11institutetext: INAF, Osservatorio Astronomico di Padova, vicolo dell’
Osservatorio 5, IT 35122, Padova, Italy, 11email<EMAIL_ADDRESS>22institutetext: Instituto de Astrofisíca de Andalucía, IAA-CSIC, Glorieta de
la Astronomia s/n 18008 Granada, Spain, 22email<EMAIL_ADDRESS>33institutetext:
Departamento de Matemática Aplicada and IUMA, Universidad de Zaragoza, E-50009
Zaragoza, Spain 33email<EMAIL_ADDRESS>44institutetext: Center for
Theoretical Physics, Polish Academy of Science 02-668, Warsaw, Poland 44email:
<EMAIL_ADDRESS>55institutetext: INAF, Osservatorio di Astrofisica e Scienza
dello Spazio, IT-40129 Bologna, Italy 55email<EMAIL_ADDRESS>66institutetext: Instituto de Astronomía, UNAM, Mexico D.F. 04510, Mexico,
66email<EMAIL_ADDRESS>77institutetext: Dipartimento di Fisica
& Astronomia “Galileo Galilei”, Università di Padova, Vicolo dell’Osservatorio
3, IT 35122 Padova, Italy 77email<EMAIL_ADDRESS>88institutetext:
Astronomical Observatory, Belgrade, Serbia 88email<EMAIL_ADDRESS>
# Black hole mass estimates in quasars

A comparative analysis of high- and low-ionization lines

Paola Marziani 11 Ascensión del Olmo 22 M. A. Martínez-Carballo 33 Mary Loli
Martínez-Aldama 44 Giovanna M. Stirpe 55 C. Alenka Negrete CONACyT research
fellow, Instituto de Astronomía, UNAM66 Deborah Dultzin 66 Mauro D’ Onofrio
INAF associate, Osservatorio Astronomico di Padova77 Edi Bon 88 Natasha Bon 88

###### Abstract

Context. The inter-line comparison between high- and low-ionization emission
lines has yielded a wealth of information on the quasar broad line region
(BLR) structure and dynamics, including perhaps the earliest unambiguous
evidence in favor of a disk + wind structure in radio-quiet quasars.

Aims. We carried out an analysis of the Civ$\lambda$1549 and H$\beta$ line
profiles of 28 Hamburg-ESO high luminosity quasars and of 48 low-$z$, low
luminosity sources in order to test whether the high-ionization line
Civ$\lambda$1549 width could be correlated with H$\beta$ and be used as a
virial broadening estimator.

Methods. We analyze intermediate- to high-S/N, moderate resolution optical and
NIR spectra covering the redshifted Civ$\lambda$1549 and H$\beta$ over a broad
range of luminosity $\log L\sim 44-48.5$ [erg s-1] and redshift ($0-3$),
following an approach based on the quasar main sequence.

Results. The present analysis indicates that the line width of
Civ$\lambda$1549 is not immediately offering a virial broadening estimator
equivalent to H$\beta$. At the same time a virialized part of the BLR appears
to be preserved even at the highest luminosities. We suggest a correction to
FWHM(Civ$\lambda$1549) for Eddington ratio (using the Civ$\lambda$1549
blueshift as a proxy) and luminosity effects that can be applied over more
than four dex in luminosity.

Conclusions.  Great care should be used in estimating high-$L$ black hole
masses $M_{\rm BH}$ from Civ$\lambda$1549 line width. However, once corrected
FWHM Civ$\lambda$1549 are used, a Civ$\lambda$1549-based scaling law can yield
unbiased $M_{\rm BH}$ values with respect to the ones based on H$\beta$ with
sample standard deviation $\approx$ 0.3 dex.

###### Key Words.:

quasars: general – quasars: emission lines – quasars: supermassive black holes
– ISM: jets and outflows – line: profiles

##  1 Introduction

Type-1 active galactic nuclei (AGN) and quasars show the same broad optical-UV
lines almost always accompanied by broad permitted Feii emission (e.g., Vanden
Berk et al., 2001). However, even among type-1 sources we face a large
diversity in observational manifestations involving line profiles, internal
line shifts as well as emission line intensity ratios (e.g., Sulentic et al.,
2000a; Bachev et al., 2004; Yip et al., 2004; Kuraszkiewicz et al., 2009;
Zamfir et al., 2010; Shen & Ho, 2014, and Sulentic & Marziani 2015 for a
recent review). Broad line measurements involving H$\beta$ line width and Feii
strength are not randomly distributed but instead define a quasar “main
sequence” (MS) (e.g., Boroson & Green, 1992; Sulentic et al., 2000a; Shen &
Ho, 2014). The MS can be traced in an optical plane defined by Feii emission
prominence and the Hydrogen H$\beta$ line width. The Feii strength is
parametrized by the intensity ratio involving the Feii blue blend at 4570 Å
and broad H$\beta$ i.e., $R_{\rm FeII}$= I(Feii$\lambda$4570)/I(H$\beta$), and
the Hydrogen H$\beta$ line width by its FWHM. Along the MS, sources with
higher $R_{\rm FeII}$ show narrower broad H$\beta$ (Population A,
FWHM(H$\beta$)$\lesssim 4000$ km s-1, Sulentic et al. 2000a). Lower $R_{\rm
FeII}$ is associated with sources with broader H$\beta$ profiles (Pop. B with
FWHM(H$\beta$)$\gtrsim 4000$ km s-1, Sulentic et al. 2011). A glossary of the
MS-related terminology is provided in Appendix A.

Studies of the Balmer lines have played a prominent role for characterizing
the MS and the properties of the broad line emitting region (BLR) in low $z$
($\lesssim$0.8) quasars with H$\beta$ providing information for the largest
number of sources (e.g., Osterbrock & Shuder, 1982; Wills et al., 1985;
Sulentic, 1989; Zamfir et al., 2010; Hu et al., 2012; Steinhardt & Silverman,
2013; Shen, 2016, for a variety of observational and statistical approaches).
A most important application of the FWHM H$\beta$ has been its use as a virial
broadening estimator (VBE) to derive black hole masses ($M_{\rm BH}$) from
single-epoch observations of large samples of quasars (e.g., McLure & Jarvis,
2002; McLure & Dunlop, 2004; Vestergaard & Peterson, 2006; Assef et al., 2011;
Shen, 2013; Peterson, 2014, and references therein). The underlying assumption
is that the H$\beta$ line width provides the most reliable VBE, which is
likely to be the case, even if with some caveats (e.g., Trakhtenbrot & Netzer,
2012, see also Shen 2013, Peterson 2014 for reviews).

Balmer lines provide a reliable VBE up $z\lesssim$ 2 (Matsuoka et al., 2013;
Karouzos et al., 2015) at cosmic epochs less than a few Gyr. The importance to
have a reliable VBE at even earlier cosmic epochs cannot be underemphasized.
The entire scenario of early structure formation is affected by inferences
from estimates of quasar black hole masses. Overestimates of $M_{\rm BH}$ by
lines whose broadening is in excess to the virial one can have implications on
the quasar mass function, and at high redshift ($z\gtrsim$ 6) when the
Universe was less than 1 billion year of age, on the formation and mass
spectrum of the seed black holes (Latif & Ferrara, 2016) that may have been
responsible, along with Pop. III stars, of the reionization of the process at
$z\sim 7-10$ (e.g., Gallerani et al., 2017, for a review).

Strong and relatively unblended Civ$\lambda$1549 has been the best candidate
for a VBE beyond $z\sim 1.5$, where H$\beta$ is shifted into the IR domain.
Civ$\lambda$1549 can be observed up to redshift $z\approx 6$ with optical
spectrometers, and in the NIR bands up to redshift $z\approx 7.5$ (Bañados et
al., 2018), and beyond. Can Civ$\lambda$1549 be used as an immediate surrogate
for H$\beta$ when H$\beta$ is invisible or hard to obtain? Before attempting
an answer to this question, two considerations are in order.

Firstly, measures of the Civ$\lambda$1549 line profiles remain of uncertain
interpretation without a precise determination of the quasar rest frame: an
accurate $z$ measurement is not easy to obtain from broad lines, and redshift
determinations at $z\gtrsim 1$ from optical survey data suffer systematic
biases as large as several hundreds km s-1 (Hewett & Wild, 2010; Shen et al.,
2016). Reliable studies tie Civ measures to a rest frame derived from H$\beta$
narrow component (+ [Oiii]$\lambda\lambda$4959,5007 whenever applicable Mejía-
Restrepo et al. 2016, e.g.,; for problems in the use of
[Oiii]$\lambda\lambda$4959,5007, see Zamanov et al. 2002; Hu et al. 2008).

Secondly, significant Civ$\lambda$1549 blueshifts are observed over a broad
range in $z$ and luminosity, from the nearest Seyfert 1 galaxies to the most
powerful radio-quiet quasars (Wills et al., 1993; Sulentic et al., 2007;
Richards et al., 2011; Coatman et al., 2016; Shen, 2016; Bischetti et al.,
2017; Bisogni et al., 2017; Vietri, 2017; Sulentic et al., 2017). Measures of
the Civ$\lambda$1549 profile velocity displacement provide an additional
dimension to a 4D “eigenvector 1” (4DE1) space built on parameters that are
observationally independent (“orthogonal”) and related to different physical
aspects (Sulentic & Marziani, 2015). Inclusion of Civ$\lambda$1549 shift as a
4DE1 parameter was motivated by the earlier discovery of internal redshift
differences between low- and high-ionization lines (Burbidge & Burbidge, 1967;
Gaskell, 1982; Tytler & Fan, 1992; Brotherton et al., 1994b; Corbin & Boroson,
1996; Marziani et al., 1996).

The current interpretation of the BLR in quasars sees the broad lines arising
in a region that is physically and dynamically composite (e.g., Collin-
Souffrin et al., 1988; Elvis, 2000; Ferland et al., 2009; Kollatschny & Zetzl,
2013; Grier et al., 2013; Du et al., 2016). Civ$\lambda$1549 is a doublet
originating from an ionic species of ionization potential (IP) four times
larger than Hydrogen (54 eV vs. 13.6 eV), and is therefore a prototypical
high-ionization line (HIL). The line is mainly produced by collisional
excitation from the ground state 2S0 to 2S${}_{\frac{1}{2},\frac{3}{2}}$ at
the temperature of photo-ionized BLR gas (T$\sim 10^{4}$K, Netzer 1990), in
the fully-ionized zone of the line emitting gas. Empirically, the line is
relatively strong (rest frame equivalent width $W\sim$ 10 – 100 Å depending on
the source location on the MS) and only moderately contaminated on the red
side (red shelf) by Heii$\lambda$1640 and Oiii] $\lambda$1663 plus weak
emission from FeiiUV multiplets (Fine et al., 2010). The Balmer line H$\beta$
assumed to be representative of the low-ionization lines (LILs, from ionic
species with IP $\lesssim 20$ eV) is instead enhanced in a partially-ionized
zone due to the strong X-ray emission of quasars and to the large column
density of the line emitting gas ($N_{\rm c}$$\gtrsim$ 1023 cm-2; Kwan &
Krolik 1981). Comparison of H$\beta$ and Civ$\lambda$1549 profiles in the same
sources tells us that they provide independent inputs to BLR models — their
profiles can be dissimilar and several properties uncorrelated (see, for
instance, Fig. C2 of Mejía-Restrepo et al., 2016).

It is possible to interpret H$\beta$ and Civ$\lambda$1549 profiles as
associated with two sub-regions within the BLR (e.g., Baldwin et al., 1996;
Hall et al., 2003; Leighly, 2004; Snedden & Gaskell, 2004; Czerny &
Hryniewicz, 2011; Plotkin et al., 2015): one emitting predominantly LILs
(e.g., Dultzin-Hacyan et al. 1999; Matsuoka et al. 2008), and a second HILs,
associated with gas outflows and winds (e.g., Richards et al., 2011; Yong et
al., 2018). This view is in accordance with early models of the BLR structure
involving a disk and outflow or wind component (Collin-Souffrin et al., 1988;
Elvis, 2000). Intercomparison of Civ$\lambda$1549 and H$\beta$ at low $z$ and
moderate luminosity provided the most direct observational evidence that this
is the case at least for radio-quiet (RQ) quasars (Corbin & Boroson, 1996;
Sulentic et al., 2007; Wang et al., 2011; Coatman et al., 2016). Modeling
involves a disk + wind system (e.g., Proga et al., 2000; Proga & Kallman,
2004; Flohic et al., 2012; Sa̧dowski et al., 2014; Vollmer et al., 2018, for
different perspectives), although the connection between disk structure and
BLR (and hence the H$\beta$ and Civ$\lambda$1549 emitting regions) is still
unclear.

There are additional caveats, as the Civ$\lambda$1549 blueshifts are not
universally detected. Their amplitude is a strong function of the location
along the MS (Sulentic et al., 2000b, 2007; Sun et al., 2018). Large
blueshifts are clearly detected in Population A, with sources accreting at
relatively high rate, and reach extreme values for quasars at the high $R_{\rm
FeII}$ end along the MS. In Pop. B, the wind component is not dominating the
line broadening of Civ$\lambda$1549 at moderate luminosity; on the converse,
the Civ$\lambda$1549 and H$\beta$ line profile intercomparison indicates that
the dynamical relevance of the Civ$\lambda$1549 blueshift is small i.e., that
the ratio between the centroid at half-maximum $c(\frac{1}{2})$ and the FWHM
is $\ll 1$ (Sulentic et al., 2007). Reverberation mapping studies indicate
that the velocity field is predominantly Keplerian (Pei et al. 2017 and
references therein for the protypical source NGC 5548, Denney et al. 2010;
Grier et al. 2013), and that the Civ$\lambda$1549 emitting region is closer to
continuum source than the one of H$\beta$ (e.g., Peterson & Wandel, 2000;
Kaspi et al., 2007; Trevese et al., 2014). The issue is complicated by
luminosity effects on the Civ$\lambda$1549 shifts that may have gone
undetected at low-$z$. Both Pop. A and B sources at $\log L\gtrsim 47$ erg s-1
show large amplitude blueshifts in Civ$\lambda$1549 (Sulentic et al., 2017;
Bisogni et al., 2017; Vietri et al., 2018). The present work considers the
trends associated with the MS as well as the luminosity effects that may
appear second-order in low-luminosity samples to provide corrections to the
FWHM of H$\beta$ and ultimately a scaling law based on Civ$\lambda$1549 FWHM
and UV continuum luminosity that may be unbiased with respect to H$\beta$ and
with a reasonable scatter.

