Polar Dust, Nuclear Obscuration,
and IR SED Diversity in Type-1 AGNs
Item Type Article
Authors Lyu, Jianwei; Rieke, George H.
Citation Jianwei Lyu and George H. Rieke 2018 ApJ 866 92
DOI 10.3847/1538-4357/aae075
Publisher IOP PUBLISHING LTD
Journal ASTROPHYSICAL JOURNAL
Rights © 2018. The American Astronomical Society. All rights reserved.
Download date 09/08/2022 23:12:22
Item License http://rightsstatements.org/vocab/InC/1.0/
Version Final published version
Link to Item http://hdl.handle.net/10150/631723
Polar Dust, Nuclear Obscuration, and IR SED Diversity in Type-1 AGNs
*
Jianwei Lyu (吕建伟) and George H. Rieke
Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA; jianwei@email.arizona.edu
Received 2018 May 15; revised 2018 August 29; accepted 2018 September 8; published 2018 October 16
Abstract
Despite the hypothesized similar face-on viewing angles, the infrared emission of type-1 active galactic nuclei
(AGNs) has diverse spectral energy distribution (SED) shapes that deviate substantially from the well-
characterized quasar templates. Motivated by the commonly seen UV-optical obscuration and the discovery of
parsec-scale mid-IR polar dust emission in some nearby AGNs, we develop semi-empirical SED libraries for
reddened type-1 AGNs built on quasar intrinsic templates, assuming low-level extinction caused by an extended
distribution of large dust grains. We demonstrate that this model can reproduce the nuclear UV to IR SED and the
strong mid-IR polar dust emission of NGC 3783, the type-1 AGN with the most relevant and robust observational
constraints. In addition, we compile 64 low-z Seyfert-1 nuclei with negligible mid-IR star formation contamination
and satisfactorily fit the individual IR SEDs as well as the composite UV to mid-IR composite SEDs. Given the
success of these fits, we characterize the possible infrared SED of AGN polar dust emission and utilize a simple but
effective strategy to infer its prevalence among type-1 AGNs. The SEDs of high-z peculiar AGNs, including the
extremely red quasars, mid-IR warm-excess AGNs, and hot dust-obscured galaxies, can be also reproduced by our
model. These results indicate that the IR SEDs of most AGNs, regardless of redshift or luminosity, arise from
similar circumnuclear torus properties but differ mainly due to the optical depths of extended obscuring dust
components.
Key words: dust, extinction – galaxies: active – galaxies: Seyfert – infrared: galaxies – quasars: general
1. Introduction
Accreting supermassive black holes (BHs) are surrounded by
copious dust and generally emit strongly in the infrared (IR).
Compared with star-forming or quiescent galaxies whose
emission is dominated by stars and H
II regions, active galactic
nuclei (AGNs) have distinctive IR spectral energy distributions
(SEDs), offering a unique window to look for obscured AGNs
(e.g., Lacy et al. 2004, 2007 ; Stern et al. 2005, 2012; Donley
et al. 2012), a critical means to explore the properties of dusty
structures around their central engines (e.g., Fritz et al. 2006;
Nenkova et al. 2008a, 2008b; Stalevski et al. 2012, 2016), and
a powerful tool to constrain their host galaxy properties by
SED decompositions (e.g., Bongiorno et al. 2007, 2012;Xu
et al. 2015; Lyu et al. 2016; Collinson et al. 2017). More
importantly, the dusty structures around the central engine,
commonly termed as “torus,” lay the foundation of AGN
unification and have physical scales that bridge the BH
accretion disk and the host galaxy (e.g., Antonucci 1993; Urry
& Padovani 1995). IR SEDs have become a crucial probe for
the AGN phenomenon and the BH-galaxy coevolution (e.g.,
Caputi 2014; Heckman & Best 2014; Netzer 2015). Never-
theless, most studies are focused on individual objects or some
particular narrowly defined AGN population. In this work, we
propose and test a model to reconcile the various IR SEDs of
type-1 AGNs over a broad range of luminosity and redshift.
The intrinsic SEDs of the most luminous AGNs, or quasars
(bolometric luminosity
L
L10
AGN,bol
11
), are observation-
ally well characterized (e.g., Elvis et al. 1994; Richards
et al. 2006; Shang et al. 2011; Krawczyk et al. 2013; Scott &
Stewart 2014; Lyu et al. 2017; Lyu & Rieke 2017; Lani et al.
