\rm
X}=\eta\dot{M}c^{2}$ and $L_{\rm X}$ is the X-ray luminosity of the AGN. Our
main results are summarized in Table 2. These measurements allow us to explore
the large-scale structure of the accretion flow and compare timescales which
dictate the flow’s dynamics, namely the cooling time $t_{\rm cool}$ and inflow
time $t_{\rm inflow}$. We conclude this section with maps of temperature,
metallicity, and pseudo-pressure to connect the small-scale physics of
accretion with the kpc-scale structure of the X-ray halo.

###  3.1 Density Profile

The top panel of Figure 4 displays the density profile for each sector.
Density increases monotonically toward smaller radii in all sectors from
$7^{\prime\prime}$ to $2^{\prime\prime}$. The scaling of density with radius
provides a direct comparison between our data and the theoretical prediction
from the adiabatic Bondi solution. We can model the observed density profile
as a simple power law,

$n_{e}(r)=n_{e,0}\left(\frac{r}{R_{\rm B}}\right)^{\alpha},$ (1)

where $n_{e,0}$ is the number density at the Bondi radius, $r=R_{\rm B}$.
Using a Markov chain Monte Carlo (MCMC; Foreman-Mackey et al., 2013) method,
we fit this power law model to the density data for each sector. We omit the
innermost data points at $1^{\prime\prime}$ in the North and South sector fits
as these points are strongly affected by the presence of Bondi radius-scale
cavities. Values of $\alpha$ are shown in Table 2.

###  3.2 Temperature Profiles

We see evidence for shock heated gas in the temperature profile of the North
sector (middle panel of Figure 4). A jump from $T~{}=~{}0.75~{}\pm~{}0.03$ keV
at $4^{\prime\prime}$ to $T=0.94^{+0.02}_{-0.04}$ keV at $3^{\prime\prime}$ in
the North sector points to the influence of the radio jet. However, this
feature appears to be exceptional rather than commonplace. We find that the
temperature profiles are relatively flat with radius, increasing only
gradually inward from 0.6 keV at 800 pc to $0.7-0.8$ keV at 100 pc.

Temperatures at the innermost points, with radial error bars crossing through
the Bondi radius, show substantial scatter from $0.6-0.8$ keV. We see that
temperature in the South begins to decrease inward at the radii where M84’s
quasi-spherical central halo begins; gas may be cooling more efficiently in
these denser regions. However, we note that the South sector, especially the
innermost point, is certainly affected by cavities.

Note that points obtained in the South sector lack the constraining power of
the other sectors due to a clear point source throughout much of the region.
In all but the South sector, there are at least 700 source counts per radial
bin; however, from $3^{\prime\prime}-6^{\prime\prime}$ in the South, the
number of counts drops below 200. Thus, while the temperature of the innermost
point in the South sector may be reliable (although certainly a cavity is
present), the decreasing trend in temperature may not be physical. Instead,
based on the other sectors, one may reasonably conclude that temperature is
relatively constant over the inner kpc, with a gradual increase toward Bondi
radius scales.

Figure 4: Deprojected electron number density $n_{e}$, gas temperature $T$,
and iron-peak element metallicity $Z$ as a function of radius for all four
sectors. Dashed lines indicate the range of our measured Bondi radii. The
presence of cavities in the South sector precludes a measurement of $n_{e}$
for the innermost point; however, because the emissivity profile drops sharply
at this radius, we use the data point at $2^{\prime\prime}$ as an upper limit.

###  3.3 Bondi Accretion Rate

The adiabatic Bondi accretion rate is given by,

$\dot{M}_{\rm
B}=0.012\>\left(\frac{T}{\mathrm{keV}}\right)^{-3/2}\left(\frac{n_{e}}{\mathrm{cm}^{-3}}\right)\left(\frac{M_{\mathrm{BH}}}{10^{9}\>\mathrm{M}_{\odot}}\right)^{2}\>\mathrm{M}_{\odot}\mathrm{yr}^{-1},$
(2)

where we have measured $n_{e}$ and $T$ at the Bondi radius (Rafferty et al.,
2006), and we use the most recent measurement of the SMBH mass from Walsh et
al. (2010), $M_{\rm BH}=8.5_{-0.8}^{+0.9}\times 10^{8}M_{\odot}$.

Central values for $\dot{M}_{\rm B}$ are calculated through Equation 2 and the
data in Table 2. Our method for computing errors is described in Appendix B.
We choose to use a Monte Carlo method, drawing samples from distributions of
$n_{e}$, $T$, and $M_{\rm BH}$ and applying Equation 2. The 1$\sigma$ errors
on $\dot{M}_{\rm B}$ are then the 16th and 84th percentiles of the resulting
distribution.

Cavities formed by the radio jet and uncertain subtraction of the AGN PSF have
a significant impact on measurements of the Bondi accretion rate. Averaging
together the East and West sectors to yield $\dot{M}_{\rm B}=(5.94\pm
0.94)\times 10^{-3}\>M_{\odot}\rm{yr}^{-1}$ perpendicular to the jet, we find
that the accretion rate in the North sector parallel to the jet, $\dot{M}_{\rm
B}=1.57^{+1.03}_{-1.01}\times 10^{-3}\>M_{\odot}\rm{yr}^{-1}$, is discrepant
at the level of $4.6\sigma$. Given the dearth of photons in the South sector,
the upper limit obtained from this sector is likely a vast over-estimate, with
the true inferred $\dot{M}_{\rm B}$ lying even below that in the North sector.
These discrepancies point not only to the difficulty of measuring
$\dot{M}_{\rm B}$, but also the importance of carefully accounting for
cavities, which will systematically suppress the measured $\dot{M}_{\rm B}$.
Traditional methods which assume spherical symmetry to compute a deprojected
$n_{e}$ in a full annulus around the AGN rather than in sectors are possibly
under-estimating the true Bondi accretion rate.

We close this section by noting that recent measurements by the Event Horizon
Telescope (EHT; Event Horizon Telescope Collaboration et al., 2019a) may
indicate that the gas dynamical measurement of M84’s SMBH mass is an
underestimate of the true value. The EHT employs an emission modeling
technique for assessing SMBH masses which, in the case of Sgr A∗ (Event
Horizon Telescope Collaboration et al., 2022) yields a value completely
consistent with stellar dynamical measurements (Ghez et al., 2008; Gillessen
et al., 2009). However, when applied to M87∗, the EHT measurement (Event
Horizon Telescope Collaboration et al., 2019b) is discrepant with previous gas
dynamical measurements (Walsh et al., 2013). Thus, the Walsh et al. (2010)
measurements of M84’s SMBH and subsequently our measurements of the Bondi
radius and Bondi accretion rate may also be underestimates of the true values.

