# Dust formation in AGN winds

Arkaprabha Sarangi  Observational Cosmology Lab, NASA Goddard Space Flight
Center, Mail Code 665, Greenbelt, MD 20771, USA  CRESST II/CUA/GSFC
<EMAIL_ADDRESS>Eli Dwek  Observational Cosmology Lab, NASA
Goddard Space Flight Center, Mail Code 665, Greenbelt, MD 20771, USA  Demos
Kazanas  Gravitational Astrophysics Lab, NASA Goddard Space Flight Center,
Mail Code 663, Greenbelt, MD 20771, USA

###### Abstract

Infrared observations of active galactic nucleus (AGN) reveal emission from
the putative dusty circumnuclear “torus” invoked by AGN unification, that is
heated up by radiation from the central accreting black hole (BH). The strong
9.7 and 18 $\mu$m silicate features observed in the AGN spectra both in
emission and absorption, further indicate the presence of such dusty
environments. We present detailed calculations of the chemistry of silicate
dust formation in AGN accretion disk winds. The winds considered herein are
magnetohydrodynamic (MHD) winds driven off the entire accretion disk domain
that extends from the BH vicinity to the radius of BH influence, of order
$\sim 1-100$ pc depending on the mass of the resident BH. Our results indicate
that these winds provide conditions conducive to the formation of significant
amounts of dust, especially for objects accreting close to their Eddington
limit, making AGN a significant source of dust in the universe, especially for
luminous quasars. Our models justify the importance of a $r^{-1}$ density law
in the winds for efficient formation and survival of dust grains. The dust
production rate scales linearly with the mass of the central BH and varies as
a power law of index between 2 to 2.5 with the dimensionless mass accretion
rate. The resultant distribution of the dense dusty gas resembles a toroidal
shape, with high column density and optical depths along the equatorial
viewing angles, in agreement with the AGN unification picture.

ISM: dust, extinction; stars: black holes, winds, outflows; galaxies: quasars:
general

††software: CLOUDY (Ferland et al., 1998)

##  1 Introduction

The presence of dust in galaxies has a profound effect on their spectral
appearance. Dust manifests itself in the absorption and scattering of
starlight, in the production of interstellar polarization, X-ray haloes, and
echoes around variable sources, and in the depletion of refractory elements
from the ISM gas (Li, 2005; Draine, 2003; Zubko et al., 2004). On a
cosmological scale, dust reprocesses a significant fraction of all
gravitational and nuclear energy releases throughout cosmic history into IR
radiation, generating the cosmic IR background (Hauser & Dwek, 2001; Dwek &
Krennrich, 2013).

Despite the many manifestations of dust, its nature, origin, and evolution in
galaxies are still poorly understood. For instance, a detailed study of the
nearby Magellanic Clouds revealed that the dust formation rate in the ejecta
of core collapse supernovae and the quiescent winds of low mass stars is
significantly lower than its destruction rate by blast waves from supernova
remnants. Therefore, the presence of dust in the LMC suggests that we are
unable to account for the total amount of dust observed in the galaxy (Temim
et al., 2015). In local galaxies, where the primary sites of dust formation
are known to be quiescent winds from the asymptotic giant branch (AGB) stars,
the disparity of the rates between dust and star formation implies also the
necessity of an additional source of dust (Dwek & Cherchneff, 2011).

A large mass of dust ($\sim$ 108 M⊙) is reported to be present in a number of
high redshift ($z>6$) quasars (Bertoldi et al., 2003), which implies a rapid
rate of dust formation in the short $\simeq 0.7$ Gyr time span between the
reionization redshift, $z_{r}\simeq 17$ and $z\simeq 6$. Supernovae, resulting
from rapidly evolving supermassive stars, are considered to be the primary
source of refractory elements and dust in the early universe.

Elvis et al. (2002) proposed that dust may be produced in outflows from the
broad line region of an AGN, noting that the conditions (density and
temperature) in the AGN Broad Line Region (BLR) clouds are similar to those in
the winds of AGB stars, where most of the galactic dust is thought to be made.
From this argument and the thermodynamic conditions of AGN outflows they
estimated a dust production rate of $\sim$ 10-2 M⊙ yr-1. More recently,
Maiolino et al. (2006) invoking the Elvis et al. (2002) estimates, suggested
that they could account for all dust in the high redshift (z = 6.4) quasar
SDSSJ1148+52, while Pipino et al. (2011) argued that dust formation would only
be important in the earlier state of AGN life.

Besides their potential role as dust sources at high redshifts, local AGN are
known to contain significant quantities of dust, considering that structures,
referred to as “dusty tori”, are thought to be ubiquitous features of the AGN
morphology, closely related to the unification of Seyfert 1,2 (Antonucci,
1993) and the Broad and Narrow Line Radio Galaxies (Barthel, 1989). According
to AGN unification, these distinct classes represent single entities,
distinguished only by their angle of inclinations with respect to the viewer’s
line of sight. Dust is presumably present in toroidal structures obstructing
direct view of the AGN central regions for observers at sufficiently large
inclination positions (Mason, 2015).

The sizes of the “dusty tori” ($R\sim 1$ pc) are much too small to resolve
angularly, even for the closest AGN. Spectroscopically, the presence of dust
in the galaxies with an AGN has been well documented over the past couple of
decades (Almeida & Ricci, 2017). The fact that the AGN IR luminosity is
similar to that of UVO - X-ray one, in low inclination objects where the
central engine is directly visible, implies that the tori subtend a
significant fraction of the continuum source luminosity. For observers at
higher inclination, the obscuration of the central regions may bring most of
the observed luminosity to the IR. In this respect, several radiative transfer
models in the past couple of decades have focused on the study of infrared
emission and absorption from this dusty toroidal gas as a function viewing
angle (Pier & Krolik, 1992; Rowan-Robinson, 1995; Fritz et al., 2006; Hönig et
al., 2006; Stalevski et al., 2012).

Silicates are known to be the primary constituent of dust in the AGN torus
(Stenholm, 1994; Srinivasan et al., 2017). In the last decade, Spitzer Space
Telescope has detected strong emission features of silicate dust present in
quasars (Hao et al., 2005; Siebenmorgen et al., 2005; Sturm et al., 2005). For
the Seyfert 2 NGC 1068, silicate features have been reported in absorption
(Mason et al., 2006) as well as emission (Lopez-Rodriguez et al., 2018a, b).
Detection of emission features in a type 2 AGN could indicate to a possible
clumpy nature of the gas in the outflows (Elitzur, 2007; Tristram et al.,
2007; Nenkova et al., 2008).

The dust present in the AGN tori can be pre-existing dust from the ISM, newly
formed dust in the winds or a combination of both. The detection of the 9.7
and 18 $\mu$m silicate features, reported in the AGN spectra (Shi et al.,
2006), are characteristic to the tetrahedral Si-O bonds which naturally form
in a high temperature condensation sequences. In as much as the formation of
these bonds requires high activation energy, then these emission features
suggest the in situ formation of silicates in the wind. Also, the
crystallinity of the dust grains in AGNs is found to be higher compared to the
upper limit in the local ISM (Sturm et al., 2005; Srinivasan et al., 2017),
arguing in the favor of the formation of new dust grains in the AGN.

In spite of recent developments in the field, the structure and origin of the
AGN tori is still uncertain, as is the origin of dust in them. Their most
vexing issue is their large vertical extent, of height $H$ comparable to their
distance to the AGN center $R$. Under hydrostatic equilibrium this implies
random velocities comparable to the Keplerian one ($V_{K}$) at distance $R\sim
1-10$ pc, $V_{K}\sim 300-1000$ km/s (for a BH mass $M\simeq 10^{8}M_{\odot}$),
which is in stark disagreement with their low gas and dust temperatures
($10-100$ K) that imply velocities $\langle V^{2}\rangle^{1/2}\sim 1$ km/s.
According to certain models (Hopkins et al., 2012) the dusty tori represent
gas inflowing from the galaxies’ larger radii. Alternatively, it has been
suggested that they represent MHD winds outflowing from the outer regions of
the AGN accretion disks (Konigl & Kartje, 1994; Elitzur & Shlosman, 2006;
Keating et al., 2012), and therefore are not subject to the constraints of
their internal velocities, implied by assuming hydrostatic equilibrium.

The MHD wind-view is strengthened by the discovery that approximately 50% of
AGN exhibit UV and X-ray _blue-shifted_ absorption line features (Crenshaw et
al., 2003), indicating the generic presence of outflows in AGN. Of particular
interest is the absorption features of the AGN X-ray spectra, because they
span a much wider range of ionization states thereby probing a wider range of
outflow conditions. At the same time, because of the high ionization states of
their plasma, they preclude winds driven by line radiation pressure akin to
those of O-star winds. These facts, therefore, argue in favor of MHD driven
winds (Blandford & Payne, 1982; Contopoulos & Lovelace, 1994). These winds,
due to their self-similar structure, span a large range of radii (and hence
possibly plasma ionization states) and do not rely on radiation pressure for
their launch. The study of the polarization of dust emission in Cygnus A also
supports the magnetic field dominated wind structure (Lopez-Rodriguez et al.,
2018b).

The wide range of ionization states of X-ray absorbers (equivalently the large
distance of absorbing plasma from the ionizing source of radiation), provides
a means of estimating the plasma density along the observers’ line of sight
(George et al., 1998), by measuring the absorbing column of each transition.
Such analyses by Holczer et al. (2007) and Behar (2009) showed that the wind
density profile is close to $n(r)\propto r^{-1}$ rather than $r^{-2}$,
generally expected in a radiation pressure driven wind. Furthermore, it should
be noted that these shallow density profiles are consistent with the winds
proposed by Contopoulos & Lovelace (1994) rather than those of Blandford &
Payne (1982) which imply $n(r)\propto r^{-3/2}$. The photoionization structure
of these MHD winds were modeled by Fukumura et al. (2010, 2018b) and was shown
to be consistent with the propositions of AGN unification.

Motivated by these developments we study the formation of silicate dust in AGN
MHD accretion disk winds. With a well defined background model for the MHD
winds that represents the AGN tori, we formulate a simple, generic dust
evolution model to study the chemistry of dust formation in these winds. Our
aim is to quantify the following: (a) the pathways of silicate dust formation
in the outflows, (b) the two dimensional distribution of dust density and
temperatures, (c) the dust grain sizes, (d) the dust column density present
along various line of sight, (e) the rate of silicate dust formation and (f)
the global dusty structure and dust formation budget of the associated AGN
tori.

The paper is arranged in the following order. In Section 2, we discuss the
geometry and input constraints on the physical model of the winds, along with
the parameters related to the central source. Following that, in Section 3 we
discuss the evolution and distribution of the gas temperature (Section 3.1),
the dust temperature (Section 3.2) and the characteristics of different
regions in the winds (Section 3.3), that are relevant to dust formation. The
rate and mechanism of nucleation of molecules and dust precursor clusters are
presented in Section 4. The estimated density of dust grains and formation of
the torus is explained in Section 5. The distribution of grain sizes, and the
dust to gas mass ratio is calculated in Section 6, while the total dust mass
and the optical depths are discussed in Section 7. In Section 8 we show how
the calculated results depend on the variation of the mass accretion rate and
the mass of the BH. Finally in Section 9, we summarize our findings and
discuss the dependencies of the model on various physical parameters.

