Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.arcetri.astro.it/~sbianchi/clumping/paper.ps Äàòà èçìåíåíèÿ: Tue Aug 24 19:52:34 1999 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 10:58:22 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: dust disk

Mon. Not. R. Astron. Soc. 000, 000--000 (0000) Printed 24 August 1999 (MN L A T E X style file v1.4)
Effects of Clumping on the Observed Properties of Dusty
Galaxies
S. Bianchi 1? , A. Ferrara 2 , J. I. Davies 1 , P. B. Alton 1
1 Department of Physics and Astronomy, Cardiff University, Cardiff, P.O. Box 913, Cardiff, CF2 3YB, UK
2 Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy
24 August 1999
ABSTRACT
We present Monte Carlo radiative transfer simulations for spiral galaxies modelled
as a stellar disk and a two­phase clumpy dust distribution. We divide the volume
occupied by the dust into a three­dimensional grid and assign each cell a clump or
smooth medium status. Cell dimension, clump dust mass and spatial distribution are
derived from the observed properties of Giant Molecular Clouds and molecular gas in
the Galaxy. We produce models for several values of the optical depth and fraction
of the interstellar medium residing in clumps. As a general result, clumpy models are
less opaque than the corresponding homogeneous models. For the adopted parameters,
the increase in the fraction of energy that escapes the disk is moderate, resulting in
surface brightness profiles that are less than one magnitude brighter than those of the
homogeneous models. The effects of clumping are larger for edge­on views of the disk.
This is in contrast with previous preliminary results for clumping in the literature. We
show how differences arise from the different parametrisation and clump distribution
adopted. We also consider models in which a fraction of the stellar radiation is emitted
within the clumps. In this case, galaxies are less transparent than in the case when
only dust is clumped. The opacity can be even higher than in the homogeneous case,
depending on the fraction of embedded stellar emission. We point out the implications
of the results for the determination of the opacity and dust mass of spiral galaxies.
Key words: dust, extinction -- ISM: cloud -- galaxies: spiral-- methods: numerical --
radiative transfer -- scattering
1 INTRODUCTION
An understanding of the effects of dust extinction in spiral
galaxies is crucial, both for deriving the intrinsic properties
of the galactic radiation field and interpreting observations
of the distant universe in the background.
While in earlier studies the opacity of a galactic disk was
often treated in a simplistic way, causing misinterpretations
and mutually exclusive results (Disney, Davies & Phillipps
1989), a number of more realistic models has been devel­
oped in the last years; the radiative transfer have been mod­
elled for appropriate galactic geometries, including multiple
scattering, via analytical approximations (Byun, Freeman &
Kylafis 1994; Silva et al. 1998) or Monte Carlo techniques
(Bianchi, Ferrara & Giovanardi 1996, hereafter BFG; De
Jong 1996; Wood 1997). Recently, the BFG code has been
exploited to produce data for a conspicuous set of models
(Ferrara et al. 1999, hereafter The Atlas).
The majority of the radiative transfer models for spiral
? Email address: Simone.Bianchi@astro.cf.ac.uk
galaxies deal with smooth distributions of dust and stars.
Instead, the interstellar medium is observed to have a com­
plex structure. It is difficult to solve the radiative trans­
fer problem for a clumpy dust distribution. Therefore most
models have been implemented for simple geometries. Ana­
lytical solutions for a clumpy plane parallel dust layer have
been provided by Boiss'e (1990) for a two­phase medium
and by Hobson & Scheuer (1993) for a generic N­phase
medium, under the assumption of isotropic scattering. Witt
& Gordon (1996) and Wolf, Fischer & Pfau (1998) performed
Monte Carlo simulations for the radiative transfer through a
spherical, two­phase dusty medium, illuminated by a central
source. They divided the sphere in a cubic lattice, assigning
each cell a low or high density status according to chosen val­
ues for the filling factor and density ratio of smooth/clump
medium. Spherical clumps inside a spherical dust distribu­
tion have been studied by V'arosi & Dwek (1999), allowing
radiation to come from a central point source, a uniform dis­
tribution of emitters, or a uniform distribution of external
sources; analytical approximations are tested against Monte
Carlo simulations.