###  3.4 The Inefficiency of Bondi Accretion

Using the central values and distributions of $\dot{M}_{\rm B}$, we can
compute the efficiency of Bondi accretion. We define this efficiency factor
through $L_{\rm Jet}+L_{\rm X}=\eta\dot{M}c^{2}$, where our combined fit to
the AGN gives an X-ray luminosity $L_{\rm X}=1.6_{-0.14}^{+0.15}\times
10^{39}$ erg/s. The jet power $L_{\rm Jet}$ is obtained by measuring the
enthalpy of M84’s cavities assuming they are in pressure equilibrium with
their surroundings, and dividing by the characteristic timescale of the
bubbles, either the sound crossing time or buoyancy timescale. Using this
method, Russell et al. (2013) found the jet power to be $L_{\rm
Jet}=1.1_{-0.4}^{+0.9}\times 10^{42}$ erg/s.

For determining the errors in $\eta$, we assume dimidiated Gaussians for
$L_{\rm X}$ and $L_{\rm Jet}$, but instead of assuming distributions for
$\dot{M}_{\rm B}$, we use the distributions computed in §3.3. The results are
shown in Table 2.

While typically $\sim$10% of the $\dot{M}c^{2}$ power is released by
gravitational infall through an accretion disk, Bondi accretion onto M84’s
SMBH is far less efficient, with $\eta$ $\sim$ $10^{-6}$ in the East and West
sectors unaffected by cavities. These low efficiencies imply that M84 hosts a
radiatively inefficient accretion flow (RIAF). A similar conclusion follows
from the Eddington ratios for Bondi accretion. Using the definition $L_{\rm
B}=\dot{M}_{\rm B}c^{2}$, and the Eddington luminosity for the SMBH (Russell
et al., 2013),

$L_{\rm Edd}=1.26\times 10^{47}\left(\frac{M_{\rm
BH}}{10^{9}M_{\odot}}\right)\>\rm{erg/s},$ (3)

we can determine the Eddington ratio for Bondi accretion, $L_{\rm B}/L_{\rm
Edd}$. In all sectors, M84’s AGN displays Eddington ratios around a few
$\times 10^{-4}$. When the Eddington ratio is based on the true accretion rate
onto the hole $\dot{M}$, the value is much lower, $\eta L_{\rm B}/L_{\rm
Edd}\sim 10^{-10}$. Accretion proceeds not through a thin, radiatively
efficient disk (Shakura & Sunyaev, 1973), but rather via a hot RIAF (Yuan &
Narayan, 2014).

Figure 5: Profiles of cooling time and inflow time in the inner
$10^{\prime\prime}$ (0.8 kpc). The cooling time $t_{\rm cool}$ gradually
decreases inward from 200 Myr at 0.5 kpc to 60 Myr near the Bondi radius
(dashed lines). Inflow times show a similar trend; however, the inflow time is
$\sim$10 Myr at the Bondi radius in all sectors. The ratio of timescales shows
a clear transition above $t_{\rm cool}/t_{\rm inflow}=1$ at $2^{\prime\prime}$
as the flow approaches the Bondi radius.

###  3.5 Timescales for Accretion

We can better elucidate the structure of the flow by measuring profiles of the
relevant timescales for accretion, namely the cooling time $t_{\rm cool}$,
free-fall time $t_{\rm ff}$, and Bondi inflow time $t_{\rm inflow}$. The
cooling time is the timescale for a gas with thermal energy density
$\frac{3}{2}n_{e}T$ to radiate its energy away, $t_{\rm
cool}=(3/2)n_{e}T/n_{e}^{2}\Lambda$. Here, $\Lambda$ represents the cooling
function for the X-ray gas and consists of bremsstrahlung continuum as well as
significant line cooling in $T\lesssim 1$ keV gas.

The free-fall time is the dynamical timescale for gas to free fall from a
radius $r$ under the gravitational influence of M84’s SMBH and dark matter
halo. In line with the Bondi (1952) solution, we assume that M84’s dark matter
distribution is described by a spherically-symmetric Hernquist (1990) profile.
We add an additional point mass potential with mass $M_{\rm BH}$ to the dark
matter potential, which is used to determine the gravitational acceleration as
a function of radius, $g(r)$. The free-fall time can then be computed using
the simple expression, $t_{\rm ff}=\sqrt{2r/g}$.

Figure 6: Left: Temperature map of M84, where brighter colors indicate higher
temperature, measured in keV. Right: Metallicity map, where darker colors
indicate higher metallicity, measured relative to solar metallicity
$Z_{\odot}$. The same $2\times 10^{-3}-0.1$ mJy VLA radio contours from Figure
1 are overlaid to emphasize the structure of the $\mathcal{H}$-shaped
filaments.

Traditionally, the ratio $t_{\rm cool}/t_{\rm ff}$ has been used as a probe of
thermal instability in galaxy clusters. At the small $\lesssim$ 1 kpc scales
where the Bondi flow originates, the free fall time is far too short to be
relevant for the structure of the flow (see Discussion). Rather, the dynamical
timescale for the accretion flow is the Bondi inflow time, $t_{\rm inflow}$.
For steady-state Bondi accreton,

$\dot{M}_{\rm B}=4\pi r^{2}\rho(r)u_{r}(r)=\rm{constant},$ (4)

where we have introduced the mass density $\rho(r)$ and the radial inflow
velocity $u_{r}(r)$. The mass density is related to our measured number
density by assuming quasi-neutrality and introducing the mean-molecular weight
$\mu$, which we take to be 0.6 such that, $\rho(r)=1.15\>m_{p}n_{e}(r)$, and
$m_{p}$ is the proton mass. We can approximate the inflow time as

$t_{\rm inflow}=\frac{r}{u_{r}(r)}=\frac{4\pi r^{3}\rho(r)}{\dot{M}_{\rm B}}.$
(5)

Note that this expression is particularly simple as the number density $n_{e}$
drops out when substituting in the expression for $\dot{M}_{\rm B}$ (Equation
2),

$t_{\rm inflow}\approx
30\left(\frac{r}{\rm{kpc}}\right)^{3}\left(\frac{T}{\rm{keV}}\right)^{3/2}\left(\frac{M_{\rm
BH}}{10^{9}\>M_{\odot}}\right)^{2}\>\rm{Gyr}.$ (6)

Figure 5 shows profiles of the timescales. The free-fall timescale is short
($t_{\rm ff}~{}<~{}1.2$ Myr) within the inner kpc and does not dictate the gas
flow. Rather, cooling is the dominant process from 200$-$800 pc. We emphasize
that this flow structure is different from a “cooling flow” (Fabian, 1994)
which involves catastrophic levels of star formation and a deluge of cold gas
from the cluster ICM onto the galaxy. Our observations probe much smaller
scales than would be relevant for cooling flows, which are suppressed by AGN
feedback (Fabian, 2012). At these small scales, gas slowly condenses and
begins to flow inward as the atmosphere cools and loses thermal pressure
support.

