# Discovery of a Planar Black Hole Mass Scaling Relation for Spiral Galaxies

Benjamin L. Davis Center for Astrophysics and Space Science (CASS), New York
University Abu Dhabi, PO Box 129188, Abu Dhabi, UAE  Zehao Jin (金泽灏) Center
for Astrophysics and Space Science (CASS), New York University Abu Dhabi, PO
Box 129188, Abu Dhabi, UAE

(Received July 11, 2023; Revised September 14, 2023; Accepted September 15,
2023)

###### Abstract

Supermassive black holes (SMBHs) are tiny in comparison to the galaxies they
inhabit, yet they manage to influence and coevolve along with their hosts.
Evidence of this mutual development is observed in the structure and dynamics
of galaxies and their correlations with black hole mass ($M_{\bullet}$). For
our study, we focus on relative parameters that are unique to only disk
galaxies. As such, we quantify the structure of spiral galaxies via their
logarithmic spiral-arm pitch angles ($\phi$) and their dynamics through the
maximum rotational velocities of their galactic disks ($v_{\mathrm{max}}$). In
the past, we have studied black hole mass scaling relations between
$M_{\bullet}$ and $\phi$ or $v_{\mathrm{max}}$, separately. Now, we combine
the three parameters into a trivariate $M_{\bullet}$–$\phi$–$v_{\mathrm{max}}$
relationship that yields best-in-class accuracy in prediction of black hole
masses in spiral galaxies. Because most black hole mass scaling relations have
been created from samples of the largest SMBHs within the most massive
galaxies, they lack certainty when extrapolated to low-mass spiral galaxies.
Thus, it is difficult to confidently use existing scaling relations when
trying to identify galaxies that might harbor the elusive class of
intermediate-mass black holes (IMBHs). Therefore, we offer our novel
relationship as an ideal predictor to search for IMBHs and probe the low-mass
end of the black hole mass function by utilizing spiral galaxies. Already with
rotational velocities widely available for a large population of galaxies and
pitch angles readily measurable from uncalibrated images, we expect that the
$M_{\bullet}$–$\phi$–$v_{\mathrm{max}}$ fundamental plane will be a useful
tool for estimating black hole masses, even at high redshifts.

Astrostatistics (1882) — Galaxy evolution (594) — Hubble classification scheme
(757) — Intermediate-mass black holes (816) — Late-type galaxies (907) —
Regression (1914) — Scaling relations (2031) — Spiral galaxies (1560) — Spiral
pitch angle (1561)

††journal: The Astrophysical Journal Letters††software:
Astropy (Astropy Collaboration et al., 2013, 2018)
Hyper-Fit (Robotham & Obreschkow, 2015, 2016)
Matplotlib (Hunter, 2007)
NumPy (Harris et al., 2020)
Pandas (McKinney, 2010)
PySR (Cranmer, 2023)
Python (Van Rossum & Drake, 2009)
SciPy (Virtanen et al., 2020)
uncertainties

citenum

##  1 Introduction

Black hole mass scaling relations (i.e., relations with central black hole
mass as the dependent variable and some physical property of its host galaxy
as the independent variable) have evolved and proliferated over the past
quarter-century, beginning with the identification of a correlation between
black hole mass ($M_{\bullet}$) and the stellar mass of its host galaxy’s
bulge (Magorrian et al., 1998). The most reliable of these scaling relations
are those that are built and calibrated upon samples of galaxies with
dynamically-measured black hole masses (e.g., Graham & Scott, 2013; Graham,
2023a, b; Graham & Sahu, 2023a, b; Savorgnan et al., 2013, 2016; Savorgnan,
2016a, b; Davis et al., 2017, 2018, 2019a, 2019b, 2019c, 2021; Sahu et al.,
2019a, b, 2020, 2022a, 2022b; Sahu, 2021, 2022; Jin & Davis, 2023).111 We note
that alleged biases purported to exist between galaxies that host directly-
measured black holes and galaxies from the general population (Shankar et al.,
2016) have been quashed (Sahu et al., 2023).  To-date, only about 150 such
supermassive black holes (SMBHs) with $M_{\bullet}\gtrsim
10^{6}\,\mathrm{M}_{\sun}$ have been directly measured in just the nearest and
most massive galaxies.222See Jin & Davis (2023) for an online database of the
sample. With SMBHs expected to reside in the hearts of most every massive
galaxy (Rees, 1984), one can take their pick of scaling relations (see Graham,
2016; D’Onofrio et al., 2021, for informative reviews) to perform black hole
mass estimates for large numbers of galaxies in surveys to construct black
hole mass functions (e.g., Graham et al., 2007; Davis et al., 2014; Mutlu-
Pakdil et al., 2016).

The accuracy of scaling relations can vary significantly based on which
independent variable is selected; some variables are less accurate or not
applicable for certain galaxy morphologies (e.g., bulge relations are useless
for bulgeless galaxies). Moreover, the prevalence of morphologically-dependent
black hole mass scaling relations (e.g., Davis et al., 2018, 2019b; Sahu et
al., 2019a) hints that particular independent variables alone might not be
sufficient to cover different galaxy morphologies. Specifically, this is
evident from the different coefficients required for the same variables when
applied to separate morphologies in isolation. As such, it is problematic to
be restricted to using only one predictor of black hole mass. Thus, we seek a
methodology that incorporates multiple mass predictors in one relation.

Dynamical measurements of smaller black holes like intermediate-mass black
holes (IMBHs) are much more difficult to obtain because the gravitational
sphere of influence radius of a black hole is directly proportional to the
mass of the black hole. As expected, an observational bias exists among the
catalog of directly-measured black holes, i.e., only black holes that are
sufficiently massive and/or nearby are measurable (Batcheldor, 2010). Our
current sample of 145 dynamically-measured black holes ranges from $4\times
10^{5}\,\mathrm{M}_{\sun}\lesssim M_{\bullet}\lesssim 2\times
10^{10}\,\mathrm{M}_{\sun}$ (Jin & Davis, 2023). Although this sample reaches
down almost to the IMBH regime ($10^{3}\leq
M_{\bullet}<10^{5}\,\mathrm{M}_{\sun}$), the sample of black holes is very top
heavy with a median mass of $\approx$$10^{8}\,\mathrm{M}_{\sun}$. Therefore,
interpolation of existing black hole mass scaling relations are incapable of
predicting IMBHs and extrapolation is heavily reliant on the largest SMBHs. As
such, several studies have instead relied on meta-analyses to combine the
predictions of multiple black hole mass scaling relations to more securely
extrapolate down into the IMBH regime (Koliopanos et al., 2017; Graham &
Soria, 2019; Graham et al., 2019; Davis & Graham, 2021; Davis et al., 2023).

