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GLOBAL PARAMETERS OF QUASARS WITH ANOMALOUS ELECTROMAGNETIC SPECTRUM

DOCTOR OF PHILOSOPHY

IN

PHYSICS

Marcin Marculewicz

Faculty of Physics

University of Białystok

PhD thesis supervisor:

dr hab. Marek Nikołajuk, prof. UwB

Białystok, June 2021

2021Marcin Marculewicz

I would like to thank my advisor, dr hab Marek Nikolajuk, prof. UwB, for
guiding and supporting me over the years.

## Acronyms

AD

    Accretion Disk
AGN

    Active Galactic Nucleus
BH

    Black Hole
BLR

    Broad Line Region
EW

    Equivalent Width
f

    virial factor
FWHM

    Full Width at Half Maximum
IGM

    Intergalactic medium
IR

    Infrared
NLR

    Narrow Line Region
NT

    Novikov-Thorne
QSO

    Quasar
SED

    Spectral Energy Distribution
SF

    Starformation
SMBH

    Supermassive black hole
SS

    Shakura-Sunayaev
UV

    Ultraviolet
WLQ

    Weak emission-line Quasar

###### Contents

  1. Preface
  2. 1 Introduction
    1. 1.1 Active Galactic Nuclei
    2. 1.2 The spectrum of AGN
    3. 1.3 Weak Emission-lines Quasars (WLQs)
    4. 1.4 Estimation of Black hole mass
    5. 1.5 Accretion disk model
  3. 2 Sample selection and data reduction
    1. 2.1 Sample selection and data preparation
      1. 2.1.1 Sample selection
      2. 2.1.2 Observed data
    2. 2.2 Data reduction
      1. 2.2.1 Dereddening
      2. 2.2.2 Photoelectric absorption in the IGM
      3. 2.2.3 Host-galaxy and its starlight
      4. 2.2.4 Torus contamination
    3. 2.3 Sample selection of type 1 QSO and calibration of method
    4. 2.4 Hypothesis testing approach
      1. 2.4.1 $\chi^{2}$ procedure
      2. 2.4.2 Bayesian inference
  4. 3 Results
    1. 3.1 Method - Novikov-Thorne model of accretion disk
    2. 3.2 Description of results
    3. 3.3 Analysis of results
      1. 3.3.1 Bayes’ theorem
      2. 3.3.2 The $M_{\mathrm{BH}}$ of the thesis vs literature - $M_{\mathrm{BH}}$ comparison
  5. 4 Discussion
    1. 4.1 The virial factor and black hole masses in Weak emission-line Quasars
    2. 4.2 Inclination dependency
    3. 4.3 Shielding gas as an explanation of the nature of WLQs
  6. 5 The corona investigation in the high mass SMBH.
    1. 5.1 Introducing the SDSS J110511.15+530806.5 quasar.
    2. 5.2 The Global Absorption
    3. 5.3 The Influence of the Corona
    4. 5.4 The Fitting Procedure for J1105
    5. 5.5 Results
  7. 6 Discussion
    1. 6.1 Calculations for the Potential Absorption Source
    2. 6.2 How far from the galaxy is the absorption cloud?
      1. 6.2.1 If We Assume the Wind is the Absorber
    3. 6.3 The column density of J1105
    4. 6.4 Final remarks
  8. 7 Conclusion
    1. 7.1 Future work/prospect of WLQs research

###### List of Tables

  1. 1.1 AGN Unification
  2. 2.1 Continuum fitting windows
  3. 2.2 Sample of Weak Emission-Line Quasars and the sources of their photometry points
  4. 2.3 Fitted temperature of tori in the sample of WLQs
  5. 3.1 Parameter values for the grid of the AD models
  6. 3.2 The best fit parameters
  7. 3.3 The Schwarzschild black hole solutions
  8. 3.4 The BH masses from the Bayesian analysis
  9. 3.5 The Full Width at Half Maximum of H$\beta$ emission-line and the $\gamma$ factor
  10. 5.1 SDSS J110511.15+530806.5 parameters.
  11. 5.2 QSOSED model with fixed parameters.
  12. 5.3 The fitting model redden*QSOSED*gabs*gabs parameters for J1105.
  13. 5.4 The fitting model redden*AGNSED*gabs*gabs parameters for J1105.
  14. 6.1 Gaussian function in the observed frame and the rest frame.
  15. 6.2 The parameters of winds of Mg 2 and C 4
  16. 6.3 The column density of J1105

###### List of Figures

  1. 1.1 Types of AGNs
  2. 1.2 The zoom picture of the vicinity of SMBH
  3. 1.3 AGN spectral energy distribution
  4. 1.4 Accretion disk as a blackbody
  5. 1.5 The mean QSO spectrum
  6. 1.6 Potential vicinity of the WLQ
  7. 1.7 The ’shielding gas scenario’
  8. 2.1 The best fit and components of SED fit for WLQ SDSS J083650.86+142539.0.
  9. 3.1 Example of the Novikov-Thorne model
  10. 3.2 Comparison of LBQS masses with results from the continuum-fitting method
  11. 3.3 The best fit and components of SED fit for WLQ SDSS J094533.98+100950.1
  12. 3.5 The best fit and components of SED fit for WLQ SDSS J123743.08+630144.9
  13. 3.7 The best fit and components of SED fit for WLQ SDSS J141730.92+073320.7
  14. 3.9 The best fit and components of SED fit for WLQ SDSS J152156.48+520238.5
  15. 3.11 The best fit and components of SED fit for WLQ PG1407
  16. 3.13 Comparison of SMBH masses of WLQs.
  17. 3.14 The $\gamma$ factor ($=M_{\mathrm{BH}}/M_{\mathrm{BH}}^{\mathrm{lit}}$) versus FWHM(H$\beta$).
  18. 4.1 The distribution of inclinations in AGNs and WLQs.
  19. 5.1 SDSS J110511.15+530806.5 spectrum.
  20. 5.2 J1105 spectrum with mean QSO spectrum
  21. 5.3 Comparison of Gaussian functions
  22. 5.4 The vicinity of the BH with AD + corona system.
  23. 5.5 The geometry of the Kubota & Done (2018) model.
  24. 5.6 The fit of the SDSSJ110511.15+530806.5. - redden*QSOSED*gabs*gabs
  25. 5.7 The fit of the SDSSJ110511.15+530806.5. - redden*AGNSED*gabs*gabs
  26. 6.1 The illustration of the two Gaussian’s
  27. 6.2 A scheme of potential absorption Mg 2 clouds in front of the quasar
  28. 6.3 A scheme of potential absorption C 4 clouds
  29. 6.4 The vicinity of the J1105
  30. 6.5 An explanation of the nature of J1105 - winds.
  31. 6.6 An explanation of the nature of J1105 - two dwarf galaxies.
  32. 7.1 Baldwin effect for WLQs.

## Preface

The thesis concerns the analysis of the Active Galactic Nuclei. These are
galaxies with an active core. The most luminous type of Active Galactic Nuclei
is Quasar. It contains the supermassive black hole at the center. One of the
least known subtype of Quasars are: Weak emission-Line Quasars. Their
recognizable feature are weak emission-lines.

The primary goal of PhD thesis is to evaluate the global parameters such as:
the black hole mass, the accretion rate, spin of the black hole, and the
inclination of weak emission-line quasars based on the continuum fit method.
This method apart from the literature black hole masses estimation methods
does not depend on the observed Full Width at Half Maximum of emission line,
which could be biased due to the weakness or lack of the emission lines in
these quasars. Using the Spectral Energy Distribution of quasars, I have
fitted the geometrically thin and optically thick accretion disk model
described by Novikov & Thorne equations. I have obtained the model of the
continuum of the accretion disk for the 10 weak emission-line quasars.

The second project concerned the description of abnormal, deep absorption of
SDSS J110511.15+530806.5 quasar. I checked the correctness of the thesis posed
that corona and warm skin concept above/around an accretion disk explain this
phenomenon.

My main motivation during my thesis is:

  * •

Examining the validity of hypotheses of the nature of the weak emission-line
quasars,

  * •

Analysis of the Broad Line Region in the vicinity of the host of weak
emission-line quasars,

  * •

Development of the virial factor for the weak emission-line quasars,

  * •

Development of the new method of the $M_{\mathrm{BH}}$ estimation of weak
emission-line quasars,

  * •

Comparison of my results with the previous literature reports,

  * •

Modeling the corona and warm skin for SDSS J110511.15+530806.5

  * •

Validate the hypothesis of the phenomena in SDSS J110511.15+530806.5

The following work is divided into Chapter 1: Introduction of the Active
Galactic Nuclei and Weak emission-line Quasars description. In Chapter 2, I
have described the sample selection and data reduction. I have collected
photometric points and spectra of 10 Weak emission-line Quasars in the broad
range (Infrared/Optic/Ultraviolet) using the WISE, 2MASS, SDSS, Galex. In
Chapter 3, I have focused on the model and results description. Chapter 4
presents a discussion of results and potential explanation of the Weak
emission-line Quasars. Chapter 5 contains the description of the SDSS
J110511.15+530806.5. Additionally, I have described the modeling and the
results of the fitting of the corona and the warm skin of SDSS
J110511.15+530806.5. I have concluded my thesis in Chapter 6.

The doctoral dissertation was created on the basis of:

  * •

M. Marculewicz and M. Nikolajuk - "Black Hole Masses of Weak Emission Line
Quasars Based on the Continuum Fit Method", the Astrophysical Journal, Volume
897, Number 2 (2020)

  * •

M. Marculewicz and M. Nikolajuk - "Models of Continuum of Weak Emission-Line
Quasars", Proceedings of the Polish Astronomical Society, Volume 10, 243-247,
ISBN: 978-83-950430-8-6 (2020)

and unpublished material described in Chapter 5 (in preparation).

My additional papers not included into this PhD thesis are:

  * •

H. Abdalla et al. (Cherenkov Telescope Array Consortium) - "Sensitivity of the
Cherenkov Telescope Array for probing cosmology and fundamental physics with
gamma-ray propagation", Journal of Cosmology and Astroparticle Physics, Issue
02, article id. 048 (2021),

  * •

A. Acharyya et al. (Cherenkov Telescope Array Consortium) - "Sensitivity of
the Cherenkov Telescope Array to a dark matter signal from the Galactic
centre", Journal of Cosmology and Astroparticle Physics, Issue 01, article id.
057 (2021),

  * •

A. Acharyya et al. (Cherenkov Telescope Array Consortium) - "Monte Carlo
studies for the optimisation of the Cherenkov Telescope Array layout",
Astroparticle Physics, 111, 35-53 (2019),

  * •

M. Ostrowski et al. - "Progress of the Cherenkov Telescope Array project in
Poland", Proceedings of the Polish Astronomical Society, vol. 7, 343-348
(2018),

  * •

M. Giedyk et al. - "Photoinduced Vitamin B12-Catalysis for Deprotection of
(Allyloxy)arenes", Organic Letters, 19, 10, 2670-2673, (2017)

##  Chapter 1 Introduction

### Abstract

An AGN (Active Galactic Nucleus) provides a unique test of fundamental physics
in regimes inaccessible on Earth and plays an important role in our
understanding of the evolution of galaxies. We describe these systems through
the "Unified Model" in which the AGN contains a central BH (Black Hole), an AD
(Accretion Disk), jets, magnetic fields, and exhibit chaotic inflow/outflow
processes. Modeling these components, therefore, requires understanding
physical processes across a wide range of scales. While the Unified Model
explains many of the peculiar behaviors of different types of AGN, they still
fail to explain some classes such as WLQ s (Weak emission-Line Quasars).

###  1.1 Active Galactic Nuclei

For hundreds of years, we have been looking at the sky and observing the
astonishing nature of the cosmos. Recent decades have shown many discoveries
about the nature of AGN s. They are powered by accretion onto SMBH and we are
able to observe their features across the full electromagnetic spectrum. From
the zoo collection of all of the types of AGN, thanks to a recent study we can
constrain them well. One of the main discrepancies/differences is the angle at
which the observer sees the object. Currently, we are certain that the line of
sight is crucial. The type of AGN is determined among others by the
inclination angle (see Fig.1.1). If we could look closer, the vicinity of the
SMBH (Supermassive black hole) would look like in Fig.1.2 which represents the
SMBH at the center. The blue cloud is the ’hot’ plasma commonly named a
’corona’. Above it is the ’warm’ absorber which might be a part of the corona.
This region is mostly dominated by X-ray emissions. A sub-parsec further one
would see the AD. Looking at the geometry and optical depth, we can
distinguish a few ADs: the geometrically thin (SS\- Shakura-Sunayaev) or thick
(slim) AD (Abramowicz et al. 1988). An optically thin AD is called radiation
inefficient accretion flow (RIAF, Yuan & Narayan, 2004) or advection dominated
accretion flow (ADAF, Narayan & Yi, 1994). Gaseous, high density clouds BLR
(Broad Line Region) ($\sim 10^{10}$ cm-3) are present roughly at the distance
of 0.01 - 1 parsec 1111 pc = 3.0857$\times 10^{16}$ m from the central engine
(Netzer 2013). Subsequently, at a distance of 0.1 - 10 parsec is a dusty
torus, often called the central torus. Moving further one would observe a low
density ($\sim 10^{10}$ cm-3), low velocity ionized gas, the NLR (Narrow Line
Region). This extends from the outside of the torus to a hundred or a thousand
parsec. A few of the AGNs include a central radio jet associated with
$\gamma$-ray emission (Netzer 2015). The jet can be stretched to a few hundred
kpc (Blandford et al. 2019).

Figure 1.1: Types of AGNs including radio, inclination, and power dependency
(Antonucci 2002). Figure 1.2: The zoom picture of the vicinity of SMBH

In the late 1970s, studies of AGNs started, long after the first discovery of
quasi-stellar objects (QSOs) in the 1960s. Three decades before that Carl
Seyfert observed one of the first galaxies. In honor of him, there are two
types of Seyfert galaxies: type 1 and type 2. Nowadays, all objects containing
active SMBHs are referred to as AGNs. The main discrepancy between these two
is in the optical-ultraviolet spectra. Seyfert 1 galaxies have strong and
broad emission lines (2000 - 10 000 $\mathrm{km\,s}^{-1}$), whereas Seyfert 2
galaxies do not exceed a width of 1200 $\mathrm{km\,s}^{-1}$ (Peterson 1997).
Differences in the broadness are assigned the discrepancy in the viewing angle
to the central source. Tab. 1.1 presents the AGN Unification (according to
Table 7.1 Peterson, 1997). The types of AGNs are LIRG – luminous infrared
galaxies; BL Lac – BL Lacertae (rapid and large-amplitude flux variability and
significant optical polarization); BLRG – broad-line radio galaxies; NLRG –
narrow-line radio galaxies; OVV – optically violently variable QSO; FR
(Fanaroff–Riley) class I and II.

Orientation |
---|---
Face-On | Edge-On | Radio properties
Seyfert 1 | Seyfert 2 | Radio Quiet
QSO | LIRG galaxy |
BL Lac | FR I | Radio Loud
BLRG | NLRG |
Quasar/OVV | FR II |

Table 1.1: AGN Unification

> Note: R - radio loudness parameter; the ratio between radio 5GHz to optical
> (B-band = 4400 Å based on Johnson-Morgan system) monochromatic luminosity. L
> (5GHz)/L (4400Å). The dividing line between radio-loud and radio-quiet AGNs
> is usually set at R = 10 (Netzer 2013)

AGNs are a stronger emitter than the nuclei of normal galaxies due to a very
energetic event – accretion onto a central SMBH. They have very high
luminosities (up to $L_{bol}\approx 10^{45}-10^{48}\mathrm{erg\,s}^{-1}$) and
can emit radiation across the whole electromagnetic spectrum. These features
provide an excellent field for the astrophysical research of their nature.

The multi-wavelength study of AGN provides many windows for observing their
properties. The infrared (IR) band is sensitive to obscuring matter and dust.
The optical and UV bands are related to the emission from the AD. The X-ray
emission is assigned mostly to the corona. According to Padovani et al. (2017)
the non-thermal emission is attributed to radio and $\gamma$ emission. Table 1
in Padovani et al. (2017) presents a list of the AGN classes.

The AGNs are some of the biggest objects in the whole Universe, although in
the description of their physical phenomena they are simpler than their size
would indicate. In summary, the previously mentioned features – orientation;
accretion rate; the absence or the presence of jets; the environment within
the host galaxy is enough to describe those behemoths well.

Figure 1.3: AGN spectral energy distribution (SED). HSP - high synchrotron
peaked blazar, LSP - low synchrotron peaked blazar (Padovani et al. 2017).
Each component is represented by different color curves (for better clarity
the authors have shifted it down).

Fig. 1.3 represents a schematic SED  (Spectral Energy Distribution) of an AGN.
Total emission is represented by the black solid curve. The intrinsic shape of
the SED in the mm-far infrared (FIR) regime is uncertain; however, it is
widely believed to have a minimal contribution (to an overall galaxy SED)
compared to SF (Starformation), except in the most intrinsically luminous
quasars and powerful jetted AGN. The primary emission from the AGN accretion
disk peaks in the UV region. The jet SED is also shown for a high synchrotron
peaked blazar (e.g. Mrk 421) and a low synchrotron peaked blazar (e.g 3C
454.3).

###  1.2 The spectrum of AGN

The AGN continuum in the optical-UV range is dominated by accretion processes.
In the simplest case, it is spherically-symmetrical accretion (Bondi 1952).
The geometrically thin and optically thick AD is described by Shakura &
Sunyaev (1973) equations, where matter is accreting onto a non-rotating BH. An
extension of this approach for a rotating BH is the Novikov & Thorne (1973)
approach.

