PASA 2024

# Refining the mass estimate for the intermediate-mass black hole candidate in
NGC 3319

Benjamin L. Davis1,2, https://orcid.org/0000-0002-4306-5950  and Alister W.
Graham1https://orcid.org/0000-0002-6496-9414  Author for correspondence: BLD,
E-mail: mailto:ben.davis<EMAIL_ADDRESS>1Centre for Astrophysics and
Supercomputing, Swinburne University of Technology, Hawthorn, VIC 3122,
Australia  2Center for Astro, Particle, and Planetary Physics (CAP3), New York
University Abu Dhabi

###### Abstract

Recent X-ray observations by Jiang et al. have identified an active galactic
nucleus (AGN) in the bulgeless spiral galaxy NGC 3319, located just $14.3\pm
1.1$ Mpc away, and suggest the presence of an intermediate-mass black hole
(IMBH; $10^{2}\leq M_{\bullet}/\mathrm{M_{\odot}}\leq 10^{5}$) if the
Eddington ratios are as high as 3 to $3\times 10^{-3}$. In an effort to refine
the black hole mass for this (currently) rare class of object, we have
explored multiple black hole mass scaling relations, such as those involving
the (not previously used) velocity dispersion, logarithmic spiral-arm pitch
angle, total galaxy stellar mass, nuclear star cluster mass, rotational
velocity, and colour of NGC 3319, to obtain ten mass estimates, of differing
accuracy. We have calculated a mass of $3.14_{-2.20}^{+7.02}\times
10^{4}\,\mathrm{M_{\odot}}$, with a confidence of 84% that it is
$\leq$$10^{5}\,\mathrm{M_{\odot}}$, based on the combined probability density
function from seven of these individual estimates. Our conservative approach
excluded two black hole mass estimates (via the nuclear star cluster mass, and
the fundamental plane of black hole activity — which only applies to black
holes with low accretion rates) that were upper limits of $\sim$$10^{5}\,{\rm
M}_{\odot}$, and it did not use the $M_{\bullet}$–$L_{\rm 2-10\,keV}$
relation’s prediction of $\sim$$10^{5}\,{\rm M}_{\odot}$. This target provides
an exceptional opportunity to study an IMBH in AGN mode and advance our
demographic knowledge of black holes. Furthermore, we introduce our novel
method of meta-analysis as a beneficial technique for identifying new IMBH
candidates by quantifying the probability that a galaxy possesses an IMBH.

###### doi:

10.1017/pas.2024.xxx

###### keywords:

black hole physics – galaxies: active – galaxies: evolution – galaxies:
individual: NGC 3319 – galaxies: spiral – galaxies: structure

Original unedited manuscript, accepted for publication by PASA, May 7, 2021.

##  1 Introduction

There is a largely-missing population of intermediate-mass black holes (IMBHs)
with masses higher than those formed by stable, single stars today
($M_{\bullet}\lesssim 100\,\mathrm{M_{\odot}}$) and less massive than the
supermassive black holes (SMBHs; $10^{5}\,\mathrm{M_{\odot}}\leq
M_{\bullet}\lesssim 10^{10}\,\mathrm{M_{\odot}}$)111The massive central object
in the quasar TON 618 is alleged to have the most massive black hole with a
mass of $6.61\times 10^{10}\,\mathrm{M_{\odot}}$, estimated from its H$\beta$
emission line and a virial $f$-factor of 5.5 (Shemmer et al., 2004; Onken et
al., 2004). known to reside at the centres of massive galaxies. Not
surprisingly, astronomers around the world have been hotly pursuing the much-
anticipated discovery of IMBHs for some time (e.g. Miller & Colbert, 2004). In
addition to providing a fundamental input to the cosmic inventory of our
Universe, the abundance, or rarity, of IMBHs has implications for the
formation of the Universe’s SMBHs (Graham, 2016b; Mezcua, 2017; Koliopanos,
2017; Inayoshi et al., 2020; Sahu et al., 2019a).

As yet, there is no consensus as to how SMBHs came to be. While the observed
extent of quasar activity over the history of our Universe has revealed that
the accretion of baryons fattened them up (e.g. Soltan, 1982; Shankar et al.,
2004), we do not know what their (potentially range of) birth masses were.
Some theories have speculated that their birth or ‘seed’ masses were
$\approx$$10^{5}\,{\rm M}_{\odot}$, thereby providing a kick-start to explain
the early-formation of the high-$z$, active galactic nuclei (AGN) with
sizeable black hole masses around $\approx$$10^{9}\,{\rm M}_{\odot}$ (e.g.
Mortlock et al., 2011; Yang et al., 2020; Mignoli et al., 2020). Theories have
included primordial black holes (e.g. Grobov et al., 2011), massive metal-free
Population III stars which subsequently collapse (or collide, e.g. Alister
Seguel et al., 2020) to form massive black holes (e.g. Madau & Rees, 2001;
Schneider et al., 2002), or the direct collapse of massive gas clouds,
effectively by-passing the stellar phase of evolution (e.g. Bromm & Loeb,
2003; Mayer et al., 2010).

The suggestion of massive seeds arose from the notion that the ‘Eddington
limit’ (Eddington, 1925) of gas accretion onto a black hole implied that
stellar-mass black holes did not have sufficient time to grow into the SMBHs
observed in the young, high-redshift AGN. However, the Eddington limit on the
accretion rate applies only to (unrealistic) spherical conditions (Nayakshin
et al., 2012; Alexander & Natarajan, 2014) and can be significantly exceeded
in real systems. For example, super-critical (super-Eddington) accretion flows
onto massive black holes can occur when the accretion flow is mostly confined
to the disk plane while most of the radiation emerges in outflows along the
rotation axis (Abramowicz et al., 1980; Jiang et al., 2014; Pezzulli et al.,
2016). Hyper-Eddington accretion rates can exist in spherically-symmetric
accretion flows when energy advection reduces radiative efficiency (Inayoshi
et al., 2016). Thus, the practicality of super-critical accretion has been
invoked to explain the early existence of SMBHs at high redshifts (Volonteri &
Rees, 2005; Volonteri, 2012; Volonteri & Bellovary, 2012; Volonteri et al.,
2015). Besides, most ultra-luminous X-ray sources are nowadays explained as
stellar-mass X-ray binaries accreting much faster than their Eddington limit
(Feng & Soria, 2011; Kaaret et al., 2017). Such accretion negates the need for
massive black hole seeds.

An additional motive for starting AGN with massive seeds was that black holes
with masses intermediate between that of stellar-mass black holes and SMBHs
had not been directly observed, and therefore seemed not to exist. However,
this may be a sample selection bias because the sphere-of-gravitational-
influence around such IMBHs, where one would directly observe a Keplerian
rotation curve, is typically too small to resolve spatially. Furthermore,
there is now a rapidly rising number of IMBH candidates based upon indirect
estimates of the black hole mass (Farrell et al., 2009; Secrest et al., 2012;
Baldassare et al., 2015; Graham et al., 2016; Kızıltan et al., 2017; Nguyen et
al., 2017; Chilingarian et al., 2018; Mezcua et al., 2018; Jiang et al., 2018;
Nguyen et al., 2019; Graham & Soria, 2019; Graham et al., 2019; Woo et al.,
2019; Lin et al., 2020). In addition, there are currently five IMBH candidates
in the Milky Way (Takekawa et al., 2020).

There is no shortage of scenarios for how a bridging population of IMBHs may
have arisen. Possible pathways include the runaway collapse of dense ‘nuclear
star clusters’ (Portegies Zwart & McMillan, 2002; Davies et al., 2011; Lupi et
al., 2014; Stone et al., 2017), especially if gas-drag and dynamical friction
are in play at the centre of a galaxy, or the gas-fuelled growth of a stellar-
mass black hole that has not yet devoured enough material to become an SMBH
(Natarajan, 2021). These ideas would place, at least some, IMBHs at the
centres of galaxies, where established black hole mass scaling relations
involving some property of the host galaxy can be applied.

Recent Chandra X-ray Observatory (CXO; Weisskopf et al., 2000) observations
(Soria, 2016, see also Chilingarian et al. 2018 and Bi et al. 2020), have
discovered IMBH candidates at the centres222Some of the off-centre X-ray
sources that were detected may also be IMBHs. Indeed, the best localised IMBH
candidate to date is an off-centre source in the galaxy ESO 243-49 (Farrell et
al., 2009), whose optical counterpart was discovered by Soria et al. (2010)
and is thought to be the nucleus of an in-falling galaxy. However, the
likelihoods of these off-centre targets being IMBHs are generally considered
to be notably lower than that of the central targets — although perhaps not
zero (e.g. Barrows et al., 2019; Bellovary et al., 2021). of several nearby,
low-mass galaxies. Long exposures have enabled the discovery of faint X-ray
point-sources (consistent with low-mass black holes accreting with low
Eddington ratios) in galaxies which have been predicted to host a central IMBH
based upon each galaxy’s velocity dispersion, luminosity, and spiral-arm pitch
angle (Koliopanos et al., 2017; Graham & Soria, 2019; Graham et al., 2019).
The high-energy X-ray photons, originating from the (not so) dead centres of
the galaxies, are likely coming from the accretion disks around black holes
because of their point-source nature, where emission favours active black
holes rather than spatially extended star formation.

Several studies have identified IMBH candidates in galaxies based on single,
or a few, black hole mass estimates. In this work, we have selected a galaxy,
NGC 3319, where we can apply a wealth of independent black hole mass
estimates. NGC 3319 is a gas-rich, bulgeless, late-type galaxy. It is a
strongly-barred spiral galaxy classified as SBcd(rs) (de Vaucouleurs et al.,
1991) and has its bar aligned with the major axis (Randriamampandry et al.,
2015). Moreover, Jiang et al. (2018) identify it as possessing a low-
luminosity AGN with a high-accretion-rate signalled by a nuclear X-ray point
source and assume a black hole mass between $3\times
10^{2}\,\mathrm{M_{\odot}}$ and $3\times 10^{5}\,\mathrm{M_{\odot}}$ based on
a high Eddington ratio of 1 to $10^{-3}$, despite a non-detection in the
radio. Using the X-ray variability, they report an estimate of $\sim$105±2 M⊙,
and using the ‘fundamental plane of black hole activity’, they reported an
upper limit of 105 M⊙ in the absence of radio data. NGC 3319 had previously
been recognised as a possible low-ionisation nuclear emission-line region
(LINER) galaxy (Heckman et al., 1980; Pogge, 1989), or at least it possessed
an uncertain H ii nucleus (Ho et al., 1997). Recently, Baldi et al. (2018)
classified its nuclear type as a LINER based on BPT (Baldwin et al., 1981)
diagram diagnostics. This classification is of significance since AGN with
black holes are suspected sources of stimulating LINER spectral emission
(Heckman, 1980).