As material approaches the Bondi radius, the dominant (shorter) timescale
becomes the Bondi inflow time $t_{\rm inflow}$. Thermal pressure support loses
relevance and the flow begins to experience the influence of the SMBH. In this
way, detection of the transition from $t_{\rm cool}/t_{\rm inflow}<1$ to
$t_{\rm cool}/t_{\rm inflow}>1$ may indicate that we are probing the beginning
of the large-scale accretion flow onto the SMBH.

###  3.6 Temperature and Metallicity Maps

The temperature and metallicity profiles presented in Figure 4 provide insight
into the structure of the beginnings of the accretion flow feeding the AGN in
M84. In this section, we “zoom out” from these small, sub-kpc scales to
explore the temperature and metallicity structure of the galactic gas.

Figure 6 shows the temperature and metallicity maps produced using the methods
described in §2.12. The maps are fit using the combined VAPEC+APEC model
described in §2.5 whenever the surface brightness of the region is larger than
$10^{-7}$ counts/s/cm2/arcsec2. Otherwise, we use an APEC model since low
surface brightness regions are likely dominated by emission from the Virgo ICM
rather than from M84’s galactic gas. This choice does not have a significant
effect on the structure of the maps; the use of an APEC+APEC model for all
regions yields similar maps.

The temperature map shows that the galactic gas is remarkably isothermal, with
the vast majority of gas occupying a narrow range of temperatures from
0.5$-$0.8 keV, similar to what is seen in the temperature profiles of the
inner kpc around the AGN. There appears to be a large-scale, although weak,
temperature gradient, with colder material located West of the radio jet and
warmer material sandwiched between the radio lobes. Gas is generally hotter
within the radio lobes; however this trend may be due to the fact that the
cavities are dominated by dim Virgo ICM emission. In the Northern radio lobe,
we see rich temperature structure, with a colder filament bridging through the
bubble. This feature may be a projection effect. Instead, a cold filament
could be wrapping around in front of or behind the bubble in three dimensional
space.

Colder temperatures trace out filaments; however, the $\mathcal{H}$-shape is
far more apparent in the metallicity map in Figure 6. In this map, higher
metallicities, with values around 0.3$-$0.5 $Z_{\odot}$ define the
$\mathcal{H}$, and metallicity appears relatively symmetric about the radio
jet. We see evidence for a drop in metallicity approaching the interior of the
galaxy, and this trend is clearly apparent in the metallicity profiles in
Figure 4. Radio cavities appear to clear out the X-ray halo of metal-enriched
material; however, rather than pulling high metallicity material into the wake
of the bubbles, the jet appears to simply push metal enriched material aside
into the $\mathcal{H}$-shaped filaments.

While jets shape the filaments, feedback is gentle and does not lead to a
substantial over-pressurization of the filaments. We can study this process
via the pseudo-pressure map (Figure 7). Pseudo-pressure is calculated by
multiplying the gas temperature in each region with the square root of the
normalization per unit area. For bremsstrahlung emission, which has a weak
$T^{1/2}$ temperature dependence, the normalization is proportional to the
emission measure, $\int n_{e}^{2}d\ell$, where $d\ell$ is the differential
path length through the cluster.

Pseudo-pressure increases only by a factor 1.5 between the Virgo ICM regions
located beyond about $55^{\prime\prime}$ East of the SMBH and the outer halo
of M84. The filaments, which begin $30^{\prime\prime}$ along the same
direction, show another factor of 1.5 increase. Such an increase cannot be
explained by the temperature, which instead decreases inward along the same
East-pointing ray. The increase is also too steep to be attributable simply to
an increase in the path length (and thus emission measure) if we assume M84’s
galactic gas is distributed quasi-spherically. Instead, gas density is
enhanced in the filaments, likely mediated by cooling in the dense, metal-rich
gas.

The crossbar of the $\mathcal{H}$ can be interpreted as a disk of dense gas
around the AGN, viewed in projection. Cooling in the dense disk leads to a
collapse into a thin, over-pressurized structure. In this way, rather than
being squeezed by the radio lobes, the crossbar may simply be condensing
through the cooling and gravitational collapse of a large-scale
($35^{\prime\prime}$ or $\sim$3 kpc) centrifugally-supported disk. Similarly,
features of the pressure map which seem to extend into the radio lobes, such
as a “fish-tail” like structure visible in the Southeast in Figure 7, can be
attributed to filaments wrapping around the radio lobes, viewed in projection.
If such filaments are uplifted with the bubbles, cooling may be encouraged,
leading to the formation of the dense, metal rich structures clinging to the
bubbles.

##  4 Discussion

The density, temperature, and metallicity profiles in Figure 4 provide a
direct comparison to the spherically-symmetric Bondi (1952) solution.
Similarly, our measurements of the profile index $\alpha$ and Bondi accretion
rate $\dot{M}_{\rm B}$ (Table 2) quantify the degree of asymmetry in the flow
for sectors aligned with the jet axis and those anti-aligned.

In this section, we discuss the origins of the observed deviations from the
Bondi solution, namely the influence of the jet on the flow. We include a
brief discussion of multiphase structure and thermal instability in M84,
presenting the entropy profiles based on our temperature and density
measurements, and discuss the lack of an observed temperature rise at the
Bondi radius. Finally, we close with an analysis of the “hot blob” of material
noted in §3.2.

Figure 7: Pseudo-pressure map, formed by multiplying the temperature map in
Figure 6 with a map of the square root of area-corrected normalization. The
scale shown is logarithmic and spans 2 orders of magnitude. Filaments are not
substantially over-pressurized relative to the surrounding cluster.

###  4.1 Asymmetry Imposed by the Jet

Within the 1$\sigma$ error bars, all density profiles are just slightly
flatter than $n_{e}\propto r^{-1}$, which is consistent with findings by HRR15
in M87. This profile however is completely inconsistent with the density
profile predicted by the Bondi (1952) solution for an adiabatic gas, which
instead would predict a density profile of $d\ln{\rho}/d\ln{r}=-0.373$ at
$r=R_{\rm B}$—a 5.5$\sigma$ discrepancy from the “All” sectors value in Table
2. This discrepancy points to the fact that the flow may be strongly
influenced by the galactic gravitational potential rather than the SMBH point
mass alone (Quataert & Narayan, 2000).

