2022

[1]Bożena Czerny

[1]Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników
32/46, Warsaw, 02-668, Poland

2]Department of Theoretical Physics and Astrophysics, Faculty of Science,
Masaryk University, Kotlářská 2, Brno, 611 37, Czech Republic

3]Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences,
Bartycka 18, Warsaw, 00-716, Poland

4]School of Physics and Astronomy, Tel Aviv University, Tel Aviv, 69978,
Israel

5]Laboratório Nacional de Astrofísica, MCTIC, R. dos Estados Unidos 154,
Itajubá, 37504-364, Brazil

6]Inter University Center for Astronomy and Astrophysics, Maharashtra, Pune,
411007, India

7]Astronomical Institute, Czech Academy of Sciences, Boční II 1401, Prague,
CZ-14100, Czech Republic

8]Departamento de Astronomía, Universidad de Chile, Camino del Observatorio
1515, Santiago, Casilla 36-D Correo Central, Chile

# Dusty plasma in active galactic nuclei

<EMAIL_ADDRESS>Michal Zajaček<EMAIL_ADDRESS>Mohammad-Hassan Naddaf
<EMAIL_ADDRESS>Marzena Sniegowska<EMAIL_ADDRESS>Swayamtrupta
Panda<EMAIL_ADDRESS>Agata Różanska<EMAIL_ADDRESS>Tek P. Adhikari
<EMAIL_ADDRESS>Ashwani Pandey<EMAIL_ADDRESS>Vikram Kumar Jaiswal
<EMAIL_ADDRESS>Vladimír Karas<EMAIL_ADDRESS>Abhijeet Borkar
<EMAIL_ADDRESS>Mary Loli Martínez-Aldama<EMAIL_ADDRESS>Raj Prince<EMAIL_ADDRESS>*  [  [  [  [  [  [  [

###### Abstract

Since many years we know that dust in the form of the dusty-molecular torus is
responsible for the obscuration in active galactic nuclei (AGN) at large
viewing angles and thus for the classification of AGN. Recently, we gained
some observational and theoretical insight into geometry of the region and the
role of the dust in the dynamics of the outflow and failed winds. We will
briefly touch on all these aspects, including our dust-based model (FRADO -
Failed Radiatively Accelerated Dusty Outflow) of the formation of the Balmer
lines in AGN.

###### keywords:

galaxies: active, quasars: emission lines, supermassive black holes, radiative
transfer

##  1 Introduction

Dust is present in all galaxies, both active and non-active, as it is one of
the constituents of the interstellar medium (see Draine, 2003, for the
review). Gas originates from the stellar winds, and vigorous winds are
characteristic of evolved stars which enriched their chemical composition
through the nuclear burning. Supernovae explosions are particularly efficient
in ejection of highly enriched material. Our Galaxy - Milky Way - also
contains considerable amount of dust (Calzetti et al, 2000; Weingartner and
Draine, 2001), so its nucleus is shielded from us by extinction of the order
of 25 mag in V-band (e.g. Nishiyama et al, 2008; Schödel et al, 2010;
Nogueras-Lara et al, 2020) since we are located at a distance of $\sim 8$ kpc
from it, roughly in the Galactic plane. However, in AGN (for a review of its
physical properties see Laor and Draine, 1993) the dust located at inner $\sim
10$ pc plays a very specific role which is directly associated with their
enhanced accretion activity in comparison with the currently inactive Milky
Way.

##  2 Basic properties of an AGN, dusty torus and standard unification scheme

Over the fifty years of the studies of quasars and even longer studies of
other types of active galaxies, such as Seyfert galaxies, radio galaxies and
blazars a basic understanding of the structure of these objects has been
achieved (see books by Krolik, 1999; Meier, 2012, for an in-depth summary).
All active galactic nuclei (AGN) consist of a supermassive black hole, and an
inflowing material which determines the level of activity.

Some AGN (about 10 %) also show strong relativistic outflows in the form of
relativistic jets, and these are classified traditionally as radio-loud
objects, or (more recently) as jetted objects (see e.g. Padovani et al, 2017,
for a review). Out of those objects, some have jets oriented closely to the
line of sight towards an observer, so the jet emission is relativistically
boosted and frequently over-shines the nuclear emission. In the later text, we
will concentrate on the radio-quiet (non-jetted) sources where we have much
better insight into processes going on close to the black hole.

Radio-quiet sources (about 30% of them, Véron-Cetty and Véron 2010; Shen et al
2011; Rakshit et al 2020) frequently show broad and narrow emission lines. The
line broadening must be caused by the dispersion in the velocity of the
emitting material since the kinematic widths of narrow lines are of the order
of a few hundred km s-1, and the broad lines have line widths of the order of
a few thousands km s-1, while the expected thermal broadening is of order of a
few km s-1 (Kollatschny and Zetzl, 2011). This last constrain comes from the
fact that Balmer lines are very strong, meaning that the hydrogen is not fully
ionized and the corresponding plasma temperature must be of the order of 10
000 K, and much higher temperature would lead to full ionization of hydrogen
and Balmer lines would not be observed (Collin-Souffrin, 1987). Thus the
standard view is that strong emission clouds come from clouds orbiting the
central black hole at a distance of a fraction of a parsec (for broad lines)
and above a parsec (for narrow lines). These regions are named broad lines
region (BLR), and narrow line region (NLR), correspondingly. The line emission
is caused by the irradiation of the plasma by the central region where most of
the accretion energy is dissipated and reemitted (Wills et al, 1985;
Osterbrock and Ferland, 2006). When broad lines are seen, narrow lines are
also present, and this shows up as a complex line shape of a single line
consisting of two separate components: one broad, and the other narrow.

The emission lines differ not only in emitter velocities but also in the
character of their formation. Balmer lines are seen both as broad and narrow
lines in a given source, but some of the narrow lines coming from other
elements are actually forbidden lines, line [OIII] at $\lambda 5009$ Å, while
all broad lines are permitted lines, which means that the density of the
emitting material is much lower in the NLR (e.g. Baskin and Laor, 2005;
Bennert et al, 2006; Müller-Sánchez et al, 2011; Kakkad et al, 2018).