In a larger study (Jin & Davis, 2023), we used modern machine learning methods
to identify higher-dimensional ($n$-D) black hole mass scaling relations that
have lower intrinsic scatters than existing two-dimensional (2-D) black hole
mass scaling relations. With 145 galaxies and as much as a hundred different
measured quantities for every galaxy, the task of checking the vast number of
permutations of possible $n$-D black hole mass scaling relations is an immense
undertaking. Therefore, we applied modern machine learning methods to find the
best scaling relations, which ideally are an optimized combination of accuracy
and simplicity. For this task, we ran symbolic regression software PySR
(Cranmer, 2023) to find the best combination of variables and mathematical
operations to describe our dataset of directly-measured SMBH masses and their
host galaxy parameters.

In this letter, we describe in detail one such solution found by of our study:
a trivariate relationship between $M_{\bullet}$, logarithmic spiral-arm pitch
angle ($\phi$), and maximum rotational velocity ($v_{\mathrm{max}}$). We will
present our sample, fit, and analysis of the planar black hole mass scaling
relation in §2. In §3, we will discuss benefits, reasons, comparisons,
implications, and utility of our $M_{\bullet}$–$\phi$–$v_{\mathrm{max}}$
fundamental plane. Finally, we provide a summary of our findings and remark on
future work (in §4). We represent black hole masses ($M_{\bullet}$) throughout
this work as logarithmic (solar) masses ($\mathcal{M}_{\bullet}$), such that
$\mathcal{M}_{\bullet}\equiv\log(M_{\bullet}/\mathrm{M}_{\sun})$. All
uncertainties are quoted at $1\,\sigma\equiv 68.3\%$ confidence intervals;
median absolute deviations are given as uncertainties associated with medians.

##  2 Data and Analysis

###  2.1 Sample

Our sample consists of all spiral galaxies with $M_{\bullet}$, $\phi$, and
$v_{\mathrm{max}}$ measurements from Davis et al. (2019c). This yields a set
of 41 galaxies (not including the Milky Way), all with dynamically-measured
black hole masses (see references compiled by Davis et al., 2017, 2019b).
Pitch angles were consistently measured by Davis et al. (2017, 2019c)333For
details regarding the measurement of galactic logarithmic spiral-arm pitch
angles, see additional reading (Davis et al., 2012; Davis & Hayes, 2014;
Davis, 2015; Shields et al., 2022). and rotational velocities compiled by
Davis et al. (2019c, see references therein). This sample that we use to
construct the planar relation is listed in Table 1.