In this study, we endeavour to constrain better the mass of the potential IMBH
in the nucleus of NGC 3319 via a meta-analysis of multiple mass estimates
based on independently measured quantities. In the numerous subsections of
Section 2, we present a detailed analysis and application of ten separate
black hole mass scaling relations and ultimately combine these estimates to
yield an overall black hole mass estimate with confidence limits. The
uncertainty on each mass estimate is used to weight every estimate before
combining the results, via standard statistical techniques, to obtain the
final mass estimate whose uncertainty is naturally less than that of the
individual mass estimates. In the final section (Section 3), we discuss the
results of our investigation, comment on the implications, and remark on the
benefit from continued study of NGC 3319.

Following Jiang et al. (2018), we adopt a redshift-independent luminosity
distance of $14.3\pm 1.1$ Mpc (Cepheid variable star distance from Sakai et
al., 1999), with a physical scale of $69\pm 5$ pc arcsec-1. All values from
the literature have been adjusted to accommodate our adopted distance to NGC
3319. Black hole masses ($M_{\bullet}$) and other masses throughout this work
are represented as logarithmic (solar) mass values, i.e.
$\mathcal{M}\equiv\log{M}$, where $M$ is mass in units of solar masses (M⊙).
All uncertainties are presented as (or have been scaled to) $1\,\sigma\approx
68.3\%$ confidence intervals. All magnitudes are given in the AB system (Oke,
1974).

##  2 Black Hole Mass Estimates

In the following subsections (2.1–2.10), we applied ten different black hole
mass scaling relations to estimate the mass of the black hole (NGC 3319*)
residing at the centre of NGC 3319. We use the latest, and thus in some
instances morphology-dependent, black hole scaling relations. Although the use
of reverberation mapping has revealed that AGN extend the
$M_{\bullet}$–$M_{\rm bulge,\star}$ relation to black hole masses of
$10^{5}\,\mathrm{M_{\odot}}$ (Graham & Scott, 2015), the paucity of confirmed
IMBHs (and thus their dearth in the construction of black hole mass scaling
relations) requires us to extrapolate these relations to reach into the IMBH
regime.333This is also the case with reverberation mapping, which assumes the
$f$-factor (used to convert virial products into virial masses) holds
constant. Albeit, we note that NGC 205 (Nguyen et al., 2019) and NGC 404
(Nguyen et al., 2017) now extend the relations down to $\sim$104 and $\sim$105
M⊙, respectively. In Section 2.11, we combine the black hole mass estimates,
accounting for the different levels of scatter in each estimate.

###  2.1 The $M_{\bullet}$–$\phi$ relation

The aesthetic beauty of ‘spiral nebulae’ has been observed for 175 years,
since Lord Rosse’s observations of the Whirlpool Galaxy (NGC 5194). However,
significant mysteries still abound between the nature of these striking
features and properties of their host galaxies (D’Onghia et al., 2013). The
seminal works that established the spiral density wave theory (Lin & Shu,
1964, 1966; Lin et al., 1969) have provided perhaps the most lucid and lasting
explanation of (grand design) spiral genesis. Indeed, the spiral density
theory has been supported by observations in numerous studies (Davis et al.,
2015; Pour-Imani et al., 2016; Yu & Ho, 2018; Peterken et al., 2019; Miller et
al., 2019; Vallée, 2019, 2020; Abdeen et al., 2020; Griv et al., 2020; Griv et
al., 2021).

In particular, Lin & Shu (1966) predicted that the geometry of spiral patterns
should be governed by two primary galactic properties: (i) the density of the
galactic disk and (ii) the central gravitational potential (mass) of the
galaxy. Specifically, the pitch angle of the spiral pattern at a distance $R$
from a galaxy’s centre should be directly proportional to the density of the
disk at $R$ and inversely proportional to the mass of the galaxy $\leq$$R$.
Davis et al. (2015) tested this prediction and found a tight trivariate
relationship between the pitch angle, the stellar bulge mass, and the neutral
atomic hydrogen density in the disk of a galaxy. Additional studies pertaining
to dark matter halos have also shown a correlation between pitch angle and the
central mass concentration, as determined by the shear of the rotation curve
of a galaxy (Seigar et al., 2006, 2014). These theoretical and observational
studies provide perhaps the best explanations of why the pitch angle
correlates with its host galaxy: the pitch angle is clearly related to the
central mass of a galaxy, of which the ‘barge’ (bar and bulge) and black hole
are integral components entwined via coevolution.

The geometry of logarithmic spirals closely matches the shape of spiral arms
in galaxies. Quantitatively, the shape (tightness of winding) of a logarithmic
spiral is governed by the absolute value of its pitch angle,444For
introductory reading on pitch angle, see section 2 from Davis et al. (2017).
$|\phi|$, as introduced by von der Pahlen (1911). Seigar et al. (2008) first
presented evidence of a strong relationship between pitch angle and the mass
of a spiral galaxy’s central black hole. As the sample of spiral galaxies with
directly-measured black hole masses grew incrementally in size over the years,
Berrier et al. (2013) and later Davis et al. (2017) presented refinements to
the $M_{\bullet}$–$\phi$ relation. A graphical representation of the relation
found by Davis et al. (2017, equation 8) is shown in Fig. 1.

Figure 1: Spiral galaxy arms with varying degrees of tightness, with the
corresponding galaxy morphological type and central black hole mass in units
of our Sun’s mass. This template can be used to estimate central black hole
masses in spiral galaxies. The outermost spiral ([0.35ex]6mm0.4mm) has
$|\phi|=26\mbox{$.\\!\\!^{\circ}$}7$, which is indicative of a central black
hole with a mass of $10^{5}\,\mathrm{M_{\odot}}$ via equation 8 from Davis et
al. (2017).

We employ equation 8 from Davis et al. (2017) to convert measured pitch angles
into black hole masses, including an intrinsic scatter of 0.33 dex (added in
quadrature with a full propagation of errors on the pitch angle measurement,
as well as errors on the slope and intercept of the relation).

The existence of an $M_{\bullet}$–$\phi$ relation has been seen not only in
observations (Seigar et al., 2008; Berrier et al., 2013; Davis et al., 2017)
but also in simulations. Mutlu-Pakdil et al. (2018) measured the pitch angles
for a random sample of 95 galaxies drawn from the Illustris simulation
(Vogelsberger et al., 2014) and recovered an $M_{\bullet}$–$\phi$ relation
that was consistent with that found from observational studies. Thus, the
nascent $M_{\bullet}$–$\phi$ relation has already garnered empirical and
theoretical (via theory and simulations) support to become a full-fledged
black hole mass scaling relation. Its progress has proliferated in only a
dozen years; future improvements in observations and sample size should add to
its established legitimacy. The search for the primary relation with black
hole mass continues, and the lack of a spiral pattern in early-type galaxies
rules out the $M_{\bullet}$–$\phi$ relation, just as the absence of bulges in
some late-type galaxies negates the $M_{\bullet}$–$M_{\rm bulge,\star}$
relation. Nonetheless, the low level of scatter in both relations make them
valuable black hole mass estimators.

Several software programs have been devised to handle the quantitative
measurement of spiral galaxy pitch angle. In this work, we utilise three of
the most prominent and robust packages to measure pitch angle: 2dfft (Davis et
al., 2012; Davis et al., 2016; Seigar et al., 2018), spirality (Shields et
al., 2015a, b), and sparcfire (Davis & Hayes, 2014). Each code uses an
independent method of measuring pitch angle, each with its unique
advantage.555For a demonstrative comparison of each software package, see the
appendix from Davis et al. (2017). Each routine measures pitch angle after the
original galaxy image (Fig. 2, left panel) has been deprojected to an
artificial _face-on_ orientation (Fig. 2, middle panel). We adopt the outer
isophote position angle ($PA_{\rm outer}$, degrees east of north) and
ellipticity ($\epsilon_{\rm outer}$) values for NGC 3319 from Salo et al.
(2015): $PA_{\rm outer}=43\mbox{$.\\!\\!^{\circ}$}0\pm
0\mbox{$.\\!\\!^{\circ}$}7$ and $\epsilon_{\rm outer}=0.435\pm 0.003$. This
ellipticity is equivalent to an inclination of the disk, $i_{\rm
disk}\equiv\cos^{-1}(1-\epsilon_{\rm outer})=55\mbox{$.\\!\\!^{\circ}$}6\pm
0\mbox{$.\\!\\!^{\circ}$}2$.

We measured the pitch angles from a Spitzer Space Telescope Infrared Array
Camera (IRAC) $8.0\,\mu{\rm m}$ image obtained from the Spitzer Heritage
Archive.666https://irsa.ipac.caltech.edu/applications/Spitzer/SHA Recent
studies (Pour-Imani et al., 2016; Miller et al., 2019) have presented
observational evidence that 8.0-$\mu{\rm m}$ light highlights the physical
location of the spiral density wave in spiral galaxies. 8.0-$\mu{\rm m}$ light
comes from the glow of warm dust around nascent natal star-forming regions
that have been shocked into existence by the spiral density wave.

####  2.1.1 sparcfire

sparcfire (Davis & Hayes, 2014) uses computer vision techniques to identify
the pixel clusters that form the architecture of spiral arms in spiral
galaxies and fits logarithmic spiral segments to the clusters. sparcfire
classifies each spiral based on its chirality: Z-wise, spirals that grow
radially in a counterclockwise direction ($\phi<0$); and S-wise, spirals that
grow radially in a clockwise direction ($\phi>0$). Based on the number and arc
lengths of the ensemble of fitted spirals, we adopted a dominant chirality for
the galaxy and ignored all spurious arcs matching the secondary chirality. We
calculated a weighted-arithmetic-mean pitch angle for the galaxy based on a
weight for each arc ($w_{i}$) such that $w_{i}\equiv s_{i}/r_{0,i}$, where
$s_{i}$ is the arc length and $r_{0,i}$ is the inner radius (from the origin
at the galactic centre) for an individual arc segment. Therefore, the highest
weighting resides with long arcs near the centre of the galaxy and short,
possibly spurious arc segments in the outer region of the galaxy, are made
insignificant.

As seen in the right panel of Fig. 2, the dominant chirality is Z-wise.