A fraction of sources shows only narrow lines, and this was a puzzle for years
why BLR in these sources is not seen. Most quasars show broad lines, but
fainter Seyfert galaxies frequently did not show BLR, so Seyfert galaxies were
further sub-classified as Seyfert 1 (with BLR), and Seyfert 2 (without BLR),
and quasars were also divided into type 1 and type 2 by analogy, although type
2 is rare. The resolution came from polarimetric observations perfomed by
Antonucci and Miller (1985) for the famous Seyfert galaxy NGC 1068 from the
original list of active galaxies selected by Seyfert in his seminal paper
(Seyfert, 1943). In ’normal’ (i.e. unpolarized) light the source did not show
BLR component but in the polarized light the broad component of the Balmer
H$\beta$ line was clearly present. The interpretation was that a thick torus
surrounds the nucleus, and the BLR is located closer in than the torus while
NLR is located further out. If the viewing angle is low (as measured with
respect to the symmetry axis) the view of the nucleus is unobscured and we see
both NLR and BLR line components but if the viewing angle is high the torus
shields the BLR component and we do not see it. It is only revealed again in
the polarized light since in this case we can see a small fraction of the
light that has been scattered by the low density material filling the almost
empty space around the torus. This torus is now known as a dusty/molecular
torus, with the inner radius a distance of about a parsec (Koshida et al,
2014). The torus itself is seen in the infrared part of the AGN spectrum,
since dust also absorbs the radiation of the central region but since the dust
sublimation temperature is of the order of 1500 K the dust emission appears
only at long wavelengths. Dust cannot be much hotter - it then evaporates and
the material is changed into (rich in heavy elements) gaseous phase
(Barvainis, 1987; Kishimoto et al, 2007). A lot of further insight came from
observing and modelling this source (e.g. Miller et al, 1991; Hönig et al,
2006; García-Burillo et al, 2014).

Thus, the dusty/molecular torus provided a key element in the unification
picture of radio-quiet AGN implying that their structure is always the same
and the apparent variety of types comes just from different orientation. The
truth is slightly more complex since another key element is the accretion rate
onto the black hole. It is conveniently parameterized by the Eddington ratio:
the ratio of the luminosity dissipated in the flow to the maximum luminosity
of the spherical system in which the radiation pressure (calculated for fully
ionized pure hydrogen) marginally balanced the gravity of the central body.
This ratio does not depend on the value of the black hole mass which in AGN
can range from a few hundreds of thousands of solar masses to ten billions of
solar masses, so it is a very convenient value to use, although an AGN is
neither spherically symmetric nor consists of pure hydrogen. In quasars the
Eddington rate is typically about 10 % (between 1 % and 100 %, see e.g. Panda
et al 2018, their Fig. 3, Martínez-Aldama et al 2020, their Fig. 3), but in
nearby AGN it is frequently much lower. Thus quasars seem to always have a
torus, while for nearby sources there may be a considerable fraction of true
Seyfert 2 galaxies, where there is no torus. The spectra of faint sources are
strongly dominated by starlight so the corresponding polarization measurements
are difficult, and broad lines are also difficult to distinguish against the
strong starlight background. On one hand, extremely inactive galaxies like the
center of the Milky Way - Sgr A* - do not have a dusty torus (its Eddington
ratio is about $10^{-9}$; see e.g. Event Horizon Telescope Collaboration et al
2022, Eckart et al 2017). On the other hand, very careful observation of
potentially true Seyfert 2 galaxy NGC 3147 with the Hubble Space Telescope
revealed the existence of broad H$\alpha$ line (Bianchi et al, 2019, 2022) so
in this case the line was not shielded, it was just difficult to detect. The
Eddington rate also influences the shape of the broad band spectrum which is
related to the character of the inner flow close to the black hole. In higher
Eddington sources (Eddington ratio above 1 % ) the inflow is in the form of a
cold optically thick geometrically thin accretion disk described rather well
by a standard model of Shakura and Sunyaev (1973) and its relativistic version
(Novikov and Thorne, 1973), and by slim disk at very high Eddington rates
(Abramowicz et al, 1988). The spectrum is dominated by the Big Blue Bump
component (e.g. Czerny and Elvis, 1987). At lower Eddington rates the inner
flow is most likely optically thin, in the form of advection-dominated
accretion flow (ADAF; Ichimaru 1977; Narayan and Yi 1994). Various extensions
of this model were proposed, particularly in the context of jetted sources,
like JED-SAD (jet emitting disk - standard accretion disk) models (Ferreira
and Pelletier, 1995; Petrucci et al, 2008) or MAD (magnetically arrested disk)
models (Narayan et al, 2003, 2022). So the unification scheme actually
requires three parameters: jet strength, orientation, and Eddington rate, but
nevertheless unification works surprisingly well over the huge range of black
hole masses, and there is also a very good analogy in spectral behaviour
between AGN and galactic black holes of masses of the order of only 10 solar
masses (e.g. Sobolewska et al, 2011; Moravec et al, 2022), apart from the
issue of the torus.

All three parameters strongly influence the broad band spectrum of an AGN, and
by observing the spectrum, including the variability, the structure of AGN was
studied since the nucleus is basically unresolved. Only in the nearby two
objects: the famous radio galaxy M87 and the center of the Milky Way - Sgr A*
- which were resolved to such an extent by the Event Horizon Telescope that
the shadow, or silhouette, of the central black hole is visible, as well as
the innermost pattern of the hot flow in semi-circular motion (Event Horizon
Telescope Collaboration et al, 2019, 2022). Observations were done in the mm
band, optimized for the resolution and the extinction, and the observed
radiation was the synchrotron emission of hot, low-density plasma. Both
sources are examples of sources at very low Eddington rates, so no cold disk
was observed.

##  3 Coexistence of the hot and cold dustless plasma

Spectral complexity of AGN - broad band continuum and various types of
emission lines - implies that the plasma in AGN has a broad parameter range
with respect to the density and ionization state. In addition, AGN show a
range of absorption lines which come from the plasma located along the line of
sight towards the black hole vicinity where most of the optical/UV/X-ray
emission originates. These lines are phenomenologically characterized by their
width, strength and the ionization of the corresponding ions. We thus have
narrow absorption lines (NAL; e.g. Wampler et al 1995; Elvis 2000), broad
absorption lines (BAL; e.g. Weymann et al 1991; Choi et al 2022), warm
absorbers (WA; e.g. Nandra and Pounds 1994; Holczer et al 2007; Adhikari et al
2019) and ultra-fast outflows (UFO; e.g. Tombesi et al 2010). The distance of
the material forming those lines is difficult to estimate since then we cannot
identify the line widths or shifts with Keplerian motion, as we do for the
BLR. It may be that some of these absorption lines are actually due to BLR or
NLR seen in absorption instead of emission. Absorption lines imply a clumpy
structure of the absorbing medium, since numerous kinematic components are
observed, particularly in UV, with different velocity shifts with respect to
the rest frame of an AGN. Also for BLR and NLR traditionally a clumpy picture
was envisioned.