Table 1: Sample of Spiral Galaxies

Galaxy | $|\phi|$ | $v_{\mathrm{max}}$ | $\mathcal{M}_{\bullet}$
---|---|---|---
| [$\arcdeg$] | $\left[\frac{\text{km}}{{\text{s}}}\right]$ | [dex]
(1) | (2) | (3) | (4)
Circinus (catalog ) | $17\fdg 0\pm 3\fdg 9$ | $153\pm 7$ | $6.25\pm 0.11$
IC 2560 (catalog ) | $22\fdg 4\pm 1\fdg 7$ | $196\pm 3$ | $6.52\pm 0.11$
Milky Way (catalog )${}^{\ast}$${}^{\ast}$footnotemark:  | $13\fdg 1\pm 0\fdg 6$ | $198\pm 6$ | $6.60\pm 0.02$
NGC 224 (catalog ) | $8\fdg 5\pm 1\fdg 3$ | $257\pm 6$ | $8.15\pm 0.16$
NGC 253 (catalog ) | $13\fdg 8\pm 2\fdg 3$ | $196\pm 3$ | $7.00\pm 0.30$
NGC 613 (catalog ) | $15\fdg 8\pm 4\fdg 3$ | $289\pm 5$ | $7.57\pm 0.15$
NGC 1068 (catalog ) | $17\fdg 3\pm 1\fdg 9$ | $192\pm 12$ | $6.75\pm 0.08$
NGC 1097 (catalog ) | $9\fdg 5\pm 1\fdg 3$ | $241\pm 34$ | $8.38\pm 0.04$
NGC 1300 (catalog ) | $12\fdg 7\pm 2\fdg 0$ | $189\pm 28$ | $7.86\pm 0.14$
NGC 1320 (catalog ) | $19\fdg 3\pm 2\fdg 0$ | $183\pm 13$ | $6.77\pm 0.22$
NGC 1365 (catalog ) | $11\fdg 4\pm 0\fdg 1$ | $198\pm 3$ | $6.60\pm 0.30$
NGC 1398 (catalog ) | $9\fdg 7\pm 0\fdg 7$ | $289\pm 7$ | $8.03\pm 0.11$
NGC 1566 (catalog ) | $17\fdg 8\pm 3\fdg 7$ | $154\pm 14$ | $6.83\pm 0.30$
NGC 1672 (catalog ) | $15\fdg 4\pm 3\fdg 6$ | $213\pm 8$ | $7.70\pm 0.10$
NGC 2273 (catalog ) | $15\fdg 2\pm 3\fdg 9$ | $211\pm 16$ | $6.95\pm 0.06$
NGC 2748 (catalog ) | $6\fdg 8\pm 2\fdg 2$ | $188\pm 27$ | $7.54\pm 0.21$
NGC 2960 (catalog ) | $14\fdg 9\pm 1\fdg 9$ | $257\pm 34$ | $7.07\pm 0.05$
NGC 2974 (catalog ) | $10\fdg 5\pm 2\fdg 9$ | $284\pm 26$ | $8.23\pm 0.07$
NGC 3031 (catalog ) | $13\fdg 4\pm 2\fdg 3$ | $237\pm 10$ | $7.83\pm 0.09$
NGC 3079 (catalog ) | $20\fdg 6\pm 3\fdg 8$ | $216\pm 6$ | $6.38\pm 0.12$
NGC 3227 (catalog ) | $7\fdg 7\pm 1\fdg 4$ | $240\pm 10$ | $7.97\pm 0.14$
NGC 3368 (catalog ) | $14\fdg 0\pm 1\fdg 4$ | $218\pm 15$ | $6.89\pm 0.11$
NGC 3393 (catalog ) | $13\fdg 1\pm 2\fdg 5$ | $193\pm 48$ | $7.49\pm 0.05$
NGC 3627 (catalog ) | $18\fdg 6\pm 2\fdg 9$ | $188\pm 7$ | $6.94\pm 0.09$
NGC 4151 (catalog ) | $11\fdg 8\pm 1\fdg 8$ | $272\pm 16$ | $7.69\pm 0.37$
NGC 4258 (catalog ) | $13\fdg 2\pm 2\fdg 5$ | $222\pm 8$ | $7.60\pm 0.01$
NGC 4303 (catalog ) | $14\fdg 7\pm 0\fdg 9$ | $214\pm 7$ | $6.78\pm 0.17$
NGC 4388 (catalog ) | $18\fdg 6\pm 2\fdg 6$ | $180\pm 5$ | $6.90\pm 0.10$
NGC 4395 (catalog ) | $22\fdg 7\pm 3\fdg 6$ | $145\pm 11$ | $5.62\pm 0.17$
NGC 4501 (catalog ) | $12\fdg 2\pm 3\fdg 4$ | $272\pm 4$ | $7.31\pm 0.08$
NGC 4594 (catalog ) | $5\fdg 2\pm 0\fdg 4$ | $277\pm 22$ | $8.81\pm 0.03$
NGC 4699 (catalog ) | $5\fdg 1\pm 0\fdg 4$ | $258\pm 7$ | $8.27\pm 0.09$
NGC 4736 (catalog ) | $15\fdg 0\pm 2\fdg 3$ | $182\pm 5$ | $6.83\pm 0.11$
NGC 4826 (catalog ) | $24\fdg 3\pm 1\fdg 5$ | $167\pm 9$ | $6.18\pm 0.12$
NGC 4945 (catalog ) | $22\fdg 2\pm 3\fdg 0$ | $171\pm 2$ | $6.13\pm 0.30$
NGC 5055 (catalog ) | $4\fdg 1\pm 0\fdg 4$ | $270\pm 14$ | $8.94\pm 0.10$
NGC 5495 (catalog ) | $13\fdg 3\pm 1\fdg 4$ | $202\pm 43$ | $7.04\pm 0.08$
NGC 5765b (catalog ) | $13\fdg 5\pm 3\fdg 9$ | $238\pm 15$ | $7.72\pm 0.05$
NGC 6926 (catalog ) | $9\fdg 1\pm 0\fdg 7$ | $246\pm 10$ | $7.68\pm 0.50$
NGC 7582 (catalog ) | $10\fdg 9\pm 1\fdg 6$ | $200\pm 9$ | $7.72\pm 0.12$
UGC 3789 (catalog ) | $10\fdg 4\pm 1\fdg 9$ | $210\pm 14$ | $7.07\pm 0.05$
UGC 6093 (catalog ) | $10\fdg 2\pm 0\fdg 9$ | $170\pm 59$ | $7.41\pm 0.03$

∗∗footnotetext: We do not include the Milky Way in our preferred determination
of the fundamental plane (see §A for further details).

Note. — This sample of 41 spiral galaxies (not including the Milky Way)
consists of all spiral galaxies with $\phi$, $v_{\mathrm{max}}$, and
$M_{\bullet}$ measurements from our larger sample of all galaxy types. The
full parent dataset of 145 galaxies is available online via Jin & Davis
(2023). Column (1): galaxy name. Column (2): absolute value of the _face-on_
(i.e., de-projected from the plane of the sky) spiral-arm pitch angle (in
degrees), from Davis et al. (2017, 2019c). Column (3): physical maximum
velocity rotation (in km s-1) corrected for inclination and compiled by Davis
et al. (2019c) from references therein. Column (4): dynamical black hole mass
(in dex, solar masses) measurement compiled by Davis et al. (2017, 2019b) from
references therein.

The sample of SMBH host galaxies exhibits a broad range in each of the three
variables. To illustrate this, we have plotted probability density functions
(PDFs) of each parameter in Figure 1. From these distributions, we normalize
the $\phi$ and $v_{\mathrm{max}}$ values about their respective medians to
minimize the covariance between the estimated coefficients during regression
analysis. Following an initial symbolic regression, we then use the outputs
from PySR (Cranmer, 2023) as our input initial guesses for Hyper-Fit (Robotham
& Obreschkow, 2015, 2016).444 The combination of PySR and Hyper-Fit is
necessary to produce a relation that considers and takes into account the
errors on individual measurements. PySR only considers the uncertainties on
the dependent variable (i.e., $M_{\bullet}$) without accounting for the
uncertainties on the independent variables (i.e., $\phi$ and
$v_{\mathrm{max}}$), and produces a best-fit relation without uncertainties on
the derived coefficients nor computing the intrinsic scatter of the relation.
Whereas, Hyper-Fit is able to refine the fit found by PySR while taking into
account errors on every measurement, producing uncertainties on each derived
coefficient, and determining the intrinsic scatter of the plane.