The occurrence of Civ$\lambda$1549 large shifts challenges the suitability of
the Civ$\lambda$1549 profile broadening as a VBE for $M_{\rm BH}$ estimates
(see e.g., Shen, 2013, for a review). Results at low-redshift suggest that the
Civ$\lambda$1549 line is fully unsuitable for part of Pop. A sources (Sulentic
et al., 2007). A similar conclusion was reached at $z\approx 2$ on a sample of
15 high-luminosity quasars (Netzer et al., 2007). More recent work tends to
confirm that the Civ$\lambda$1549 line width is not straightforwardly related
to virial broadening (e.g., Mejía-Restrepo et al., 2016). However, the
Civ$\lambda$1549 line is strong and observable up to $z\approx 6$ with optical
spectrometers. It is so highly desirable to have a consistent VBE up to the
highest redshifts that various attempts (e.g., Brotherton et al., 2015) have
been done at rescaling Civ$\lambda$1549 line width estimators to the width of
LILs such as H$\beta$ and Mgii$\lambda$2800\. Several conflicting claims have
been recently made on the valid use of Civ$\lambda$1549 width in high-$z$
quasars (e.g., Assef et al., 2011; Shen & Liu, 2012; Denney et al., 2012;
Karouzos et al., 2015; Coatman et al., 2017; Mejía-Restrepo et al., 2018b).

From the previous outline we infer that a proper approach to testing the
suitability of the Civ$\lambda$1549 line width as a VBE is to compare
Civ$\lambda$1549 and H$\beta$ profiles along the quasar MS, and to extend the
luminosity range including intermediate-to-high $z$ ($\gtrsim 1.4$) sources
when H$\beta$ is usually not covered by optical observations. A goal of this
paper is to analyse the factors yielding to large discrepancies between the
$M_{\rm BH}$ estimates from H$\beta$ and Civ$\lambda$1549, with a focus on the
aspect and physical factors affecting the broadening of the two lines.

The quasar sample used in the present paper joins two samples with both
H$\beta$ and Civ$\lambda$1549 data, one at low luminosity and $z$ ($\lesssim
0.7$, Sulentic et al. 2007), and one at high luminosity, in the range
$1.5\lesssim z\lesssim 3$ presented and analyzed by Sulentic et al. (2017,
hereafter Paper I). The sample provides a wide coverage in luminosity, and
Eddington ratio (Sec. 2); H$\beta$ line coverage for each Civ$\lambda$1549
observation; consistent analysis of the line profiles of both Civ$\lambda$1549
and H$\beta$ (Sec. 3). Our approach is intended to overcome some of the
sample-dependent difficulties encountered by past studies. Results involve the
reduction of the measured Civ$\lambda$1549 line width to a VBE (Sec. 4) with a
correction factor dependent on both shift amplitude and luminosity. They are
discussed in terms of BLR structure (Sec. 5.2), and specifically of the
interplay between broadening associated with the outflow (very relevant for
Civ$\lambda$1549) and with orientation effects (which are dominating for
H$\beta$). Finally, a new $M_{\rm BH}$ scaling law with line width and
luminosity (§5.4) is presented. The new Civ$\lambda$1549 scaling law, which
considers different corrections for Pop. A and B separately, may provide an
unbiased estimator of black hole masses derived from H$\beta$ over a wide
range in luminosity (Sect. 5.5).

##  2 Sample

###  2.1 High-luminosity VLT data for Hamburg-ESO quasars

The high-$L$ quasars considered in the present study are 28 sources identified
in the HE survey (Wisotzki et al., 2000, hereafter the HE sample), in the
redshift range $1.4\lesssim z\lesssim 3.1$. All satisfy the conditions on the
absolute B magnitude $M_{\mathrm{B}}\lesssim-27.5$ and on the bolometric
luminosity $\log L\gtrsim 10^{47.5}$ erg s-1. They are therefore among the
most luminous quasars ever discovered in the Universe, and a relatively rare
population even at $z\approx 1-2$ when luminous quasars were more frequent
than at low-$z$ (the luminosity function at $M_{\mathrm{B}}\approx-27.5$ is
$\Phi(M_{\mathrm{B}})\sim 10^{-8}$ Mpc-3 mag-1 compared to $\sim 10^{-6}$
Mpc-3 mag-1 at $M_{\mathrm{B}}\approx-25$, corresponding to the “knee” of the
Boyle et al. 2000 luminosity function).

The Civ$\lambda$1549 data were obtained with the FORS1 spectrograph at VLT and
Dolores at TNG; the matching H$\beta$ observations with the ISAAC spectrometer
were analyzed in detail in Sulentic et al. (2006b). The resolution at FWHM of
the Civ$\lambda$1549 data is $\lesssim 300$ km s-1 and $\lesssim$ 600 km
s-1for FORS1 and Dolores, respectively; the H$\beta$ resolution is $\approx$
300 km s-1 (Sulentic et al., 2004). Typical S/N values are $\gtrsim 50$.

Resolution and S/N are adequate for a multicomponent nonlinear fitting
analysis using the IRAF routine specfit (Kriss, 1994), involving an accurate
deconvolution of H$\beta$, [Oiii]$\lambda\lambda$4959,5007, Feii,
Heii$\lambda$4686 in the optical, and of Civ$\lambda$1549 and
Heii$\lambda$1640 in the UV. The Civ$\lambda$1549 and H$\beta$ data and the
immediate results of the specfit analysis were reported in Paper I.

###  2.2 Low-luminosity Civ$\lambda$1549 and H$\beta$ data

We considered a Faint Object Spectrograph (FOS) sample from Sulentic et al.
(2007) as a complementary sample at low-$L$ and low-$z$. For the sake of the
present paper, we restrict the FOS sample to 29 Pop. A and 19 Pop. B RQ (48 in
total) sources covering the Civ$\lambda$1549 blend spectral range and with
previous measures for the H$\beta$ profile and $R_{\rm FeII}$ (Marziani et
al., 2003). The list of sources can be obtained by the cross-correlation of
the Sulentic et al. (2007) RQ sources (Kellermann’s ratio
$\log$$R{\rm{}_{K}}$$<1.8$) and the Marziani et al. (2003) catalog on Vizier.
We excluded NGC 4395 and NGC 4253 whose luminosities are $\log L\approx$ 40.4
and 41.7 [erg s-1] respectively, outlying with respect to the $L$ distribution
of the FOS sample. The FOS high-resolution grisms yielded an inverse
resolution $\lambda/\delta\lambda\sim 1000$, equivalent to typical resolution
of the Marziani et al. (2003)’s data. The S/N is above $\gtrsim 20$ for both
the optical and UV low-$z$ data. The FOS sample has a typical bolometric
luminosity $\log L\sim 45.2$ [erg s-1] and a redshift $z\lesssim 0.5$.

###  2.3 Joint HE+FOS sample

The HE+FOS sample has therefore 76 sources, of which 43 are Pop. A and 33 Pop.
B. The distribution of $\log L$ for the 76 sources of the joint sample
(derived from the rest-frame luminosity at 1450 Å, assuming a constant
bolometric correction equal to 3.5) uniformly covers the range 44 – 48.5, with
similar distributions for Pop. A and B (lower panel of Fig. 1; a K-S test
confirms that the two distributions are not significantly different). The
Eddington ratio ($L/L_{\rm Edd}$) covers the range 0.01 – 1 which means
complete coverage of $L/L_{\rm Edd}$ range where most sources in optically-
selected samples are found.

Figure 1: Cumulative distribution of bolometric luminosity $L$ of the FOS+HE
sample (black), Pop. A (blue) and Pop. B (red).

##  3 Emission line profile analysis for the FOS+HE sample

###  3.1 Line modeling and measured parameters

In the following we consider the merit of H$\beta$ and Civ$\lambda$1549 as
VBEs. Previous work has shown that the H$\beta$ and Mgii$\lambda$2800 profiles
are almost equally reliable estimators of the “virial” broadening in samples
of moderate-to-high luminosity (e.g., Wang et al., 2009; Trakhtenbrot &
Netzer, 2012; Shen & Liu, 2012; Marziani et al., 2013b, excluding the
Mgii$\lambda$2800 extreme Population A that is significantly broadened by a
blueshifted component, Marziani et al. 2013b). However, the broad H$\beta$
line full profile is often affected by asymmetries toward the line base and by
significant line centroid shifts. Typically, the H$\beta$ line profiles are
characterized by two main asymmetries, differently affecting sources in
spectral types along the MS (Appendix A provides the definition of spectral
types):

  * •

Pop. A: a blueshifted excess, often modeled with a blueward asymmetric
Gaussian component (BLUE) related to the outflows strongly affecting the
Civ$\lambda$1549 and [Oiii]$\lambda\lambda$4959,5007 line profiles (e.g.
Negrete et al., 2018, Paper I, and references therein).

  * •

Pop. B: a redward asymmetry modeled with a broader redshifted (FWHM $\sim
10000$ km s-1, $c(\frac{1}{2})$$\sim 2000$ km s-1) Gaussian. The very broad
Gaussian is meant to represent the innermost part of the BLR, providing a
simple representation of the radial stratification of the BLR in Pop. B
suggested by reverberation mapping (e.g., Snedden & Gaskell, 2007). This
component (hereafter the very broad component, VBC) has been associated with a
physical region of high-ionization virialized and closest to the continuum
source (Peterson & Ferland, 1986; Brotherton et al., 1994a; Sulentic et al.,
2000c; Snedden & Gaskell, 2007; Wang & Li, 2011). While the properties of the
Very Broad Line Region (VBLR) remains debatable, a decomposition of the full
H$\beta$ profile into a symmetric, unshifted H$\beta$ component (H$\beta_{\rm
BC}$) and a H$\beta_{\rm VBC}$ provides an excellent fit to most H$\beta$ Pop.
B profiles (Sulentic et al., 2002; Zamfir et al., 2010).

Figs. 4 and 5 of Paper I show the H$\beta$ and Civ$\lambda$1549 profiles of
the HE sample, and their multicomponent interpretation. To extract a
symmetric, unshifted component that excluded the blueshifted excess and the
VBC, we considered a model of the broad H$\beta$ and Civ$\lambda$1549 line
with the following components (see also Appendix A):

  * •

Pop. A H$\beta$ and Civ$\lambda$1549: an unshifted Lorentzian profile
(H$\beta_{\rm BC}$) + one or more asymmetric Gaussians to model the blueward
excess (BLUE).

  * •

Pop. B H$\beta$ and Civ$\lambda$1549: an unshifted Gaussian (H$\beta_{\rm
BC}$) + a redshifted VBC for H$\beta$ (H$\beta_{\rm VBC}$). In the H$\beta$
case, there is no evidence of a blueward excess even at the highest
luminosity. However, among Pop. B sources of the HE sample, a prominent
Civ$\lambda$1549 BLUE appears, implying an intensity ratio
Civ$\lambda$1549/H$\beta$$\gg 1$ in the BLUE component. The Civ$\lambda$1549
BLUE is usually fainter in the low-luminosity FOS sample (Paper I).

In the fits, narrow components of both H$\beta$ (H$\beta_{\rm NC}$) and
Civ$\lambda$1549 (Civ$\lambda$1549NC) were included. In the case of
Civ$\lambda$1549, separation of the broad and narrow component is subject to
significant uncertainty, so that the effect of the Civ$\lambda$1549NC needs to
be carefully considered (see discussion in Sect. 3.2).

The decomposition approach summarized above has a heuristic value, as the
various components are not defined on the basis of a physical model, even if
the assumptions on line shapes follows from MS trends. The distinction between
BC and VBC might be physically motivated (the emitting region associated with
the BC is the one emitting most of all Feii), but the decomposition into two
symmetric Gaussians is a crude approximation at best. Full profile measures
are added to avoid any exclusive dependence of the results on the profile
decomposition. The full profiles of H$\beta$ and Civ$\lambda$1549 are
parameterized by the FWHM, an asymmetry index (AI) and centroid at fractional
intensity at 1/2 and 1/4 of peak, $c(\frac{1}{2})$ and $c(\frac{1}{4})$. The
definition of centroids and A.I. follows Zamfir et al. (2010):

$c(\frac{i}{4})=\frac{\lambda_{\mathrm{B}}(\frac{i}{4})+\lambda_{\mathrm{R}}(\frac{i}{4})}{2\lambda_{0}}c,\,i=1,2,3;\frac{i}{4}=0.9$
(1)

$A.I.=\frac{\lambda_{\mathrm{B}}(\frac{1}{4})+\lambda_{\mathrm{R}}(\frac{1}{4})-2\lambda_{\mathrm{P}}}{\lambda_{\mathrm{B}}(\frac{1}{4})+\lambda_{\mathrm{B}}(\frac{1}{4})}$
(2)

where $\lambda_{\mathrm{P}}$ is the peak wavelength, and
$\lambda_{\mathrm{B}}$ and $\lambda_{\mathrm{R}}$ are the wavelengths on the
blue and red side of the line at the $i/4$ fractional intensities. The
centroids are referred to the quasar rest frame, while the AI is referred to
the peak of the line that may be shifted with respect to rest-frame. A proxy
to $\lambda_{\mathrm{P}}$ which will be used in this paper is
$\tilde{\lambda_{\mathrm{P}}}\approx\lambda_{0}(1+c(0.9)/c)$.

We assume that the symmetric and unshifted H$\beta_{\rm BC}$ and
Civ$\lambda$1549BC are the representative line components of the virialized
part of the BLR. It is expedient to define a parameter $\xi$ as follows:

$\xi_{\mathrm{line}}=\frac{{\rm FWHM}_{\mathrm{vir}}}{\rm FWHM}$ (3)

where the FWHMvir is the FWHM of the “virialized” component, in the following
assumed to be H$\beta_{\rm BC}$, and the FWHM is the FWHM measured on the full
profile (i.e., without correction for asymmetry and shifts). The $\xi$
parameter is a correction factor that can be defined also using components of
different lines, for instance Civ$\lambda$1549 full profile FWHM and
H$\beta_{\rm BC}$, where H$\beta_{\rm BC}$ is assumed to be a reference VBE.

###  3.2 The Civ$\lambda$1549 narrow component in the HE sample and its role
in FWHM Civ$\lambda$1549BC estimates

In only two cases does Civ$\lambda$1549NC contribute to the total
Civ$\lambda$1549 flux of the HE Pop. B sources by more than 10%: HE2202-2557
and HE2355-4621 (Pop. B, Fig. 5 of Paper I). There is no evidence for a strong
NC in the HE Pop. A sources except for HE0109-3518 where
I(Civ$\lambda$1549NC)$\lesssim$ 0.09 of the total line flux and whose
Civ$\lambda$1549 profile resembles the ones of low-$z$ sources that are 2-3
dex less luminous (the HE0109-3518 Civ$\lambda$1549 profile is shown in Fig. 4
of Paper I).