The low-order approximation of the SED of AGNs can be described as a power law
of the form $F_{\nu}\propto\nu^{-\alpha}$, where $\alpha$ called a spectrum
index. The AD is described as a perfect blackbody emission with multi-
temperature components. Each of the components is assigned as a single ring
with a specific temperature. The single ring radiates as a blackbody (see
Fig.1.4).

Figure 1.4: The continuum of the accretion disk. The green line is the sum of
the black lines. Each black line represents the one temperature blackbody.

The short-band (optics/UV) SED of a quasar continuum approximately also takes
the form of power dependency expressed mainly with $0\leq\alpha\leq 1$ from
the observer’s point of view. Dependency of the SED may have the form:
$L_{\lambda}=\lambda^{-\beta}$, where $\beta$ is the wavelength spectral
index, $\beta=2-\alpha$ (Netzer 2013). For example, the observed 1200-6000 Å
continuum of many luminous AGNs is described by $F_{\nu}\approx\nu^{-0.5}$.
This single power law approximation fails for wavelengths below 1200 or above
6000 Å.

In the AGN, bolometric luminosity can be seen mainly from the UV part of the
maximum peak emission called the big blue bump. The inner part of it is
generally agreed to be thermal in origin, although it is not clear whether it
is optically thick (blackbody) or optically thin (free-free) emission
(Peterson 1997). Many proponents of the optically thick interpretation ascribe
the big blue bump to the accretion disk.

The continuum of the object is enriched by the existence of emission and
absorption lines. The mean quasar spectrum of the Large Bright Quasar Survey
(Fig. 1.5) presents prominent emission lines such as Lyman $\alpha$, Si IV, C
IV, Mg II, and Balmer lines. There is a weaker feature superimposed on the big
blue bump between $\sim$ 2000-4000 Å. This feature is attributable to a
combination of Balmer continuum emission and blends of Fe II emission lines
arising in the broad-line region (Peterson 1997). The AGN spectrum contains
thermal and non-thermal emissions. Non-thermal emission comes from particles
whose velocities are not described by a Maxwell-Boltzmann distribution (e.g.
particles which give rise to Compton scattering (corona) or synchrotron power-
law spectra (jets)).

The secondary instance is the emission by gas which receives its energy from
the previous processes and re-radiates the emission. This is the free-free
emission from photoionized or collisionally ionized gas i.e. reprocessing and
reflection (García et al. 2013). Isotropy plays an important role in
determining the type of AGN. Let’s focus on the non-thermal emission of jets.
If a line of sight is close to the axis of the jet, the spectrum might be
dominated by it since the emission is highly beamed in the direction of the
electron motion. However, the total amount of energy in the beam may be far
less than that emitted by thermal processes, which is radiated isotropically.

Figure 1.5: The mean QSO spectrum of 700 objects from the Large Bright Quasar
Survey. Source: Fig 2.Francis et al. (1991), Fig 2.2 Peterson (1997).

The influence of the new direction on the differentiation of AGN types can be
seen in the example of Seyfert galaxies The main observational difference
between Seyfert 1 and 2 galaxies is their different optical-ultraviolet
spectra (Krolik 1999; Netzer 2013). Seyfert 1 shows strong, very broad (2000 -
10000 km s-1) permitted and semiforbidden emission lines, whereas the broadest
lines in Seyfert 2 have widths that do not exceed 1200 km s-1. Such
differences are now interpreted as arising from different viewing angles to
the centers of such sources and from a large amount of obscuration along the
line of sight. Type 2 are heavily obscured along the line of sight that
extinguishes basically all the optical-UV radiation from the inner parsec.
Sources with unobscured lines of sight to their centers are called Type 1.
Véron-Cetty & Véron (2001) adopted classification introduced by Winkler (1992)
accordingly, where $R_{sey}:=\frac{H{\small\beta}}{[OIII]}$:

S 1.0 | 5.0 < $R_{sey}$
---|---
S 1.2 | 2.0 < $R_{sey}$ < 5.0
S 1.5 | 0.333 < $R_{sey}$ < 2.0
S 1.8 |  $R_{sey}$ < 0.333, broad components visible in $H\alpha$ and H$\beta$
S 1.9 | broad components visible in $H\alpha$ but not H$\beta$
S 2.0 | no broad component visible

For example, the broad components are very weak in Seyfert 1.8 galaxies, but
detectable at H$\beta$ as well as $H\alpha$. Indeed, the AGN continuum is
usually so weak in Seyfert 2 galaxies that it is very difficult to
unambiguously isolate it from the stellar continuum.

###  1.3 Weak Emission-lines Quasars (WLQs)

WLQs are an unsolved puzzle in the model of active galactic nuclei (AGN).
There are $\sim$ 200 confirmed sources or candidates (Diamond-Stanic et al.
2009; Shemmer et al. 2010; Hryniewicz et al. 2010; Wu et al. 2011; Plotkin et
al. 2015; Ni et al. 2018; Timlin et al. 2020). The typical EW (Equivalent
Width) of the C 4 emission line is extremely weak ($\lesssim
10{\textup{\AA}}$) compared to normal quasars and very weak or absent in
Ly$\alpha$ emission (Fan et al. 1999; Diamond-Stanic et al. 2009). WLQs are
type 1, radio-quiet quasars with weak or no emission lines; X-ray emission is
also suppressed.

Diamond-Stanic et al. (2009) note that WLQs have optical continuum properties
similar to normal quasars, although Ly$\alpha$+N 5 line luminosities are
significantly weaker, by a factor of 4. An explanation for the weak or absent
emission lines has not been found so far. The results of Diamond-Stanic et al.
(2009) support the idea of WLQs as intrinsically weak UV emission-lines
quasars.

Figure 1.6: Potential vicinity of the WLQ (cigar-shaped area - AD, red clouds
- the BLR, and blue rectangle - ’shielding gas’).

Fig. 1.6 represents the potential image of the vicinity of the WLQ. On the
upper panel of Fig. 1.6 the cigar-shaped area represents AD, red clouds refer
to the BLR, and the blue rectangle is the ’shielding gas’. Below I present
hypotheses that describe approaches to explain the nature of quasars.

The explanation of the nature of the WLQs can be divided into two main
divisions. First, something is happening with the photoionized photons
illuminating the BLR. Photoionized photons (soft X-ray/hard UV) are produced
by the close vicinity of SMBH. Second idea, something is happening with the
BLR itself.

Each of the explanations can still be divided into subsections.

Photoionized photons illuminating the BLR:

  * ✔

an extremely high accretion rate where the continuum is too soft and the
photoionized flux is inefficient (Leighly et al. 2007a, b) (Fig. 1.6 the b
right panel)

  * ✔

a cold accretion disk with a small accretion rate (Laor & Davis 2011)(Fig. 1.6
the b left and b’ lower panels)

  * ✔

Wu et al. (2011) postulate the presence of a shielding gas between the
accretion disk and the broad line region (BLR), which could absorb the high-
energy ionizing photons from the accretion disk (Fig. 1.6 the a panel).

  * ✔

a radiatively inefficient accretion flow (Yuan & Narayan 2004)

The BLR itself

  * ✔

an unusual BLR structure i.e., anemic in construction and/or weak gas
abundance (Shemmer et al. 2010; Nikołajuk & Walter 2012)

  * ✔

the idea that WLQs can also be in an early stage of AGN evolution (Hryniewicz
et al. 2010; Liu & Zhang 2011; Bañados et al. 2014; Meusinger & Balafkan 2014)

Leighly et al. (2007a, b) suggest that the combination of large
$M_{\mathrm{BH}}$ and a high accretion rate might be at plausible explanation
for this weakness, which could be due to the relative deficiency in high-
energy photons in the SED. It is worth pointing out that the photoionized flux
is too soft in energy and not efficient enough to produce strong emission
lines.

In this approach the prominent high-ionization emission line, e.g. C 4 is
suppressed to the low-ionization line H$\beta$ (Shemmer et al. 2010). This
effect is a plausible explanation of the nature of PHL 1811. This is a high
accretion rate quasar with EW(C 4) = 6.6 Å, whereas H$\beta$ is more typical,
but still weak with EW = 50 Å (Leighly et al. 2007a; Diamond-Stanic et al.
2009). Similar to PHL 1811 is PG1407+265 described by McDowell et al. (1995).

Laor & Davis (2011) postulate that the cold AD provides a good explanation for
SDSS J094533.99+100950.1. The ionizing continuum in such an object may
originate in the X-ray power-law component, which produces an extended
partially ionized region in the illuminated gas, which cools mostly through
low-ionization lines. Additionally, I would like to note that this approach
allows them to place an upper limit on the BH spin ($a_{*}\leq 0.3$).

The analogs of PHL 1811 from Wu et al. (2011) confirm previous assumptions.
They have a blue UV/optical continua without detectable broad absorption lines
or dust reddening. PHL 1811 analogs seem to be an X-ray weak subset of WLQs
with strong UV Fe emission and C 4 blueshift. Wu et al. (2011) claim that
unusual UV properties might be explained by the intrinsically X-ray weak SED.
However, a confirming hypothesis is still missing. It cannot be ruled out that
WLQs are heavy X-ray absorption objects. The radio-quiet study of PHL 1811
analogs suggests the connection of very high-ionization absorbers (e.g. O VI)
to the UV and X-ray properties of WLQs.

Luo et al. (2015) measured a relatively hard $\Gamma={1.16^{+0.32}_{-0.37}}$
effective power-law photon index222It is worth noting $N(E)\propto
E^{-\Gamma}$, where N(E) is the amount of photons and E is the energy. For
example $\Gamma\geq 2.0$ indicates soft X-ray source (e.g. Haardt & Maraschi
1991; Cao 2009) for a stack of the X-ray weak subsample, suggests X-ray
absorption. If WLQs and PHL 1811 analogs have very high Eddington ratios, the
inner disk could be significantly puffed up to resemble a slim disk. This
suggests that the shielding gas could be described as a geometrically thick
inner accretion disk that shields the broad-line region from the ionizing
continuum (Fig. 1.7). Shielding of the broad emission-line region by a
geometrically thick disk may have a significant role in setting the broad
distributions of C IV EW and blueshift for quasars more generally (Luo et al.
2015).

Several works about X-ray properties of WLQs have recently appeared (e.g. Wu
et al. 2011, 2012a; Ni et al. 2018; Marlar et al. 2018). The conclusion
arising from these works is that WLQs are more likely to be X-ray weaker
(about half of them) than normal quasars. For example, Ni et al. (2018)
mention that 7 of the 16 WLQs in their sample are X-ray weak. Luo et al.
(2015) suggests that this weakness may be caused by the shielding gas which
prevents the observer from seeing the central X-ray emitting region.

Figure 1.7: A schematic diagram of the geometrically thick disk scenario for
WLQ. PHL 1811 analogs are denoted as X-ray weak WLQ (Luo et al. 2015).

Now, let’s move to the second type, which is the BLR itself. Shemmer et al.
(2009); Nikołajuk & Walter (2012) claim that abnormally broad emission line
region properties – e.g. a significant deficit of line-emitting gas in BLR –
impact the weakness. Nikołajuk & Walter (2012) postulate that the weakness or
absence of emission lines in WLQs does not seem to be caused by their
extremely soft ionizing continuum but rather by the low covering factor
($\Omega/4\pi$) of their BLR, their low-ionization emission lines are weak.
Hryniewicz et al. (2010) hypothesize an early stage of formation of QSO. WLQ
would be a quasar where BLR is just forming. In this case, where the BLR is
just created the covering factor might be lower than in normal AGNs. According
to Nikołajuk & Walter (2012) the low value of $\Omega$ means that the BLR in
the WLQ could have fewer clouds. However, they found that the $\Omega$ of
regions responsible for producing CIV and Ly $\alpha$ are similar in WLQs and
QSOs, whereas the covering factor of high-ionization lines to low-ionization
lines are lower in WLQ than in QSOs (this result was observed only in four
sources of their sample).

As I mentioned above, WLQs have stunning X-ray properties. An X-ray to optical
power-law slope parameter ($\alpha_{OX}$)333$\alpha_{OX}\equiv$
log$(L_{2keV}/L_{2500})$/log$(\nu_{2keV}/\nu_{2500})$ correlates with the
luminosities at 2500 Å in the typical radio-quiet quasar without broad
absorption lines (BALs) (e.g. Shen et al. 2011a). However, about half of the
WLQs have notably lower X-ray luminosities compared to the expectation of
$\alpha_{OX}-L_{2500\textup{\AA}}$ relation (Nikołajuk & Walter 2012; Luo et
al. 2015). These WLQs populations have a high apparent level of intrinsic
X-ray absorption, Compton reflection, and scattering (Ni et al. 2018, 2020).
It is worth noting that the other half of the WLQ population that is not X-ray
weak indicates high Eddington ratios (Luo et al. 2015). The shielding
mechanism represents the thick, inner accretion disk that prevents radiation
to reach the BLR. WLQs generally have not been associated with extreme X-ray
variability before. However, recently Ni et al. (2020) reported a dramatic
increase in the X-ray of SDSS J1539+3954 by a factor of > 20\. At the same
time, they found that the overall UV continuum and emission-line properties do
not show significant changes compared with previous observations.

###  1.4 Estimation of Black hole mass

Knowledge about the values of BH masses and accretion rates is crucial in
understanding the phenomena of accretion flows. The most robust technique is
the reverberation mapping method (RM, Blandford & McKee 1982; Peterson 1993,
2014; Fausnaugh et al. 2017; Bentz & Manne-Nicholas 2018; Shen et al. 2019).
This method is based on the study of the dynamics surrounding the black hole
gas. In this way, we are able to determine the supermassive black hole (SMBH)
mass:

$M_{\mathrm{BH}}=\frac{v^{2}_{\mathrm{BLR}}R_{\mathrm{BLR}}}{G}=f\frac{\mathrm{\textit{FWHM}}^{2}R_{\mathrm{BLR}}}{G}$
(1.1)

where $M_{\mathrm{BH}}$ is the black hole mass, G is the gravitational
constant, $R_{\mathrm{BLR}}$ is the distance between the SMBH and a cloud in
the BLR, and $v_{\mathrm{BLR}}$ is the velocity of the cloud around the SMBH.
This speed is unknown and we express our lack of knowledge in the form of the
FWHM of an emission-line and $f$ – the virial factor, which describes the
distribution of BLR clouds. In the RM the $R_{\mathrm{BLR}}$ is determined as
the time delay between the continuum change and the BLR response. This
technique requires a significant number of observations.

A modification of this method is the single-epoch virial BH mass estimator
(see Shen 2013, for review). The correlation between $R_{\mathrm{BLR}}$ and
the continuum luminosity ($\nu L_{\nu}$) is observed (Kaspi et al. 2005; Bentz
et al. 2009) and incorporated in Eq. (1.1): $R_{BLR}=$coeff.$\times(\nu
L_{\nu})^{0.5}$, where the coefficient may differ depending on the geometry of
the BLR (e.g $32.9^{+2.0}_{-1.9}$ in Kaspi et al. 2000). Thus, the method is
powerful and eagerly used because of its simplicity (Kaspi et al. 2000;
Peterson et al. 2004; Vestergaard & Peterson 2006; Shen 2013; Plotkin et al.
2015; Wang et al. 2019).

The non-dynamic method, which is the spectra disk-fitting method (see chapter
2), is based on a well-grounded model of emission from an AD surrounded black
hole (e.g. Shakura & Sunyaev 1973; Novikov & Thorne 1973). The most important
parameter in such models is the mass of the black hole and the accretion rate.
The spin of the black hole and the viewing angle are also taken into account.
In this technique, the observed SED of an AGN is fitted to the theoretical
model and one can constrain these four parameters (Marculewicz & Nikolajuk
2020). More advanced disk spectra models, which take into account the
irradiation effect, limb-darkening/brightening effects, the departure from a
blackbody due to radiative transfer in the disk atmosphere, and the ray-
tracing method to incorporate general relativity effects in light propagation,
can be fitted (e.g. Hubeny et al. 2000; Loska et al. 2004; Sadowski et al.
2009; Czerny et al. 2011; Laor & Davis 2011; Czerny et al. 2019).

###  1.5 Accretion disk model

The spectrum of the Keplerian disk is a perfect blackbody approximation. The
first approximation of the AD is the disk in the Newtonian gravitational
potential. Energy generated in accretion is transported vertically to the
plane of the symmetry of the disk and radiated outside. The disk flux radiated
at radius $r$ is expressed as:

$F(r)=\frac{3GM\dot{M}}{8\pi r^{3}}(1-\frac{R_{in}}{r}),$ (1.2)

where $\dot{M}$ is the accretion rate, $R_{in}$ the beginning of the disk.
Inner boundary condition for the Schwarzschild ($a_{*}=0$, non-rotating) BH is
$R_{in}=3R_{Schw}$ and for Kerr ($a_{*}\neq 0$, rotating) BH is
$\frac{1}{2}R_{Schw}\leq R_{in}\leq 3R_{Schw}$.

