# Characterising SMSS J2157–3602, the most luminous known quasar, with
accretion disc models

Samuel Lai (赖民希),1 ID Christian Wolf,1,2 ID Christopher A. Onken,1,2 ID and
Fuyan Bian (边福彦)3 ID
1Research School of Astronomy and Astrophysics, Australian National
University, Canberra, ACT 2611, Australia
2Centre for Gravitational Astrophysics, Research Schools of Physics, and
Astronomy and Astrophysics, Australian National University
3European Southern Observatory, Alonso de Córdova 3107, Casilla 19001,
Vitacura, Santiago 19, Chile
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We develop an accretion disc (AD) fitting method, utilising thin and slim disc
models and Bayesian inference with the Markov-Chain Monte-Carlo approach,
testing it on the most luminous known quasar, SMSS J215728.21-360215.1, at
redshift $z=4.692$. With a spectral energy distribution constructed from near-
infrared spectra and broadband photometry, the AD models find a black hole
mass of $\log(M_{\rm{AD}}/M_{\odot})=10.31^{+0.17}_{-0.14}$ with an
anisotropy-corrected bolometric luminosity of
$\log{(L_{\rm{bol}}/\rm{erg\,s^{-1}})}=47.87\pm 0.10$, and derive an Eddington
ratio of $0.29^{+0.11}_{-0.10}$ as well as a radiative efficiency of
$0.09^{+0.05}_{-0.03}$. Using the near-infrared spectra, we estimate the
single-epoch virial black hole mass estimate to be
$\log(M_{\rm{SE}}/M_{\odot})=10.33\pm 0.08$, with a monochromatic luminosity
at 3000Å of $\log{(L(\rm{3000\text{\AA}})/\rm{erg\,s^{-1}})}=47.66\pm 0.01$.
As an independent approach, AD fitting has the potential to complement the
single-epoch virial mass method in obtaining stronger constraints on
properties of massive quasar black holes across a wide range of redshifts.

###### keywords:

galaxies: active – galaxies: high-redshift – quasars: emission lines

††pubyear: 2022††pagerange: Characterising SMSS J2157–3602, the most luminous
known quasar, with accretion disc models–B

##  1 Introduction

Active galactic nuclei (AGNs) with black holes weighing up to 10 billion solar
masses have been discovered in the early Universe, less than a billion years
after the Big Bang (Ghisellini et al., 2015; Wu et al., 2015). These luminous
high-redshift quasars (QSOs) hosting massive black holes present significant
challenges to theoretical models of supermassive black hole growth.
Discovering and characterising luminous QSOs allows us to better understand
the massive seeds and super-Eddington accretion scenarios necessary for the
black holes to reach the observed masses (Bromm & Loeb, 2003; Pacucci et al.,
2015; Amarantidis et al., 2019; Zubovas & King, 2021). Furthermore, luminous
QSOs at high-redshift are also useful for mapping our Universe’s cosmic
reionisation history (e.g. Bañados et al., 2018; Davies et al., 2018).

As QSOs are powered by rapid accretion onto supermassive black holes (SMBHs),
the thermal emission from an accretion disc typically peaks in the rest-frame
ultraviolet (UV) and lends a degree of homogeneity to samples of QSO spectra.
Complete samples of QSOs, however, have a tail towards redder UV-optical
colours, usually attributed to dust obscuration (e.g. Webster et al., 1995;
Richards et al., 2003; Glikman et al., 2007; Klindt et al., 2019; Fawcett et
al., 2022). But also, accretion discs of more massive black holes are colder
and thus their intrinsic spectra peak at longer wavelengths (Laor & Davis,
2011). This is because the innermost stable orbits of discs scale with the
black hole mass $M_{\rm BH}$, while the thin accretion disc radius for a given
temperature scales with $\sim M_{\rm BH}^{2/3}$ at fixed Eddington ratio. The
most extreme SMBHs in QSOs might thus appear red in the rest-frame UV due to a
large BH mass instead of dust. Disentangling the effects of BH masses and dust
on QSO colours then relies on observing the disc over a sufficiently broad
wavelength range.

A number of different methods have been employed in recent years to estimate
black hole masses ($M_{\rm{BH}}$) including: reverberation mapping (e.g.
Blandford & McKee, 1982; Peterson, 1993; Netzer & Peterson, 1997; Wandel et
al., 1999; Kaspi et al., 2000; Peterson et al., 2004), velocity-delay maps
(Grier et al., 2013; De Rosa et al., 2018), gravitational micro-lensing (e.g.
Irwin et al., 1989; Sluse et al., 2011; Guerras et al., 2013), polarization of
broad emission-lines (e.g. Savić et al., 2018, 2021; Capetti et al., 2021;
Popović et al., 2022), single-epoch virial estimates (e.g. Vestergaard, 2002;
McLure & Jarvis, 2002; McLure & Dunlop, 2004; Greene & Ho, 2005; Vestergaard &
Peterson, 2006), accretion disc fitting (e.g. Malkan, 1983; Wandel &
Petrosian, 1988; Sun & Malkan, 1989; Laor, 1990; Rokaki et al., 1992; Tripp et
al., 1994; Calderone et al., 2013; Capellupo et al., 2015; Campitiello et al.,
2018; Mejía-Restrepo et al., 2018; Cheng et al., 2019), dynamical estimates
(e.g. Davies et al., 2006; Onken et al., 2007; Hicks & Malkan, 2008; Greene et
al., 2010; Raimundo et al., 2019), and gravitational redshift of the Fe iii
line (Mediavilla et al., 2018; Mediavilla et al., 2019). These methods are
widely applied based on the features of the system being studied and the
availability of data on various observables. In particular, the single-epoch
virial mass estimate is frequently used on large samples across a wide range
of redshifts (e.g. Shen et al., 2011). It is also the most common method
applied for QSOs found at the highest redshifts (e.g. Wu et al., 2015;
Mazzucchelli et al., 2017; Reed et al., 2019; Wang et al., 2021) due to the
relative ease of obtaining single-epoch spectra compared to spatially or
temporally resolved observations.

In this study, we characterise SMSS J215728.21$-$360215.1 (hearafter
J2157–3602), the most luminous known quasar, with a spectroscopically measured
redshift of $z=4.692$ (Wolf et al., 2018; Onken et al., 2020). Using both
survey photometry and spectra, we construct a spectral energy distribution
(SED) from the rest-frame infrared to ultraviolet. We describe further
developments to the accretion disc fitting technique by Bayesian inference of
black hole mass with the Markov-Chain Monte-Carlo sampling approach and
compare our measurements against the single-epoch virial method. Constraining
the peak of the accretion disc emission proves to be effective at measuring
the black hole mass (Campitiello et al., 2018). For the peak accretion disc
emission to be located at wavelengths longer than Ly$\alpha$, the black hole
needs to be sufficiently high mass. Our results show that AD fitting is a
viable independent method for characterising the highest mass QSO black holes,
which can be applied even in the absence of spectra, making it suitable for
large samples. This creates an opportunity to obtain more stringent
constraints on black hole properties and their growth rates, particularly for
high-redshift QSOs where other techniques are impractical.

The content of this paper is organised as follows: in Section 2, we describe
the photometric and spectroscopic data obtained for J2157–3602. In Section 3,
we present the two complementary methods we use in this study to characterise
black hole mass. In Section 4, we discuss measurements of the black hole mass
and bolometric luminosity, in addition to estimates of the Eddington ratio and
radiative efficiency. We conclude with Section 5. Throughout the paper, we
adopt a flat $\Lambda$CDM cosmology with H${}_{0}=70$ km s-1 Mpc-1 and
$\left(\Omega_{\rm m},\Omega_{\Lambda}\right)=\left(0.3,0.7\right)$. All
referenced emission-line wavelengths are measured in vacuum.

##  2 Spectral Energy Distribution

We use spectroscopic observations and publicly available survey photometry to
construct the SED of J2157–3602.

###  2.1 Photometric Data

We crossmatch J2157–3602, positioned at RA = 329.36762∘ and Dec = -36.03756∘
(J2000), with AllWISE (Wright et al., 2010; Cutri et al., 2013), the VISTA
Hemisphere Survey (VHS; McMahon et al., 2013) DR6, Two Micron All-Sky Survey
(2MASS; Skrutskie et al., 2006), SkyMapper Southern Survey (SMSS; Onken et
al., 2019) DR3, and NOIRLab Source Catalog (NSC; Nidever et al., 2021) DR2 to
collect photometry from the infrared to optical passbands. The W1 and W2
magnitudes from CatWISE2020 (Marocco et al., 2021) are not appreciably
different from that of AllWISE. Properties of J2157–3602, including its
photometry, can be found in the discovery paper (Wolf et al., 2018). We also
obtain the transmission profile of all broadband filters using the SVO Filter
Profile Service (Rodrigo & Solano, 2020).

###  2.2 Spectroscopic Data

Comprehensive details of the spectroscopic observation and data description
are presented in Onken et al. (2020). Briefly, observations were obtained from
two medium resolution and wide-band spectrographs: the Near-Infrared
Echellette Spectrometer (NIRES) instrument (Wilson et al., 2004) at Keck
Observatory and the X-shooter instrument (Vernet et al., 2011) at the Very
Large Telescope. The full observed wavelength coverage is from 3000 Å to
nearly 2.5 $\mu$m, although little of the source flux is transmitted shortward
of Ly$\alpha$ at an observed wavelength of $\sim 6920$ Å. The data are reduced
using PypeIt (Prochaska et al., 2020) and a stacked spectrum is created by
scaling the NIRES spectrum to the X-shooter data in the overlapping wavelength
region. The absolute flux calibration of the combined spectrum is based on the
VHS DR6 J-band photometry. From the Mg ii line, the systemic redshift is
measured to be $z=4.692$ and the median signal-to-noise (SNR) per $\sim$50 km
s-1 velocity dispersion bin measured between rest-frame 2700$-$2900Å is nearly
200.

###  2.3 Galactic Extinction

For Galactic extinction, we use $R_{\rm{v}}=3.1$ and the Schlegel, Finkbeiner
& Davis (SFD; Schlegel et al., 1998) extinction map to apply a correction in
the observed frame. We also utilise a 14% re-calibration factor
$E(B-V)=0.86\times E(B-V)_{\rm{SFD}}$ (Schlafly & Finkbeiner, 2011), which is
informed by the Sloan Digital Sky Survey (SDSS; York et al., 2000) data and an
analysis of the blue tip of the stellar locus (Schlafly et al., 2010). We find
the Galactic extinction to be small for J2157–3602, confirmed by
$E(B-V)=0.013$, and due to its extreme luminosity, we assume no host galaxy
extinction. However, we briefly discuss the effect of host galaxy extinction
on the $M_{\rm{BH}}$ measurement in Section 4.2.1.

##  3 Black Hole Characterisation

We discuss two complementary methods for estimating the black hole mass of
J2157–3602 from photometric and spectroscopic data.

###  3.1 Single-Epoch Virial Mass

We begin by describing the single-epoch (SE) virial mass method, in order to
later examine use of the virial mass estimate to inform the Bayesian priors of
the accretion disc fitting. The SE technique is routinely applied to QSO
spectra (e.g. Vestergaard, 2002; McLure & Jarvis, 2002; McLure & Dunlop, 2004;
Greene & Ho, 2005; Vestergaard & Peterson, 2006). The dynamics of the line-
emitting gas is assumed to be virialised with the gravitational potential of
the black hole. The velocity-broadened emission-line profile measures the gas
velocity, and the continuum luminosity is used to infer the radius of the
broad-line region (BLR) through the radius-luminosity (R-L) relation, which is
empirically derived from reverberation mapping experiments (e.g. Kaspi et al.,
2000, 2005; Bentz et al., 2006; Bentz et al., 2013). While most reverberation
mapping experiments calibrate the R-L relation using the H$\beta$ line, the Mg
ii line profile is found to correlate with H$\beta$ and can be used as its
substitute (e.g. Salviander et al., 2007; Shen et al., 2008; Wang et al.,
2009; Shen & Liu, 2012). This is convenient for the J2157–3602 spectrum where
the H$\beta$ line is redshifted out of the X-shooter NIR coverage into
wavelengths that are more difficult to observe from the ground. Single-epoch
virial mass estimates take on the following form,

$\left(\frac{M_{\rm{SE}}}{M_{\odot}}\right)=10^{\rm{a}}\left[\frac{\lambda
L_{\lambda}}{10^{44}\,\rm{erg\,s^{-1}}}\right]^{b}\left[\frac{\rm{FWHM_{\rm{line}}}}{1000\,\rm{km\,s^{-1}}}\right]^{c}\,,$
(1)

where $\lambda L_{\lambda}$ is the monochromatic luminosity at a particular
wavelength or an emission line luminosity and FWHM is the full-width half-
maximum of a broad emission line. In this study, we use the Shen et al. (2011)
Mg ii calibration with exponents (a, b, c) calibrated to the values (6.74,
0.62, 2.0), based on a high-luminosity local AGN subsample. The differences
between sets of calibrations, such as those from Vestergaard & Osmer (2009),
are approximately 0.1 dex, but the overall statistical uncertainties of the SE
virial mass estimate can be up to 0.5 dex, due to QSO variability and the
propagated uncertainties from reverberation mapping masses (e.g. Krolik, 2001;
Woo et al., 2010; Steinhardt & Elvis, 2010; Shen, 2013; Kozłowski, 2017; Dalla
Bontà et al., 2020). In this study, we present the SE virial mass estimates
with their measurement uncertainties, but acknowledge that the 0.5 dex
statistical uncertainty is often the dominant error.