Figure 1: Left-panel: A schematic representation of the helical outflow of the
gas from the accretion disk, controlled by the magnetic field of the central
BH, is shown in the figure. Assuming azimuthal symmetry, the projection of the
motion of the wind in the x-z plane is shown in blue. The arrowheads indicate
the direction of flow. For a more accurate presentation of this figure, we
refer to Fukumura et al. (2018a) (Fig. 3). Right-panel: The projection of the
accretion disk (AD) is shown along with the outflowing winds (in blue) in the
upper-hemisphere. The motion of the outflows along this plane can be
represented by parabolic equations of motion. The luminosity of the system is
attributed to a central X-ray UV source. The dotted line (in red) represents
the sublimation radius and the dust-free zone.

##  2 The MHD wind model

The MHD wind model consists of a Keplerian, geometrically thin disk around a
supermassive BH of mass $M_{\rm BH}$, threaded by a poloidal magnetic field.
The Keplerian disk rotation induces a toroidal field component and an ensuing
magneto-centrifugal outflow of the disk plasma (Blandford & Payne, 1982). The
axisymmetry of the problem allows the treatment in 2.5D, $i.e.$ the spatial
description in the 2D poloidal plane, while keeping track of all three
velocity components.

The mathematical details of the problem can be found in Contopoulos & Lovelace
(1994) and Konigl & Kartje (1994), while more detailed applications to the
ionization structure of the resulting wind and to X-ray observations can be
found in Fukumura et al. (2010). The field treads the disk across its entire
extent, which spans many decades in radius, and so does the wind’s lateral
extent. This is consistent with the the X-ray observations that demand such a
large lateral wind size (Behar et al., 2003; Holczer et al., 2007; Fukumura et
al., 2010), and justifies the self-similar treatment of the MHD wind models
implemented in the above references (Contopoulos & Lovelace, 1994; Konigl &
Kartje, 1994).

Following that, we consider the presence of an underlying, steady-state, self-
similar magnetic field driven MHD wind with an $r^{-1}$ density law, along any
given line of sight, while also maintaining a steep dependence on the
inclination angle. These are consistent with the compilation of density
profiles obtained by Behar (2009), while we followed the models of these winds
detailed in by Fukumura et al. (2010, 2018b).

Figure 2: The luminosity of the central source $L_{\nu}(\theta)$ is presented
for an inclination of $\theta=75^{\circ}$, defined in Equation (2). The shape
of the spectra is determined by $\alpha_{\mathrm{{ox}}}$ = -1.5. For all the
inclination angles, the shape of the spectra remain unchanged whereas the
total luminosities vary with $f(\theta)$. The material around the AGN is
solely heated by the radiation from the central source.

These winds can either be dusty or dust free in nature. In this study, we have
assumed that the outflows start with an initial dust-free gas of solar
composition. In reality, the actual chemical composition of the outflowing
winds is not well known, as it depends on several complex factors related to
the nature of the stars around the accretion disk and the underlying physics.
We then follow the physical conditions and chemical evolution of the gas along
the flow. Our choice of an initially dust free composition sets an upper limit
on the production rate of new dust in AGNs.

The formation of dust can be affected by the possibility of clump formation in
the flows, as indicated by similar studies of supernova ejecta by Sarangi &
Cherchneff (2015). However, in order to keep our models sufficiently simple
and manageable, considering the complexity of addressing the chemistry in a 2D
geometry, we have assumed a homogeneous flow with $r^{-1}$ density law over
the entire region. The possible effect of clumpiness is discussed in the
discussion.

The outflow in the form of winds from the accretion disk is shown as a
schematic diagram in Figure 1. Figure 1(left-panel) shows the upper hemisphere
of the accretion disk with the azimuthally symmetric helical winds flowing
outwards from the disk-plane. The blue lines in the figure show the wind-path
and flow-line along the 2D xz-plane. Figure 1(right) further illustrates the
flow-lines in 2D geometry with various starting or launch radius on the
surface of the accretion disk. The polar coordinates ($r,\theta$) of a given
point and the axes are shown on Figure 1(right-panel), where $r$ is the radial
distance from the centre of the accretion disk and $\theta$ is the inclination
angle with the z-direction.

As detailed in Contopoulos & Lovelace (1994), the poloidal field lines can be
approximated in a self-similar model by a set of homothetic parabolas (Fig. 1
right-panel) with the gravitating center (BH) at their focus. The similarity
is broken by the BH horizon and the Innermost stable circular orbit (ISCO) at
radii a few times the Schwarzschild radius $R\rm_{sch}$ ($R\rm_{sch}\sim
3\times 10^{5}$ $M_{0}$ cm; $M_{0}=M_{\rm BH}/M_{\odot}$). However, herein we
consider radial scales many orders of magnitude larger than $R\rm_{sch}$,
warranting the self-similar treatment of the problem. The poloidal flow of the
plasma then takes place along each such line. Their detailed shape of these
flow lines is given by the solution of the corresponding transfield balance
equations (the Grad-Safranov equation). In the present work we refrain from
solving explicitly these equations and present only approximate kinematics and
geometry, based on the numerical integration of the equations given in
Fukumura et al. (2010).

In this 2D geometry the particles at $\theta\mathrel{\raise
2.15277pt\hbox{$>$}\mkern-14.0mu\lower 2.58334pt\hbox{$\sim$}}75^{\circ}$ move
almost in Keplerian fashion, with their motion becoming progressively
poloidal, especially beyond the Alfvén point to follow the parabolic magnetic
field lines (Blandford & Znajek, 1977; Meier, 2012). Placing the central
source at the focus (of these parabolas), we consider the following parabolic
equation for their shape,

$r=r_{0}/(1-cos\theta)$ (1)

where, $r_{0}$ is the radius where any of these parabolas (magnetic field
lines) crosses the accretion disk and denotes the radius at which the wind
plasma flow is launched along each such field line.

The luminosity of the central UV-Xray source is given in terms of the
Eddington luminosity as,

$L(\theta)=\eta\dot{m}L_{\mathrm{{Edd}}}(M_{8})f(\theta)$ (2)

where, $\eta$ ($\sim$ 0.1-1) is the radiative efficiency of an accreting BH,
$\dot{m}$ is the normalized mass accretion rate given by
$\dot{m}=\dot{M}/\dot{M}_{\mathrm{Edd}}$, $L_{\mathrm{Edd}}$(M8) and
$\dot{M}_{\mathrm{Edd}}$ (=$L_{\mathrm{Edd}}$/$c^{2}$) are the Eddington
luminosity and Eddington accretion rate for a 108 M⊙ BH, $M_{8}$ is the mass
of the central BH normalized by 108 M⊙ and $f(\theta)$ is the anisotropic
coefficient. For $M_{8}$ = 1, $L_{\mathrm{Edd}}$(M8) = 1.3$\times$1046 ergs
s-1. The radiative efficiency was taken as 0.3. The anisotropic coefficient,
describing the angular distribution of radiation emitted by a thin disk are
given by (Netzer, 1987; Stalevski et al., 2016),

$f(\theta)=(1/3)\cos\theta(1+2\cos\theta).$ (3)

where $\theta$ is the inclination angle $\theta$ shown in Figure 1.

The calculations presented in the following sections assume a BH mass of
$10^{7}$ M⊙ or $M_{8}=0.1$, the radiative efficiency $\eta\sim 0.3$ and the
wind-mass rate is equal to the accretion on the BH, $i.e.$, $\dot{m}$ = 1
(Fukumura et al., 2010). The Eddington luminosity for a 107 M⊙ BH is $\sim$
1.3 $\times$ 1045 ergs s-1. Because the wind mass flux depends on the radius
(generally increases with increasing disk radius), we assumed that its mass
flux at the smallest disk radius, $R_{\rm ISCO}\simeq 3R_{\rm sch}$, is equal
to the mass accretion rate onto the BH.

We assume that the material around the AGN is solely heated by the radiation
from the central source. The shape of the spectrum employed in our
calculations is given in Fig. 2, properly normalized for the inclination angle
of 75∘. The AGN spectra are usually characterized by a power law with an index
$\alpha_{\mathrm{{OX}}}$ between 2100 Å and 2 keV ($\sim 0.2$ and
6$\times$10-4 $\mu$m). The values of $\alpha_{\mathrm{{OX}}}$ range between
$-1$ and $-2$; the spectrum presented in Fig. 2 has
$\alpha_{\mathrm{{OX}}}=-1.5$. For other inclination angles, the same shape of
the source spectra is retained and the effective luminosity is calculated
using $L(\theta)$ from Equations 2 and 3.

Figure 3: The colormap of physical quantities in polar ($r,\theta$)
coordinates are shown. The lines (in black) on the figure represents the
parabolic flow-lines with various starting/launch radius on the accretion
disk, $r_{0}$. The accretion disk is located along the $\theta=90^{\circ}$
plane, $i.e.$, the lower-horizontal axis of the figures. Top-left: Gas
velocities in the z-direction in terms of the Keplerian velocities (VK) are
presented (Equation (LABEL:eq_vz)). Top-right: The time (in years) taken by
the gas to reach a given distance on the $r$-$\theta$ plane is presented
(Equation (LABEL:eq_time)). Bottom-left: The distribution of gas gas density
$n(r,\theta)$ in polar coordinate is presented, assuming solar composition of
the gas. Bottom-right: The colormap shows the gas temperature in polar
($r,\theta$) coordinates, assuming solar composition of the gas. The line in
yellow demarcates the region lower than 4000 K, in the $r,\theta$ plane, where
formation of molecular clusters and seeds are supported. Importantly to note
however, the formation of dust grains from these seeds happens only when the
temperature is lower than 2000 K.

Our region of interest encompasses a cylindrical volume above and below the
accretion disk bound by the gravitational sphere of influence of the central
BH. The radius of influence ($R_{\mathrm{infl}}$) is approximately the
distance on the accretion disk where the total mass of the stars equals the
mass of the central BH (Thomas et al., 2016) and is determined by the velocity
of dispersion ($\sigma$) of the stars on the disk (Gebhardt et al., 2000;
Ferrarese & Merritt, 2000; McConnell & Ma, 2013). It is given by the equation,

$R_{\mathrm{infl}}\sim R_{\mathrm{sch}}(c/\sigma)^{2}$ (4)

where, $R_{\mathrm{sch}}$ is the Schwarzschild radius of the central BH and
$c$ is the speed of light. We take $\sigma$ = 200 km s-1 for this study
(Forbes & Ponman, 1999). An inclination of $\theta=20^{\circ}$ is taken as the
upper edge of the cylindrical volume as the wind have already crossed the
Alfvén point at the altitude (Fukumura et al., 2010). Beyond the corresponding
latitude, its $v_{z}$ becomes super-Keplerian (Fukumura et al., 2010), and as
such, we do not expect dust formation beyond this point. For a 107 M⊙ BH,
$R_{\mathrm{sch}}$ $\sim$ 2.94 $\times$ 1012 cm. Therefore,
$R_{\mathrm{infl}}$ $\sim$ $r_{\mathrm{0,max}}$ $\sim$ 6.6 $\times$ 1018 cm.
The outer radius of this cylindrical volume is $R_{\mathrm{max}}$ $\sim$
$R_{\mathrm{out}}$ $\sim$ $r_{\mathrm{0,max}}$/$sin\theta_{\mathrm{min}}$ =
1.9 $\times$ 1019 cm.