Figure 2: Left (Original) – Spitzer $8.0\,\mu{\rm m}$ image of NGC 3319. Here,
the image has been aligned, pointing the top of the image in the direction of
the galaxy’s position angle ($43\mbox{$.\\!\\!^{\circ}$}0$ east of north), and
the image has been cropped into a square that is $5^{\prime}\times 5^{\prime}$
($20.7\,{\rm kpc}\times 20.7\,{\rm kpc}$). Middle (Deprojected) – here, the
original image has been deprojected to an artificial _face-on_ orientation,
achieved by stretching the X-axis by a factor of $a/b\equiv(1-\epsilon_{\rm
outer})^{-1}=1.77$, where $a$ is the semi-major axis length, and $b$ is the
semi-minor axis length of the outer isophotes (Salo et al., 2015). Right
(Spiral Arcs) – the spiral arcs measured by sparcfire (Davis & Hayes, 2014)
are overlaid upon the deprojected image. Fitted lines depict: (used) Z-wise
spiral arcs ([0.35ex]6mm0.4mm), (ignored) S-wise spiral arcs
([0.35ex]6mm0.4mm), and the galactic bar ([0.35ex]6mm0.4mm). The reported
pitch angle, $31\mbox{$.\\!\\!^{\circ}$}7\pm 4\mbox{$.\\!\\!^{\circ}$}5$, is
the weighted-mean pitch angle of the dominant-chirality red spiral arcs (see
Section 2.1.1).

We computed the pitch angle and converted it to a black hole mass prediction
via the $M_{\bullet}$–$\phi$ relation as follows, rCl
|ϕ|_sparcfire&=31$.\\!\\!^{\circ}$7±4$.\\!\\!^{\circ}$5→
M_∙(|ϕ|_sparcfire)=4.15±0.86.

####  2.1.2 2dfft

2dfft (Davis et al., 2012; Davis et al., 2016; Seigar et al., 2018) is a two-
dimensional fast Fourier transform software package that decomposes a galaxy
image into logarithmic spirals. It computes the amplitude of each Fourier
component by decomposing the observed distribution of light in an image into a
superposition of logarithmic spirals as a function of pitch angle, $\phi$, and
harmonic-mode, $m$, i.e. the order of rotational symmetry (e.g. two-fold,
three-fold, and higher-order symmetries). For the _face-on_ view of NGC 3319
(Fig. 2, middle panel), the maximum amplitude is achieved with $m=2$ (i.e. two
spiral arms) and

$|\phi|_{\textsc{2dfft}}=28\mbox{$.\\!\\!^{\circ}$}4\pm
3\mbox{$.\\!\\!^{\circ}$}5\rightarrow\mathcal{M}_{\bullet}(|\phi|_{\textsc{2dfft}})=4.72\pm
0.70.$ (1)

####  2.1.3 spirality

spirality (Shields et al., 2015a, b) is a template-fitting software. Given a
_face-on_ image of a spiral galaxy, it computes a library of spiral coordinate
systems with varying pitch angles. For NGC 3319 (Fig. 2, middle panel), the
best-fitting spiral coordinate system has a pitch angle of rCl
|ϕ|_spirality&=24$.\\!\\!^{\circ}$4±4$.\\!\\!^{\circ}$1→
M_∙(|ϕ|_spirality)=5.40±0.74.

####  2.1.4 Weighted-mean pitch angle and black hole mass

The $M_{\bullet}$–$\phi$ relation is a tight relation, with intrinsically low
scatter. However, the slope of the relation is relatively steep, and thus
small changes in pitch angle equate to large changes in black hole mass.
Specifically, a change in pitch angle of only $5\mbox{$.\\!\\!^{\circ}$}8$ is
associated with a 1.0 dex change in black hole mass. For late-type spiral
galaxies like NGC 3319, their open spiral structures often feature inherent
flocculence and asymmetries amongst individual spiral arms. Furthermore, due
to the diminished total masses of these galaxies (as compared to early-type
spiral galaxies), galaxy harassment and tidal interactions are more impactful
in disrupting their spiral structures.

The average uncertainty amongst our equations 2–2.1.3 is
$4\mbox{$.\\!\\!^{\circ}$}0$ (a difference of 0.68 dex in black hole mass).
Nonetheless, all three of the pitch angle measurements possess overlapping
error bars. To produce a more robust pitch angle measurement, we combine all
three measurements (equations 2–2.1.3) to yield a weighted-arithmetic-mean
pitch angle,

$\bar{\phi}=\frac{\sum_{i=1}^{N}w_{i}\phi_{i}}{\sum_{i=1}^{N}w_{i}},$ (2)

and associated uncertainty,

$\delta\bar{\phi}=\frac{\sqrt{\sum_{i=1}^{N}(w_{i}\delta\phi_{i})^{2}}}{\sum_{i=1}^{N}w_{i}}=\sqrt{\frac{1}{\sum_{i=1}^{N}w_{i}}},$
(3)

with a weight for each measurement that is inversely proportional to the
square of the uncertainty of its measurement, i.e. inverse-variance weighting,
$w_{i}=\left(\delta\phi_{i}\right)^{-2}$. This yields

$|\bar{\phi}|=28\mbox{$.\\!\\!^{\circ}$}0\pm
2\mbox{$.\\!\\!^{\circ}$}3\rightarrow\mathcal{M}_{\bullet}(|\bar{\phi}|)=4.79\pm
0.54.$ (4)

Our use of the independent black hole mass scaling relations, and their
reported $\pm 1\,\sigma$ scatter, assumes a normal distribution for each.
Assuming a normal distribution for our weighted-mean, we can then calculate
the probability of having an IMBH. Given a mass estimate for a black hole and
its associated error ($\delta\mathcal{M}_{\bullet}$), we can compute the
probability that the black hole is less-than-supermassive
($\mathcal{M}_{\bullet}\leq 5$) as follows,

$P(\mathcal{M}_{\bullet}\leq 5)=\frac{1}{2}\left[1+{\rm
erf}\left(\frac{5-\mathcal{M}_{\bullet}}{\delta\mathcal{M}_{\bullet}\sqrt{2}}\right)\right]$
(5)

(Weisstein, 2002). Doing so for the mass estimate from equation (4), we find
$P(\mathcal{M}_{\bullet}\leq 5)=65\%$. We have additionally checked the pitch
angle in alternative imaging that also traces star formation in spiral arms,
by using the Galaxy Evolution Explorer (GALEX) far-ultraviolet (FUV) passband
(1350–1750 Å). We found that the pitch angle from GALEX FUV imaging,
$27\mbox{$.\\!\\!^{\circ}$}5\pm 3\mbox{$.\\!\\!^{\circ}$}9$, is highly
consistent with that from 8.0-$\mu{\rm m}$ imaging.

###  2.2 The $M_{\bullet}$–$M_{{\rm gal},\star}$ relation

For our second estimate, we used the total stellar mass of NGC 3319 as a
predictor of the black hole mass at its centre. We began by obtaining Spitzer
images and masks for NGC 3319 from the S4G catalogue (Sheth et al.,
2010).777https://irsa.ipac.caltech.edu/data/SPITZER/S4G/index.html We elected
to use the $3.6\,\mu{\rm m},\star$ stellar image from Querejeta et al. (2015).
The $3.6\,\mu{\rm m},\star$ image has been created after determining the
amount of glowing dust present (by analysing the empirical $3.6\,\mu{\rm m}$
and $4.5\,\mu{\rm m}$ images) and subsequently subtracting the dust light from
the $3.6\,\mu{\rm m}$ image. Thus, the $3.6\,\mu{\rm m},\star$ image shows
only the light emitted from the stellar population, and its luminosity can be
directly converted into a stellar mass. We adopted a $3.6\,\mu{\rm m}$ stellar
mass-to-light ratio, $\Upsilon_{3.6\,\mu{\rm m},\star}=0.60\pm 0.09$ from
Meidt et al. (2014),888The $3.6\,\mu{\rm m}$ bandpass has a low uncertainty
for the stellar mass-to-light ratio, with $\Upsilon_{\star}$ from 0.40 to 0.55
(Schombert et al., 2019). This is consistent with the observed (i.e. with dust
glow) $\Upsilon_{3.6\,\mu{\rm m},\star,{\rm obs}}=0.453\pm 0.072$ value
derived by Davis et al. (2019a, section 2.8), which is equivalent to the dust-
corrected $\Upsilon_{3.6\,\mu{\rm m},\star}=0.60\pm 0.09$ from Meidt et al.
(2014). along with a solar absolute magnitude, $\mathfrak{M}_{3.6\,\mu{\rm
m},\odot}=6.02$ mag (AB), at $3.6\,\mu{\rm m}$ (Oh et al., 2008).

To model the light from NGC 3319, we utilised the isophotal fitting and
modelling software routines isofit and cmodel (Ciambur, 2015), respectively.
After masking extraneous light sources, we ran isofit on the $3.6\,\mu{\rm
m},\star$ image (Fig. 3, left panel) and used cmodel to extract, and create a
representation of, the galaxy (Fig. 3, second panel). The quality of the
extraction can be seen in the residual images presented in the right two
panels of Fig. 3.

Figure 3: Left (Original) – Spitzer $3.6\,\mu{\rm m},\star$ image of NGC 3319.
Here, the image has been aligned so that the top of the image is pointing in
the direction of the galaxy’s position angle ($43\mbox{$.\\!\\!^{\circ}$}0$
east of north), and the image has been cropped, so it is $5^{\prime}\times
7^{\prime}$ ($20.7\,{\rm kpc}\times 28.98\,{\rm kpc}$). The black pixels
indicate no intensity, and white pixels (pixel intensity of 8.2 MJy sr-1)
indicate $\mu_{3.6\,\mu{\rm m},\star}\leq 18.188$ mag arcsec-2. Second from
Left (Model) – model produced by isofit and cmodel (Ciambur, 2015), which
includes a sky background of 0.0180 MJy sr-1 (Salo et al., 2015). Second from
Right (Residual) – residual image, such that Residual $\equiv$ Original $-$
Model. Right (Division) – division image, such that Division $\equiv$ Residual
$\div$ Original. The division image depicts the relative difference between
the original and the residual image. Pixel values are between zero (black) and
one (white), representing maximal and minimal change, respectively.