####  3.1.1 Spectral fitting

For the SE mass estimate, we fit the stacked spectrum from Keck/NIRES and
VLT/X-shooter. One of the primary difficulties in obtaining the pure velocity-
broadened profile of Mg ii is the treatment of the broad Fe ii emission, which
forms a pseudo-continuum in QSO rest-frame UV and optical spectra. Following
after similar studies of QSO spectra (e.g. Wang et al., 2009), we model the
underlying continuum with three components: a power-law, Balmer continuum, and
blended Fe ii flux. Our power-law is normalised to 3000Å while the Balmer
continuum is modeled as a Planck blackbody with a uniform electron temperature
$T_{\rm{e}}$ attenuated by an optical depth $\tau_{\lambda}$ (e.g. Grandi,
1982; Dietrich et al., 2002; Wang et al., 2009; Kovačević et al., 2014). Both
parameters, along with the overall normalisation, are free parameters of the
continuum fit. The Balmer continuum is often not well constrained
independently of the power-law and Fe ii continuum, so we do not suggest a
physical interpretation of the temperature and optical depth.

The Mg ii line profile is sensitive to the underlying Fe ii emission features.
Adopting only one Fe ii model can induce a bias in the resulting FWHM (e.g.
Schindler et al., 2020), so we consider a variety of empirical and semi-
empirical Fe ii templates. Our Vestergaard & Wilkes (2001, VW01) template is
spliced with Salviander et al. (2007), which extrapolates under the Mg ii line
from the rest-frame 2200$-$3090Å; it is the same version of VW01 used in other
spectral fitting codes such as PyQSOFit (Guo et al., 2018). We also use the
Tsuzuki et al. (2006, T06), Bruhweiler & Verner (2008, BV08), and Mejía-
Restrepo et al. (2016, M16) templates.

The full pseudo-continuum is uniquely defined by eight free parameters. All
components of the continuum are simultaneously fit to selected windows in
close proximity to the Mg ii line: 2200$-$2740Å, 2840$-$3300Å, and
3500$-$3650Å in the rest-frame. The emission-line is fit to the continuum-
subtracted spectrum between the rest-frame wavelengths 2710$-$2930Å, which
overlaps with the continuum windows in order to force the Mg ii flux
contribution to converge to zero at the wings. Similar results (within
0.5$\sigma$) are obtained without the overlap. The broad line is fit with a
maximum of three Gaussian components, each with an independent wavelength
shift of up to $\pm$30Å. Although the decomposition of the Mg ii line is not
necessarily unique, we obtain the FWHM from the total line profile, which is
less sensitive to the particulars of the decomposition.

###  3.2 Accretion Disc Fitting

The QSO accretion disc emission can be modeled by a superposition of
blackbodies with a wide range of effective temperatures, where the spatial
temperature profile is derived from the disc emissivity under classical
accretion disc theory (Shakura & Sunyaev, 1973; Novikov & Thorne, 1973). Under
the standard framework of assuming large optical thickness, gas radiates
isotropically as a blackbody with a local effective temperature, and flux is
integrated across the whole disc surface down to the innermost edge. For a set
of black hole properties, its mass and spin, alongside an observed inclination
angle and accretion rate or disc luminosity, one can produce predicted SEDs of
the observed thermal emission by ray-tracing null geodesics from an observer
placed at infinity to the vicinity of the black hole. The steady-state
accretion disc structure is solved by semi-analytical or numerical models. The
accretion disc fitting method (hereafter AD fitting) can be used to recover
black hole properties from observed spectra (e.g. Kawaguchi et al., 2004).
Specifically, we use AD fitting to estimate the black hole mass,
$M_{\rm{AD}}$, where the uncertainty of the measurement originates from a
space of degenerate solutions due to the unknown black hole spin and
orientation.

####  3.2.1 Spectral hardening

Realistic accretion disc emission is more complex than a superposition of
blackbodies at various temperatures, motivating the addition of a colour
correction (hardening) factor, $f_{\rm{col}}\geq 1$, to capture the combined
effects of Compton scattering, absorption opacity, optical depth, and plasma
density structure (e.g. Shimura & Takahara, 1993; Ebisawa et al., 1993;
Shimura & Takahara, 1995). In practice, $f_{\rm{col}}$ approximates the
departure of a more realistic disc from a multitemperature blackbody model.
The black hole mass, $M_{\rm{AD}}$, is degenerate with the colour correction,
following the relation $M_{\rm{AD}}\propto f_{\rm{col}}^{2}$ (Ebisawa et al.,
1993; Shimura & Takahara, 1995) for a Keplerian thin disc with all other
intrinsic properties being equal (Li et al., 2005, Eq. 16-17). Thus, any black
hole mass measurements from AD fitting will depend sensitively on the
hardening factor.

The value of $f_{\rm{col}}$ is most sensitive to the accretion disc’s maximum
effective temperature $T_{\rm{max}}$ (Davis et al., 2005; Done et al., 2012;
Davis & El-Abd, 2019), where $f_{\rm{col}}$ converges toward unity for
$T_{\rm{max}}\leq 3\times 10^{4}\,\rm{K}$ (Davis & Laor, 2011). At
sufficiently low temperatures, helium is largely in a neutral state and the
enhanced neutral hydrogen fraction increases absorption opacity, supporting
the complete thermalisation of the accretion disc while minimising scattering
and Comptonisation effects from free electrons. The maximum effective
accretion disc temperature is proportional to accretion rate as
$\dot{M}^{1/4}$ and to black hole mass as $M_{\rm{BH}}^{-1/2}$ (Laor & Davis,
2011). This suggests that the $f_{\rm{col}}$ correction is high for low-mass
black holes accreting at super-Eddington rates, where electron scattering is
dominant. In contrast, for supermassive black holes, such as J2157–3602, where
$\log(M_{\rm{BH}}/M_{\odot})>8$ and Comptonisation has little effect (Hubeny
et al., 2001), the totality of the accretion disc radiation is likely
dominated by thermal emission.

In contrast to the typical value of $f_{\rm{col}}=1.7$, which was developed
for X-ray binaries (XRBs) (Shimura & Takahara, 1995), similar studies of AGN
with supermassive black holes adopt $f_{\rm{col}}=1$ (Vasudevan & Fabian,
2007, 2009; Done et al., 2012; Calderone et al., 2013; Campitiello et al.,
2018; Mejía-Restrepo et al., 2018; Cheng et al., 2019). However, we caution
that some models predict an increased hardening factor with black hole mass
(e.g. Davis & El-Abd, 2019). Other approaches and theoretical integrated
spectra of AGN accretion discs around supermassive BHs predict that the
hardening factor should be close to unity (Hubeny et al., 2000, 2001; Czerny
et al., 2011). Empirically, if the hardening factor was significant, AD
fitting studies that assume a value of unity should find their measurements of
$M_{\rm{BH}}$ to be systematically underestimated compared to independent
measurements, but significant overestimation is sometimes observed instead
(e.g. Calderone et al., 2013; Campitiello et al., 2020). In this study, we
assume $f_{\rm{col}}=1$ to remain consistent with similar AD fitting
approaches developed for AGN with supermassive BHs.

We consider both geometrically thin (kerrbb; Li et al., 2005) and slim
(slimbh; Sadowski, 2011; Straub et al., 2011) accretion disc emission models,
where the pre-calculated spectral tables are accessed through XSPEC 12.12.0
(Arnaud, 1996), packaged as part of Sherpa (Freeman et al., 2001), the
modeling and fitting suite of Chandra Interactive Analysis of Observations
(CIAO) v4.14. Both models are ray-traced numerical solutions to steady-state
accretion disc models with general relativistic effects and they have been
found to be consistent with synthetic spectra created from numerical general
relativistic radiative magnetohydrodynamic simulations of puffy discs (Wielgus
et al., 2022). To tackle the inherent degeneracies in the problem, we use
Markov-Chain Monte-Carlo techniques to infer the black hole mass from its
posterior probability distribution.

####  3.2.2 Thin disc

The kerrbb model is an optically thick and geometrically thin numerical
Keplerian accretion disc model originally designed for black hole X-ray binary
spectra (Li et al., 2005). Ray-tracing is used to calculate the spectrum,
under the assumption that emission from every point in the accretion disc is
locally blackbody-like. A limitation is that the kerrbb model assumes a
perfectly flat disc with no vertical thickness, flaring, or warping (Li et
al., 2005). Nevertheless, Campitiello et al. (2018), hereafter C18, used
kerrbb to develop analytic expressions approximating the black hole mass
$M_{\rm{BH}}$ and accretion rate $\dot{M}$ from fitting the QSO SED, finding
that $M_{\rm{AD}}$ can be constrained by the peak of the accretion disc
spectrum, $\nu_{\rm{p}}$, and the peak luminosity,
$\nu_{\rm{p}}L_{\nu_{\rm{p}}}$. However, even for a fixed inclination
$\theta_{\rm{inc}}$, the peak frequency and luminosity are degenerate with
$M$, $\dot{M}$, and $a$, such that any accretion disc SED can be reproduced by
a carefully selected family of solutions. The parameter space covered by the
degenerate solutions can be minimised by making reasonable assumptions about
the spin and inclination (C18). Despite the degeneracies involved, we discuss
how we derive the maximum likelihood black hole mass in Section 3.2.5.

Comparisons of reverberation mapping and SE virial mass estimates showed that
the analytical approximations of C18 can measure $M_{\rm{AD}}$ to a precision
of 0.45 dex (Campitiello et al., 2020, hereafter C20). They further showed
that the kerrbb-based mass estimates are systematically larger (up to 0.4 dex
depending on black hole mass and spin) than masses estimated through SE
methods for their sample, which spans
$\log{(\rm{M_{BH}}/\rm{M_{\odot}})}=7.5-9.5$.

In our application of kerrbb, we toggle on the effects of self-irradiation,
whereby radiation can be gravitationally deflected, illuminating another part
of the accretion disc, and include limb-darkening effects. We also set the
torque at the inner boundary of the accretion disc to zero, following the
standard theory of accretion discs. The effect of a nonzero torque enhances
the disc power by boosting returning radiation, particularly at higher
energies. However, nearly indistinguishable spectra can be created by
adjusting the effective mass accretion rate even for very high torques (Li et
al., 2005). We show in Section 4.2 an alternative $M_{\rm{AD}}$ estimate if
the disc power from torque is comparable to the disc emission.

####  3.2.3 Slim disc

The thin disc approximation breaks down for Eddington ratios $\geq 0.3$ (e.g.
Laor & Netzer, 1989), motivating the relaxation of the thin disc assumption.
The XSPEC spectral model, slimbh, is a fully relativistic, optically thick
slim disc accretion model with numerical ray-tracing originally designed to
fit the X-ray continuum of black hole X-ray binaries (Sadowski, 2011; Straub
et al., 2011). In contrast to standard thin disc models, slimbh relaxes the
imposed Keplerian angular momentum condition and accounts for advective
cooling (Sadowski, 2011). Campitiello et al. (2019) derived analogous
analytical approximations showing that, as with the thin disc model, the black
hole mass can be constrained by the peak flux and wavelength of the accretion
disc SED. However, in the slimbh model, the mass accretion rate $\dot{M}$ is
replaced with the disc luminosity $L_{\rm{disc}}$ parameterised in terms of
the Eddington ratio $\lambda_{\rm{Edd}}$, which introduces a black hole mass
dependence. Additionally, the vertical structure is only evaluated up to the
Eddington limit. In the low Eddington regime, slimbh spectra converges towards
the geometrically thin kerrbb spectra, making it a more general model, except
that zero torque is enforced at the inner boundary of the accretion disc.
Another constraint is that slimbh is restricted to positive black hole spins
only, whereas kerrbb is also calculated for negative spins.

In our application of slimbh, we toggle on the effects of limb darkening and
the heightened surface profile, such that ray-tracing can be performed from
the disc photosphere rather than from the equatorial plane. We assume a
constant $f_{\rm{col}}=1$ and do not utilise BHSPEC disc atmosphere models
(Davis et al., 2005). We also set the $\alpha$-viscosity parameter to 0.01. In
Section 4.2, we discuss how the $M_{\rm{AD}}$ estimate is not sensitive to the
$\alpha$-viscosity parameter. We present sample slim disc spectra created
using slimbh for various black hole masses, disc luminosities, inclinations,
and spins in Figure 1.

Figure 1: Accretion disc spectra created from slimbh slim disc models, with
each panel varying one of the four parameters: (from left to right) black hole
mass, disc luminosity, inclination, and spin. The default parameters are
listed in black within the rightmost panel and all SEDs are labelled with the
modified value corresponding to the parameter on the top-right of each panel.
The black arrows indicate the direction that the SEDs evolve towards as the
parameter increases for this test case.

####  3.2.4 Accretion disc SED

We consider three datasets for fitting the AD models: synthetic photometry
created from the stacked spectra, survey photometry, and modified photometry
with QSO template subtraction. In all cases, we apply the Galactic extinction
correction from Section 2.3. We fit accretion disc emission in a region
bounded by rest-frame 1 $\mu$m and 1600Å, due to the increasing hot dust
contribution at the red end and the intergalactic medium Ly$\alpha$ absorption
at the blue end. Because of the wide wavelength coverage of each of the
photometric data and broad C iv visible in the spectrum, characteristic of a
high-ionisation broad absorption line (HiBAL) QSO, we leave a sizable buffer
between the blue threshold and the Ly$\alpha$ forest. Very broad photometry,
such as those from Gaia, are ill-suited to the accretion disc fitting approach
because they can hide many details of the underlying spectral shape that are
crucial to obtaining a good fit.

  1. 1.

Synthetic Photometry (SynPhot)

The presence of broad emission lines, the Balmer continuum, and Fe ii flux
implies that a significant fraction of the structure observable in spectra is
not required for fitting the underlying continuum emission. Instead of fitting
the spectrum directly, we create synthetic photometric data points with a flat
transmission profile for contiguous sections of accretion disc emission that
are free of emission-line contribution. The emission-line-free windows are
created by masking all broad lines and identifying regions of the Gaussian-
broadened Bruhweiler & Verner (2008, BV08) Fe ii template that fall below a
flux threshold, which we set to 70% of the median template flux. Both the
Gaussian broadening of the Fe ii template and the width of the broad line mask
are scaled to a velocity dispersion value, which we set to 4500 km s-1. The
size of the mask is three times the velocity dispersion. We found that our
results are not sensitive to reasonable values chosen for the Fe ii template
flux threshold and width of the Gaussian broadening kernel. This procedure
creates synthetic photometry from the stacked spectrum, which we use in place
of the photometry from the same wavelength region.