The fluid velocity at low latitudes (close to the plane of the disk) is mainly
in the toroidal direction at the Keplerian value, with the toroidal magnetic
and velocity fields assuming a corkscrew geometry as shown in Figure 1. At
higher latitudes ($\theta$ small), the flow passes beyond the Alfvén point and
the magnetic field cannot enforce corotation. The magnetic field lines bend
backward and angular momentum conservation forces the toroidal velocity (and
field) to fall-off like $1/r\sin\theta$ while the dominant fluid velocity is
in the poloidal direction (see e.g Fukumura et al., 2010).

For the velocity of the plasma in the $z$-direction, we use an approximation
of the expression derived numerically in Fukumura et al. (2010), valid up to
the Alfvén radius, i.e. to an inclination angle of $\theta\simeq 20^{\circ}$,
an approximation sufficient for our purposes since the plasma is much too
dilute at larger latitudes to be of any significance in the formation of dust.
Therefore, we fit the poloidal velocities calculated by Fukumura et al. (2010)
and estimate the velocity in polar direction ($v_{z}$) at all points.
Considering the helical motion, the velocity of Keplerian rotation (VK) is
given by $(GM_{\mathrm{BH}}/x)^{1/2}$. The best fit case of $v_{z}$ at a given
coordinate ($r_{1},\theta_{1}$) is presented as,

$\begin{split}&v_{z}=\beta\Big{(}\frac{z}{r_{0}}\Big{)}^{p}V_{K},\ \ \ \
\beta=0.45,\ \ \ p=0.5\\\
&v_{z}(r_{1},\theta_{1})=\beta\Big{(}\frac{GM_{\mathrm{BH}}}{r_{0}}\frac{cos\theta_{1}}{sin\theta_{1}}\Big{)}^{1/2}\end{split}$
(5)

where, $\beta$ and $p$ are fitting parameters, $r_{1}$ =
$r_{0}/(1-cos\theta_{1})$. The equation of the flow (and poloidal magnetic
field) along the line (Eqn. 1), combined with the expression for the
$z$-component of the velocity flow (Eqn. LABEL:eq_vz) below, can determine the
motion of the gas in the direction, most importantly the time spent at each
latitude, as this should be sufficiently long for the formation of dust.

Figure 3 shows the different physical conditions in the flow as a function of
the polar coordinates. The top left panel shows the variation of $v_{z}$ as a
function of the Keplerian velocities (VK) in the polar coordinate ($r\theta$)
plane. As seen in the figure, initially when a parcel of gas is closer to the
plane of the accretion disk, the polar velocities are small. As the gas parcel
flows further away (smaller $\theta$), the velocities continue to increase,
until it crosses the Alfvén radius.

The top right panel show the flow time along the poloidal magnetic field
lines. The flow-time is the time ($t$) taken by the wind to reach a given
point ($r_{1},\theta_{1}$) on the $r\theta$-plane. It is calculated using the
equations of motion, given by Eqn. 1 and LABEL:eq_vz, as shown in the equation
below. Dust formation requires that the timescales for chemical reactions and
nucleation processes be shorter than the flow time.

$\begin{split}&z=\frac{r_{0}\ cos\theta}{1-cos\theta},\ \ \ \
\frac{\mathrm{d}z}{d\theta}=-\frac{r_{0}sin\theta}{(1-cos\theta)^{2}}\\\
&\frac{\mathrm{d}z}{dt}=\frac{\mathrm{d}z}{d\theta}\frac{\mathrm{d}\theta}{dt}=-\frac{r_{0}sin\theta}{(1-cos\theta)^{2}}\frac{\mathrm{d}\theta}{dt}=\beta\Big{(}\frac{GM_{\mathrm{BH}}}{r_{0}}\frac{cos\theta}{sin\theta}\Big{)}^{\frac{1}{2}}\\\
&dt=-\frac{r_{0}^{3/2}}{\beta(GM_{\mathrm{BH}})^{1/2}}\frac{\
sin^{3/2}\theta}{(1-cos\theta)^{2}cos^{1/2}\theta}d\theta\\\
&t(r_{1},\theta_{1})=-\int^{\theta_{1}}_{\theta>\pi/2}\frac{r_{0}^{3/2}}{\beta(GM_{\mathrm{BH}})^{1/2}}\frac{\
sin^{3/2}\theta}{(1-cos\theta)^{2}cos^{1/2}\theta}d\theta\\\ \end{split}$ (6)

The flow-time is proportional to the launch radius $r_{0}$ to its 3/2 power,
as befits the free-fall time in a gravitational field. Therefore, at a larger
launch radius $r_{0}$, the local wind velocities are slower. To give an
example, outflows starting from the inner regions ($r_{0}$ $\sim$ 1016 cm) of
the accretion disk reaches an inclination angle of $\theta=20^{\circ}$ within
an year, where as in the outer regions ($r_{0}$ $\sim$ 1018 cm) reaches at the
same height in more than 104 years.

The bottom left panel shows the gas density in the $r-\theta$ plane. For the
gas density, including its angular distribution, we have employed a fit to the
expression given in Fukumura et al. (2010)(Fig. 2a), who integrated the wind
equations choosing the poloidal current in a way that the radial density
dependence be proportional to $r^{-1}$, in agreement with the observations of
Holczer et al. (2007). As in Fukumura et al. (2010), we normalize the radius
$r$ in units of the Schwarzschild radius, $R_{\rm sch}=3\times 10^{13}M_{8}$
cm, where $M_{8}$ is BH mass normalized to $10^{8}$ solar masses, and the
dimensionless unit of length $X=r/R_{\rm sch}$. For the angular distribution
we simply use a fitting formula to the results of Fukumura et al. (2010) (fig.
2a). The density is normalized so that the Thomson depth over the BH
Schwarzschild radius be equal to the dimensionless accretion rate $\dot{m}$,
i.e. $n_{M}\,\sigma_{\rm T}\,R_{\mathrm{sch}}=\dot{m}$ or $n_{M}\simeq
n_{0}(\dot{m}/M_{8})$, where $n_{0}$ = $5\times 10^{10}$. So the density has
the form,

$n(r,\theta)=n_{0}\frac{\dot{m}}{M_{8}}X^{-1}e^{\frac{5}{2}(\theta-\frac{\pi}{2})}$
(7)

As seen in the figure, the gas densities are higher at smaller radii and large
inclinations ($r$ small, $\theta$ large). Therefore, the density of the gas in
the outflows gradually decrease along its path, as the wind moves outwards and
towards higher latitudes (low $\theta$).

In this study, $M\rm_{BH}$ is 107 M⊙, $M_{8}$ is 0.1, $R_{\mathrm{sch}}$
$\sim$ 2.94 $\times$ 1012 cm and $\dot{m}$ = 1 (explained with Equation (2)).
Using Equation (1), we have,

$n(r,\theta)=\frac{C_{0}}{r_{0}}(1-cos\theta)e^{\frac{5}{2}(\theta-\frac{\pi}{2})},\
\ C_{0}=\frac{n_{0}\dot{m}R\rm_{sch}}{M_{8}}$ (8)

The bottom right panel shows the gas temperature distribution derived in
Section 3.1 below.

Figure 4: The figure shows the ratio of the luminosities transmitted through
the column of outflowing gas from the accretion disk, compared to the source
luminosities, for two inclinations, $\theta=30^{\circ},75^{\circ}$. Owing to
the larger column densities and high densities along high inclination angles,
a larger fraction of the source luminosity is absorbed by the gas.

##  3 The thermodynamic conditions for the formation of dust

###  3.1 The gas temperature

The formation of dust proceeds through a series of chemical processes, such as
chemical reactions, nucleation, coagulation and gas accretion onto dust, which
involves atoms, ions, molecules, stable clusters (seeds) and solid grains
(Sarangi & Cherchneff, 2015; Gobrecht et al., 2016). The collective process
requires high gas densities in order to ensure adequate number of collisions
within a flow timescale. It also requires moderately high gas temperatures to
sustain higher thermal velocities needed to overcome energy barriers where
necessary.

The gas temperature is determined by the balance between its radiative and
adiabatic expansion cooling and its heating by the illumination by the central
source. We have calculated the gas temperature employing heat balance as a
function of inclination angle. The central source with luminosity $L(\theta)$,
given by Equation (2), and the shape of the spectra shown in Figure 2, is
taken as the heating source. The gas density of the medium is taken as
$n(r,\theta)$ from Equation (7). The radiative transfer through various
inclination planes is computed using the spectral synthesis code CLOUDY
(Ferland et al., 1998) to determine the local absorption coefficient and the
spectral flux at each point of the flow. We assume a gas of solar composition
(Grevesse & Sauval, 1998).

Figure 4 (left) presents the ratio of the transmitted luminosity to the
intrinsic source luminosity, at the outer radius of the cylindrical volume of
gas, $R_{\mathrm{out}}$ (defined in Section 2), for inclination angles
$\theta=30^{\circ},75^{\circ}$. For both the cases, the soft X-rays
($\sim$10-3-10-1 $\mu$m) are heavily attenuated by the gas, while the hard
X-rays ($\lambda$$<$10-3 $\mu$m) and non-ionizing radiations at wavelengths
larger than 0.1 $\mu$m are only partially absorbed. It is evident that at
$\theta=75^{\circ}$, the attenuation of the source radiation is much higher
compared to the attenuation at $\theta=30^{\circ}$. This is mainly because of
a larger gas densities and larger total column density along higher
inclination angles (larger $\theta$), as shown in Figure 3 (bottom-left). The
rate of radiative cooling was found to be dominant over adiabatic cooling
rates for all densities in our region of interest.

Figure 3 (bottom-right) shows the resultant distribution of gas temperatures
in the $r$-$\theta$ plane. The flow-lines shown (in black) on the figure help
us to read the gas temperatures along the path of the flow. The gas
temperatures range between 103 to 107 K within this region of interest. The
temperatures along lower inclination directions ($\theta$ small) are found to
be much higher compared to the inclination planes which are closer to the
accretion disk ($\theta$ large, close to the disk). Moreover, the figure also
confirms that the gas is cooler towards the outer radii because of the decline
of the incident flux with radius. Radiative cooling rate of a parcel of gas in
proportional to the square of gas density (Dalgarno & McCray, 1972). Due to
high densities close to the accretion disk ($\theta$ large), the cooling is
faster, which leads to a lower temperature of the gas at equilibrium.
Moreover, the luminosities are also lower along high inclinations (Eqn. 2),
which aid the lowering of gas temperatures.

As a parcel of gas flows along the flow-line, it moves toward larger $r$ and
smaller $\theta$. Along its path, its temperature is determined by radiative
and adiabatic cooling and heating by the continuum source. Because adiabatic
cooling is always subdominant in the cases of interest, the gas temperature is
determined by its photoionization parameter $\xi=L/n(r)r^{2}$, provided to us
by the code CLOUDY (Ferland et al., 1998) once the value of $\xi$ is given.
Because the particle density drops faster than the photon density with
latitude, the gas temperature increases with decreasing $\theta$.