Both jet-aligned sectors show steeper radial profiles compared with the
shallow profiles perpendicular to the jet. Because the points most affected by
the presence of cavities were removed when fitting for $\alpha$, the cavities
do not account for this steepening in the density profile. Instead, jet-
inflated bubbles may entrain dense material from the core of the galaxy in
their wakes, buoyantly lifting this gas to larger radii. The result is a
dearth of material at the Bondi radius along the jet and density enhancement
at larger radii (Churazov et al., 2001; Fabian et al., 2003). Alternatively,
because the highest densities in the North sector ($n_{e}=0.26\pm 0.009$
$\rm{cm}^{-3}$ at $3^{\prime\prime}$) are coincident with the “hot blob” of
shocked gas, the density enhancement may be due to compression in the shocked
region itself.

###  4.2 Mechanical Feedback on the Accretion Flow

The Bondi accretion rates $\dot{M}_{\rm B}$ aligned and anti-aligned with the
jet axis are discrepant at a level $>4\sigma$. This discrepancy may be
attributed to small-scale cavities formed by radio jets blasting through and
clearing out halo gas. However, whether or not this discrepancy indicates that
jets modify the true accretion rate of material through the Bondi radius
remains an open question. Certainly, cavities complicate measurements of
$\dot{M}_{\rm B}$. Spherical symmetry does not apply in the presence of a jet
and deprojection is no longer well-posed if surface brightness decreases
inward.

Still, the radio jet may have a negligible impact on the accretion flow
itself. Relativistic jets with small opening angles can impart substantial
energy to the accretion flow via shock heating (see §4.6) which impedes gas
cooling and introduces further asymmetry to the flow. However, these jets
impact a relatively small fraction of the accreting gas. While a large number
of simulations have been able to explore self-regulation of AGN in cluster
environments (Cattaneo & Teyssier, 2007; Sijacki et al., 2007; Dubois et al.,
2010; Gaspari et al., 2012; Li & Bryan, 2014; Prasad et al., 2015; Yang &
Reynolds, 2016; Bourne & Sijacki, 2017), these simulations lack the dynamic
range to study black hole feeding at scales below $R_{\rm B}$.

Recently, Ressler et al. (2018) and Ressler et al. (2020) demonstrated a
calculation of black hole feeding for the RIAF in Sgr A∗ which evolved the
origins of the flow fed by stellar winds down to the black hole horizon. A
similar procedure has been undertaken by Guo et al. (2022) for the AGN in M87,
with a heating prescription standing in for jetted AGN feedback. In all of
these works, angular momentum plays a crucial role. Thermal instability,
turbulence, stellar winds, and cloud-cloud or cloud-filament interactions set
the angular momentum distribution of accreting gas. High angular momentum gas
which is unable to shed angular momentum through collisions of turbulent
transport (Narayan & Fabian, 2011), is flung away as it encounters the
centrifugal barrier of the SMBH. Yet, low angular momentum gas has the
possibility of settling into the observed accretion flow.

These works predict a suppression of the Bondi accretion rate with the scaling
$\dot{M}\sim(r/R_{\rm B})^{1/2}\dot{M}_{\rm B}$. M84’s $M_{\rm
BH}=8.5^{+0.9}_{-0.8}\times 10^{8}\>M_{\odot}$ black hole has an innermost
stable circular orbit (ISCO) with a radius $R_{\rm ISCO}\approx 7.6\times
10^{14}\>\rm{cm}$ (assuming no black hole spin). For the Bondi radius based on
all sectors, $R_{\rm B}(\rm{All})=48.1^{+5.3}_{-4.7}\>\rm{pc}$, the predicted
accretion rate at the ISCO using the scaling inferred from simulations is
$\dot{M}=(R_{\rm ISCO}/R_{\rm B})^{1/2}\dot{M}_{\rm B}\approx 8.5\times
10^{-6}\>M_{\odot}\rm{yr}^{-1}$. If the flow liberates $\eta\sim 10\%$ of the
$\dot{M}c^{2}$ energy which reaches the hole, the inferred power is $L_{\rm
ISCO}\approx 5\times 10^{40}\>\rm{erg/s}$. This power is well short of the
Gyr-averaged jet power $L_{\rm Jet}=1.1^{+0.9}_{-0.4}\times
10^{42}\>\rm{erg/s}$. Thus, if the $r^{1/2}$ scaling obtains in M84, there
must be additional sources of accreting gas beyond the hot phase material
inferred from X-ray observations alone.

Understanding the interaction between jets and the accretion flows powering
them remains an open problem. Self-consistently evolving the sub-parsec scales
responsible for launching jets with the $\sim$50 pc scales of the Bondi radius
requires resolving gas thermodynamics, inflows, and outflows over 5 orders of
magnitude in scale. We expect that a combination of increased computational
power and deep observations of molecular gas, enabled by observatories like
the Atacama Large Millimeter Array (ALMA), will serve to better elucidate how
jets and bubbles affect the distribution of mass and angular momentum in the
gas fueling RIAFs in massive elliptical galaxies.

###  4.3 Cold vs. Hot Mode Accretion

Large-scale accretion at scales comparable to the Bondi radius can be broadly
divided into two classes, similar to those invoked in the galaxy formation
community (Kereš et al., 2005): cold mode and hot mode accretion. Bondi
accretion of $\sim$keV X-ray gas represents the hot mode of accretion. As we
have shown by our measurements of density and temperature at the Bondi radius,
Bondi accretion alone is more than sufficient to power the central AGN in M84.
However, if multiphase gas, particularly components much colder than what we
are studying in the X-rays, is present, the cold mode of accretion may be
equally if not more important.

In cold mode accretion, thermally unstable (Field, 1965) gas cools, condenses,
and precipitates out of the hot medium, forming dense structures such as
“clouds” (or “blobs”) and filaments. As long as these cold structures possess
a minimal amount of angular momentum, or can shed angular momentum via cloud-
cloud, cloud-filament, etc. collisions, they can chaotically “rain down” onto
the central SMBH, providing a gas supply even in excess of that provided by
Bondi accretion alone (Pizzolato & Soker, 2005; Gaspari et al., 2012).

While this picture of accretion is straightforward in principle, in practice,
a number of challenges remain. In cluster environments, buoyancy acts to
negate thermal instability, at least at the level of linear theory (Defouw,
1970; Cowie et al., 1980; Nulsen, 1986; Balbus & Soker, 1989). Idealized
nonlinear simulations with heating and cooling globally balanced, as carried
out by McCourt et al. (2012) in plane parallel geometry, Sharma et al. (2012)
in spherical coordinates, and with jetted feedback as in the simulations by
Gaspari et al. (2012), argue that the existence of multiphase gas depends
sensitively on the minimum of the cooling to free-fall time ratio, $t_{\rm
cool}/t_{\rm ff}$. Subsequently, simulations by Li & Bryan (2014) and Meece et
al. (2015), as well as observational efforts by Voit & Donahue (2015) and Voit
et al. (2015) have further solidified the importance of the ratio of these
timescales in the literature.