For the red end of the SED outside of the wavelength coverage of the stacked
spectrum, we scale the Selsing X-shooter template (Selsing et al., 2016) to
match the flux of J2157–3602 photometry, where each photometric bandpass is
treated independently. The cutouts of the Selsing template, each with the
width of one bandpass, are used to create synthetic photometry using the same
method as above. In this way, we replace the W1 and W2 photometric points,
avoiding flux contribution from bright emission lines, such as H$\alpha$.
Because each photometric point is treated independently, differences in the
global continuum slope between our target and the Selsing template are
minimised. We test this by creating variations of the synthetic photometry for
deviations in the Selsing template continuum slope, parameterised by an
$F_{\lambda}$ power-law slope of $\alpha=-1.70$ (Selsing et al., 2016). We
test variations of $1\,\sigma_{\alpha}$, where we take $\sigma_{\alpha}=0.6$
from the Rakshit et al. (2020) catalogue, and find median flux variations of
5$-$7% in the synthetic photometry We find the flux variations to have no
significant effect on our measurements as described in Section 4.2.

  2. 2.

Photometry-Only

For comparison to the synthetic photometry, we present results obtained from
fitting only the survey photometry in Section 4.2.

  3. 3.

Template-Corrected Photometry

In a further modification to the photometry-only models, we apply the same
approach as for the long-wavelength data of the SynPhot model and scale the
Selsing template to the observed photometry and use its continuum model to
correct for flux contribution from emission lines.

These two alternative datasets to the synthetic photometry demonstrate how the
black hole mass estimate from AD fitting is affected if spectroscopic data is
unavailable.

####  3.2.5 Markov-Chain Monte-Carlo

The free parameters in AD fitting are the black hole mass $M_{\rm{BH}}$,
luminosity, spin $a$, and observer’s inclination $\theta_{\rm{inc}}$ with
respect to the symmetry axis of the accretion disc (equivalent to the spin
axis), where the luminosity is parameterised in terms of the mass accretion
rate $\dot{M}$ for kerrbb and the Eddington ratio for slimbh. The inclination
angle corresponds to $\theta_{\rm{inc}}=0^{\circ}$ when viewed face-on and
$\theta_{\rm{inc}}=90^{\circ}$ when viewed edge-on. These parameters are
inescapably degenerate in the absence of independent constraints. We utilise
Bayesian Markov-Chain Monte-Carlo (MCMC) methods, as implemented in the Python
emcee module (Foreman-Mackey et al., 2013), to probe the multi-dimensional
parameter space and obtain a posterior distribution of $M_{\rm{BH}}$.

We adopt an uninformative, flat prior probability distribution for all free
parameters, except for the black hole mass. There, we investigate three
probability distributions: a flat prior, a SE virial mass prior, and a black
hole mass function prior. The SE virial mass prior is a Gaussian distribution
centred around the SE mass estimate with a width set to a statistical
uncertainty of 0.5 dex. The black hole mass function prior is based on a
double power-law parameterisation of the QSO luminosity function (Onken et
al., 2022), which is transformed from 145nm absolute magnitude to black hole
mass through the Runnoe et al. (2012) bolometric correction and an assumed
fixed mean Eddington ratio. For mean Eddington ratios $>0.1$, such as 0.25
(Kollmeier et al., 2006) or 0.6 (Trakhtenbrot et al., 2011), the slope of the
mass function on the bright-end, which determines the relative likelihood
between black hole masses, is constant across the range of black hole masses
being considered and the exact Eddington ratio assumed is inconsequential. We
use double power-law parameters derived from Onken et al. (2022) with faint-
end constraints from Niida et al. (2020) and the redshift evolution of the
normalisation from Fan et al. (2001) for the sample with a median redshift of
$z=4.83$. The location of the knee in the mass function for fixed mean
Eddington ratios of 0.25 and 0.6 are $\log(M_{\rm{BH}}/M_{\odot})=9.28$ and
$8.90$, respectively. For bright QSOs similar to J2157–3602, the slope of the
mass function is a conservative prior which disfavours spuriously high black
hole masses.

In Section 4.2, we discuss the effect of the prior on our black hole mass
measurement. The black hole spin parameter is allowed to explore the space of
all positive spins, 0 < $a$ < 0.998, and we exclude edge-on cases for the
inclination, requiring $\theta_{\rm{inc}}<65^{\circ}$, which is our assumed
opening angle of the obscuring torus. We consider edge-on orientations to be
intrinsically less likely given the extreme luminosity of J2157–3602. We also
check the convergence of our model using ChainConsumer (Hinton, 2016) and
confirm that our $M_{\rm{BH}}$ posterior is independent of the initial
parameters. The final $M_{\rm{AD}}$ estimates are taken from the maximum
likelihood of the posterior distribution, with uncertainties determined by the
68% iso-likelihood line. If a parameter could not be constrained this way such
as when the distribution is strongly skewed towards one end of the allowed
parameter space, then the median of the distribution is used instead, with
uncertainties determined by the 16th and 84th percentiles.

##  4 Results and Discussion

In this section, we present the results of both methods for determining
$M_{\rm{BH}}$ from the combined photometric and spectroscopic data. We compare
these methods against each other and to previous results obtained for the
black hole of J2157–3602 (Onken et al., 2020). We discuss bolometric
corrections to the observed luminosity at 3000Å with anisotropy correction and
use these quantities to further derive the QSO Eddington ratio and radiative
efficiency. Finally, we discuss the potential of the AD fitting method.

###  4.1 Black Hole Mass from SE Virial Method

Previous estimates of the black hole in J2157–3602 based on a similar analysis
of the Mg ii line found
$\log(M_{\rm{SE}}/M_{\odot})_{\rm{Mg\,\textsc{ii}}}=10.53\pm 0.08$ (Onken et
al., 2020), using the Shen et al. (2011) calibration. This value, however,
depends on the FWHM measured above a continuum defined with the VW01 template,
which can result in systematically overestimated FWHMs compared to other Fe ii
templates (Schindler et al., 2020).

Figure 2 presents fits to the Mg ii emission feature of J2157–3602 with
various Fe ii templates. Each template is broadened by a Gaussian kernel with
a dispersion between 3300–5100 km s-1 depending on the template and the Mg ii
FWHM is measured in the range 4200–5150 km s-1.

We consider two sources of uncertainty in the measurement: the statistical
uncertainty from the choice of Fe ii template and the measurement uncertainty.
We estimate the statistical uncertainty by independently fitting the emission
feature with each of the four Fe ii templates and measuring variance in the
resulting FWHMs. To minimise bias towards any particular template, the mean
from the four fits is used as the final estimate of the FWHM. We also consider
an additional measurement uncertainty which is estimated from fitting 100
synthetic spectra, where the flux at each wavelength bin is resampled
according to a Gaussian distribution with a standard deviation equivalent to
the flux uncertainty. The variance in the resulting fits is added to the final
uncertainty of the FWHM. Because of the high SNR of the spectra, the
statistical uncertainty from the Fe ii templates is over 5$\times$ the
measurement uncertainty, but the final error in $M_{\rm{SE}}$ remains
significantly below the overall statistical error of the virial mass
estimator.

Figure 2: Fits of the Mg ii emission feature of J2157–3602 with the combined
pseudo-continuum from the power-law, Balmer, and Fe ii components. The fits
differ from one another by the applied Fe ii template (see legend). The
resulting continuum for each template is plotted in solid lines and the fitted
line profile in dashed lines, while the data is shown in black. The BV08
template is not well-matched to features on the red end of the Mg ii line.
Individual emission line fits are offset to $1.6\times 10^{-16}$ erg s-1 cm-2
Å-1 and the FWHM between fits with different Fe ii models range from 4200 to
5150 km s-1. The Mg ii FWHM mean and standard deviation are 4550 $\pm$ 400 km
s-1.

From the Galactic extinction-corrected and stacked spectrum, we remeasure the
FWHM to be 4550 $\pm$ 400 km s-1 with fits to the Mg ii feature, relative to
5720 $\pm$ 570 km s-1 from Onken et al. (2020). The monochromatic luminosity
at 3000Å measured from the power-law continuum is
$\log{(L(\rm{3000\text{\AA}})/\rm{erg\,s^{-1}})}=47.66\pm 0.01$. Therefore,
the adjusted black hole mass calculated from the SE virial mass estimate is
$\log(M_{\rm{SE}}/M_{\odot})_{\rm{Mg\,\textsc{ii}}}=10.33\pm 0.08$, which is
0.2 dex less massive than the estimate by Onken et al. (2020).

###  4.2 Black Hole Mass from AD Fitting Method

We present the results from the AD fitting method in Table 1 for three choices
of the prior probability distribution: a flat uninformative prior, a
distribution based on the SE virial mass estimate, and another based on the
QSO mass function. We also present fits to three datasets: the synthetic
photometry (SynPhot), template-subtracted photometry (Template), and
photometry-only (Photometry). Descriptions of these three datasets can be
found in Section 3.2.4.

In Figure 3, we present a corner plot for the parameter chain from the slimbh
model fits to synthetic photometry using the SE virial mass estimate with 0.5
dex width as the prior, which is our default model for future discussions,
unless otherwise stated. A black hole mass estimate of
$\log(M_{\rm{AD}}/M_{\odot})=10.31^{+0.17}_{-0.14}$ is measured from the
$M_{\rm{BH}}$ posterior, which has a log-normal shape and a slightly extended
high-mass tail. It’s clear that neither the black hole spin $a$ nor
inclination $\theta_{\rm{inc}}$ are well-constrained by the data, but lower
spin and face-on orientation solutions are preferred. Notably, while the slim
disc model solutions are clearly skewed towards face-on orientations which we
consider intrinsically more probable, the thin disc inclination posteriors are
more biased towards edge-on. This further supports that, for extremely
luminous QSOs such as J2157–3602, the slim disc model is a more realistic
representation of the accretion disc spectra. The convergence between the
emergent spectra of the two models suggest that the thin disc model is a good
approximation for black holes with low Eddington ratios. To test the
sensitivity of our result to the synthetic photometry, we consider variations
to the data created by changing the Fe ii template, the Fe ii flux threshold,
or the power-law continuum slope applied to the Selsing template, and find no
appreciable effect on the resulting black hole mass estimate.

From fitting the photometry alone, we find a 0.05$-$0.10 dex greater
$M_{\rm{AD}}$ compared to synthetic photometry, because the additional flux
from broad emission-lines boosts the perceived luminosity of accretion disc
models. The results from the template-corrected photometry are intermediate
between that of the observed photometry and synthetic photometry, with a small
but consistent correction of $\Delta M_{\rm{AD}}\sim 0.05$ dex over results
from synthetic photometry. However, depending on the properties of an
individual QSO with respect to the template, this adjustment may be sufficient
to offset the higher mass bias from photometry. The kerrbb thin disc model
estimates are also systematically greater than slimbh slim disc models,
suggesting that the thin disc assumption may be partially responsible for an
overestimation of $M_{\rm{AD}}$ in C20, although the difference shown for this
target is minimal. It remains to be seen whether the discrepancy is greater at
lower black hole masses as was the case in C20.

Between prior probability distributions, the SE virial mass estimate with its
0.5 dex uncertainty is not a strong prior, producing results similar to the
uninformative prior. In contrast, the steep high-mass end of the QSO mass
function favours $M_{\rm{AD}}$ estimates over 0.10 dex lower. Measurements
presented in Table 1 imply that the rms uncertainty of $M_{\rm{AD}}$ from an
unconstrained spin and inclinations below $65^{\circ}$ is less than 0.2 dex.
Relaxing the inclination angle limitation or enforcing a more stringent
inclination constraint produces black hole mass estimates within 0.6$\sigma$,
which implies that our result is not sensitive to the assumed opening angle.

Despite the indications of the spin posterior distribution, one might expect
that the spin of the J2157–3602 black hole is fairly high due to its inferred
history of prolonged, ordered accretion, in order for it to have grown to its
observed mass in roughly 1 Gyr (e.g. Volonteri et al., 2005; Barausse, 2012;
Walton et al., 2013). If we used a prior of $a>0.6$, the $M_{\rm{AD}}$
estimate would be consistent with the unconstrained case, at
$\log(M_{\rm{AD}}/M_{\odot})=10.39^{+0.19}_{-0.10}$. Independent constraints
on degenerate parameters, such as an estimate of spin from the Fe-K$\alpha$
broad lines in the X-ray (e.g. Reynolds & Fabian, 2008; Reynolds, 2021), have
the potential to improve the black hole mass estimates from AD fitting.
However, as estimated in the later Section 4.4, the radiative efficiency is
fairly low, implying lower spins ($a\lesssim 0.5$) in concordance with the
MCMC result and that the growth history of J2157–3602 may have involved
periods of chaotic accretion, suppressing its spin and radiative efficiency,
while maintaining high mass accretion rates (Zubovas & King, 2021). In this
study, we remain agnostic about the spin distribution, preferring to keep the
spin prior uniform across the entire positive spin parameter space, which
produces a conservative estimate of the precision by which the black hole mass
can be measured.

For the results of the thin disc kerrbb model, if the disc power from the
torque at the inner boundary of the accretion disc were assumed to be
comparable to the disc emission, then the resulting mass estimate using
synthetic photometry with the SE prior would be
$\log(M_{\rm{AD}}/M_{\odot})=10.61^{+0.15}_{-0.21}$, which is 0.18 dex higher
than with zero torque. As the kerrbb mass estimates are already higher than
the virial and slimbh masses, including the effect of a nonzero torque further
exacerbates the disparity. We also consider an alternative slim disc slimbh
model with the $\alpha$-viscosity parameter set to 0.1 instead of 0.01, which
is a high estimate for AGN accretion discs (e.g. Starling et al., 2004). The
mass estimate with the higher $\alpha$-viscosity is
$\log(M_{\rm{AD}}/M_{\odot})=10.31^{+0.16}_{-0.15}$, virtually
indistinguishable from the original estimate. This is expected as the
viscosity parameter has little effect on the emergent spectrum aside from
boosting the spectrum to higher energies through electron scattering
(Kawaguchi, 2003). However, as discussed previously, Compton effects are
unlikely to be significant in the regime of supermassive black holes accreting
at sub-Eddington rates. To test this assumption, we utilise our Bayesian MCMC
AD fitting method with the slim disc model to constrain the hardening factor,
finding $f_{\rm{col}}=1.00^{+0.25}_{-0.00}$ from the posterior distribution.
Therefore, the hardening factor is consistent with unity as we have assumed.