That scenario might pose a problem of the cloud stability since a dense, non-
self-gravitating cloud in an empty space would expand on the dynamical
timescale and would get eventually dispersed. However, if cooler denser clouds
are actually embedded in much hotter low density fully ionized plasma, they
can be stabilized. This is well justified theoretically. As pointed out by
Krolik et al (1981) in the context of AGN, irradiated plasma is subject to
thermal instability, and at a certain range of the gas pressure, two solutions
for the gas temperature can co-exist: low density hot plasma, at the
temperature of the order of $10^{7}$ K, which heats and cools by the Compton
process, and colder, denser plasma at temperature of order of $10^{4}$ K,
which mostly heats and cools by atomic processes. In general, all radiative
processes are important in partially ionized plasma, and they include
Comptonization (heating and cooling), free-free transitions, bound-free as
well as bound-bound transitions. Thus, plasma at two extremely different
temperatures can coexist. The equilibrium curves are most conveniently showed
using the ionization parameter $\Xi$ introduced by Krolik et al (1981)

$\Xi={F_{ion}\over nkTc},$ (1)

where $F_{ion}$ is the ionizing flux between 1 and $10^{3}$ Rydberg, $n$ is
the local number density, $T$ is the medium temperature, $k$ is the Boltzmann
constant, and $c$ is the speed of light. This parameter is dimensionless, and
it basically measures the ratio of the radiation pressure to gas pressure.

The colder clouds can be in equilibrium when they are located on the
equilibrium curve at the same value of $\Xi$ on the lower and upper branch.
The intermediate branch is unstable, and under small perturbation the plasma
rapidly evolves on the thermal timescale towards one of the stable branches.

The coexistence of the two plasma states with a sharp discontinuity is
actually more complex. Clouds out of equilibrium can reach equilibrium but
they can also be slowly growing through condensation or disappearing by
evaporation. Computations of clouds out of equilibrium require the inclusion
of the additional process: electron conduction, which is important at the
sharp border between the two phases, where the temperature gradient is very
large. This was recognized by Field (1965) and subsequently studied in a
number of papers in the context of AGN, both for the transition between the
colder disk and the hotter plasma at the disk corona, and for warm absorber
(WA) and BLR clouds (e.g. Begelman and McKee, 1990; Różańska, 1999; Chevallier
et al, 2007). These computations also require proper treatment of the
radiative processes, and for that complex radiative transfer codes are used,
such as CLOUDY (Ferland et al, 2017), which allow to calculate the ionization
state of the material, its temperature and actually perform the radiative
transfer. The conduction term is frequently neglected if only the stationary
solution is of interest, but then there are problems with uniqueness of the
solution under the constant pressure.

There are numerous papers which use the equilibrium curve for the discussion
of the properties of the multiphase medium in the context of AGN. For example,
Pittard et al (2001) used it in the context of cloud formation in the
colliding AGN-supernovae winds. Several authors (e.g. Krolik and Kriss, 2001;
Gupta et al, 2013; Adhikari et al, 2015, 2019) considered the pressure balance
in different phases of the WA medium. The issue further complicates if we
consider outflowing medium (Dannen et al, 2020; Waters et al, 2021), or even
supersonic outflow velocities (Waters et al, 2022).

The scenario of the coexistence of the colder clouds and the intercloud medium
was also formulated as the radiation pressure confinement (RPC) of the clouds
(see Stern et al, 2014; Baskin and Laor, 2021, and the references therein). In
any case, the rough pressure balance and approximate thermal equilibrium allow
for cloud formation and the relatively long cloud existence. There are,
however, mechanisms destroying such clouds (see e.g. a discussion in Müller et
al, 2022) e.g. in cloud collisions, action of tidal forces, Kelvin-Helmholtz
instability so the clumpy medium is necessarily always dynamical. The same is
true for the interstellar medium, although timescales in AGN are much shorter
due to smaller sizes of the clouds and much higher local densities.

##  4 Coexistence of the hot and cold dusty plasma

While the issue of the coexistence of the hot $T\sim 10^{7}$ K and warm $\sim
10^{4}$ K plasma was frequently discussed in the context of AGN (see Section
3), the coexistence of the third phase, cold dust $T<1500$ K with the other
two phases is less well studied from the physical point of view. On the other
hand, we know that the dust is there, within the central 1 pc as the
dusty/molecular torus (see Section 2). In the advanced torus models, the
clumpiness of the medium is included (e.g. Nenkova et al, 2008a, b; Stalevski
et al, 2012; Siebenmorgen et al, 2015) since otherwise the more distant parts
of the torus are not adequately illuminated and the model does not reproduce
well the observed silicate features in absorption or emission. However, the
intercloud medium in these models is usually neglected.

The first, extremely influencial paper which addressed the issue of the dusty
plasma interaction with the incident radiation was published by Netzer and
Laor (1993). Their study of the dusty plasma and its efficiency of the
production of emission lines showed that the presence of dust strongly
suppressed the line emissivity, and thus they showed that the gap between the
BLR and NLR (see Section 2) is actually caused by the presence of the dust.
Thus one of the key puzzles in AGN was explained.

However, later a new sub-type of AGN was introduced - Narrow Line Seyfert 1
galaxies (Osterbrock and Pogge, 1985). These sources were different from type
2 sources, they had broad lines, but much narrower than in typical Seyfert
galaxies, and their forbidden lines were weak, unlike in Seyfert 2 objects. A
formal definition included the maximum width of the lines of 2000 km s-1.
Later, for quasars which typically have much higher masses, a similar new
class was introduced, type A (with the limit of the line width of 4000 km
s-1), while typical quasars were classified as type B, with broader lines
(Sulentic et al, 2000). Subsequent studies showed that these sources actually
accrete close to the Eddington limit, and they form a high accretion rate tail
of all AGN (Panda et al, 2019; Panda and Marziani, 2022). In addition, these
objects do not show two-component shapes of lines, and their line profiles
were well fitted by a single Lorentzian shape (e.g. Sulentic et al, 2000;
Kollatschny and Zetzl, 2013; Zajaček et al, 2020). The explanation of this
lack of the BLR - NLR gap in these sources was solved by Adhikari et al (2016)
who showed that if the medium density is high enough, then the dust does not
compete efficiently for photons with the partially ionized plasma, so the gap
postulated by Netzer and Laor (1993) does not form, and indeed objects
accreting at high Eddington rate do have higher local densities in the BLR
than less vigorously accreting sources (Panda et al, 2018).