The extracted galaxy was then analysed by the surface brightness profile
fitting software profiler (Ciambur, 2016). This works by convolving the galaxy
model with the Spitzer (IRAC channel 1) point spread function (PSF) with a
full width at half maximum (FWHM) of $1\mbox{$.\\!\\!^{\prime\prime}$}66$ for
the cryogenic
mission999https://irsa.ipac.caltech.edu/data/SPITZER/docs/irac/iracinstrumenthandbook/5/
until an optimal match is achieved.101010profiler uses an unweighted least-
squares Levenberg-Marquardt (Marquardt, 1963) algorithm (via python package
lmfit; Newville et al., 2016) to minimise the total rms scatter,
$\Delta_{rms}=\sqrt{(n-f+1)^{-1}\sum_{i=R_{\rm min}}^{R_{\rm
max}}\left(\mu_{{\rm data},i}-\mu_{{\rm model},i}\right)^{2}}$, with surface
brightnesses of the data (obtained from isofit) and model, each at radial bin,
$i$, where $n$ is the number of radial bins (inclusive) between the minimum
($R_{\rm min}$) and maximum ($R_{\rm max}$) user-selected radii, and $f$ is
the number of free parameters (i.e. the number of user-selected components);
profiler adjusts the model (summation of user-selected components) until a
global minimum is reached. Additionally, a residual profile,
$\Delta\mu(R)=\mu_{\rm data}(R)-\mu_{\rm model}(R)$, is provided in the output
plots of profiler to demonstrate the quality of the fit as a function of
galactocentric radius ($R$). We present the resulting galaxy surface
brightness profiles and multi-component fits for both the major axis (Fig. 4,
left two panels) and the geometric mean axis, equivalent to a circularised
representation of the galaxy (Fig. 4, right two panels).

Figure 4: Surface brightness profile decompositions produced by profiler
(Ciambur, 2016). Panels (from left to right): linear major-axis, log major-
axis, linear equivalent-axis, and log equivalent-axis profiles; $R_{\rm
eq}=\sqrt{ab}=R_{\rm maj}\sqrt{1-\epsilon}$ and $R_{\rm maj}\equiv a$.
Subplots (from top to bottom): surface brightness profile ($\bullet$) and
model ([0.35ex]6mm0.5mm) built from the summation of the following components:
PSF ([0.35ex]6mm0.4mm), bar ([0.35ex]6mm0.4mm), disk ([0.35ex]6mm0.4mm), and
spiral arms ([0.35ex]6mm0.4mm), the faint outer spiral arm (at $R_{\rm
maj}\approx 140^{\prime\prime}\equiv R_{\rm eq}\approx 85^{\prime\prime}$)
lies below the plotted region; residual profile with total rms scatter
($\Delta_{rms}$); ellipticity profile; position angle profile; and fourth-
order cosine Fourier harmonic coefficient, $B_{4}$ ($B_{2}$, $B_{3}$, $B_{6}$,
$B_{8}$, and $B_{10}$ harmonics are also fit and contribute to the model).

We confirm that NGC 3319 is a bulgeless galaxy and does not require a
traditional Sérsic bulge component (Sérsic, 1963; Ciotti, 1991; Graham &
Driver, 2005). Instead, we generate a convincing fit that adequately captures
all of the light of the galaxy (with a total rms scatter, $\Delta_{rms}<0.11$
mag) using five components: a Ferrers bar (Ferrers, 1877); an exponential
disk; two Gaussian components to capture spiral arm crossings of the major
axis; and a point source at the centre. We calculate a total integrated
$3.6\,\mu{\rm m},\star$ apparent magnitude of $12.42\pm 0.11$ mag (AB).
Additional component magnitudes are tabulated in Table 1. Based on its
distance ($14.3\pm 1.1$ Mpc), we determine an absolute magnitude of $-18.37\pm
0.20$ mag for the galaxy at $3.6\,\mu{\rm m},\star$. Applying
$\Upsilon_{3.6\,\mu{\rm m},\star}=0.60\pm 0.09$ (Meidt et al., 2014) and
$\mathfrak{M}_{3.6\,\mu{\rm m},\odot}=6.02$ mag (AB) yields a total
logarithmic stellar mass of $\mathcal{M}_{{\rm gal},\star}=9.53\pm 0.10$ (cf.
Georgiev et al., 2016, $\mathcal{M}_{{\rm gal},\star}=9.53\pm 0.16$) for NGC
3319.

Table 1: NGC 3319 component magnitudes and masses. Columns: (1) Surface brightness profile component. (2) $3.6\,\mu{\rm m},\star$ apparent magnitude (AB). (3) $3.6\,\mu{\rm m},\star$ absolute magnitude (AB). (4) Logarithmic (solar) mass.  Component | $\mathfrak{m}_{3.6\,\mu{\rm m},\star}$ | $\mathfrak{M}_{3.6\,\mu{\rm m},\star}$ | $\mathcal{M}_{\star}$
---|---|---|---
| (mag) | (mag) | (dex)
(1) | (2) | (3) | (4)
PSF (NC) | $20.22\pm 0.32$ | $-10.57\pm 0.36$ | $6.41\pm 0.16$
Bar | $14.70\pm 0.53$ | $-16.09\pm 0.56$ | $8.62\pm 0.23$
Disk | $12.67\pm 0.10$ | $-18.12\pm 0.20$ | $9.43\pm 0.10$
Inner Spiral | $15.20\pm 0.10$ | $-15.59\pm 0.20$ | $8.42\pm 0.10$
Outer Spiral | $17.63\pm 0.29$ | $-13.16\pm 0.34$ | $7.45\pm 0.05$
Total | $12.42\pm 0.11$ | $-18.37\pm 0.20$ | $9.53\pm 0.10$

Savorgnan et al. (2016) discovered a distinct red and blue sequence for early-
and late-type galaxies in the $M_{\bullet}$–$M_{{\rm gal},\star}$ diagram,
forming a revision to the core-Sérsic (giant early-type galaxies) and Sérsic
(spiral and low-mass early-type galaxy) sequence from Graham (2012), Graham &
Scott (2013), and Scott et al. (2013). van den Bosch (2016) subsequently
showed this separation including additional galaxies, albeit with less
reliable black hole masses, while Terrazas et al. (2016) captured it in terms
of star formation rate. Here, we apply the latest relation established for
spiral galaxies with directly-measured black hole masses. Applying equation 3
(with $\upsilon\equiv 1$) from Davis et al. (2018), this total galaxy stellar
mass predicts a central black hole mass as follows,

$\mathcal{M}_{{\rm gal},\star}=9.53\pm
0.10\rightarrow\mathcal{M}_{\bullet}(M_{{\rm gal},\star})=3.38\pm 1.02,$ (6)

with $P(\mathcal{M}_{\bullet}\leq 5)=94\%$.

As can be seen in the images and from the ellipticity profile, there is no
mistaking that NGC 3319 possesses a strong bar that accounts for most of the
light from the inner $R_{\rm maj}\lesssim 30^{\prime\prime}$ ($\lesssim$2.1
kpc) region of the galaxy. There is no obvious evidence of a bulge (spheroid)
component; thus, NGC 3319 is considered to be a bulgeless galaxy. Even if one
were to describe the bar as a pseudobulge mistakenly, its logarithmic ‘bulge’
mass would only be $\mathcal{M}_{\rm bulge,\star}=8.62\pm 0.23$ (see Table 1).
If applied to the $M_{\bullet}$–$M_{\rm bulge,\star}$ relation from Davis et
al. (2019a, their equation 11), this would still comfortably predict an IMBH
of $\mathcal{M}_{\bullet}=3.73\pm 0.91$, with $P(\mathcal{M}_{\bullet}\leq
5)=92\%$.

###  2.3 The $M_{\bullet}$–$M_{{\rm NC},\star}$ relation

From our surface brightness profile decomposition of NGC 3319, we extracted a
central point source apparent magnitude of $\mathfrak{m}_{3.6\,\mu{\rm
m},\star}=20.22\pm 0.32$, yielding an absolute magnitude of
$\mathfrak{M}_{3.6\,\mu{\rm m},\star}=-10.57\pm 0.36$. We will assume that
this luminosity is due to the nuclear cluster (NC) of stars. Of course, some
contribution of flux will come from the AGN. Therefore, we estimate an upper
limit to the nuclear star cluster mass using $\Upsilon_{3.6\,\mu{\rm
m},\star}=0.60\pm 0.09$ and $\mathfrak{M}_{3.6\,\mu{\rm m},\odot}=6.02$ mag
(AB), to give $\mathcal{M}_{\rm NC,\star}\leq 6.41\pm 0.16$. We deem this to
be a reasonable estimate since it lies between the recent estimates of
$\mathcal{M}_{\rm NC,\star}=6.24\pm 0.07$ (Georgiev & Böker, 2014; Georgiev et
al., 2016) and $\mathcal{M}_{\rm NC,\star}=6.76\pm 0.07$ (Jiang et al., 2018),
both from Hubble Space Telescope imaging of NGC 3319.

Using the new $M_{\bullet}$–$M_{{\rm NC},\star}$ relation of Graham
(2020),111111See also Graham (2016a) and equation 12 from Graham et al.
(2019). given by rCl M_∙(M_NC,⋆)=&(2.62±0.42)log(MNC,⋆107.83M⊙)+
(8.22±0.20), and with an intrinsic scatter of 1.31 dex. However, due to AGN
contamination, we treat this as an upper-limit black hole mass estimate.
Therefore, we predict the following black hole mass,

$\mathcal{M}_{\rm NC,\star}\leq 6.41\pm
0.16\rightarrow\mathcal{M}_{\bullet}(M_{\rm NC,\star})\leq 4.51\pm 1.51,$ (7)

with $P(\mathcal{M}_{\bullet}\leq 5)\geq 63\%$.

###  2.4 The $M_{\bullet}$–$v_{\rm rot}$ relation

From HyperLeda121212http://leda.univ-lyon1.fr/ (Paturel et al., 2003), we
adopted their apparent maximum rotation velocity of the gas, $v_{\rm
max,g}=84.33\pm 1.80\,{\rm km\,s^{-1}}$ (homogenised value derived from 24
independent measurements), which is the observed maximum rotation velocity
uncorrected for inclination effect. We then converted this to a maximum
physical velocity rotation corrected for inclination ($v_{\rm rot}$) via

$v_{\rm rot}\equiv\frac{v_{\rm max,g}}{\sin{i_{\rm disk}}}=102.21\pm
2.20\,{\rm km\,s^{-1}}.$ (8)

Application of equation 10 from Davis et al. (2019b) gives

$v_{\rm rot}=102.21\pm 2.20\,{\rm
km\,s^{-1}}\rightarrow\mathcal{M}_{\bullet}(v_{\rm rot})=3.90\pm 0.59,$ (9)

with $P(\mathcal{M}_{\bullet}\leq 5)=97\%$.