2017). The average SEDs of unobscured quasars, whether the
parent sample is (mainly) optically selected (e.g., Elvis et al.
1994; Krawczyk et al. 2013), or combined with mid-IR
selection (e.g., Richards et al. 2006), or X-ray selected (e.g.,
Polletta et al. 2007; Elvis et al. 2012), have been found to be
strikingly similar. The classical Elvis et al. (1994)-like template
has proven to be a realistic representation of the IR emission by
hot dust in most quasars ( Lyu et al. 2017). The exceptions,
including hot-dust-free or hot-dust-poor objects (e.g., Hao et al.
2010, 2011; Jiang et al. 2010), are part of the quasar intrinsic
IR SED diversity seen at all redshifts. For unobscured quasars
at z∼0–6, the AGN IR SEDs can be generally grouped into
three basic types; besides the normal AGNs described by the
Elvis-like template, there are also the warm-dust-deficient
(WDD) AGNs and the hot-dust-deficient (HDD) AGNs (Lyu
et al. 2017). Despite the variations in the near-IR or mid-IR, the
shapes of quasar far-IR intrinsic SEDs have an identical pattern
in a statistical sense (Lyu & Rieke 2017), which drops quickly
at λ20μ m (Xu et al. 2015; Lyu & Rieke 2017; see also
Lani et al. 2017).
By comparison, the behavior of the intrinsic emission from
relatively faint AGNs (L
AGN,bol
∼ 10
8
–10
11
L
e
) is elusive
because of the significant galaxy contamination as well as the
possible line-of-sight (LOS ) extinction. A frequent hypothesis
is that the intrinsic SEDs of these Seyfert nuclei are identical to
those of quasars, thus one single template for AGNs with a
very broad luminosity range is assumed (e.g., Hopkins
et al. 2007; Assef et al. 2010; Donley et al. 2012). However,
notable differences seem to exist, particularly in the IR. The
AGN emission of many Seyfert-1 galaxies is peaked at
λ10 μm and the near- to mid-IR broad-band SEDs are
commonly described by a single power law (e.g., Spinoglio
et al. 1995; Alonso-Herrero et al. 2003; Prieto et al. 2010).On
the other hand, normal quasars exhibit a prominent SED bump
peaked in the UV and emit relatively less strongly in the IR
with an obvious SED jump due to emission by hot dust starting
The Astrophysical Journal, 866:92 (28pp), 2018 October 20 https://doi.org/10.3847/1538-4357/aae075
© 2018. The American Astronomical Society. All rights reserved.
*
The AGN template library developed in this work can be obtained on github
(https://github.com/karlan/AGN_templates).
1
at 1.3 μm and an almost flat (
n
nµ
n
F
0
) mid-IR SED at
∼3–20 μm (e.g., Sanders et al. 1989; Elvis et al. 1994).As
suggested by, e.g., Prieto et al. (2010), such different behavior
might be caused by obscuration. However, frequently only the
reddening of the UV-optical SED is considered, while the
accompanying effect on the IR emission is ignored.
In fact, an equatorial optically thick torus is only a first-order
approximation of the circumnuclear dust environment around
an AGN. Evidence for polar dust at ∼10
2
pc scale has been
suggested since the early 1990s (e.g., Braatz et al. 1993;
Cameron et al. 1993; Bock et al. 2000) and the AGN
obscuration is known to happen at a range of different scales
(e.g., see review by Bianchi et al. 2012). In theory, the AGN
torus could form during the gas accretion of the central BH and
have material exchanges with the ambient environment ( e.g.,
Hopkins et al. 2012). Although it cannot survive very close to
the central engine, dust is expected to be found in many other
AGN components, e.g., narrow-line regions (NLRs)(e.g.,
Groves et al. 2006; Mor et al. 2009) and/or AGN-driven
outflows (e.g., Fabian 1999; Murray et al. 2005). As argued by
many authors, the AGN infrared SED might be easily reshaped
by these extended dusty structures (e.g., Sturm et al. 2005;
Groves et al. 2006; Hönig et al. 2012, 2013; Hönig &
Kishimoto 2017).
Mid-IR interferometric observations have become available
for some nearby Seyfert nuclei, allowing a direct investigation
of the geometry of their nuclear IR structures at parsec scales.