In general, considering H$\beta_{\rm BC}$ as a reference for Pop. B sources,
and comparing FWHM H$\beta_{\rm BC}$ to FWHM Civ$\lambda$1549 with and without
removing the Civ$\lambda$1549NC(i.e., to FWHM Civ$\lambda$1549BC and FWHM
Civ$\lambda$1549BC\+ Civ$\lambda$1549NC), the Civ$\lambda$1549NC removal
improves the agreement with FWHM H$\beta_{\rm BC}$ in 5 cases out of 6 when
Civ$\lambda$1549NC has an appreciable effect on the line width (in the other
eight cases there is no effect because Civ$\lambda$1549NC is too weak). The
FWHM measured on the Civ$\lambda$1549 profiles without removing the
Civ$\lambda$1549NC (i.e., FWHM Civ$\lambda$1549BC+Civ$\lambda$1549NC) are, on
average, $\approx$ -4 % and -11 % of the FWHM Civ$\lambda$1549, for Pop. A and
B respectively. Therefore, (1) subtracting the Civ$\lambda$1549NC improves the
agreement between H$\beta$ and Civ$\lambda$1549 FWHM; (2) the average effect
is too small to affect our inferences concerning on the Civ$\lambda$1549 line
width as a VBE in the HE sample. The Civ$\lambda$1549NC has been always
included as an independent component in the line profile fitting of Paper I,
following an approach consistently applied for the low-$z$ FOS sample and
described by Sulentic et al. (2007).

##  4 Results

###  4.1 H$\beta$ in the HE sample

We considered several different measures of the H$\beta$ width following
empirical corrections derived from previous work on low-$z$ samples:

  * •

substitution of the H$\beta_{\rm BC}$ extracted through the specfit analysis
in place of the full H$\beta$ profile.

In principle, extraction of the H$\beta_{\rm BC}$ should be the preferred
approach, and the FWHM H$\beta_{\rm BC}$ the preferred VBE. To test the
reliability of the FWHM values, we performed Monte-Carlo repetitions of the
H$\beta$ fit for Pop. B sources with the broadest lines (FWHM H$\beta_{\rm
BC}$$\sim$ 7000 km s-1, and FWHM H$\beta_{\rm VBC}$$\sim$ 11000 km s-1), under
the assumption of S/N $\approx$20,111 S/N is measured per pixel on the
continuum. weak and relatively broad [Oiii]$\lambda\lambda$4959,5007, changing
noise pattern and initial values of the fitting. The values of FWHM
H$\beta_{\rm BC}$ and H$\beta_{\rm VBC}$ were chosen to represent the broadest
lines, where FWHM H$\beta_{\rm BC}$ measures might be affected by a degeneracy
in the BC+VBC decomposition. The dispersion of the Monte Carlo FWHM
distribution is almost symmetric, and implies typical FWHM H$\beta_{\rm BC}$
uncertainties $\approx 10$% at 1$\sigma$ confidence level. Therefore, the
blending should not be a source of strong bias or of large uncertainties in
the H$\beta_{\rm BC}$ and H$\beta_{\rm VBC}$ FWHM.222If S/N is relatively high
$(\gtrsim 20$) only in some peculiar cases the uncertainty might be
significantly larger. For example, if the H$\beta$ profile is composed for a
narrower core and a broader base, the FWHM measure is unstable, and may
abruptly change depending on continuum placement. However, we still expect
that in the case of very broad profiles, and low S/N or low dispersion, the
decomposition of the H$\beta$ profile into H$\beta_{\rm BC}$ and H$\beta_{\rm
VBC}$ is subject to large uncertainties difficult to quantify. To retrieve
information on the H$\beta_{\rm BC}$ we introduce several corrections that can
be applied to the full H$\beta$ profile without any multicomponent fitting
(which makes the results also model-dependent).

  * •

symmetrization of the full profile: FWHMsymm = FWHM – 2 $c(\frac{1}{2})$ (symm
in Fig. 2). The physical explanation behind the symmetrization approach
involves an excess radial velocity on the red side that may be due to gas with
a radial infall velocity component, with velocity increasing toward the
central black hole (e.g., Wang et al., 2017, and references therein).
Generally speaking, redward displacements of line profiles have been explained
by invoking a radial infall component plus obscuration (Hu et al., 2008;
Ferland et al., 2009);

  * •

Substitution of the FWHM H$\beta_{\rm BC}$ with the FWHM measured on the full
broad profile of H$\beta$, corrected according to its spectral type. The
spectral types have been assigned following Sulentic et al. (2002, for a
conceptually equivalent approach see ). The correction are as defined from the
analysis of the H$\beta$ profile in a large SDSS-based sample at $0.4\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}z\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}0.7$ (labeled as st in Fig. 2). In
practice, this means to correct H$\beta$ for Pop. B sources by a factor
$\xi_{\mathrm{H\beta}}\approx 0.8$ (Marziani et al., 2013a) and extreme
population A sources ($R_{\rm FeII}$$\geq 1$) by a factor
$\xi_{\mathrm{H\beta}}\approx 0.9$. On average, spectral types A1 and A2 show
symmetric profiles for which $\xi_{\mathrm{H\beta}}\approx 1$. Recent work
confirmed that the effect of a blueshifted excess on the full profile of
H$\beta$ is small at half-maximum, $0.9\lesssim\xi_{\mathrm{H\beta}}\lesssim
1.0$ (Negrete et al., 2018). We assume $\xi_{\mathrm{H\beta}}=0.9$ as an
average correction. The ratio we derive between BC and full profile FWHM of HE
Pop. B H$\beta$ is $\approx 0.82\pm 0.09$, consistent with the same ratio
estimates at moderate luminosity (Marziani et al., 2013a). The st correction
can be summarized as follows:

ST | $\xi_{\mathrm{H\beta}}$
---|---
A3-A4 | 0.9
A1-A2 | 1.0
B1-B1+ | 0.8

  * •

correction of the width of the full broad H$\beta$ profile based on the one
derived at low $z$ by pairing the observed full broad H$\beta$ FWHM to the
best width estimator from reverberation mapping, following the relation
FWHM${}_{\mathrm{c}}\approx 1.14$ FWHM$-601-0.0000217$FWHM2 derived by
Sulentic et al. (2006b, labeled corr);

Fig. 2 shows that these corrections all provide similar results if applied to
the HE sample FWHM H$\beta$. Error bars of Fig. 2 were estimated propagating
the uncertainty values reported in Paper I for the full profiles, and the ones
derived from specfit for the line components (assuming a minimum error of
10%).

The middle panels of Fig. 2 show the ratios of corrected FWHM measures as a
function of the FWHM of the full H$\beta$ profile. The low $\chi^{2}_{\nu}$
indicates that the $\chi_{\nu}^{2}$ associated with the ratios between BC and
symm, and BC and st not significantly different from unity. In the case of BC
and corr the two measurements are different but only at 1$\sigma$ confidence
level. $F$ tests do not exclude that BC, symmetrization, st, and reverberation
corrections can be equivalent at a minimum confidence level of $2\sigma$. The
bottom panel of Fig. 2 shows the behavior of the full and corrected FWHM
versus the “symmetrized” H$\beta$ FWHM. We consider the symmetrization, as it
is relatively easy to apply (once the quasar rest frame is known), and the st
correction (that does not even require the knowledge of the rest frame) as
reference corrections. We remark again that these corrections are relatively
minor but still significant: a 20% correction translates into a factor 1.44
correction in $M_{\rm BH}$. They do not undermine the value of the full line
width of H$\beta$ as a useful VBE (with the caveats discussed in Sec. 5.2),
since the H$\beta$ full line width remains preferable to the uncorrected
Civ$\lambda$1549 width for most objects.

Figure 2: Virial broadening estimators based on H$\beta$, with several
corrections applied to the HE sample. Top square panel: FWHM H$\beta_{\rm BC}$
(blue (Pop. A) and red (Pop. B)) with error bars, symmetrized FWHM H$\beta$
(symm, black), and FWHM H$\beta$ corrected according to spectral type (st,
aquamarine and dark orange for Pop. A and B respectively) versus FWHM of the
full H$\beta$ profile. The grey line traces the correction (corr) following
the relation of Sulentic et al. (2006b) reported in §4. Middle panels: ratios
of FWHM after various corrections vs. full profile FWHM. First panel from top:
BC/symm, BC/st, BC/corr, symm/ st, corr/st. Average values, standard deviation
and normalized $\chi^{2}_{\nu}$ are reported in the upper right corner of the
panels. Bottom square panel: BC, spectral type st and reverberation corrected
(corr, open symbols) FWHM values vs. symmetrized FWHM.

###  4.2 Civ$\lambda$1549 in the full HE+FOS sample

The results on the HE Civ$\lambda$1549 profiles do not bode well for the use
of Civ$\lambda$1549 FWHM as a VBE, as also found by Sulentic et al. (2007) and
other workers (Sect. 5.1 for a brief critical review). The presence of very
large blueshifts in both Pop. A and B makes the situation even more critical
than at low $L$. Fig. 3 (top panel) shows that there is no obvious relation
between the FWHM of Civ$\lambda$1549 and the FWHM of H$\beta$ if FOS+HE data
are considered together.

For the Pop. A sources in the HE+FOS sample, Civ$\lambda$1549 is broader than
H$\beta$ save in two cases in the HE sample, and FWHM(Civ$\lambda$1549) shows
a broad range of values for similar FWHM H$\beta$ i.e., FWHM(Civ$\lambda$1549)
is almost degenerate with respect to H$\beta$. The Civ$\lambda$1549 line FWHM
values are so much larger than the ones of H$\beta$ to make it possible that
the $M_{\rm BH}$ derived from FWHM Civ$\lambda$1549 might be higher by even
more than one order of magnitude. Formally, the Pearson’s correlation
coefficient $r\approx$ 0.52 is highly significant for a sample of $n=43$, with
significance at an $\approx 4.5\sigma$ confidence level. A weighted least
square fits yields FWHM(Civ$\lambda$1549) = (1.822 $\pm$ 0.204) FWHM(H$\beta$)
+( -624 $\pm$ 677) km s-1, with a significant scatter, rms $\approx$1959 km
s-1. Unfortunately it is not possible to apply a simple Civ$\lambda$1549
symmetrization as done for H$\beta$: subtracting 2 $\cdot$ $c(\frac{1}{2})$ to
FWHM Civ$\lambda$1549 leads to corrections that are unrealistically large.

If we combine the Pop. B FOS and HE samples, FWHM Civ$\lambda$1549 and
H$\beta$ become loosely correlated (the Pearson’s correlation coefficient is
$\approx 0.4$, significant at $P\approx 98$% for a sample of 33 objects). A
weighted least-square fit yields FWHM(Civ$\lambda$1549) $\approx(0.764\pm
0.165)$ FWHM(H$\beta$) $+(810\pm 1030)$ km s-1, and rms $\approx 1090$ km s-1,
with a significant deviation from the 1:1 relation. In the case of Pop. B
sources, the trend implies FWHM Civ$\lambda$1549$\sim$ FWHM H$\beta$, and even
a slightly narrower FWHM Civ$\lambda$1549 with respect to H$\beta$.

The large scatter induced by using uncorrected Civ$\lambda$1549 line FWHM may
have contributed to the statement that line width does not contribute much to
$M_{\rm BH}$ determinations (Croom, 2011).

###  4.3 Practical usability of Civ$\lambda$1549BC

The fitting procedure scaled the H$\beta$ profile to model the red side of
Civ$\lambda$1549 so that the FWHM Civ$\lambda$1549BC estimate is not
independent from FWHM H$\beta_{\rm BC}$. The FWHM values of the two BCs are in
agreement because of this enforced condition.

The Civ$\lambda$1549BC extraction is very sensitive to the assumed rest frame,
and also requires that the Civ$\lambda$1549 line is cleaned from contaminant
such as Feii (weak) and Heii$\lambda$1640 (moderate, but flat topped and
gently merging with the Civ$\lambda$1549 red wing; Marziani et al. 2010; Fine
et al. 2010; Sun et al. 2018). Without performing a line profile
decomposition, one can consider the width of the red side with respect to rest
frame as the half-width half maximum (HWHM) of the virial component. Again
this requires (1) an accurate redshift that can be set, in the context of high
$z$ quasars, either by using the H$\beta$ narrow component or by the
[Oii]$\lambda$3727 doublet (Eracleous & Halpern, 2004; Hu et al., 2008), and
(2) the decomposition from Heii$\lambda$1640 emission blended on the
Civ$\lambda$1549 red side. If [Oii]$\lambda$3727 is covered, then
Mgii$\lambda$2800 is also likely to be covered. As mentioned in Sect. 3, the
Mgii$\lambda$2800 line width is a reliable VBE for the wide-majority of type-1
AGN. The same is not true for Civ$\lambda$1549\. For spectra where
Civ$\lambda$1549 is conveniently placed at $z\gtrsim 1.45$, the
[Oii]$\lambda$3727 line is shifted beyond 9000 Å, a domain where intense sky
emission makes it difficult to analyze a relatively faint narrow line. The
extraction of Civ$\lambda$1549BC is therefore not a viable solution if single-
epoch Civ$\lambda$1549 observations are available without the support of at
least a narrow LIL that may set a reliable rest frame. This is unlikely to
occur on the same optical spectra. An alternative strategy for $M_{\rm BH}$
estimation using Civ$\lambda$1549 FWHM should consider the origin of the
Civ$\lambda$1549 non-virial broadening.

###  4.4 Reducing Civ$\lambda$1549 to a VBE estimator consistent with H$\beta$

The main results of Paper I suggest a strong dependence of the
Civ$\lambda$1549 blueshift on $L/L_{\rm Edd}$, especially above a threshold
value $L/L_{\rm Edd}$$\approx 0.2\pm 0.1$ (Sulentic et al., 2014, and
references therein). A correlation between Eddington ratio and the
FWHM(Civ$\lambda$1549) to FWHM(H$\beta$) ratio (i.e., $1/\xi_{\mathrm{CIV}}$,
c.f., Saito et al., 2016) is detected at a high significance level (Pearson’s
correlation coefficient $r\approx 0.55$) joining all FOS RQ sources of
Sulentic et al. (2007, Fig. 4). In this context, $L/L_{\rm Edd}$ was computed
from the $M_{\rm BH}$ scaling law of Vestergaard & Peterson (2006), using the
FWHM of H$\beta$ and $\lambda L_{\lambda}$(5100). A bisector best fit with
SLOPES (Feigelson & Babu, 1992a) yields

$\log\frac{1}{\xi_{\mathrm{CIV}}}\approx(0.426\pm
0.043)\frac{L}{L_{\mathrm{Edd}}}+(0.401\pm 0.035).$ (4)

An $L/L_{\rm Edd}$ – dependent correction is in principle a valid approach.
However, it is not obvious how to calculate $L/L_{\rm Edd}$ from UV spectra
without resorting to H$\beta$ observations. In addition FWHM H$\beta$ is
strongly affected by orientation and yields biased values of $L/L_{\rm Edd}$
(Sect. 5.2.1). Both FWHM(H$\beta_{\rm BC}$)/FWHM(Civ$\lambda$1549) and
$c(\frac{1}{2})$ are both correlated with Eddington ratio. Consistently, the
Civ$\lambda$1549 blueshift is correlated with FWHM Civ$\lambda$1549 (Paper I,
Coatman et al., 2016), and accounts for the broadening excess in the
Civ$\lambda$1549 FWHM. Measures of the Civ$\lambda$1549 blueshift or the
FWHM(H$\beta_{\rm BC}$)/FWHM(Civ$\lambda$1549) can be used as proxies for
$L/L_{\rm Edd}$. At the same time, Paper I reveals a weaker correlation with
$L$, which is expected in the case of a radiation driven wind. If the
correction factor is $\xi_{\mathrm{CIV}}=$FWHM(H$\beta_{\rm
BC}$)/FWHM(Civ$\lambda$1549), then it should include a term in the form
$1/\zeta(L,$ $L/L_{\rm Edd}$).