Formation and nucleation of molecules are the first steps towards forming dust
grains (Cherchneff & Lilly, 2008; Cherchneff & Dwek, 2009; Cherchneff, 2013).
Many of the molecular reactions are neutral-neutral processes and they are
only efficient when the ambient gas temperatures fall below $\sim$ 4000 K
(Sarangi & Cherchneff, 2013). In Figure 3 (bottom-right) we have demarcated
the region (yellow line), below which the temperature is lower than the
threshold temperature. We call the cooler zone below the demarcation boundary
as the nucleation zone. The figure indicates that the volume of gas in the
nucleation zone is larger towards outer radii and the time spent by the winds
in this zone is also longer. The formation of molecules and stable clusters,
that act as the seed/monomers to dust grains, takes place in this zone. Once
the winds move out of the nucleation zone, formation of molecules and clusters
is no more feasible.

Figure 5: The heating rates of a 10 Å silicate grain as a function of the
radial distance is shown for two inclination angles,
$\theta=30^{\circ},75^{\circ}$, subjected to radiative heating by the central
source and collisional heating by the ambient gas. For both the cases the
figure shows that, owing to the low densities of the gas, the collisional
heating is found to be negligible, compared to the rate of radiative energy
absorption.  Figure 6: The colormap shows the dust temperature in polar
($r,\theta$) coordinates for a 10 Å grain of silicate. The lines in white on
the figure represents the parabolic flow-lines with various starting/launch
radius on the accretion disk, $r_{0}$. The brown line demarcates the region
where dust temperatures are lower than the the sublimation temperature, taken
to be $\sim$ 2000 K. Hence, the region in the left-side of the line is the
sublimation zone, which is free of dust grains.

###  3.2 The dust temperature

Formation of dust grains proceeds through the condensation of stable molecular
clusters. For a gas of solar composition, silicates are expected to be the
primary dust component. We consider a 10 Å grain of astronomical silicate as
the prototype of the smallest dust grain. Once formed, these grains grow by
coagulation and accretion thereafter.

Dust grains are radiatively heated by the incident radiation from the central
source and collisionally heated by the ambient gas. The collisional heating
rates are estimated as a function of gas densities and gas temperatures using
the analysis by Dwek (1987) and Hollenbach & McKee (1979), which are recently
applied by Sarangi et al. (2018). The collisional heating rate is compared to
the rate of radiative heating by the central source. Figure 5 shows the rate
of heating of a 10 Å silicate grain as a function of radial distance from the
source, for two inclination angles. It is clear from the figure that for both
the cases radiation is the primary heating source, as the collisional heating
rates are much lower. In such environments, gas and dust temperatures are
individually controlled by radiative heating. At the high densities the gas
and dust temperatures are coupled but at different values.

The dust temperatures are calculated using the heating and cooling balance of
the individual grains (Dwek, 1987), subjected to the radiation from the
central source. Most of the incident radiation is found to be in the range of
0.1-1 $\mu$m wavelength, as evident in Figures 2 and 4. The mass absorption
coefficient, $\kappa(a_{\mathrm{i}},\lambda)$, of astronomical silicates is
taken from the study by Weingartner & Draine (2001). The sublimation
temperature is taken as $\sim$ 2000 K. Figure 6 shows the distribution of dust
temperature in polar coordinates for a hypothetical 10 Å grain. The
sublimation zone (Tdust $\geq$ 2000 K) and the condensation zone (Tdust $<$
2000 K) are separated by the demarcation line (in brown). Newly formed grains
as well as pre-existing dust can exist when the gas flows through the
condensation zone (on the right of the line).

Figure 7: The figure shows the colormap of different regions in the outflows which are are characterized by the following features: (A) region where seeds do not form and dust grains do not survive, (B) region where new seeds form but do not grow (C) region where new seeds form and also simultaneously grow through accretion and coagulation, (D) region where seeds grow via accretion and coagulation but new seeds do not form in the gas Table 1: Various zones in the $r$-$\theta$ plane Zone | Nucleation | Condensation | Sublimation
---|---|---|---
A | ✗ | ✗ | ✔
B | ✔ | ✗ | ✔
C | ✔ | ✔ | ✗
D | ✗ | ✔ | ✗

###  3.3 Regions where molecules and dust can form and survive

Based on the temperatures of the gas and dust, various zones of the outflowing
winds can be categorized in light of their role in molecule and dust
synthesis. The combined contours defined from Figure 3 and 6 mark the
boundaries of these zones. Figure 7 shows the regions labeled by A, B, C and D
on the $r$-$\theta$ plane, which are characterized as nucleation, condensation
and sublimation zones in Table 1.

Region A comprises of gas that is inside the sublimation zone and outside the
nucleation zone. In this region, the hypothetical dust temperatures are too
high to sustain the stable grains and also the ionization state of the gas
prevents the synthesis of stable seeds, which if formed quickly evaporate
because of their high temperature. Region B is defined by the part of the
nucleation zone which is inside the sublimation area. Here, gas phase
reactions can proceed towards forming stable seeds yet formation of dust from
these seeds will not commence when the wind flows through this zone. Region C
is the overlapping area of the nucleation and the condensation zone. This the
most suitable zone for molecules and dust formation. In region D, temperatures
are too high to support the formation of new seeds, however the already formed
seeds can grow by coagulation and by accretion of refractory metals. Important
to note, these regions are defined solely for a stationary gas. In an
outflowing wind, the formation of molecules and dust also depends on the
relative timescales of the outflow and the relevant reactions and the time-
dependent diminishing concentration of condensible elements in the gas.

Figure 8: The colormap shows the number density of dust precursors (seeds) in
polar coordinates ($r,\theta$) coordinates, prior to condensation. The white
region is free of seeds as the temperatures are too hot to sustain stable
clusters along the course of the flow. The black lines on the figure
represents the parabolic flow-lines with various starting/launch radius on the
accretion disk, $r_{0}$.

##  4 Kinetics of seed nucleation

We assume the winds are launched dust-free from the surface of the accretion
disk and follow the path given by Equation (1). Nucleation of silicates
transpires through a set of chemical processes involving atoms and ions such
as Si, O, Mg, Fe and SiO, H2O, O2 etc. (Goumans & Bromley, 2012; Sarangi &
Cherchneff, 2013; Gobrecht et al., 2016; Bromley et al., 2016). Other than the
dimerization of SiO molecules, all processes in this chain are energetically
favored, $i.e.$, the product is more stable than the reactants. This leads to
the formation of stable dust precursors or seeds which in this case will have
a form $\sim$ (MgxFe2-xSiO4)2 (Bromley, 2017). We consider this as the seed-
nucleus, which is the end-product of the nucleation process. The reaction rate
of the bottle-neck for reaction for nucleation of (SiO-Mg-Fe) clusters is
$\sim$ 10-14 cm3 s-1 (Sarangi & Cherchneff, 2013) and for simplicity we take
this as the general rate of nucleation, $\zeta\rm_{SiO-Mg}$ (= $\zeta$).

The rate coefficient $\zeta$ is a function of the metal density,
$n_{z}(r,\theta)$ = $nf_{z}$, where $f_{z}(r,\theta)$ is the fractional
abundance of any metal (which is a constituent of the seed nuclei) at a given
$r,\theta$. In case of silicate nucleation, we take it as the abundance of Si
atoms in the gas, as the efficiency of the seed formation is limited by the
availability of Si atoms. Let $f_{z0}$ be the fractional abundance at solar
composition, $i.e.$, of the dust free gas. We calculate the number of seed
nuclei in a co-moving unit-volume, $Y_{\mathrm{s}}$, (unit volume at $r_{0}$)
and the seed number density, $n_{\mathrm{s}}$ in cm-3, at a given coordinate
$r_{1},\theta_{1}$.

$\begin{split}&Y_{\mathrm{s}}(r_{1},\theta_{1})=\int\mathrm{d}Y_{\mathrm{s}}=\int^{\theta_{1}}_{\frac{\pi}{2}-\delta}\zeta\
f^{2}_{\rm{z}}(r,\theta)\ n^{2}(r,\theta)dt\\\
&f_{\rm{z}}(r,\theta)=f_{z0}-\frac{n_{s}(r,\theta)}{n(r,\theta)}\\\
&Y_{\mathrm{s}}(r_{1},\theta_{1})=-\int^{\theta_{1}}_{\frac{\pi}{2}-\delta}\frac{\zeta\
C_{0}^{2}\
f^{2}_{\rm{z}}(r,\theta)}{\beta(r_{0}GM_{\mathrm{BH}})^{1/2}}\frac{sin^{3/2}\theta}{cos^{1/2}\theta}e^{5(\theta-\pi/2)}\mathrm{d}\theta\\\
&n_{\mathrm{s}}(r_{1},\theta_{1})=\int\frac{n(r_{1},\theta_{1})}{n(r,\theta)}\
\ \mathrm{d}Y_{\mathrm{s}}(r,\theta)\end{split}$ (9)

Here we have used the ‘$\mathrm{d}t/\mathrm{d}\theta$’ relation from Equation
(LABEL:eq_time) and the gas density from Equation (8). In the lower limit of
the integral, the infinitesimal angular width, $\delta$ ($\sim$
$\Delta\theta$) represents the scale width over the surface of the accretion
disk from which the outflow starts. In our study, we assumed a scale width of
$h/R\sim$ 10-2, which translates to an approximate inclination angle of $\sim$
89.3∘ where the outflow launches.

Figure 9: The colormap shows the number density of 100 Å silicate grains in
polar coordinates ($r,\theta$) coordinates. The white region is free of dust
as it is the boundary of the sublimation region. The lines in black on the
figure represents the parabolic flow-lines with various starting/launch radius
on the accretion disk, $r_{0}$.

The number density ($n_{\mathrm{s}}$) and the number of seeds in the co-moving
volume ($Y_{\mathrm{s}}$) are related by the dilution in density along the
line of the flow. In other words, $Y_{\mathrm{s}}$ gives the total number of
seed-nuclei formed at a given $r_{1},\theta_{1}$, from all the metals that
were present in a unit volume of gas, when the flow started ($r_{0}$,
$\theta\sim\pi/2$), whereas $n_{\mathrm{s}}$ calculates the number density of
these seeds at $r_{1},\theta_{1}$.

Figure 8 shows the the number density ($n_{\mathrm{s}}$) of the seed-nuclei in
polar coordinates. The colormap clearly shows a high density of these dust
precursors (seeds) near the large inclination-planes ranging between
$\theta$$\sim$70∘ to $\theta$$\sim$85∘. As the winds move further outwards,
there is a rapid decline in the number densities of these seeds caused by the
(a) movement of the gas out of the nucleation zone, so formation of seeds is
no longer viable; (b) decrease in the number density of the gas, leading to
the thinning in the concentration of seeds; and (c) depletion of a large
fraction of the refractory elements into molecules and seeds, so the rate of
formation drops due to the lack of available condensible elements.

The maximum density of the seeds is attained within the radial range of 1016
and 1017 cm. The drop in number density of the seeds towards the outer radius
is mainly due to the decrease in gas density, which also reduces the rate of
nucleation. The white region on the figure represents the lack of any stable
molecules or seeds. The wind in this region never passes through the
nucleation zone, and hence always remains free of seed-clusters.