The part in parentheses specifies the internal boundary condition in the
Newtonian approximation. When we consider a case involving a more real
situation, namely the Kerr instance, we consider the local flux Eq. (1.2) with
Novikov & Thorne (1973) correction:

$F(r)=\frac{3GM\dot{M}}{8\pi r^{3}}\frac{L}{BC^{1/2}},$ (1.3)

where:

$\displaystyle B=1+\frac{a_{*}}{x^{3}}$ (1.4) $\displaystyle
C=1-\frac{3}{x^{2}}+2\frac{a_{*}}{x^{3}}$ $\displaystyle
x=\sqrt{\frac{r}{GM/c^{2}}},$

The L is an integral function that depends on the radius of the circular orbit
where the accreting matter is. For a stationary disk the L function takes the
form (Page & Thorne 1974):

$L=\frac{1+a_{*}x^{-3}}{\sqrt{1-3x^{-2}+2a_{*}x^{-3}}}\frac{1}{x}\left[x-x_{0}-\frac{3}{2}a_{*}\ln\left(\frac{x}{x_{0}}\right)-\frac{3(x_{1}-a_{*})^{2}}{x_{1}(x_{1}-x_{2})(x_{1}-x_{3})}\ln\left(\frac{x-x_{1}}{x_{0}-x_{1}}\right)-\frac{3(x_{2}-a_{*})^{2}}{x_{2}(x_{2}-x_{1})(x_{2}-x_{3})}\\\
\ln\left(\frac{x-x_{2}}{x_{0}-x_{2}}\right)-\frac{3(x_{3}-a_{*})^{2}}{x_{3}(x_{3}-x_{1})(x_{3}-x_{2})}\ln\left(\frac{x-x_{3}}{x_{0}-x_{3}}\right)\right],$
(1.5)

where $x_{1},x_{2},$ and $x_{3}$ are the three zeros of the equation
$x^{3}-3x+2a_{*}=0$.

$\displaystyle x_{1}=2\cos\left(\frac{\arccos(a_{*})-\pi}{3}\right)$ (1.6)
$\displaystyle x_{2}=2\cos\left(\frac{\arccos(a_{*})+\pi}{3}\right)$
$\displaystyle x_{3}=-2\cos\left(\frac{\arccos(a_{*})}{3}\right),$

In this approach, we assume that the disk radiated as a perfect blackbody.
Using the temperature relation:

$\sigma T_{eff}^{4}(r)=F(r),$ (1.7)

where $\sigma$ is the Stefan-Boltzmann constant, we are able to describe flux
and temperature dependency.

In the case of a disk as a black-body, the Stefan-Boltzmann law holds, hence
the intensity of the radiation as a function of frequency is expressed as the
Planck spectrum:

$B_{\nu}(T)=\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{\frac{h\nu}{kT_{eff}(r)}}-1},$
(1.8)

Putting Eq. (1.7) into Eq. (1.8) we are able to calculate the $F_{\nu}$ (see
Eq. 1.9). For the AD and the radiation flux that reaches the observer in the
distance ($D$) and the disc observed inclined at an angle i is equal:

$F_{\nu}=\frac{\cos
i}{D^{2}}\int_{R_{out}}^{R_{in}}B_{\nu}\left[T(r)\right]\times 2\pi RdR,$
(1.9)

where $F_{\nu}$ is a monochromatic flux observed on the frequency $\nu$;
$R_{out}$ and $R_{in}$ are the end radius of a disk and the beginning of the
radiated disk respectively. $2\pi r$ because we are integrated rings.

Based on Eq. (1.9) we are able to estimate $M_{\mathrm{BH}}$ $\dot{M}$,
$a_{*}$, and $i$ parameters. For example, in the case $a_{*}=0$ after solving
Eq. (1.9) we get:

$F_{\nu}\sim\nu^{\frac{1}{3}}\times\frac{\cos
i}{D^{2}}\times(M\dot{M})^{\frac{2}{3}},$ (1.10)

Based on it and using the observations we are able to constrain the $M$,
$\dot{M}$, and $cos$ $i$.

##  Chapter 2 Sample selection and data reduction

### Abstract

An unexpected population of quasars display exceptionally weak or completely
missing broad emission lines in the ultraviolet (UV) rest-frame (Plotkin et
al. 2015). There are above 200 objects known as WLQs or WLQs candidates
(Diamond-Stanic et al. 2009; Shemmer et al. 2010; Hryniewicz et al. 2010; Wu
et al. 2011; Plotkin et al. 2015; Ni et al. 2018; Timlin et al. 2020). Only
over a dozen have the estimated masses of a black hole by the single-epoch
method. Many contributing factors were taken into consideration. In this
chapter, I describe the influence of the most impactful ones on the data.

In this thesis I compute luminosity distances using the standard cosmological
model ($H_{0}$ = 70 km s-1 Mpc-1, $\Omega_{\Lambda}$ = 0.7, and $\Omega_{\rm
M}$ = 0.3 Spergel et al. 2007).

###  2.1 Sample selection and data preparation

####  2.1.1 Sample selection

In Sec. 1.3 I described the properties of WLQs and their history. From nearly
200 known WLQ candidates I have chosen 10. The selection was dictated by the
calculation of their masses by the single-epoch virial BH mass method.
According to Shemmer et al. (2010); Hryniewicz et al. (2010); Wu et al.
(2011); Plotkin et al. (2015), only SMBH in 10 of the WLQs were well estimated
by this method.

The sample contains WLQs whose positions cover a wide range of redshift from
0.2 to 3.5 (see Tab. 2.2). Four objects, namely SDSS J083650.86+142539.0
(hereafter J0836), SDSS J141141.96+140233.9 (J1411), SDSS J141730.92+073320.7
(J1417), and SDSS J144741.76-020339.1 (J1447) were analysed by Shen et al.
(2011a) and Plotkin et al. (2015). Three other sources – SDSS
J114153.34+021924.3 (J1141) and SDSS J123743.08+630144.9 (J1237) were studied
by Diamond-Stanic et al. (2009), and SDSS J094533.98+100950.1 (J0945) by
Hryniewicz et al. (2010). The quasar SDSS J152156.48+520238.5 (J1521) was
inspected by Just et al. (2007) and Wu et al. (2011). The number of the above
objects in the sample taken from the SDSS campaign (Abazajian et al. 2009) has
been increased by the two next WLQs: PG 1407+265 and PHL 1811. The first
object named PG1407+265 (hereafter PG1407) is the first observed WLQ in
history and was intensively examined by McDowell et al. (1995). PHL 1811 is
the low redshift source classified also as the NLS1 galaxy (Leighly et al.
2007a, b).

####  2.1.2 Observed data

Observational data was collected from various catalogs. The infrared range was
provided by the Wide-field Infrared Survey Explorer (WISE) and the Extended
Source Catalog of the Two Micron All Sky Survey (2MASS). The first one is a 40
cm diameter telescope in Earth orbit. The second project is related to the
U.S. Fred Lawrence Whipple Observatory on Mount Hopkins, Arizona, and at the
Cerro Tololo Inter-American Observatory in Chile.

The optical data was taken from the Sloan Digital Sky Survey (SDSS) 2.5-m
wide-angle optical telescope located at Apache Point Observatory in New
Mexico, United States. Fluxes in B (4361 Å)1111 Å= 0.1 nm = $10^{-10}m$ and R
(6407 Å) colors were performed by the Dupont 2.5 m telescope at Las Campanas
Observatory (LCO) in the southern Atacama Desert of Chile. The ultraviolet
range was taken from the Galaxy Evolution Explorer (GALEX), an orbiting
ultraviolet space telescope (NUV – 2267 Å, FUV – 1516 Å). Photometric points
of WLQs at visible wavelengths are collected based mainly on the SDSS optical
catalog Data Release 7 (Abazajian et al. 2009). It contains the u (3551 Å), g
(4686 Å) , r (6166 Å), i (7480 Å), and z (8932 Å) photometry. In the case of
PHL 1811, the measurements of fluxes have been based on B and R colors
(Prochaska et al. 2011). The flux at U (3465 Å) band was observed by the UVOT
telescope on-board the Swift satellite (Page et al. 2014). Near-infrared
photometry in the W1-W4 bands have been taken from the WISE Preliminary Data
Release (Wright et al. 2010; Wu et al. 2012b) 222Band W1 –
$3.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$, Band W2 –
$4.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$, Band W3 –
$12\text{\,}\mathrm{\SIUnitSymbolMicro m}$, Band W4 –
$22\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Those data were supplied by
photometry in the J ($1.25\text{\,}\mathrm{\SIUnitSymbolMicro m}$), H
($1.65\text{\,}\mathrm{\SIUnitSymbolMicro m}$), and KS
($2.17\text{\,}\mathrm{\SIUnitSymbolMicro m}$) colors obtained from the 2MASS
(Skrutskie et al. 2006). Crucial points for the project are those detected in
near- and far-ultraviolet (NUV, FUV, respectively) wavelengths. They are
provided by the Galex Catalog Data Release 6 (Bianchi et al. 2017).

I have used the photometric points to cover wide SED. However, the photometric
point, which lays on an emission line, may represent the flux of this line not
of the continuum. Therefore I have used the spectra when possible. I have used
the spectra observed by SDSS, to check photometric data positions in regard to
the spectrum. Eight out of ten WLQs spectra have been taken from the SDSS
catalog. In the case of PHL 1811 and PG 1407+265 the spectra have been taken
from Leighly et al. (2007a) and McDowell et al. (1995), respectively. To check
photometric points, I used continuum fitting windows (see Tab. 2.1 adapted
from Forster et al. 2001). I have taken the spectrum of each WLQ, have set the
range of each window, and have binned the spectral data to get the point
representing the mean flux in the window. This is my new photometric point.
Then I checked whether these points are comparable with photometric points
taken from the mentioned catalogs and added them to the photometric set.

Rest frame wavelength range (Å)
---
Continuum
1140 - 1150
1275 - 1280
1320 - 1330
1455 - 1470
1690 - 1700
2160 - 2180
2225 - 2250
3010 - 3040
3240 - 3270
3790 - 3810
Table 2.1: Continuum fitting windows (Forster et al. 2001)

The spectra and photometric points are in observed frame. Eventually, I have
shifted them to a source rest frame based on wavelength relation:

$1+z_{spec}=\frac{\lambda_{obs}}{\lambda_{emit}}$ (2.1)

where $z_{spec}$ is mainly emission redshift of the source; $\lambda_{obs}$ is
the observed wavelength of a source by telescopes; $\lambda_{emit}$ is emitted
by the source.

Basic observational properties of the sample of WLQs and the sources of their
photometry points are listed in Tab. 2.2.

Table 2.2: Sample of Weak Emission-Line Quasars and the sources of their
photometry points

Name | RA | Dec | $z_{spec}$ | $A_{V}$ | Telescope/satellite
---|---|---|---|---|---
| (degrees) | (degrees) |  |  | (electromagnetic bands)
| (J2000.0) | (J2000.0) |  |  |
(1) | (2) | (3) | (4) | (5) | (6)
SDSS J083650.86+142539.0 | 129.211935 | +14.427527 | 1.749 | 0.129 | WISE (W1,W2,W3), SDSS (u,g,r,i,z), Galex (NUV)
SDSS J094533.98+100950.1 | 146.391610 | +10.163912 | 1.683 | 0.062 | 2MASS (J,H,KS), SDSS (u,g,r,i,z), Galex (NUV,FUV)
SDSS J114153.34+021924.3 | 175.472251 | +02.323508 | 3.550 | 0.065 | WISE (W1,W2,W3,W4), SDSS (u,g,r,i,z)
SDSS J123743.08+630144.9 | 189.429435 | +63.029141 | 3.490 | 0.032 | WISE (W1,W2,W3,W4), SDSS (u,g,r,i,z)
SDSS J141141.96+140233.9 | 212.924908 | +14.042742 | 1.754 | 0.064 | WISE (W1,W2,W3,W4), SDSS (u,g,r,i,z), Galex (NUV,FUV)
SDSS J141730.92+073320.7 | 214.378855 | +07.555744 | 1.716 | 0.084 | WISE (W1,W2,W3,W4), SDSS (u,g,r,i,z), Galex (NUV,FUV)
SDSS J144741.76-020339.1 | 221.924048 | -02.060986 | 1.430 | 0.163 | WISE (W1,W2), SDSS (u,g,r,i,z), Galex (NUV,FUV))
SDSS J152156.48+520238.5 | 230.485324 | +52.044062 | 2.238 | 0.052 | WISE (W1,W2,W3,W4), 2MASS (J,H,KS), SDSS (u,g,r,i,z)
PHL 1811 | 328.756274 | -09.373407 | 0.192 | 0.133 | WISE (W1,W2,W3,W4), 2MASS (J,H,KS), LCO (B,R), Swift (U),
|  |  |  |  | Galex (NUV,FUV)
PG1407+265 | 212.349634 | +26.305865 | 0.940 | 0.043 | WISE (W1,W2,W3), 2MASS (J,H,KS), SDSS (u,g,r,i,z)

> The coordinates (Col. 2 and 3), spectral redshift (Col. 4), and foreground
> Galactic extinction measured at the V color (Col. 5) are taken from the
> NASA/IPAC Extragalactic Database (NED). The column (6) contains references
> to the names of the relevant catalogs and photometric points.

###  2.2 Data reduction

Due to contamination either by internal or external effects such as the dust
in our Galaxy, the influence of the intergalactic medium (IGM), starlight from
the host galaxy, or the dusty torus in the AGN, the observational data has to
be corrected. Let’s imagine a beam of photons that are moving toward an
observer. It has traveled a long way through many obstacles, mediums, and
materials. Firstly, let’s consider the beginning of that travel: the AD
radiate and this light I want to observe. One of the biggest impacts in the
vicinity of the black hole is different absorbers like a molecular torus,
winds, or a warm absorber. During calculation, I have to take into account the
radiation of them. Relatively close to the system is the host-galaxy region,
which I have to take into account as well. Next, the light has to come through
the intergalactic stellar medium.

Last, but not least, is our mother galaxy – the Milky Way and the absorption
in our atmosphere as well, which is mostly taken into consideration within
observations and calibrated before each observation (Cardelli et al. 1989).

####  2.2.1 Dereddening

Dust scatters, absorbs light, and contains heavy elements produced by the
nuclear burning of the stars. It is important to take this into account to get
data as close as possible to the truth. The first correction of the Spectral
Energy Distribution (SED) of all objects was the Galactic reddening with an
extinction law. Interstellar extinction influences the whole range of spectra,
especially in the ultraviolet. The influence of our Galaxy is well understood.
Interstellar medium were studied by Cardelli et al. (1989); Schlegel et al.
(1998); Fitzpatrick (1999); Hutchings & Giasson (2001); Czerny et al. (2004a).
This interstellar reddening can be written as (Zombeck 2007):

$E(B-V):=(m_{B}-m_{V})_{observed}-(m_{B}-m_{V})_{intrinsic}$ (2.2)

Where $B\equiv m_{B}$ and $V\equiv m_{V}$ are the apparent brightness measured
in blue and visible color (e.g. at 4000 Å and 5500 Å respectively in the
Johnson-Morgan system, in magnitude). Equation (2.2) is equal to

$E(B-V)=A_{B}-A_{V}$ (2.3)

where $A_{\lambda}=m_{\lambda}^{observed}-m_{\lambda}^{intrinsic}$ and
$\lambda$ is for B and V.

Based on the Pogson relation we know that each $A_{\lambda}$ is equal to:

$A_{\lambda}=2.5\log\frac{F_{\lambda}^{intrinsic}}{F_{\lambda}^{observed}}$
(2.4)

In order to calculate proper $F_{\lambda}^{intrinsic}$ for an object (e.g.
WLQ), we must know $A_{\lambda}$ of it. Cardelli et al. (1989) express Eq.
(2.3) in the form :

$A_{\lambda}=E(B-V)(a(\lambda)\times R_{V}+b(\lambda)),$ (2.5)

where $a(\lambda)$ and $b(\lambda)$ are functions calculated for each
wavelength $\lambda$.

$R_{V}$ is defined as $A_{V}/E(B-V)$. $a(\lambda)$ and $b(\lambda)$ are given
by the authors. $E(B-V)$ and $A_{V}$ values for a specific direction are taken
from NED 333The NASA/IPAC Extragalactic Database (NED): ned.ipac.caltech.edu
database, based on the dust map of the Milky Way created by Schlegel et al.
(1998).

Generally for a normal region $R_{V}=3.2\pm 0.2$; however, it varies from 2.6
to 5.5 in the measurements of the diffuse ISM (Fitzpatrick 1999).

In Cardelli et al. (1989) the extinction curve has a cutoff at 1250 Å and for
a few photometric points of WLQs I have used $\lambda<1200$Å. Nevertheless,
the extinction law examined in the range of 900-1200 Å seems to follow the
Cardelli et al. law (Hutchings & Giasson 2001). In this way, I extrapolate the
curve down to 900 Å for my FUV photometric points by using the same formula.
$A_{V}$ values in the direction of the examined WLQ are listed in Tab. 2.2.

####  2.2.2 Photoelectric absorption in the IGM

In the case of high-z quasars, the UV fluxes are very sensitive to
photoelectric absorption in the IGM. We do not know the attenuation of the
flux by the IGM along each line of sight. Therefore, following Castignani et
al. (2013), I have used the effective optical depth, $\tau_{\rm eff}(\nu,z)$,
averaged over all possible directions. The correction for the observed
intensity is: $I_{\nu,intrinsic}=I_{\nu,observed}\exp(\tau_{\rm eff}(\nu,z))$,
based on the values of $\tau_{\rm eff}$ collected in Castignani et al. (their
Table 1). Following this formula, I recalculate the fluxes in the Galex FUV
and NUV, the Swift U band, and the SDSS u, g filters. The effective optical
depth in the three other SDSS filters (i.e. r, i, and z) vanishes for the
redshift range considered there. For weak active galactic nuclei such as WLQ,
correction in the UV range is crucial for the calculation of the contribution
from the AGN.