Figure 3: Top panel: FWHM(Civ$\lambda$1549) vs FWHM(H$\beta$) (full profiles)
for the FOS+HE sample. Data points are color-coded according to sample and
population. HE Pop. A: blue circles ($\bullet$), HE Pop. B: red squares ( ),
FOS Pop. A: aquamarine circles ($\bullet$), FOS Pop. B: golden squares ( ).
Best fitting lsq lines (dashed) are shown in the blue for all Pop. A and red
for all Pop. B. The black dot dashed line is the equality line. The continuous
grey line is the expected FWHM following the correction of Sulentic et al.
(2006b, corr). Lower panel: ratio between FWHM Civ$\lambda$1549 and FWHM
H$\beta$ as a function of FWHM (H$\beta$).  Figure 4: Relation between the
logarithm of the FWHM ratio of Civ$\lambda$1549 to H$\beta$ and the logarithm
of the Eddington ratio $L/L_{\rm Edd}$. The vertical dot-dashed line traces
the Eddington limit. The colors and shape of symbols are as in Fig. 3. The
dashed line is an unweighted least squares fit, the filled line was obtained
with the bisector method (Feigelson & Babu, 1992b).  Figure 5: Top panel:
bisector flux estimator of Coatman et al. (2017) vs $c(\frac{1}{4})$, in km
s-1. The dot-dashed line is the equality line. Bottom panel: Percentage
residuals. The shaded areas indicate the average error of measurement at a
$2\sigma$ confidence level computed from Table 4 of Paper I.

###  4.5 Calibrating empirical corrections on FWHM Civ$\lambda$1549

Coatman et al. (2017) introduced a non-parametric measure of the
Civ$\lambda$1549 blueshift associated with the wavelength that splits the line
flux in equal parts on its blue and red side (flux bisector). The flux
bisector is strongly correlated with $c(\frac{1}{2})$ and $c(\frac{1}{4})$,
and $c(\frac{1}{2})$ and $c(\frac{1}{4})$ are correlated among themselves in
the FOS+HE sample (Pearson’s $r\approx 0.95$): $c(\frac{1}{2})=(0.773\pm
0.307)c(\frac{1}{4})-(58\pm 65)\,\,\mathrm{km\,s^{-1}}$. The bisector
correlation is stronger with $c(\frac{1}{4})$ (Pearson’s correlation
coefficient $r\approx 0.98$), with flux bisector $\sim(0.98\pm
0.04)c(\frac{1}{4})+(220\pm 110)\,\,\mathrm{km\,s^{-1}}$ (Fig. 5). The lower
panel of Fig. 5 shows a few objects with difference between $c(\frac{1}{4})$
and flux bisector $\gtrsim 20$%; these sources are either with small shifts
(within the measurement uncertainties; shaded area of Fig. 5), or sources
strongly affected by broad absorptions, for which a measure of blueshift is
tricky regardless of the method employed. Therefore it is possible to apply
Eq. 4 of Coatman et al. (2017) substituting the $c(\frac{1}{4})$ to the flux-
bisector blueshift measurements:

$\xi_{\mathrm{CIV,0}}=\frac{1}{a\left(-\frac{c(\frac{1}{4})}{1000}\right)+b}$
(5)

with $a=0.41\pm 0.02$ and $b\approx 0.62\pm 0.04$ (the minus sign is because
Coatman et al. (2017) assumed blueshifts to be positive), to correct the FWHM
Civ$\lambda$1549 of the FOS+HE sample. The resulting trend is shown in Fig. 6.
The Eq. 4 of Coatman et al. (2017) undercorrects both Pop. A and B sources at
low $L$ (the FOS sample) and provides a slight overcorrection for the HE
sources. The correction of Coatman et al. (2017) does not yield FWHM
(Civ$\lambda$1549) in agreement with the observed values of FWHM(H$\beta$).
This does not necessarily mean that the FWHM(Civ$\lambda$1549) values are
incorrect, as FWHM(H$\beta$) is likely more strongly affected by orientation
effects than FWHM(Civ$\lambda$1549) (see the discussion in Sect. 5.2).

A correction dependent on luminosity reduces the systematic differences
between the various samples in the present work but it has to be separately
defined for Pop. A and B (Fig. 7). The following expression:

$\xi_{\mathrm{CIV,1}}=\frac{1}{b\left(a-\log\lambda
L_{\lambda}(1450)\right)\cdot\left(|\frac{c(\frac{1}{2})}{1000}|\right)+c}$
(6)

provides a suitable fitting law, with $a$, $b$, $c$ different for Pop. A and
B. Here we consider the $c(\frac{1}{2})$ because of its immediate connection
with the FWHM, and because it is highly correlated with $c(\frac{1}{4})$ (Sec.
5.5). Eq. 6 is empirical: it entails a term proportional to shift and one to
the product of $\log L_{1450}$ and shift. Multivariate, nonlinear lsq results
for Eq. 6 are reported in the first rows of Table 1. For Pop. A the correction
is rather similar to the one of Coatman et al. (2017), and is driven by the
large blueshifts observed at high $L/L_{\rm Edd}$. The luminosity-dependent
factor accounts for low-luminosity sources that are not present in Coatman et
al. (2017) sample. The use of the absolute value operator provides an
improvement with respect to the case in which blueshifts are left negative.
There are only three objects for which $c(\frac{1}{2})$ is positive. The
improvement is understandable if one consider any Civ$\lambda$1549 shift as
affecting the difference between the FWHM of Civ$\lambda$1549 and H$\beta$. An
A(+) sample was defined from the Pop. A sample minus three objects with
positive $c(\frac{1}{2})$, i.e., all A(+) sample sources show blueshifts. No
significant improvement was found with respect to Eq. 6.

Figure 6: Top panel: FWHM(Civ$\lambda$1549) C16 i.e., corrected following
Coatman et al. (2016) vs FWHM(H$\beta$) st for the FOS+HE sample. Meaning of
symbols is the same as for Fig. 3. The black dot dashed line is the equality
line. The bottom panel shows the residuals. Average ratio, dispersion and
$\chi^{2}_{\nu}$ refer to all sources.

The correction for Pop. B is less well defined, considering the uncertainty in
$a$, and the low value of $b$ (Table 1). The corrections for Pop. B still
offer an improvement because they remove a significant bias (as evident by
comparing Fig. 3 and Fig. 7). In practice, for Pop. B, at low $L$ the
$\xi_{\mathrm{CIV}}$ could be considered constant to a zero-order
approximation, with $\xi_{\mathrm{CIV}}\sim 1$. In other words, when the
velocity field is predominantly virial, and no prominent blueshifted component
affects the line width, Civ$\lambda$1549 may be somewhat broader than H$\beta$
as expected for the stratification revealed by reverberation mapping of lines
from different ionic species (Peterson & Wandel, 1999, 2000). The fit is
consistent with $\xi_{\mathrm{CIV}}$ depending on shift but only weakly on
luminosity. The Pop. B correction is so ill-defined that larger samples are
needed for a better determination of its coefficients.

Slightly different fitting laws

$\xi_{\mathrm{CIV,2}}=\frac{1}{a+b\log\lambda
L_{\lambda}(1450)+c\frac{c(\frac{1}{2})}{1000}}$ (7)

which considers a linear combination $\log\lambda L_{\lambda}(1450)$ and
shift, and

$\xi_{\mathrm{CIV,3}}=\frac{1}{a-b\log\lambda
L_{\lambda}(1450)\cdot\frac{c(\frac{1}{2})}{1000}}$ (8)

which assume a dependence from the product $\log\lambda L_{\lambda}(1450)$ and
shift, provide consistent results, with fitting parameters, their $1\sigma$
confidence level associated uncertainty, and rms residuals of
$\xi_{\mathrm{CIV}}$ reported in Table 1. The fitting relations yield a lower
residual scatter in $\xi_{\mathrm{CIV}}$ than assuming no luminosity
dependence. For instance, using Eq. 7 we obtain a scatter in
$\xi_{\mathrm{CIV,2}}$ that is a factor 1.86 lower than if Eq. 5 is used. We
also considered the A.I. in place of $c(\frac{1}{2})$ in Eq. 6 (without the
absolute value operator; bottom rows of Table 1). The A.I. has a non-
negligible advantage to be independent from the choice of the rest frame. The
A.I. is correlated with both $c(\frac{1}{4})$ and $c(\frac{1}{2})$, and shows
higher correlation with $c(\frac{1}{4})$ (Pearson’s $r\approx 0.66$). However,
the scatter in $\xi_{\mathrm{CIV,A.I.}}$ is unfortunately large, and would
imply a scatter $\approx 1.5$ higher in $M_{\rm BH}$ estimates than in the
case Eq. 6 is considered for Pop. A.

If Pop. A and B are considered together, the final scatter in
$\xi_{\mathrm{CIV}}$ is close for the different fitting function (Eq. 6 yields
a slightly better result) but much higher than if Pop. A and B are kept
separated. It is therefore necessary to distinguish between Pop. A and B as
the intrinsic structure of their BLR may be different (e.g., Goad & Korista,
2014; Wang et al., 2014b). In Pop. B, at low $L/L_{\rm Edd}$, the lines are
mainly broadened following a virial velocity field (Peterson & Wandel, 2000).
The relative prominence of the blueshifted to the virialized component (ratio
BLUE over BC), a consequence of the low $L/L_{\rm Edd}$ for Pop. B sources.
Both properties are expected to contribute to the overall consistency between
H$\beta$ and Civ$\lambda$1549 profiles in Pop. B sources. At any rate,
$\xi_{\mathrm{CIV}}$ should always be $\lesssim 1$, with
$\xi_{\mathrm{CIV}}\approx 1$ for Pop. B at low-$L$, and
$\xi_{\mathrm{CIV}}\ll 1$ in case of very large shifts, as in Pop. A at
high-$L$.

It is possible, in most cases, to distinguishing between Pop. A and B from the
UV spectrum emission blend, making the correction applicable at least to a
fraction of all quasars in large samples. Several criteria were laid out by
Negrete et al. (2014): (1) broad line width; (2) evidence of a prominent red
wing indicative of a VBC; (3) prominence of Ciii]$\lambda$1909\. Population B
sources show a Civ$\lambda$1549 red wing and strong Ciii]$\lambda$1909 in the
1900 Å. Extreme Population A (xA) sources are easy to recognize; they show
strong Aliii$\lambda$1860 in 1900 Å blend and low $W$(Civ$\lambda$1549). A
prototypical composite spectrum of xA sources is shown by Martínez-Aldama et
al. (2018). However, some intermediate cases along the MS (i.e., spectral type
A1) may be easier to misclassify. Also, with only the UV spectral range
available the redshift estimate may be subject to large errors. Negrete et al.
(2014) provide a helpful recipe; however, their recipe applied to three of
their 8 sources allowed for a precision $\sim 100$ km s-1 in the rest frame,
but the remaining 5 had an uncertainty on average $\gtrsim$ 500 km s-1.

Table 1: Fits of $\xi_{\mathrm{CIV}}$

Sample | $a\pm\delta a$ | $b\pm\delta b$ | $c\pm\delta c$ | rmsξ | d.o.f.
---|---|---|---|---|---
$\xi_{\mathrm{CIV,1}}\approx 1/(b*(a-x)*|y|+c)$
A | -0.3093 0.1581 | 0.3434 0.0881 | 1.0763 0.0949 | 0.198 | 40
B | 3.9224 14.1170 | 0.0206 0.0568 | 0.9845 0.0602 | 0.187 | 30
A+B | -0.1805 0.2603 | 0.1978 0.0584 | 1.0117 0.0641 | 0.227 | 73
$\xi_{\mathrm{CIV,1}}\approx 1/(b*(a-x)*y+c)$
A | -0.4161 0.3325 | -0.1825 0.0693 | 1.3245 0.0919 | 0.222 | 40
B | 28.967 496.591 | -0.0030 0.0486 | 1.0182 0.0449 | 0.187 | 30
A+B | -0.1346 0.4991 | -0.1202 0.0467 | 1.1508 0.0510 | 0.238 | 73
$\xi_{\mathrm{CIV,2}}\approx 1/(a+b*x+c*y)$
A | 0.356 0.160 | -0.3480 0.0580 | -0.351 0.0814 | 0.199 | 40
B | 1.080 0.126 | 0.0223 0.0416 | -0.0764 0.0515 | 0.186 | 30
A+B | 0.7605 0.125 | -0.1460 0.0452 | -0.2204 0.0532 | 0.237 | 73
$\xi_{\mathrm{CIV,3}}\approx 1/(a-b*x*y)$
A | 1.3036 0.0884 | 0.1408 0.0481 | … | 0.222 | 41
B | 1.0343 0.0436 | -0.0463 0.0255 | … | 0.187 | 31
A+B | 1.1470 0.0488 | -0.1116 0.0288 | … | 0.237 | 74
$\xi_{\mathrm{CIV,A.I.}}\approx 1/(b*(a-x)*z+c)$
A | -1.7004 0.5528 | -1.0575 0.4703 | 1.5515 0.1183 | 0.240 | 40
B | -2.346 0.3153 | 0.7571 0.3026 | 1.1057 0.0413 | 0.184 | 30
A+B | 5.147 21.138 | -0.1121 0.3292 | 1.2474 0.0501 | 0.257 | 73
333$x=\log\lambda L_{\lambda}(1450)-48$, $y=$ $c(\frac{1}{2})$/1000, $z$ =
A.I.