###  2.5 The $M_{\bullet}$–$\sigma_{0}$ relation

We obtained the central stellar velocity dispersion from Ho et al. (2009) and
utilised equation 2 from Sahu et al. (2019b) to predict a black hole mass as
follows,

$\sigma_{0}=87.4\pm 9.2\,{\rm
km\,s^{-1}}\rightarrow\mathcal{M}_{\bullet}(\sigma_{0})=6.08\pm 0.67,$ (10)

with $P(\mathcal{M}_{\bullet}\leq 5)=5\%$. This black hole mass estimate is
the highest of all our estimates; it is our only discrete mass estimate of NGC
3319* with $\mathcal{M}_{\bullet}>5.2$.

Ho et al. (2009) presented a catalogue of pre-existing velocity dispersions,
observed sometime between 1982 and 1990 (Ho et al., 1995). The measurements
were weighted-mean dispersions from the blue- and red-side of the Double
Spectrograph (Oke & Gunn, 1982) mounted at the Cassegrain focus of the Hale
5.08 m telescope at Palomar Observatory. However, Ho et al. (2009) found that
the blue-side spectral resolution is insufficient to reliably measure
dispersions for most of the later-type galaxies in their sample, as was the
case for NGC 3319. Ho et al. (2009) only presented a red-side velocity
dispersion for NGC 3319. Moreover, Ho et al. (1995) noted that for their
observations of NGC 3319 ‘the continuum shape of its spectrum may be uncertain
because of imperfect correction for spatial focus variations’.

Although Jiang et al. (2018) do present a spectrum of NGC 3319 (see their
figure 5) from the Sloan Digital Sky Survey (SDSS), they do not report on the
velocity dispersion. The SDSS Data Release 12 (Alam et al., 2015)
states131313https://www.sdss.org/dr12/algorithms/redshifts/ that ‘best-fit
velocity-dispersion values $\lesssim$100 km s-1 are below the resolution limit
of the SDSS spectrograph and are to be regarded with caution’. Nonetheless, we
have attempted to measure the velocity dispersion from the SDSS spectrum (Fig.
5) and found $\sigma_{0}=99\pm 9\,{\rm km\,s^{-1}}$ (given the aforementioned
resolution limit, this is likely an upper limit), albeit with a discrepant
estimate of its recessional velocity. We found $cz=860\pm 6\,{\rm
km\,s^{-1}}$, which is markedly different from the SDSS value ($cz=713\pm
5\,{\rm km\,s^{-1}}$), or even the mean heliocentric radial velocity from
HyperLeda ($cz=738\pm 7\,{\rm km\,s^{-1}}$). Although $\sigma_{0}\lesssim 100$
km s-1, and is thusly suspicious, our measurement of $\sigma_{0}=99\pm 9\,{\rm
km\,s^{-1}}$ is consistent with the value from (Ho et al., 1995). Better
spectral resolution should provide greater clarity as to the velocity
dispersion of this galaxy, which might also be influenced by the nuclear star
cluster.

Figure 5: Fit to the SDSS spectrum of NGC 3319 by ppxf (Cappellari, 2017). The
relative flux of the observed spectrum ([0.35ex]6mm0.4mm) is over-plotted by
the ppxf fit ([0.35ex]6mm0.4mm) to the spectrum. The residuals to the fit
($\blacklozenge$) are at the bottom (normalised about the arbitrary horizontal
black line) along with residuals from the masked emission features
([0.35ex]6mm0.4mm), while grey vertical bands delineate the masked regions not
included in the $\chi^{2}$ minimisation of the fit. The fit is consistent with
$\sigma_{0}=99\pm 9\,{\rm km\,s^{-1}}$ and $cz=860\pm 6\,{\rm km\,s^{-1}}$.

###  2.6 The $M_{\bullet}$–$L_{\rm 2-10\,keV}$ relation

Mayers et al. (2018) studied a sample of 30 AGN ($z\leq 0.23$, $L_{\rm
2-10\,keV}\geq 10^{40.8}\,{\rm erg\,s^{-1}}$, and $\mathcal{M}_{\bullet}\geq
5.45$), with black hole masses estimated via Bentz & Katz (2015) and reported
a trend between black hole mass and X-ray luminosity. From 2-10 keV CXO
observations of the nuclear point source in NGC 3319, Jiang et al. (2018)
calculated a luminosity of $L_{\rm 2-10\,keV}=10^{39.0\pm 0.1}\,{\rm
erg\,s^{-1}}$. We applied the $M_{\bullet}$–$L_{\rm 2-10\,keV}$ relation of
Mayers et al. (2018, extracted from their figure 11), rCl M_∙(L_2-10 keV) =
&(0.58±0.05)log(L2-10 keV2×1043erg s-1)+
(7.46±0.34), with a scatter of 0.89 dex, such that rCl L_2-10 keV&=10^39.0±0.1
erg s^-1→
M_∙(L_2-10 keV)=4.97±0.98, with $P(\mathcal{M}_{\bullet}\leq 5)=51\%$.
However, given that the Eddington ratio will vary over time, as the AGN duty
cycle turns the AGN on and off, this is unlikely to be a stable mass
estimate.141414Woo & Urry (2002) found that the Eddington ratio for a given
black hole can vary, spanning a range of up to three orders of magnitude. In
order to be a stable relation, the $M_{\bullet}$–$L_{\rm 2-10\,keV}$ relation
would require the time-varying distribution of Eddington ratios for a given
black hole to resemble a normal distribution; several studies have found
supportive evidence for a peaked distribution (Kollmeier et al., 2006;
Steinhardt & Elvis, 2010; Lusso et al., 2012). The fundamental plane of black
hole activity (Section 2.9) can offer additional insight, with its
counterbalance from the waxing/waning radio emission.151515Although, unmatched
(in the radio) X-ray variability (typically not more than a factor of
$\approx$3; Timlin et al., 2020) can possibly contribute to the scatter in the
relation.

In what follows (Sections 2.7–2.9) are three black hole mass estimates from
Jiang et al. (2018), which are explicitly described here.

###  2.7 The $M_{\bullet}$–$\sigma_{\rm NXS}^{2}$ relation

From the light curves obtained by the CXO observations, Jiang et al. (2018)
estimated the X-ray variability, represented as the (10 ks) normalised excess
variance ($\sigma_{\rm NXS}^{2}$). They found $\sigma_{\rm NXS}^{2}=0.093\pm
0.088$. By applying the $M_{\bullet}$–$\sigma_{\rm NXS}^{2}$ relation from Pan
et al. (2015, figure 4), Jiang et al. (2018) obtained

$\sigma_{\rm NXS}^{2}=0.093\pm
0.088\rightarrow\mathcal{M}_{\bullet}(\sigma_{\rm NXS}^{2})=5.18\pm 1.92,$
(11)

with $P(\mathcal{M}_{\bullet}\leq 5)=46\%$. Clearly, with an upper 1$\sigma$
estimate for the black hole mass of $\sim$107 M⊙, on its own this is not
evidence for an IMBH.

###  2.8 Eddington ratio

Based upon the median radio-quiet quasar spectral energy distribution (SED) of
Elvis et al. (1994), Jiang et al. (2018) determined a bolometric luminosity of
$L_{\rm bol}=(3.6\pm 1.1)\times 10^{40}\,{\rm erg\,s^{-1}}$ for NGC 3319*, by
scaling the SED to the CXO luminosity ($L_{\rm 2-10\,keV}=10^{39.0\pm
0.1}\,{\rm erg\,s^{-1}}$) and integrating the entire SED. Using XMM-Newton and
CXO observations of NGC 3319*, Jiang et al. (2018) also determined a hard
X-ray photon index of $\Gamma=2.02\pm 0.27$. Following Jiang et al. (2018), we
converted this into an Eddington ratio, $\log(L_{\rm bol}/L_{\rm
Edd})=-0.56\pm 0.99$, with an Eddington luminosity, $L_{\rm Edd}\equiv
1.26\times 10^{38}\,M_{\bullet}(\rm M_{\odot}^{-1}\,{erg\,s}^{-1})$, via
equation 2 from Shemmer et al. (2008). Therefore, $L_{\rm Edd}=10^{41.12\pm
1.00}\,{\rm erg\,s^{-1}}$.

From this point in the calculation, Jiang et al. (2018) arbitrarily selected
$L_{\rm bol}/L_{\rm Edd}=0.1^{+0.9}_{-0.099}$, implying
$M_{\bullet}=3^{+297}_{-2.7}\times 10^{3}\,\mathrm{M_{\odot}}$. Thus, Jiang et
al. (2018) broadened the mass estimate to a range from $M_{\bullet}=3\times
10^{2}$ to $3\times 10^{5}\,\mathrm{M_{\odot}}$ for arbitrary Eddington ratios
ranging from 1 to $10^{-3}$, a range of 3 dex. For our purposes, we will
remain with the calculated $\log(L_{\rm bol}/L_{\rm Edd})=-0.56$ with $L_{\rm
Edd}=10^{41.12}\,{\rm erg\,s^{-1}}$, but will follow Jiang et al. (2018)’s
conservative 3 dex range of uncertainty by broadening our estimate to

$L_{\rm Edd}=10^{41.12\pm 1.50}\,{\rm
erg\,s^{-1}}\rightarrow\mathcal{M}_{\bullet}(L_{\rm Edd})=3.02\pm 1.50,$ (12)

with $P(\mathcal{M}_{\bullet}\leq 5)=91\%$.