####  2.2.3 Host-galaxy and its starlight

Contamination by starlight plays an important role in the low-luminosity AGNs
(e.g. Greene & Ho 2004; Netzer & Trakhtenbrot 2014). However, the contribution
to the SED from stars in the QSO host galaxy is likely to be negligible (Shen
et al. 2011b; Collinson et al. 2015). This is a common assumption for quasars
at $z>0.5$ (Shen et al. 2011b). According to Shen et al. (2011b) the host
contamination for the luminosity of a quasar is substantial at $log$L(5100 Å)
< 44.5, and becomes negligible toward higher luminosities. The WLQs which I
have used all had luminosities higher than $10^{44.5}$ erg s-1at 5100 Å.

Nevertheless, to check its contribution, I have determined the level of
starlight for each of the objects individually. Following Collinson et al., a
5 Gyr-old elliptical galaxy template444SWIRE Template Library: Polletta et al.
(2007); www.iasf-milano.inaf.it/$\sim$polletta/templates/ which is not AGN,
have been used as the stars’ contamination to the fluxes. I have estimated the
level of starlight in each WLQ using the $M_{\mathrm{BH}}$–$L_{\rm bulge}$
relation (DeGraf et al. 2015), where $L_{\rm bulge}$ is the bulge luminosity
in the V-band, and $M_{\mathrm{BH}}$ means the mass of SMBH in WLQ. The
$M_{\mathrm{BH}}$ of WLQs was taken from previous literature studies (see
Tab.3.2). I assumed that they are correct to estimate the level of starlight
in WLQs, even if their masses are biased due to the inappropriate value of the
FWHM. For this approach only the order of a mass was important.

I have taken the $M_{\mathrm{BH}}$–$L_{\rm bulge}$ relation given by DeGraf et
al. (2015):

$\frac{M_{\mathrm{BH}}}{{\rm
M_{\odot}}}=10^{\alpha}\left(\frac{x}{x_{0}}\right)^{\beta}$ (2.6)

and parameters $\alpha$ and $\beta$ from their Table 2 and Table 5, x =
$L_{\rm bulge,V}$ and $x_{0}=10^{10.5}{\rm L_{\odot}}$. For each of the
objects, I found the best host galaxy starlight contribution to the SED. The
level of starlight for WLQ SDSS J0836 is shown by cyan line in Fig. 2.1. I
have concluded that the host galaxy contribution to the total SED continuum is
small in all the objects.

Starlight has a bigger contribution to infrared WISE points than to the
optical/UV data and adding the contamination of the starlight to that data
helps refine the fitting procedure.

Figure 2.1: The best fit of WLQ SDSS J083650.86+142539.0. Summary of all
components. Black points with errors show photometry data and the grey line
represents spectrum. The violet line shows the contribution of the AD model.
Tori with temperatures 450K and 1400K, respectively, are displayed by orange
and green lines. Cyan line points to the level of starlight. The sum of the
components, and thus the best fit, is shown by the red line.

####  2.2.4 Torus contamination

Torus is the dense, dusty molecular gas present in the tens of parsec away
from a black hole. The contribution of the torus can be prominent for type 2
AGNs (obscured, Seyfert 2). To describe the torus contamination, I have used
one or two single-temperature black-bodies (BB). It describes the thermal
emission of tori visible at IR data, as a hot and cold component (see Fig.
2.1, orange and green lines). I have used the Planck function:

$B_{\nu}(T)=\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{\frac{h\nu}{kT}-1}}$ (2.7)

where $\nu$ is the frequency, T \- absolute temperature, k is the Boltzmann
constant, h \- the Planck constant and c is the speed of light.

This approach describes the contribution of torus well. The fitted effective
temperatures are collected in Tab. LABEL:tab:Temperature, the mean values of
them for the two BB components are 1110 K and 460 K, respectively. Those
values are close to those referred to ’hot’ and ’warm’ BB components in AGNs
(1100-2200K vs. 300-700K); Collinson et al. (2017). The temperatures of the
’hot’ component are also similar to those seen in WLQs
($\mathrm{870\,K<T<1240\,K}$); Diamond-Stanic et al. (2009). Finally, the
photometric points and spectra of all WLQs have been corrected (see Fig. 2.1)
and prepared for the next step of study based on the fitting method.

$2$ Name Temperature [K] J0836 450, 1400 J0945 - J1141 450, 970 J1237 430, 970
J1411 410, 970 J1417 430, 950 J1447 410, 1250 J1521 750 PHL 1811 450, 1100 PG
1407 650, 1240

Table 2.3: Fitted temperature of tori in the sample of WLQs

###  2.3 Sample selection of type 1 QSO and calibration of method

To check if my disk fitting method works concerning WLQs correctly, I am
running a fitting method on a sample of normal type 1 quasars. For this
purpose, I have selected a sample of objects taken from the Large Bright
Quasar Survey (LBQS) (Hewett et al. 1995, 2001). It is one of the largest
published spectroscopic surveys of optically selected quasars at bright
apparent magnitudes. It contains data including the positions and spectra of
1067 quasars. Additionally, Vestergaard & Osmer (2009) give black hole masses
and Eddington accretion rate estimates of 978 LBQS (see their Table 2). The
disk fitting method gives results that we can trust as long as the spectrum in
the ultraviolet is visible. My numerical program needs to see the UV pivot
point in the Big Blue Bump regime of the SED to calculate $M_{\mathrm{BH}}$
and the accretion rate with errors. For this reason, I have chosen 27 quasars
with the presence of a well visible big blue bump. The sample of the quasars
is observed at redshifts between 0.254 and 3.36. Their supermassive black hole
masses are in the range 8.09–10.18 in $\log M_{\rm BH}$ (M⊙) and luminosities,
$\log L_{\rm bol}$ (erg s-1) = 45.25–47.89. The photometric points of the
selected quasars come from the same catalogs mentioned earlier. I have
performed the data correction process in the same way as for WLQs.

###  2.4 Hypothesis testing approach

####  2.4.1 $\chi^{2}$ procedure

In my fitting technique, I have used a $\chi^{2}$ procedure to find the best-
fit model and evaluate the quality of the fit. In order to obtain the best
method for hypothesis testing approach I followed methods that has been
applied in statistical tests. It is based on directly matching the observed
photometric points to the AD model (E, which I will generate). In this
approach, I have calculated
$\chi^{2}=\sum_{i=1}^{n}(O_{i}-E_{i})^{2}/\sigma_{i}$ for SED of each quasar,
$O_{i}$ is the $i$th photometric point of the corrected flux, $E_{i}$ is the
continuum level taken from model. Modeled monochromatic flux $F_{i}$ which
correspond to the $i$th photometric point, $\sigma_{i}$ is the observed error,
and $n$ is the total number of observed data for the quasar (Cochran 1952).
The $\chi^{2}$ test determines if data sample matches with the model. If the
$\chi^{2}$ statistic value is high (e.g. > 10) it means that a data does not
fit with the model, whereas a small $\chi^{2}$ value (< 5) tells us how
observed data fits with a model which I represent. It valid how well
observations one can define by a model description.

####  2.4.2 Bayesian inference

An alternative inference technique is to treat unknown parameters as if they
were random variables. We start with an a priori distribution of the
parameters and then use the collected data and Bayes’ theorem to obtain an
updated a posteriori distribution of the parameters. Instead of determining
the probability associated with the test, we determine the probability of the
parameters themselves. For this reason to use the information on yet
determined data (e.g. black hole masses, accretion rates in quasar sample) one
can carry out an analysis using the Bayesian method.

According to Bayes’ theorem the posterior probability can be assigned as
(Sivia & Rawlings 2009) :

$P(H|D,I)=P(D|H,I)\frac{P(H,I)}{P(D,I)},$ (2.8)

where $H$ is a hypothesis, which could be affected by the data $D$. In our
case the hypothesis is a particulate model $m$. $I$ is any prior information
about $H$. The information $I$ corresponds to the observed values e.g. for the
black hole masses, accretion rates, black hole spins that influence the model.
$P(H,I)$ is the prior probability of an uncertain quantity that express one’s
beliefs before calculation of the posterior probability. The $P(D|H,I)$ is the
probability of observing the measured data $D$ if the hypothesis $H$ was true.
It measures the goodness of a fit of the model to data. In other word it
refers to the likelihood of model $m$. Note that this likelihood function is
an evidence function, whereas the posterior probability is a hypothesis
function. $P(D,I)$ is the probability of $D$ occurring.

To sum up, prior probability, in Bayesian statistical inference, is the best
assessment of the probability of an outcome based on the current knowledge
before a new observation/experiment is performed and collection of new data.
The posterior probability is the revised probability of an event occurring
after taking into consideration new information. The posterior probability is
calculated by updating the prior probability by using Bayes’ theorem. In
statistical terms, the posterior probability is the probability of the event
$H$ occurring given that event $D$ has occurred (Gelman et al. 2004).

##  Chapter 3 Results

###  3.1 Method - Novikov-Thorne model of accretion disk

The primary goal of this work is to fit the SED of quasars by the
geometrically thin and optically thick accretion disk (AD) model described by
the Novikov & Thorne (NT) equations. In the simplest approach, the AD
continuum can be illustrated by the Shakura & Sunyaev model, nevertheless this
attitude does not include a non-zero spin. The solution to this problem has
resulted in the NT equations that I use in my numerical code. The code was
written in the FORTRAN language. Equations (1.3 \- 1.6) were included and the
model output of the continuum of the AD were produced. An example of the model
is shown on Fig. (3.1). As the spin of the black hole increases, the innermost
stable circular orbit (ISCO) decreases and the disk produces more high-energy
radiation. The output continuum of the NT model is fully specified by four
parameters, which I determine. These 4 parameters are: the black hole mass –
$M_{\mathrm{BH}}$, the mass accretion rate – $\dot{M}$, the dimensionless
spin111$a_{\ast}=\frac{cJ}{GM^{2}}$ – $a_{\ast}$, and the inclination – $i$ at
which an observer looks at the AD. The mass of the black hole is expressed in
units of mass of the Sun (${\rm M_{\odot}}$)2221 ${\rm M_{\odot}}$= 1.99
$\times 10^{30}$ kg, and the accretion rate in the form of the dimensionless
Eddington rate, i.e. $\dot{m}$333$\dot{m}:=\frac{\dot{M}}{\dot{M}_{Edd}}$,
where $\dot{M}_{Edd}=\frac{4\pi GM_{\mathrm{BH}}\mu m_{p}}{\sigma_{T}c\eta}$,
where: G - gravitational constant, c - speed of light, $m_{p}$ \- mass of
proton, $\mu$ \- mean molecular weight, $\sigma_{T}$ \- Thomson cross-section
$\eta$ \- radiative efficiency.

Figure 3.1: Example of the Novikov-Thorne model: $M_{\mathrm{BH}}=2.5\times
10^{9},\dot{m}=0.15,a_{*}=0.1,i=0.52.$

I construct a grid of 366000 models of AD, for evenly spaced values of
$M_{\mathrm{BH}}$, $\dot{m}$, $a_{\ast}$, and $i$. The $\log M_{\mathrm{BH}}$
range is from 6.0 to 12.0, the Eddington accretion rate covers the band 0–1,
and the dimensionless spin $0\leq a_{\ast}\leq 0.9$ with the step 0.1. The
inclination is fixed for 6 values that cover a range from 0∘ to 75∘ in steps
of 15∘ (see Tab. LABEL:tab:grid).

Table 3.1: Parameter values for the grid of the AD models Parameter | $\Delta$ | min-max values
---|---|---
$\log$ $M_{\mathrm{BH}}$ | 0.1 | 6–12
$\dot{m}$ | 0.01 | 0–1
$a_{\ast}$ | 0.1 | 0–0.9
$i$ | 15∘ | 0∘–75∘

It is important to determine the radiative efficiency, $\eta$, in the SED
fitting method. There are many approaches to estimate this. The $\eta\simeq
0.057$ is analytical calculation (i.e. taken from theory) for non-rotating BH.
Shankar et al. (2009) suggest $\eta=0.05-0.1$ in relation to AGNs based on
observations of the power emitted by them. Observational constraints on growth
of BHs made by Yu & Tremaine (2002) give us reasonable argument that $\eta$
should be $\gtrsim 0.1$. Even more, Cao & Li (2008) proposed $\eta=0.18$ for
QSOs with BH masses above $10^{9}$ ${\rm M_{\odot}}$.

My performed analysis (i.e. calibration on LBQS sources) allows us to conclude
that $\eta$ should be in the range of $0.15-0.20$. Those values are required
by me to obtain the conformity of SMBH masses in LBQS if I use my SED fitting
and single-epoch virial methods. I adopt the value of $\eta=0.18$ in relation
to both types of quasars – LBQS and WLQs in my PhD thesis.

###  3.2 Description of results

For the initial analysis, I have used 27 quasars from the LBQS survey (see
Sec. 2.3). I have fitted their photometric points to the generated models of
the ADs. I would like to note that I use the same grid of 366,000 models. Fig.
3.2 shows us a comparison of the supermassive black hole masses determined by
Vestergaard & Osmer (2009) – $M_{\mathrm{BH}}^{\mathrm{lit}}$ (on the y-axis),
to those obtained by us – $M_{\mathrm{BH}}$ (on the x-axis). Both masses are
given in mass units of the Sun. The solid violet line is a 1:1 identity line.
Vestergaard & Osmer (2009) used the black hole mass determination based on
their formula (1), which is proportional to FWHM(line) and luminosity $\nu
L_{\nu}$. The compliance of masses and relatively small distribution of errors
means that the continuum fitting method applies to quasars.

Figure 3.2: Comparison of LBQS masses (on the y-axis) with $M_{\mathrm{BH}}$
from my models based on the continuum fitting method (on the x-axis). Violet
solid line is identity 1:1 line.

Next, I switch to the main project task – determining the global parameters of
10 WLQs. The sample of Weak emission-line Quasars contains 10 objects. Fig.
2.1 and 3.4 \- 3.11 show in detail how the fitting procedure works. The
different lines show the individual components: accretion disk, tori,
starlight level (the solid red line shows the sum of those components). Fig.
3.12 presents the best fits of disk continua that match the quasars SED. On
the x-axis of the figures is the logarithmic value of frequency in Hertz,
while on the y-axis is the logarithmic value of $\nu L_{\nu}$ in erg s-1. The
accretion disk continuum is marked with a solid purple line, the photometric
data are shown by black crosses and blue points with errors. The blue points
observed in UV bandpass of 5 WLQs (J0836, J1141, J1411, J1417, and J1447 (Fig.
3.12) and in optical range (J0836) could suggest an absorption seen in some
quasars (e.g. KVRQ 1500-0031 Heintz et al. 2018, SDSS J080248.18+551328.9 Ji
et al. 2015; Liu et al. 2015). The absorption is caused by intrinsic gas in
the host galaxy and/or the greater influence of the assumed UV photoelectric
absorption. The blue points are outliers and for this reason, I model the best
fits in both cases: taking into consideration all points with and without
outliers ($\chi^{2}$ values in parenthesis of Tab. 3.2).

In the next step, I would like to compare the obtained SMBH and $\dot{m}$ with
values from the literature. Satisfactory fits are defined as those showing
reduced $\chi^{2}\lesssim 5.5$. The values of BH masses and the accretion
rates ($M_{\mathrm{BH}}^{\mathrm{lit}}$ and $\dot{m}_{\mathrm{lit}}$,
respectively) of 9 WLQs were collected by different authors (see Tab. 3.2,
Col. 9 for references).

Those values for PG 1407 were determined below because there is a lack of
them. I estimate its SMBH mass based on the equation (7.27) from Netzer
(2013), which states:

$M_{\mathrm{BH}}(H{\small\beta})=1.05\times
10^{8}\left[\frac{L_{5100}}{10^{46}\mathrm{erg\,s}^{-1}}\right]^{0.65}\times\left[\frac{\mathit{FWHM}(H{\small\beta})}{10^{3}\mathrm{km\,s}^{-1}}\right]^{2}{\rm
M_{\odot}},$ (3.1)

The spectrum and the level of continuum at 5100 Å is given by McDowell et al.
(1995). This level is $\nu L_{\nu}=3.16\times 10^{46}$ erg s-1. Unfortunately,
the H$\beta$ emission-line is almost undetectably weak (McDowell et al. 1995).
For this reason, I estimate the FWHM of the Mg 2 line as follows. Using the Fe
2 template taken from Vestergaard & Wilkes (2001), I subtract contribution of
the Fe 2 pseudo-continuum from the Mg 2 presented in the spectrum and fit a
Gauss function to the emission line. The FWHM(Mg 2) calculated in this way is
equal to $4300^{+1400}_{-530}$ km s-1. Next, I have converted the width of the
magnesium to appropriate the hydrogen based on the equation (6) by Wang et al.
(2009):

$\log\left[\frac{\mathit{FWHM}(\mathrm{Mg}\
\text{\small{$\textrm{II}$}})}{1000\mathrm{km\,s}^{-1}}\right]=(0.81\pm
0.02)\times\log\left[\frac{\mathit{FWHM}(H{\small\beta})}{1000\mathrm{km\,s}^{-1}}\right]+(0.05\pm
0.01),$ (3.2)

Thus, FWHM(H$\beta$)$=5400^{+2240}_{-810}$ km s-1. Finally, the calculated BH
mass is $M_{\mathrm{BH}}^{H{\small\beta}}=(2.62^{+2.61}_{-0.73})\times 10^{9}$
${\rm M_{\odot}}$. I take this value as $M_{\mathrm{BH}}^{\mathrm{lit}}$ (Tab.
3.2). Additionally, I have calculated BH mass based on the Mg 2 line following
the similar procedure. I have used equation (7.28) from Netzer (2013):

$M_{\mathrm{BH}}(\mathrm{Mg}\ \text{\small{$\textrm{II}$}})=8.9\times
10^{7}\left[\frac{L_{3000}}{10^{46}\mathrm{erg\,s}^{-1}}\right]^{0.58}\times\left[\frac{\mathit{FWHM}(\mathrm{Mg}\
\text{\small{$\textrm{II}$}})}{10^{3}\mathrm{km\,s}^{-1}}\right]^{2}{\rm
M_{\odot}},$ (3.3)

As a result of Eq. (3.3) I have obtained
$M_{\mathrm{BH}}^{\mathrm{Mg(II)}}=(3.69^{+2.79}_{-0.85})\times\ 10^{9}$ ${\rm
M_{\odot}}$. The accretion rate in PG 1407 is calculated using the
relationship (see equation 2 in Plotkin et al. 2015):

$\frac{L_{bol}}{L_{Edd}}=0.13f(L)\times\left[\frac{L_{5100}}{10^{44}\mathrm{erg\,s}^{-1}}\right]^{0.5}\times\left[\frac{\mathit{FWHM}(H{\small\beta})}{10^{3}\mathrm{km\,s}^{-1}}\right]^{-2},$
(3.4)

where $f(L)$ is the luminosity-dependent bolometric correction; it equals to
5.7 (Shemmer et al. 2010). Eventually, $\dot{m}_{\mathrm{lit}}=0.45$ based on
calculated FWHM(H$\beta$) and luminosity at 5100 Å in PG 1407. The reason for
calculating the $M_{\mathrm{BH}}$ based on H$\beta$ in all WLQs is due to the
fact that, in the WLQ spectrum, neither Mg 2 nor C 4 lines are well built.
Still, weaker H$\beta$ than in normal quasars is a better choice. Due to the
weak emission-lines, the measurement of the proper FWHMvalue of the line might
be biased. It seems that the FWHMvalue of H$\beta$ in the WLQ is biased but
with lower contamination; however, Mg 2 and especially C 4 are very different
compared to the normal, well-build lines in the quasar. The continuum fit
method is an unbiased way to describe the global parameters.