Interestingly, these studies show that the mid-IR warm dust
emission in some systems is largely distributed along the AGN
polar direction, instead of from an equatorial torus (e.g., Raban
et al. 2009; Hönig et al. 2012, 2013; Tristram et al. 2014;
López-Gonzaga et al. 2016; Leftley et al. 2018). These
observations have motivated the developments of increasingly
sophisticated dust models to explain the few best-studied cases
(Hönig & Kishimoto 2017; Stalevski et al. 2017). In the case of
NGC 3783, it is proposed that both the torus and the polar dust
are composed of optically thick clouds and that the polar dust is
composed of large carbon grains (Hönig & Kishimoto 2017).
In comparison, the Circinus Galaxy is modeled with a parsec-
scale optically thick dusty disk and an IR optically thin cone
following the structure of the NLR out to a distance of ∼40 pc
(Stalevski et al. 2017). The success of these different
approaches indicates that these complex models could be
highly degenerate for most AGNs due to the lack of detailed
observational constraints. We will address whether it is
possible to develop a much simpler model that can be applied
uniformly and is still of sufficient fidelity to provide useful
insights.
With the success of our intrinsic templates to reproduce the
AGN IR emission of bright quasars at different redshifts (Lyu
et al. 2017), it is of considerable interest to explore whether
they also apply to relatively low-luminosity AGNs. The
possible existence of low-optical-depth dust in the vicinity of
the AGN nucleus, as outlined above, motivates us to develop a
new library of reddened AGN templates. We take the Lyu et al.
(2017) empirical quasar templates as givens for polar-dust-free
AGNs and investigate the extent to which the addition of a low
optical depth but extended dust component with reasonable
assumptions for the dust grain properties and their large-scale
distribution can yield IR SED shapes consistent with those
observed. In Section 2, we introduce this model and validate it
by fitting the detailed observations of NGC 3783, the
archetypal example of a type-1 AGN with its mid-IR emission
dominated by polar dust.
Although the extended dust distribution may have a range of
morphologies, any mid-infrared emission by low-optical-depth
dust should be roughly isotropic and hence detectable from any
view angle. We therefore focus on whether the model trained
for NGC3783 can be generally applied to match the infrared
SEDs of those AGNs where standard quasar templates fail,
assuming the choice of intrinsic AGN template and the optical
depth of the obscuration as the only free parameters for the
SED shape. This analysis is carried out on 64 low-z Seyfert
nuclei with negligible mid-IR star formation contamination in
Section 3. In addition, if the polar dust emission is a common
phenomenon for all populations of moderate-luminosity AGNs,
a consequence is that Seyfert-1 nuclei should be moderately
obscured on a statistical basis. We build composite SEDs of
Seyfert-1 SEDs and confirm this prediction.
We find that the deviations from the quasar-like SED
templates can indeed be explained to first order by the
combination of extinction and infrared emission by polar dust.
In Section 4, we characterize the SED features of the polar dust
emission and discuss the prevalence of polar dust in a sample
of AGNs much larger than those that can currently be explored
in any detail through mid-infrared interferometry. We also
demonstrate that one single semi-empirical template can
describe the influence on the AGN SED by the polar dust
emission for most objects. A consistency check of the results
from our SED analysis and those from morphology-based
identification for AGN polar dust emission is also carried out.
Various AGN populations with peculiar SED features that
cannot be easily matched by the classical AGN templates have
been reported at high z. Some notable examples are extremely
red quasars (ERQs)(Ross et al. 2015; Hamann et al. 2017),
AGNs with mid-IR warm-excess emission (e.g., Xu et al.
2015), and the hot dust-obscured galaxies (hot DOGs; e.g.,
Eisenhardt et al. 2012
; Wu et al. 2012). The success of our
model at low z encourages us to explore if these peculiar SED
features can be explained in a similar way. These studies are
presented in Section 5.
Section 6 provides a discussion of the implications of these
results for interpreting the AGN IR emission, the relation
between X-ray obscuration and polar dust extinction, and the
AGN unification scheme. We propose a tentative picture of the
different circumnuclear dusty environments among AGNs that
lead to their diverse IR properties. Section 7 is a final summary.
We adopt the cosmology
W=0.27
m
,
W=
L
0.73
and
=
H
0
7
1
kms
−1
Mpc
−1
(Bennett et al. 2003)throughout this paper.