Figure 8: Entropy profiles of all 4 sectors. There is a clear inward decrease
in entropy within the inner kpc of M84, which should indicate a dearth of
multiphase gas based on the $t_{\rm cool}/t_{\rm ff}$ ratio at these small
radii.

Voit et al. (2017) adds motivation for the minimum $t_{\rm cool}/t_{\rm ff}$
ratio in clusters, arguing that the ratio sets a critical slope for the
entropy profiles in clusters. For entropy profiles steeper than this
threshold, multiphase gas cannot precipitate from the hot phase since it is
subject to buoyant oscillations and thus strong buoyancy damping. Yet, when
the slope is flattened by an injection of high entropy material into the
center of the cluster via feedback, thermal instability can proceed and cold
mode accretion is once again relevant. In this way, the minimum $t_{\rm
cool}/t_{\rm ff}$ ratio alone is not the only crucial parameter in clusters.
Rather, this ratio must be compared to the entropy gradient to predict the
presence of multiphase gas.

We note that while there is some observational support for these models, the
idea of a critical $t_{\rm cool}/t_{\rm ff}$ ratio setting the conditions for
the formation of multiphase gas is by no means settled physics. Buoyantly
rising bubbles may stimulate cooling and multiphase gas formation via
adiabatic uplift (McNamara et al., 2016), as may be indicated by our
temperature, metallicity, and psuedo-pressure maps (Figures 6-7). In addition,
observational biases may over-emphasize the importance of $t_{\rm cool}/t_{\rm
ff}$ (Hogan et al., 2017; Pulido et al., 2018). Our work is focused on AGN
fueling at the Bondi radius. Thus, rather than wade headlong into rather
subtle questions of thermal instability in galaxy clusters, we present a
simple test for multiphase gas based on the measured entropy profiles in M84’s
X-ray halo.

Figure 8 presents the radial entropy profiles in each of the four sectors.
Following Voit et al. (2017), we define the dimensionless entropy gradient as
$\alpha_{K}\equiv\boldsymbol{r}\cdot\nabla\ln{(K)}$, where $K\equiv
Tn_{e}^{-2/3}$ is the gas entropy in units of keV cm2. Equation 22 of Voit et
al. (2017) provides a condition on the entropy gradient for nonlinear
condensation,

$\alpha_{K}<\alpha_{K,\rm{crit}}\equiv\frac{3(2-\lambda)^{2}}{40}\left(\frac{t_{\rm
ff}}{t_{\rm cool}}\right)^{2},$ (7)

where $\lambda\equiv d\ln{\Lambda}/d\ln{T}$ parameterizes the cooling function
$\Lambda$, and for the relevant cooling mechanisms in clusters lies in the
range $-1\lesssim\lambda\lesssim 0.5$. Thus, entropy profiles steeper than the
critical value $\alpha_{K,\rm{crit}}$ should result in gradually damped
buoyant oscillations of cooling gas, rather than the condensation necessary to
fuel cold mode accretion. Because of the short free fall times so close to the
Bondi radius ($t_{\rm ff}\sim$ 0.2 Myr) and comparatively long cooling times
($t_{\rm cool}\sim 0.1$ Gyr; Figure 5) at these small scales, the ratio
$t_{\rm ff}/t_{\rm cool}\sim 2\times 10^{-3}$ implies that the critical
entropy gradient is essentially flat. Figure 8 indicates that in all sectors,
the entropy gradient is far too steep to admit condensation and the formation
of multiphase gas. If multiphase gas is in fact present, this material must
have been sent toward the Bondi radius from much larger scales.

While we searched for multi-temperature gas in the central kpc as an
indication of gas cooling out of the ionized phase of the X-ray emitting
plasma, we were unable to find evidence of a second temperature component in
the X-ray band. Adding in a second VAPEC component provided no constraint on a
second temperature. The likely reason is that Chandra can only distinguish
temperatures separated by $\sim 0.5$ keV in energy space. Because M84’s gas is
cold ($0.6-0.7$ keV), a colder component would appear at $0.2-0.3$ keV, below
the detector’s sensitive energy range. A hotter component may be detectable;
however, if the AGN was over-subtracted rather than under-subtracted, this
potentially weak signal may be lost. Thus, consistent with the conclusion from
the entropy profiles, we find no evidence for the presence of multiphase X-ray
emitting gas.

We note that the strict criterion presented in Equation 7 may not be
applicable to Bondi radius scales. The central assumption underpinning the
importance of $t_{\rm ff}/t_{\rm cool}$ in cluster environments is that
heating and cooling is globally balanced. While such a balance may apply
globally within the Virgo Cluster, locally, at the small sub-kpc scales probed
in our analysis, heating cannot offset cooling. AGN jet energy is thermalized
on 10s of kpc length scales comparable to or larger than the bubbles, via weak
shocks and sound waves (Fabian et al., 2003; Sanders & Fabian, 2007; Graham et
al., 2008; Fabian et al., 2017; Bambic & Reynolds, 2019), turbulence driven by
g-modes (Churazov et al., 2002, 2004; Zhuravleva et al., 2014; Zhang et al.,
2018), mixing of high entropy bubble material with cluster gas (Hillel &
Soker, 2016, 2018), cosmic rays (Guo & Oh, 2008; Pfrommer, 2013; Ruszkowski et
al., 2017; Ehlert et al., 2018; Yang et al., 2019; Kempski & Quataert, 2020),
etc. Near the Bondi radius of M84 where the cooling time is $\sim$0.1 Gyr,
these heating mechanisms operate inefficiently.

For this reason, the traditional comparison of the free-fall and cooling
timescales should give way to a comparison of the cooling and Bondi inflow
timescales (see §3.5). Figure 5 indicates a transition from a “cooling-
dominated” flow to an “inflow-dominated” flow at scales comparable to $R_{\rm
B}$. In this way, we see that the accretion flow around the Bondi radius
should not be regarded as a static equilibrium defined by the interplay of AGN
heating and radiative cooling, but a dynamic inflow of material under the
influence of the SMBH.

###  4.4 Metallicity Structure

When the iron-peak element metallicity is free to vary in the fit, we see a
clear metallicity gradient. Metallicity decreases inwards in all sectors, with
the exception being the cavity-affected point in the South. This gradient
(referred to in the literature as a central abundance drop) is common in
cluster environments and has been observed in more than 8 objects (Panagoulia
et al., 2013, 2015; Lakhchaura et al., 2019; Liu et al., 2019). Though dust
from old stars should be increasing the central metallicity, if this dust is
locked into filaments, it can be lifted buoyantly to larger radii by jet-
inflated bubbles. Thus, the same processes which shape the density gradients
along the jet axis may be responsible for the metallicity gradient.