Figure 10: Left-panel: The colormap shows the total number density of dust
grains comprising of sizes ranging between 10 Å and $\sim$ 200 Å in polar
($r,\theta$) coordinates. The black lines on the figure represents the
parabolic flow-lines with various starting/launch radius on the accretion
disk, $r_{0}$. As seen in the figure, for each radius, the number density of
grains increases at higher inclinations, $i.e.$, close to the plane of the
accretion disk at $\theta\sim 90^{\circ}$. The figure shows that the dust
forming region has a toroidal morphology. The three horizontal lines
correspond to the inclination angles, for which the dust densities are shown
in the figure on the right in the same color code. Right-panel: The number
density of dust grains is presented as a function of radial distance $r$ for
inclination angles $\theta=30^{\circ},75^{\circ}$ and $85^{\circ}$. The
presence of a dust toroidal region at high inclination angles are further
visible in the figure.

##  5 The number density of dust grains after seed coagulation

The number density and final size of dust grains are determined by the
coagulation and accretion of the seeds.

Figure 11: The dust to gas number density ratio along three different
inclinations, $\theta=30^{\circ},75^{\circ}$ and $85^{\circ}$, are presented
as function of the radial distance from the central source, with reference to
the gas number density shown in Figure 3 (bottom-left) and dust number density
in Figure 10.

Coagulation of seeds (dust precursors) and intermediate grains, and accretion
of remaining metals on the grain surface are the two efficient routes of dust
formation from molecules (Dwek & Cherchneff, 2011; Sarangi & Cherchneff,
2015). We consider the growth of seeds through coagulation using a volume
conserved formalism described by Jacobson (2005) and used by Sarangi &
Cherchneff (2015). The rate-coefficient for coagulation, $\xi_{\mathrm{c}}$
(cm3 s-1), between two particles of radii $a_{i}$ and $a_{j}$, in such
environments is given by,

$\xi_{\mathrm{c}}(i,j)=\pi(a_{i}+a_{j})^{2}\times\Big{[}\frac{8k}{\pi}\Big{(}\frac{T_{i}}{m_{i}}+\frac{T_{j}}{m_{j}}\Big{)}\Big{]}^{0.5}W_{ij}$
(10)

where, $T$ and $m$ are the kinetic temperatures and masses of the particles
and $W_{ij}$ is the Van der Waal’s force coefficient between them. $W_{ij}$
ranging between 2 to 10, we adopt an average value of 5 for this study
(Jacobson, 2005; Sarangi & Cherchneff, 2015). We consider the growth of seeds
until they reach a size of 100 Å, after which their growth proceeds through
accretion.

For simplicity, we use an average coagulation rate as function of temperature.
It corresponds to the coagulation between two 10 Å silicate grains, with
internal density of 3.2 g cm-3, and express the rate coefficient as
$\xi_{\mathrm{c}}\sim\eta\ T^{0.5}$. The dust temperature, depending on its
size and the intensity of the incident radiation, can be kinetic or radiative
in nature. For the coagulating grain, we have considered the kinetic
temperature of the gas, given by Figure 3, as the temperature of the
coagulating seeds.

The first step is marked by the coagulation between two seed grains of size
$a_{0}$, whose yields was calculated in Equation (9). The number of grains in
a co-moving unit-volume, $Y_{\mathrm{d}}(a_{1})$, and the number density,
$n_{\mathrm{d}}(a_{1})$ in cm-3, of the resultant grain of size $a_{1}$ is
given as,

$\begin{split}&Y_{\mathrm{d}}(a_{1},r_{1},\theta_{1})=\int\mathrm{d}Y_{\mathrm{d}}\\\
&=\int^{\theta_{1}}_{\frac{\pi}{2}-\delta}\eta\
T^{0.5}_{\mathrm{g}}(r,\theta)\ [Y_{s}^{\prime}(a_{0},r,\theta)]^{2}dt\\\
&Y_{s}^{\prime}(a_{0},r,\theta)=Y_{\mathrm{s}}(a_{0},r,\theta)-2Y_{\mathrm{d}}(a_{1},r,\theta)\\\
&n_{\mathrm{d}}(a_{1},r_{1},\theta_{1})=\int\frac{n(r_{1},\theta_{1})}{n(r,\theta)}\
\ \mathrm{d}Y_{\mathrm{d}}(a_{1},r,\theta)\end{split}$ (11)

where $Y_{s}^{\prime}(a_{0})$ is the number of the remaining seeds of size
$a_{0}$ in that co-moving volume of gas, at a given $r,\theta$, after the
coagulation step. The factor 2 in the equation stands for the fact that each
coagulation removes two seeds from the system. The relation between
$n_{\mathrm{d}}$ and $Y_{\mathrm{d}}$, and the limits of the integration,
follows the same analogy given in Section 4.

Similar formalism was used to calculate the number density of grains of any
size $a_{\mathrm{i}}$, until they reach 100 Å in radius. For coagulation
between grains of different sizes, the $Y_{d}^{2}(a_{i})$ in Equation (11) is
replaced by $Y_{d}(a_{i})Y_{d}(a_{j})$.

Figure 9 shows the colormap for the distribution of the 100 Å grains, in polar
coordinates. The number density of dust grains in a given parcel of gas at
($r,\theta$), depends on the number density of the seeds present in the gas
and on the timescales of coagulation compared to the time of the flow through
a small volume around ($r,\theta$). Owing to the high density of seeds and the
slower flow velocities near the large inclination angles, the density of 100 Å
grains are higher between $\theta$ $\sim$ 60∘ and 80∘.

The distribution of the total dust density, $n_{\mathrm{dust}}$ = $\sum
n_{\mathrm{d}}(a_{\mathrm{i}})$, is shown in Figure 10 where grains range
between 10 and 100 Å. The left-panel shows the colormap, where the dust number
densities are found to be higher near the large-$\theta$ planes as expected.
Importantly, the density distribution represents the combined result of
simultaneous nucleation and coagulation processes, so dust density is higher
where there is a higher concentration of seeds and slower rate of coagulation,
while density is lower in the opposite case. The density of the dust grains
declines rapidly with radius which is primarily due to efficient coagulation
resulting from slower flow velocities towards outer radii. The white region
represents the sublimation zone, which is free of dust.

Only coagulation, however does not alter the dust to gas mass ratio, it only
changes the size distribution of the grains. The accretion of the remaining
metals on the surface of these grains, discussed in the following section
alters the mass ratio of gas and dust.

Figure 10 (right panel) shows dust densities as a function of radial distance
for three inclination angles. Both the figures confirm the presence of a dense
toroidal region surrounding the accretion disk, with a scale height ranging up
to $\theta$ $\sim$ 70∘. We identify this region with the observed dusty ‘tori’
surrounding the AGN.

The ratio of number densities of dust to gas, at three different viewing
angles are presented in Figure 11 as a function of the distance from the
central source. Close to the central BH lies the sublimation zone, where the
ratio is zero due to the absence of dust. In the condensation zone, the dust
to gas number density ratio is of the order of 10-8. Along any inclination
shown on the figure, the ratio increases fast along radial direction, and then
gradually declines. The lowering of the dust to gas number ratio at larger
radii indicates the lower efficiency of seed formation, owing to low
densities. The ratio also reduces as the gas moves from high to low
inclination planes (high to low $\theta$). This is mainly attributed to the
condensation of small grains over time, into larger grains, thereby reducing
the dust to gas number density ratio.

Figure 12: Left-panel: The figure shows the column density of dust grains as
function of the inclination angle $\theta$. The dust column densities are
larger at high inclinations, as seen in the figure. The steep drop in column
densities towards low inclinations (low $\theta$, high-z) is attributed to the
lowering of the gas densities and the scarcity of seeds owing to the
coagulation of multiple seeds into one grain. Right-panel: The column
densities of the gas, the dust precursors (seeds) and the dust grains are
presented as a function of the inclination angle $\theta$.

Taking the central source as the origin, the column density
($N_{\mathrm{dust}}$) of dust grains along various lines of sight is given by,

$N_{\mathrm{dust}}(\theta)=\int n_{\mathrm{dust}}(r,\theta)\mathrm{d}r$ (12)

Figure 12 (left-panel) shows the dust column density as a function of
inclination angles. The column densities also reflect the shape of the
toroidal dusty zone, as the viewing angles along the equatorial directions
(high-$\theta$, small-z) have a very high column density of dust which drops
steeply with the decrease in inclination, $i.e.$, when viewed along the polar
directions. Figure 12 (right-panel) shows the column densities of the gas,
dust-precursors or seeds and the dust grains, as the function of inclination
angles. All the processes related to dust formation, $i.e.$, nucleation of
molecules and coagulation of grains, are most efficient when the wind is still
at high inclination planes (zone C, Figure 7). In this low-inclination
regions, the rate of coagulation drops due to the lack of available seeds.
Moreover, the density of coagulating grains also declines because multiple
seeds coagulate to form single grains. Therefore, the dust density in the wind
is found to simply follow the behavior of gas density.

##  6 Size distribution following accretion

Once formed, the grains grow in size via accretion of the remaining metals on
their surface. The efficiency of accretion is higher for large grain sizes as
the grain surface area is larger and the sticking coefficient is close to
unity. We consider the smallest size of the accreting grains as 100 Å, which
is the terminal size considered in the coagulation process.

A parcel of gas at a certain coordinate ($r,\theta$) comprises of grains of
various sizes. The distribution of grain sizes in that parcel depend on the
following initial conditions: (i) physical conditions of the gas (ii) relative
abundance of free metals compared to the accreting grains (iii)
coordinates/epoch of formation of a certain grain along the path of the flow.

The mass density of dust grains of radius $a$ is related to the number density
of the grains by $\rho_{\rm{d}}(a)$ = $m_{\rm{gr}}(a)n_{\rm{d}}(a)$, where
$m_{\rm{gr}}(a)$ is the mass of the individual grains of radius $a$. The rate
of increase in mass density ($\rho_{\mathrm{d}}$) of grains of radius $a$,
located at ($r,\theta$), due to accretion (Dwek & Cherchneff, 2011), and the
timescale of accretion, $\tau_{\rm{acc}}(r,\theta)$, is given by,

$\begin{split}\frac{\mathrm{d\rho_{\rm{d}}}}{dt}&=\alpha_{\mathrm{s}}\ \pi
a^{2}\ \mu_{z}f_{z}\ n\
n_{\rm{d}}\Big{(}\frac{8kT_{\mathrm{gas}}}{\pi\mu_{z}}\Big{)}^{\frac{1}{2}}\\\
&\tau_{\rm{acc}}^{-1}=\frac{1}{\rho_{\rm{d}}}\frac{\mathrm{d}\rho_{\rm{d}}}{dt}\\\
\end{split}$ (13)

where $\alpha_{\mathrm{s}}$ is the sticking coefficient, $n$ and
$n_{\mathrm{d}}$ is the number density of the gas and the dust grains,
$T_{\mathrm{g}}$ is the gas temperature, $\mu_{z}$ is the mean molecular
weight of the accreting metals. The fraction of refractory elements remaining
in the gas phase, $f_{z}(r,\theta)$, is given by Equation 9.

Figure 13: The colormap shows the maximum grain sizes at a given region
($r,\theta$) where the grains have grown by accretion of free metals in the
gas along the flow-lines. The lines in black on the figure represents the
parabolic flow-lines with various starting/launch radius on the accretion
disk, $r_{0}$.