Figure 1:  The distributions of $\phi$ (_left_), $v_{\mathrm{max}}$
(_middle_), and $M_{\bullet}$ (_right_) for our sample of 41 spiral galaxies.
The smoothed histograms ([0.35ex]8mm1pt1mm) are generated from the summation
of all 41 respective measurements with their uncertainties for each galaxy
(i.e., kernel density estimations). The PDFs ([0.35ex]8mm0.4mm) are skew-
kurtotic-normal distributions fit to each histogram. Left: mean = $13\fdg 7\pm
5\fdg 4$, median = $13\fdg 2\pm 3\fdg 6$, and peak probability at $12\fdg 1\pm
0\fdg 9$. Middle: mean = $217\pm 44\,\mathrm{km\,s}^{-1}$, median = $211\pm
32\,\mathrm{km\,s}^{-1}$, and peak probability at $213\pm
7\,\mathrm{km\,s}^{-1}$. Right: mean = $7.30\pm 0.75$ dex, median = $7.27\pm
0.51$ dex, and peak probability at $7.28\pm 0.12$ dex.

###  2.2 Finding the Plane via Machine Learning

Symbolic Regression is a sub-field of machine learning that aims to find
mathematical expressions that best fit a given set of data. Symbolic
Regression searches over equations made of possible selections and
combinations of variables, operators, and constants, and judges these
equations with a score defined by both accuracy and simplicity. In this work,
we adopt the symbolic regression package PySR (Cranmer, 2023), which conducts
the equation search through a multi-population evolutionary algorithm. The
accuracy is defined by the mean squared error loss, and the simplicity is
characterized by a complexity score, where each use of variables, operators,
and constants adds some pre-defined complexity. The final score of an equation
aims to maximize the accuracy and penalize the complexity with a parsimony
constant.

The variable pool that we input to PySR includes all of the data from Davis et
al. (2017, 2019c), parameters modeled by the bulge/disk decompositions of
Davis et al. (2019b), and their derived spheroid stellar density properties
(Sahu et al., 2022b). These variables included, but are not limited to: pitch
angle, central stellar velocity dispersion, maximum rotational velocity,
galaxy stellar mass, and several properties of the spheroid, including Sérsic
index, half-light radius, stellar mass, and densities (apparent, projected,
and de-projected). We also included all available measurements from HyperLeda
(Makarov et al., 2014), e.g., colors, diameters, etc. The variable pool also
included multiple copies of each variable in different forms of natural
numbers, their logarithms, and trigonometric functions (e.g., $|\phi|$ or
$\tan|\phi|$ and $v_{\mathrm{max}}$ or $\log{v_{\mathrm{max}}}$). The
arithmetic operator pool is simply $+$, $-$, $\times$, and $\div$, with
additional $\log_{10}$, power, and exponentiation in rare cases. Based upon a
search on these criteria, PySR found an optimal correlation between
$\mathcal{M}_{\bullet}$, $\tan|\phi|$, and
$\log(v_{\mathrm{max}}/\mathrm{km}\,\mathrm{s}^{-1})$. A presentation and
discussion of other interesting scaling relations found by PySR are beyond the
intended scope of this letter, they will be addressed in a more comprehensive
work (Jin & Davis, 2023).

The functional form and initial parameters were identified by PySR and refined
via Hyper-Fit. The final fitted equation for the
$M_{\bullet}$–$\phi$–$v_{\mathrm{max}}$ relationship is lCl M_∙∼
&N[μ=α(tan—ϕ—-0.24)
\+ βlog(vmax211 km s-1) +γ,
σ=0.22±0.06], with $\alpha=-5.58\pm 0.06$, $\beta=3.96\pm 0.06$,
$\gamma=7.33\pm 0.05$, and intrinsic scatter ($\sigma$) in the
$\mathcal{M}_{\bullet}$-direction.555Hyper-Fit minimizes the intrinsic scatter
orthogonal to the plane and then performs a transformation from normal to
Cartesian coordinates and outputs the intrinsic scatter along the axis of the
dependent variable. We present a 3-D plot of the resulting plane in Figure 2.
The orientation of the plane intuitively matches the expectation of the
extreme cases:

  * •

the most massive black holes reside in host galaxies with tightly wound spiral
arms _and_ high rotational velocities,

  * •

the least massive black holes are found in galaxies with loosely wound spiral
arms _and_ low rotational velocities,

  * •

no black holes are found in galaxies with tightly wound spiral arms _and_ low
rotational velocities, and

  * •

no black holes are found in galaxies with loosely wound spiral arms _and_ high
rotational velocities.

For additional analyses and discussions, see Jin & Davis (2023) for higher-
dimensional relations featuring all galaxy types.666 In the symbolic
regression analysis of the spiral galaxies in our sample, we investigated and
considered higher-dimensional versions of Equation 2.2 that incorporated
additional quantities such as colors and bulge-to-total ratios. However, none
of our higher-dimensional combinations improved upon the optimization of
Equation 2.2 with the added expense of increased complexity and error
propagation. We find the planar relation is valid because it is built upon
parameters ($\phi$ and $v_{\mathrm{max}}$) that are unique to disk galaxies.
In our forthcoming work (Jin & Davis, 2023), we will present higher-
dimensional relations that we were able to find due to the larger combined
sample of late-type and early-type galaxies with their more varied ranges of
colors, bulge-to-total ratios, etc., as compared to our sample of just spiral
galaxies in the current work.

Figure 2:  The three-dimensional plot (viewed from four different vantage
points) of the planar $M_{\bullet}$–$\phi$–$v_{\mathrm{max}}$ relationship
(Equation 2.2). Onto the surface ($\blacklozenge$), we show the locations of
the 41 spiral galaxies ($\bullet$) from Jin & Davis (2023) used to define the
plane. The fainter gray planes above and below the darker gray middle plane
depict the intrinsic scatter bounds ($\pm 0.22$ dex in the
$\mathcal{M}_{\bullet}$-direction). This plot illustrates that our galaxies
are dispersed over the area of the plane, demonstrating a lack of degeneracy
between the parameters by the apparent embedding of the two-dimensional
manifold (i.e., surface) in three-dimensional space. For an animation of this
plot (showing also the intrinsic scatter bounds above and below the plane),
see the following link, http://surl.li/iggdg.