While buoyancy may explain some of the metallicity gradient along the jet
axis, the challenge remains to explain the central abundance drops in sectors
perpendicular to the jet axis. Without a modulation of the SNe Ia rate with
radius in the galaxy, this gradient is difficult to account for. Rather than
arising from a physical process, the drop in metallicity may point to an
unresolved second temperature component and thus, gas cooling en route to a
multiphase structure. While we found no evidence for such multi-temperature
structure (§4.3), future, deeper observations free of the constraints on soft
energy response which afflict Chandra are necessary to tease out the existence
of this cooler material.

We close by noting that absorption, specifically “intrinsic” absorption due to
interlaced cold and hot phase gas may be obstructing our view of the gas
cooling which is responsible for fueling M84’s AGN. Such a “hidden” cooling
flow (Fabian, 1994; Fabian et al., 2022) may be present within M84. Indeed,
there is evidence from XMM-Newton observations that an intrinsic absorption
model may describe M84’s galactic gas (Fabian et al. 2022 in prep.). However,
high spectral resolution, far beyond what can be achieved by Chandra, is
required to tease out the parameters of this model. Thus, XMM-Newton
observations, which probe much larger scales than Chandra (comparable with the
extent of M84’s $\mathcal{H}$-shaped filaments) are unable to constrain an
accretion rate for a “hidden” cooling flow at Bondi radius scales.

###  4.5 No Temperature Rise at the Bondi Radius

We find no evidence for a temperature rise approaching the Bondi radius, a
phenomenon that has been proposed as evidence for the transition from the
galactic potential to that of the SMBH. This conclusion may be a consequence
of the changing metallicity, which decreases by nearly a factor of 4 over the
inner few hundred pc in all but the South sector (although the cavity and
limited numbers of counts may be playing a role).

When we fix the metallicity to the radially averaged value ($\sim$0.3
$Z_{\odot}$), we see signs of a temperature jump, with the innermost points in
the East and West sectors reaching 1.5 and 1.2 keV respectively. While
temperature does increase with fixed iron-peak element metallicity, so also
does the reduced C-statistic, indicating that a temperature rise may not truly
be present. Instead, there may have been an under-subtraction of the AGN which
provides an excess of hard photons, enough to over-estimate the temperature
when metallicity is not a free parameter.

The lack of an observed temperature rise may not be surprising. Observations
of the temperature profiles in M87 by HRR15 find a similar absence. In some
respects this is to be expected: the analytical Bondi solution at radii
comparable to $R_{\rm B}$ shows a relatively flat temperature profile, with
the majority of the adiabatic heating occurring at small scales, well below
the Bondi radius.

However, the notable absence of the temperature rise may be a result of biases
inherent to observations of multiphase gas. Indeed, Guo et al. (2022) have
recently performed simulations of Bondi accretion from tens of kpc scales down
to accretion flow scales well below the Bondi radius which include gas at a
wide range of temperatures. They find that their simulations tend to predict a
flat emission-weighted temperature profile in the $0.5-7$ keV band, even at
scales an order of magnitude below $R_{\rm B}$, where adiabatic heating of hot
phase gas becomes significant. Our X-ray observations may be biased by the
energy band accessible to Chandra, and even future missions which probe scales
below $R_{\rm B}$ may similarly never detect a temperature rise.

###  4.6 Shock Heating by the Jet or Nonthermal Emission?

While there is no evidence for a temperature rise at the Bondi radius, we do
see a clear (3.8$\sigma$) temperature increase from $4^{\prime\prime}$ to
$3^{\prime\prime}$ in the North sector which we refer to as a “hot blob.” This
temperature increase at $2^{\prime\prime}-3^{\prime\prime}$ from the AGN is at
the same angular separation from the AGN as a knot of radio emission detected
by VLA in the 5 and 8.5 GHz bands, and ALMA in the 97 and 236 GHz bands (see
Knot B in Figures 1-3 of Meyer et al. (2018)). This “blob” or “knot” of X-ray
emission, first detected by Harris et al. (2002), stands out clearly in
X-rays, even with the limited exposure time ($\approx$29 ks) of Chandra’s
first observation of M84.

The $0.5-7$ keV X-ray spectrum of the region containing the “hot blob” is well
described (reduced C-statistic of 1.08 and 1.06 for the North Sector at 2′′
and 3′′ respectively) by a VAPEC model in our analysis, indicating thermal
X-rays. However, the radio emission is far more complicated. Early results
favored a synchrotron origin for the emission (Harris et al., 2002); however,
Meyer et al. (2018) argue that the radio and X-ray spectra of the knots cannot
both be explained by standard models for jet emission. In their analysis, the
X-ray emission is modeled as both a power law representing the jet and an APEC
component representing the thermal gas.

Motivated by these works, we re-fit the spectra from the North sector at 2′′
and 3′′ using our M84 model (§2.8), with the addition of a red-shifted power
law (zpowerlw) component meant to represent the X-ray jet detected by Meyer et
al. (2018). When fitting the 2′′ and 3′′ North spectra with the extra zpowerlw
“jet” component, the resulting temperature and metallicity of the VAPEC
component are unconstrained (in the case of the 3′′ point, temperature is
constrained but metallicity is not). We thus proceeded to leave the
temperature and zpowerlw normalization free in the fit, but fix the VAPEC
metallicity to three different values of metallicity: 0.1 (consistent with our
measurements), 0.2, and 0.3 solar (consistent with that used by Meyer et al.
(2018)). For each metallicity, we scan zpowerlw photon indices from
$\Gamma=1-3$, which we fix in the fit.

This procedure yields improved C-statistics over the 1.08 and 1.06 found
initially, in some cases comparable to or better than the fit to the full
annulus spectrum including all sectors at 2′′ and 3′′. Reasonable photon
indices near $\Gamma\approx 2$ yield good fits. As expected, the corresponding
VAPEC temperature is lower when the jet component is included; however, rather
intriguingly, the fit temperature is lower than all other sectors save that in
the West at the same radii. The temperature never exceeds 0.7 keV for all
photon indices and metallicities tested. For the metallicity of 0.1 solar
consistent with what was found in our profiles (Figure 4), the fit finds VAPEC
temperatures below 0.6 keV for the point at 2′′. When using the metallicitity
of 0.3 solar assumed in Meyer et al. (2018), the fits settle around $T=0.65$
keV, which is just below the temperature at a radius of 2′′ in the East Sector
($T=0.67^{+0.04}_{-0.03}$ keV). Note that all of these temperatures are well
below the $3$ keV thermal model for the X-ray emission proposed as an
alternative to synchrotron emission in Harris et al. (2002).