When the limb darkening effect is disabled, we find
$\log(M_{\rm{AD}}/M_{\odot})=10.33^{+0.15}_{-0.12}$ using the slimbh model
with the SE prior. The black hole mass estimate is marginally increased and
the mass posterior distribution is slightly narrower. However, the difference
between the mass estimates with and without limb darkening depends on the
shape of the inclination posterior. For J2157–3602, we find face-on solutions
are generally favoured by the slim disc model and the thin disc model favours
inclinations closer to edge-on. In this scenario, the kerrbb mass estimates
experience a marginal decrease when limb darkening is disabled and the slimbh
mass estimates increase slightly.

Table 1:  Black hole masses, $\log{(\rm{M_{BH}}/\rm{M_{\odot}})}$, measured by the AD fitting method for thin and slim accretion disc models and three different prior probability distributions. The three sets of measurements are distinguished by the dataset to which the AD model is fit. “SynPhot" refers to the synthetic photometry created from spectroscopic data, “Template" refers to the photometric data scaled to the continuum model of an average QSO template (Selsing et al., 2016), and “Photometry" refers to the photometry-only data. The Galactic extinction correction is applied to all datasets. Our default AD fitting model, marked in bold font and referenced throughout this paper, is the slimbh model fit to synthetic photometry using the Single-Epoch prior. Model (Data) / Prior | Flat | Single-Epoch | Mass Function
---|---|---|---
kerrbb (SynPhot) | $10.44^{+0.11}_{-0.25}$ | $10.43^{+0.12}_{-0.21}$ | $10.29^{+0.17}_{-0.15}$
slimbh (SynPhot) | $10.32^{+0.17}_{-0.15}$ | $\mathbf{10.31^{+0.17}_{-0.14}}$ | $10.21^{+0.18}_{-0.10}$
kerrbb (Template) | $10.49^{+0.12}_{-0.24}$ | $10.47^{+0.12}_{-0.22}$ | $10.31^{+0.20}_{-0.13}$
slimbh (Template) | $10.37^{+0.17}_{-0.17}$ | $10.35^{+0.17}_{-0.15}$ | $10.27^{+0.16}_{-0.13}$
kerrbb (Photometry) | $10.51^{+0.13}_{-0.23}$ | $10.51^{+0.10}_{-0.24}$ | $10.35^{+0.19}_{-0.14}$
slimbh (Photometry) | $10.39^{+0.18}_{-0.16}$ | $10.36^{+0.20}_{-0.14}$ | $10.27^{+0.19}_{-0.11}$
Figure 3: Corner plot for the accretion disc fit to J2157–3602 synthetic
photometry using the slimbh model. The frequency is presented in the rest-
frame. The SE virial mass estimate is used as a prior probability distribution
on the black hole mass. The final estimate of each parameter is determined by
the maximum likelihood of the posterior distribution, with uncertainties
determined by the 68% iso-likelihood line. The top-right panel shows the
stacked spectrum in dark grey, the synthetic photometry in blue square points,
photometry in black, and masked data in light grey. The red model is the
highest likelihood model and the one-sigma spread from a random sample of
posterior models is similar in width to the red model line. Neither black hole
spin $a$ or inclination $\theta_{\rm{inc}}$ are constrained by the data, but
the black hole mass is estimated to be
$\log(M_{\rm{AD}}/M_{\odot})=10.31^{+0.17}_{-0.14}$.

####  4.2.1 Dust extinction

The average QSO with HiBAL features typically exhibits some reddening due to
host galaxy dust (e.g. Brotherton et al., 2001; Reichard et al., 2003). The
dust extinction inferred from QSOs at $z<2.2$ is consistent with the SMC
extinction curve (Richards et al., 2003; Hopkins et al., 2004). At higher
redshifts, the extinction curve has a tendency to flatten at $\lambda\leq
2000$Å, indicating a different dust production mechanism or a different
physical mechanism by which dust is processed into the ISM (Gallerani et al.,
2010), although an SMC extinction law is still consistent with the data in
most cases (e.g. Krawczyk et al., 2015).

In order to quantify the effects of host galaxy dust attenuation, we assume
the empirically inferred mean extinction curve from Gallerani et al. (2010)
with an SMC-like R${}_{\rm{V}}=2.7$ (Bouchet et al., 1985) The mean reddening
measured for QSOs with HiBAL features is $\Delta$E${}_{\rm{B-V}}\sim 0.02$
relative to an ensemble average QSO composite, which is itself reddened by
$\Delta$E${}_{\rm{B-V}}\sim 0.004$ compared to a non-BAL composite (Reichard
et al., 2003). However, individual measurements show a wide distribution, with
a sizable fraction (30–40%) of HiBALs less reddened than a similar sample of
non-BALs (Reichard et al., 2003). Thus, we do not refer to the mean case to
infer host galaxy reddening for an individual target, especially not for a
target as extreme as J2157–3602. However, to test how significant dust
reddening would affect our measurements, we choose EB-V = 0.08, which is
fairly substantial dust extinction for a QSO (Krawczyk et al., 2015). This
value was also chosen to keep the disc luminosity sub-Eddington and
constrained by the slim disc model.

An AD fit to the de-reddened synthetic photometry results in a black hole mass
estimate of $\log(M_{\rm{AD}}/M_{\odot})=10.15^{+0.14}_{-0.19}$, which is
$\Delta\log(M_{\rm{AD}}/M_{\odot})=-0.16$ for $\rm{E}_{\rm{B-V}}=0.08$.
Increasing the host galaxy extinction has two competing effects on the AD fit:
the true luminosity of the continuum rises and the intrinsic continuum becomes
bluer. The former favours higher black hole masses, while the latter implies
higher accretion disc temperatures, which favours lower black hole masses.
Overall, the influence from a bluer continuum is stronger, resulting in an AD
model with a less massive black hole, but higher luminosity and Eddington
ratio.

The SE virial black hole mass estimate with host galaxy extinction would be
increased by the boosted 3000Å monochromatic luminosity, which implies a
larger BLR radius. Assuming the measured FWHM does not change appreciably, the
extinction law implies

$d\log(M_{\rm{SE}}/M_{\odot})/d(\Delta\rm{E}_{\rm{B-V}})=0.248\,\rm{R}_{\rm{V}}\left(A_{\textrm{3000\AA}}/A_{\rm{V}}\right)\,,$
(2)

where $\left(A_{\textrm{3000\AA}}/A_{\rm{V}}\right)$ is evaluated from the
mean extinction curve and the extinction of the object is measured relative to
the mean extinction of the SE calibration sample, meaning $\Delta
E_{\rm{B-V}}=E_{\rm{B-V,obj}}-E_{\rm{B-V,cal}}$. Using the Gallerani et al.
(2010) curve with $\left(A_{\textrm{3000\AA}}/A_{\rm{V}}\right)=1.86$, we find
$d\log(M_{\rm{SE}}/M_{\odot})_{\rm{SE}}/d(\Delta\rm{E}_{\rm{B-V}})=+1.2$.
Thus, increasing host galaxy extinction causes our two independent
$M_{\rm{BH}}$ measurement methods to diverge, which implies our source is
likely to be minimally extinguished. In further discussions, we use the
$M_{\rm{BH}}$ estimates without host galaxy extinction, specifically
$\log(M_{\rm{AD}}/M_{\odot})=10.31^{+0.17}_{-0.14}$ from the default slim AD
fitting model.

###  4.3 Bolometric Luminosity

Using the myriad of accretion disc models from the MCMC posterior, we
integrate all of the resulting SEDs with an anisotropy correction to derive a
bolometric luminosity of $\log{(L_{\rm{bol}}/\rm{erg\,s^{-1}})}=47.87\pm
0.10$, implying a bolometric correction factor of $k_{3000\text{\AA}}\sim
1.62$, following $L_{\rm{bol}}=k_{\lambda}L(\lambda)$. This differs from
empirically derived bolometric correction factors, such as
$k_{3000\text{\AA}}\sim 5.15$ (Richards et al., 2003), $k_{3000\text{\AA}}\sim
3.33$ (Runnoe et al., 2012), $k_{3000\text{\AA}}\sim 3.2$ (Trakhtenbrot &
Netzer, 2012), and $k_{3000\text{\AA}}\sim 1.84$ (Netzer, 2019), from which we
estimate $\log{(L_{\rm{bol}}/\rm{erg\,s^{-1}})}=48.17\pm 0.16$ using all four
correction factors. We provide a detailed description of the bolometric
luminosity measurement in Appendix A, and compare it with commonly used
empirical bolometric corrections. We also derive and present mean flux
anisotropy corrections for samples of QSOs in Appendix A.3.

###  4.4 Eddington Ratio and Radiative Efficiency

The Eddington ratio, $\epsilon$, is calculated by the ratio of the bolometric
luminosity and the Eddington luminosity for a particular black hole mass,
$L_{\rm{bol}}=\epsilon L_{\rm{Edd}}$, where $L_{\rm{Edd}}\approx 1.26\times
10^{38}(M_{\rm{BH}}/M_{\odot})$ erg s-1. From the AD mass estimate,
$\log(M_{\rm{AD}}/M_{\odot})=10.31^{+0.17}_{-0.14}$, we measure
$\epsilon=0.29^{+0.11}_{-0.10}$. As discussed in Onken et al. (2020), the
relatively low Eddington ratio suggests that the extreme luminosity of
J2157–3602 is a consequence of the size of its black hole.

Other studies of radiative efficiency for individual QSOs, defined as $\eta$
in $L_{\rm{bol}}=\eta\dot{M}c^{2}$, are also based on accretion disc models
(e.g. Davis & Laor, 2011; Trakhtenbrot, 2014; Trakhtenbrot et al., 2017). In
order to address the degeneracy with inclination, these studies assumed
$\cos({\theta_{\rm{inc}}})=0.8$, functionally a Dirac $\delta$-function prior
in the MCMC context. In the absence of an independent constraint on the
inclination, we use the black hole mass prior determined by the SE virial mass
estimate. The kerrbb thin disc model with four parameters ($M_{\rm{BH}}$,
$\dot{M}$, $a$, $\theta_{\rm{inc}}$) is considered, because the disc
luminosity is parameterised in terms of the mass accretion rate. From the fit
to synthetic photometry, we determine the MCMC posterior mass accretion rate
to be $\dot{M}=144^{+84}_{-48}\,M_{\odot}\,\rm{yr}^{-1}$, suggesting that the
radiative efficiency is $\eta=0.09^{+0.05}_{-0.03}$, which is consistent with
estimates based on the Sołtan argument (e.g Yu & Tremaine, 2002).

###  4.5 Potential of AD Fitting

Recently, the potential for AD fitting to complement other black hole mass
measurements, such as the SE virial mass method, has been confirmed by several
studies (Calderone et al., 2013; Capellupo et al., 2015; Mejía-Restrepo et
al., 2018; Campitiello et al., 2020). Here we find that although neither the
black hole spin or observed inclination are constrained by the data, we can
obtain approximately log-normal posterior distributions of the black hole mass
with widths under 0.2 dex. We are also able to use the model SED to estimate
anisotropy-corrected bolometric luminosities within 0.1 dex precision.

As an independent method of estimating black hole masses, the AD fitting
method can be used to disentangle cases for which SE virial mass estimates
between different broad lines are in disagreement. For instance, the QSO SDSS
J102325.31+514251.0, hereafter J1023$+$5142, is measured to host a black hole
of mass $\log(M_{\rm{SE}}/M_{\odot})_{\rm{Mg\,\textsc{ii}}}=10.52\pm 0.08$ or
$\log(M_{\rm{SE}}/M_{\odot})_{\rm{H\textsc{$\beta$}}}=9.58\pm 0.14$, as
estimated from the Mg ii or H$\beta$ line, respectively (Zuo et al., 2015). We
collect photometric data by crossmatching J1023$+$5142 with AllWISE, 2MASS,
and PanSTARRS DR1. We also use the SFD extinction map with the 14%
recalibration (Schlafly & Finkbeiner, 2011), finding $E(B-V)=0.013$,
coincidentally matching that of J2157–3602. Assuming no host galaxy extinction
and by fitting only photometry, we find
$\log(M_{\rm{AD}}/M_{\odot})=9.55^{+0.20}_{-0.17}$ using the AD fitting
method, which strongly disfavours the Mg ii-based estimate. As the measured
black hole mass scales inversely to extinction, additional host galaxy
reddening corrections will not boost the mass estimate to become consistent
with $\log(M_{\rm{SE}}/M_{\odot})_{\rm{Mg\,\textsc{ii}}}$. A more
comprehensive description of the fits to J1023$+$5142 photometry can be found
in Appendix B alongside a discussion of its spectral properties, measured
bolometric luminosity, Eddington ratio, and radiative efficiency.

From the Shen et al. (2011) QSO catalogue of 105,783 QSOs, we find 1599
targets with $\log(M_{\rm{SE}}/M_{\odot})>8$ and a difference between the
H$\beta$ and Mg ii-based virial mass estimate of over 0.5 dex. Applying the
same criteria to the Rakshit et al. (2020) catalogue, we find 1122 such QSOs.
In a forthcoming study, we will use AD fitting to investigate targets for
which the SE mass estimates are discrepant.

We showed in Section 4.2 that a host galaxy extinction correction has
diverging effects on the SE and AD fitting mass estimate. Hence, they can be
used together to constrain the magnitude of the host galaxy extinction. A
unique advantage of AD fitting is that it can also be used to constrain the
radiative efficiency of individual QSOs, as shown in Section 4.4.