###  2.9 Fundamental plane of black hole activity

Baldi et al. (2018) obtained high-resolution
($\leq$$0\mbox{$.\\!\\!^{\prime\prime}$}2$) 1.5 GHz-radio images of the core
in NGC 3319 but failed to detect a source; therefore, establishing an upper
limit to the luminosity, $L_{1.5\,{\rm GHz}}\leq 10^{35.03}\,{\rm
erg\,s}^{-1}$.161616Baldi et al. (2018) presented $L_{1.5\,{\rm GHz}}\leq
10^{34.84}\,{\rm erg\,s}^{-1}$ for NGC 3319, based on their adopted distance
of 11.5 Mpc. This radio luminosity can be applied to the fundamental plane of
black hole activity (Merloni et al., 2003; Falcke et al., 2004; Gültekin et
al., 2009; Plotkin et al., 2012; Dong & Wu, 2015; Liu et al., 2016; Nisbet &
Best, 2016), which demonstrates an empirical correlation between the continuum
X-ray, radio emission, and mass of an accreting black hole. This fundamental
plane applies to supermassive, as well as stellar-mass black holes; therefore,
it should also be suitable for the intervening population of IMBHs (e.g.
Gültekin et al., 2014). Using the fundamental plane of black hole activity,
Jiang et al. (2018) reported a black hole mass estimate of
$\leq$$10^{5}\,\mathrm{M_{\odot}}$. However, it is typically the 5 GHz, not
the 1.5 GHz luminosity as we have, that is employed in the fundamental plane
relation. Therefore, we follow the radiative flux density,
$S_{\nu}\propto\nu^{\alpha_{R}}$, conversion of Qian et al. (2018) by adopting
$\alpha_{R}=-0.5\pm 0.1$ as the typical radio spectral index of bright (high
Eddington ratio) AGN. Doing so, this predicts an associated 5 GHz luminosity
of $L_{5\,{\rm GHz}}\leq 10^{34.77\pm 0.05}\,{\rm erg\,s}^{-1}$. Using this
value along with $L_{\rm 2-10\,keV}=10^{39.0\pm 0.1}\,{\rm erg\,s^{-1}}$
(Section 2.6), we applied the relation of Gültekin et al. (2019, equation 8)
to predict the following upper limit to the black hole mass, rCl L_FP&≡(L_2-10
keV,L_5 GHz)
= (10^39.0±0.1,≤10^34.77±0.05) erg s^-1→
M_∙(L_FP)≤5.62±1.05, with $P(\mathcal{M}_{\bullet}\leq 5)\geq
28\%$.171717Given the connection between the black hole mass estimates from
the Eddington ratio (equation 12) and the fundamental plane (equation 16) we
also check that the former ($L_{\rm 2-10\,keV}=10^{39.0\pm 0.1}\,{\rm
erg\,s^{-1}}$ and $\mathcal{M}_{\bullet}(L_{\rm Edd})=3.02\pm 1.50$) is
consistent with no radio detection ($L_{5\,{\rm GHz}}\leq 10^{34.77\pm
0.05}\,{\rm erg\,s}^{-1}$). Using equation 19 from Gültekin et al. (2019),
with the radio luminosity as the dependent variable in their regression, we
find $L_{5\,{\rm GHz}}=10^{33.76\pm 1.41}\,{\rm erg\,s}^{-1}$. Thus, the
inverse prediction is consistent with no radio detection.

However, two issues make this particular prediction problematic. The first is
that the radio and X-ray data were not obtained simultaneously, and the
timescale for variations in flux will be short for IMBHs given that it scales
with the size of the ‘event horizon’ and thus with the black hole mass. The
second issue is that the ‘fundamental plane of black hole activity’ is
applicable to black holes with low accretion rates (Merloni et al., 2003,
their final paragraph of section 6), and NGC 3319* is considered to have a
high accretion rate (Jiang et al., 2018, see their section 3.2). Therefore, we
do not include this estimate in our derivation of the black hole mass.

###  2.10 The $M_{\bullet}$–$\mathcal{C}_{\text{FUV,tot}}$ relation

Dullo et al. (2020) present a relationship between the black hole mass and its
host galaxy’s UV$-3.6\,\mu{\rm m}$ colour181818See also the dependence of
black hole mass on the colour of its host galaxy presented by Zasov &
Cherepashchuk (2013). from their study of 67 galaxies with directly-measured
black hole masses. From table D1 in Dullo et al. (2020), the predicted black
hole mass for NGC 3319* is $\mathcal{M}_{\bullet}=5.36\pm 0.85$, based on its
FUV$-3.6\,\mu{\rm m}$ colour ($\mathcal{C}_{\text{FUV,tot}}$).191919Dullo et
al. (2020) also supply an $M_{\bullet}$–$\mathcal{C}_{\text{NUV,tot}}$
relation, but given the similarity with the
$M_{\bullet}$–$\mathcal{C}_{\text{FUV,tot}}$ relation, we prefer to use the
FUV relation due to its smaller uncertainty on the slope and intercept.
However, we can further refine this prediction by accounting for the internal
dust extinction in NGC 3319. Given that NGC 3319 is bulgeless, we treat it as
being all disk. Using our adopted inclination, $i_{\rm
disk}=55\mbox{$.\\!\\!^{\circ}$}6\pm 0\mbox{$.\\!\\!^{\circ}$}2$, and applying
equations 2 and 4 from Dullo et al. (2020, see also ), we find that these
corrections make NGC 3319 $0.57\pm 0.16$ mag brighter in the ultraviolet and
$0.05\pm 0.02$ mag brighter at $3.6\,\mu{\rm m}$.202020We note the caveat that
the relations of Dullo et al. (2020) are based on bulge plus disk magnitudes,
not total galaxy magnitudes. In contrast, the colours from Bouquin et al.
(2018), which were used to predict black hole masses in table D1 of Dullo et
al. (2020), are derived from asymptotic magnitudes that may include additional
fluxes from bars, rings, and nuclear components. For NGC 3319, we assume the
bar and disk have the same colour and require the same dust correction because
bars are just the inner parts of disks that have changed their orbits. Thus,
the change in colour will be $0.52\pm 0.14$ mag bluer, which updates the
FUV$-3.6\,\mu{\rm m}$ colour from Bouquin et al. (2018) to an internal-dust-
corrected $\mathcal{C}_{\text{FUV,tot}}=1.16\pm 0.14$ mag. Using the BCES
bisector (Akritas & Bershady, 1996)
$M_{\bullet}$–$\mathcal{C}_{\text{FUV,tot}}$ relation for late-type galaxies
with a slope of $1.03\pm 0.13$ from table 2 in Dullo et al. (2020), we obtain
a black hole mass which is $1.03\pm 0.13\times 0.52\pm 0.14=0.53\pm 0.16$ dex
smaller than reported in table D1. This revision reduces the tabulated
estimate of $\mathcal{M}_{\bullet}$ from $5.36\pm 0.85$ to $4.83\pm 0.87$.
Thus, rCl C_FUV,tot&=1.16±0.14 mag→
M_∙(C_FUV,tot)=4.83±0.87, with $P(\mathcal{M}_{\bullet}\leq 5)=58\%$.

###  2.11 Probability density function

With such a multitude of mass estimates and a hesitancy to place confidence in
one measurement alone, we combined the aforementioned mass estimates (except
for that from equation 2.6) to yield a single black hole mass estimate for NGC
3319*. We did so by analysing the probability density function (PDF) of the
distribution of mass estimates (see Fig. 6). For our seven selected black hole
mass estimates (equations 4, 6, 9, 10, 11, 12, and 20), we let a normal
distribution represent each estimate with their respective means
($\mathcal{\bar{M}}_{\bullet}$) and standard deviations
($\delta\mathcal{\bar{M}}_{\bullet}$). We then added the seven Gaussians
together to produce a combined summation. To ensure the area of the summation
is equal to one, we divided each of the seven Gaussian addends by seven so
that the area under each Gaussian equalled $1/7$.

Figure 6:  Determination of the PDF of the black hole mass estimates for NGC
3319*. The PDF ([0.35ex]8mm0.4mm) is the best-fit skew-kurtotic-normal
distribution to the Sum ([0.35ex]8mm1pt1mm) of each of the seven selected
black hole mass estimates’ normal distributions;
[0.35ex]8mm1pt1mm = [0.35ex]8mm1pt1pt \+ [0.35ex]8mm1pt1pt \+
[0.35ex]8mm1pt1pt \+ [0.35ex]8mm1pt1pt \+ [0.35ex]8mm1pt1pt \+
[0.35ex]8mm1pt1pt \+ [0.35ex]8mm1pt1pt. The solid vertical line
([0.35ex]8mm0.4mm) indicates the position of
$\mathcal{\widehat{M}}_{\bullet}$. The dotted vertical lines
([0.35ex]8mm1pt1pt) demarcate
$\mathcal{\widehat{M}}_{\bullet}-\delta^{-}\mathcal{\widehat{M}}_{\bullet}$
(left) and
$\mathcal{\widehat{M}}_{\bullet}+\delta^{+}\mathcal{\widehat{M}}_{\bullet}$
(right, overlapping with the dashed line). The dashed vertical lines
([0.35ex]8mm1pt1mm) demarcate the upper- and lower-bound mass definitions of
an IMBH.

We fit a skew-kurtotic-normal distribution to the summation and measured the
peak (mode) black hole mass of the PDF as

$\mathcal{\widehat{M}}_{\bullet}\equiv\mathcal{M}_{\bullet}(\max{P})=4.50_{-0.52}^{+0.51},$
(13)

where $\mathcal{M}_{\bullet}(\max{P})$ is the black hole mass when the
probability ($P$) reaches its maximum ($\max{P}=0.272$). We quantify its
standard error as rCl δ^+^M_∙&≡ RWHM2Nln2 = 0.51 and
δ^-^M_∙≡ LWHM2Nln2 = 0.52, with right width at half max ${\rm RWHM}=1.59$ dex,
left width at half max ${\rm LWHM}=1.63$ dex, the number of predictors $N=7$,
and $P(\mathcal{\widehat{M}}_{\bullet}\leq 5)=84\%$. For a complete comparison
of all the mass estimates, see Table 2 and Fig. 7.