The takeaway message from this analysis is clear: a model with nonthermal
emission from an X-ray jet and colder ($T\approx 0.65$ keV) galactic gas
describes the “hot blob” as well as a purely thermal emission model of
$T\gtrsim$ 0.9 keV gas shock heated by the jet. Because of Chandra’s limited
spectral resolution, we cannot discriminate between these models. However,
given the complexity of M84’s jet emission in the radio band and the co-
spatial temperature increase observed in our X-ray observations, a deeper
study of M84’s jet which can harness the full power of our 840 ks data set is
merited.

###  4.7 Comparison to Other Measurements

Bondi accretion is incredibly inefficient. The Bondi accretion rate
$\dot{M}_{\rm B}$ is measured to be a few
$\times\>10^{-3}M_{\odot}\rm{yr}^{-1}$ in each of the sectors in this work.
Thus, the accretion flow in M84 need only liberate $\eta\sim 10^{-6}$ of the
$\dot{M}_{\rm B}c^{2}$ fuel provided by Bondi accretion to power the galaxy’s
relativistic jets and X-ray AGN.

We are not the first group to arrive at this conclusion in M84. The first
measurement of the Bondi accretion rate can be attributed to Allen et al.
(2006), who found an accretion rate of $\dot{M}_{\rm
B}=8.5^{+8.4}_{-4.1}\times 10^{-3}M_{\odot}\rm{yr}^{-1}$ by measuring the
temperature and density of the full annulus around the AGN, i.e. including all
sectors at the innermost “Bondi radius” point. While this work was in
preparation, another measurement of $\dot{M}_{\rm B}$ was performed by Plšek
et al. (2022) which leveraged the data from our new campaign, presented here
and publicly available. They found $\dot{M}_{\rm B}=2.4^{+1.9}_{-1.5}\times
10^{-3}M_{\odot}\rm{yr}^{-1}$, again using all sectors. If we compare these
values with our “All” sector measurement of $\dot{M}_{\rm
B}=3.74^{+1.05}_{-0.89}\times 10^{-3}M_{\odot}\rm{yr}^{-1}$, then there is
strong agreement among all published values of the Bondi accretion rate in
M84, with notably tighter error bars in the more recent values enabled by an
extra $\gtrsim 750$ ks provided by the new campaign.

##  5 Conclusion

We have presented the deepest Chandra X-ray observations to date of M84, a
jetted elliptical galaxy in the Virgo Cluster. These observations, which
comprise over 840 ks of Chandra data, have enabled a detailed study of the
temperature, density, and metallicity structure of the galaxy, from kiloparsec
scales to $\approx 50$ pc scales just inside the Bondi radius of the galaxy’s
SMBH. New images of M84 have been presented, emphasizing the intricate
structure of the soft X-ray filaments, formed into an $\mathcal{H}$ morphology
by the action of powerful ($L_{\rm Jet}=1.1^{+0.9}_{-0.4}\times 10^{42}$
erg/s; Russell et al., 2013) radio jets.

Density and temperature measurements obtained through spectra extracted from
the innermost $0.5^{\prime\prime}-1.5^{\prime\prime}$ bin allowed us to
compute Bondi accretion rates for each of 4 sectors around the central AGN.
“All” sectors are fit together to allow comparison to previous works. The main
conclusions of our analysis are as follows:

  1. 1.

Radial profiles of deprojected electron number density $n_{e}$ are consistent
with $n_{e}\propto r^{-1}$, but slightly flatter (Figure 4 and Table 2). This
profile is in tension at the level of $5.5\sigma$ with the expectation of
Bondi accretion, which predicts $d\ln{\rho}/d\ln{r}=-0.373$ at $r=R_{B}$ and
an $r^{-3/2}$ scaling at $r\ll R_{B}$.

  2. 2.

The radial profile indices $\alpha$ are statistically consistent; however, we
see that the profiles are steeper along the jet axis than perpendicular to the
jet (Table 2). This violation of spherical symmetry is counter to the
assumptions of the Bondi solution.

  3. 3.

There is a discrepancy in the Bondi accretion rate depending upon which sector
is used to measure $\dot{M}_{B}$ (Table 2). This discrepancy between jet-
aligned and mis-aligned sectors is at the level of $4.6\sigma$, which is
statistically significant. While the discrepancy may point to the influence of
the jet on the large-scale accretion flow, the disparity likely arises due to
the presence of cavities (Figures 3 and 4) or uncertainties in modeling the
AGN emission at Bondi radius scales (see Appendix A).

  4. 4.

Temperatures do not vary widely throughout the galaxy (Figure 6) and only
increase gradually from 0.6 keV to 0.7 keV over the inner kpc approaching the
Bondi radius. The exception is that in the North sector, we see evidence for a
temperature increase at points $2^{\prime\prime}-3^{\prime\prime}$ from the
AGN (Figure 4). We refer to this feature as a “hot blob” of gas. The physical
origin of this “hot blob” remains an open question. Shock heating by the radio
jet or nonthermal emission from knots in an unresolved X-ray jet (Meyer et
al., 2018) are both plausible explanations; however, Chandra lacks the
spectral resolution to rule out either of these two models.

  5. 5.

We detect no temperature rise at the Bondi radius, consistent with findings by
HRR15 in M87.

  6. 6.

By comparing the Bondi inflow time $t_{\rm inflow}$ to the cooling time as a
function of radius, we observe evidence for a transition from a “cooling-
dominated” flow to an “inflow-dominated” flow at scales of
$1^{\prime\prime}-2^{\prime\prime}$, providing support to the conclusion that
we have resolved M84’s Bondi radius.

## Acknowledgements

The data presented in this work was obtained through Chandra Proposal
#19800344: “Fueling and self-regulation of AGN feedback at the Bondi radius of
M84” and is publically available on the Chandra archive. We are grateful to
the Chandra X-ray Center for support, not only with the initial observations
and pointing of the telescope, but also with use of the ChaRT tool. This
manuscript was improved thanks to the careful reading and suggestions of an
anonymous referee. CJB thanks Andy Goulding and Jeremy Sanders for technical
support throughout this project. This work benefited from stimulating
discussions with Eliot Quataert and Minghao Guo at Princeton and Eileen Meyer
at the Texas Symposium on Relativistic Astrophysics. CJB thanks the graduate
students of the Institute of Astronomy, Cambridge and Princeton University for
their many insights as well. This work is possible through the financial
support of the Churchill Foundation of the United States and CJB continues to
be supported by a National Science Foundation (NSF) Graduate Research
Fellowship. Early stages of this work were performed at the Multiscale
Phenomena in Plasma Astrophysics program at KITP in Santa Barbara, CA,
research supported in part by the NSF under Grant No. NSF PHY-1748958. HRR
acknowledges support from an STFC Ernest Rutherford Fellowship and an Anne
McLaren Fellowship provided by the University of Nottingham. CSR thanks the
STFC for support under the Consolidated Grant ST/S000623/1, as well as the
European Research Council (ERC) for support under the European Union’s Horizon
2020 research and innovation programme (grant 834203).