Having explored the AD fitting approach extensively on a single target,
J2157–3602, we will apply the method to much larger samples in future studies.
Samples of luminous QSOs with high-quality spectra, such as XQ-100 (López et
al., 2016) and XQR-30 (e.g. Zhu et al., 2021; Lai et al., 2022), will enable
us to produce comparisons between AD fitting and the more established SE
virial mass estimate, particularly for supermassive
($\log(M_{\rm{BH}}/M_{\odot})>8$) black holes at high redshift.

##  5 Conclusions

In this study, the survey photometry and high-quality medium-resolution
spectra of the extremely luminous QSO SMSS J215728.21$-$360215.1 enabled us to
compare two complementary methods of estimating black hole masses. The main
results are as follows:

  * •

An updated black hole mass of
$\log(M_{\rm{SE}}/M_{\odot})_{\rm{Mg\,\textsc{ii}}}=10.33\pm 0.5$ was obtained
from the Mg ii line using multiple empirical and semi-empirical templates to
constrain the surrounding Fe ii emission. This estimate is 0.2 dex lower than
an earlier analysis for J2157–3602 (Onken et al., 2020).

  * •

The best performing model from fitting the accretion disc using MCMC suggested
a black hole mass of $\log(M_{\rm{AD}}/M_{\odot})=10.31^{+0.17}_{-0.14}$, in
concordance with the SE estimate.

  * •

We measured the anisotropy-corrected bolometric luminosity to be
$\log{(L_{\rm{bol}}/\rm{erg\,s^{-1}})}=47.87\pm 0.10$ from the SEDs of the
accretion disc models. We also measured the monochromatic luminosity at 3000Å
to be $\log{(L(\rm{3000\text{\AA}})/\rm{erg\,s^{-1}})}=47.66\pm 0.01$, from
which we inferred $\log{(L_{\rm{bol}}/\rm{erg\,s^{-1}})}=48.17\pm 0.16$, using
bolometric corrections from Richards et al. (2003), Runnoe et al. (2012),
Trakhtenbrot & Netzer (2012), and Netzer (2019). We use the bolometric
luminosity measured from the accretion disc SED to further derive the
Eddington ratio and radiative efficiency.

  * •

The Eddington ratio was calculated to be $\epsilon=0.29^{+0.11}_{-0.10}$,
based on the AD model estimates of the BH mass and bolometric luminosity.

  * •

We also estimated the radiative efficiency to be $\eta=0.09^{+0.05}_{-0.03}$,
measured from the thin disc mass accretion rate posterior distribution.

We demonstrated the utility of fitting accretion disc models in characterising
basic properties of high-mass black holes in QSOs. As an independent estimate
of black hole mass, this method can be used to complement the SE virial mass
estimate, especially at high redshift where other methods are not as easily
accessible. The key findings of the AD fitting method are the following:

  * •

Both black hole mass $M_{\rm{BH}}$ and the anisotropy-corrected bolometric
luminosity $L_{\rm{bol}}$ can be constrained without prior knowledge of the
black hole spin or the observed inclination angle. The uncertainties of these
measurements are 0.2 dex and 0.1 dex on the mass and luminosity, respectively.
Independent constraints on either spin or inclination would improve
measurement precision.

  * •

Quantities such as the Eddington ratio and radiative efficiency can be derived
from comparing observed properties of the black hole to its intrinsic
properties, parameterised as part of the disc models.

  * •

It is possible to obtain a reasonable estimate of black hole mass with only
broadband photometric data. The offset is found to be 0.1 dex for the target
examined in this study, but spectroscopic data can further refine these
results.

  * •

Host galaxy dust extinction has diverging effects on single-epoch and AD
fitting mass estimates. There is potential to exploit this idea to constrain
host galaxy extinction for luminous QSOs and the shape of the extinction
curve.

We plan future studies to investigate a larger sample of luminous QSOs with
high-quality spectra, comparing black hole mass estimates between measurements
of broad emission-line profiles and the large-scale accretion disc continuum
emission.

## Acknowledgements

We thank the anonymous referee for their constructive comments and suggestions
which have improved this manuscript. Additionally, we thank Samuele
Campitiello for useful discussions as well as Xiaohui Fan, Feige Wang, and
Jinyi Yang for supplying the final reduced spectrum of J2157–3602. We also
thank the authors of Vestergaard & Wilkes (2001), Tsuzuki et al. (2006),
Bruhweiler & Verner (2008), and Mejía-Restrepo et al. (2016) for producing and
sharing their Fe ii emission templates.

S.L. is grateful to the Research School of Astronomy & Astrophysics at
Australian National University for funding his Ph.D. studentship.

CAO was supported by the Australian Research Council (ARC) through Discovery
Project DP190100252.

This paper is based on observations made with ESO Telescopes at the La Silla
Paranal Observatory under programme ID 0104.A-0410(A).

The national facility capability for SkyMapper has been funded through ARC
LIEF grant LE130100104 from the Australian Research Council, awarded to the
University of Sydney, the Australian National University, Swinburne University
of Technology, the University of Queensland, the University of Western
Australia, the University of Melbourne, Curtin University of Technology,
Monash University and the Australian Astronomical Observatory. SkyMapper is
owned and operated by The Australian National University’s Research School of
Astronomy and Astrophysics. The survey data were processed and provided by the
SkyMapper Team at ANU. The SkyMapper node of the All-Sky Virtual Observatory
(ASVO) is hosted at the National Computational Infrastructure (NCI).
Development and support of the SkyMapper node of the ASVO has been funded in
part by Astronomy Australia Limited (AAL) and the Australian Government
through the Commonwealth’s Education Investment Fund (EIF) and National
Collaborative Research Infrastructure Strategy (NCRIS), particularly the
National eResearch Collaboration Tools and Resources (NeCTAR) and the
Australian National Data Service Projects (ANDS).

This publication makes use of data products from the Wide-field Infrared
Survey Explorer, which is a joint project of the University of California, Los
Angeles, and the Jet Propulsion Laboratory/California Institute of Technology,
and NEOWISE, which is a project of the Jet Propulsion Laboratory/California
Institute of Technology. WISE and NEOWISE are funded by the National
Aeronautics and Space Administration.

Based on observations obtained as part of the VISTA Hemisphere Survey, ESO
Progam, 179.A-2010 (PI: McMahon)

This research uses services or data provided by the Astro Data Lab at NSF’s
NOIRLab. NOIRLab is operated by the Association of Universities for Research
in Astronomy (AURA), Inc. under a cooperative agreement with the National
Science Foundation.

This publication makes use of data products from the Two Micron All Sky
Survey, which is a joint project of the University of Massachusetts and the
Infrared Processing and Analysis Center/California Institute of Technology,
funded by the National Aeronautics and Space Administration and the National
Science Foundation.

The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been
made possible through contributions by the Institute for Astronomy, the
University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society
and its participating institutes, the Max Planck Institute for Astronomy,
Heidelberg and the Max Planck Institute for Extraterrestrial Physics,
Garching, The Johns Hopkins University, Durham University, the University of
Edinburgh, the Queen’s University Belfast, the Harvard-Smithsonian Center for
Astrophysics, the Las Cumbres Observatory Global Telescope Network
Incorporated, the National Central University of Taiwan, the Space Telescope
Science Institute, the National Aeronautics and Space Administration under
Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA
Science Mission Directorate, the National Science Foundation Grant No.
AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the
Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation.

Software packages used in this study include Numpy (van der Walt et al.,
2011), Scipy (Virtanen et al., 2020), Astropy (Astropy Collaboration et al.,
2013), Specutils (Earl et al., 2022), Matplotlib (Hunter, 2007), emcee
(Foreman-Mackey et al., 2013), corner (Foreman-Mackey, 2016) and ChainConsumer
(Hinton, 2016).