###  2.3 Error Analysis

One sign of a robust multi-parameter relationship is when different variables
contribute equitably. That is, one variable should not have an overly-dominant
influence on the relationship. Therefore, we need to check the relative change
in $M_{\bullet}$ when there is a proportional change in $\phi$ or
$v_{\mathrm{max}}$. To check this, we test Equation 2.2 with equivalent 10%
variations in the median values of $\phi$ or $v_{\mathrm{max}}$. Doing so, we
find that a change of 10% in $\phi$ leads to a 37.39% change in $M_{\bullet}$
or a 10% change in $v_{\mathrm{max}}$ leads to a 48.59% change in
$M_{\bullet}$. Ergo, in terms of overall weight, $v_{\mathrm{max}}$ accounts
for a slight majority (56.51%) of the variation in $M_{\bullet}$ as compared
to a similarly-sized variation in $\phi$. Thus, neither variable has an
outsized influence on $M_{\bullet}$.

We used the Hyper-Fit routine to robustly fit the equation of the plane to the
($\phi$, $v_{\mathrm{max}}$, $M_{\bullet}$) variable set, with consideration
of the individual uncertainties on all three parameters and accounting for
intrinsic scatter in the relation. As suggested by its name, Hyper-Fit is
uniquely designed to fit “linear models to multi-dimensional data with multi-
variate Gaussian uncertainties.” Additionally, Hyper-Fit calculates the
intrinsic scatter of a scaling relation, which can be considered as the root-
mean-square deviation in the observed data from the fitted function in the
case of zero measurement error.777For a more detailed description of intrinsic
scatter and its determination in galaxy scaling relations, see Stone et al.
(2021). Therefore, intrinsic scatter is the ideal parameter to judge and
compare the accuracy of various scaling relations.

Our determination of a fundamental plane of black hole mass in spiral galaxies
is ultimately advantageous because of its combination of the
$M_{\bullet}$–$\phi$ ($\sigma=0.33\pm 0.08$ dex) and
$M_{\bullet}$–$v_{\mathrm{max}}$ ($\sigma\sim 0.45$ dex) relations, reducing
the intrinsic scatter down to $\sigma=0.22\pm 0.06$ dex in the
$\mathcal{M}_{\bullet}$-direction. Previously, the $M_{\bullet}$–$\phi$
relation had the lowest level of intrinsic scatter among black hole mass
scaling relations for spiral galaxies. This reduction down to $\sigma=0.22\pm
0.06$ dex (now below a factor of $\log{2}\approx 0.3$ dex) with the planar
relation significantly improves upon the previous accuracy of the
$M_{\bullet}$–$\phi$ relation, which was already well below intrinsic scatters
available for black hole mass scaling relations built from samples of late-
and early-type galaxies. With such a low-level of intrinsic scatter, this
makes the $M_{\bullet}$–$\phi$–$v_{\mathrm{max}}$ relation the preeminent
scaling relation for black hole mass in spiral galaxies.

##  3 Discussion

###  3.1 The Benefit of Combining Two Relations

Because the $M_{\bullet}$–$\phi$–$v_{\mathrm{max}}$ relation is a combination
of the $M_{\bullet}$–$\phi$ and $M_{\bullet}$–$v_{\mathrm{max}}$ relations
(see the bottom panels of Figure 2 for projections of these relations), we
begin by comparing our 3-D relation to each of the 2-D relations (see Figure
3). First (in the left column of plots in Figure 3), we compare the
fundamental plane with the $M_{\bullet}$–$\phi$ relation (Davis et al., 2017,
equation 8). From a glance, we find that both relations display tight
correlations without significant outliers. For a more complete comparison, we
have included subplots that break down the performance of each relation versus
the planar relation across eight bins. One subplot shows root mean square
error ($\Delta_{\mathrm{rms}}$) in each bin and the other subplot shows the
mean absolute scatter ($\bar{\Delta}$) in each bin. As we can see, the
$M_{\bullet}$–$\phi$–$v_{\mathrm{max}}$ relation generally equals or
outperforms the $M_{\bullet}$–$\phi$ relation in all except for the most
massive bin. Although, the planar relation does tend to be biased towards
slightly over-massive black holes in the middle bins, as compared to the
$M_{\bullet}$–$\phi$ relation.

Second (in the right column of plots in Figure 3), we compare the fundamental
plane with the $M_{\bullet}$–$v_{\mathrm{max}}$ relation (Davis et al., 2019c,
equation 10). We can see that the planar relation is significantly more
accurate than the $M_{\bullet}$–$v_{\mathrm{max}}$ relation in the four most
massive bins. Here, the plane is only slightly lopsided towards over-massive
black hole predictions in the central bins, relative to the
$M_{\bullet}$–$v_{\mathrm{max}}$ relation. Overall, these comparisons to each
of the 2-D relations demonstrates that the fundamental plane performs well,
particularly so at the low-mass end, which should make it advantageous for
extrapolating toward lower mass black holes, i.e., IMBHs.

Figure 3:  Plots between the dynamically-measured $M_{\bullet}$ on the
$x$-axis and the $M_{\bullet}$ predicted by Equation 2.2 ($\bullet$) on the
$y$-axis, compared with the $M_{\bullet}$–$\phi$ relation from Davis et al.
(2017, equation 8) in the _left column_ of plots and the
$M_{\bullet}$–$v_{\mathrm{max}}$ relation from Davis et al. (2019c, equation
10) in the _right column_ of plots (both depicted with $\blacksquare$). The
subplots show histograms (spread across eight bins, each 0.41 dex wide) for
the $\Delta_{\mathrm{rms}}$ scatter (top subplots) and $\bar{\Delta}$ average
residual (bottom subplots) about the 1:1 line for both the planar relation
([0.35ex]8mm0.4mm) and the $M_{\bullet}$–$\phi$ (_left column_) or
$M_{\bullet}$–$v_{\mathrm{max}}$ (_right column_) relations (both depicted
with [0.35ex]8mm0.4mm). The dashed line ([0.35ex]8mm1pt1mm) depicts the 1:1
correlation between observed and predicted masses in the main plots and
$\bar{\Delta}=0.0$ dex in the lower subplots. For clarity, predicted errors
(along the $y$-axis) are not shown because they are directly proportional to
(and always greater than) the errors along the $x$-axis.