We use the term type-1 to describe AGNs showing broad
emission lines without distinguishing if the line of sight
(LOS) is dust-obscured or not. Since the word obscured
is frequently reserved to describe type-2 (or narrow-line)
AGNs, we adopt the n ame reddened type-1 AGNs to denote
the broad-line AGNs with some extinction along the LOS
(aka the polar direction), even if the real reddening might be
insignificant because of a very flat extinction curve (e.g.,
Gaskelletal.2004). Lastly, the word unobscured means no
extinction, neither from the torus nor from any exte n ded du st
distribution, along the LOS to the central engine.
2
The Astrophysical Journal, 866:92 (28pp), 2018 October 20 Lyu & Rieke
2. A Semi-empirical SED Model for
Reddened Type-1 AGNs
We introduce a relatively simple framework to produce a
library of reddened type-1 AGN templates, which will be used
to fit the SEDs of Seyfert-1 nuclei in Section 3. This model is
based on two major assumptions:
1. Seyfert nuclei have a circumnuclear optically thick torus
whose SED variations from a face-on viewpoint can be
described by the intrinsic AGN templates of unobscured
quasars;
2. Besides the torus, there could exist an extended dust
component with some power-law density profile that is
dominated by large dust grains heated by the AGN.
We describe the motivations as well as the details of these
assumptions in Sections 2.1 and 2.2. The model and its
behavior are presented in Section 2.3. In Section 2.4, we test
our approach by fitting the observations of NGC3783.
2.1. Accretion Disk and Dusty Torus
The continuum SED of an AGN is contributed mostly by the
UV-optical emission from the accretion disk around the BH
and the near-IR to mid-IR emission emerges from the
surrounding dusty structures. To reduce the uncertainties, we
adopt well-tested extinction-free empirical templates to repre-
sent this AGN intrinsic emission.
The UV to mid-IR SEDs of most luminous type-1 quasars
are well described by the Elvis et al. (1994)-like AGN template,
regardless of the redshift (e.g., see discussion in Lyu et al.
2017). An AGN-heated dusty structure in type-1 quasars is
revealed by the broad IR emission bump at λ∼1.3–40 μm. In
addition, broad emission lines have been detected in the optical
polarized spectra of type-2 quasars (Zakamska et al. 2005).
Under the precepts of AGN unification (Antonucci 1993; Urry
& Padovani 1995), these observations support the existence of
some equatorial optically thick dusty structures, which cause
the nuclear photons to preferentially escape along the polar
direction.
As demonstrated by Lyu et al. (2017), the diversity of
intrinsic IR emission among type-1 quasars at z∼0–6 can be
characterized by three distinct templates derived for (1) normal
AGNs, (2
) WDD AGNs, and (3) HDD AGNs. These templates
are unlikely to be affected significantly by dust extinction since
their derivations are based on the study of optically blue
quasars that are not obscured. In other words, there should
be no significant dust distribution along the polar direction. We
suggest these AGN templates describe the emission from the
unobscured accretion disk plus a face-on view of the dusty
torus.
1
At λ<0.1μm, current observations do not give good
constraints. Following Stalevski et al. (2016), we assume a
broken power law, where
n
lmlm
lmlm
µ
<<
<<
n
⎧
⎨
⎩
()F
0.01 m 0.1 m
0.001 m 0.01 m.
1
0
1.2
Nevertheless, our study will not be influenced by the assumed
X-ray to UV SED shape. In fact, considering the likely
dominance of large dust grains along the face-on direction (see
Section 2.2.1), the extinction at these wavelengths is small and
will not contribute to the IR SEDs.
2.2. Extended Polar Dust Component
Besides the torus component characterized by the intrinsic
AGN templates discussed above, we introduce another dust
component to provide relatively low-level obscuration for the
nucleus as well as the additional IR emission from the absorbed
energy.
We suggest that the dust size distribution in this component
is dominated by very large particles, which will be character-
ized by grain size cutoffs, a
max
and a
min
. In real situations,
different grain species sublimate at different temperatures and
could have a broad range of dust sublimation zones. However,
we find that the calculated SEDs do not change significantly
between T
sub
=2000 and T
sub
=1500, indicating that the
introduction of separate values appropriate for carbon and
silicates would not change our results. For simplicity, we adopt
the same dust sublimation temperature T
sub
for all the grain
compositions.
For the large-scale structure, we assume a density profile
parameterized as a power law in radius with slope α:
r µ<<
a-
() ()rr r rr,,2
in out
where the inner radius r
in
is equal to the dust sublimation radius
R
sub
set by T
sub
as well as the light source luminosity, and the
outer radius r
out
is a free parameter. We introduce another
parameter, the outer-to-inner radius ratio Y=r
out
/r
in
,to
describe r
out
. Given the nature of this model, the geometry
will not influence the dust emission SED so that there is no
need to introduce more free parameters.