Figure 1:  Stability curves for the circumnuclear medium for two different
heavy element abundances: upper panels show ISM abundances, while bottom
panels show solar abundances, with increasing radii from left to right: 11.5,
55, and 180 pc. Each panel contains calculations with and without dust. Each
point in the curve represents the model computed for one starting value of
density. The range of computed densities is given in each panel. The SED
luminosity is always kept at $L_{bol}=10^{43}$ erg s-1 (from Borkar et al
2021).

The issue of the coexistence of the three phases in AGN were finally studied
in some detail by Borkar et al (2021) using the code CLOUDY in the context of
high resolution observational data for Centaurus A from ALMA and Chandra. We
constructed the equilibrium curves for dusty plasma irradiated by hard X-rays
from the central region (see Figure 1). Here the temperature $T$ is the plasma
temperature. We see that without dust the temperature saturates at values just
below $10^{4}$ K even for a low value of the ionization parameter $\Xi$, only
at the distance of $\sim 200$ pc, the temperature finally drops below $10^{3}$
K. At the highest values of $\Xi$ the plasma temperature saturates at the
Inverse Compton temperature value, about $10^{8}$ K for the adopted shape of
the radiation spectrum. We clearly see the unstable branch, corresponding to
the negative slope of the stability curve. This corresponds to the possibility
of the coexistence of the warm and hot plasma. However, in the presence of the
dust, the temperature drops to much lower values for high $\Xi$, forming there
an additional branch of solutions with an instability present at $\Xi\sim
10^{-5}$. There we can have a coexistence of the cold and warm plasma. For
high values of $\Xi$ the temperature of the dusty plasma is much lower, due to
the interaction with dust grains. There, at $\Xi\sim 100$ there is a narrow
range of the coexistence of the warm and hot plasma. Note that the temperature
of the dusty plasma can be as high as $10^{7}$ K. The dust there can still
exist, and its temperature is lower than the sublimation temperature, as
reported by the CLOUDY code but the dust temperature is not plotted. Thus the
dusty medium at high $\Xi$ is a two-temperature medium, with individual cold
dust grains embedded in the hot plasma. This is possible due to the low
density of the hot plasma.

The computations thus show two ranges of $\Xi$ at which clumpy medium can
form, but all three phases cannot coexist. Cold dusty cloud cannot be in
pressure equilibrium with the hot plasma, at least in the presented
computations that consider optically thin medium. In the optically thick
medium such coexistence is perhaps possible but the code CLOUDY cannot be used
to describe such a scenario. Computations performed by Borkar et al (2021)
also did not address the processes taking place closer to the black hole,
within the BLR medium.

During the photoionization process by intense X-rays near an active nucleus
the dust grains develop electric charge that is attached to their surface, and
thus a complex dusty plasma is formed (Draine and Salpeter, 1979; Weingartner
et al, 2006). Collective effects between the dust and gaseous components are
essential in this environment, and the dust grain properties are different
from those observed in the local interstellar medium. It has been argued that
in AGN the dust grains suffer desctruction by charging (e.g. Tazaki et al,
2020, and further references cited therein) and the dust can be depleted.
Also, the equilibrium structure of dusty tori may be significantly altered by
the inteprlay of gravitational and electromagnetic forces in contrast with
classical test-fluid solutions (Trova et al, 2016).

##  5 Coexistence of hot and warm dusty plasma in the Galactic Center

The Galactic center serves as the nearest prototype of a low-luminosity
galactic nucleus. The bolometric luminosity of the compact variable source Sgr
A*, which is associated with the supermassive black hole of $M_{\bullet}\sim
4\times 10^{6}\,M_{\odot}$ (Eckart et al, 2017; Genzel, 2022; de Laurentis et
al, 2022), is $L_{\rm SgrA*}\lesssim 10^{37}\sim 2600\,L_{\odot}\,{\rm
erg\,s^{-1}}$ (Narayan et al, 1998), while the comparison with the Eddington
luminosity $L_{\rm Edd}=5.03\times 10^{44}(M_{\bullet}/4\times
10^{6}\,M_{\odot})\,{\rm erg\,s^{-1}}$ for a given mass yields the Eddington
ratio of $\lambda_{\rm Edd}=L_{\rm SgrA*}/L_{\rm Edd}\lesssim 2\times
10^{-8}$. However, the source is variable with order-of-magnitude near-
infrared and X-ray flares taking place a few times per day (see e.g. Witzel et
al, 2021, and references therein), which appears to be related to the hot
plasma components with the size of $\sim 4$ gravitational radii and the
temperature of $\sim 10^{10}\,{\rm K}$, the electron density of $4\times
10^{7}\,{\rm cm^{-3}}$, and the magnetic field of $\sim 10\,{\rm G}$ orbiting
close to the innermost stable circular orbit in the form of “hot spots”
(GRAVITY Collaboration et al, 2018; Wielgus et al, 2022). There are several
signatures that the accretion rate was significantly larger on the timescale
of several hundred to million years, which has been revealed by the
reprocessing of the central X-ray emission by the surrounding molecular clouds
(Sunyaev et al, 1993; Sunyaev and Churazov, 1998) and by the large-scale
$\gamma$-ray emitting Fermi bubbles and the X-ray emitting eRosita bubbles (Su
et al, 2010; Predehl et al, 2020).

Because of the current low Eddington ratio, the accretion flow surrounding Sgr
A* is hot, diluted with a low radiative efficiency. It can be modelled in the
framework of advection-dominated accretion flows (or ADAFs) with a large
scale-height to radius ratio approaching unity. On the larger scale, the flow
is Bondi-like with power-law temperature and density profiles of the hot
plasma with the asymptotic values of $n\sim 26\,{\rm cm^{-3}}$ and $T\sim
1.5\times 10^{7}\,{\rm K}$ at the Bondi radius (Baganoff et al, 2003), which
can be estimated as follows,

$\displaystyle R_{\rm B}$ $\displaystyle=\frac{2GM_{\bullet}}{c_{\rm
s}^{2}}\simeq\frac{2GM_{\bullet}\mu m_{\rm H}}{kT}\,$ $\displaystyle\sim
0.14\,\left(\frac{M_{\bullet}}{4\times
10^{6}\,M_{\odot}}\right)\left(\frac{T}{1.5\times 10^{7}\,{\rm
K}}\right)^{-1}{\rm pc}\,,$ (2)

where $c_{\rm s}$ is the asymptotic sound speed in the plasma. The Bondi
radius expressed by Eq. (2) corresponds to $R_{\rm B}\sim 7.3\times 10^{5}$
gravitational radii. However, the X-ray surface brightness is better described
by a flatter density profile than for the standard Bondi inflow, i.e.
$n\propto r^{-3/2+s}$, where $s\sim 1$, which implies the presence of an
outflow (Wang et al, 2013b).