## Data Availability

The Chandra data described in this work are available in the Chandra data
archive (https://cxc.harvard.edu/cda/). Processed data products detailed in
this paper will be made available on reasonable request to the author.

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## Appendix A: Modeling AGN Contamination

ChaRT and MARX simulations of the AGN require an input energy spectrum, which
was obtained by fitting the spectrum extracted from a $1^{\prime\prime}$
circle about the AGN. While this spectrum is dominated by the AGN “source,” it
includes “background” contributions from M84’s galactic gas, the Virgo screen,
and unresolved XRB/AB/CV stars. The Virgo and unresolved point source
backgrounds were determined in §2.4 and §2.6 respectively; however, the
treatment of the underlying galactic gas emission required more care.

Figure 9: Top: Hard band ($4-7$ keV) surface brightness profiles extracted
from merged event file and ChaRT+MARX AGN simulation, with 7% boost applied to
simulated profiles. Bottom: Subtraction of hard band simulation profile from
hard band data profile. The hard band, where only AGN emission and background
from the Virgo screen and unresolved point sources remains, is more than an
order of magnitude subdominant to the total SB. By applying a 7% boost to the
simulation, we achieve a flattening of the AGN-subtracted profile within the
inner $2^{\prime\prime}$, indicating that the AGN has been properly subtracted
from the data. All that remains at PSF scales are negligible contributions
from the spatially uniform Virgo screen and unresolved point sources. The
point source emission appears to be subdominant given the smoothness of the
AGN-subtracted hard band SB from $1^{\prime\prime}-3^{\prime\prime}$.

To set the background galactic gas component at $1^{\prime\prime}$, we fit the
spectrum of an annulus from $2^{\prime\prime}-4^{\prime\prime}$ circumscribing
the AGN where AGN emission is negligible. This procedure provided a
normalization for the VAPEC background component at 3′′ from the AGN. Then, we
fit the surface brightness (SB) distribution with the assumption of spherical
symmetry using a simple power law in radius, assuming that the power law
extrapolation from $3^{\prime\prime}$ to $1^{\prime\prime}$ accurately
described the SB at $1^{\prime\prime}$ from the central AGN. The ratio of the
SB determined at $1^{\prime\prime}$ and $3^{\prime\prime}$ from this power law
was taken as a “boost” factor multiplied on to the previously fit-for VAPEC
normalization. In the case of M84, we found this boost factor to be 3.98.

Then, the parameters of the input energy spectrum passed to ChaRT were
determined by fitting the full spectrum (source + background) extracted from
the $1^{\prime\prime}$ circle with a fixed background VAPEC component and only
the zpowerlw parameters (meant to model the AGN source) left free. The
normalization of this fixed VAPEC component is simply the normalization of the
$2^{\prime\prime}-4^{\prime\prime}$ annulus, multiplied by the boost factor
(3.98) and corrected by the ratio of the annulus to $1^{\prime\prime}$ circle
areas. Once the parameters for the zpowerlw source component were determined,
a clean spectrum including only phabs(zphabs(zpowerlw)) parameters was
produced and passed to the ChaRT tool as the input spectrum. This source
spectrum is representative of the AGN without contributions from the
background.

Because the normalization of the input spectrum was determined by assuming a
model for the galactic gas spectrum, the flux of the simulation may not be an
accurate representation of the true AGN. To test the accuracy of the AGN
modeling, we choose an energy band where the AGN completely dominates and
which is free of the small-scale variations (on scales comparable to the PSF)
imposed by bright, lumpy, soft emission from galactic gas. In this case,
following HRR15, we choose the hard $4-7$ keV band. At these energies, the
only contributions to the hard band SB should be from the AGN source, Virgo
ICM background (which should be spatially uniform), and unresolved point
source background. If point source emission is substantial, the hard band SB
profile should display discontinuities and rapid spatial variations on scales
of the PSF.

The top panel of Figure 9 shows a comparison of the hard band profiles for the
data (black) and simulation (red). By forward modeling the AGN, we are working
to subtract the AGN contribution at Bondi radius scales from the spectra
extracted in each sector. The blue points in the lower panel of Figure 9 show
a subtraction of the simulation’s hard band SB profile from that of the data.
If we were to overestimate the AGN flux and thus over-subtract the AGN from
the data, we should expect a drop in hard band SB at $1^{\prime\prime}$, i.e.
the scales of the PSF. Alternatively, if we under-subtract the AGN, we should
expect a hard band excess in the bottom panel of Figure 9.

We find that our AGN simulation based on modeling the galactic gas background
leaves a 7% excess in the hard ($4-7$ keV) band. Without compensating for this
excess, we would under-subtract the AGN and possibly bias our temperature
measurements with excess hard AGN photons. Thus, we boost the overall AGN
simulation normalization by 7%, which means that the difference profile in the
bottom panel of Figure 9 flattens at the scales of the PSF. When computing
errors on temperature, we do not simply report the statistical uncertainties
determined by XSPEC. Rather, we boost the AGN normalization by 5% (on top of
the 7% compensation) and decrease the normalization by 5% (shown as the red
and black points in the bottom panel of Figure 9 respectively), to marginalize
over uncertainties in the AGN modeling. As a result, errors in temperature and
metallicity are larger at the Bondi radius. Finally, because the hard band SB
is relatively continuous at PSF scales, we conclude that unresolved point
sources are properly accounted for.

## Appendix B: Ascertaining Errors on $\dot{M}_{\rm B}$ and $\eta$

For computing errors on $\dot{M}_{\rm B}$, we use a Monte Carlo method,
drawing $10^{7}$ samples from distributions of $n_{e}$, $T$, and $M_{\rm BH}$
and applying Equation 2. Because of asymmetric error bars in $T$ and $M_{\rm
BH}$, we model the distributions of these variables as “dimidiated Gaussians”
(Barlow, 2003), two Gaussians centered on the same mean with different
standard deviations above and below the mean based on the 1$\sigma$ upper and
lower error bars. For equal positive and negative error bars, the dimidiated
Gaussian is equivalent to a normal distribution. We model the underlying
distribution of $n_{e}$ as a log-normal with mean and standard deviation based
on the central value and $1\sigma$ error bar respectively. This choice ensures
strict positivity of $n_{e}$ but only has significance for the point in the
North—a Gaussian yields a similar error bar for all other points. We take the
1$\sigma$ errors on $\dot{M}_{\rm B}$ to be the 16th and 84th percentile of
the resulting distribution.