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

## References

  * Amarantidis et al. (2019) Amarantidis S., et al., 2019, MNRAS, 485, 2694
  * Antonucci (1993) Antonucci R., 1993, ARA&A, 31, 473
  * Arnaud (1996) Arnaud K. A., 1996, in Jacoby G. H., Barnes J., eds, Astronomical Society of the Pacific Conference Series Vol. 101, Astronomical Data Analysis Software and Systems V. p. 17
  * Astropy Collaboration et al. (2013) Astropy Collaboration et al., 2013, A&A, 558, A33
  * Bañados et al. (2018) Bañados E., et al., 2018, Nature, 553, 473
  * Barausse (2012) Barausse E., 2012, MNRAS, 423, 2533
  * Bentz et al. (2006) Bentz M. C., Peterson B. M., Pogge R. W., Vestergaard M., Onken C. A., 2006, ApJ, 644, 133
  * Bentz et al. (2013) Bentz M. C., et al., 2013, ApJ, 767, 149
  * Blandford & McKee (1982) Blandford R. D., McKee C. F., 1982, ApJ, 255, 419
  * Bouchet et al. (1985) Bouchet P., Lequeux J., Maurice E., Prevot L., Prevot-Burnichon M. L., 1985, A&A, 149, 330
  * Bromm & Loeb (2003) Bromm V., Loeb A., 2003, ApJ, 596, 34
  * Brotherton et al. (2001) Brotherton M. S., Tran H. D., Becker R. H., Gregg M. D., Laurent-Muehleisen S. A., White R. L., 2001, ApJ, 546, 775
  * Bruhweiler & Verner (2008) Bruhweiler F., Verner E., 2008, ApJ, 675, 83
  * Calderone et al. (2013) Calderone G., Ghisellini G., Colpi M., Dotti M., 2013, MNRAS, 431, 210
  * Campitiello et al. (2018) Campitiello S., Ghisellini G., Sbarrato T., Calderone G., 2018, A&A, 612, A59
  * Campitiello et al. (2019) Campitiello S., Celotti A., Ghisellini G., Sbarrato T., 2019, A&A, 625, A23
  * Campitiello et al. (2020) Campitiello S., Celotti A., Ghisellini G., Sbarrato T., 2020, A&A, 640, A39
  * Capellupo et al. (2015) Capellupo D. M., Netzer H., Lira P., Trakhtenbrot B., Mejía-Restrepo J., 2015, MNRAS, 446, 3427
  * Capetti et al. (2021) Capetti A., Laor A., Baldi R. D., Robinson A., Marconi A., 2021, MNRAS, 502, 5086
  * Cheng et al. (2019) Cheng H., Yuan W., Liu H.-Y., Breeveld A. A., Jin C., Liu B., 2019, MNRAS, 487, 3884
  * Cutri et al. (2013) Cutri R. M., et al., 2013, Explanatory Supplement to the AllWISE Data Release Products, Explanatory Supplement to the AllWISE Data Release Products
  * Czerny et al. (2011) Czerny B., Hryniewicz K., Nikołajuk M., Sądowski A., 2011, MNRAS, 415, 2942
  * Dalla Bontà et al. (2020) Dalla Bontà E., et al., 2020, ApJ, 903, 112
  * Davies et al. (2006) Davies R. I., et al., 2006, ApJ, 646, 754
  * Davies et al. (2018) Davies F. B., et al., 2018, ApJ, 864, 142
  * Davis & El-Abd (2019) Davis S. W., El-Abd S., 2019, ApJ, 874, 23
  * Davis & Laor (2011) Davis S. W., Laor A., 2011, ApJ, 728, 98
  * Davis et al. (2005) Davis S. W., Blaes O. M., Hubeny I., Turner N. J., 2005, ApJ, 621, 372
  * De Rosa et al. (2018) De Rosa G., et al., 2018, ApJ, 866, 133
  * Dietrich et al. (2002) Dietrich M., Appenzeller I., Vestergaard M., Wagner S. J., 2002, ApJ, 564, 581
  * Done et al. (2012) Done C., Davis S. W., Jin C., Blaes O., Ward M., 2012, MNRAS, 420, 1848
  * Earl et al. (2022) Earl N., et al., 2022, astropy/specutils: V1.7.0, doi:10.5281/zenodo.6207491, https://doi.org/10.5281/zenodo.6207491
  * Ebisawa et al. (1993) Ebisawa K., Makino F., Mitsuda K., Belloni T., Cowley A. P., Schmidtke P. C., Treves A., 1993, ApJ, 403, 684
  * Elvis et al. (1994) Elvis M., et al., 1994, ApJS, 95, 1
  * Fan et al. (2001) Fan X., et al., 2001, AJ, 121, 54
  * Fawcett et al. (2022) Fawcett V. A., Alexander D. M., Rosario D. J., Klindt L., Lusso E., Morabito L. K., Calistro Rivera G., 2022, MNRAS, 513, 1254
  * Foreman-Mackey (2016) Foreman-Mackey D., 2016, The Journal of Open Source Software, 1, 24
  * Foreman-Mackey et al. (2013) Foreman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013, PASP, 125, 306
  * Freeman et al. (2001) Freeman P., Doe S., Siemiginowska A., 2001, in Starck J.-L., Murtagh F. D., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 4477, Astronomical Data Analysis. pp 76–87 (arXiv:astro-ph/0108426), doi:10.1117/12.447161
  * Gallerani et al. (2010) Gallerani S., et al., 2010, A&A, 523, A85
  * Ghisellini et al. (2015) Ghisellini G., Tagliaferri G., Sbarrato T., Gehrels N., 2015, MNRAS, 450, L34
  * Glikman et al. (2007) Glikman E., Helfand D. J., White R. L., Becker R. H., Gregg M. D., Lacy M., 2007, ApJ, 667, 673
  * Grandi (1982) Grandi S. A., 1982, ApJ, 255, 25
  * Greene & Ho (2005) Greene J. E., Ho L. C., 2005, ApJ, 630, 122
  * Greene et al. (2010) Greene J. E., et al., 2010, ApJ, 721, 26
  * Grier et al. (2013) Grier C. J., et al., 2013, ApJ, 764, 47
  * Guerras et al. (2013) Guerras E., Mediavilla E., Jimenez-Vicente J., Kochanek C. S., Muñoz J. A., Falco E., Motta V., 2013, ApJ, 764, 160
  * Guo et al. (2018) Guo H., Shen Y., Wang S., 2018, PyQSOFit: Python code to fit the spectrum of quasars, Astrophysics Source Code Library, record ascl:1809.008 (ascl:1809.008)
  * Hicks & Malkan (2008) Hicks E. K. S., Malkan M. A., 2008, ApJS, 174, 31
  * Hinton (2016) Hinton S. R., 2016, The Journal of Open Source Software, 1, 00045
  * Hopkins et al. (2004) Hopkins P. F., et al., 2004, AJ, 128, 1112
  * Hubeny et al. (2000) Hubeny I., Agol E., Blaes O., Krolik J. H., 2000, ApJ, 533, 710
  * Hubeny et al. (2001) Hubeny I., Blaes O., Krolik J. H., Agol E., 2001, ApJ, 559, 680
  * Hunter (2007) Hunter J. D., 2007, Computing in Science and Engineering, 9, 90
  * Irwin et al. (1989) Irwin M. J., Webster R. L., Hewett P. C., Corrigan R. T., Jedrzejewski R. I., 1989, AJ, 98, 1989
  * Kaspi et al. (2000) Kaspi S., Smith P. S., Netzer H., Maoz D., Jannuzi B. T., Giveon U., 2000, ApJ, 533, 631
  * Kaspi et al. (2005) Kaspi S., Maoz D., Netzer H., Peterson B. M., Vestergaard M., Jannuzi B. T., 2005, ApJ, 629, 61
  * Kawaguchi (2003) Kawaguchi T., 2003, ApJ, 593, 69
  * Kawaguchi et al. (2004) Kawaguchi T., Pierens A., Huré J. M., 2004, A&A, 415, 47
  * Klindt et al. (2019) Klindt L., Alexander D. M., Rosario D. J., Lusso E., Fotopoulou S., 2019, MNRAS, 488, 3109
  * Kollmeier et al. (2006) Kollmeier J. A., et al., 2006, ApJ, 648, 128
  * Kovačević et al. (2014) Kovačević J., Popović L. Č., Kollatschny W., 2014, Advances in Space Research, 54, 1347
  * Kozłowski (2017) Kozłowski S., 2017, ApJS, 228, 9
  * Krawczyk et al. (2015) Krawczyk C. M., Richards G. T., Gallagher S. C., Leighly K. M., Hewett P. C., Ross N. P., Hall P. B., 2015, AJ, 149, 203
  * Krolik (2001) Krolik J. H., 2001, ApJ, 551, 72
  * Lai et al. (2022) Lai S., et al., 2022, MNRAS, 513, 1801
  * Laor (1990) Laor A., 1990, MNRAS, 246, 369
  * Laor & Davis (2011) Laor A., Davis S. W., 2011, MNRAS, 417, 681
  * Laor & Netzer (1989) Laor A., Netzer H., 1989, MNRAS, 238, 897
  * Li et al. (2005) Li L.-X., Zimmerman E. R., Narayan R., McClintock J. E., 2005, ApJS, 157, 335
  * Liu et al. (2021) Liu H., Luo B., Brandt W. N., Brotherton M. S., Gallagher S. C., Ni Q., Shemmer O., Timlin J. D. I., 2021, ApJ, 910, 103
  * López et al. (2016) López S., et al., 2016, A&A, 594, A91
  * Malkan (1983) Malkan M. A., 1983, ApJ, 268, 582
  * Marconi et al. (2004) Marconi A., Risaliti G., Gilli R., Hunt L. K., Maiolino R., Salvati M., 2004, MNRAS, 351, 169
  * Marocco et al. (2021) Marocco F., et al., 2021, ApJS, 253, 8
  * Mazzucchelli et al. (2017) Mazzucchelli C., et al., 2017, ApJ, 849, 91
  * McLure & Dunlop (2004) McLure R. J., Dunlop J. S., 2004, MNRAS, 352, 1390
  * McLure & Jarvis (2002) McLure R. J., Jarvis M. J., 2002, MNRAS, 337, 109
  * McMahon et al. (2013) McMahon R. G., Banerji M., Gonzalez E., Koposov S. E., Bejar V. J., Lodieu N., Rebolo R., VHS Collaboration 2013, The Messenger, 154, 35
  * Mediavilla et al. (2018) Mediavilla E., Jiménez-Vicente J., Fian C., Muñoz J. A., Falco E., Motta V., Guerras E., 2018, ApJ, 862, 104
  * Mediavilla et al. (2019) Mediavilla E., Jiménez-vicente J., Mejía-restrepo J., Motta V., Falco E., Muñoz J. A., Fian C., Guerras E., 2019, ApJ, 880, 96
  * Mejía-Restrepo et al. (2016) Mejía-Restrepo J. E., Trakhtenbrot B., Lira P., Netzer H., Capellupo D. M., 2016, MNRAS, 460, 187
  * Mejía-Restrepo et al. (2018) Mejía-Restrepo J. E., Lira P., Netzer H., Trakhtenbrot B., Capellupo D. M., 2018, Nature Astronomy, 2, 63
  * Nemmen & Brotherton (2010) Nemmen R. S., Brotherton M. S., 2010, MNRAS, 408, 1598
  * Netzer (2019) Netzer H., 2019, MNRAS, 488, 5185
  * Netzer & Peterson (1997) Netzer H., Peterson B. M., 1997, in Maoz D., Sternberg A., Leibowitz E. M., eds, Astrophysics and Space Science Library Vol. 218, Astronomical Time Series. p. 85 (arXiv:astro-ph/9706039), doi:10.1007/978-94-015-8941-3_8
  * Nidever et al. (2021) Nidever D. L., et al., 2021, AJ, 161, 192
  * Niida et al. (2020) Niida M., et al., 2020, ApJ, 904, 89
  * Novikov & Thorne (1973) Novikov I. D., Thorne K. S., 1973, in Black Holes (Les Astres Occlus). pp 343–450
  * Onken et al. (2007) Onken C. A., et al., 2007, ApJ, 670, 105
  * Onken et al. (2019) Onken C. A., et al., 2019, Publ. Astron. Soc. Australia, 36, e033
  * Onken et al. (2020) Onken C. A., Bian F., Fan X., Wang F., Wolf C., Yang J., 2020, MNRAS, 496, 2309
  * Onken et al. (2022) Onken C. A., Wolf C., Bian F., Fan X., Hon W. J., Raithel D., Tisserand P., Lai S., 2022, MNRAS, 511, 572
  * Pacucci et al. (2015) Pacucci F., Volonteri M., Ferrara A., 2015, MNRAS, 452, 1922
  * Peterson (1993) Peterson B. M., 1993, PASP, 105, 247
  * Peterson et al. (2004) Peterson B. M., et al., 2004, ApJ, 613, 682
  * Popović et al. (2022) Popović L. Č., Shablovinskaya E., Savić D., 2022, Astronomische Nachrichten, 343, e210089
  * Prochaska et al. (2020) Prochaska J., et al., 2020, The Journal of Open Source Software, 5, 2308
  * Raimundo et al. (2019) Raimundo S. I., Pancoast A., Vestergaard M., Goad M. R., Barth A. J., 2019, MNRAS, 489, 1899
  * Rakshit et al. (2020) Rakshit S., Stalin C. S., Kotilainen J., 2020, ApJS, 249, 17
  * Reed et al. (2019) Reed S. L., et al., 2019, MNRAS, 487, 1874
  * Reichard et al. (2003) Reichard T. A., et al., 2003, AJ, 126, 2594
  * Reynolds (2021) Reynolds C. S., 2021, ARA&A, 59, 117
  * Reynolds & Fabian (2008) Reynolds C. S., Fabian A. C., 2008, ApJ, 675, 1048
  * Richards et al. (2003) Richards G. T., et al., 2003, AJ, 126, 1131
  * Richards et al. (2006) Richards G. T., et al., 2006, ApJS, 166, 470
  * Rodrigo & Solano (2020) Rodrigo C., Solano E., 2020, in XIV.0 Scientific Meeting (virtual) of the Spanish Astronomical Society. p. 182
  * Rokaki et al. (1992) Rokaki E., Boisson C., Collin-Souffrin S., 1992, A&A, 253, 57
  * Runnoe et al. (2012) Runnoe J. C., Brotherton M. S., Shang Z., 2012, MNRAS, 427, 1800
  * Sadowski (2011) Sadowski A., 2011, arXiv e-prints, p. arXiv:1108.0396
  * Salviander et al. (2007) Salviander S., Shields G. A., Gebhardt K., Bonning E. W., 2007, ApJ, 662, 131
  * Savić et al. (2018) Savić D., Goosmann R., Popović L. Č., Marin F., Afanasiev V. L., 2018, A&A, 614, A120
  * Savić et al. (2021) Savić Đ. V., Popović L. Č., Shablovinskaya E., 2021, ApJ, 921, L21
  * Schindler et al. (2020) Schindler J.-T., et al., 2020, ApJ, 905, 51
  * Schlafly & Finkbeiner (2011) Schlafly E. F., Finkbeiner D. P., 2011, ApJ, 737, 103
  * Schlafly et al. (2010) Schlafly E. F., Finkbeiner D. P., Schlegel D. J., Jurić M., Ivezić Ž., Gibson R. R., Knapp G. R., Weaver B. A., 2010, ApJ, 725, 1175
  * Schlegel et al. (1998) Schlegel D. J., Finkbeiner D. P., Davis M., 1998, ApJ, 500, 525
  * Selsing et al. (2016) Selsing J., Fynbo J. P. U., Christensen L., Krogager J. K., 2016, A&A, 585, A87
  * Shakura & Sunyaev (1973) Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337
  * Shang et al. (2011) Shang Z., et al., 2011, ApJS, 196, 2
  * Shen (2013) Shen Y., 2013, Bulletin of the Astronomical Society of India, 41, 61
  * Shen & Liu (2012) Shen Y., Liu X., 2012, ApJ, 753, 125
  * Shen et al. (2008) Shen Y., Greene J. E., Strauss M. A., Richards G. T., Schneider D. P., 2008, ApJ, 680, 169
  * Shen et al. (2011) Shen Y., et al., 2011, ApJS, 194, 45
  * Shimura & Takahara (1993) Shimura T., Takahara F., 1993, ApJ, 419, 78
  * Shimura & Takahara (1995) Shimura T., Takahara F., 1995, ApJ, 445, 780
  * Skrutskie et al. (2006) Skrutskie M. F., et al., 2006, AJ, 131, 1163
  * Sluse et al. (2011) Sluse D., et al., 2011, A&A, 528, A100
  * Starling et al. (2004) Starling R. L. C., Siemiginowska A., Uttley P., Soria R., 2004, MNRAS, 347, 67
  * Steffen et al. (2006) Steffen A. T., Strateva I., Brandt W. N., Alexander D. M., Koekemoer A. M., Lehmer B. D., Schneider D. P., Vignali C., 2006, AJ, 131, 2826
  * Steinhardt & Elvis (2010) Steinhardt C. L., Elvis M., 2010, MNRAS, 406, L1
  * Straub et al. (2011) Straub O., et al., 2011, A&A, 533, A67
  * Sun & Malkan (1989) Sun W.-H., Malkan M. A., 1989, ApJ, 346, 68
  * Trakhtenbrot (2014) Trakhtenbrot B., 2014, ApJ, 789, L9
  * Trakhtenbrot & Netzer (2012) Trakhtenbrot B., Netzer H., 2012, MNRAS, 427, 3081
  * Trakhtenbrot et al. (2011) Trakhtenbrot B., Netzer H., Lira P., Shemmer O., 2011, ApJ, 730, 7
  * Trakhtenbrot et al. (2017) Trakhtenbrot B., Volonteri M., Natarajan P., 2017, ApJ, 836, L1
  * Tripp et al. (1994) Tripp T. M., Bechtold J., Green R. F., 1994, ApJ, 433, 533
  * Tsuzuki et al. (2006) Tsuzuki Y., Kawara K., Yoshii Y., Oyabu S., Tanabé T., Matsuoka Y., 2006, ApJ, 650, 57
  * Vasudevan & Fabian (2007) Vasudevan R. V., Fabian A. C., 2007, MNRAS, 381, 1235
  * Vasudevan & Fabian (2009) Vasudevan R. V., Fabian A. C., 2009, MNRAS, 392, 1124
  * Vernet et al. (2011) Vernet J., et al., 2011, A&A, 536, A105
  * Vestergaard (2002) Vestergaard M., 2002, ApJ, 571, 733
  * Vestergaard & Osmer (2009) Vestergaard M., Osmer P. S., 2009, ApJ, 699, 800
  * Vestergaard & Peterson (2006) Vestergaard M., Peterson B. M., 2006, ApJ, 641, 689
  * Vestergaard & Wilkes (2001) Vestergaard M., Wilkes B. J., 2001, ApJS, 134, 1
  * Virtanen et al. (2020) Virtanen P., et al., 2020, Nature Methods, 17, 261
  * Volonteri et al. (2005) Volonteri M., Madau P., Quataert E., Rees M. J., 2005, ApJ, 620, 69
  * Walton et al. (2013) Walton D. J., Nardini E., Fabian A. C., Gallo L. C., Reis R. C., 2013, MNRAS, 428, 2901
  * Wandel & Petrosian (1988) Wandel A., Petrosian V., 1988, ApJ, 329, L11
  * Wandel et al. (1999) Wandel A., Peterson B. M., Malkan M. A., 1999, ApJ, 526, 579
  * Wang et al. (2009) Wang J.-G., et al., 2009, ApJ, 707, 1334
  * Wang et al. (2021) Wang F., et al., 2021, ApJ, 907, L1
  * Webster et al. (1995) Webster R. L., Francis P. J., Petersont B. A., Drinkwater M. J., Masci F. J., 1995, Nature, 375, 469
  * Wielgus et al. (2022) Wielgus M., et al., 2022, MNRAS, 514, 780
  * Wilson et al. (2004) Wilson J. C., et al., 2004, in Moorwood A. F. M., Iye M., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 5492, Ground-based Instrumentation for Astronomy. pp 1295–1305, doi:10.1117/12.550925
  * Wolf et al. (2018) Wolf C., Bian F., Onken C. A., Schmidt B. P., Tisserand P., Alonzi N., Hon W. J., Tonry J. L., 2018, Publ. Astron. Soc. Australia, 35, e024
  * Woo et al. (2010) Woo J.-H., et al., 2010, ApJ, 716, 269
  * Wright et al. (2010) Wright E. L., et al., 2010, AJ, 140, 1868
  * Wu et al. (2015) Wu X.-B., et al., 2015, Nature, 518, 512
  * York et al. (2000) York D. G., et al., 2000, AJ, 120, 1579
  * Yu & Tremaine (2002) Yu Q., Tremaine S., 2002, MNRAS, 335, 965
  * Zhu et al. (2021) Zhu Y., et al., 2021, ApJ, 923, 223
  * Zubovas & King (2021) Zubovas K., King A., 2021, MNRAS, 501, 4289
  * Zuo et al. (2015) Zuo W., Wu X.-B., Fan X., Green R., Wang R., Bian F., 2015, ApJ, 799, 189
  * van der Walt et al. (2011) van der Walt S., Colbert S. C., Varoquaux G., 2011, Computing in Science and Engineering, 13, 22

##  Appendix A Bolometric Luminosity

Several bolometric corrections have been proposed to transform monochromatic
luminosities into QSO bolometric luminosities. At times, a fixed correction is
used, $L_{\rm{bol}}=k_{\lambda}L(\lambda)$, where $k_{\lambda}$ is the
bolometric correction factor and $L(\lambda)$ represents the monochromatic
luminosity, $\lambda L_{\lambda}$ at rest wavelength $\lambda$. From the
power-law continuum model of J2157–3602, we measured
$\log{(L(\rm{3000\text{\AA}})/\rm{erg\,s^{-1}})}=47.66\pm 0.01$.