Figure 7: Top left panel: $\xi_{\mathrm{CIV}}$FWHM(Civ$\lambda$1549) i.e.,
FWHM(Civ$\lambda$1549) after correction for blueshift and luminosity
dependence following Eq. 6 vs FWHM(H$\beta$) cm for the FOS+HE sample. The
black dot dashed line is the equality line, meaning of color code is the same
as in Fig. 3. Top right: same, but with $\xi_{\mathrm{CIV}}$ computed from Eq.
7.

##  5 Discussion

Recently, the problems outlined in earlier works by Sulentic et al. (2007) and
Netzer et al. (2007) have been ascribed to a “bias” in the Civ$\lambda$1549
$M_{\rm BH}$ estimates (Denney et al., 2016). The Civ$\lambda$1549 $M_{\rm
BH}$ bias is dependent on the location in the 4DE1 quasar MS: Fig. 2 and 3
clearly show the different behavior for Pop. A and B. By the same token, an
$L/L_{\rm Edd}$ – dependent correction is in principle a valid approach, as
$L/L_{\rm Edd}$ is probably one of the main drivers of the MS (Boroson &
Green, 1992; Sulentic et al., 2000a; Sun & Shen, 2015). Unfortunately, several
recent works still ignore 4DE1-related effects (or, in other words, MS
trends). For instance, scaling laws derived from the pairing of the virial
products for all sources with reverberation mapping data should be viewed with
care (as shown by the reverberation mapping results of Du et al., 2018).

###  5.1 Civ$\lambda$1549 and H$\beta$ as $M_{\rm BH}$ estimators: input from
recent works

Attempts at using the Civ$\lambda$1549 as a VBE have been renewed in the last
few years, not last because Civ$\lambda$1549 can be observed in the optical
and NIR spectral ranges over which high-redshift quasars have been discovered
and are expected to be discovered in the near future. The large
Civ$\lambda$1549 blueshifts indicate that part of the BLR gas is under
dynamical conditions that are far from a virialized equilibrium. At high
Eddington ratio ionized gas may escape from the galactic bulge, and even be
dispersed into the intergalactic medium, as predicted by numerical simulations
(e.g., Debuhr et al., 2012), and at high luminosity ($\log L\gtrsim 47$ [erg
s-1]) might have a significant feedback effect on the host galaxy (Marziani et
al., 2016).

A firm premise is that the disagreement between H$\beta$ and Civ$\lambda$1549
mass estimates is not a matter of S/N (Denney et al., 2013). The
Civ$\lambda$1549 line width suffers of systematic effects which emerge more
dramatically at high S/N i.e., when it is possible to appreciate the
complexity of the Civ$\lambda$1549 profile. Given this basic result, recent
literature can be tentatively grouped into three main strands: (1) low-$z$
studies, involving FOS and Cosmic Origin Spectrograph (COS) spectra to cover
Civ$\lambda$1549; (2) high-$z$ studies, where the prevalence of large
Civ$\lambda$1549 shifts is high; (3) studies attempting to correct the
Civ$\lambda$1549 FWHM and reduce it to an equivalent of H$\beta$, some of them
employing results that are directly connected to the MS contextualization of
quasar properties.

Figure 8: Black hole mass computed from the fiducial relation of Vestergaard &
Peterson (2006) based on FWHM H$\beta$ vs. the one computed from the
Civ$\lambda$1549 FWHM following Park et al. (2013), for Pop. A (blue) and B
(red) HE sources. $M_{\rm BH}$ values obtained from the Vestergaard & Peterson
(2006) Civ$\lambda$1549 scaling law after the correction suggested by
Brotherton et al. (2015, small dots) are shown by small open circles. The
lower panel shows residuals as a function of $M_{\rm BH}$. The average and the
scatter reported for Pop. A and B refer to the Park et al. (2013) scaling
laws.

##### Low-$z$ studies

A large systematic analysis of the Civ$\lambda$1549 profiles paired to
H$\beta$ emission was carried out using HST/FOS (in part used for the present
work) and optical observations (Sulentic et al., 2007). The results of this
study emphasized the role of the Civ$\lambda$1549 line width in the $M_{\rm
BH}$ estimates. The Fig. 6 of Sulentic et al. (2007) clearly shows the
importance of placing sources in an E1 context: estimates of the masses could
be easily overestimated by a factor $\lesssim$ 100 for extreme Pop. A sources
such as I Zw 1, while for Pop. B Civ$\lambda$1549 and H$\beta$ $M_{\rm BH}$
estimates appeared more consistent albeit with a large scatter. The line width
(let it be the FWHM or the velocity dispersion $\sigma$) remains a major
factor in Civ$\lambda$1549 vs H$\beta$ $M_{\rm BH}$ determinations since
broadening enters squared in the scaling laws (Kelly & Bechtold, 2007).
Similar warnings on using Civ$\lambda$1549 FWHM were issued by Netzer et al.
(2007). Low-$z$ samples are less affected by the Eddington ratio bias that is
cutting low-Eddington ratio sources at a given $M_{\rm BH}$ for a fixed flux
limit (Sulentic et al., 2014). Therefore, it may not be surprising to find
studies based on excellent spectra that find an overall consistency between
H$\beta$ and Civ$\lambda$1549 $M_{\rm BH}$ estimates. Intrinsic scatter is
probably high if full line width without any correction are used: Tilton &
Shull (2013) find $\approx 0.5$ dex from COS observations of low-$z$ quasars.
Denney et al. (2013) claim to be able to reduce the disagreement between
H$\beta$ and Civ$\lambda$1549 derived $M_{\rm BH}$ to $\approx$ 0.24 dex by
using the velocity dispersion of the Civ$\lambda$1549 line. Since the
Civ$\lambda$1549 profile in the Denney et al. (2013) sample almost never shows
large blueshifts which may be associated to a velocity shear in outflowing
gas, these results appear consistent with the Pop. B properties of the FOS
sample.

##### High-$z$ studies

generally concur that the Civ$\lambda$1549 FWHM is poorly correlated with the
Balmer line FWHM. Shen & Liu (2012) describe the scatter between
Civ$\lambda$1549 and H$\beta$ FWHM as due to an irreducible part
($\approx$0.12 dex), and a part that correlates with the blueshift of the
Civ$\lambda$1549 centroid relative to that of H$\beta$. They propose scaling
laws in which the virial assumption is abandoned i.e., with the exponent of
the line FWHM significantly different from 2. For Civ$\lambda$1549, this means
to correct for the overbroadening associated with the non-virial component.
The scaling law introduced by Park et al. (2013) is consistent with the Shen &
Liu (2012) approach and implies $M_{\rm BH}$$\propto$FWHM0.5 i.e., a FWHM
dependence that is very different from the one expected from a virial law
($M_{\rm BH}$$\propto$FWHM2). As shown in Fig. 8, the scaling law suggested by
Park et al. (2013) applied to the HE sample properly corrects for the
overbroadening of Pop. A sources, but overcorrects the width of Pop. B,
yielding a large deviation from the H$\beta$-derived $M_{\rm BH}$ values (on
average $\approx 0.28$ dex).

##### Studies exploiting MS trends

The results reported in §4 and in Paper I indicate that any solution seeking
to bring Civ$\lambda$1549 $M_{\rm BH}$ estimates in agreement with the ones
from H$\beta$ cannot exclude the strong $L/L_{\rm Edd}$ dependence of the
Civ$\lambda$1549 blueshift that is in turn affecting the Civ$\lambda$1549 FWHM
(Fig. 4). The discussion in Sect. 3 and in Sect. 4.5 identifies the
Civ$\lambda$1549 blueshift as an expedient $L/L_{\rm Edd}$ proxy. Any
parameterization of the blueshifted amplitude such as $c(\frac{1}{2})$, the
flux bisector of Coatman et al. (2016) or the ratio
FWHM(Civ$\lambda$1549)/FWHM(H$\beta$) is taking into account the MS trends in
Civ$\lambda$1549 properties, due to the $L/L_{\rm Edd}$ and Civ$\lambda$1549
blueshift correlation. Another $L/L_{\rm Edd}$ proxy may involve the
Siiv$\lambda$1397/Civ$\lambda$1549 peak ratio: at low
Siiv$\lambda$1397/Civ$\lambda$1549 the $M_{\rm BH}$ is underestimated with
respect to H$\beta$, at high Siiv$\lambda$1397/Civ$\lambda$1549 the mass is
overestimated (Brotherton et al., 2015). Since the ratio
Siiv$\lambda$1397/Civ$\lambda$1549 is a known 4DE1 correlate (Wills et al.,
1993; Bachev et al., 2004), these results confirm that FWHM Civ$\lambda$1549
leads to overestimate $M_{\rm BH}$ for Pop. A (as Pop. A outflows produce blue
shifted emission that significantly broadens the line (Fig. 3; cf. Denney et
al. 2012). In our sample, however, applying the correction suggested by
Brotherton et al. (2015), $\delta\log M\approx-1.23\log\frac{1400}{CIV}-0.91$
to the masses derived from the Park et al. (2013) would move the $M_{\rm BH}$
of Pop. B further down, leading to a further increase of the overcorrection,
and also destroying the agreement for Pop. A sources: on top of the
$\propto$FWHM0.5 law, the additional correction is $\delta\log M\approx-0.91$
if $\frac{1400}{CIV}\sim 1$, as for extreme Pop. A. The correction is lower
but still negative for most Pop. B sources where $\frac{1400}{CIV}\sim 0.3$,
exacerbating the disagreement between the H$\beta$ and Civ$\lambda$1549
derived masses. A better consistency is achieved if the correction of
Brotherton et al. (2015) is applied to the Vestergaard & Peterson (2006)
scaling law for Civ$\lambda$1549\. In this case (shown in Fig. 8 by small open
circles) the correction for Pop. A still imply non-negligible systematic
residuals $\delta\log M=\log M_{\mathrm{BH}}$(H$\beta_{\rm BC}$) - $\log
M_{\mathrm{BH}}$(Civ$\lambda$1549) $\approx 0.23$. The average residual is
higher for Pop. B $M_{\rm BH}$, with $\delta\log M\approx 0.27$, and scatter
$\approx 0.39$ dex.

Assef et al. (2011) used a sample of $\approx$10 quasars with optical spectra
covering Civ$\lambda$1549 and near-IR spectra covering H$\beta$ or H$\alpha$
and show that $M_{\rm BH}$ estimates can be made consistent. The approach of
Assef et al. (2011) may be also understood as a correction related to the MS.
Assef et al. (2011) suggest that much of the dispersion in their virial mass
is caused by the poor correlation between $\lambda L_{\lambda}$ at 5100 Å and
at 1350 Å rather than between their line widths. Their Figs. 14 and 15 shows
that the FWHM Civ$\lambda$1549 over H$\beta$ ratio depends on the flux ratio
at 1350 Å and 5100 Å, which is an MS correlate (Laor et al., 1997; Shang et
al., 2011). The Assef et al. 2011 sample of gravitationally-lensed quasars
might have lowered the Eddington ratio bias described by Sulentic et al.
(2014), leading to a preferential section of Pop. B quasars, and better
agreement between H$\beta$ and Civ$\lambda$1549 line width.

###  5.2 A virialized component

A systematic increase in line width in the HE sample is expected if the line
broadening is predominantly virial: Fig. 5 of Paper I shows that there are no
FWHM H$\beta$$\lesssim$ 3000 km s-1at $\log L\gtrsim 47$ [erg s-1]. Fig. 9
shows that a similar increase in FWHM as a function of luminosity is occurring
in the FOS+HE sample for both H$\beta$ and Civ$\lambda$1549\. The FWHM ratio
between Civ$\lambda$1549 and H$\beta$ does not instead appear strongly
influenced by $L$, suggesting the interpretation that the broadening of both
lines – even if the Civ$\lambda$1549 centroid measurements are significantly
affected by an outflowing component – may be mostly related to the
gravitational effects of the supermassive black hole (as further discussed
below). As mentioned, Balmer lines provide a VBE up $z\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}$ 2, and the results of (Paper I)
extended this finding to the highest luminosities. For Civ$\lambda$1549, Fig.
9 and the correlation FWHM – $c(\frac{1}{2})$ justify the assumption of a
virial broadening component coexisting with a non virial one (Wang et al.,
2011).

Figure 9: H$\beta$ and Civ$\lambda$1549 profile parameter comparison as a
function of luminosity. Top panels: behavior of FWHM Civ$\lambda$1549 and
H$\beta$ (upper half) and of the ratio FWHM(Civ$\lambda$1549)/FWHM(H$\beta$)
as a function of $L$ (lower half), for FOS (golden and pale blue) and HE
sample (red and blue). The yellow band identifies the region where
FWHM(Civ$\lambda$1549)/FWHM(H$\beta$)=1 within the errors. Middle panels:
$c(\frac{1}{2})$ of H$\beta$ and Civ$\lambda$1549 (upper half), and difference
$\delta(\frac{1}{2}$) as a function of $L$ (lower half). Square symbols
indicate Pop. B, circles Pop. A. Lines trace an unweighted lsq fit for the
Pop. A (black) and for Pop. B (red) sources. Bottom panels: same for
$c(\frac{1}{4})$ and $\delta(\frac{1}{4}$). The vertical dotted lines join
H$\beta$ and Civ$\lambda$1549 parameters for the same object (e.g., they are
not error bars).

####  5.2.1 Orientation effects on H$\beta$

A large part of the Civ$\lambda$1549 – H$\beta$ scatter is expected to be due
to orientation effects. The issue of orientation effects remains open for RQ
sources, and orientation effects are most-likely strongly affecting the FWHM
of H$\beta$ (Mejía-Restrepo et al., 2018b), even if it remains hard to
distinguish them from other physical factors (such as $M_{\rm BH}$ and
$L/L_{\rm Edd}$). A clue is given by the 4DE1 predictions at extreme
orientations: objects observed near the disk rotation axis (i.e., nearly pole-
on) have the smallest FWHM H$\beta$, the strongest Feii and Caii intensities
(Dultzin-Hacyan et al., 1999), the largest soft X excess, and the largest
Civ$\lambda$1549 blue shifts/asymmetries. These predictions are motivated by
the physical scenarios involving an accretion disk - wind system. From a pole-
on orientation we should see the smallest Doppler broadening of virially
dominated H$\beta$ emitting clouds, and the strongest intensity of Feii and
other LILs if they are emitted from clouds in the outer part of the disk
(Martínez-Aldama et al., 2015). We should also observe the largest
contribution of the soft X excess if it is related to disk emission (Wang et
al., 1996; Boller et al., 1996; Wang et al., 2014a). And finally, if a wind is
associated with an optically thick disk, and its dynamics is dominated by
radiation pressure, HILs such as Civ$\lambda$1549 emitted in the wind would
show the largest blue shifts (if the receding part of the flow is shielded
from view). The case of I Zw 1 provided a prototypical case in which a
flattened LIL emitting systems and a radial outflow could be seen at small
inclination (e.g., Marziani et al., 1996; Leighly, 2004).