In the following, we outline the motivations behind these
configurations.
2.2.1. Grain Properties
There are strong reasons to suggest that the classical dust
properties are altered by the harsh environment of the direct
exposure to an AGN. As suggested by Aitken & Roche (1985),
small grains (a ∼ 10
−3
μm) can be easily destroyed out to
several hundred parsecs in a typical Seyfert-1 nucleus, on a
timescale of less than a few years. Physically, the majority of
dust destruction mechanisms around an AGN are relatively less
significant for large dust grains (Laor & Draine 1993). For
example, large grains are expected to exist close to the torus
inner part since they have smaller R
sub
than small grains. The
torus itself can be dynamically unstable and material exchanges
with the surrounding environments through various mechan-
isms are expected from simulations (e.g., Hopkins et al. 2012).
Baskin & Laor (2018) have analyzed the effects of sublimation
on both carbon and silicate grains near an AGN. They found
that only large (a > 0.1 μm) carbon grains can survive at the
outer edge of the broad-line region out to about 20 times this
radius. They also concluded that the silicate grain size
distribution will be skewed toward large sizes to significantly
greater distances.
Conditions for grain growth may exist in the circumnuclear
tori, where the densities are high and grains are shielded from
the X-ray and UV output of the central engine (e.g., Maiolino
et al. 2001a). As suggested by, e.g., Hönig et al. (2012, 2013),
some dust in the torus can be uplifted into the polar direction by
1
We use the word torus to describe the polar-dust-free obscuration structures
as in optically blue quasars and do not make assumptions on the geometry or
boundaries.
3
The Astrophysical Journal, 866:92 (28pp), 2018 October 20 Lyu & Rieke
AGN winds. It may also be possible for these large dust grains
to form in situ. Elvis et al. (2002) suggested that dust can form
in AGN-driven winds where conditions are similar to those in
the winds of late-type stars. By these mechanisms, grains in the
0.1–1μm range are plausible (Höfner 2008). However,
determining observational constraints on the grain sizes around
an AGN can be quite difficult. This can be seen from the
diverse AGN UV-optical extinction curves reported that range
from steeply rising SMC-like laws (e.g., Hall et al. 2002;
Richards et al. 2003) to flat or gray laws (e.g., Czerny et al.
2004; Gaskell et al. 2004; Gaskell & Benker 2007). As argued
by, e.g., Baskin & Laor (2018), it is possible that the grain
properties depend on the observing angle.
Nonetheless, despite various uncertainties, there are observa-
tional indications of relatively large dust grains around AGNs,
including (1) the lower ratios A
V
/N
H
and E(B − V )/N
H
in
intermediate-type Seyfert galaxies (Maiolino et al. 2001b,
2001a; but see Weingartner & Murray 2002); (2) the lower
ratios between A
V
and the mid-IR silicate absorption strength,
Δτ
9.7
, in type-2 AGNs (Lyu et al. 2014; Shao et al. 2017);
(3) successful fittings of the silicate emission profile in quasars
and Seyfert galaxies with micron-sized grain models (Xie
et al. 2017); and (4) the smaller observed torus inner radius
from near-IR interferometry of nearby Seyfert nuclei compared
with the expectations for classical dust grains (e.g.,Kishimoto
et al. 2007, 2009; Burtscher et al. 2013; Hönig et al. 2013; but
see Kawaguchi & Mori 2010).
As shown in Appendix A, for classical interstellar medium
(ISM) properties at low optical thickness, the mid-IR silicate
emission feature at λ∼10 μm would be very prominent with a
sharp peak (see also, e.g., Fritz et al. 2006; Nenkova
et al. 2008a) that is not commonly seen among Seyfert nuclei
(Hao et al. 2007). Instead, the lack of such detections could be
an expected consequence of large grains. For example, dust
grains with size a0.3 μm would reduce the strength of
silicate features effectively (Laor & Draine 1993).
Due to the difficulties for setting direct constraints on the
dust properties around the AGN, we minimize departures from
standard ISM grain models and assume only the grain sizes, as
characterized by the grain size cuts,
a
max
and a
min
, are altered in
the vicinity of an AGN.