###  A.1 Bolometric corrections

Using composite SEDs from 259 SDSS QSOs, Richards et al. (2006) calculated an
average bolometric correction factor of $k_{3000\text{\AA}}=5.62\pm 1.14$,
although a fixed median value of $k_{3000\text{\AA}}=5.15$ is more commonly
used in practice (e.g. Shen et al., 2011). However, this value likely
overestimates the bolometric luminosity for luminous QSOs, motivating a
luminosity-dependent correction which flattens to $k_{3000\text{\AA}}\sim 3.2$
for $L(3000\text{\AA})>10^{46}$ erg s-1 (Marconi et al., 2004; Trakhtenbrot &
Netzer, 2012), which is derived from QSO spectral templates.

Another bolometric correction from Runnoe et al. (2012) is derived from the
NIR$-$Xray SED atlas of 63 QSOs (Shang et al., 2011). Using the best-fitting
linear correction with non-zero intercepts and the recommended 25% luminosity
suppression based on the sample’s orientation bias (Nemmen & Brotherton, 2010,
hereafter N10), the luminosity-dependent bolometric correction for J2157–3602
is then $k_{3000\text{\AA}}\sim 3.33$.

A more recent study based on theoretical calculations of optically thick and
geometrically thin accretion discs (Netzer, 2019) estimated
$k_{3000\text{\AA}}=25(L(3000\text{\AA})/10^{42}\rm{erg\,s^{-1}})^{-0.2}$,
which is approximately $k_{3000\text{\AA}}\sim 1.84$ for J2157–3602. This
study assumed that accretion through the disc is the only energy production
mechanism and that the accretion rate is sufficiently sub-Eddington for the
disc to remain geometrically thin. Furthermore, the X-ray luminosity from a
corona heated to temperatures above that of the AD is assumed to be drawn from
the same gravitational energy source and does not affect $L_{\rm{bol}}$.

The collection of estimated bolometric luminosities with correction factors of
(1.84, 3.2, 3.33, 5.15) are $\log{(L_{\rm{bol}}/\rm{erg\,s^{-1}})}=$ (47.93,
48.17, 48.19, 48.38). The mean bolometric luminosity and the standard
deviation is then $\log{(L_{\rm{bol}}/\rm{erg\,s^{-1}})}=48.17\pm 0.16$. This
measurement is well-matched to the previous estimate of
$\log{(L_{\rm{bol}}/\rm{erg\,s^{-1}})}\approx 48.2$ (Onken et al., 2020),
which was derived with the Runnoe et al. (2012) correction. However, the
result from independent bolometric corrections spans 0.45 dex and the
individual studies assume a mean SED or that a straightforward luminosity
dependence is sufficient to extrapolate the full SED shape for every QSO. We
adopt an approach in the following section that is independent of the assumed
mean SED shape and sample properties used to calibrate the bolometric
corrections.

###  A.2 Integrated SED

The kerrbb thin disc and slimbh slim disc parameterise disc luminosity in
terms of the mass accretion rate and Eddington ratio, respectively. However,
while the model parameters describe intrinsic properties of the black hole and
its accretion disc, the transformation from the model parameterisation to the
observed luminosity is not trivial, due to additional effects considered by
general relativistic ray-tracing. The mass accretion rate parameter from
kerrbb can be converted to luminosity, but the radiative efficiency of the
thin disc does not correspond to a standard Keplerian disc (Li et al., 2005),
especially if the torque at the inner boundary is non-zero or limb darkening
effects are included. For the slim disc, the Eddington luminosity
parameterised in the model does not consider general relativistic effects, and
thus only roughly corresponds to the total output disc luminosity (Sadowski,
2011). Therefore, we circumnavigate these potential issues by integrating the
model SED to derive the bolometric luminosity.

Here, we estimate the bolometric luminosity by drawing 1000 random models out
of the MCMC posterior distribution from the kerrbb thin disc model fits to
synthetic photometry and integrating the SED. We use the thin disc model
because the mass accretion rate is a fitted parameter, allowing us to
constrain the radiative efficiency and correct for anisotropy. The integrated
luminosity, defined by

$L_{\rm{iso}}=\int_{0}^{\infty}L_{\nu}d\nu=\int_{-\infty}^{\infty}\ln(10)\nu
L_{\nu}d\log(\nu)\,,$ (3)

is measured to be $\log{(L_{\rm{iso}}/\rm{erg\,s^{-1}})}=47.99\pm 0.01$, where
$L_{\rm{iso}}$ is the luminosity evaluated under the assumption of isotropy.
Although a slim disc is arguably a better model of the accretion disc
structure, the difference in the integrated luminosity is only
$\log{(L_{\rm{thin}}/L_{\rm{slim}})}=0.02$. In general, the isotropic-
equivalent luminosity, $L_{\rm{iso}}$, is evaluated by

$\displaystyle L_{\rm{iso}}$
$\displaystyle=\int_{0}^{2\pi}\int_{0}^{\pi}F_{\rm{obs}}d_{L}^{2}\sin\theta
d\theta d\phi\,,$ (4) $\displaystyle=4\pi d_{L}^{2}F_{\rm{obs}}\,,$

where $d_{L}^{2}$ is the luminosity distance and $F_{\rm{obs}}$ is the
observed flux density. The isotropic-equivalent luminosity would only be
equivalent to the total bolometric luminosity, integrated over all solid
angles, if the emission is isotropic or viewed at an intermediate angle where
the isotropy assumption is reasonable. If $F_{\rm{obs}}$ is a function of the
inclination angle, then we would need to determine a correction for the
anisotropy. As the spin and orientation of each thin disc model are known, we
can calculate an anisotropy correction to the isotropic luminosity, defined as
a ratio of the total bolometric radiative efficiency, $\eta_{\rm{bol}}$, to
the isotropic-equivalent radiative efficiency, $\eta_{\rm{iso}}$, such that
the total bolometric luminosity, $L_{\rm{bol}}$, is measured by

$L_{\rm{bol}}=\frac{\eta_{\rm{bol}}}{\eta_{\rm{iso}}}L_{\rm{iso}}=\dot{M}c^{2}\int_{0}^{1}\eta(\theta,a)d(\cos\theta)\,,$
(5)

where we construct analytical approximations to estimate $\eta(\theta,a)$
defining,

$\displaystyle\eta(\theta,a)=\sum_{i=0}^{\mathcal{N}_{1}}\mathcal{F}_{i}\cos(\theta)^{i}\,,$
(6)
$\displaystyle\mathcal{F}_{i}(a)=\sum_{j=0}^{\mathcal{N}_{2}}\zeta_{j}\log^{j}(1-a)\,,$

and the $\zeta_{j}$ coefficients are presented in Table 2, updating the
analytical approximation in C18 to include the effects of limb darkening. We
caution that our parameterisation differs from that of C18, but we find this
model to produce smoother fits. Coefficients have been provided up to
$\mathcal{N}_{1}=5$ and $\mathcal{N}_{2}=6$, which are sufficient to estimate
$\eta(\theta,a)$ in Equation 6 to $\sim 0.1\%$ accuracy.

In Figure 4, we compare the observed radiative efficiency and normalised 3000Å
monochromatic luminosities with and without limb darkening using the kerrbb
thin disc model. With limb darkening, the peak observed luminosity is always
at face-on orientations, but without limb darkening, the peak luminosity can
be at intermediate viewing angles for close to maximally spinning black holes.
Over a large range of viewing angles up to 50–60∘ depending on spin, the
models with limb darkening are brighter than similar models without limb
darkening. The relationship between the observed radiative efficiency and the
viewing angle depends sensitively on spin, but the response of the
monochromatic luminosity to orientation is similar at 3000Å regardless of
black hole spin. However, the shape of this curve can be quite different at
shorter wavelengths, especially beyond the peak of the SED.

Having derived analytical approximations of $\eta(a,\theta)$, we then define
the anisotropy correction, $f(\theta,a)$ and a fractional error, $\Delta L/L$,
in the isotropic luminosity as,

$\displaystyle
f(\theta,a)=\frac{\eta_{\rm{bol}}}{\eta_{\rm{iso}}}=\frac{1}{\eta_{\rm{iso}}(\theta,a)}\int_{0}^{1}\eta(\theta^{\prime},a)d(\cos\theta^{\prime})\,,$
(7) $\displaystyle\frac{\Delta
L}{L}\equiv\frac{L_{\rm{iso}}-L_{\rm{bol}}}{L_{\rm{bol}}}=\frac{1-f(\theta,a)}{f(\theta,a)}\,,$

where $\Delta L/L$ estimates the error incurred between the observed and total
bolometric luminosity of a QSO if the observed emission is assumed to be
isotropic. A fractional error of 0 implies no correction and occurs at
intermediate viewing angles, such as 60∘ in the Newtonian case or in specific
general relativistic scenarios, 56∘ (Netzer, 2019) or 66∘ (N10) depending on
the spin of the system.

In Figure 5, we show the fractional error derived from the kerrbb thin disc
model across inclination angles for the full range of spins from $a=-1$ to
$a=0.998$. We contrast our model with the Newtonian case and the average
fractional error from N10. Much of the parameter space is covered by the
highest 10% of spins. If all spins are uniformly probable, then the $a=0.4$
line best represents the average fractional error. It is also well-
approximated by a Newtonian model, except for near face-on and edge-on
orientation. If only positive spins are considered, then the $a=0.65$ curve
best represents the mean fractional error. These single-spin curves can
reproduce the values of their respective mean curves to better than 1%
precision. In Appendix A.3, we continue the discussion on the anisotropy
correction and derive an average correction for QSO samples with any maximum
orientation angle, $\theta_{\rm{max}}$. We create analytical approximations
for the correction, $f(\theta_{\rm{max}},a)$, so that they can be derived for
any combination of parameters without evaluating and integrating thin disc
model spectra.

For each of the 1000 models drawn from the MCMC posterior, we generate a
unique curve from its black hole spin and evaluate its anisotropy correction
for its inclination as in Figure 5. After applying each individual correction,
we find $\log{(L_{\rm{bol}}/\rm{erg\,s^{-1}})}=47.87\pm 0.10$, which implies a
bolometric correction factor of $k_{3000\text{\AA}}\sim 1.62$.
Methodologically, this procedure is similar to Netzer (2019), albeit with the
orientation and spin-dependent anisotropy correction instead of a fixed
viewing angle assumption of $\sim 56^{\circ}$, and the estimated luminosity is
slightly lower as expected due to the shallower mean viewing angle of $\sim
45^{\circ}$.

Compared to the empirically derived bolometric corrections from Section A.1,
the integrated AD SED includes only thermal emission from the accretion disc.
Only a small fraction of the total radiative output of QSOs is emitted in the
X-rays and this fraction also decreases with the UV luminosity of the AGN (the
$\alpha_{\rm{OX}}$ correlation; e.g. Elvis et al., 1994; Steffen et al., 2006;
Vasudevan & Fabian, 2007, 2009; Liu et al., 2021). Naturally, we also make the
assumption that the thin disc spectra adequately model the observed optical-UV
continuum of QSOs. By integrating the locally-blackbody emission, we avoid
double-counting reprocessed emission in the infrared or hard X-ray regime.
Additionally, we do not depend upon congruence between properties of our
target and the mean properties of the samples used to calibrate the bolometric
corrections, which can create an order of magnitude difference in the
measurement.

Figure 4: Comparison of the observed radiative efficiency (left) and
normalised 3000Å monochromatic luminosities (right) at various viewing angles
with and without limb darkening, for selected black hole spins from $a=-1$ to
$a=1$. The solid (dotted) lines show the result with (without) limb darkening
and the Newtonian model, which represents $L\propto\cos\theta$, is shown with
the blue dashed lines. On the right panel, the monochromatic luminosities are
normalised to the brightest model. With the 3000Å monochromatic luminosity,
the shape of the curves remains consistent for the full range of black hole
spins. Figure 5: Fractional error incurred between the isotropic luminosity
and the total bolometric luminosity of a QSO if the observed emission is
assumed to be isotropic. The secondary axis (right) shows the isotropic-
equivalent luminosity as a fraction of the bolometric luminosity. Our
estimates are derived from the kerrbb thin disc model and contrasted against
N10 and the Newtonian case. The mean fractional error is similar to $a=0.4$ if
all spins are uniformly probable and to $a=0.65$ if only positive spins are
considered.