The number density of the grains does not change via accretion and we simplify
Equation (13) as,

$\frac{\mathrm{d}a}{dt}=\frac{\alpha_{\mathrm{s}}f_{z}n}{\rho_{\mathrm{gr}}}\Big{(}\frac{k\mu_{z}}{2\pi}\Big{)}^{\frac{1}{2}}[T_{g}(r,\theta)]^{\frac{1}{2}}$
(14)

where $\rho_{\mathrm{gr}}$ is the density of each grain in g cm-3 (3.2 g cm-3
for silicates).

However, the remaining metals in an unit volume of gas are accreted uniformly
among all the dust grains present in that volume. So the rate of growth of a
single grain is corrected by the grain density as,

$\frac{\mathrm{d}a}{dt}=\frac{\alpha_{\mathrm{s}}f_{z}n}{\rho_{\mathrm{gr}}||n_{d}||}\Big{(}\frac{k\mu_{z}}{2\pi}\Big{)}^{\frac{1}{2}}[T_{g}(r,\theta)]^{\frac{1}{2}}$
(15)

where $||n_{d}||$ is the dimensionless quantity representing the grain
density.

Using the values of $n$ and $\mathrm{d}t$, from Equation (8) and
LABEL:eq_time, the final radius of a grain of initial size $a_{c}$ is,

$\begin{split}&a(r_{1},\theta_{1})=\int^{a}_{a_{c}}da=\\\
&-\int^{\theta_{1}}_{\frac{\pi}{2}-\delta}\frac{\alpha_{\mathrm{s}}C_{0}f_{z}(r,\theta)}{\rho_{\mathrm{gr}}\beta||n_{d}||}\Big{(}\frac{k\mu_{z}r_{0}}{2\pi
GM_{\mathrm{BH}}}\Big{)}^{\frac{1}{2}}\frac{sin^{3/2}\theta}{(1-cos\theta)cos^{1/2}\theta}\\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
[T_{g}(r,\theta)]^{\frac{1}{2}}e^{\frac{5}{2}(\theta-\pi/2)}\mathrm{d}\theta\\\
\end{split}$ (16)

where $a_{c}$ = 100 Å.

The largest radii of the dust grains, $a_{max}$($r,\theta$) in presented in
Figure 13, resulting from accretion of the remaining metals. The largest
grains are found to be around $\sim$ 220 Å in radius, which is present in the
outer regions with radius larger than 1018 cm. The figure indicates in general
that the grains are larger towards the longer radii, outwards of 1018 cm.
Inefficient nucleation in this region leads to a larger abundance of residual
metals in the gas. This in turn make accretion more efficient. Slower flow-
velocities (Figure 3 (top-left)) further aid the grain growth in these
regions. However, grain sizes are limited to the amount of metals available
for accretion. Hence, as the wind moves further away from the accretion disk
(along the flow), most of the metals are already locked up into dust and the
accretion rate slows down significantly. Overall, the newly formed dust grains
range in radii between 10 to 220 Å.

Figure 14: The colormap shows the dust to gas mass ratio in polar coordinates
($r,\theta$). The white region is the sublimation zone, and the black lines
indicates flow lines with various launch radius $r_{0}$. The mass ratio seems
to be larger towards the longer radii, outwards of 3$\times$1018 cm, where the
ratio is $\sim$ 10-2.

Summing over grains of all sizes present in an unit volume of gas at a given
($r,\theta$), the mass density of the dust grains is calculated. Figure 14
shows the ratio of dust to gas mass densities in polar coordinates. The ratio
is shown to vary between 10-5 to 10-2 where the maximum attained around $r$
$\sim$ 6$\times$1018 cm and $\theta\sim$ 60∘. Moreover, the dust to gas mass
ratio along the 2D plane follows a similar distribution as that of $a_{max}$
(Figure 13), thereby verifying a direct connection between the efficiency of
accretion and the mass density of dust grains.

Figure 15: Left-panel: The mass column density of the dust grains is shown as
a function of inclination angle $\theta$. The column density of dust masses
increases rapidly from the time of launch at $\sim$ $90^{\circ}$ and reaches
its peak around $\sim$ $80^{\circ}$. Following that, as the gas flows towards
low inclinations (low $\theta$, high-z) the number density of dust declines
and mass column density also likewise. This further confirms that new dust
formation in these regions are negligible. Right-panel: The optical depth
$\tau(\lambda)$ of the medium, resulting from the dust formation, is presented
as a function of the inclination angle $\theta$, for four wavelengths,
$\lambda$ = 0.01, 0.1, 1 and 10 $\mu$m.  Figure 16: The mass absorption
coefficient of astronomical silicates, derived from Weingartner & Draine
(2001), is shown for three different grain radii, $a$ = 0.01, 0.1 and 1
$\mu$m, which is the range in which majority of the grains are present.

##  7 The dust masses and the optical depths

In this section we calculate the dust masses at any given epoch, the rate of
dust formation and the optical depths at various wavelengths.

Assuming the winds to be in steady state, the total amount of dust present at
any given epoch is just the integrated sum over all grain sizes.

Taking the spherical volume encompassed by the radius of the black hole
influence and assuming azimuthal symmetry, the total mass of dust at any given
epoch is calculated as,

$M_{\mathrm{tot}}=4\pi\sum_{i}\iint
n_{\mathrm{d}}(a_{\mathrm{i}},r,\theta)m_{\mathrm{d}}(a_{\mathrm{i}})r^{2}sin\theta\
\mathrm{d}r\ \mathrm{d}\theta$ (17)

where $m_{\mathrm{d}}(a_{\mathrm{i}})$ is the mass of a dust grain of radius
$a_{\mathrm{i}}$. Integrating over the $r$ and $\theta$ coordinates, the total
mass of dust present within the sphere of influence at a given epoch is found
to be $\sim$ 1.5 $\times$ 104 M⊙ for the values of the black hole mass and
dimensionless accretion rates employed throughout ($M_{\rm BH}=10^{7}$M⊙ and
$\dot{m}=1$; see section 2). The column density in mass (g cm-2), along any
given inclination is calculated as,

$\Sigma_{\mathrm{dust}}(\theta)=\sum_{i}\int
n_{\mathrm{d}}(a_{\mathrm{i}},r,\theta)m_{\mathrm{d}}(a_{\mathrm{i}})\mathrm{d}r$
(18)

Figure 15 (left-panel) shows the mass column densities as a silicate grains
function of the inclination angle. The behavior of $\Sigma_{\mathrm{dust}}$
resembles that of the column densities ($N_{\mathrm{dust}}$) shown in Figure
12. The maximum of the $\Sigma_{\mathrm{dust}}$ is around $\sim$ 85∘ while for
$\Sigma_{\mathrm{dust}}$ the same is around $\sim$ 80∘. This small variation
is caused by the growth of grains through accretion as the winds flow out
towards lower inclination planes.

The rate of dust formation ($\dot{M}_{\mathrm{dust}}$) is calculated by
integrating the rate of formation of the different sized grains over the
entire volume. Using Equations LABEL:eq_time and 17, $\dot{M}_{\mathrm{dust}}$
is given by,

$\dot{M}_{\mathrm{dust}}=4\pi\sum_{i}\iint\frac{\mathrm{d}n_{\mathrm{d}}(a_{\mathrm{i}},r,\theta)}{\mathrm{d}t}m_{\mathrm{d}}(a_{\mathrm{i}})r^{2}sin\theta\mathrm{d}r\mathrm{d}\theta$
(19)

In this case, the dust formation rate is calculated to be $\sim$ 0.37 M⊙ yr-1
(for $M_{\rm BH}=10^{7}$ M⊙ and $\dot{m}=1$). Assuming the lifetime of the AGN
to be around 1 Gyr, therefore it will lead to the formation of $\sim$ 108 M⊙
of dust. However, one has to bear in mind that over this timescale, both the
mass of the black hole and the dimensionless accretion/wind rates may change
considerably. Moreover, the wind can actually contain pre-existing dust which
gets reprocessed in the winds, instead of new dust formation. While the above
number could be an absolute upper limit, further insights on the evolution and
the dynamics of the galaxy is necessary to determine how much dust will be
formed in the AGN winds over the entire galactic lifetime.

Should the accreting gas contain already a certain amount of dust, the winds
likely be reaching dust saturation quickly, $i.e.$, all available metals being
aggregated onto dust grains, a situation with likely observable consequences.
On the other hand, a lower abundance of available metals could reduce the rate
of chemical processes considerably, which would therefore lead to inefficient
formation of new dust.

For the specific conditions considered in producing the figures of dust
distribution given above, we can compute the winds’ optical depth, $\tau$, for
different wavelengths $\lambda$ and viewing angle $\theta$. This is given by
the expression,

$\tau(\lambda,\theta)=\sum_{i}\kappa(a_{\mathrm{i}},\lambda)\int
n_{\mathrm{d}}(a_{\mathrm{i}},r,\theta)m_{\mathrm{d}}(a_{\mathrm{i}})\mathrm{d}r$
(20)

where $\kappa(a_{\mathrm{i}}\lambda)$ is the mass absorption coefficient (cm2
g-1) of astronomical silicates, derived from Weingartner & Draine (2001).
Figure 15 (right-panel) shows the variation of $\tau(\lambda,\theta)$ for
$\lambda$ = 0.01, 0.1, 1 and 10 $\mu$m, as a function of the inclination angle
$\theta$. The absorption coefficients for astronomical silicates is presented
in Figure 16, to show the variation of $\kappa(a,\lambda)$ as functions of
grain size and wavelength. Since the optical depth is the integral of
$\kappa(a_{\mathrm{i}}\lambda)\times\rho_{d}(r,\theta)$ along the observer’s
line of sight ($\rho_{d}$ is the mass density), the wavelength dependence of
the absorption coefficient plays significant role in shaping the optical
depth. Figure 16 shows that for longer wavelengths ($\lambda>2\mu$m), the
absorption coefficient is almost independent of the grain size, while in case
of shorter wavelengths the resultant size distribution would make a
difference. The mass absorption coefficient for silicates has local maxima at
9.7 and 18 $\mu$m, a feature reflected in the larger optical depth at 10
$\mu$m in Figure 15 (right-panel) and also in the AGN spectra.

The optical depth increases with $\theta$, which is expected, given the
increase of the wind column at higher inclinations. More significant is the
depth increase with decreasing wavelength $\lambda$, which makes these winds
at $\lambda\mathrel{\raise 1.29167pt\hbox{$<$}\mkern-14.0mu\lower
2.58334pt\hbox{$\sim$}}1\;\mu$m highly opaque even for $\theta\simeq
20^{\circ}$. The optical depth decreases almost monotonically for
$\lambda>1\;\mu$m with exception of the region around the silicate transitions
at $\lambda\simeq 9.7,18\;\mu$m, implying the possibility of discerning these
features in emission or absorption depending on the observer inclination angle
and hence with the classification of an AGN as Syefert 1 or 2. This notion
fits well with the analysis of these features in both types of Seyferts by Shi
et al. (2006).