Table 3.2: The best fit parameters

Name | $M_{\mathrm{BH}}$ | $\dot{m}$ | $a_{\ast}$ | $i$ |  $\chi^{2}$/d.o.f | $M_{\mathrm{BH}}^{\mathrm{lit}}$ | $\dot{m}_{\rm lit.}$ | ref.
---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
J0836 | $(1.30^{+5.00}_{-0.30})\times 10^{9}$ | ${0.38^{+0.28}_{-0.25}}$ | ${0.00^{+0.30}_{-0.00}}$ | ${0.26^{+0.52}_{-0.26}}$ | 2.48 (1.91) | $(3.89^{+13.11}_{-1.75})\times 10^{8}$ | ${0.87^{+1.36}_{-0.65}}$ | 1/1
J0945 | $(2.00^{+4.30}_{-0.50})\times 10^{9}$ | $0.47^{+0.19}_{-0.16}$ | ${0.00^{+0.30}_{-0.00}}$ | $0.52\pm 0.26$ | 3.73 | $(1.12^{+0.43}_{-0.19})\times 10^{9}$ | $0.51\pm 0.15$ | 1/1
J0945 |  |  |  |  |  | $(3.08\pm 0.27)\times 10^{9}\dagger$ |  | 2
J1141 | $(6.30^{+13.70}_{-1.30})\times 10^{9}$ | $0.56^{+0.28}_{-0.36}$ | ${0.10^{+0.40}_{-0.10}}$ | $0.26\pm 0.26$ | 4.58 (2.67) | $(3.16^{+1.41}_{-0.97})\times 10^{9}$ |  $0.38^{+0.20}_{-0.10}$ * | 3/3
J1237 | $(5.00^{+6.00}_{-1.80})\times 10^{9}$ | $0.30^{+0.29}_{-0.21}$ | ${0.10}^{+0.60}_{-0.10}$ | ${0.26^{+0.52}_{-0.26}}$ | 6.64 (3.93) | $(2.00^{+1.39}_{-0.65})\times 10^{9}$ |  $0.40^{+0.21}_{-0.16}$ * | 3/3
J1411 | $(2.50^{+0.70}_{-1.20})\times 10^{9}$ | $0.31^{+0.44}_{-0.08}$ | ${0.80^{+0.10}_{-0.40}}$ | $0.26^{+0.52}_{-0.26}$ | 5.58 (1.25) | $(5.25^{+3.87}_{-2.23})\times 10^{8}$ | $0.34^{+0.42}_{-0.17}$ | 1/1
J1417 | $(3.20^{+0.70}_{-1.90})\times 10^{9}$ | $0.54^{+0.19}_{-0.41}$ | $0.20^{+0.30}_{-0.10}$ | $0.00^{+0.52}_{-0.00}$ | 2.88 | $(3.55^{+3.53}_{-2.15})\times 10^{8}$ | $0.92\pm 0.50$ | 1/1
J1447 | $(1.30^{+1.20}_{-6.30})\times 10^{9}$ | $0.35^{+0.41}_{-0.07}$ | $0.80\pm 0.10$ | $0.26\pm 0.26$ | 5.37 (4.41) | $(1.12^{+2.51}_{-1.25})\times 10^{8}$ | $1.30^{+0.17}_{-0.78}$ | 1/1
J1521 | $(2.00^{+0.50}_{-5.70})\times 10^{10}$ | $0.48^{+0.19}_{-0.35}$ | ${0.80^{+0.10}_{-0.20}}$ | $0.52^{+0.26}_{-0.52}$ | 3.68 | $(6.20^{+1.73}_{-1.51})\times 10^{9}$ |  $0.81^{+0.27}_{-0.17}$ * | 4/4
PHL 1811 | $(7.90^{+2.10}_{-3.90})\times 10^{8}$ | $0.34^{+0.50}_{-0.03}$ | $0.00^{+0.10}_{-0.00}$ | $0.00^{+0.26}_{-0.00}$ | 1.87 | $(1.14^{+2.57}_{-2.57})\times 10^{8}$ |  $1.30^{+0.03}_{-0.02}$ * | 5/5
PG 1407 | $(7.90^{+5.10}_{-2.90})\times 10^{9}$ | $0.26^{+0.28}_{-0.09}$ | $0.90^{+0.00}_{-0.10}$ | $0.78\pm 0.26$ | 1.42 | $(2.62^{+2.61}_{-0.73})\times 10^{9}$ |  $0.45^{+0.17}_{-0.23}$ * | a/a
PG 1407 |  |  |  |  |  | $(3.69^{+2.79}_{-0.85})\times 10^{9}\dagger$ |  | a

> Black hole masses, Eddington accretion rates, spins, and fitted inclinations
> are in Col. (2)-(5), respectively. Col. (6) contains the normalized
> $\chi^{2}$ values in two cases: numbers without parenthesis – all
> photometric points are taken into account, numbers in parenthesis – data
> with removed outliers (only black points in Fig. 3.12, 3.12, 3.12) are
> considered. The values in Col. (2)-(5) refer to the case where all points
> are fitted. The values of the parameters in the absence of outliers are the
> same as before within the errors. Black hole masses and accretion rates
> taken form literature are in Col.(7)-(8). They are based on FWHM(H$\beta$)
> measurements (values without $\dagger$). $\dagger$ – $M_{\mathrm{BH}}$ is
> based on Mg 2 line. $M_{\mathrm{BH}}$ and $M_{\mathrm{BH}}$,lit. are in
> units of $M_{\odot}$. * – errors of $\dot{m}_{\rm lit.}$ are estimated by
> us. Numbers refer to articles: 1) Plotkin et al. (2015), 2) Hryniewicz et
> al. (2010), 3) Shemmer et al. (2010), 4) Wu et al. (2011), 5) Leighly et al.
> (2007b), 6) McDowell et al. (1995), a) this work.

The results are collected in Tab. 3.2. The black hole masses, accretion rates,
spins, inclinations of each WLQs, and $\chi^{2}$ values are in Columns
(2)-(6). Degeneration of solutions due to the spin parameter takes place. Two
groups of the best fit for zero and non-zero spin are difficult to distinct
(Sun & Malkan 1989). Thus, I also perform an additional fit with the fixed
$a_{\ast}$ equals 0 (Tab. LABEL:tab:Schwarzschild). In this approach, the
photometric points without outliers are taken into account (only the black
points in Fig. 3.12).

Table 3.3: The Schwarzschild black hole solutions ($a_{\ast}$=0)

Name | $M_{\mathrm{BH}}$ | $\dot{m}$ | $i$ |  $\chi^{2}$/d.o.f
---|---|---|---|---
J0836 | $(1.30^{+5.00}_{-0.30})\times 10^{9}$ | ${0.38^{+0.28}_{-0.25}}$ | ${0.26^{+0.52}_{-0.26}}$ | (1.91)
J0945 | $(2.00^{+4.30}_{-0.50})\times 10^{9}$ | $0.47^{+0.19}_{-0.16}$ | $0.52\pm 0.26$ | (3.73)
J1141 | $(7.90^{+2.10}_{-2.90})\times 10^{9}$ | ${0.84^{+0.03}_{-0.28}}$ | ${1.30^{+0.00}_{-0.52}}$ | (4.11)
J1237 | $(4.00^{+2.30}_{-0.80})\times 10^{9}$ | ${0.30^{+0.06}_{-0.29}}$ | ${0.00^{+0.26}_{-0.00}}$ | (4.54)
J1411 | $(7.90^{+3.40}_{-1.60})\times 10^{9}$ | ${0.50^{+0.03}_{-0.11}}$ | ${0.00^{+0.26}_{-0.00}}$ | (1.43)
J1417 | $(2.50^{+0.70}_{-0.50})\times 10^{9}$ | ${0.66^{+0.19}_{-0.22}}$ | ${1.30^{+0.00}_{-0.26}}$ | (1.98)
J1447 | $(5.00^{+2.90}_{-1.00})\times 10^{9}$ | ${0.49^{+0.20}_{-0.05}}$ | ${0.78^{+0.52}_{-0.26}}$ | (4.96)
J1521 | $(1.30^{+0.70}_{-3.40})\times 10^{10}$ | ${0.86^{+0.11}_{-0.63}}$ | ${1.30^{+0.00}_{-0.52}}$ | (3.99)
PHL 1811 | $(7.90^{+2.10}_{-3.90})\times 10^{8}$ | $0.34^{+0.50}_{-0.03}$ | $0.00^{+0.26}_{-0.00}$ | (1.87)
PG 1407 | $(2.50^{+0.70}_{-0.50})\times 10^{9}$ | ${0.85^{+0.06}_{-0.19}}$ | $0.78\pm 0.26$ | (0.93)

Note. – The normalized $\chi^{2}$ values in parenthesis means the data with
removed outliers

Figure 3.3: The best fit of WLQ SDSS J094533.98+100950.1. Summary of all
components. Black points with errors show photometry data and the grey line
represents spectrum. The violet line shows the contribution of the AD model.
Cyan line points to the level of starlight. The sum of the components, and
thus the best fit, is shown by the red line. Only in this case I did not need
to model tori. A violet and a red lines are very close to each other.

Figure 3.4: The best fit of WLQ J114153.34+021924.3. Summary of all
components. Black points with errors show photometry data and the grey line
represents spectrum. The violet line shows the contribution of the AD model.
Tori with temperatures 450K and 970K, respectively, are displayed by orange
and green lines. Cyan line points to the level of starlight. The sum of the
components, and thus the best fit, is shown by the red line.

Figure 3.5: The best fit of WLQ SDSS J123743.08+630144.9. Summary of all
components. Black points with errors show photometry data and the grey line
represents spectrum. The violet line shows the contribution of the AD model.
Tori with temperatures 430K and 970K, respectively, are displayed by orange
and green lines. Cyan line points to the level of starlight. The sum of the
components, and thus the best fit, is shown by the red line.

Figure 3.6: The best fit of WLQ SDSS J141141.96+140233.9. Summary of all
components. Black points with errors show photometry data and the grey line
represents spectrum. The violet line shows the contribution of the AD model.
Tori with temperatures 410K and 950K, respectively, are displayed by orange
and green lines. Cyan line points to the level of starlight. The sum of the
components, and thus the best fit, is shown by the red line.

Figure 3.7: The best fit of WLQ SDSS J141730.92+073320.7. Summary of all
components. Black points with errors show photometry data and the grey line
represents spectrum. The violet line shows the contribution of the AD model.
Tori with temperatures 430K and 970K, respectively, are displayed by orange
and green lines. Cyan line points to the level of starlight. The sum of the
components, and thus the best fit, is shown by the red line.

Figure 3.8: The best fit of WLQ SDSS J144741.76-020339.1. Summary of all
components. Black points with errors show photometry data and the grey line
represents spectrum. The violet line shows the contribution of the AD model.
Tori with temperatures 410K and 1250K, respectively, are displayed by orange
and green lines. Cyan line points to the level of starlight. The sum of the
components, and thus the best fit, is shown by the red line.

Figure 3.9: The best fit of WLQ SDSS J152156.48+520238.5. Summary of all
components. Black points with errors show photometry data and the grey line
represents spectrum. The violet line shows the contribution of the AD model.
Tori with temperatures 750K displayed by orange line. Cyan line points to the
level of starlight. The sum of the components, and thus the best fit, is shown
by the red line.

Figure 3.10: The best fit of WLQ PHL1811. Summary of all components. Black
points with errors show photometry data and the grey line represents spectrum.
The violet line shows the contribution of the AD model. Tori with temperatures
450K and 1100K, respectively, are displayed by orange and green lines. Cyan
line points to the level of starlight. The sum of the components, and thus the
best fit, is shown by the red line.

Figure 3.11: The best fit of WLQ PG1407. Summary of all components. Black
points with errors show photometry data and the grey line represents spectrum.
The violet line shows the contribution of the AD model. Tori with temperatures
650K and 1240K, respectively, are displayed by orange and green lines. Cyan
line points to the level of starlight. The sum of the components, and thus the
best fit, is shown by the red line.

(a)

(b)

(c)

(d)

Figure 3.12: The best fit of SED to photometric points of 10 WLQs. Black
crosses and blue points with errors show corrected observational data, a grey
line – spectra (optic/UV). The solid violet line represents the theoretical
curve of the AD continuum model.

(e)  (f)

(g)

(h)

Figure 3.12: The best fit of SED to photometric points of 10 WLQs. Black
crosses and blue points with errors show corrected observational data, grey
line – spectra (optic/UV). The solid violet line represents the theoretical
curve of the AD continuum model.

(i)

(j)

Figure 3.12: The best fit of SED to photometric points of 10 WLQs. Black
crosses and blue points with errors show corrected observational data, grey
line – spectra (optic/UV). The solid violet line represents the theoretical
curve of the AD continuum model.

###  3.3 Analysis of results

####  3.3.1 Bayes’ theorem

I determine the best fits based on the $\chi^{2}$ procedure. Additionally, I
would like to estimate the parameters in more sophisticated way using Bayes’
theorem. The posterior probability, $P(H|D,I)$ is calculated using Eq. (2.8).
The hypothesis is a model $m=m$($M_{\mathrm{BH}}$, $\dot{m}$, $a$, $i$). As
mentioned earlier, the model is affected by the data –- the SED and spectra of
quasars. For each model $m$, I derive its likelihood $P(D|m,I)\propto
exp(-\chi^{2}/2)$. The information, $I$, should be $M^{obs}_{BH}$,
$\dot{m}_{obs}$, $a_{obs}$, $i_{obs}$ which are observed. Nevertheless, I do
not have those real parameters and therefore, $I$, is based on earlier
estimated $M^{lit}_{BH}$, $\dot{m}_{lit}$, $a_{lit}$, $i_{lit}$, and their
errors (i. e. $\dot{m}_{obs}=\dot{m}_{lit}$, etc.).

Assuming a Gaussian probability distribution for
$M_{\mathrm{BH}}^{\mathrm{lit}}$ with standard deviations equal to
$\sigma_{M}$, the prior can be written as
$P(H|M_{\mathrm{BH}}^{\mathrm{lit}})\propto\exp{(-(M_{\mathrm{BH}}-M_{\mathrm{BH}}^{\mathrm{lit}})^{2}/2\sigma_{M}^{2})}$.
The prior probability related to $\dot{m}$ takes a similar Gaussian form. We
do not have prior knowledge of either BH spin or inclination. I have assumed a
delta function probability distribution for both parameters.

Finally, the posterior probability is determined for each of the 366,000
models, as the product of the likelihood ($\propto exp(-\chi^{2}/2))$ and the
priors on $M_{\mathrm{BH}}$ and $\dot{m}$ (for details see Capellupo et al.
2015, Appendix A). This is given by:

$P\left(H|D,I\right)=N\times\exp{\left(\frac{-\chi^{2}}{2}\right)}\times\exp{\left(-\frac{(M_{\mathrm{BH}}-M_{\mathrm{BH}}^{\mathrm{lit}})^{2}}{2\sigma_{M}^{2}}\right)}\times\exp{\left(-{\frac{(\dot{m}-\dot{m}_{\mathrm{lit}})^{2}}{2\sigma^{2}_{\dot{m}}}}\right)},$
(3.5)

where $N$ is the normalization constant.

The Bayesian analysis is helpful for saying which model has the highest
probability of explaining the observed SED (Capellupo et al. 2015).

For Bayesian analysis, it is needed to determine the errors of the accretion
rates of those WLQs, which the literature does not provide. I have used the
mentioned Eq. 3.4 (see Plotkin et al. 2015), the upper and lower limits of
FWHM(H$\beta$) and $L_{5100{\textup{\AA}}}$. Errors of
$\dot{m}_{\mathrm{lit}}$ are listed in Tab. 3.2.