In a more modern perspective, there are several indications that the low-
ionization BLR is a flattened system (Mejía-Restrepo et al., 2017; Mejía-
Restrepo et al., 2018a; Negrete et al., 2017, 2018). We see the clearest
evidence at the MS extrema: extreme Pop. B sources radiating at very-low
$L/L_{\rm Edd}$ frequently show LIL profiles consistent with a geometrically
thin accretion disk profiles (e.g., Chen & Halpern, 1989; Strateva et al.,
2003; Storchi-Bergmann et al., 2017), which may be hidden in the majority of
Pop. B sources (Bon et al., 2007, 2009). A highly flattened LIL-BLR is also
suggested in blazars, which are also Pop. B low-radiatiors (Decarli et al.,
2011), by comparing the virial product to mass estimates obtained from the
correlation between $M_{\rm BH}$ the host galaxy luminosity. At the other end
of the MS, extreme Pop. A quasars show deviations from virial luminosity
estimates consistent with the effect of orientation on the line width, if the
emitting region is highly flattened (Negrete et al., 2018). A flattened low-
ionization BLR is also suggested by comparing the virial product to mass
estimates obtained from accretion disk fits to the SED (Mejía-Restrepo et al.,
2017; Mejía-Restrepo et al., 2018a).

The effect of orientation on the FWHM and on $M_{\rm BH}$ and $L/L_{\rm Edd}$
estimates can be computed by assuming that we are observing randomly-oriented
samples of quasars whose line emission arises from a flattened structure –
possibly the accretion disk itself. The probability of viewing the structure
with an isotropic velocity broadening $\delta v_{\mathrm{iso}}$ at an angle
$\theta$ between line-of-sight and the symmetry axis of a flattened structure
is $P(\theta)=\sin(\theta)$. The radial velocity spread (in the following we
use the FWHM as a measure, $\delta v_{\mathrm{obs}}=$FWHM) can be written as

$\frac{\mathrm{FWHM}^{2}}{4}=\delta v^{2}_{\mathrm{iso}}+\delta
v^{2}_{\mathrm{K}}\sin^{2}\theta,$ (9)

which implies that

$\frac{M_{\mathrm{BH,obs}}}{M_{\mathrm{BH,K}}}=\frac{\delta
v^{2}_{\mathrm{obs}}}{\delta
v^{2}_{\mathrm{K}}}=4\cdot(\kappa^{2}+\sin^{2}\theta),$ (10)

where $\kappa=\delta v_{\mathrm{iso}}/\delta v_{\mathrm{K}}$. From Eq. 9 one
can estimate the ratio $\delta v_{\mathrm{obs}}$ $\over$ intrinsic velocity
$\delta v_{\mathrm{K}}$ either by computing a most probable value of $\theta$
or by deconvolving the observed velocity distribution from $P(\theta)$. The
calculations are described in Appendix B. The average ratio is
$<\frac{M_{\mathrm{BH,obs}}}{M_{\mathrm{BH,K}}}>\approx 1.1$ if $\kappa=0.1$.
If the FWHM of the H$\beta$ line is used, the $M_{\rm BH}$ suffers of a small
bias, if the LIL emitting region is highly flattened. If $\kappa=0.5$ (a “fat”
emitting region), then the bias is much larger
$<\frac{M_{\mathrm{BH,obs}}}{M_{\mathrm{BH,K}}}>\approx 2.1$. The $M_{\rm BH}$
dispersion in the case of $\kappa=0.1$ was estimated for large samples
($10^{6}$ replications) with $\theta$ distributed according to $P(\theta)$
(Appendix C), and was found to be $\sigma_{M_{\mathrm{BH}}}\approx 0.33$ dex.
Therefore, even if $P(\theta)$ strongly disfavor cases with $\theta\rightarrow
0$, the viewing angle can account for a large fraction of the dispersion in
the $M_{\rm BH}$ scaling laws with line width and luminosity.

###  5.3 A wind component

The interpretation of the Civ$\lambda$1549 profile (and H$\beta$ profile
differences) rests on the main results of Paper I: the Civ$\lambda$1549 shifts
are dependent on $L/L_{\rm Edd}$ and, to a lesser extent on $L$; the
Civ$\lambda$1549 broadening is due to a blueshifted component whose strength
with respect to a virialized component increases with $L/L_{\rm Edd}$ and $L$.

At $\frac{1}{4}$ and $\frac{1}{2}$ fractional intensity the difference in the
line centroid radial velocity of H$\beta$ and Civ$\lambda$1549 i.e.
$c(\frac{1}{2})$(H$\beta$) – $c(\frac{1}{2})$(Civ$\lambda$1549) and
$c(\frac{1}{4})$(H$\beta$) - $c(\frac{1}{4})$(Civ$\lambda$1549) are almost
always positive, and can reach 7000 km s-1 and 4000 km s-1 in the HE sample
and FOS respectively mainly because of the large Civ$\lambda$1549 blueshifts
(Fig. 9). A luminosity dependence of $\delta(\frac{1}{2})$ =
$c(\frac{1}{2})$(H$\beta$) – $c(\frac{1}{2})$(Civ$\lambda$1549) and
$\delta(\frac{1}{4})$ = $c(\frac{1}{4})$(H$\beta$) -
$c(\frac{1}{4})$(Civ$\lambda$1549) is illustrated in Fig. 9. The centroid
separations are correlated with $L$, with a similar slope at both
$\frac{1}{2}$ and $\frac{1}{4}$ fractional intensity (Fig. 9): for
$c(\frac{1}{2})$ of Pop. A,

$\delta(\frac{1}{2})\approx(648\pm 121)\log L-(28530\ \pm\
5600)\,\mathrm{kms}^{-1},$ (11)

in the range $44\lesssim\log L\lesssim 48.5$.

The trends of Fig. 9 suggest that the Civ$\lambda$1549 broadening is however
affected by $M_{\rm BH}$, as both the H$\beta$ and Civ$\lambda$1549 widths
steadily increase with luminosity, and their ratio shows no strong dependence
on luminosity. This may be the case if the outflow velocity is a factor $k$ of
the virial velocity ($k=\sqrt{2}$ would correspond to the escape velocity).
The correlation between shift and FWHM of Paper I indicates that we are seeing
an outflow component “emerging” on the blue side of the BC. If we assume that
line emission arises from a flattened structure with velocity dispersion
$v_{\mathrm{iso}}$ (i.e., as in Eq. 9), and that the outflowing component from
the accretion disk contributes to an additional broadening term proportional
to $\cos\theta$ (the projection along the line of sight of the outflow
velocity), then the observed Civ$\lambda$1549 broadening can be written as

$\mathrm{FWHM}^{2}_{\mathrm{CIV}}=4(\delta v^{2}_{\mathrm{iso}}+\delta
v^{2}_{\mathrm{K}}\sin^{2}\theta)+\gimel^{2}\delta
v^{2}_{\mathrm{K}}\mathcal{M}\frac{L}{L_{\mathrm{Edd}}}\cos^{2}\theta,$ (12)

where $\gimel$ is a proportionality constant, and $\cal{M}$ the force
multiplier. It follows that the total broadening can easily exceed $\delta
v_{\mathrm{K}}$ for a typical viewing angle $\theta=\pi/6$, provided that the
factor $\mathcal{Q}=\gimel^{2}\mathcal{M}\frac{L}{L_{\mathrm{Edd}}}$ is larger
than 1. The factors $\gimel$ and $\cal{M}$ depend on physical properties
(density, ionization level) and should be calculated in a real physical model
linking ionization condition and dynamics. The factor $\mathcal{Q}$ encloses
the dependence of wind properties on radiation forces, opacity, etc. (Stevens
& Kallman, 1990) along with the dependence on ionization. For example, in the
case of optically thick gas being accelerated by the full absorption of the
ionizing continuum, the force multiplier is ${\cal
M}=\frac{\alpha}{\sigma_{\mathrm{T}}N_{\mathrm{c}}}\approx 7.5$ for column
density $N_{\mathrm{c}}=10^{23}$cm-2, and $\alpha=0.5$ ($\alpha$ is the
fraction between the ionizing and bolometric luminosity, Netzer & Marziani
2010). If $L/L_{\rm Edd}$$\rightarrow 1$, and $\gimel\sim 1$, implying
$\mathcal{Q}\sim O(10)$, the FWHMCIV can exceed by up to a factor of several
the virial broadening, as indeed observed in the most extreme radiators from
the comparison between Civ$\lambda$1549 and H$\beta$.

Eqs. 9 and 12 account for the consistent increase in broadening of
Civ$\lambda$1549 and H$\beta$ (Fig. 9). In the context of the present sample
covering a wide range in luminosity, $L$ can be considered a proxy for the
increase in $M_{\rm BH}$ ($L\propto$ $M_{\rm BH}$, with a scatter set by the
$L/L_{\rm Edd}$ distribution) and therefore in Keplerian velocity. The top and
middle panels of Fig. 9 show a consistent increase of the centroid, and of the
centroid difference $\delta(\frac{1}{2})$ and $\delta(\frac{1}{4})$ with $L$.
This result motivated the introduction of a luminosity-dependent correction to
the line width. The centroid difference can be written as:

$\delta(\frac{i}{4})\sim\frac{1}{2}\delta
v_{\mathrm{K}}(\frac{i}{4})\cos\theta\left(-\gimel\left(\mathcal{M}\frac{L}{L_{\mathrm{Edd}}}\right)^{\frac{1}{2}}+{f}\right),i=1,2,\\\
$ (13)

with $f\equiv 0$ for Pop. A, and $f$ defined by the infall velocity
$v_{\mathrm{inf}}=f\delta v_{\mathrm{K}}$ as a fraction of the radial free-
fall velocity for Pop. B. Eq. 13 and 12 imply that FWHM${}^{2}_{\mathrm{CIV}}$
= FWHM${}_{\mathrm{H\beta}}^{2}+4\delta^{2}(\frac{1}{2})$, if $f=0$.

If we ascribe the redward displacement of the H$\beta$ wing in Pop. B sources
to gravitational and transverse redshift (e.g., Corbin, 1990; Bon et al.,
2015),

$\displaystyle\delta(\frac{i}{4})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\delta
v_{\mathrm{K}}(\frac{i}{4})\gimel\left(\mathcal{M}\frac{L}{L_{\mathrm{Edd}}}\right)^{\frac{1}{2}}\cos\theta+\frac{3}{2}cz_{\mathrm{g}}(\frac{i}{4})\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2}\delta
v_{\mathrm{K}}(\frac{i}{4})\left(-\gimel\left(\mathcal{M}\frac{L}{L_{\mathrm{Edd}}}\right)^{\frac{1}{2}}\cos\theta+\frac{3}{2}\frac{\delta
v_{\mathrm{K}}(\frac{i}{4})}{c}\right)$

where $cz_{\mathrm{g}}\sim c{GM_{\mathrm{BH}}}/{c^{2}r}$ is the
$c(\frac{1}{2})$ or $c(\frac{1}{4})$ of H$\beta$, which usually can be 0 (Pop.
A) or $\geq$0 (Pop. B), and where we have used the weak field approximation
for the gravitational redshift.

Eqs. 13 and 5.3 account for the steady increase of the centroid difference
$\delta$ with luminosity. The H$\beta$ centroid displacement in Pop. B may be
associated with free fall or gravitational redshift. The amplitude of
blueshift depends on luminosity in Pop. A. The point is that both H$\beta$
redward displacement and blueshift of Civ$\lambda$1549 (and hence their
differences) are proportional to the $\delta v_{\mathrm{K}}$ and hence to the
$M_{\rm BH}$.

The $\xi_{\mathrm{CIV}}$ factor can be written as:

$\displaystyle\xi_{\mathrm{CIV}}$ $\displaystyle=$
$\displaystyle\left(\frac{\delta v^{2}_{\mathrm{iso}}+\delta
v^{2}_{\mathrm{K}}\sin^{2}\theta}{4(\delta v^{2}_{\mathrm{iso}}+\delta
v^{2}_{\mathrm{K}}\sin^{2}\theta)+\gimel^{2}\delta
v^{2}_{\mathrm{K}}\mathcal{M}\frac{L}{L_{\mathrm{Edd}}}\cos^{2}\theta}\right)^{\frac{1}{2}}$
$\displaystyle=$
$\displaystyle\left(\frac{1}{1+\frac{{\mathcal{Q}}\cos^{2}\theta}{4(\kappa^{2}+\sin^{2}\theta)}}\right)^{\frac{1}{2}}$

Figure 10: Top: parameter $\xi_{\mathrm{CIV}}$ behavior as a function of
viewing angle as a function of $\theta$ for a “thin” emitting region with
$\kappa=0.1$, for different $\mathcal{Q}$ values (0.4, 1.6, 4.0, 12.0; black
lines). Red line: $\xi_{\mathrm{CIV}}$ behavior for a thick emitting region
$\kappa=0.5$, for the same $\mathcal{Q}$ values. Bottom: same as above for
$\kappa=0.1$, with $\mathcal{Q}=12$ . The thin lines are the $\tilde{\xi}$
values for H$\beta$ (black) and Civ$\lambda$1549 (blue). See text for more
details. The thick line is their ratio (also shown in the top panel).

Figure 11: Top left: Observed distribution of $\xi_{\mathrm{CIV}}$ for the
full FOS+HE sample (shaded histogram), and distribution of
$\xi_{\mathrm{CIV}}$ for $\mathcal{Q}$=0.4 (grey),0.8 (magenta), 2.0 (blue),
7.8 (green), assuming $\kappa=0.1$, for randomly-oriented synthetic samples.
Top right: same as in top left panel, for $\kappa=0.5$. Bottom left:
distribution of $\xi_{\mathrm{CIV}}$ restricted to Pop. A sources for
$\mathcal{Q}$=1.0 (grey), 2.0 (magenta), 4.0 (blue), 12.0 (green), for
$\kappa=0.1$. Bottom right: same for Pop. B sources, with $\mathcal{Q}$=0.
(grey),0.2 (magenta), 0.4 (blue), 1.6 (green), for $\kappa=0.1$. See text for
more details.