The ADAF is located inside the Bondi radius, which is well within the ionized
central cavity with the radius of $\sim 1-1.5$ parsecs filled mostly with hot
X-ray emitting optically thin plasma. In this region, which is also referred
to as Sgr A West, there are distinct thermal streamers known as the
Minispiral, whose kinematics is consistent with three bundles of Keplerian
orbits around Sgr A* corresponding to the Northern Arm, Eastern Arm, and the
Western Arc (Zhao et al, 2009, 2010). The minispiral is well detected in mid-
infrared domain (Bhat et al, 2022), which reveals the dust emission, as well
as in the submm-, mm-, and radio domains. The electron density of the
Minispiral is in the range of $3-21\times 10^{4}\,{\rm cm^{-3}}$ and the
electron temperature is $5\,000-13\,000\,{\rm K}$ (Zhao et al, 2010). The
largest densities of $21\times 10^{4}\,{\rm cm^{-3}}$ are coincident with the
IRS 13 region, while the highest temperature of $1.3\times 10^{4}\,{\rm K}$ is
in the Bar region where the Northern and the Eastern Arms appear to collide at
$\sim 0.1-0.2$ pc south of and behind Sgr A* (Zhao et al, 2010). The dust
density traces the ionized gas density along the three arms, which leads to
the emission enhancement in the mid-infrared domain in the clumps along the
Minispiral as well as at the places where supersonic stars plow through the
ambient plasma and create denser bow shocks (Tanner et al, 2002, 2003; Rauch
et al, 2013; Sanchez-Bermudez et al, 2014). This shows that dust is generally
coupled to gas via gas drag like in the standard interstellar medium and the
dust-to-gas mass ratio of 1:100 (Bohlin et al, 1978) is usually adopted for
basic estimates.

Towards the Galactic center, the near-infrared emission (2.2 ${\rm\mu m}$) is
moderately polarized with the mean degree of $\sim 4\%$ and the position angle
of $\sim 30^{\circ}$ along the Galactic plane. This foreground polarization is
apparently due to anisotropic scattering of infrared radiation on elongated
dust grains that are aligned along the Galactic spiral arms between the Earth
and the Galactic center (Eckart et al, 1995; Ott et al, 1999). The presence of
dust grains that are dynamically coupled to the sheared gas along the orbiting
Minispiral arms can also be revealed via the polarized light. Aitken et al
(1991) and Aitken et al (1998) detected the polarized mid-infrared emission
($12.4\,{\rm\mu m}$) of ionized Minispiral filaments, which corresponds to the
thermal emission of elongated dust grains aligned along the magnetic field
lines that are approximately parallel to the bulk streaming motion of the
Minispiral gas. This indicates that the magnetic field with the strength of
$\gtrsim 2$ mG is an inherent property of the sheared ionized gas of the
Minispiral. Roche et al (2018) confirmed that the mid-infrared polarization is
a general property of the diffused ionized gas in the Galactic center. They
found the largest polarization degree of $\sim 12\%$ at 12.5 ${\rm\mu m}$ in
the Northern Arm. The polarized emission of embedded bow-shock sources is
generally enhanced due to the compressed magnetic field and the polarization
degree also increases from near-infrared to mid-infrared domains, which
implies that thermal emission by aligned dust grains, i.e. the reprocessed UV
emission of the central star, is responsible for the polarization properties
(Buchholz et al, 2011, 2013). For the most prominent bow-shock sources, such
as IRS 21 and IRS 1W, the degree of polarization of the integrated infrared
emission falls in the interval of $\sim 10-30\%$, which can be explained by
the Mie-scattering on dust grains whose density is enhanced in asymmetric
gaseous-dusty shells surrounding the supersonic stars (Eckart et al, 1995; Ott
et al, 1999).

The ionized central cavity ends quite sharply at $\sim 1.5$ parsecs. Its
radius is determined by the ambient UV field of the nuclear star cluster,
which is concentrated in the inner parsec. Further away, from about 1.5 pc to
$\sim 3-4$ parsecs, dense ring-like circumnuclear disk (CND) is located that
is composed of neutral and molecular gas and dust with the equilibrium
temperature between 20 and 80 K (Becklin et al, 1982; Mezger et al, 1989). The
gas number density falls into the range $\sim 10^{5}-10^{8}\,{\rm cm^{-3}}$,
being the densest in numerous clumps along the ring (Christopher et al, 2005)
that have typical sizes of $\sim 0.25\,{\rm pc}$ and masses of a few
$10^{4}\,M_{\odot}$. The whole CND has a mass of $\sim 10^{6}\,M_{\odot}$,
being about three to four orders of magnitude more massive than the
Minispiral, whose ionized mass is $\sim 350\,M_{\odot}$ (Zhao et al, 2010).
Because of its higher mass and several tens of dense clumps, it has a
potential to provide the material for the in-situ star formation in the
nuclear star cluster (Trani et al, 2018), though there is no evidence of
ongoing star formation; most likely turbulence and magnetic field prevent most
clumps from collapsing to form stars (see e.g. Hsieh et al, 2021). The CND and
the Minispiral streamers are interestingly not coplanar. While the Northern
Arm and the Western Arc reside in nearly the same plane, which seems to be
coincident with the plane of the CND, the Eastern Arm is nearly perpendicular
to them (Vollmer and Duschl, 2000; Zhao et al, 2010). In this regard, the
Northern Arm and the Western Arc might have originated in the inner rim of the
CND, where they got detached by the combined effect of photoionization and the
interaction with the outflow from the nuclear star cluster. The Minispiral is
therefore a transient, disappearing feature with the dynamical timescale of
$\sim 10^{4}$ years. At larger distances from the Sgr A complex, there are
several massive molecular clouds, such as “$+20\,{\rm km\,s^{-1}}$” and
“$+50\,{\rm km\,s^{-1}}$” giant molecular clouds (see Mezger et al, 1996, for
a review).