###  3.2 Explanation of the Fundamental Plane

The existence of a tight $M_{\bullet}$–$\phi$–$v_{\mathrm{max}}$ relation
built upon $M_{\bullet}$–$\phi$ and $M_{\bullet}$–$v_{\mathrm{max}}$ relations
is not a revelation, but rather an expected consequence of numerous prior
studies. It first goes back to the so-called “Hubble” tuning fork diagram
(Jeans, 1928; Hubble, 1936)888For an account of the complicated provenance of
the tuning fork, we suggest further reading (e.g., Block et al., 2004; Block &
Freeman, 2015; Graham, 2019, 2023c)., which established a clear and
understandable sequence that organized spiral galaxies into morphological
classes based upon the prominence of their bulges and the winding geometry of
their spiral arms. To simplify morphological trends, we can use the Hubble
sequence morphological stage number, $T$, where spiral galaxies are defined as
$T>0$ and higher numbers are considered to be “later” types. In this way, the
Hubble sequence qualitatively establishes $T\propto|\phi|$ and $T\propto
M_{\star,\mathrm{sph}}$, where $M_{\star,\mathrm{sph}}$ is the stellar mass of
the spheroid (bulge) component of a spiral galaxy. Decades after dissemination
of the Hubble sequence, many studies conducted quantitative studies showing
that indeed $|\phi|\propto T$ (Kennicutt, 1981; Seigar & James, 1998; Ma et
al., 1999; Baillard et al., 2011; Yu et al., 2018; Díaz-García et al., 2019;
Yu & Ho, 2019, 2020) and Davis et al. (2019b) showed that $|\phi|\propto
M_{\star,\mathrm{sph}}$.999The $\phi$–$M_{\star,\mathrm{sph}}$ relation is
actually a projection of the fundamental plane of spiral structure in disk
galaxies (Lin & Shu, 1966; Davis et al., 2015). It follows from these
correlations that there should be a correlation between $\phi$ and
$M_{\bullet}$ (Seigar et al., 2008; Berrier et al., 2013; Davis et al., 2017)
as both are strongly correlated to bulge mass.

As for uncovering the $M_{\bullet}$–$v_{\mathrm{max}}$ relation, we can look
first at the correlation between $v_{\mathrm{max}}$ and $T$ identified by
Roberts (1978). By substituting $|\phi|$ as a proxy for $T$, we then arrive at
the $\phi$–$v_{\mathrm{max}}$ relation (Kennicutt, 1981; Davis et al., 2019c).
Armed with the knowledge of both $\phi$–$M_{\bullet}$ and
$\phi$–$v_{\mathrm{max}}$ relations, Davis et al. (2019c) produced an
$M_{\bullet}$–$v_{\mathrm{max}}$ relation that is informed by, and consistent
with, the Tully-Fisher relation (Tully & Fisher, 1977; Tiley et al., 2019).
Thus, we now arrive at a unified $M_{\bullet}$–$\phi$–$v_{\mathrm{max}}$
relation that is a manifestation of the gravitational potential well of a
spiral galaxy. Ergo, in more massive galaxies with deeper potential wells, we
find more massive black holes, more tightly-wound spiral patterns, and higher
rotational velocities.

###  3.3 Fundamental Planes with Black Hole Mass

There have been a couple prior attempts at obtaining a fundamental plane
scaling relation for SMBHs. The most relevant example is the trivariate
relation between $M_{\bullet}$–$\sigma_{e}$–$R_{e}$, where $R_{e}$ is a
galaxy’s half-light radius and $\sigma_{e}$ is the stellar velocity dispersion
inside an aperture equal to $R_{e}$ (Marconi & Hunt, 2003; van den Bosch,
2016). For the $M_{\bullet}$–$\sigma_{e}$–$R_{e}$ relation, van den Bosch
(2016) found an intrinsic scatter of $\sigma=0.49\pm 0.03$ dex in the
$\mathcal{M}_{\bullet}$-direction. However, this is insignificant because van
den Bosch (2016) also found an identical intrinsic scatter for the
$M_{\bullet}$–$\sigma_{e}$ relation, meaning that the addition of the third
parameter, $R_{e}$, serves no purpose, and in practice makes things worse
because it introduces another variable that contributes to error propagation.
Moreover, van den Bosch (2016) utilized a large sample of 230 black hole mass
measurements that is “very heterogenous” because they are derived from a
variety of measurement methods, most of which are indirect and not from
dynamical methods.

The other notable example is the so-called fundamental plane of black hole
activity (Merloni et al., 2003; Falcke et al., 2004). This plane is one
between $M_{\bullet}$–$L_{\mathrm{R}}$–$L_{\mathrm{X}}$, where
$L_{\mathrm{R}}$ and $L_{\mathrm{X}}$ are radio and X-ray luminosity,
respectively. The fundamental plane of black hole activity is based upon the
interpretation of scale invariant disk–jet coupling manifesting as an
empirical relation between jet power probed by radio and mass accretion rate
via X-rays. In order to perceive the intrinsic disk–jet coupling mechanism,
this requires simultaneity of radio and X-ray observations to account for the
duty-cycle of active galactic nuclei. However, given that the processes that
govern the $M_{\bullet}$–$L_{\mathrm{R}}$–$L_{\mathrm{X}}$ relation are
highly-secular, this leads to an intrinsic scatter of $\sigma=0.96\pm 0.13$
dex in the $\mathcal{M}_{\bullet}$-direction (Gültekin et al., 2019),
“indicating a large amount of unexplained variance.” Additionally, Gültekin et
al. (2022) caution against using the fundamental plane of black hole activity
without additional constraints beyond just straightforward X-ray and radio
observations. All together, the noted problems with either of the
aforementioned planar relations further cements the
$M_{\bullet}$–$\phi$–$v_{\mathrm{max}}$ relation’s superiority as a best-in-
class black hole mass scaling relation for spiral galaxies.

###  3.4 Implications

One advantageous application we envision for the
$M_{\bullet}$–$\phi$–$v_{\mathrm{max}}$ relation is to use it to construct
black hole mass functions (BHMFs) from surveys of spiral galaxies. Already,
the $M_{\bullet}$–$\phi$ relation has been utilized to model the local BHMF
derived from spiral galaxies (Davis et al., 2014; Fusco et al., 2022). The
simple addition of $v_{\mathrm{max}}$, which is widely available for many
spiral galaxies, could better aid in modeling the shape of the BHMF with lower
scatter. This is particularly useful as the BHMF is well known at the high-
mass end, but lacking clarity at the low-mass end, which is the purview of
spiral galaxies.