The $\xi_{\mathrm{CIV}}$ behavior as a function of the viewing angle $\theta$
is described in of Fig. 10. Fig. 10 shows the dependence in case of a flat
$\kappa=0.1$ (black) or fat $\kappa=0.5$ (red) for four values of ${\cal Q}$.
The bottom panel of Fig. 10 shows the behaviour of the ratios
$\tilde{\xi}_{\mathrm{H\beta}}=1/[4(\kappa^{2}+\sin^{2}\theta)]^{1/2}$ and
$\tilde{\xi}_{\mathrm{CIV}}=1/(4\kappa^{2}+4\sin^{2}\theta+\mathcal{Q}\cos^{2}\theta)^{1/2}$.
The $\tilde{\xi}$ are the ratios between the $\delta v_{\mathrm{K}}$ and the
observed FWHM. At low $\theta$, the FWHM(H$\beta$) underestimates the $\delta
v_{\mathrm{K}}$ by a large factor, while the overestimation of $\delta
v_{\mathrm{K}}$ by the FWHM(Civ$\lambda$1549) is almost independent of
$\theta$ and a factor $\approx 2$.

The panels of Fig. 11 compare the observed distribution of
$\xi_{\mathrm{CIV,1}}$ (shaded histogram) with the prediction of synthetic
samples randomly oriented, at different ${\cal Q}$. We are not seeking a fit
of the observed distribution especially around $\xi_{\mathrm{CIV}}\approx 1$
because of the many biases affecting our sample and of the problem raised by
$\xi_{\mathrm{CIV}}>1$ (see below), but a qualitative consistency in the
distribution of $\xi_{\mathrm{CIV}}<1$.

If we focus the analysis of Fig. 11 mainly on large shifts, the presence of
low $\xi_{\mathrm{CIV}}$ values and their higher frequency favors a highly
flattened low-ionization BLR, as well as high ${\cal Q}$ for the full sample.
A fat $\kappa=0.5$ BLR is unable to reproduce the largest shift amplitudes.
The scatter in $\xi_{\mathrm{CIV}}$ linear values at ${\cal Q}\gtrsim 2$ is
$\approx 0.2$, implying a dispersion in the $M_{\rm BH}$ of $\approx$ 0.15
dex. If we separate Pop. A and B, more extreme values of ${\cal Q}\gtrsim 2$
are required to fit the large shift distribution in Pop. A, with ${\cal Q}\sim
10$. The distribution of $\xi_{\mathrm{CIV}}$ for Pop. B is more peaked around
$\xi_{\mathrm{CIV}}\approx 1$, and the $\xi_{\mathrm{CIV}}$ distribution can
be qualitatively accounted for if ${\cal Q}\lesssim 2$.

The $\xi_{\mathrm{CIV}}$ observed distribution includes values $>1$. These
values are not possible following our model: the FWHM(Civ$\lambda$1549) should
be always in excess or comparable to FWHM(H$\beta_{\rm BC}$). In the case
${\cal Q}\gg 1$, the Civ$\lambda$1549 line is broadened by an outflowing
component; if ${\cal Q}\rightarrow 0$, $\xi_{\mathrm{CIV}}\lesssim 1$. In the
latter case, the excess broadening may come from the smaller emissivity-
weighted distance expected for Civ$\lambda$1549 in a virial velocity field.
The existence of cases with FWHM Civ$\lambda$1549$<$ FWHM H$\beta$ was already
noted by Mejía-Restrepo et al. (2018b), so it is not unique to the FOS+HE
sample. The bottom right panel of Fig. 11 shows that such cases are relatively
frequent among Pop. B. Inspection of the HE spectra in Paper I reveals that
the Civ$\lambda$1549 profile is significantly affected by semi-broad
absorptions such as the ones often found in mini-BAL quasars (Vestergaard,
2003; Sulentic et al., 2006a). Since mini-BALs cluster around the line core,
it is most likely that these Population B sources would satisfy the condition
FWHM Civ$\lambda$1549$\gtrsim$ FWHM H$\beta$ if the effect of the absorptions
could be removed.

Figure 12: $M_{\rm BH}$ computed from the fiducial relation of Vestergaard &
Peterson (2006) based on FWHM H$\beta$ vs. the one computed from the
Civ$\lambda$1549 FWHM corrected following Eq. 6. Error bars include luminosity
uncertainties estimated by the scatter in $L$ derived from the UV and the
visual spectral ranges along with errors on FWHM propagated quadratically. The
lower panel shows the residuals for the three cases. Meaning of color code is
the same as in the previous Figures.

###  5.4 $M_{\rm BH}$ scaling laws dependent on $L/L_{\rm Edd}$ and $L$

The goal is to obtain an $M_{\rm BH}$ estimator based on Civ$\lambda$1549 that
is consistent with the scaling law derived for H$\beta$. In this context, a
second-order dependence on luminosity of FWHM Civ$\lambda$1549 cannot be
ignored especially if samples encompass a broad range in luminosity. This will
be the case in deep, forthcoming surveys. Considering the corrections to FWHM
Civ$\lambda$1549 of §4, the $M_{\rm BH}$ scaling law is derived in the form
$\log M_{\mathrm{BH}}=\alpha\cdot\log L+2\cdot\log{\mathrm{F}WHM}+\gamma$ by
minimizing the scatter and any systematic deviation of $M_{\rm BH}$ estimated
from Civ$\lambda$1549 with respect to the H$\beta$-derived masses: the
unweighted least square fit of Fig. 12 yields

$M_{\mathrm{BH}}(\mathrm{CIV})\approx(0.99\pm
0.04)M_{\mathrm{BH}}(\mathrm{H\beta})+(0.11\pm 0.39).$ (16)

The Civ$\lambda$1549 scaling law takes the form:

$\displaystyle\log M_{\mathrm{BH,1}}\,\mathrm{CIV}$ $\displaystyle\approx$
$\displaystyle(0.64^{+0.045}_{-0.025})\log L_{1450}+$ $\displaystyle
2\log\left(\xi_{\mathrm{CIV,1}}\mathrm{FWHM}(\mathrm{CIV})\right)+(0.525^{+0.22}_{-0.18})$

for the FWHM correction using Eq. 6. Applying Eq. 7, the scaling law does not
change appreciably, and uncertainties in the coefficients are only slightly
different.

$\displaystyle\log M_{\mathrm{BH,2}}\,\mathrm{CIV}$ $\displaystyle\approx$
$\displaystyle(0.63^{+0.045}_{-0.035})\log L_{1450}+$ $\displaystyle
2\log\left(\xi_{\mathrm{CIV,2}}\mathrm{FWHM}(\mathrm{CIV})\right)+(0.525^{+0.275}_{-0.19})$

The scaling law parameter uncertainties have been estimated following the
standard approach in Bevington & Robinson (2003, p. 210ff), with the constrain
that unbiased consistency between $M_{\rm BH}$ from H$\beta$ and
Civ$\lambda$1549 (Eq. 16) is satisfied within the $1\sigma$ uncertainties. The
rms scatter is $\sigma\approx 0.33$ for the Eq. 6 and to $\sigma\approx 0.35$
for Eq. 7. Assuming a single correction for both Pop. A and B significantly
worsens the fit quality, and no scaling law is reported.

An application of the bisector fitting technique using SLOPES (Feigelson &
Babu, 1992a) yields :

$\displaystyle\log M_{\mathrm{BH,1}}\,\mathrm{CIV}$ $\displaystyle\approx$
$\displaystyle(0.5925^{+0.0275}_{-0.030})\log L_{1450}+$ $\displaystyle
2\log\left(\xi_{\mathrm{CIV,1}}\mathrm{FWHM}(\mathrm{CIV})\right)+(0.62\pm
032)$

$\displaystyle\log M_{\mathrm{BH,2}}\,\mathrm{CIV}$ $\displaystyle\approx$
$\displaystyle(0.572^{+0.0285}_{-0.032})\log L_{1450}+$ $\displaystyle
2\log\left(\xi_{\mathrm{CIV,2}}\mathrm{FWHM}(\mathrm{CIV})\right)+(0.64\pm
035)$

The Vestergaard & Peterson (2006) H$\beta$ scaling laws suffer from a
significant scatter (see the discussion in their paper) that can be explained
on the basis of the scatter induced by orientation (0.33 dex at 1$\sigma$)
according to the results of Appendix B. The Civ$\lambda$1549 and H$\beta$
relation should be considered equivalent. The luminosity exponent ($\approx
0.64$) is in agreement with previous observations (Peterson et al., 2005). It
is slightly above the exponent of the Civ$\lambda$1549 radius dependence on
luminosity found in more recent reverberation mapping studies ($\approx
0.52-0.55$ Kaspi et al., 2007; Lira et al., 2017, 2018).

Fig. 12 suggests the presence of a well-behaved distribution with a few
outlying points. It is possible to reduce the scatter to $\sigma\approx 0.25$
applying a $\sigma$ clipping algorithm (i.e., eliminating all sources
deviating more than $\pm$ 2$\sigma$), with no significant change in the best
fitting parameters. This selective procedure is however unwarranted: as shown
in Appendix C, outlying points are expected right because of the possible
occurrence of low-probability viewing angles. The residual rms can be largely
accounted for by orientation effects if $\cal{Q}$ is small, and by the
combination of orientation effects and outflow prominence if $\cal{Q}$ is much
larger than 1 (Appendix C).

###  5.5 Application to a large sample with H$\beta$ and Civ$\lambda$1549 data

The Coatman et al. (2017) data provides a different sample for the testing of
the scaling law of Eq. 5.4. The flux bisector can be converted into
$c(\frac{1}{4})$ (Sect. 4.5). An application of Eq. 5.4 to Pop. A and B
(applying the luminosity-dependent separation as in Paper I) is yielding
agreement with the expectation of an unbiased $M_{\rm BH}$ estimator with
respect to the H$\beta$ $M_{\rm BH}$ estimates using the scaling law of
Vestergaard & Peterson (2006): the slope of an unweighted lsq fit is $\approx
0.958\pm 0.065$ (using SLOPES, Feigelson & Babu 1992a, Fig. 13).

##  6 Conclusion

The present investigation was focused on the Civ$\lambda$1549 relations to
H$\beta$ over a broad range of luminosity ($\log L\sim 43-48$, including very
high luminosities $\log L\gtrsim$ 47) in the Eigenvector 1 context, with the
goal of testing the Civ$\lambda$1549 suitability as virial broadening
estimators when low-ionization lines observations are not available. The
Eigenvector 1 context means that the quasar main sequence is considered to
properly interpret first-order Eddington ratio effects and luminosity effects
that appear to be second order in low-$z$ samples.

The main conclusions reached in this paper are as follows:

  1. 1.

Within the limits of our sample size, and of our UV spectral coverage, it does
not appear that the Civ$\lambda$1549 FWHM can be used as a reliable virial
broadening estimator without significant corrections. There is a large scatter
between FWHM measurements on H$\beta$ and Civ$\lambda$1549 that seems to defy
the definition of a meaningful trend.

  2. 2.

The Civ$\lambda$1549NC removal improves the agreement between Civ$\lambda$1549
and H$\beta$ measures in the HE sample, but Civ$\lambda$1549NC is not a major
factor hampering the definition of a Civ$\lambda$1549 VBE consistent with
H$\beta$.

  3. 3.

Corrections to FWHM Civ$\lambda$1549and Civ$\lambda$1549-based $M_{\rm BH}$
estimates that vary systematically along the 4DE1 sequence and are strongly
dependent on Eddington ratio are promising and should be further explored.

  4. 4.

Following the results of Paper I, we define a correction to the
FWHM(Civ$\lambda$1549) based on the full-profile Civ$\lambda$1549
$c(\frac{1}{2})$ (a proxy for $L/L_{\rm Edd}$) and on the luminosity at 1450
Å. Given the intrinsic differences between Pop. A and B, and their “threshold”
separation dependent on a critical $L/L_{\rm Edd}$, two different correction
laws were considered for the two populations. We remark that the correction
for Pop. B as derived from the FOS+HE sample is highly uncertain.

  5. 5.

The $M_{\rm BH}$ scaling law (Eq. 5.4) associated with the corrected
FWHM(Civ$\lambda$1549) following Eq. 6, as explained in Sect. 5.4, allows for
the preservation of the virial dependence on line broadening. Its practical
usefulness rests on the ability to distinguish Pop. A and B quasars. This can
be achieved in a large fraction of quasars following the guideline set forth
by Negrete et al. (2014).

  6. 6.

We constructed a toy model that helped the interpretation of the scatter.
Orientation effects induce scatter $\approx 0.3-0.4$ dex in mass estimates
that account for a large fraction of the dispersion in the landmark scaling
law of Vestergaard & Peterson (2006). A physical model of the disk + wind
system might allow to recover the viewing angle $\theta$ for individual
quasars.

Figure 13: Estimates of $M_{\rm BH}$ using corrected Civ$\lambda$1549 FWHM as
a VBE versus $M_{\rm BH}$ computed from H$\beta$ FWHM using the scaling-law of
Vestergaard & Peterson (2006). The luminosity and FWHM data from Coatman et
al. (2017) with Pop. A (blue) and B sources kept separated (see Sect. 5.5 for
more details), and with $M_{\rm BH}$ computed according to Eq. 5.4. The dot-
dashed line traces the 1:1 relation between $M_{\rm BH}$ from Civ$\lambda$1549
and H$\beta$; the black line is an unweighted lsq fit for the A+B $M_{\rm BH}$
estimates.

###### Acknowledgements.

The authors acknowledge the contribution of Jack Sulentic to the development
of this paper which is based on observations made with ESO Telescopes at the
Paranal Observatory under programme 082.B-0572(A), and with the Italian
Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the
Fundación Galileo Galilei of the INAF at the Spanish Observatorio del Roque de
los Muchachos of the Instituto de Astrofísica de Canarias.. PM acknowledges
the Programa de Estancias de Investigación (PREI) No. DGAP/DFA/2192/2018 of
UNAM, where this paper was completed. The relevant research is part of the
projects 176001 “Astrophysical spectroscopy of extragalactic objects” and
176003 ”Gravitation and structure of universe on large scales” supported by
the Ministry of Education, Science and Technological Development of the
Republic of Serbia. MLMA acknowledges a CONACyT postdoctoral fellowship, and
wishes to thank the IAA for support during her postdoctoral stay from February
2016 to July 2018. MLMA acknowledges a CONACyT postdoctoral fellowship and
National Science Centre, Poland, grant No.2017/26/A/ST9/00756 (Maestro 9). AdO
and MLMA acknowledge financial support from the Spanish Ministry of Economy
and Competitiveness through grant AYA2016-76682-C3-1-P. AdO acknowledges
financial support from the State Agency for Research of the Spanish MCIU
through the “Center of Excellence Severo Ochoa” award for the Instituto de
Astrofísica de Andalucía (SEV-2017-0709). MLMA, PM and MDO acknowledge funding
from the INAF PRIN-SKA 2017 program 1.05.01.88.04. DD and AN acknowledge
support from CONACyT through grant CB221398. DD also thanks for support from
grants IN108715 and IN113719 PAPIIT, DGAPA, UNAM. MAMC was partially supported
by the Spanish Research project MTM2015-64095-P and by Diputación General de
Aragón, Group E24-17R. The scientific results reported in this article are
also based on publicly available HST spectra.