2.2.2. Large-scale Geometry
Currently we do not have strong observational constraints
about the geometry of the dust responsible for the low-level
obscuration in type-1 AGNs. Nevertheless, AGN outflows
(e.g., Crenshaw et al. 2003; Piconcelli et al. 2005) could be a
natural mechanism to distribute the dust around the nucleus.
Physically, it has been found that the radiation pressure on
resonant absorption lines alone cannot explain the outflow
rates. The radiation feedback on dust within the clouds could be
an effective mechanism (e.g., Roth et al. 2012). Based on a
study of ∼3000 type-1 AGNs, Zhang et al. (2013) found that
the relative strength of the mid-IR to the optical flux of these
objects is correlated with the strength of outflows.
Several teams have tried to explore the origin of polar dust,
showing the outflow scenario is a promising solution. For
example, Hönig et al. (2012, 2013) proposed that the dusty
outflows could be launched from the surface of the inner torus
and the Hönig & Kishimoto (2017) model motivated by this
picture successfully explained the behavior of NGC3783, a
Seyfert-1 nucleus with a firm detection of polar dust emission.
From an analysis of high-spatial mid-IR images of 149 nearby
Seyfert galaxies, Asmus et al. (2016) found that elongated polar
dust emission is cospatial with the direction of AGN outflows
for 18 objects.
Little is known about the exact density profile of the gas
outflows. As a result, analytic analyses of self-similar solutions
are typically pursued. We introduce a power-law density
profile, ρ (r)∝r
− α
, to approximate the real situations.
Physically, we do not expect the outflow solution retains the
memory of initial conditions on large scales. As suggested by
Faucher-Giguère & Quataert (2012), to reach a finite-free
expansion radius, the gas density profile should have profiles
with α2. Observationally, various values of α have been
derived for materials in the outflows. For example, Behar
(2009) derived α∼1.0–1.3 for five nearby Seyfert nuclei by
analyzing the X-ray absorption spectra. Feruglio et al. (2015)
suggested an r
−2
profile for the ultra-luminous IR galaxy
Mrk 231. Revalski et al. (2018) derived the electron density
of the NLRs in the Seyfert-2 nucleus Mrk 573, finding
n
e
∝r
−0.4
–r
−0.6
. Additionally, a constant density absorber
(α=0) is quite unlikely the real case. Assuming that the dust
and gas are well mixed with a constant dust-to-gas ratio, we
suggest the dust density profile should satisfy 0<α2.
Based on mid-IR interferometry observations, AGN polar
dust emission is found to be elongated (e.g., Hönig et al.
2012, 2013; Tristram et al. 2014; López-Gonzaga et al. 2016)
and possibly distributed along the edges of the ionization cone
(Stalevski et al. 2017). In our model, however, the likely
uneven distribution of the polar dust component will not
influence its IR emission SED. This is a direct consequence of
optically thin dust emission in the IR, especially for large
grains (e.g., Laor & Draine 1993; Ivezic & Elitzur 1997).
The modest levels of face-on extinction in type-1 AGNs
correspond to a small value of τ
V
. Since the dust opacity is a
strong function of wavelength that decreases rapidly toward the
infrared, the extinction for such IR-reprocessed dust emission is
likely to be close to zero. In other words, the IR emission of
any dusty structures with a low τ
V
is highly transparent: the
emission from a single geometry element at some given
location, where the included dust grains can be considered in
local thermodynamic equilibrium (LTE), would share the same
SED from different viewing angles and this SED would transit
through other surrounding dusty structures without any notable
changes. Consequently, the total integrated IR SED can be
described as a summation of the dust emission from individual
LTE geometry elements at all possible locations.
At the same distance r, the temperature of each LTE element
would be the same as is the SED, B
λ
(r). The total emission
from all the dust at the same distance is linearly scaled by the
total numbers of LTE elements at the corresponding radius,
ρ(r), and has little to do with their possible uneven distribution.
The total integrated SED, F
λ
, can be approximated by adding
the contributions of all the elements at various radii, or
ò
r
ll
() () ()FrBrdr.3
r
r
in
out
Thus, if the average radial profile, ρ(r), is similar, the dust
distribution at small scales, whether it is smooth, clumpy or
filamentary, will not influence the SED.
As long as the integrated optical depth, τ
V
, is low, the effects
of asymmetries in the dust large-scale structure and/or
illumination along the radial directions would proportionally
4
The Astrophysical Journal, 866:92 (28pp), 2018 October 20 Lyu & Rieke