The BHMF is virtually unknown at $M_{\bullet}<10^{5}\,\mathrm{M}_{\sun}$
because of a dearth of observational evidence of IMBHs. Extrapolating the
$M_{\bullet}$–$\phi$ (Davis et al., 2017) and $M_{\bullet}$–$v_{\mathrm{max}}$
(Davis et al., 2019c) relations down to the low-mass end predicts
$M_{\bullet}<10^{5}\,\mathrm{M}_{\sun}$ IMBHs at $|\phi|>26\fdg 8\pm 2\fdg 3$
and $v_{\mathrm{max}}<130\pm 15\,\mathrm{km\,s}^{-1}$. Using these
aforementioned values as inputs to Equation 2.2, we similarly find a line
across the plane defining the upper-limit at $\mathcal{M}_{\bullet}<5.0\pm
0.4$ dex. Thus, these values of $|\phi|$ and $v_{\mathrm{max}}$ serve as a
sort of midline path down the plane into the IMBH regime. However, because of
the flexibility of the plane, these need not be hard and fast values for
identifying potential IMBH-hosting galaxies. That is, a galaxy with a slightly
smaller $|\phi|$ in conjunction with a slightly larger $v_{\mathrm{max}}$, or
vice versa, could still lie below $M_{\bullet}=10^{5}\,\mathrm{M}_{\sun}$ on
the fundamental plane. Particularly, in a forthcoming work (Davis et al.,
2023), we use the fundamental plane to identify strong candidates for IMBH
hosts among a sample of late-type spiral galaxies.

##  4 Conclusions

Arguably, spiral galaxies are the most interesting galaxies. This is not just
because of their intrinsic beauty, but also because they are galactic
laboratories of ongoing star formation, growth, and evolution. Moreover, the
most interesting discoveries await in the realm of low-mass spiral galaxies as
potential hosts of elusive IMBHs. However, these interesting characteristics
make them more difficult to analyze than their more massive, simpler, and
older cousins, i.e., early-type galaxies. Indeed, commonly-used black hole
mass scaling relations like the $M_{\bullet}$–$\sigma_{0}$ (central/bulge
stellar velocity dispersion) or $M_{\bullet}$–$M_{\star,\mathrm{sph}}$
relations are far less accurate for late-type galaxies than early-type
galaxies. Specifically, the $M_{\bullet}$–$\sigma_{0}$ relation has an
intrinsic scatter of $\sigma=0.32$ dex for early-type galaxies (Sahu et al.,
2019b), cf. $\sigma=0.57$ dex for late-type galaxies (Davis et al., 2017; Sahu
et al., 2019b); the $M_{\bullet}$–$M_{\star,\mathrm{sph}}$ relation has an
intrinsic scatter of $\sigma=0.41$ dex for early-type galaxies (Sahu et al.,
2019a), cf. $\sigma=0.48$ dex for late-type galaxies (Davis et al., 2019b;
Sahu et al., 2019a). Judged upon these criteria, the
$M_{\bullet}$–$\phi$–$v_{\mathrm{max}}$ relation (with $\sigma=0.22\pm 0.06$
dex) is not only more accurate than other late-type black hole mass scaling
relations, but also those for early-type galaxies.

Verily, one might expect that a tighter relationship would exist for single-
component galaxies like elliptical galaxies, rather than multi-component
spiral galaxies. However, if we consider the evolution of these systems and
the increase in entropy from spiral to elliptical galaxies, it becomes evident
that this initial assumption may not hold. Although late-type galaxies may be
_complex_ , early-type galaxies are _complicated_ ; the distinction being that
complexity implies many understandable components and complication implying
less components, but more chaos and disorder. This can be understood by
tracking the impact of merger histories and the genesis of morphologically-
dependent black hole mass scaling relations (Graham, 2023a; Graham & Sahu,
2023a; Graham, 2023b; Graham & Sahu, 2023b). As such, the unique merger
history of a galaxy can effectively muddle the ordered rotationally-supported
disk galaxies by transforming them into dispersion-supported elliptical
galaxies. Moreover, this helps explain why we find that the strongest
correlation with black hole mass is not via bulge properties, which are
similar to the disordered spheroids of elliptical galaxies and may be the
result of disk cloaking (Hon et al., 2022).

The fact that the $M_{\bullet}$–$\phi$–$v_{\mathrm{max}}$ relation, which
correlates the black hole mass with global properties of its host galaxy’s
disk rather than bulge properties, as in the $M_{\bullet}$–$\sigma_{0}$ and
$M_{\bullet}$–$M_{\star,\mathrm{sph}}$ relations, shows that BH–galaxy
coevolution is active over large scales. Indeed, recent work (Davies et al.,
2019; Oppenheimer et al., 2020; Sanchez et al., 2023) has shown that
$M_{\bullet}$ is inversely correlated with the fraction of baryons in the
circumgalactic medium of its host galaxy. This is thought to be because more
massive black holes are more energetic, and thus transport more baryons beyond
their host galaxies’ virial radii, all while reducing gas accretion and star
formation over time. This is clear evidence that processes of a central SMBH
are capable of affecting change on scales over eleven orders of magnitude
larger than extent of their event horizons!101010The Milky Way’s Sgr A∗ has a
shadow with a radius of $0.21\pm 0.01$ AU (Event Horizon Telescope
Collaboration et al., 2022a) and the virial radius of the Galaxy is 258 kpc
(Klypin et al., 2002), which is an astounding difference in scale of $(2.52\pm
0.12)\times 10^{11}$. For comparison, there is similar difference in scale
between the width of a human hair and the radius of Earth.