Important to note, most of the observed spectra from the AGNs are not
spatially resolved. Therefore, the observed spectra are generally an average
over the entire cross-section of the region that we have modeled in this
study. Detailed radiative transfer calculations, taking into account the
absorption and re-emission of secondary photons, are required to model spectra
which can be compared to the observations. We will present such radiative
transfer calculations in a future article.

Figure 17: Left-panel: The dust mass flux in solar masses per year, as a
function of the (normalized) accretion rate, $\rm\dot{m}$, onto the BH
(assumed equal to the wind mass flux at the smallest disk radii) for two
different BH masses, $M=10^{7}$ M⊙ (red) and $10^{8}$ M⊙ (blue). For
$\dot{m}\simeq 1$ the dust mass flux is comparable to the accretion rate onto
the BH (we remind the reader that the wind mass flux increases like $\simeq
X^{1/2}$ with radius, so the local mass flux in the wind at $X\sim
10^{5}-10^{6}$ is much higher than that needed to power the BH emission).
Right-panel: The figure shows the relation between the dimensionless length
($X$) and the normalized mass accretion rate $\dot{m}$ to identify the
limiting conditions which are necessary for dust formation and survival. The
green line depicts the line where the ratio of the timescales of chemical
reactions and the flow time of the wind is equal to unity. The dotted lines
represent the limiting conditions for dust survival, based on the sublimation
temperature, for different BH masses. The region in white, therefore,
represents the parameter space, where the values of $\dot{m}$ and $X$ together
support the formation and survival of dust grains.

##  8 Dependance on accretion rate and blackhole mass

To examine the dependance of the results on the mass accretion rate and the BH
mass, we have computed six cases in total, with mass accretion rates $\dot{m}$
= 0.01, 0.1 and 1, and BH mass of 107 and 108 M⊙. The changes are manifested
in the gas density and the luminosity of the source, while the spectral shape
of the source is kept unaltered.

In Figure 17 (left-panel), we show the dust mass flux ($\dot{M}_{\rm{dust}}$)
in solar masses per year as a function of $\dot{m}$ for these two different BH
masses. For a given value of $\dot{m}$, the dust mass flux was found to scale
proportional to the BH mass, just like the mass flux in the wind, in agreement
with the qualitative arguments given above. However, its normalization has a
stronger dependence on the dimensionless wind flux, which we estimated as
$\dot{m}_{\rm dust}\propto\dot{m}^{b}$; where $b$ is between 2 and 2.5. One
should note that the ordinate of this figure is given in solar mass per year
rather than in dimensionless units; as such it is proportional to the BH mass,
as the calculations indicate and the figure shows. In case of the dust mass
production rate in dimensionless units, the figure would be independent of the
BH mass.

The steep dependance on $\dot{m}$ is due to the fact that dust production
requires the formation of seeds, whose number density is originally too small
for the dust growth time scale to compete with the plasma flow time scale, as
$\dot{m}$ decreases (remember that the ordinate in Figure is in absolute and
not normalized values of $\dot{m}$). Moreover, the formation of dust grains
depends of a series of interactions, the rate of which are proportional to the
square of the density $n^{2}$, while $n$ is linearly proportional to
$\dot{m}$. The relation between $\dot{M}_{\rm{dust}}$ and $\dot{m}$ in Figure
17 therefore is a curve of slope $\sim$ 2 in log-scales, which confirms the
linear relation between gas density and $\dot{m}$.

Incidentally, near $\dot{m}\simeq 1$, the value of $\dot{M}_{\rm dust}$ can be
(for the conditions considered) larger than the mass accretion rate that
powers the BH luminosity. This may sound at first sight impossible, however,
we would like to remind the reader that under the wind scalings assumed in our
calculations, especially its radial dependence, $n(r)\propto 1/r$, the wind
mass flux has the following dependence $\dot{M}\propto
r^{2}V_{K}(r)n(r)\propto\dot{m}MX^{1/2}$ ($X$ =$R/R\rm_{sch}$), i.e. increases
with distance from the black hole (as discussed elsewhere, this is because the
accreting matter “peels-off” into the wind before approaching the black hole
with only a small fraction of it being eventually accreted). So at the outer
disk regions ($X\sim 10^{5}-10^{6}$) the wind mass flux can significantly
larger ($\mathrel{\raise 2.15277pt\hbox{$>$}\mkern-14.0mu\lower
2.58334pt\hbox{$\sim$}}100$ times) than the accretion rate needed to produce
the BH emission, a fact supported by X-ray spectroscopy in a few AGN and
Galactic BH. If the dust production is saturated at these large distances,
i.e. $\dot{M}/\dot{M}_{\rm dust}\simeq 10^{-2}$, we can get the dust mass
production rate to exceed that accreted into the black hole.

To investigate further the necessary conditions imposed on the flow parameters
(i.e. $M,\dot{m},X$) by the formation of dust grains, we compare the
timescales for chemical interactions ($t\rm_{chem}$) and the flow-timescale
($t\rm_{flow}$) as a function of the dimensionless distance $X$ from the
central source. The formation of dust in the outflowing wind medium, just like
any species that is the product of reactions of the wind constituents, depends
on the balance between the flow and the interaction time scales. Therefore one
can provide a qualitative view of the parameter space where production is
viable and significant.

Considering that the local averaged, chemical dust-producing reaction rate is
$\zeta\sim\langle\sigma_{\rm chem}\ v_{\rm th}\rangle$, the chemical evolution
time for dust production at a distance $X$ is given by the following
expression,

$\begin{split}t_{\rm chem}&\simeq 1/(n_{z}(X)\sigma_{\rm chem}v_{\rm
th})\simeq n_{z}(X)^{-1}\zeta^{-1}\\\ &\simeq
f_{z}^{-1}n_{0}^{-1}\dot{m}^{-1}XM_{8}\ \zeta^{-1}\end{split}$ (21)

where $n_{z}$ is the metal density introduced in Section 4 and we have used
the density term from Equation 7. Here we have considered that the reaction
rate has a weak dependence on the (dimensionless) distance $X$ from the BH and
ignored the $\theta$ dependance of $n$ for simplicity. For this study, as
explain in Section 4, we have assumed a temperature averaged constant reaction
rate $\zeta$ for all nucleation processes.

Similarly, ignoring the $\theta$ dependance, the flow-time $t\rm_{flow}$,
using the expression for velocity from Eqn. LABEL:eq_vz, is given by,

$\begin{split}t_{\rm flow}\sim R/v\sim R/(\epsilon V_{K})\sim\epsilon
X^{3/2}R_{\rm sch}/c\simeq AX^{3/2}M_{8}\end{split}$ (22)

where for the expression of velocity we have used Equation LABEL:eq_vz, which
shows that the flow-velocity is slower than the local Keplerian velocity. Here
we have approximated it by an empirical factor $\epsilon\sim 0.1\beta\simeq
0.05$, to correct the velocity. The Schwarzschild radius was expressed as
3$\times$1013 $M_{8}$, and the constant A therefore is $\sim 3\times
10^{13}\epsilon/c$.

Condition for dust formation demands that the timescale for reaction should be
shorter than the flow timescale, and hence the ratio should be greater than
equal to 1. Using Eqn. 21 and 22 we can therefore derive the relation,

$t_{\rm flow}/t_{\rm chem}\geq 1\ \ \rightarrow\ \ \ C_{1}\dot{m}X^{1/2}\geq
1$ (23)

where $C_{1}$ is the normalization constant appropriate for the time scales
and reaction rates introduced in our detailed calculations. Using, $f_{z}\sim
10^{-4}$, $\zeta\sim 10^{-13}$, $n_{0}\sim 5\times 10^{10}$, we derive that
$C_{1}\simeq 10^{-4}$. Importantly, this signifies that when the gas density
normalization is proportional to $\dot{m}_{0}\,M_{\rm BH}^{-1}$ as in our
case, the necessary condition for dust formation is independent of the mass of
the BH.

Survival of the dust grains at a given $X$ depends on the dust temperature.
From the equation for luminosity given by Equation 2, we can write, $L\simeq
C_{2}\dot{m}M\rm_{BH}$, where $C_{2}$ is a constant that takes into account
the Eddington luminosity and the efficiency $\eta$, ignoring the dependence of
luminosity on the inclination angle. For empirical understanding, if we equate
this to the blackbody temperature of dust emission, and consider the
sublimation temperature for silicate dust as 2000 K, we can derive a limiting
condition for the survival of dust grains by,

$C_{2}\dot{m}M_{\rm{BH}}\simeq 4\pi X^{2}R_{\rm{Sch}}^{2}\sigma
T_{\rm{dust}}^{4}\ \rightarrow\
C_{2}^{\prime}\dot{m}X^{-2}M_{\rm{BH}}^{-1}\leq 2000^{4}\\\ $ (24)

where $\sigma$ is the Stefan - Boltzmann constant. Therefore, the condition
necessary for the survival of dust grains does depend on the BH mass
$M\rm_{BH}$.

Figure 17 (right-panel) presents these two constraints ($i.e.$, survival and
formation of dust) in terms of $X$ and $\dot{m}$, directly from Eqns. 23 and
24.The parameter space shown in white color on the figure corresponds to the
combination of $X$ and $\dot{m}$ that satisfies both these conditions. It is
evident from the figure that with decreasing the mass accretion/outflow rate
$\dot{m}$, dust formation becomes feasible at increasingly larger values of
$X$, that is toward the outer boundaries of the accretion disk. Also, the
dependence of the black hole mass in these constraints implies that a larger
black hole mass makes a larger fraction of $X$-space available to dust
formation. As a consequence, a BH of only 10 M⊙, limits the allowed values of
$X$ to larger than 107, (in physical units $R\sim 10^{13}$ cm), which is
generally beyond the outer boundary of the accretion disk, suggesting the
absence of dust formation in the black holes of the Galactic binaries.

The radial dependence of the winds (taken at present to be proportional to
$(\dot{m}/M)X^{-1}$) affects significantly the rate of dust production in two
ways: First by limiting the $\dot{m}-X$ parameter space where dust formation
becomes possible and second by reducing the amount of dust production rate
over the volume of the wind. For instance, for $n(X)\propto X^{-3/2}$
(Blandford & Payne, 1982), the ratio of the flow/reaction timescales (the
green line of Fig. 17b) would be independent of $X$, i.e. it would be a
vertical line at some value of $\dot{m}$. Finally, a radiation pressure driven
wind of $n(X)\propto X^{-2}$ would limit even further the parameter space,
with the green line having a positive slope, $X^{1/2}\leq(\dot{m}\zeta)$, as
one can easily deduce.

The rate of dust production over the wind volume $V$ is proportional to
$\dot{M}_{\mathrm{dust}}\propto\int n^{2}(X)\mathrm{d}V(X)$ where the density
is expressed as $n(X)\propto(\dot{m}/M)X^{-p}$. For $p\simeq 1$, $i.e.$, the
value taken in our model, we have
$\dot{M}_{\mathrm{dust}}\propto\dot{m}^{2}MX_{\rm max}$ ($X_{\rm max}$ is the
outer boundary of the volume in dimensionless units), a scaling consistent
with the result of the detailed calculations depicted in Fig. 17 (left-panel).
For $p=3/2$ (Blandford & Payne, 1982),
$\dot{M}_{\mathrm{dust}}\propto\dot{m}^{2}M\log X_{\rm max}$ while for $p=2$,
$\dot{M}_{\mathrm{dust}}\propto\dot{m}^{2}M(1/X_{\rm max})$, indicating
significant reduction in dust production for the larger values of the
parameter $p$, as one intuitively anticipates.