The results for the model with the highest posterior probability are shown in
Tab. LABEL:tab:Bayes. I have found that the values of BH masses with a
satisfactory fit are the same using the $\chi^{2}$ or Bayes’ theorem, within
their errors. No significant differences in black hole masses calculated from
both the $\chi^{2}$ and the Bayesian analysis are indicated. This suggests
that the determined global parameters are correct and describe the overall SED
shape of these objects. I take the $M_{\mathrm{BH}}$ calculated from the
$\chi^{2}$ from Tab. 3.2 for further analysis.

Table 3.4: The BH masses from the Bayesian analysis ($a_{\ast}$$\neq$0)

Name |  $M_{\mathrm{BH}}^{\mathrm{Bay}}$ [${\rm M_{\odot}}$] | $\dot{m}_{Bay}$
---|---|---
J0836 | $(2.00^{+4.30}_{-1.00})\times 10^{9}$ | $(0.32^{+0.31}_{-0.19})$
J0945 | $(3.20^{+3.10}_{-1.20})\times 10^{9}$ | $(0.25^{+0.22}_{-0.18})$
J1141 | $(7.90^{+4.10}_{-2.90})\times 10^{9}$ | $(0.39^{+0.45}_{-0.19})$
J1237 | $(6.30^{+3.70}_{-1.30})\times 10^{9}$ | $(0.27^{+0.33}_{-0.26})$
J1411 | $(1.30^{+1.90}_{-1.30})\times 10^{9}$ | $(0.48^{+0.51}_{-0.28})$
J1417 | $(2.50^{+2.50}_{-1.20})\times 10^{9}$ | $(0.32^{+0.41}_{-0.31})$
J1447 | $(1.00^{+1.50}_{-6.00})\times 10^{9}$ | $(0.37^{+0.50}_{-0.23})$
J1521 | $(1.30^{+1.20}_{-5.00})\times 10^{10}$ | $(1.27^{+0.22}_{-0.82})$
PHL 1811 | $(7.90^{+3.10}_{-3.90})\times 10^{8}$ | $(0.40^{+0.14}_{-0.05})$
PG 1407 | $(7.90^{+3.40}_{-2.90})\times 10^{9}$ | $(0.29^{+0.25}_{-0.13})$

Note. – I have calculated spins and inclination but the errors were so large
($\geq$ 60 %) that they were within the entire range of the mesh

####  3.3.2 The $M_{\mathrm{BH}}$ of the thesis vs literature -
$M_{\mathrm{BH}}$ comparison

In order to compare my $M_{\mathrm{BH}}$ and $\dot{m}$ with those values taken
from the literature, Fig. 3.13 shows the mass distribution of black holes.
Identity 1:1 line is marked as a solid purple line. The presented mass
comparison suggests that the literature determinations of black hole masses,
$M_{\mathrm{BH}}^{\mathrm{lit}}$, based on FWHM(H$\beta$), are generally
underestimated. I also determine the difference between
$M_{\mathrm{BH}}^{\mathrm{lit}}$ and my $M_{\mathrm{BH}}$ values. The $\gamma$
factor is calculated
($\gamma:=M_{\mathrm{BH}}/M_{\mathrm{BH}}^{\mathrm{lit}}$). In Fig. 3.14, I
present the relationship between the logarithmic value of FWHM(H$\beta$) in km
s-1 (see Tab. LABEL:tab:fwhm_and_gamma) and the logarithmic value of the
$\gamma$ factor. The green solid line shows the best fit between those
variables. The fit is made using the nonlinear least-squares (NLLS) Marquardt-
Levenberg algorithm, which takes into account errors in both the $x$ and $y$
directions. The relationship is:

$\log\gamma=\left(-1.338\pm
0.366\right)\times\log\left(\frac{\mathit{FWHM}(H_{\beta})}{10^{3}\,\mathrm{km\,s}^{-1}}\right)+\left(1.294\pm
0.234\right).$ (3.6)

Figure 3.13: Comparison of SMBH masses ($M_{\mathrm{BH}}^{\mathrm{lit}}$) of
WLQs based on FWHM(H$\beta$) estimations with $M_{\mathrm{BH}}$, which come
from spectral fitting method. The solid violet line is identity 1:1 line.
Table 3.5: The Full Width at Half Maximum of H$\beta$ emission-line and the
$\gamma$ factor

Name |  FWHM(H$\beta$) [km s-1] | $\log\gamma$ | ref.
---|---|---|---
J0836 | $2880^{+1877}_{-1069}$ | $0.52^{+0.18}_{-0.31}$ | 1
J0945 | $4278\pm 598$ | $0.25^{+0.11}_{-0.16}$ | 1
J1141 | $5900^{+1000}_{-1100}$ | $0.30^{+0.14}_{-0.20}$ | 2
J1237 | $5200^{+1500}_{-1000}$ | $0.40^{+0.17}_{-0.29}$ | 2
J1411 | $3966\pm 1256$ | $0.68^{+0.22}_{-0.44}$ | 1
J1417 | $2784\pm 759$ | $0.96^{+0.27}_{-0.82}$ | 1
J1447 | $1923^{+933}_{-164}$ | $1.06^{+0.18}_{-0.32}$ | 1
J1521 |  $5750\pm 750$ * | $0.51^{+0.36}_{-0.44}$ | 3
PHL 1811 | $1943\pm 19$ | $0.84^{+0.3}_{-0.52}$ | 4
PG 1407 |  $5400^{+2240}_{-810}$ $\sharp$ | $0.47^{+0.17}_{-0.26}$ | a

Note. – $\gamma$ = $M_{\mathrm{BH}}$/$M_{\mathrm{BH}}^{\mathrm{lit}}$. * –
errors are estimated by us. $\sharp$ H$\beta$ is weak and almost not visible,
its FWHM is estimated based on FWHM(Mg 2). Numbers refer to FWHM sources: 1)
Plotkin et al. (2015), 2) Shemmer et al. (2010), 3) Wu et al. (2011), 4)
Leighly et al. (2007b), a) this work.

Figure 3.14: The $\gamma$ factor
($=M_{\mathrm{BH}}/M_{\mathrm{BH}}^{\mathrm{lit}}$) versus FWHM(H$\beta$). The
best fit (i.e. $\gamma\propto\mathit{FWHM}^{-1.34}$) is shown by the solid
green line. Points show data of 10 WLQs. The blue dash line represents the fit
of the virial factor (i.e. $f\propto\mathit{FWHM}^{-1.17}$) to 37 AGNs made by
Mejía-Restrepo et al. (2018a). In my case $\gamma=\mathrm{const}\times f$,
where const. = 0.75. The dash line is shifted down for better visibility.

For a better assessment of my calculations, I have determined the Spearman
coefficient, which is $r_{s}=-0.806$ and the linear correlation coefficient,
$r=-0.82$. It can be seen that three of the ten objects (J0945, J1141, and
J1237) are close to the 1:1 line and the $\gamma$ factor is $\lesssim 2.5$.
The masses of three other sources (PHL 1811, J1417, J1447) should be
multiplied by $\gamma>7$. The mean $\gamma$ factor is 4.7, and the median is
3.3. The dashed blue line in Fig. 3.14 represents the best fit obtained by
Mejía-Restrepo et al. (2018a). They used a sample of 37 Type I AGNs, which lie
in the range of redshifts $\sim 1.5$. Eq. (3.6) allows us to correct the black
hole masses $M_{\mathrm{BH}}^{\mathrm{lit}}$ which has been determined so far
and based on the FWHM values of the H$\beta$ line.

$M_{\mathrm{BH}}=M_{\mathrm{BH}}^{\mathrm{lit}}\times
10^{1.294}\times\left(\frac{\mathit{FWHM}(H{\small\beta})}{10^{3}\mathrm{km\,s}^{-1}}\right)^{-1.1338}$
(3.7)

##  Chapter 4 Discussion

###  4.1 The virial factor and black hole masses in Weak emission-line Quasars

The virial factor (f in Eq. 1.1) is often assumed to be constant, with values
of 0.6-1.8 (e.g. Peterson 2004; Onken et al. 2004; Nikołajuk et al. 2006),
where 3/4 corresponds to the spherical geometry of the BLR (Blandford et al.
1990). Generally, $f$ dependents on non-virial velocity components such as
winds, the relative thickness $(H/R_{\mathrm{BLR}})$ of the Keplerian BLR
orbital plane, the line-of-sight inclination angle ($i$) of this plane, and
the radiation pressure, and should be a function of those phenomenon (Wills &
Browne 1986; Gaskell 2009; Denney et al. 2009, 2010; Shen & Ho 2014; Runnoe et
al. 2014). The analysis carried out by Mejía-Restrepo et al. (2018a) indicates
the low influence of radiation pressure on the $f$ factor; however, this
mechanism cannot be excluded. Whether I omit the influence of radiation
pressure influence or not, the line-of-sight of a gas in a planar distribution
of the BLR plays an important role in calculating black hole mass. The
measured FWHM of emission lines depends on the velocity of the radiated
gas/source. Unfortunately, the nature of the velocity component responsible
for the thickness of the BLR, and thus its geometry, is unclear (e.g. Done &
Krolik 1996; Collin et al. 2006; Czerny et al. 2016; for a recent review, see
Czerny 2019).

My results support those gotten by Mejía-Restrepo et al. (2018a). Instead of
quasars, they study 37 AGNs at redshifts $\sim 1.5$. They based of different
analysis that performed by me find similar behavior. They fitted the AD model
to the spectra of AGNs. Their AD model was the standard, geometrically thin,
optically thick with general relativistic and disc atmosphere corrections.
They examined alternative scenarios, including radiation pressures and
luminosity dependency (for details see, Mejía-Restrepo et al. 2018a). Similar
to my approach, Mejía-Restrepo et al. (2018a) method and modeling is
independent of the BLR geometry and kinematics. The authors indicate the
dependency of the virial factor on the observed FWHM of the broad emission-
line (such as $H\alpha$, $H\beta$, Mg 2, C 4) in the form of an anti-
correlation,

$M_{\mathrm{BH}}\propto f(\mathit{FWHM})\times
R_{BLR}\times\mathit{FWHM}^{2},$ (4.1)

where $f(\mathit{FWHM})\propto\mathit{FWHM}^{-a}$, $a$ is some positive index
(for the H$\beta$ line $a$ = 1.17).

This implies that the BH mass estimations based on the reverberation or the
single-epoch virial BH mass method are systematically overestimated for AGN
systems with larger FWHM (e.g. $\gtrsim 4000$ km s-1 for $H\alpha$) and
underestimated for systems with small $\mathit{FWHM}(H\alpha)\lesssim 4000$km
s-1. It is worth to note that the opposite rule applies to the Eddington
accretion rates (because $\dot{m}\propto M_{\mathrm{BH}}^{-1}$). I have found
a similar underestimation of $M_{\mathrm{BH}}^{\mathrm{lit}}$ values (taken
from the literature) in the sample of WLQs (see Fig. 3.14). The SMBH masses of
WLQs which show FWHM(H$\beta$)$\simeq 5000\ \mathrm{km\,s}^{-1}$ need to be
multiplied by a small factor of 1.5, while the rest of them (i.e.
FWHM(H$\beta$) > 7000 km s-1) requires a larger factor up to 12. This means
that the masses of about 50-60% of WLQs are underweight based on
FWHM(H$\beta$) values. This fact is also seen in Fig. 3.13 where correction of
the SMBH masses based on FWHM estimation is needed. I modify the equation used
to calculate $M_{\mathrm{BH}}^{\mathrm{lit}}$ (Equation 1 in Plotkin et al.
2015):

$\frac{M_{\mathrm{BH}}^{\mathrm{lit}}}{10^{6}{\rm
M_{\odot}}}=5.05\left(\frac{\nu
L_{\nu}(5100)}{10^{44}\mathrm{erg\,s}^{-1}}\right)^{0.5}\left(\frac{\mathit{FWHM}(H_{\beta})}{10^{3}\mathrm{km\,s}^{-1}}\right)^{2}$
(4.2)

using the definition of the $\gamma$ factor ($M_{\mathrm{BH}}=\gamma\times
M_{\mathrm{BH}}^{\mathrm{lit}}$) and Eq. (3.6). The corrected formula for the
SMBH masses in WLQs is:

$\frac{M_{\mathrm{BH}}}{10^{7}{\rm
M_{\odot}}}=(9.94^{+7.09}_{-4.13})\times\left(\frac{\nu
L_{\nu}(5100)}{10^{44}\mathrm{erg\,s}^{-1}}\right)^{0.5}\times\left(\frac{\mathit{FWHM}(H_{\beta})}{10^{3}\mathrm{km\,s}^{-1}}\right)^{0.66\pm
0.37}$ (4.3)

The weaker dependence on FWHM(H$\beta$) (index 2 vs 0.66) can be realized when
the BLR is elongated and parallel to the accretion disk. There is accumulated
evidence in the literature favoring a disk-like geometry for the BLR (Wills &
Browne 1986; Laor et al. 2006; Decarli et al. 2008; Pancoast et al. 2014; Shen
& Ho 2014; Mejía-Restrepo et al. 2018b; Wang et al. 2019). On the other hand,
the BLR may be also dominated by outflows, which are perpendicular to the
line-of-sight. This scenario will favor the idea of quasar reactivation. The
outflow could rebuild the H$\beta$ region (Hryniewicz et al. 2010; Li et al.
2020; Andika et al. 2020). The systematic underestimation of FWHM (and
$M_{\mathrm{BH}}^{\mathrm{lit}}$) may also be caused by the strong influence
of the Fe 2 pseudo-continuum in optics. In this case observed FWHM of line is
smaller. Such phenomena are noticed by Plotkin et al. (2015) for their sample
of WLQs, which have larger $R_{\rm opt,FeII}$ and narrower H$\beta$ than most
reverberation mapped quasars.

Mejía-Restrepo et al. (2018a) dependence of $M_{\mathrm{BH}}$ on the observed
FWHM of the Balmer lines for AGNs is close to linear
($M_{\mathrm{BH}}$$\propto$ FWHM(H$\beta$)0.82±0.11, when $f$ is a function of
the Full Width) rather than quadratic ($M_{\mathrm{BH}}$$\propto$ FWHM2 when
$f=$ const). In my case, this relationship for WLQs is a bit weaker
($M_{\mathrm{BH}}$$\propto$ FWHM(H$\beta$)0.66±0.37, when $f$ depends on
FWHM), but still compatible with the Mejía-Restrepo et al. result within the
1$\sigma$ error. The similar or even same behavior of normal AGNs and WLQs
suggests that both kinds of sources have the same dim nature of the velocity
component and similar geometry of the BLR.

The relationship between the widths of Mg 2 and H$\beta$ emission-lines can be
noticed in the AGNs (see Shen et al. 2008; Wang et al. 2009, Eq.3.2). The
estimation of BH masses based on those lines is consistent with each other.
However, a bias between the estimation of C 4 and Mg 2 mass suggests that the
C 4 estimator is severely affected by an outflow (Baskin & Laor 2005;
Trakhtenbrot & Netzer 2012; Kratzer & Richards 2015). The authors argue that
using H$\beta$ or Mg 2 is better for BH mass estimation than C 4. A similar
note is made by Shen et al., whose results are based on 58,643 quasars from
the SDSS catalog and who claim that the bias may be too large for individual
objects when using a CIV estimator, but it is still consistent with Mg 2 and
H$\beta$ on average. On the other hand, Wang et al. (2009) have found that
FWHM(Mg 2) is systematically smaller than FWHM(H$\beta$), and the BH masses
based on the Mg 2 estimator show subtle deviations from those commonly used.
Referring to WLQs, Plotkin et al. (2015) suggest that using the Mg 2 line
could cause bias to the mass measurements due to the large contribution of the
Fe 2 pseudo-continuum. Thus, in this work, I base the estimations of BH masses
on FWHM(H$\beta$) by trying to avoid C 4 and Mg 2.

In this thesis, I have fixed the value of the radiative efficiency
$\eta=0.18$. However, I would like to check its influence on the results. I
again perform simulations and compare values of the best fits of BH masses and
accretion rates in two cases, when $\eta=0.18$ and $\eta=0.36$. The accretion
rates are 40-70% higher and the BH masses are on average 20-30 % less massive
for higher $\eta$ (for the first order approximation I have
$M_{\mathrm{BH}}^{2}\dot{m}/\eta\simeq$ const). Please note that such a
decline in BH masses cannot explain the underestimation of
$M_{\mathrm{BH}}^{\mathrm{lit}}$ and the choice of solutions with $\eta\nsim
0.18$ increases the $\chi^{2}$.

###  4.2 Inclination dependency

The agreement for LBQS and discrepancy for WLQ quasars between their virial
and SED masses may suggest that FWHM is a biased indicator of the virial
velocity, due to the inclination of the emission region of the H$\beta$ line.
The BLR in WLQs could be less face-on (i.e. seen more in parallel with the
accretion disk) than those in LBQS and other quasars. I compare the
inclinations of 10 WLQs and LBQS quasars in my sample with those derived from
44 SDSS quasars (Wildy et al. 2018), 17 PG quasars, and 7 Seyfert 1 galaxies
(Bian & Zhao 2002). Generally, the main inclinations in LBQS are located in
the range 0-15∘, whereas WLQs are shifted toward higher values with the main
peak at 15-30∘ (Fig. 4.1). I point out the same average inclination values in
SDSS and LBQS quasars. It is worth mentioning that fitted errors are
significant. Thus, inclinations may be the same in all quasars. According to
Collin & Kawaguchi (2004) (see their equation 11
($M_{real}/M_{obs}=1/[(H/R_{BLR})^{2}+sin(i)^{2}]$) and figure 8), the bigger
inclination in WLQs (e.g. $i_{LBQS}=5^{\circ}\rightarrow i_{WLQ}=20^{\circ}$)
together with the assumption of at flat BLR geometry
($H/R_{\mathrm{BLR}}\lesssim 0.3$, where H is the height of the BLR, and
$R_{BLR}$ \- the distance from BH) causes the underestimation of FWHM by a
factor $\gtrsim 2.5$. If we assume that the widths with values in the order of
2000 km s-1are produced in the disk-like geometry (see 3.14), then BH masses
should be 6-10 times heavier. This is comparable with my analysis. Higher
widths, like those in LBQS quasars, could be formed in the BLR with spherical
geometry. In this case, the calculated BH masses do not require correction. It
seems that the flatness of the BLR in weak emission-line quasars plays a more
important role than the inclination.