Extrapolation of black hole mass scaling relations down into the IMBH range is
important for future studies, including the design and predictions for space-
based gravitational-wave interferometers (Amaro-Seoane et al., 2023).
Therefore, we anticipate that the fundamental plane will be advantageous for
estimating the demographics of IMBHs hosted by spiral galaxies. With more than
one parameter, there is redundancy built into the planar relationship that
makes it more resilient to abnormalities in a single parameter, helping it to
be more robust. This adds a degree of confidence when using it to predict
black holes masses below the limits of our sample (NGC 4395 with
$\mathcal{M}_{\bullet}=5.62\pm 0.17$ dex). Moreover, spiral-arm pitch angle is
straightforward enough to measure that it could be accomplished most basically
with just an uncalibrated image and a protractor. What is more,
$v_{\mathrm{max}}$ values are readily-available from large 21-cm line width
surveys and easily-accessible in online archives. Therefore, we hope that the
$M_{\bullet}$–$\phi$–$v_{\mathrm{max}}$ relation will facilitate new and
impactful studies and influence further advancements in black hole mass
scaling relations and galaxy evolution.

The authors are grateful for stimulating discussions with Andrea Macciò,
Joseph Gelfand, Ingyin Zaw, and Ivan Katkov. This material is based upon work
supported by Tamkeen under the NYU Abu Dhabi Research Institute grant CASS.
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).

## ORCID iDs

Benjamin L. Davis *B https://orcid.org/0000-0002-4306-5950

Zehao Jin (金泽灏) *B

https://orcid.org/0009-0000-2506-6645

##  Appendix A The Milky Way

Sgr A∗ in our Galaxy has been robustly studied by many independent methods,
determining its black hole mass with incredible accuracy and precision. We
adopt the mass determined by the multi-star orbit analysis of Boehle et al.
(2016), but many such other studies present consistent masses, most notably
the black hole mass determined by the size of its shadow (Event Horizon
Telescope Collaboration et al., 2022b). Therefore, we strongly intended to
include the Milky Way in our sample, just as it was included in the
determination of the $M_{\bullet}$–$\phi$ and $M_{\bullet}$–$v_{\mathrm{max}}$
relations. However, our home galaxy stands out as a significant outlier below
the $M_{\bullet}$–$\phi$–$v_{\mathrm{max}}$ plane, i.e., its dynamically-
measured $M_{\bullet}$ is under-massive with respect to that predicted by the
plane. Specifically, the fundamental plane predicts
$\mathcal{M}_{\bullet}=7.26\pm 0.24$ dex, whereas Sgr A∗ is highly-constrained
to $\mathcal{M}_{\bullet}=6.60\pm 0.02$ dex. You can see how the Milky Way
stands out in Figure 4.

Figure 4:  This figure is identical to Figure 3, except it includes the Milky
Way (marked with $\star$). For an animation of Figure 2 that also includes the
Milky Way, see the following link, http://surl.li/iggks.

In absolute terms, this does not necessarily make the Milky Way the most
extreme outlier in the sample, but the accuracy of its dynamically-measured
$M_{\bullet}$ means that it is weighted heavily in the regression of the
fundamental plane. The coefficients for Equation 2.2, with the Milky Way
included, changes to $\alpha=-5.57\pm 0.06$, $\beta=3.95\pm 0.06$,
$\gamma=7.31\pm 0.06$, and intrinsic scatter $\sigma=0.28\pm 0.06$ dex in the
$\mathcal{M}_{\bullet}$-direction. This represents a small change in the
predicted black hole mass; for a galaxy with the median $\phi$ and
$v_{\mathrm{max}}$, the plane without the Milky Way yields
$M_{\bullet}=(2.16\pm 1.13)\times 10^{7}\,\mathrm{M}_{\sun}$ and including the
Milky Way it becomes $M_{\bullet}=(2.06\pm 1.36)\times
10^{7}\,\mathrm{M}_{\sun}$. Thanks to its position relatively near the balance
point of the fundamental plane, its affect does not noticeably tug the plane
off in any direction. However, it does diminish the accuracy of the
fundamental plane, as evidenced by the 0.06 dex increase in the intrinsic
scatter due entirely to our Galaxy. Thus, one can choose to use the
alternative coefficients for Equation 2.2 that considered the Milky Way and
yield highly-consistent black hole mass predictions with only a small decrease
in accuracy, which is still more accurate than either the $M_{\bullet}$–$\phi$
or $M_{\bullet}$–$v_{\mathrm{max}}$ relations alone.

Since the mass of Sgr A∗ is practically unassailable, the fault in our Galaxy
must lie in its $\phi$ and/or $v_{\mathrm{max}}$. Of course, both quantities
are difficult to measure from inside the Galaxy; even with modern data, it is
hard to truly represent the geometric shape that astronomers from the
Andromeda Galaxy would see or what $v_{\mathrm{max}}$ they would measure from
long-slit spectroscopic observations of the Milky Way. As shown in Figure 4,
the $M_{\bullet}$–$\phi$ relation predicts a less accurate black hole mass,
whereas the $M_{\bullet}$–$v_{\mathrm{max}}$ relation actually predicts a more
accurate black hole mass than the fundamental plane. Indeed, rearranging
Equation 2.2 to solve for $\phi$ yields a prediction of $|\phi|=19\fdg 3\pm
2\fdg 1$ for the Galaxy, which is not far-fetched to envision. Our adopted
value of $13\fdg 1\pm 0\fdg 6$ (Vallée, 2015) is the median pitch angle
derived from a meta-analysis of 50 studies with a range of
$3\arcdeg\leq|\phi|\leq 28\arcdeg$. Moreover, with our intimate vantage of the
Milky Way, observations can be overwhelmed with small-scale structures, such
as a high pitch angle structure in the Sagittarius Arm (Kuhn et al., 2021),
that can complicate determinations of the global Galactic pitch angle.

For a final consideration, it could be that Sgr A∗ is simply under-massive.
Indeed, Oppenheimer et al. (2020) point out that Sgr A∗ is under-massive with
respect to other SMBHs in galaxies with similarly-sized halos as the Milky
Way. In fact, it is thought that an under-massive Sgr A∗ could be conducive to
supporting the genesis of life in the Milky Way (e.g., Lingam et al., 2019).
Thus, the Milky Way, and its central black hole, must be at least consistent
with the anthropic principle (Dicke, 1957).

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