Figure 2: Illustration of the central region of the Galactic center (S
cluster, 1”$\sim$0.04 pc). Dust-embedded D/G sources are characterized by a
higher dust temperature of $\sim 500$ K with the photospheric radius of
$R_{\rm D}\sim 1$ AU, while coreless Minispiral streamers have the dust
temperature of $\sim 200-300$ K. Main-sequence stars within the S cluster are
predominantly of spectral type B with the effective temperature of
$20-30\times 10^{3}$ K. The whole region is filled with optically thin, X-ray
emitting plasma with the number density of $n\sim 10\,{\rm cm^{-3}}$ and
$T\sim 10^{7}$ K. Redrawn from Peißker et al (2020).

In the central cavity, dust is present and observed not only within the
Minispiral, but also closer in within the S cluster, i.e. within 1”$\sim
0.04\,{\rm pc}$ from Sgr A*. Coreless gaseous-dusty filaments within the
Minispiral are characterized by warm dust with the temperature of $\sim
190-265$ K, which was inferred based on the 12.5/20.3 ${\rm\mu m}$ colour-
temperature map (Cotera et al, 1999). In the S cluster, fast and compact dusty
D or G objects with emission lines of hydrogen and helium have been identified
(Gillessen et al, 2012; Eckart et al, 2013; Peißker et al, 2020; Ciurlo et al,
2020; Peißker et al, 2021) that orbit Sgr A* on eccentric orbits similar to S
stars. They are characterized by the optically thick envelopes with the
photospheric radius of $\sim 1\,{\rm AU}$ and the black-body temperature of
$\sim 500-600$ K (Peißker et al, 2020), which can be attributed to the dust
thermal emission. The higher temperature of dust in comparison with dusty
streamers can be explained by the model of a dust-enshrouded star (Zajaček et
al, 2014, 2017), i.e. an optically thick dusty shell is heated by a central
star, see Fig. 2 for an illustration. Although the nature of D/G sources is
disputed, the pericenter passages of G2/DSO and G1 sources can be used to put
an upper limit on the dust sublimation radius around Sgr A*, $R_{\rm sub}\sim
137\,{\rm AU}=3469\,r_{\rm g}$ and $R_{\rm sub}\sim 298\,{\rm AU}=7577\,r_{\rm
g}$ for G2/DSO and G1 pericenter distances (Witzel et al, 2017; Peißker et al,
2021), respectively. Because of the current low-luminous state of Sgr A*, it
can be shown that dust can approach Sgr A* within the innermost 100
gravitational radii if it is not destroyed before in shocks and by the UV
emission of numerous OB stars. However, it cannot approach or form and linger
as close as the innermost stable circular orbit if it is not shielded. We
estimate the corresponding sublimation radius $r_{\rm sub}$ for graphite dust
following Barvainis (1987), assuming the sublimation temperature of $T_{\rm
sub}\approx 1500\,{\rm K}$, the zero optical depth $\tau_{\rm UV}=0$, and the
UV luminosity of Sgr A* with the upper limit given by its current bolometric
luminosity $L_{\rm UV}\lesssim L_{\rm bol}\approx 100\,L_{\odot}$ (Melia and
Falcke, 2001). The sublimation radius scaled to the gravitational radius can
then be expressed as,

$\frac{r_{\rm sub}}{r_{\rm g}}=38.07\left[\left(\frac{L_{\rm
UV}}{100\,L_{\odot}}\right)\exp{(-\tau_{\rm
UV})}\right]^{\frac{1}{2}}\left(\frac{T_{\rm sub}}{1500\,{\rm
K}}\right)^{-2.8}\left(\frac{M_{\bullet}}{4\times
10^{6}\,M_{\odot}}\right)^{-1}\,.$ (3)

The distribution of dust temperature $T_{\rm dust}$ up to the sublimation
temperature of $1500$ K as a function of the distance from Sgr A* (in
gravitational radii) is plotted in Fig. 3, following the simple model of
Barvainis (1987) that takes into account only the central luminosity source.
We consider the optical depth in the range $\tau_{\rm UV}\in(0,10)$ and the UV
luminosity of Sgr A* $L_{\rm UV}=100\,L_{\odot}$, which is approximately
achieved during daily infrared/X-ray flares (see e.g. Yuan and Narayan, 2014).
It is clear that dust cannot exist close to the innermost stable orbit (dashed
gray vertical line for a non-rotating black hole; $6\,r_{\rm g}$) unless it is
shielded by an optically thick envelope. Conditions for survival of the clouds
drifting in/out or forming in a changing conditions are more complex since
then the cooling timescales must be included (e.g. Różańska et al, 2014, 2017;
Bhat et al, 2022).

Figure 3: Dust temperature distribution as a function of the distance from a
low-luminous galactic nucleus (in gravitational radii) and the UV optical
depth. The calculation considers the central source UV luminosity of $L_{\rm
UV}=100\,L_{\odot}$. The white region depicts the dust evaporation zone. The
gray dashed vertical stands for the location of the innermost stable circular
orbit (ISCO) around a Schwarzschild black hole ($6\,r_{\rm g}$). The white
solid lines represent dust temperatures of 200, 500, and 1000 K.

Overall, the Galactic center stands for a unique environment where in the
gravitational sphere of influence of the supermassive black hole ($\sim 2$ pc)
one can detect cold and dense molecular gas ($T\sim 50$ K and $n\sim
10^{8}\,{\rm cm^{-3}}$) as well as plasma close to Sgr A* that is dense and
hot ($T\sim 10^{10}\,{\rm K}$ and $n\sim 10^{7}\,{\rm cm^{-3}}$). At
intermediate scales of $\sim 0.1-1\,{\rm pc}$, warm ($\sim 10^{4}$ K) and hot
gas phases ($\sim 10^{7}$ K) coexist in the Minispiral region alongside the
warm dust with the temperature of a few 100 K. Unfortunately, we do not have
such high resolution multi-wavelength images of other AGN.

##  6 Nuclear dust properties in AGN from spectral and time delay studies

In highly accreting objects, the total emission is dominated by the optical/UV
radiation from the cold disk, and there is an additional bump in the near-IR
(Neugebauer et al, 1979) which comes from the dust reprocessing (Barvainis,
1987; Pier and Krolik, 1992; Fritz et al, 2006). The basic geometry was set by
the discovery of Antonucci and Miller (1985) but subsequent spectral studies
shed more light onto the dust content, including the chemical composition.
Observations reveal spectral (mostly silicate) features in emission (e.g.
Siebenmorgen et al, 2005; Hatziminaoglou et al, 2015) and in absorption (e.g.
Aitken and Roche, 1985; Roche et al, 2007; Tsuchikawa et al, 2021), depending
on the viewing angle in the specific source. Polycyclic Aromatic Hydrocarbons
(PAH) well observed in the interstellar medium are frequently weak or not
visible in AGN since they are most likely destroyed by shocks (e.g Zhang et
al, 2022a; Xie and Ho, 2022). The issue of the extinction curve which should
be used to correct the intrinsic spectra of AGN with some level of extinction
is still an open issue. The standard extinction curve from the Milky Way
(Cardelli et al, 1989) cannot be applied since the 2175 Å feature is generally
not observed but various specific shapes were proposed (e.g. Czerny et al,
2004; Gaskell et al, 2004, 2022). This would favor amorphous carbon grains. In
some objects, however, the 2175 Å feature seems to be present (Zhang et al,
2022b, in FeLoBAL objects).