Figure 4.1: The distribution of inclinations in AGNs. My sample of 10 WLQs and
25 LBQS quasars (the top panel and below). Blue histogram indicates SDSS
quasars (Wildy et al. 2018). The lowest panel shows PG quasars (orange
histogram), 3C and Seyfert 1 galaxies (grey color). The data of them come from
Bian & Zhao (2002).

Collin & Kawaguchi (2004) use equations with factors suitable for normal
quasars which show strong lines and broad FWHM. However, this is not true for
many WLQs. For this reason, both values $M_{\mathrm{BH}}^{\mathrm{lit}}$ and
$\dot{m}_{\mathrm{lit}}$ could have been wrongly calculated.

###  4.3 Shielding gas as an explanation of the nature of WLQs

Shielding of the BLR by a geometrically thick disk is thus perhaps generally
applicable to quasars with lower Eddington ratios (which supports my result,
see Tab. 3.2). Luo et al. (2015) analysis indicate that my results of the
accretion rate values i.e. 0.3-0.6 could be explained as a shielding gas
phenomena of the weak emission-lines. In my opinion, the shielding gas is a
plausible explanation for the WLQs. Apart from the explanation that WLQs are
QSOs in a reactivation phase, along with the development of the BLR, the
shielding gas seems like a good explanation of the nature of WLQs.

Based on our geometrically thick disk scenario for PHL 1811 analogs and WLQs,
these extreme and rare quasars are likely not a distinct population but are
instead extreme members of the continuous population of quasars. The shielding
effect from a puffed-up inner disk likely exists beyond these extreme objects,
and it might be applicable at a milder level to quasars with lower Eddington
ratios. In either the slim-disk model or the Jiang et al. (2014) simulations,
the disk is unlikely to be as thin and flat as what a standard disk model
describes as long as $L_{Bol}/L_{Edd}\leq 0.3$. When the Eddington ratio is
smaller than those of PHL 1811 analogs and WLQs, the radius and scale height
of the puffed-up disk decreases, and its covering factor to the BLR also
decreases, leading to a larger C IV EW than those of the PHL 1811 analogs and
WLQs.

It is likely the blue outliers in the UV range might be related to the
shielding gas (see Fig. 3.12). The absorption is caused by intrinsic gas in
the host galaxy and/or the bigger influence of the assumed UV photoelectric
absorption.

##  Chapter 5 The corona investigation in the high mass SMBH.

I have scoured the Shen et al. (2011c) catalog for the most massive SMBH. The
catalog represents properties of the 105,783 quasars in the SDSS Data Release
7 quasar catalog (Abazajian et al. 2009). It contains the continuum and
emission line measurement of the $H\alpha$, $H\beta$, Mg 2 and C 4.

I have found a few unusual and promising candidates. In the following chapter,
I will describe one of them, SDSSJ110511.15+530806.5 quasar with the abnormal,
broad absorption.

###  5.1 Introducing the SDSS J110511.15+530806.5 quasar.

SDSS J110511.15+530806.5 (thereafter J1105) lays in the Ursa Major
constellation at a distance of $8574.8\pm 600.3$ Mpc from us. Its redshift
($z_{spec}$) equals to $1.9386\pm 0.0015$. Galactic Extinction toward J1105 is
$A_{V}$ = 0.027 mag and $A_{K}$ = 0.003 mag111NASA/IPAC Extragalactic Database
(NED). The quasar’s parameters are listed in Tab. 5.1. The spectrum of J1105
is shown in Fig. 5.1.

Table 5.1: SDSS J110511.15+530806.5 parameters.

RA | Dec | $z_{spec}$ | $A_{V}$ | log$M_{\mathrm{BH}}$ | log $\dot{m}$
---|---|---|---|---|---
(J2000.0) | (J2000.0) |  | [mag] | [log ${\rm M_{\odot}}$] |
(degrees) | (degrees) |  |  |  |
166.296486 | 53.135186 | $1.9386\pm 0.0015$ | 0.027 | $11.205\pm 0.216$ | -4.713

> In Shen et al. (2011a) quasar catalog is postulated that the virial BH mass
> is NOT true mass. They strongly recommend caution and pointed out that it
> may be biased.

Figure 5.1: SDSS J110511.15+530806.5 spectrum. Figure 5.2: J1105 spectrum with
mean QSO spectrum by Richards et al. (2003). Both spectra were correlated
regarding the Mg 2.

###  5.2 The Global Absorption

From Fig. 5.1 and Fig. 5.2 one can see the significant contribution of an
absorption. Therefore, my first attempt concerned the extinction in the Milky
Way as an explanation of the unusual, broad absorption. The first and the
simplest explanation is the Cardelli et al. (1989) extinction law.
Unfortunately, it does not work in such deep absorption. Further, the mean
extinction curve was taken from the Czerny et al. (2004b). Using Czerny’s
code, I was able to include a more sophisticated extinction law. They assumed
different carbon grain and silicate dust temperatures. Their extinction mainly
takes the amorphous carbon grains into account as an explanation of the red
quasars from SDSS. This extinction implicates that the grain radius may
explain various extinction curves for AGNs. Progressing further, I have
rebuilt the code to implement Wickramasinghe & Hoyle’s approach (1998), which
postulated that microdiamonds may imply on the interstellar medium, thus the
excess of the interstellar extinction at the UV can be observed.
Unfortunately, none of the laws of extinction work for the absorption
explanation.

This suggests the existence of intervening systems (e.g. a galaxy or IGM that
lies on the line of sight to the quasar) or intrinsic/associated systems (e.g.
large-scale outflows, accretion wind)(Stone & Richards 2019). In other words,
the second system is related to absorption from a matter associated with the
central engine and/or host galaxy. Additionally, if we focus on absorption
lines alone, we can divide quasars into the broad absorption lines (BALs)
sources associated with outflowing matter, and the narrow absorption lines
(NALs) assigned with the host galaxy or the intervening systems.

The NALs are associated with the EW value of just a few hundred km s-1(e.g.
Wild et al. 2008); BAL QSOs have those values larger (Stone & Richards 2019).
Around 60 % of QSOs spectra prevail the NALs (e.g. Vestergaard 2003), whereas
the BAL are seen in only 20 % (e.g. Knigge et al. 2008). Moreover, Stone &
Richards (2019) suggest the division of NALs with regard to velocity of the
absorbed matter. They distinguish between intervening - regarding the number
density per unit redshift ($dN/dz$) - or intrinsic - regarding the number
density per unit velocity ($dN/d\beta$, $\beta=\frac{V_{abs}}{c}$). The idea
that a NAL is represented by intrinsic or intervening material is still under
debate.

Let’s come back to our quasar. I model the global absorption in J1105 by the
Gaussian function. Fig. 5.3 shows the normalized spectrum with three fitted
Gaussian curves. EW of the top is 561 Å, the middle - 751 Å and the bottom -
1002 Å. $\lambda_{min}$ for all Gaussian functions $\sim$ 1500 Å. The sigma
values are: 280 Å, 300 Å, 400 Å, respectively. The bottom line is the most
plausible fit to the data based on one Gauss function. However, we see that
the multiple Gaussian functions could fit the data in a better way.

Figure 5.3: Comparison of three Gaussian’s, the most bottom is the best. Its
EW is 1002 Å.

###  5.3 The Influence of the Corona

The above considerations about extinction as an explanation of broad
absorption were not satisfied. Moreover, the BAL and NAL do not explain such
global absorption. Therefore I decided to make a more complex exploration
including an influence of the corona and warm skin on the AD continuum (Fig.
5.4).

Figure 5.4: The vicinity of the BH with AD + corona system. The corona
(yellow), AD (green), and warm skin (dark yellow)

The corona (hot medium) has $kT_{e,hot}$ $\sim$ 40 - 100 keV with Thomson
optical depth in the range of 1-2, whereas the warm skin is an optically thick
area with $kT_{e,warm}$ $\sim$ 0.1 - 1 keV and $\tau$ $\sim$ 2-25 (Różańska et
al. 2015; Kubota & Done 2018). The corona/warm skin has interesting
properties. One is that it is a notably ionized medium. It is mainly seen in
the observer’s line of sight towards the source. The observations indicate the
presence of narrow absorption lines and absorption continua in the soft X-ray
(Czerny et al. 2003).

There are two mechanisms concerning warm skin: a back-scattering of the light
that emerges from a disk and the Comptonization inside it (a change in the
spectrum of light due to scattering from electrons) (Nikolajuk et al. 2004).
The concerns of the Czerny et al. (2003) work might be implied for the case of
J1105, namely if the disk is dominated by radiation pressure, the warm skin
has stabilized it.

###  5.4 The Fitting Procedure for J1105

For the next step, I included the influence of the corona and warm skin on the
AD spectrum. Furthermore, I add clouds as a global absorption explanation. One
of the most plausible models to do so is the work of Kubota & Done (2018). The
model is based on the outer standard accretion disk, the inner warm
Comptonising region which produces an excess of soft X-rays, and a hot corona.
This assumption was taken for further examination.

To model the J1105 spectrum I use XSPEC tools, a command-driven, interactive,
X-ray spectral-fitting program, designed to be completely detector-independent
so that it can be used for any spectrometer (Arnaud 1996). The XSPEC package
offers a few dozen models.

In this project I have used my combination of XSPEC models:

  * •

redden*QSOSED*gabs*gabs

  * •

redden*AGNSED*gabs*gabs

The input is the observed spectrum.

The redden model is the reddening in the Milky Way described by Cardelli et
al. (1989). The E(B-V) for J1105 quasar is 0.008. The second component is the
QSOSED/AGNSED (Kubota & Done 2018). It models the intrinsic spectrum of the
QSO. The SED model has three characteristic regions: the outer standard disc
region (AD), the warm Comptonising region (warm skin), and the inner hot
Comptonising region (corona)(see Fig. 5.5). The AD is optically thin and
geometrically thick described by NT equations. The corona is a hot medium, the
skin of the AD is a warm gas. The QSOSED model assumptions are listed below:

Table 5.2: QSOSED model with fixed parameters.

Parameter | Value |  | Parameter | Value |  | Parameter | Value
---|---|---|---|---|---|---|---
$M_{\mathrm{BH}}$(${\rm M_{\odot}}$) | free |  | cos $i$ | free |  |  $R_{hot}$ ($r_{g}$) | **
D (Mpc) | free |  |  $kT_{e,hot}$ ( keV) | 100 |  |  $R_{warm}$ ($r_{g}$) | $=2R_{hot}$
$\dot{m}$ | free |  |  $kT_{e,warm}$ ( keV) | 0.2 |  |  $R_{out}$ ($r_{g}$) | 100000
$a_{\ast}$ | free |  | $\Gamma_{hot}$ | * |  |  $H_{tmax}$ ($r_{g}$) | 100
E(B-V) | free |  | $\Gamma_{warm}$ | 2.5 |  | z | free

> * - $\Gamma_{hot}$ is calculated via Kubota & Done (2018) Eq.(6)
> ** - $R_{hot}$ is calculated to satisfy $L_{diss,hot}$ = 0.02 $L_{Edd}$

where: $kT_{e,hot}$ is electron temperature for the corona; $kT_{e,warm}$ \-
electron temperature for the warm skin; $\Gamma_{warm}$ \- the spectral index
of the warm skin component; $\Gamma_{hot}$ \- the spectral index of the corona
component; $R_{hot}$ \- outer radius of the corona component in $r_{g}$;
222$r_{g}=\frac{GM_{\mathrm{BH}}}{c^{2}}$ $R_{warm}$ \- outer radius of the
warm skin component in $r_{g}$; $R_{out}$ \- the outer radius of the disc in
units of $r_{g}$; $H_{tmax}$ \- the upper limit of the scaleheight for the hot
Comptonisation component in $r_{g}$ unit;

For the warm Comptonising region, this model adopts the passive disc scenario
tested by Petrucci et al. (2018). Here, the flow is assumed to be completely
radially stratified, emitting as a standard disc blackbody from $R_{out}$ to
$R_{warm}$, as warm Comptonisation from $R_{warm}$ to $R_{hot}$, and then
makes a transition to the hard X-ray emitting hot Comptonisation component
from $R_{hot}$ to $R_{ISCO}$. The warm Comptonisation component is optically
thick, so it is associated with the material in the disc.

At a radius below $R_{hot}$, the energy is emitted in the hot Comptonisation
component. This has a much lower optical depth, so it is not the disc itself.
In the model, the albedo is fixed at a = 0.3, and the seed photon temperature
for the hot Comptonisation component is calculated internally. This model does
not take the color temperature correction into account (Arnaud 1996; Kubota &
Done 2018). The geometry of the model is shown in Fig. 5.5.

Figure 5.5: The geometry of the Kubota & Done (2018) model. It represents
geometry for hot inner flow (blue), warm Compton emission (green), and outer
AD (magenta).

The third component gabs*gabs contains two absorptions Gaussian’s and
represents deep absorption in J1105. Each Gaussian function is represented by
three parameters: the line energy in keV, line width (sigma) in keV, and line
depth.

$M(E)=exp\left(-\left(\frac{strength}{\sqrt{2\pi}\sigma}\right)exp\left(-0.5\left(\frac{(E-E_{line})}{\sigma}\right)^{2}\right)\right)$
(5.1)

###  5.5 Results

The fitting of the spectrum of J1105 by redden*QSOSED*gabs*gabs model and the
parameters are presented below.

The free parameters in the QSOSED model are black hole mass, accretion rate,
the spin of the black hole, and the inclination angle. Additionally, the
Gaussian positions were free throughout the modeling. The Tab. 5.4 shows the
best fit parameters for modeling QSOSED with two Gaussian functions. Fig. 5.6
represents the best fit of the J1105 using QSOSED by the XSPEC. The reduced
$\chi^{2}$ for the fit is: 2.80 with 15 degrees of freedom.

Table 5.3: The fitting model redden*QSOSED*gabs*gabs parameters for J1105. Model | Parameter | Value
---|---|---
redden | E(B-V) | 0.08 (fixed)
QSOSED |  $M_{\mathrm{BH}}$(${\rm M_{\odot}}$) | $(3.524\pm 0.012)\times 10^{9}$
$\dot{m}$ | $0.276\pm 0.002$
cos $i$ | $0.836\pm 0.008$
$a_{*}$ | $0.35\pm 0.02$
distance (Mpc) | 5075 (fixed)
redshift | 1.929 (fixed)
normalization | $0.4299\pm 0.0014$
First Gaussian | Line E(keV) | $(2.791\pm 0.012)\times 10^{-3}$
$\sigma$ (keV) | $(5.840\pm 0.065)\times 10^{-4}$
strength | $(4.697\pm 0.096)\times 10^{-3}$
Second Gaussian | Line E(keV) | $(2.125\pm 0.022)\times 10^{-3}$
$\sigma$ (keV) | $(3.875\pm 0.120)\times 10^{-4}$
strength | $(7.023\pm 0.365)\times 10^{-4}$
Figure 5.6: The fit of the SDSSJ110511.15+530806.5. Top panel: the best fit of
J1105 using the redden*QSOSED*gabs*gabs model. Bottom panel: residuum of
$\chi^{2}$. The $\chi^{2}$/ndf = 2.80.

It is worth noting that I have tried a similar approach by using only one
Gaussian function to describe the deep absorption. However, the reduced
$\chi^{2}$ was not promising and the values were very high (a few hundred).

In the next step, I have improved the fit. The QSOSED model is limited. I have
used the AGNSED code to improve the fit and to have free parameters describing
the corona, the warm skin, and their respective arrangements (Tab. 5.4).
Regarding the free parameters of the corona, warm skin, and disk I was able to
constrain the nature of the deep absorption well. The reduced $\chi^{2}$ for
the fit is: 2.056 for 10 degrees of freedom.

Table 5.4: The fitting model redden*AGNSED*gabs*gabs parameters for J1105.

Model | Parameter | Value
---|---|---
redden | E(B-V) | 0.08 (fixed)
AGNSED |  $M_{\mathrm{BH}}$(${\rm M_{\odot}}$) | $(3.522\pm 0.010)\times 10^{9}$
$\dot{m}$ | $0.274\pm 0.001$
cos $i$ | $0.841\pm 0.003$
$a_{*}$ | $0.36\pm 0.02$
$kT_{e,hot}$ ( keV) | 100 (fixed)*
$kT_{e,warm}$ ( keV) | $0.356^{+4.876}_{-0.350}$
$\Gamma_{hot}$ | $1.300^{+1.545}_{-1.300}$
$\Gamma_{warm}$ | $2.722\pm 0.153$
$R_{hot}$ ($r_{g}$) | $58.54\pm 3.36$
$R_{warm}$ ($r_{g}$) | $430.0^{+50.0}_{-11.0}$
$r_{out}$ ($r_{g}$) | 100000 (fixed)
$H_{tmax}$ ($r_{g}$) | = $R_{hot}$
distance (Mpc) | 5075 (fixed)
redshift | 1.0 (fixed)
| normalization | $0.5063\pm 0.0017$
First Gaussian | Line E(keV) | $(2.792\pm 0.015)\times 10^{-3}$
$\sigma$ (keV) | $(5.842\pm 0.069)\times 10^{-4}$
strength | $(4.686\pm 0.098)\times 10^{-3}$
Second Gaussian | Line E(keV) | $(2.125\pm 0.022)\times 10^{-3}$
$\sigma$ (keV) | $(3.874\pm 0.117)\times 10^{-4}$
strength | $(7.025\pm 0.367)\times 10^{-4}$

> * - at the beginning this parameter was free. However, it had large errors
> (>200 %). For this reason, I followed Kubota & Done (2018) and fixed this
> parameter according to the author’s assumptions
> NOTE: reprocess is ON

Figure 5.7: The fit of the SDSSJ110511.15+530806.5. Top panel: the best fit of
J1105 using the redden*AGNSED*gabs*gabs model. Bottom panel: residuum of
$\chi^{2}$. The $\chi^{2}$/ndf = 2.056.