##  Appendix A  The MS / 4DE1 formalism: a glossary

Figure 14:  Schematic representation of the optical plane of the quasar MS,
with the subdivisions identifying spectral types along the sequence. The main
components that are blended in the H$\beta$ and Civ$\lambda$1549 profile are
listed in each spectral bin. The shaded area shows the approximate occupation
of low-$z$ quasars sample in the plane.

In the optical plane of the quasar main sequence, spectral types are isolated
following Sulentic et al. (2002). Fig. 14 provides a sketch with the spectral
types identification in the optical plane of the MS. Here we provide a
glossary of the MS-related terms and acronyms employed in the paper, along the
order of the quasar main sequence. A more thorough description can be found in
Sulentic et al. (2011) and Marziani et al. (2018). The rational for two type-1
quasar populations (A and B) was originally given by Sulentic et al. (2000a).

ST |  Definition
---|---
|
Population B: FWHM(H$\beta$) $\gtrsim$ 4000 km s-1. Virial-dominated with
redward-asymmetric profiles in H$\beta$ and Mgii$\lambda$2800\. Also described
a disk-dominated (Richards et al., 2011).
B1++ |  12000 km s-1$\leq$ FWHM(H$\beta$) $<$ 16000 km s-1,$R_{\rm FeII}$$<$0.5
B1+ |  8000 km s-1$\leq$ FWHM(H$\beta$) $<$ 12000 km s-1$R_{\rm FeII}$$<$0.5
B1 |  4000 km s-1$\leq$ FWHM(H$\beta$) $<$ 8000 km s-1 $R_{\rm FeII}$$<$0.5
|
Population A: FWHM(H$\beta$) $\lesssim$ 4000 km s-1. Sources frequently show
Civ$\lambda$1549 blueshifts, and H$\beta$ Lorentzian-like profiles (Du et al.,
2016). Includes a range of FeII emission.
A1 |  FWHM(H$\beta$) $<$ 4000 km s-1; $R_{\rm FeII}$$<$0.5
A2 |  FWHM(H$\beta$) $<$ 4000 km s-1; $\leq$ 0.5 $R_{\rm FeII}$$<$1
A3 [xA] |  FWHM(H$\beta$) $<$ 4000 km s-1; $\leq$ 1 $R_{\rm FeII}$$<$1.5
A4 [xA] |  FWHM(H$\beta$) $<$ 4000 km s-1; $\leq$ 1.5 $R_{\rm FeII}$$<$2

Spectral types A3 and A4 are are also indicated as extreme Population A, with
$R_{\rm FeII}$$\gtrsim 1$. They are the highest radiators per unit mass and
possibly super-Eddington accretors (Wang et al., 2014a).

The 4D Eigenvector-1 formalism was introduced to limit the set of MS-
correlated parameters to the four ones that are most relevant for the MS
physical interpretation. In addition to FWHM H$\beta$, $R_{\rm FeII}$, and
$c(\frac{1}{2})$ Civ$\lambda$1549, the soft X-ray photon index
$\Gamma_{\mathrm{soft}}$ is also considered (Sulentic et al., 2000b). The four
parameters were meant to represent the velocity dispersion of the LIL-emitting
part of the BLR, the physical condition within the LIL-BLR ($R_{\rm FeII}$),
the dynamical condition of the HIL emitting gas, and the accretion state of
the black hole. $\Gamma_{\mathrm{soft}}>2$ implies a soft X-ray excess that is
exclusive of Population A (Wang et al., 1996; Boller et al., 1996; Shen & Ho,
2014; Bensch et al., 2015).

Line components assumed in the decomposition along the MS are defined as
follows (see also the sketch of Fig. 14).

Line component |  Definition
---|---
|
Low-ionization lines (LILs): HI H$\beta$
H$\beta_{\rm VBC}$ |  Gaussian FWHM $\sim 10000$ km s-1, redshifted by $1-2000$km s-1 (defining property of Pop. B; absent in Pop. A)
H$\beta_{\rm BC}$ |  Lorentzian, FWHM $\sim 1-4000$ km s-1 (Pop. A); Gaussian FWHM H$\beta$$\gtrsim$ 4000 km s-1 (Pop. B)
H$\beta$ BLUE |  Asymmetric Gaussian which models an excess of emission on the blue side of H$\beta$; usually weak save in ST A3 and A4
H$\beta$ |  Sum of H$\beta_{\rm BC}$, H$\beta_{\rm VBC}$, and BLUE (when applicable); full broad H$\beta$ profile
H$\beta_{\rm NC}$ |  H$\beta$ narrow component
High-ionization lines (HILs): Civ$\lambda$1549
Civ$\lambda$1549VBC |  Gaussian FWHM $\sim 10000$ km s-1, redshifted by $1-2000$km s-1
Civ$\lambda$1549BC |  Lorentzian, if FWHM(H$\beta$)$\lesssim$ 4000 km s-1(Pop. A); Gaussian FWHM $\gtrsim 4000$ km s-1(Pop. B)
Civ$\lambda$1549 BLUE |  Asymmetric Gaussian which models an excess of emission on the blue side of Civ$\lambda$1549; detected in most quasar and most prominent in spectral types A3 and A4
Civ$\lambda$1549 |  Sum of Civ$\lambda$1549BC, Civ$\lambda$1549VBC, and Civ$\lambda$1549 BLUE; full broad Civ$\lambda$1549 profile
Civ$\lambda$1549NC |  Civ$\lambda$1549 narrow component, prominent in Pop. B at low-$z$; almost absent in most Pop. A sources

BLUE becomes detectable as a blueward excess in the H$\beta$ profile mainly in
A3 and A4 (more infrequently in A2). For Civ$\lambda$1549, BLUE increases in
prominence along the sequence from B1++, to A4. In B1++ and B1+ is weak and
often undetectable, while in A3 and A4 it may dominate Civ$\lambda$1549
emission.

##  Appendix B Effect of orientation on H$\beta$ $M_{\rm BH}$ estimates

From the inversion of Eq. 9, we obtain an expression for the viewing angle
$\theta=\arcsin{\sqrt{x^{2}/4-\kappa^{2}}}$, where $x=v_{\mathrm{obs}}/\delta
v_{\mathrm{K}}$ and $\kappa=\delta v_{\mathrm{iso}}/\delta v_{\mathrm{K}}$.
The probability to observe $v_{\mathrm{obs}}$ for a given $\delta
v_{\mathrm{K}}$ is then

$\centering
P(x)=\sqrt{x^{2}/4-\kappa^{2}}\frac{d\theta}{dx}=\frac{x/4}{\sqrt{\kappa^{2}-x^{2}/4+1}}\@add@centering$
(21)

The black hole mass is $\propto x^{2}$. Therefore the average effect can be
written as

$<\frac{M_{\mathrm{BH,obs}}}{M_{\mathrm{BH,K}}}>=\int_{x(v_{\mathrm{iso}})}^{x(\mathrm{edge})}x^{2}P(x)dx{/}\int_{x(v_{\mathrm{iso}})}^{x(\mathrm{edge})}P(x)dx$
(22)

The integrals of Eq. 22 can be computed analytically:

$\int
x^{2}P(x)dx=(-\frac{4}{3}\sqrt{4\kappa^{2}-x^{2}+4})-\frac{4}{3}\kappa^{2}\sqrt{4\kappa^{2}-x^{2}+4}-\frac{1}{6}(x^{2}\sqrt{4\kappa^{2}-x^{2}+4})$
(23)

$\int P(x)dx=-\sqrt{1+\kappa^{2}-x^{2}/4}$ (24)

Note that the integration limits in $x$ (which correspond to $\theta=0$ and
$\theta=45$) are different for $\kappa=0.1$ and 0.5. For $\kappa=0.1$,
$x_{\mathrm{iso}}=0.2$ and $x_{\mathrm{edge}}\approx 1.43$ In the latter case,
$\theta=0$ corresponds to $x_{min}$ = 1 and $\theta=45$ to
$x_{\mathrm{edge}}\approx 1.73$: $M_{\rm BH}$ will be always overestimated,
for every possible $\theta$ value larger than 0.

If we consider the Eddington ratio, we obtain:

$\int
P(x)/x^{2}dx=-\arctan{\mathrm{h}}\left(\frac{\sqrt{\kappa^{2}+1}}{\sqrt{\kappa^{2}-x^{2}/4+1}}\right)/4{\sqrt{\kappa^{2}+1}}$
(25)

Eq. 25 implies a significant effect on $L/L_{\rm Edd}$ for both $\kappa=0.1$
and $\kappa=0.5$. In the first case, the $L/L_{\rm Edd}$ will be overestimated
by a factor $\approx 1.8$. In the second case, the $L/L_{\rm Edd}$ will be
underestimated by a factor $\approx$2, due to the systematic overestimation in
$M_{\rm BH}$.

##  Appendix C Origin of scatter in scaling laws

We considered synthetic samples of $\sim 10000$ objects obtained from random
variates with distribution $P(\theta)\propto\sin\theta$
($0\leq\theta\leq\pi/4$), and “true” $M_{\rm BH}$ uniformly distributed in the
range $10^{7}M_{\odot}\leq M_{\mathrm{BH}}\leq 10^{9}M_{\odot}$. The
dependence on orientation of the H$\beta$ FWHM is assumed to follow Eq. 9,
with $\kappa=0.1$. The effect of orientation on FWHM is such that, in a
randomly oriented synthetic sample, the $M_{\rm BH}$ estimated from H$\beta$
deviates from the true $M_{\rm BH}$ as in Fig. 15 (top panel). The dispersion
is $\approx 0.35$, which is comparable to the uncertainty in the scaling-law
$M_{\rm BH}$ estimate following Vestergaard & Peterson (2006, $\approx$ 0.5 at
1$\sigma$ confidence level).

The estimates of $M_{\rm BH}$ Civ$\lambda$1549 show a significant scatter if
plotted against $M_{\rm BH}$ H$\beta$ (Fig. 12). The origin of the scatter is
in part related to orientation, in part to the outflow component. The second
panel from top of Fig. 15 shows the $M_{\rm BH}$ Civ$\lambda$1549 vs H$\beta$
for a synthetic sample to which no correction has been applied. FWHM of
Civ$\lambda$1549 and H$\beta$ are expected to be related to the Keplerian
velocity by Eqs. 12 and 10, respectively. The distribution of $L/L_{\rm Edd}$
has been assumed Gaussian, peaking at $\log$$L/L_{\rm Edd}$$=-0.3$, and
$\sigma\approx 0.5$. Typical values of $\xi_{\mathrm{CIV}}$ and ${\cal Q}$ are
appropriate for Pop. A sources. There is a strong bias (0.5 dex) and a
standard deviation of the mass ratios of $\approx 0.6$ dex. In some rare
instances (a combination of face-on orientation and large outflow velocity)
the ratio between $M_{\rm BH}$ from H$\beta$ and Civ$\lambda$1549 can reach a
factor $\sim 10^{2}$, as actually found by Sulentic et al. (2007).

The third and fourth panels from top shows the same configuration, but after
applying a correction factor $\xi_{\mathrm{CIV}}$ in the form $1/\zeta(L,$
$L/L_{\rm Edd}$) $=1/(1+kL^{a}$($L/L_{\rm Edd}$)${}^{b})$, with $a\approx
0.1$, $b\approx 1$ and no dependence on orientation, intended to mimic the
correction actually applied to the data in Sect. 4. The orientation effect is
changing the Civ$\lambda$1549 FWHM following Eq. 12, and therefore displacing
the $M_{\rm BH}$ from the true mass also after correcting the FWHM. Note that
the outlying blue points are due to H$\beta$ underestimates of the mass by
more than 0.33 dex because of low values of the viewing angle $\theta$. On the
converse, the relatively high value of ${\mathcal{Q}}$ in the simulation
produces a significant fraction of $M_{\rm BH}$ Civ$\lambda$1549 with are
overestimating the $M_{\rm BH}$ by more than 0.33 dex (red points). The
dispersion is however reduced with respect to the case with no correction,
with an rms $\approx 0.3$. The bottom panel is a realization of the synthetic
sample for a number of sources $\lesssim 100$, comparable to the size of the
FOS+HE sample, and Gaussian distribution of $L/L_{\rm Edd}$ as for case shown
and correction in the second panel from top.

In all of these cases save the one of the second panel from top the dispersion
remains $\sim 0.3$, comparable to the one measured for the scaling laws of Eq.
5.4 and Eq. 5.4. It is interesting to note that, in the framework of the toy
model, if ${\cal Q}=0$ the orientation-induced scatter is the same for
H$\beta$ and Civ$\lambda$1549; if ${\cal Q}\approx 4$, Civ$\lambda$1549
becomes an almost perfect VBE, with all the scatter being due to H$\beta$, in
a plot $M_{\rm BH}$ Civ$\lambda$1549 vs $M_{\rm BH}$ H$\beta$. This results
may be consistent with no strong dependence on orientation of the
Civ$\lambda$1549 line shift in RL quasars (Runnoe et al., 2014).444However, it
is not clear whether the results of Runnoe et al. (2014) are applicable to
radio quiet quasars: RL sources show no strong evidence of large blueshifts
(Sulentic et al., 2007; Richards et al., 2011) as the disk outflow properties
may be strongly affected by the powerful radio ejecta (e.g. Punsly, 2010;
Punsly & Zhang, 2011; Sulentic et al., 2015, and references therein).

A uniform distribution of ${\cal Q}$ between 0 and 1.6, a situation more
appropriate for Pop. B, was also considered. Results are similar with smaller
dispersion and biases.

Figure 15: Top panel: $M_{\rm BH}$ with effect of orientation via Eq. 10 vs
“true” $M_{\rm BH}$, for a synthetic sample of 10000 sources. Second from top:
$M_{\rm BH}$ Civ$\lambda$1549 vs $M_{\rm BH}$ H$\beta$ estimated for an
Eddington ratio distribution as described in the text and no correction. Third
from top: The FWHM $M_{\rm BH}$ Civ$\lambda$1549 has been corrected because of
outflow broadening using a correction factor $\xi$. The blue dots identify the
$M_{\rm BH}$ H$\beta$ estimates that are under 0.33 dex the true $M_{\rm BH}$;
the red ones are for overestimates by more than 0.33 dex. The grey dots
represent mass estimates within $-$0.33 and +0.33 dex from the true value.
Fourth from top: same, with color coding referring to Civ$\lambda$1549 $M_{\rm
BH}$. Bottom: a synthetic sample with $n\lesssim 80$ sources, as in the FOS+HE
sample. See text for more details. In all panels, the dot-dashed line is the
equality line; the filled line traces an unweighted least square fit.

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