Some constraints for the dust location came first from the simple
consideration of the dust sublimation temperature (Barvainis, 1987; Nenkova et
al, 2008a), but later reverberation measurements (RM) - time delays of the IR
emission with respect to the optical emission gave more specific constraints
(e.g. Oknyanskij, 1999; Koshida et al, 2014; Guise et al, 2022). The distance
of the dusty emitters from the central black hole is effectively by a factor
of 5 larger than the distance of the BLR, as measured by RM in the optical
band. The dust distance in a given source responds to the changes of the
radiation flux, and as suggested by the data for the source NGC 4151, the
recovery of the dust after the episode of a high luminosity state of the
source takes a few years (Oknyanskij et al, 2008).

##  7 Problems with the standard torus model and development of the most
recent models

The simple idea of the torus, as introduced by Antonucci and Miller (1985) and
presented in many general reviews of AGN structure and properties (e.g. Urry
and Padovani, 1995) is a major oversimplification. First, the static torus
shape cannot be supported by the thermal motion, so it should be some sort of
the dynamical structure (inflow or outflow), and wind outflow seems to be the
most likely option, considered in detail in a number of papers (e.g. Konigl
and Kartje, 1994; Elitzur and Shlosman, 2006; Dorodnitsyn et al, 2008;
Dorodnitsyn and Kallman, 2021; Gallagher et al, 2015). As we now know from
detailed spectral modelling, the torus must be clumpy, otherwise the broad
band spectrum and the absorption/emission features cannot be explained
(Nenkova et al, 2008a, b; Stalevski et al, 2012; Siebenmorgen et al, 2015).

Most recent high resolution spatial observations revealed another problem.
Expectations were that the high resolution IR maps will show predominantly the
dusty/molecular torus. However, the images brought a surprise: a lot of dust
emission actually came from the direction elongated in the direction of the
symmetry axis. Hönig et al (2013) performed interferometric mapping of the
unobscured (type 1) AGN in NGC 3783. Fitting the visibility plane they have
got the following result: most of the (mid-IR) emission comes from the polar
direction. Specifically, the ratio of the polar to equatorial emission in this
source was a rising function of the wavelength. Such observations have led to
a picture of the dust distribution in a form of an empty cone suggested by
Hönig (2019). In the cross section, it looks like a narrow stream of escaping
dusty material, with the launching radius at the usual location of the
beginning of the torus, about 1 pc from the black hole.

##  8 Dust within the Broad Line Region and FRADO model

The standard unification scheme locates the dust much farther away than the
BLR. However, actually the possible presence of dust in the broad-line regions
of AGN has been a subject of debate for years, as already stressed by Goodrich
(1995). It is well known that the ratios of the broad hydrogen Balmer emission
lines are not consistent with simple predictions, and dust can contribute to
this issue (e.g. Ilić et al, 2012, and the references therein). Medium fully
exposed to the central flux cannot remain dusty at distances smaller than the
torus inner radius, but the disk in AGN is geometrically thin, and its inner
part - or winds - may in addition shield the disk allowing for the dust
survival close in.

Interesting measurements of the time lag between the non-polarized continuum
and polarized broad line variability in the object Mrk 6 implied the presence
of the scatterer much closer than the torus but further away than the BLR
(Shablovinskaya et al, 2020).

There is also a fact that broad lines appear in coexistence with cold disk
located much closer in than torus, where the effective temperature of the
accretion disk dropping below the order of 1000 K implies formation of low
ionized material containing dust (see Czerny et al, 2004; Loska et al, 2004).
This fact is the basis of a model developed by Czerny and Hryniewicz (2011)
based on the dust-driving mechanism in which the radiation pressure of the
disk is responsible for the formation of low ionized BLR due to acceleration
of dusty material. The model predicts formation of failed winds through
frequent rise up of dusty material, irradiated then by intense central disk
radiation, and subsequent fall back of dustless material onto the disk, thus
known as FRADO (Failed Radiatively Accelerated Dusty Outflow). FRADO based on
dust-driving mechanism provides the most attractive purely theoretically-
motivated explanation for the natural source of low ionized broad emission
lines. The studies of measured time delays (Wandel et al, 1999; Peterson and
Wandel, 1999; Grier et al, 2013; Zajaček et al, 2021) show that Low Ionization
Lines (LIL) such as H$\beta$ and MgII originate from cooler and denser medium
located farther with respect to the central black hole compared to the other
population of broad emission lines, i.e. High Ionization Lines (HIL) such as
CIV and HeII for which the line-driving mechanism is more likely favored
(Murray et al, 1995; Proga et al, 2000; Waters et al, 2021).

The new advanced numerical 2.5D FRADO computations opened up a new window to
the dynamical modelling of low Ionized part of BLR. First preliminary tests
showed that the 2.5D model can interestingly explain the observed location and
dispersion in the radius-luminosity relation based on the reverberation-known
location of the LIL BLR (Peterson et al, 2004; Bentz et al, 2009, 2013), on
the basis of accretion rate (Naddaf et al, 2020). Moreover, the 2.5D FRADO
pictures a rather complex pattern of motion (Naddaf et al, 2021; Naddaf and
Czerny, 2022), strongly depending on the main global parameters including
accretion rate, black hole mass, and also the metallicity of material. For low
black hole mass and accretion rate, the pattern is rather simple resembling
the model of static puffed-up disk (Baskin and Laor, 2018) with some
turbulence, while otherwise an outflow structure is formed similar in
structure to the one proposed by Elvis (2000) and Hönig (2019). Metallicity
amplifies the radiative force, thus intensifies the outflow for the same
parameters of accretion rate and black hole mass, that manifests itself in the
shape of broad emission lines (Naddaf and Czerny, 2022).