For both approaches QSOSED and AGNSED (as a core of the modeling) I have used
the same distance to the quasar and E(B-V) parameters. I was able to make a
better fitting procedure in the second case and obtained the reduced
$\chi^{2}$ = 2.056. Not much improvement regarding the global parameters of
the objects was observed.

##  Chapter 6 Discussion

###  6.1 Calculations for the Potential Absorption Source

The origin of such abnormal deep absorption in the J1105 is interesting. In
this paragraph, I have focused on two possible scenarios for the origin of the
absorber. I have calculated two of the most plausible explanations. I have
presented the two Gaussian function descriptions (Fig. 6.1). The values of
each of them were converted from the XSPEC fitting to wavelength in rest frame
interpretation. The Gaussian function observed in the rest frame at 1516 Å is
the one that dominates the fit. However, all two of them are needed for the
goodness of fit.

Figure 6.1: The illustration of the two Gaussian’s, with $\sigma=332$ Å, EW = 714 Å; $\sigma=375$ Å, EW = 124 Å. Table 6.1: Comparison of the observed frame energies and sigmas (keV) establish from the fitting of Gaussian functions with converted wavelength to the rest frame in Å. |  | observed frame (keV) | rest frame (Å)
---|---|---|---
First Gaussian | Line E | $(2.792\pm 0.015)\times 10^{-3}$ | $1516\pm 10$
$\sigma$ | $(5.842\pm 0.069)\times 10^{-4}$ | $332\pm 69$
Second Gaussian | Line E | $(2.125\pm 0.022)\times 10^{-3}$ | $1992\pm 20$
$\sigma$ | $(3.874\pm 0.117)\times 10^{-4}$ | $375\pm 64$

###  6.2 How far from the galaxy is the absorption cloud?

The minima of the Gauss functions (Fig. 6.1) for J1105 in the rest frame are
at $1516\pm 10$ Å, $1992\pm 20$ Å with $\sigma$ $332\pm 69$ and $375\pm 64$,
receptively. If we assume that the global absorption in J1105 is caused by an
extended cloud that absorbs Mg 2 (2796 Å, 2803 Å), then from equation
$z=\frac{\lambda_{obs}}{\lambda_{em}}-1=\frac{\lambda_{abs}}{2800\small\textup{\AA}}-1$
we are able to establish its $z_{1,\mathrm{MgII}}=-0.4585$,
$z_{2,\mathrm{MgII}}=-0.2885$. This is the distance of this cloud/absorber
from the host galaxy. However, if we assume that the cloud absorbs C 4 (1548
Å, 1551 Å), then its $z_{1,\mathrm{CIV}}=-0.0213$,
$z_{2,\mathrm{CIV}}=+0.2859$.

Figure 6.2: A scheme of potential absorption Mg 2 clouds in front of the
quasar Figure 6.3: A scheme of potential absorption C 4 clouds

####  6.2.1 If We Assume the Wind is the Absorber

An absorption line in the spectrum of a NAL or BAL quasar is redshifted or
blueshifted according to its lab position. Assuming the shift is caused by the
Doppler effect and the absorber is a large-scale outflow, we can estimate the
velocity of the absorbing material.

$\nu_{obs}=\nu_{em}\sqrt{\frac{1-\beta}{1+\beta}},$ (6.1)

where $\beta=\frac{V}{c}$ and $V$ is velocity of a wind, then after
rearrangement:

$\lambda_{em}=\lambda_{abs}\sqrt{\frac{1-\beta}{1+\beta}}=>\frac{1-\beta}{1+\beta}=\left(\frac{\lambda_{em}}{\lambda_{abs}}\right)^{2}$
(6.2)

From definition of redshift we know that $\lambda_{obs}=\lambda_{em}(1+z)$
therefore:

$\frac{1-\beta}{1+\beta}=\frac{1}{(1+z)^{2}}=>(1-\beta)(1+z)^{2}=(1+\beta)$
(6.3) $(1+z)^{2}-1=\beta(1+(1+z)^{2})$ (6.4)

and final equation for $\beta$ looks like:

$\beta=\frac{(1+z)^{2}-1}{(1+z)^{2}+1}$ (6.5)

The final parameters for this section look like:

| redshift | $\beta$ | V (km/s)
---|---|---|---
Mg II | $-0.4585\pm 0.0032$ | $-0.3605^{+0.0018}_{-0.0021}$ | $-108135^{+555}_{-615}$
$-0.2885\pm 0.0071$ | $-0.2482^{+0.0051}_{-0.0053}$ | $-74447^{+1543}_{-1577}$
CIV | $-0.0213\pm 0.0058$ | $-0.0211^{+0.0055}_{-0.0057}$ | $-6330^{+1698}_{-1710}$
$+0.2859\pm 0.0013$ | $0.2463^{+0.0094}_{-0.0096}$ | $73887^{+2827}_{-2871}$
Table 6.2: The parameters of winds of Mg 2 and C 4

The negative velocity indicates that the wind is blowing in our direction.

Stone & Richards (2019) classified the Quasar absorption lines systems in view
of their velocity distribution. Structures within $V\leq$ 3000 km s-1 are
called associated absorption lines. Systems with velocity separation 3000 - 12
000 km s-1 are dominated by intrinsicality or outflows from the QSO. Systems
with velocity $\geq$ 12 000 km s-1 are disconnected from the QSO (see
reference therein).

According to Stone & Richards (2019), our first case (i.e. Mg 2 absorption
clouds) points out the intervening system. In the second case (i.e C 4), the
outflow is the first Gaussian with $V=-6330^{+1698}_{-1710}$ km s-1.

###  6.3 The column density of J1105

In this section, I have estimated the ion column densities, $N$, of the
absorbers. Recent results of Saez et al. (2021) about wind properties in PG
2112+059 was taken into account. Equation (9), (10), and (12) in Saez et al.
(2021) indicated the relation of EW and $N_{j}$ for the single Gaussian.
However, in the case of J1105, I have to use two doublets of absorption,
namely the C 4 and Mg 2 lines. Using their Equation (12) for line doublet I
was able to calculate $N_{j}$ of each Gaussian function in the optically thin
regime for a fixed value of the ionization parameter and metallicity.

$N_{j}\approx\frac{m_{e}c^{2}}{\pi
e^{2}(f_{b}\lambda^{2}_{b}+f_{r}\lambda^{2}_{r})}EW,$ (6.6)

where b and r stands for parameters of the blue and red doublet respectively.
$f$ is an oscillator strengths taken from Griesmann & Kling (2000); Morton
(2003). The results are presented in the Tab. 6.3.

|  | log$(N_{j})$ | log$(N_{j})$
---|---|---|---
| EW(Å)∗ | C 4($cm^{-2}$) | Mg 2($cm^{-2}$)
First Gaussian | $714^{+145}_{-147}$ | $17.027^{+0.080}_{-0.010}$ | $16.003^{+0.080}_{-0.010}$
Second Gaussian | $124^{+20}_{-21}$ | $16.266^{+0.065}_{-0.080}$ | $15.242^{+0.065}_{-0.080}$

Table 6.3: The column density of J1105

> * EW in the rest frame

Saez et al. (2021) indicate that the logarithmic values of column densities of
CIV in BAL QSO PG 2112+059 vary during 13 years from 15.38 to 15.81. If they
assumed the covering factor $\sim$ 60 % then log $N_{CIV}$ = 16.12.

Dobrzycki et al. (2007) have determined the ion column densities of the
individual components in HS1603+3820 quasar. For a mini-BAL system in this
quasar, the mean C 4 column density is $15.265\pm 0.135$. For system B, the
mean Mg 2 column density is $15.106\pm 0.155$. The C 4 column densities vary
through systems of the quasar from $13.200\pm 0.035$ to $15.471\pm 0.085$.

###  6.4 Final remarks

In Fig. 6.4 I present the potential idea of an accretion disk and warm skin
concept, which might be seen in J1105. Czerny et al. (2003) represent the
concept of the bright AGN with a warm skin above the AD. A similar concept of
warm skin, which wraps around the inner part of AD is presented here.

Figure 6.4: My view how the vicinity of the J1105 might look like

The AGNSED is a more sophisticated code with the same approach as QSOSED, but
with all the free parameters presented in Tab. 5.4. However, as a first order
approach, the warm skins from Czerny et al. (2003); Nikolajuk et al. (2004)
and Kubota & Done (2018) model, look good enough to examine the nature of
J1105.

Primary I have fitted the QSOSED model with two Gaussian functions. The first
Gaussian is the most dominant. The second seems to be crucial. The QSOSED
model assumes the fixed parameters of the corona, the warm skin, and the disk
size. I have chosen the AGNSED model with free parameters to get to know J1105
nature better. Moreover, I tried to find the best $kT_{e,hot}$ parameters. In
this case, I have also tried many variants. It turned out that the best value
is fixed to 100 keV, same as Kubota & Done (2018) approach. I was able to make
an even better fit procedure and obtained the reduced $\chi^{2}$ from 2.800
to: 2.056. Not much improvement regarding the global parameters of the objects
was observed. SMBH from AGNSED is $M_{\mathrm{BH}}$$\simeq 3.5\times
10^{9}{\rm M_{\odot}}$, wheres $M_{\mathrm{BH}}^{\mathrm{lit}}$$\simeq
1.6\times 10^{11}{\rm M_{\odot}}$ (Shen et al. 2011c).

In the first order of approximation of the modeling procedure, I have fitted
only one Gaussian function, the reduced $\chi^{2}$ was 7.0912. It seems that
this object in particular needed more functions that described the absorption.
Another Gaussian’s were needed. However, when I have tried to narrow the
Gaussian’s down I obtained a not well fitted model. The reduced $\chi^{2}$ of
the worse fit was > 10.

Besides the improved value of the reduced $\chi^{2}$, one of the most crucial
take away is that the AGNSED provide a better understanding of the physics and
boundaries of the corona, the warm skin, and the disk size. The mentioned
parameters are within the assumptions of Czerny et al. (2003) and Kubota &
Done (2018) approaches (for more details please see their Table 1 and Table 2
respectively).

In this section, I would like to compare parameters from the literature. I
would like to remind the $\dot{m}$ of J1105 is equal to $0.274\pm 0.001$. The
corona/warm skin concept of Mrk 359 by Czerny et al. (2003) indicate the
$\dot{m}$= 0.3. Additionally, Kubota & Done (2018) studied three close by
objects ($z<0.16$). One of them PG 1115+407 has a similar accretion rate
$\sim$ 0.4.

Moreover, the $\Gamma_{warm}$ = $2.722\pm 0.153$ of J1105 is similar to one
obtained for PG 1115+407; $\Gamma_{warm}$ = 3.06 and Mrk 359 with
$\Gamma_{warm}$ = 3.32.

Comparing study with QSOSED vs AGNSED as a main component of modeling, I have
found that the $kT_{e,warm}$ = $0.356^{+4.876}_{-0.350}$ (keV). In Czerny’s
the value is fixed at 0.2 keV and Kubota & Done (2018) give $kT_{e,warm}$ 0.50
keV (fixed). Regarding the corona/warm skin size, J1105 has $R_{hot}$ =
$58.54\pm 3.36$ $r_{g}$ and $R_{warm}$ = $430.0^{+50.0}_{-11.0}$ $r_{g}$,
comparing to $R_{warm}$ of Mrk 359 = $300R_{Schw}$ ($R_{Schw}\sim 2r_{g}$).

The difference may be caused due to the divergence between $M_{\mathrm{BH}}$.
Czerny et al. (2003) and Kubota & Done (2018) objects do not contain such
massive SMBH as J1105. In their studies, the $M_{\mathrm{BH}}$ are $\sim$
$10^{8}{\rm M_{\odot}}$. However, the physics and corona/warm skin/disk
description should not vary very much. Nevertheless, the description of the
corona/warm skin for J1105 is well established and comparable with literature
studies.

Contrary to the massive black holes, Zhang et al. (2000) modeled stellar BH:
GRS 1915+105 and GRO J1655-40. Their study revealed a three-layered
atmospheric structure in the inner region of AD. They found out a warm skin
with T = 1.0-1.5 keV and an optical depth $\sim$ 10\. They sum up, the
temperatures of the cold disk of $\sim 10^{6}K$, the warm skin $\sim 10^{7}K$,
and corona $\sim 10^{9}K$ indicated the complicated phenomena around such
objects (i.e. corona/warm skin explanation). However, they focused on the AD
of the stellar mass black holes. GRO J1655-40 has a black hole $\sim 7$ ${\rm
M_{\odot}}$, GRS 1915+105 $\sim 12$ ${\rm M_{\odot}}$.

Meusinger et al. (2016) study of J1105 indicated that it has reddening steeper
than in the Small Magellanic Cloud, perhaps even steeper than for IRAS
14026+4341 galaxy. They could not either find a good reddening/extinction
solution for the case of J1105. They classify it as a strongly reddened
narrow-line quasar. Jiang et al. (2013) argued that such an unusual reddening
law is based on a speculative assumption of an exotic dust grain size
distribution that lacks large grains.

My result indicated that the deep absorption in J1105 could be explained by
two Gaussian functions which may describe two different clouds. The C 4 as an
absorbing material is more plausible due to the fact that $\lambda_{min}$ of
the main Gaussian is 1516 Å whereas the laboratory line of C 4 lies at 1548Å
and 1551 Å. The clouds or even a wind contains mostly C 4 which absorbs
radiation from an AD/corona of the J1105. The most dominant Gaussian with
$\sigma=375$ Å has a strength of nearly 90% of the whole recess. Regarding the
redshifts of the potential clouds, one can see that for Mg 2 case two clouds
along the line of sight of observer occurred. Surprisingly in the C 4 case
(Fig. 6.3) one of the clouds is ’behind’ the J1105! It seems that it is the
wind component or Failed Radiatively Accelerated Dusty Outflow (FRADO)
observed by Czerny’s group. This wind falls to the AD.

Using Weymann et al. (1979) approach, Stone & Richards (2019) distinguish
three components of the velocity distribution of NALs: the uniform
distribution of very high velocity components which is unassociated with the
QSO; the influence of the virialized material clustered around the QSO itself,
or associated with material ejected from the QSO. Regarding the values from
Sec. 6.1, the plausible explanation of the origin of the absorption in J1105
is the strong outflow represented by the first Gaussian function and the
intervening material e.g. dwarf galaxy along the line of sight seen as the
second Gaussian curve in the spectrum. Nevertheless, the intervening material
may explain both Gaussian functions. Regarding my results, I cannot exclude
either NAL or BAL phenomena as an explanation of the nature of J1105. However,
such wide absorption has not been observed in the sources of BALs (C 4 column
density $\sim$ 17 cm${}^{-}2$). Further investigation was needed. The
explanation of the abnormal properties of J1105 might lay at the root of the
warm skin concept.

In summary of the nature of the SDSSJ110511.15+530806.5 quasar I have
presented two visual explanation of this phenomena (Fig. 6.5, Fig. 6.6).

This chapter will be concluded as a paper soon. I have found more objects with
very high $M_{\mathrm{BH}}$ and abnormal features with their spectra. This
work is in progress and I will examine it further.

Figure 6.5: An explanation of the nature of J1105 - winds. Figure 6.6: An
explanation of the nature of J1105 - two dwarf galaxies.

##  Chapter 7 Conclusion

I have studied the accretion disk continua of 10 WLQs. The SMBH masses of
those objects were estimated previously based on the single-epoch virial
$M_{\mathrm{BH}}$ method ($M_{\mathrm{BH}}^{\mathrm{lit}}$). Generally, the RM
and the single-epoch virial BH mass method are inadequate for BH mass
estimation in WLQs, due to the weakness of emission-lines in these objects. I
have created a grid of 366,000 accretion disk models using the Novikov–Thorne
formulas. The phenomena of the WLQs were expressed by four parameters
($M_{\mathrm{BH}}$, $\dot{m}$, spin of BH, and the line-to-sight inclination)
to describe the observed SED.

My main findings are as follows:

  1. 1.

Using the pure Novikov–Thorne model, we can describe very well the SED of
WLQs,

  2. 2.

I have compared obtained BH masses with those obtained from the literature.
The SMBH masses of WLQs, which are estimated based on FWHM(H$\beta$), are
underestimated. On average, the masses are undervalued by 4–5 times. The
median of this correction factor is 3.3,

  3. 3.

I have proposed the new formula to estimate $M_{\mathrm{BH}}$ in WLQs based on
their observed FWHM(H$\beta$) and luminosities at 5100 Å (Equation 4.3). This
equation helps to calculate proper weight of BH masses,

  4. 4.

My results suggest that selected WLQs have accretion rates in the range of
$\sim$ 0.3–0.6 (see Tab. 3.2),

  5. 5.