###  A.3 Anisotropy Correction

In the previous section, we defined an anisotropy correction, $f(\theta,a)$,
and its fractional error for any combination of dimensionless black hole spin,
$a$, and inclination, $\theta_{\rm{inc}}$. The fractional error estimates the
error incurred between the isotropic luminosity and the total bolometric
luminosity if the observed emission is assumed to be radiated isotropically.
We used the anisotropy correction to estimate the total bolometric luminosity
from posterior model SEDs of one QSO with particular spins and observed
inclination angles. In this section, we derive the mean anisotropy correction
for a sample of QSOs. We first consider the general case of an anisotropy
correction that is a function of spin (allowing application to arbitrary spin
distributions, e.g., from simulations), then corrections for particular mean
spin values.

Under the orientation classification model for Seyfert galaxies and QSOs
(Antonucci, 1993), there’s an assumed opening angle of the obscuring torus,
where shallower viewing angles enable the observation of the BLR, rendering it
a Type 1. Large samples of Type 1 QSOs, where all orientations shallower than
the opening angle, $\theta_{\rm{max}}$, are equally likely, will need to
consider an average anisotropy correction to transform the observed
luminosities into the total bolometric luminosities of the AGN sample. We
define this correction, $f(\theta_{\rm{max}},a)$, as

$\displaystyle f(\theta_{\rm{max}},a)$
$\displaystyle=\frac{\int_{0}^{1}\eta(\theta,a)d(\cos\theta)}{1-\cos(\theta_{\rm{max}})}\int_{\cos\theta_{\rm{max}}}^{1}\eta(\theta,a)^{-1}d(\cos\theta)\,,$
(8)

which is a mean anisotropy correction for a particular black hole spin, $a$,
weighted by its probability of being observed at all orientations with
$\theta\leq\theta_{\rm{max}}$. As before, the radiative efficiency,
$\eta(\theta,a)$, is estimated using Equation 6.

We again derive an analytic expression with the following form to estimate the
anisotropy correction without evaluating and integrating thin disc model SEDs,

$\displaystyle
f(\theta_{\rm{max}},a)=C_{\rm{0}}(\theta_{\rm{max}})+\sum_{i=1}^{N_{1}}C_{\rm{i}}(\theta_{\rm{max}})\log\left(i-a\right)\,,$
(9) $\displaystyle
C_{\rm{i}}(\theta_{\rm{max}})=\sum_{j=0}^{N_{2}}D_{\rm{j}}\cos(\theta_{\rm{max}})^{j}\,,$

where we have provided coefficients, $C_{\rm{i}}$ and $D_{\rm{j}}$, up to
$N_{1}=6$ and $N_{2}=8$ in Table 3. The provided coefficients are sufficient
to estimate $f(\theta_{\rm{max}},a)$ to within 1% error for most combinations
of dimensionless spin, $a$, and maximum inclination, $\theta_{\rm{max}}$,
except for near maximal spin, where the error can be up to 3%.

In Figure 6, we show the mean anisotropy correction for various black hole
spins on the left panel and for various maximum inclination angles on the
right panel. In the left panel, each of the curves represent a different value
of $\theta_{\rm{max}}$. The range of correction factors increases dramatically
as $\theta_{\rm{max}}$ approaches edge-on. However, all mean correction
factors converge to near $f(\theta_{\rm{max}},a)\sim 1.0$ for the highest
spins. At $60^{\circ}\leq\theta_{\rm{max}}\leq 80^{\circ}$, the mean
correction factor is fairly flat with relatively little deviation across all
spins, except for near-maximal spin. On the right panel, the mean anisotropy
correction is shown for the range of maximum inclination angles. As with
Figure 5, most of the parameter space is covered by the highest 10% of spins.

If all black hole spins in a sample of QSOs are assumed to be uniformly
probable, the mean curve is best represented by the $a=0.4$ spin case, and the
$a=0.65$ curve best reflects the mean curve for all positive spins. These
single-spin curves can reproduce the values of their respective mean curves to
better than 1% precision. For instance, if a sample of QSOs was observed with
unknown spins and orientations, but an opening angle of
$\theta_{\rm{max}}=65^{\circ}$ is assumed as in this study, then the right
panel of Figure 6 can be used to evaluate a mean anisotropy correction factor
for this sample of $\langle f(65^{\circ},a)\rangle_{a}=0.75$, which should
then be applied to the isotropic luminosity of QSOs in this sample following
Equation 5. This correction factor is similar to the 25% bolometric luminosity
suppression adopted by Runnoe et al. (2012) to correct for their orientation
bias.

Table 2: Coefficients of $\zeta_{j}$ used to solve for $\mathcal{F}_{i}(a)$ and applied to Equation 6 to estimate the observed radiative efficiency $\eta(\theta,a)$ for any combination of observed inclination, $\theta$, and black hole spin, $a$. | $\zeta_{0}$ | $\zeta_{1}$ | $\zeta_{2}$ | $\zeta_{3}$ | $\zeta_{4}$ | $\zeta_{5}$ | $\zeta_{6}$
---|---|---|---|---|---|---|---
$\mathcal{F}_{0}$ | 0.0034 | -0.0102 | 0.0121 | 0.0091 | 0.0156 | 0.0057 | 0.0006
$\mathcal{F}_{1}$ | 0.0760 | -0.1092 | 0.0445 | -0.1146 | -0.0167 | 0.0307 | 0.0076
$\mathcal{F}_{2}$ | 0.0698 | -0.0942 | 0.1499 | -0.0735 | -0.6514 | -0.3703 | -0.0578
$\mathcal{F}_{3}$ | -0.0192 | 0.0691 | -0.3491 | 0.9902 | 1.7997 | 0.8101 | 0.1131
$\mathcal{F}_{4}$ | -0.0235 | 0.0603 | 0.1863 | -1.2190 | -1.6659 | -0.6792 | -0.0894
$\mathcal{F}_{5}$ | 0.0106 | -0.0397 | -0.0237 | 0.4420 | 0.5295 | 0.2039 | 0.0258
Figure 6: Mean anisotropy correction factor for different spins and uniform inclination up to $\theta_{\rm{max}}$ (left) and for maximum inclination angle with uniform spin distribution (right). On the right panel, the $a=0.4$ (or $a=0.65$) spin case best represents the mean curve if all spins (all positive spins) are uniformly probable. For any QSO sample with randomly distributed spins and an estimated opening angle of the obscuring torus, the right panel can be referenced to estimate a mean anisotropy correction. Table 3:  Coefficients of $D_{\rm{j}}$ used to solve for $C_{\rm{i}}(\theta_{\rm{max}})$ and applied to Equation 9 to estimate the anisotropic correction, $f(\theta_{\rm{max}},a)$, for any sample with black hole spin, $a$, and maximum inclination angle, $\theta_{\rm{max}}$. | $D_{0}$ | $D_{1}$ | $D_{2}$ | $D_{3}$ | $D_{4}$ | $D_{5}$ | $D_{6}$ | $D_{7}$ | $D_{8}$
---|---|---|---|---|---|---|---|---|---
$C_{0}$ | 825839.11 | -825654.54 | -12727.77 | 133489.26 | -618406.28 | 1542748.35 | -2134722.89 | 1540498.01 | -452008.97
$C_{1}$ | -145.85 | 145.92 | -0.92 | 3.70 | -11.32 | 22.76 | -27.71 | 18.39 | -5.09
$C_{2}$ | 19815.54 | -19601.21 | -942.79 | -2700.28 | 27793.58 | -86347.80 | 132556.84 | -101623.32 | 31026.11
$C_{3}$ | 852776.56 | -855518.06 | -15310.58 | 350278.39 | -1856395.08 | 4895185.38 | -6978921.44 | 5130863.11 | -1524798.56
$C_{4}$ | 19495660.09 | -19487281.21 | 176344.12 | -2554055.17 | 12705285.15 | -32688850.21 | 46008156.13 | -33560153.75 | 9920635.72
$C_{5}$ | 6247439.48 | -6256063.92 | -400372.22 | 5131351.82 | -24930987.93 | 63518064.50 | -88927307.96 | 64653620.82 | -19069014.39
$C_{6}$ | 8396532.21 | -8393904.69 | 249288.84 | -3018336.82 | 14486023.68 | -36713644.08 | 51252728.94 | -37195437.91 | 10956809.26

##  Appendix B QSO J102325.31$+$514251.0

The QSO J102325.31$+$514251.0, $z=3.477$, was measured to host a black hole of
mass $\log(M_{\rm{SE}}/M_{\odot})_{\rm{Mg\,\textsc{ii}}}=10.52\pm 0.08$ using
the SE virial mass estimate with the Mg ii line (Zuo et al., 2015), which
would make it one of the most massive black holes ever measured. However, that
estimate is in contention with the H$\beta$-based virial mass estimate in the
same study which suggested a black hole mass of
$\log(M_{\rm{SE}}/M_{\odot})_{\rm{H\textsc{$\beta$}}}=9.58\pm 0.14$ (Zuo et
al., 2015). These cases where the virial mass estimates from different
emission lines differ by more than 0.5 dex are rare, constituting about 3% of
QSOs in the Shen et al. (2011) QSO catalogue and about 1% in the Rakshit et
al. (2020) catalogue.

As an independent method of estimating black hole masses, AD fitting is useful
for disentangling these ambiguous cases. Even if only limited to cases for
which $\log(M_{\rm{SE}}/M_{\odot})>8$, where the $f_{\rm{col}}=1$ assumption
is reasonable (see Section 3.2), we find $>1000$ QSOs in each of the Shen et
al. (2011) and Rakshit et al. (2020) QSO catalogues. Here, we show how AD
fitting can be used to measure the black hole mass of J1023$+$5142, using only
publicly available photometric data.

We crossmatch J1023$+$5142 with AllWISE, 2MASS, and PanSTARRS DR1, obtaining
data from the optical to infrared. Following the procedure from Section 3.2,
we fit data between rest-frame 1 $\mu$m to 1600Å. Like J2157–3602, we
determine the Galactic extinction based on the rescaled SFD extinction map to
be $E(B-V)=0.013$ and we assume no host galaxy extinction. We adopt the slimbh
slim accretion disc models and flat uninformative prior probability
distributions for all free parameters: black hole mass $M_{\rm{BH}}$, disc
luminosity $L_{\rm{disc}}$, spin $a$, and inclination $\theta_{\rm{inc}}$.
Figure 7 presents the corner plot of the MCMC posterior distributions.
Although neither spin nor inclination are constrained by the data, the black
hole mass estimate is $\log(M_{\rm{AD}}/M_{\odot})=9.55^{+0.20}_{-0.17}$,
using the maximum likelihood and uncertainties determined by the 68% iso-
likelihood line. This result strongly disfavors the Mg ii-based virial mass
estimate, suggesting nearly 1 dex lower mass. If the H$\beta$ mass estimate is
unavailable and the Mg ii SE mass estimate is used as the mass prior, the AD
fitting mass estimate is $\log(M_{\rm{AD}}/M_{\odot})=9.71_{-0.20}^{+0.19}$,
which is sufficient to exclude the Mg ii result at $>4\sigma$ even if there is
no alternative SE $M_{\rm{BH}}$ estimate.

We calculate the bolometric luminosity of J1023$+$5142 following the methods
outlined in Appendix A, finding
$\log{(L_{\rm{bol}}/\rm{erg\,s^{-1}})}=47.47\pm 0.17$, 0.3 dex lower than in
Zuo et al. (2015), which used a fixed correction of $k_{5100\text{\AA}}=9.26$
(Richards et al., 2006). Instead, we find our bolometric luminosity to be
consistent with $k_{5100\text{\AA}}=4.68$ and $k_{3000\text{\AA}}=4.27$.
Higher mass BHs such as J1023$+$5142 are more likely to have lower bolometric
correction factors (e.g. Trakhtenbrot & Netzer, 2012) than those derived from
mean SEDs of lower-mass black holes. With the newly measured black hole mass
bolometric luminosity using AD fitting, we find an Eddington ratio of
$\epsilon=0.7^{+0.4}_{-0.4}$. We also estimate the mass accretion rate from
the kerrbb thin disc model following Section 4.4, finding
$\dot{M}=52^{+42}_{-23}\,M_{\odot}\,\rm{yr}^{-1}$, which implies a radiative
efficiency of $\eta=0.10^{+0.08}_{-0.05}$, consistent with the Sołtan argument
(e.g Yu & Tremaine, 2002).

Similar to J2157–3602, the SDSS spectra of J1023$+$5142 shows characteristics
of a HiBAL QSO, which are, on average, found to be only slightly reddened by
host galaxy dust compared to a non-BAL sample (e.g. Brotherton et al., 2001;
Reichard et al., 2003). We find that, even with photometry alone, increasing
the assumed host galaxy extinction decreases the estimated black hole mass
from the AD fitting method, so $M_{\rm{AD}}$ could not be boosted to the
$\log(M_{\rm{SE}}/M_{\odot})_{\rm{Mg\,\textsc{ii}}}$ mass through an intrinsic
reddening correction. Because our $M_{\rm{AD}}$ estimate is not overestimated
compared to the H$\beta$ SE virial mass, we find that our assumption of no
internal reddening for luminous QSOs is not contradicted.

This result motivates a future study to investigate cases for which the virial
mass estimates are discrepant in additional detail using AD fitting. It also
demonstrates that AD fitting can be used to discover cases for which the SE
virial mass estimate may have resulted in erroneous measurements.

Figure 7: Corner plot for the accretion disc fit to QSO J102325.31+514251.0
photometry using the kerrbb model. The frequency is presented in the rest-
frame. The final estimate of each parameter is determined by the maximum
likelihood of the posterior distribution, with uncertainties determined by the
68% iso-likelihood line. The top-right panel shows the photometry in black,
and masked data in light grey. The red model is the highest likelihood model
and the one-sigma spread from a random sample of posterior models is similar
in width to the red model line. The black hole mass is estimated to be
$\log(M_{\rm{AD}}/M_{\odot})=9.55^{+0.20}_{-0.17}$ and we show each of the SE
virial mass estimates in green.