Table 2: NGC 3319* mass predictions. Columns: (1) Black hole mass scaling relation predictor. (2) Logarithmic black hole (solar) mass. (3) Probability that NGC 3319* is $\leq$$10^{5}\,\mathrm{M_{\odot}}$, via equation (5).  Predictor | $\mathcal{M}_{\bullet}\pm\delta\mathcal{M}_{\bullet}$ | $P(\mathcal{M}_{\bullet}\leq 5)$
---|---|---
| (dex) | (%)
(1) | (2) | (3)
$|\bar{\phi}|=28\mbox{$.\\!\\!^{\circ}$}0\pm 2\mbox{$.\\!\\!^{\circ}$}3$ | $4.79\pm 0.54$ | 65
$\mathcal{M}_{{\rm gal},\star}=9.53\pm 0.10$ | $3.38\pm 1.02$ | 94
$\mathcal{M}_{\rm NC,\star}\leq 6.41\pm 0.16$ |  $\leq$$4.51\pm 1.51^{\dagger}$ |  $\geq$63
$v_{\rm rot}=102.21\pm 2.20\,{\rm km\,s^{-1}}$ | $3.90\pm 0.59$ | 97
$\sigma_{0}=87.4\pm 9.2\,{\rm km\,s^{-1}}$ | $6.08\pm 0.67$ | 5
$L_{\rm 2-10\,keV}=10^{39.0\pm 0.1}\,{\rm erg\,s^{-1}}$ | $4.97\pm 0.98^{\dagger}$ | 51
$\sigma_{\rm NXS}^{2}=0.093\pm 0.088$ | $5.18\pm 1.92$ | 46
$L_{\rm Edd}=10^{41.12\pm 1.50}\,{\rm erg\,s^{-1}}$ | $3.02\pm 1.50$ | 91
${L_{\rm FP}}\equiv\left(L_{\rm 2-10\,keV},L_{5\,{\rm GHz}}\right)=(10^{39.0\pm 0.1},$$\leq$$10^{34.77\pm 0.05})\,{\rm erg\,s}^{-1}$ |  $\leq$$5.62\pm 1.05^{\dagger}$ |  $\geq$28
$\mathcal{C}_{\text{FUV,tot}}=1.16\pm 0.14$ mag | $4.83\pm 0.87$ | 58
PDF | $4.50_{-0.52}^{+0.51}$ | 84
† Excluded from the black hole mass PDF.
Figure 7:  Forest plot of the ten different black hole mass estimates of NGC
3319*. Seven discrete mass estimates ($\blacksquare$) are used to generate the
$\mathcal{\widehat{M}}_{\bullet}$ estimate ($\bullet$) at the bottom of the
figure. The black hole mass estimate from the X-ray luminosity,
$\mathcal{M}_{\bullet}(L_{2-10\,{\rm keV}})$, is plotted ($\square$), but not
included in the calculation of $\mathcal{\widehat{M}}_{\bullet}$. Nor are the
upper-limit black hole mass estimates included in the calculation of
$\mathcal{\widehat{M}}_{\bullet}$. The black hole mass estimates from the
nuclear star cluster mass, $\mathcal{M}_{\bullet}(\mathcal{M}_{\rm
NC,\star})$, and the fundamental plane of black hole activity,
$\mathcal{M}_{\bullet}(L_{\rm FP})$, are upper-limit black hole mass estimates
depicted by left-pointing open triangles ($\triangleleft$). The vertical lines
are equivalent to those found in Fig. 6.

##  3 Discussion

We have presented multiple mass estimates for NGC 3319*, eight of which are
discrete estimates, and two are upper limits (Sections 2.3 and 2.9). The non-
detection of a nuclear source in the radio observations places an upper limit
that is indeed higher than most of our other mass estimates. This missing
radio detection begs for future deep, high spatial resolution radio (along
with simultaneous X-ray) observations to provide an improved mass estimate for
NGC 3319* via the fundamental plane of black hole activity. Nonetheless, the
upper-limit mass estimate from the fundamental plane of black hole activity
(equation 16) is in agreement with our other mass estimate derived from X-ray
measurements (equation 2.6).

Amongst our numerous mass estimates, it is perhaps the most well-known black
hole mass scaling relation ($M_{\bullet}$–$\sigma_{0}$) that produces the
highest mass estimate. Indeed, equation (10) provides the only mass estimate
that is not consistent with $\mathcal{M}_{\bullet}\leq 5$. It would be of
interest to obtain a suitably high-spectral resolution measurement of
$\sigma_{0}$ for NGC 3319 to confirm or revise the solitary measurement that
is (now) at least 31 years old. Although, it is not unprecedented to find a
black hole that is anomalously under-massive with respect to the
$M_{\bullet}$–$\sigma_{0}$ relation (Zaw et al., 2020).

We have used the latest refinement of the $M_{\bullet}$–$\sigma_{0}$ relation
by Sahu et al. (2019b) to estimate the black hole mass. Building on Davis et
al. (2017), Sahu et al. (2019b) have determined that
$M_{\bullet}\propto\sigma_{0}^{5.82\pm 0.75}$ from an analysis of 46 spiral
galaxies with central velocity dispersion measurements and directly-measured
black hole masses. However, none of these galaxies have black hole masses
below $\sigma_{0}\approx 100$ km s-1
($M_{\bullet}=10^{6}\,\mathrm{M_{\odot}}$). Sahu et al. (2019b, their figure
2) have revealed a tendency for galaxies with central velocity dispersions
less than $\sim$100 km s-1 to reside above the $M_{\bullet}$–$\sigma_{0}$
relation defined by the galaxies with higher velocity dispersions and
directly-measured black hole masses (i.e. spatially resolved kinematics, not
reverberation mapping nor single-epoch spectra coupled with a constant virial
$f$-factor). Therefore, should the velocity dispersion of NGC 3319 be lower
than $\sigma_{0}\approx 100$ km s-1, a shallower $M_{\bullet}$–$\sigma_{0}$
relation than used here will be required.

Baldassare et al. (2020) demonstrated that extrapolations of the shallow
$M_{\bullet}$–$\sigma_{0}$ relation for ‘classical bulges’ from Kormendy & Ho
(2013) appears (perhaps superficially) valid down to black hole masses of
$10^{5}\,\mathrm{M_{\odot}}$, with black hole mass estimates derived from
single-epoch spectroscopic (virial; with assumption of an $f$-factor to
account for the unknown broadline region geometry) masses. If we exclude the
$M_{\bullet}$–$\sigma_{0}$ mass estimate altogether, our
$\mathcal{\widehat{M}}_{\bullet}$ black hole mass estimate for NGC 3319*
(equation 13) becomes $\mathcal{\widehat{M}}_{\bullet}=4.14_{-0.49}^{+0.50}$,
with $P(\mathcal{\widehat{M}}_{\bullet}\leq 5)=96\%$, based on the remaining
six discrete measures used in Fig. 6. Additionally, if we treat the nuclear
star cluster upper-limit mass estimate as a discrete estimate, we arrive at
$\mathcal{\widehat{M}}_{\bullet}=4.19_{-0.47}^{+0.48}$, also with
$P(\mathcal{\widehat{M}}_{\bullet}\leq 5)=96\%$. This is based again on seven
measures, except now excluding the $M_{\bullet}$–$\sigma_{0}$ and
$M_{\bullet}$–$L_{X}$ relation estimates, as well as the fundamental plane
estimate.

###  3.1 Similarity to the IMBH in LEDA 87300

The IMBH candidate in LEDA 87300 (RGG 118) has been proclaimed the ‘smallest’
reported in a galaxy nucleus212121The first strong IMBH candidate is an off-
centre ultra-luminous X-ray point source that is too bright to be an accreting
stellar-mass black hole (Farrell et al., 2009). (Baldassare et al., 2015,
2017; Graham et al., 2016). We adopt the same redshift ($z=0.02647\pm
0.00026$) as Graham et al. (2016), but instead invoke the latest cosmographic
parameters ($H_{0}=67.66\pm 0.42$ km s-1 Mpc-1, $\Omega_{\Lambda}=0.6889\pm
0.0056$, and $\Omega_{\rm m}=0.3111\pm 0.0056$) from Planck Collaboration et
al. (2020, equation 28) to calculate a Hubble flow (comoving radial) distance
of $116.6\pm 1.3$ Mpc (Wright, 2006). This adjustment yields a mass of
$\mathcal{M}_{\bullet}=4.48^{+0.52}_{-0.69}$ for the IMBH (LEDA 87300*) in
LEDA 87300, as determined by Graham et al. (2016), with
$P(\mathcal{M}_{\bullet}\leq 5)=84\%$. This was based on a virial $f$-factor
of 2.8 (Graham et al., 2011) and the assumption that the
$M_{\bullet}$–$\sigma_{0}$ relation for AGN and quiescent galaxies can be
extrapolated below $10^{6}$ M⊙. Thus, the masses of NGC 3319* and LEDA 87300*
are nearly identical, $3.14_{-2.20}^{+7.02}\times 10^{4}\,\mathrm{M_{\odot}}$
and $3.00_{-2.38}^{+6.93}\times 10^{4}\,\mathrm{M_{\odot}}$, respectively.
However, given the overlapping error bars associated with both black holes,
the best we can conclude at this time is that their masses may be similar.

###  3.2 Environment and secular evolution

NGC 3319 is a relatively isolated galaxy in a group of four galaxies: NGC
3104, 3184, 3198, and 3319 (Tully, 1988). Its nearest neighbour at present is
most likely NGC 3198. NGC 3198 is at a distance ($d$) from us of $14.5\pm 1.3$
Mpc (Cepheid variable star distance from Kelson et al., 1999), J2000 right
ascension ($\alpha$) of $10^{\rm h}19^{\rm m}55^{\rm s}$, and J2000
declination ($\delta$) of $+45^{\circ}33^{\prime}09^{\prime\prime}$, while NGC
3319 is at $d=14.3\pm 1.1$ Mpc, $\alpha=10^{\rm h}39^{\rm m}09.\\!\\!^{\rm
s}8$, and $\delta=+41^{\circ}41^{\prime}15\mbox{$.\\!\\!^{\prime\prime}$}9$.
Based on the heliocentric spherical coordinates of each galaxy, the physical
distance between galaxies is rCl ∥ →d_1-→d_2 ∥ &≡
d_1^2+d_2^2-2d_1d_2cos(α_1-α_2)
-2d_1d_2sinα_1sinα_2[cos(δ_1-δ_2)-1]. The physical separation between NGC 3198 and NGC 3319 is thus $1.3\pm 0.2$ Mpc.

With this level of isolation, NGC 3319 will likely experience many gigayears
of relative tranquillity, without any significant galaxy mergers. If so, NGC
3319* should continue to coevolve along with its host galaxy via secular
accretion and feedback. There is no telling evidence that NGC 3319 has
experienced a recent major merger. However, we do note that Moore & Gottesman
(1998) detected a small system ($4.2\times 10^{7}\,\mathrm{M_{\odot}}$), just
$11^{\prime}$ ($46\pm 4$ kpc) south of NGC 3319. Moore & Gottesman (1998)
postulate that tidal interactions between this object and NGC 3319 likely
explain the distorted spiral structure, H i tail, and velocity perturbations
in the southern half of the galaxy.