To conclude, we would like to remind the reader that the relations between
$\dot{m}$ and $X$, shown in this section are meant to indicate the broader
trends associated with dust production in AGN winds; they ignore the details
of the wind structure and the chemistry involved; however, these broader
trends are not in disagreement with the more detailed results shown in fig. 17
for some specific cases of dust production as a function of $\dot{m},M$, for
$n(X)\propto X^{-1}$.

##  9 Discussion

This work presents the first attempt to a comprehensive theory of the so-
called AGN “dusty” tori, in that it includes, besides the structure of the
tori, in terms of MHD accretion disk winds, also the production of ‘new dust’
in these winds. To date, the presence of any dust has been assumed,
constrained only by its absence interior to the dust sublimation line.

The primary objective of the present study has been to determine whether the
winds blown off the accretion disks in an AGN can provide an environment that
is suitable for new dust formation.

We would like to clarify again that, the term ‘new dust’ is generally
associated with the synthesis of new metals in stellar environments through
nuclear burning. However, the outflows of AGN accretion disks are not
associated directly with any nuclear burning phase. Therefore, in this case,
by ‘new dust’ we are referring to the formation of molecules, seeds and dust
grains from an initial fully or partly dust-free wind carrying metals
synthesized elsewhere, or a high-efficiency of reformation of the dust that
has been destroyed by the radiation from the central BH.

The essence of this work is the intimate relation between the nature and
structure of the tori themselves, namely the accretion disk MHD winds (Konigl
& Kartje, 1994), and the formation of the dust they apparently incorporate. To
the best of our knowledge this is the first work to employ the details of
molecule nucleation and condensation and grain coagulation to study the
formation and growth of dust in AGN winds. As discussed in detail, significant
amounts of dust can be produced only for winds whose density dependence is
sufficiently shallow, i.e. proportional to $X^{-1}$ (Contopoulos & Lovelace,
1994; Fukumura et al., 2010, 2018b).

Assuming the winds have the requisite density profile, one of the main results
of our analysis is the quantitative calculation of the (dimensionless) dust
mass-flux in these winds, $\dot{m}_{\rm d}$; this is shown to be $\dot{m}_{\rm
d}\simeq\dot{m}^{2}$ (i.e. the dimensionless mass flux rate onto the black
hole), which for $\dot{m}\simeq 1$, becomes comparable to the mass flux needed
to power the black hole emission. This scaling provides a convenient and utile
measure of the mass and mass flux of dust in the AGN ‘dusty tori’ in terms of
the AGN global properties. As such, it also provides a measure of the rate by
which they enrich the AGN interstellar or intergalactic (should they be able
to escape the galaxy) medium. Because our models assume no dust present
initially in these flows, should observations indicate the presence of more
dust than suggested by the above scaling, one would have to surmise that the
original flow was not dust-free. The same scaling can also prove useful in
following the relative contribution of AGN and star formation to the cosmic
dust abundance and therefore to their relative contribution to the cosmic
infrared background.

While the winds we have invoked in this work are thought to extend to the
black hole vicinity, the inner radius of their dusty segment is much larger,
set by the dust sublimation distance (Eq. 24); their outer radius is obtained
assuming that the disk terminates at the distance where its Keplerian velocity
$V_{K}(X)$ is a few, say $\lambda$-times, ($\lambda>1$), the dispersion
velocity of the overlying spheroid $\sigma$, i.e.
$V_{K}(X_{\mathrm{max}})\sim\lambda\sigma$. Considering that $V_{K}/c\simeq
X^{-1/2}$, the maximum normalized distance is
$X_{\mathrm{max}}\simeq(c/\lambda\sigma)^{2}$. Assuming further that the BH
obeys the $M-\sigma$ relation, i.e. $M\sim 10^{7}M_{\odot}(\sigma_{2})^{4}$ or
$M_{7}\simeq\sigma_{2}^{4}$ ($\sigma_{2}=\sigma/100\,{\rm km\,s}^{-1})$, we
obtain for the case under consideration $R_{\mathrm{max}}\simeq 3\times
10^{12}M_{7}X_{\mathrm{max}}\,{\rm cm}\simeq 10\,M_{7}^{1/2}\lambda^{-2}\,{\rm
pc}$. This value is consistent with the size of the torus of Cyg A of
$\mathrel{\raise 2.15277pt\hbox{$>$}\mkern-14.0mu\lower
2.58334pt\hbox{$\sim$}}100$ pc, whose BH mass is $M_{8}\mathrel{\raise
2.15277pt\hbox{$>$}\mkern-14.0mu\lower 2.58334pt\hbox{$\sim$}}10$, obtained
directly by VLA imaging (Carilli et al., 2019).

Our calculations have also yielded the dust to gas mass ratio and the size
distribution of grains. As shown in the previous section, in the condensation
region, the dust to gas mass ratio varies between $10^{-5}$ and $10^{-2}$,
increasing with the dimensionless distance $X$. Freshly formed dust is
dominated by small size silicate grains, though some grains grow to a few
hundred Angstroms. The preponderance of large grains, however, increases with
the fraction of initial dust present in the wind.

The presence of grains in these winds should lead to features in the IR AGN
spectra. One of the most important such features is the 9.7 $\mu$m silicate
feature. This feature should be in emission or absorption, depending on the
observer inclination angle, in accordance with the AGN unification scheme
(Antonucci, 1993; Urry & Padovani, 1995). Such a dependence has been
apparently observed in a large sample of a wide range of AGN types (PG QSOs,
FR II radio galaxies, Seyfert 1 and 2, LINERS, BAL QSOs etc.) (Shi et al.,
2006; Mason, 2015), where the inclination angle was estimated by the absorbing
X-ray column $N_{H}$. For small values of $N_{H}$ ($\log N_{H}<21$) the
feature appears in emission, while it becomes increasingly prominent in
absorption as $\log N_{H}$ increases beyond $\log N_{H}>22$.

An open issue with the AGN dust emission is that of the presence of clumps. It
is generally assumed that the AGN tori are composed of high density clouds
similar to those associated with the AGN broad line emission. It is our
understanding that this view is sustained by the need of the broad
distribution of dust temperatures needed to reproduce the AGN IR spectra: The
source covering factor has a radial dependence, which leads to clouds of
different temperatures so that they reproduce the observed distribution
between 1 and 1000 $\mu$m. It is generally considered that a uniform wind
(e.g. Pier & Krolik (1992)) will produce dust emission too hot to be
consistent with observations. However, the density profile of our winds allows
for a well defined distribution of dust temperatures, given their $1/r$
density distribution, a fact that has been adopted in some models already
(Rowan-Robinson, 1995).

Finally, an issue we have not touched upon at all in our model is that of the
dynamical effects of radiation pressure on the dust, considering that its
cross section is much larger than that of Thomson.

The radiation pressure force on the dust would exceed that of gravity at
luminosities smaller ($\sim 10^{-2}$) than those of Eddington. One could argue
that this would modify completely the wind dynamics. While this issue requires
much more detailed modeling, we would like to note that the self-similar shape
of the poloidal magnetic field lines along which the wind flows, is eventually
determined by the pressure of gas in the spheroid into which the BH is
located. Its mass is much larger than that of the BH (by a factor of $\sim
10^{3}$) and it could help the wind maintain its general shape, despite the
increased radiation pressure. One, of course, has to bear in mind that our
description is confined to the regions of the BH sphere of influence, which is
of order of a few parsec. Once driven (by the magnetic fields) beyond this
region, the winds are subject to the effects of radiation pressure and would
likely expand on all directions in accordance with the influence of the
latter. However, it is important that the winds are lift off the accretion
disk to heights $H\simeq R_{max}$ before radiation pressure becomes important.
This cannot be achieved by the disk radiation pressure near $R_{max}$ whose
radiation field momentum is minuscule. The fact that the momentum flux in some
of these winds is larger than that of the AGN radiation field, maybe due to
the fact that they are indeed driven, at the smaller scales, by the disk
magnetic field, as suggested herein.

Other than that, a host of other issues (spectra, variability etc.) arising
with the presence of such detailed models of the dust content of AGN tori,
will be addressed in future works.

##  10 Summary

The results of our paper can be summarized as follows:

  1. 1.

We have calculated the dust production rate and the total mass of dust
produced in MHD winds driven by a centrally accreting black hole (BH)
characterized by a $M_{\rm BH}$, and a dimensionless accretion rate $\dot{m}$.

  2. 2.

We assumed that the accreted gas is dust free, and solar composition, and that
the AGN luminosity is the only radiative source controlling the temperatures
of the gas and dust in the outflow. The formation of dust starts with chemical
reactions that form seed nuclei which grow by coagulation and accretion
throughout the flow. Since the inflowing gas is oxygen rich (C/O $<$ 1), the
seed nuclei consist of various magnesium and silicate oxides, which grow by
accretion and coagulation to silicate grains.

  3. 3.

We identify four regions around the AGN, shown in Figure 7: (a) sublimation
zone where the temperature of any seed nuclei exceeds the sublimation
temperature; (b) a region where seeds can form but do not accrete or coagulate
to form dust grains; (c) a region where seeds can form and grow; and (d) a
region where no new seeds can form, but existing seeds can grow by accretion
and coagulation.

  4. 4.

We show that the resultant distribution of newly formed dust resembles a
toroidal volume of dusty gas, which we then identify as the dusty torus.
Morever the size of the torus matches well with that estimated from VLA
imaging. We emphasize that the dusty torus is therefore not a static region of
dusty gas surrounding the AGN, but instead a dynamic phenomena produced by the
outflowing winds.

  5. 5.

Formation of new dust, characterized by the high column density and opacities
along equatorial lines of sight relative to the polar ones, are in agreement
with the role of dust associated with the AGN unification theory.

  6. 6.

We justify the importance of $r^{-1}$ density law in the winds, for the
efficient formation and survival of dust grains, supporting the MHD models
where this density law was adopted.

  7. 7.

We find that the dust production rate can be approximated by a law such as
$\dot{M}_{\mathrm{dust}}\simeq A\ \dot{m}^{2.25}M_{8}$, where is $A$ constant
given by $\sim$ 3.5 yr-1. This is an upper limit.

  8. 8.

The production rates and dust yields presented above assume that the material
accreted by the BH is initially dust free. If a fraction of the condensable
material, $f_{d}$, was initially locked up in dust then the net dust
production rates need to be multiplied by $1-f_{d}$. In that case, the AGN may
be an important processor of any pre-existing dust. Their sublimation and
reformation will alter their composition and size distribution.

  9. 9.

The relative contribution of AGNs and star forming galaxies to the cosmic
abundance of dust can be related to their relative contribution to the cosmic
infrared background, and will require further detailed studies.

  10. 10.

The dust blown out of the galaxy by the AGN may be injected to the
intergalactic medium, and contribute to the diffuse IR emission observed in
galaxies.

We acknowledge NASA’s 16-ATP2016-0004 grant for supporting this project. We
thank the anonymous referee for the valuable suggestions that have helped us
in finalizing the manuscript of this paper.

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