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We summarise here a few of the properties derived for
clumpy distributions of dust. As a general rule, a clumpy
medium has a higher transparency with respect to a ho­
mogeneous one with the same mass of dust. The effect of
scattering is reduced, because light entering a dense clump
has a low probability of escaping: from the point of view of
the radiative transfer, a dense clump therefore behaves sim­
ilarly to a dust grain with smaller albedo than the one char­
acterising single grains in a smooth medium. For a clumpy
distribution the amount of energy absorbed varies less with
the wavelength of radiation than for a homogeneous dis­
tribution: this concurs, together with geometric effects (see
The Atlas), in an apparent extinction, or more appropriately
attenuation, curve flatter than the actual one.
However, the details of radiative transfer in a clumpy
medium depend not only on the parameters used for the
clumping itself, but also on the distribution of dust with
respect to the stars. In this paper we present a study of
the radiative transfer through a clumpy dust distribution
for geometries typical of a spiral galaxy. Although the ISM
may have a wide range of densities (Hartmann 1994; Heiles
1998), we adopt a simple two­phase scheme for the dust dis­
tribution: a smooth medium associated with the low­density
diffuse atomic gas and clumps associated to high density Gi­
ant Molecular Clouds (GMCs). The distribution of clumps is
derived from observations of the molecular gas in the Galaxy.
Only a few works include clumping in a radiative trans­
fer model appropriate for galactic geometries. Kuchinski et
al. (1998) use preliminary results from a Monte Carlo model
(based on Witt & Gordon 1996 formalism) to derive opaci­
ties of edge­on galaxies from their colour gradients. Although
a few aspects of the inclusion of clumping are presented in
that paper, the authors defer a detailed discussion to a forth­
coming paper. Clumping has been also included by Sauty,
Gerin & Casoli (1998) in their model of the FIR and C +
emission from the spiral galaxy NGC 6946; however, it is
difficult to single­out the effect of clumping from the results
shown.
Since star formation occurs in the dense phase of the in­
terstellar medium, it is justified to assume that a fraction of
the stellar emission occurs within clumps of dust. This may
change the effects of clumping. In their model for a cen­
trally illuminated dust distribution, V'arosi & Dwek (1997)
find that the absorption efficiency may increase, rather than
decrease, when clumps have a fractal distribution: being a
fractal distribution a more connected set with respect to a
uniform random distribution of spherical clumps, the fractal
cloud can behave like a shield in front of the source. Em­
bedded stars are included in the radiative transfer model of
Sauty et al. (1998) and Silva et al. (1998). In this paper we
will also consider the effects of a clumpy distribution for the
radiation sources.
The paper is organised as follows. In x 2 we present a
detailed description of the parameters used to describe the
stellar emission and the clumpy dust distribution, referring
to BFG and The Atlas for the general treatment of the ra­
diative transfer; x 3 is dedicated to the effect of clumping
for radiation coming from a smooth stellar distribution; x 4
shows the results when the possibility of sources embedded
in dust clumps is taken into account. Finally, x 5 contains a
summary and a discussion of the application of the results
obtained in this work. Throughout the rest of this paper
we will use the notation CM to refer to models including
clumping and HM for homogeneous models.
2 THE MODEL
The results are obtained using an adapted version of the
Monte Carlo code for the radiative transfer in dusty galaxies
described in BFG. The original code has been simplified: a
single Henyey­Greenstein scattering phase function (Henyey
& Greenstein 1941; see also BFG) is used, with empirically
derived values for the albedo ! and of the asymmetry pa­
rameter g. We use here the values ! = 0:6 and g = 0:6 y ,
appropriate for radiation in the V­band (Gordon, Calzetti
& Witt 1997). The polarisation part of the radiative transfer
code has been omitted. The same approach has been used
for The Atlas.
In this paper we restrict ourselves to a single stellar spa­
tial distribution, a three­dimensional disk with exponential
behaviour in both horizontal and vertical directions. The
horizontal and vertical scale lengths have been chosen to
match the observed values for the old disk population of
the Galaxy, ff ? = 3 kpc (Kent, Dame & Fazio 1991; Fux &
Martinet 1994) and fi ? = 0:26 kpc, respectively. Note that
in BFG and in The Atlas we used ff ? = 4 kpc (Bahcall &
Soneira 1990): while in the previous models only the ratio
between the scale lengths was relevant, for the model of this
paper the absolute values must be specified, since the phys­
ical dimensions of the clumps are required (see later). The
ratio between the horizontal and vertical scale lengths used
in this paper is the same as in BFG and the Atlas.