###  3.3 Direct measurements of NGC 3319*

Stellar remnant black holes are thought to exist between the Tolman-
Oppenheimer-Volkoff limit of $\approx$$2.17\,\mathrm{M_{\odot}}$ for cold,
non-rotating neutron stars (Tolman, 1939; Oppenheimer & Volkoff, 1939;
Margalit & Metzger, 2017; Shibata et al., 2017; Ruiz et al., 2018; Rezzolla et
al., 2018) and $\lesssim$60–$80\,\mathrm{M_{\odot}}$ from the collapse of
massive stars estimated from evolutionary models (Belczynski et al., 2010;
Woosley, 2017; Spera & Mapelli, 2017). Recent observations have found the
least massive known black hole ($\approx$$3.3\,\mathrm{M_{\odot}}$; Thompson
et al., 2019).222222See also the recent $3.04\pm 0.06\,\mathrm{M}_{\odot}$
black hole candidate (Jayasinghe et al., 2021). Over the past couple of years,
black holes have been discovered that begin to surpass the low-mass definition
of IMBHs: $84.4^{+15.8}_{-11.1}\,\mathrm{M_{\odot}}$ (Abbott et al., 2019a)
and $98^{+17}_{-11}$ M⊙ (Zackay et al., 2019). The gravitational-wave signal
GW190521 (Abbott et al., 2020a) is consistent with the BH-collisional-creation
of a $142^{+28}_{-16}$ M⊙ IMBH. Its properties and astrophysical implications
(Abbott et al., 2020b) are further remarkable given the high confidence that
at least one of its progenitors lay in the mass gap predicted by pair-
instability supernova theory (Woosley, 2017).232323Alternatively, Roupas &
Kazanas (2019) propose that black holes between 50 and 135
$\mathrm{M_{\odot}}$ can form via rapid gas accretion in primordial dense
clusters.

The dwarf elliptical galaxy NGC 205 (M110), which is a satellite of the
Andromeda Galaxy (M31), is presently the least massive nuclear black hole
measured via direct methods. Nguyen et al. (2019) estimated a black hole mass
of $\mathcal{M}_{\bullet}=3.83_{-0.60}^{+0.43}$ via stellar dynamical
modelling. Furthermore, this galaxy seemingly confirms the extrapolation of
scaling relations into the IMBH regime. Explicitly, its black hole mass is
consistent with the prediction, $\mathcal{M}_{\bullet}(\sigma_{0})=3.86\pm
0.55$, of the $M_{\bullet}$–$\sigma_{0}$ relation (Sahu et al., 2019b,
equation 1) with $\sigma_{0}=33.1\pm 4.8\,{\rm km\,s^{-1}}$ from HyperLeda.

In order to dynamically estimate the mass of NGC 3319*, it is necessary to
resolve motions within its sphere of influence (SOI). According to Peebles
(1972), the gravitational SOI of a black hole residing at the centre of a
galaxy has a radius,

$r_{h}\equiv\frac{GM_{\bullet}}{{\sigma_{0}}^{2}}.$ (14)

Based on its (questionably high) velocity dispersion (equation 10), its
$\mathcal{\widehat{M}}_{\bullet}$ black hole mass estimate (equation 13), and
distance, we obtain $r_{h}=17.7_{-12.2}^{+40.8}\,{\rm
mpc}=255_{-176}^{+591}\,\mu{\rm as}$ for NGC 3319*.242424Because we question
the discrepantly high $\sigma_{0}$ value from equation 10, we alternatively
can use the mass prediction of
$\mathcal{\widehat{M}}_{\bullet}=4.14_{-0.49}^{+0.50}$ (which does not
consider equation 10) to predict $\sigma_{0}$ from the
$M_{\bullet}$–$\sigma_{0}$ relation. Reversing the relation from Sahu et al.
(2019b, equation 2), we find that $\sigma_{0}=46.8\pm 16.9\,{\rm km\,s^{-1}}$.
Using this value now instead of the observed $\sigma_{0}$, equation 14 yields
$r_{h}=27.4_{-16.5}^{+79.6}\,{\rm mpc}=395_{-237}^{+1151}\,\mu{\rm as}$ for
NGC 3319*.

The Atacama Large Millimeter Array (ALMA) is useful for probing the gaseous
cores of galaxies, including the rotating, torus-shaped, circumnuclear rings
of molecular gas that enable measurements of the central black hole mass (e.g.
García-Burillo et al., 2014; Yoon, 2017; Combes et al., 2019; Davis et al.,
2020). ALMA currently has an impressive FWHM spatial resolution of 20 mas at
230 GHz. The East Asian VLBI Network (EAVN; see Wajima et al., 2016; Hada et
al., 2017; An et al., 2018) has achieved a spatial resolution of 0.55 mas (550
$\mu$as) at 22 GHz. Similar milliarcsecond-scale resolution can be expected
from the Long Baseline Array (LBA; Edwards & Phillips, 2015) and the European
VLBI Network (EVN; e.g. Radcliffe et al., 2018). The Very Long Baseline Array
(VLBA) could likely resolve the SOI of NGC 3319*, with its spatial resolution
of 0.12 mas (120 $\mu$as) by utilising its longest baseline at 3 mm, currently
between Mauna Kea, Hawaii and North Liberty,
Iowa.252525https://science.nrao.edu/facilities/vlba/docs/manuals/oss/ang-res
The Event Horizon Telescope (EHT) can also resolve the SOI of NGC 3319*, with
its PSF FWHM of 20 $\mu$as. The EHT was able to resolve the emission ring,
showing the event horizon, surrounding the SMBH M87* with a diameter of $42\pm
3\,\mu{\rm as}$ (Event Horizon Telescope Collaboration et al., 2019).

Due to the difficulty of obtaining a direct measurement of the mass of NGC
3319*, it would be prudent to first study the AGN in NGC 3319 via
reverberation mapping (RM) methods. In this respect, the bulgeless spiral
galaxy NGC 4395 is the prototype. NGC 4395 possesses one of the least massive
nuclear black holes that has ever been measured via direct methods. den Brok
et al. (2015) obtained a black hole mass estimate of $4.0_{-1.0}^{+2.7}\times
10^{5}\,\mathrm{M_{\odot}}$ via gas dynamical modelling; Brum et al. (2019)
similarly obtained $2.5_{-0.8}^{+1.0}\times 10^{5}\,\mathrm{M_{\odot}}$ via
gas kinematics. These direct measurements were preceded by informative RM
black hole mass estimates of $(3.6\pm 1.1)\times 10^{5}\,\mathrm{M_{\odot}}$
(Peterson et al., 2005) and $(4.9\pm 2.6)\times 10^{4}\,\mathrm{M_{\odot}}$
(Edri et al., 2012, see also Cho et al. 2020 and Burke et al. 2020). Likewise,
NGC 3319* could greatly benefit from further study by RM campaigns, or at
least single-epoch spectra mass estimates.

###  3.4 Implications

The abundance, or scarcity, of black holes in this new mass domain of IMBHs,
has a broad array of implications. These include:

  * •

Using low-mass AGN to extend the black hole scaling relations for predicting
black hole masses in galaxies with quiescent low-mass black holes.

  * •

IMBHs will enable further refinement of the $M_{\bullet}$–$M_{\rm
bulge,\star}$ and $M_{\bullet}$–$M_{\rm gal,\star}$ diagrams (e.g. Davis et
al., 2018, 2019a; Sahu et al., 2019a), further facilitating the advancement of
BH/galaxy coevolution theories (e.g. Kauffmann & Haehnelt, 2000; Croton et
al., 2006; Schaye et al., 2015).

  * •

Establishing the black hole mass function from stellar to SMBHs, and then
revising the black hole mass density of the Universe should IMBHs prove
abundant (Aller & Richstone, 2002; Graham et al., 2007; Shankar et al., 2009;
Davis et al., 2014; Mutlu-Pakdil et al., 2016).

  * •

Increased understanding of the build-up of galaxies in our hierarchical
Universe via merger events, including IMBH mergers; searches for and
constraints of merger rate densities for IMBH binaries (Abbott et al., 2019b;
Jani et al., 2019).

  * •

Connections with nuclear star clusters and ultra-compact dwarf galaxies
(Graham & Spitler, 2009; Neumayer & Walcher, 2012; Georgiev et al., 2016;
Nguyen et al., 2018; Graham, 2020; Neumayer et al., 2020) and predictions for
space-based gravitational wave detections involving longer wavelength
gravitational radiation than ground-based interferometers can detect
(Portegies Zwart, 2007; Mapelli et al., 2012; Fragione & Silk, 2020). The
much-anticipated Laser Interferometer Space Antenna (LISA; Amaro-Seoane et
al., 2017) will have a designed observational requirement of detecting the
coalescence of unequal mass black hole binaries of total intrinsic mass
$10^{4}$–$10^{6}\,\mathrm{M_{\odot}}$ at $z<3$. The merging of such black
holes (similar to NGC 3319*), each embedded in their nuclear star cluster,
should coalesce within a Hubble time due to dynamical friction (Ogiya et al.,
2020). LISA and the next generation of gravitational wave observatories should
also be able to find IMBHs in Milky Way globular clusters and the Local Volume
(Arca-Sedda et al., 2020).

  * •

The violent tidal disruption of white dwarf stars by IMBHs can trigger
calcium-rich supernovae, spurring the nucleosynthesis of iron-group elements,
and are capable of generating observable electromagnetic and gravitational-
wave energies (Rees, 1988; Haas et al., 2012; MacLeod et al., 2016; Andreoni
et al., 2017; Andreoni et al., 2020; Kuns et al., 2020; Anninos et al., 2019;
Malyali et al., 2019).

IMBHs represent the grail lemma, needed to fill the void in our demographic
knowledge of black holes, and tie up our inadequate theoretical understanding
of BH/galaxy coevolution, feedback, and the growth of the Universe’s most
massive black holes. Increased future study of NGC 3319* promises to yield
direct confirmation of the existence of an IMBH in AGN mode and offer
immediate and lasting scientific advancement.

## Acknowledgements

We are grateful to Jonah S. Gannon, who provided valuable expertise with the
spectroscopic analysis. BLD thanks David Nelson for the use of his secluded
office space during the COVID-19 pandemic. This research was supported by the
Australian Research Council’s funding scheme DP17012923. Parts of this
research were conducted by the Australian Research Council Centre of
Excellence for Gravitational Wave Discovery (OzGrav), through project number
CE170100004. This material is based upon work supported by Tamkeen under the
NYU Abu Dhabi Research Institute grant CAP3. This research has made use of
NASA’s Astrophysics Data System, and the NASA/IPAC Extragalactic Database
(NED) and Infrared Science Archive (IRSA). We acknowledge the use of the
HyperLeda database (http://leda.univ-lyon1.fr). We made use of the DS9
visualization tool (Joye & Mandel, 2003), part of NASA’s High Energy
Astrophysics Science Archive Research Center (HEASARC) software.

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