Last but not least, although Hönig (2019) proposes an outflow structure
initiated at radii typical for the onset of torus (of around 10-100 times the
predictions of 2.5D FRADO), thus representing the torus as a separate nested
structure, there are studies indicating a likely connection between the BLR
and torus. The presence of ample amount of dust of high column densities
extending from equatorial plane to high altitudes obscuring the central disk
at high inclinations (Antonucci and Miller, 1985) implies the torus structure
to be not likely static, as shortly discussed in Section 7. Moreover, the
systematic study of the absorption events caused by the dusty clouds by
Markowitz et al (2014) may entail that the BLR and torus overlapping
structures. Dust geometries proposed recenctly at the basis of the
observational data (e.g. Elvis, 2000; Kawaguchi and Mori, 2010, 2011; Goad et
al, 2012; Sobrino Figaredo et al, 2020) are consistent with the outflow
structure in theoretically motivated 2.5D FRADO model (Naddaf et al, 2021;
Naddaf and Czerny, 2022). These altogether hint that the BLR and dusty torus
may be interrelated (Wang et al, 2013a), not separate identities.

##  9 Dust scenario in CL AGN

In the standard unification AGN scheme, the AGN type is determined for a given
source by its viewing angle and should not change. Also in X-rays, the amount
of the obscuration towards the nucleus in a given source should be set by
geometry and remain unchanged. However, examples of drastic changes of the
X-ray obscuration (Matt et al, 2003) and of appearing or disappearing of the
broad lines in some sources (Alloin et al, 1985) were reported. Such objects
are now referred to as Changing Look AGN (CL AGN) phenomenon. The effect was
considered as rather rare in the past, however, with recently increasing
number of follow-up observations, the sample of known CL AGN is growing (eg.
Yang et al, 2018; Graham et al, 2020; Panda and Śniegowska, 2022).

Two different physical scenarios were tested to explain spectral and
luminosity changes in CL AGN: obscuration mechanism and the variation in
accretion state. Most of changing-look phenomena seem to be caused rather by
accretion variation than by variable obscurer. The temporary obscuration or
intrinsic dimming scenarios were also tested using spectropolarimetry, since
low level of polarization argues against the obscuration scenario (Hutsemékers
et al, 2019). LaMassa et al (2015) discussed different obscuration scenarios
for CL AGN SDSS J015957.64+003310.5., which is also reported as first CL QSO.
The possible obscuration scenarios include the transported dust from the area
of torus that could cover the BLR emission or the formation of the dust.
However, for both scenarios the timescales are longer than observed changes in
the source. Timescales disagreement is the most robust argument for most of CL
AGN against obscuration scenario. The possible presence of the dust within
BLR, and the atypical extinction properties of AGN dust may open some paths to
use obscuration as a plausible CL mechanism. In the case of intrinsic changes,
also timescales are much shorter than normal viscous timescales in the
standard accretion disks and appropriate model modifications have to be made
to accommodate the observational data (e.g. Sniegowska et al, 2020; Śniegowska
et al, 2022; Kaur et al, 2022; Pan et al, 2022). Further studies are clearly
needed.

##  10 Dust origin in AGN

Dust in the BLR and dust playing the role of the dusty/molecular torus can
either come from the interstellar medium, or can form in situ. The answer is
not clear. Always solar, and frequently supersolar metallicity even at high
redshift quasars (see e.g. Śniegowska et al, 2021; Panda and Skorek, 2021, and
the references therein) is not consistent with expected low metallicity of the
high redshift galaxies. On the other hand, vigorous star formation actually
precedes the quasar activity, and multiple supernovae in the circumnuclear
stellar cluster creating heavy elements enrich the material which later
accretes (see e.g. Fiore et al, 2017, for the most recent plot of the history
of star formation rate and of quasar activity).

The ’in situ’ option is also interesting. Outer parts of the disk can be
gravitationally unstable, leading to vigorous star forming, with predominantly
fast evolving massive stars (e.g. Jermyn et al, 2021; Cantiello et al, 2021).
What is more, the standard disk in the region of the effective temperature
$\sim$ 1500 K allows to form dust directly in the disk atmosphere, in similar
way as in stars, and this mechanism was already discussed by Elvis et al
(2002).

Local production can possibly supply the dust much faster than the external
source, if dust is temporarily destroyed during the bright stage of an AGN,
but more studies of this issue like the tentative reports by Oknyanskij et al
(2008) are needed.

Acknowledgments This project has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
program (grant agreement No. [951549]). The project was partially supported by
the Polish Funding Agency National Science Centre, project 2017/26/A/ST9/00756
(MAESTRO 9), and MNiSW grant DIR/WK/2018/12. MLM-A acknowledges financial
support from Millenium Nucleus NCN$19\\_058$ (TITANs). VK acknowledges the
Czech Science Foundation project No. 21-06825X “Accreting black holes in the
new era of X-ray polarimetry missions”. SP acknowledges financial support from
the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)
Fellowship (164753/2020-6). MS acknowledges the financial support of Polish
Funding Agency National Science Centre 2021/41/N/ST9/02280 (PRELUDIUM 20). MZ
acknowledges the financial support of the GAČR EXPRO grant No. 21-13491X
“Exploring the Hot Universe and Understanding Cosmic Feedback”. The authors
also acknowledge the Czech-Polish mobility program (MŠMT 8J20PL037 and NAWA
PPN/BCZ/2019/1/00069) and the OPUS-LAP/GAČR-LA bilateral project
(UMO-2021/43/I/ST9/01352 and GF22-04053L).

## Declarations

  * •

Funding

The funding details are listed in Acknowledgement section.

  * •

Conflict of interest/Competing interests (check journal-specific guidelines
for which heading to use)

The authors declare they have no financial interests.

  * •

Ethics approval

Not applicable

  * •

Consent to participate

All authors contributed to the work and approved sending the paper for
publication.

  * •

Consent for publication

All authors agree for the work to be published in the European Physical
Journal D, Topical Issue: "Physics of Ionized Gases and Spectroscopy of
Isolated Complex Systems: Fundamentals and Applications".

  * •

Availability of data and materials

Data can be available on request.

  * •

Code availability

Codes can be available on request.

  * •

Authors’ contributions

All authors contributed to the study conception and design. B.C. was the
leading author in writing the text. MZ, MHN, MS, and VK and wrote some parts
of the text. TPA provided a better version of Fig. 1. SP and V.K.J. helped
considerably with text formatting. MS, M.L.M.-A. provided a number of
references. All authors sent their comments to the first draft. Finally, all
authors carefully reviewed and approved the final version of the text.

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