Some specified fraction of the total dust mass is dis­
tributed in a smooth exponential disk, similar to the stellar
one, with scale lengths ff d = ff ? and fi d = 0:4fi? (BFG and
Atlas), while the remaining mass is distributed in clumps.
The total mass of dust (smooth medium + clumps) is
defined in terms of the optical depth of the model ÜV , i.e.
the V­band face­on optical depth through the centre of the
disk for a HM. For the exponential disk, the mass is easily
computed using the formula
Mdust = ÜV ff 2
d
8ú
3
affi
QV
\Theta
1 \Gamma (n + 1)e \Gamman \Lambda
= 6:8 \Theta 10 6 ÜV [M fi ] ;
where a = 0:1¯m and ffi = 3g cm \Gamma3 are the grain radius and
density, QV = 1:5 the extinction efficiency in the V­band
(Hildebrand 1983) and n = 4:6 is the radial truncation of
the disk (in scale length units; see later). In the following,
when a CM is said to have been produced for a value of ÜV ,
it means that the CM has the same dust mass as a HM with
a face­on V­band optical depth ÜV . Therefore, only in the
case of a HM ÜV is a real optical depth, while for CMs is
mainly an indicator of the mass of dust in the model, the
effective opacity depending in a complex way on ÜV and the
y The values for ! and g are measured in Galactic reflection neb­
ulae, by comparing the intensity of scattered light with radiative
transfer models. Since a clumpy medium has a lower effective
albedo, neglecting clumping in the model may bias the measure
of ! towards lower values, especially for optically thick clouds and
stronger forward scattering (Witt & Gordon 1996).
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Clumpy dust disks 3
Table 1. Parameters of the clumpy distributions.
fc 0.25 0.50 0.75 M dust
Ü V Ü s
V Nc Ü s
V Nc Ü s
V Nc 10 6 M fi
0.1 0.075 255 0.05 511 0.025 784 0.7
0.5 0.375 1277 0.25 2554 0.125 3831 3.4
1.0 0.75 2554 0.5 5108 0.25 7662 6.8
2.0 1.5 5108 1.0 10215 0.5 15476 13.6
5.0 3.75 12769 2.5 25539 1.25 38308 34.0
10.0 7.5 25539 5.0 51077 2.5 76616 68.0
For each value of the fraction of gas in clumps fc and of the optical depth Ü V , the
optical depth of the smooth medium Ü s
V and the total number of clumps Nc are listed.
The final column gives the total dust mass of the model.
other parameters for the clumpy dust distribution. Models
have been produced for optical depths ÜV = 0.1, 0.5, 1, 2, 5,
10.
Using a Galactic gas­to­dust ratio of 150 (Devereux &
Young 1990; Sodroski et al. 1994), the gas mass can be de­
rived from the dust mass. A fraction fc of the total gas mass
is then attributed to the molecular component. In this pa­
per we explore three different values, fc = 0.25, 0.5, 0.75.
These values are representative of actual ratios observed in
late type galaxies (from Scd to Sb, respectively; Young &
Scoville 1991). All of the molecular gas (and the associated
mass of dust) is supposed to be distributed in clumps. Our
choice is appropriate, most of the molecular gas (ú 90% in
mass) in the Galaxy being in the form of GMCs (Combes
1991).
As for the clumpy structure, we use the same two­phase
formalism as in Witt & Gordon (1996). The space occupied
by dust is divided into cubic cells of the dimension of Galac­
tic GMCs. Blitz (1991) quotes a typical diameter of 45 pc:
a cubic cell of 36 pc will therefore have the same volume as
a molecular cloud.
First, each cell is assigned a value corresponding to the
local absorption coefficient (i.e. the inverse of the light mean
free path through dust) for the smooth distribution of dust.
If ÜV is the optical depth of the model, a fraction (1­fc) of
the dust mass is distributed in the smooth medium. Thus,
the smooth medium opacity is defined by a face­on optical
depth Ü s
V = ÜV \Lambda (1: \Gamma fc ). The local absorption coefficient
for the smooth medium is then computed from the assumed
double­exponential distribution, using Ü s
V .
Second, we distribute the clumps. The position of each
clump is derived from the distribution of molecular gas in
the Galaxy, using the Monte Carlo method. We have used
the CO observations of the first Galactic quadrant described
by Clemens, Saunders & Scoville (1988). The radial density
distribution has been derived from the plot of the mass of
molecular hydrogen vs. galactocentric radius in their Fig. 11;
we have adopted a gaussian vertical distribution, as param­
eterised in their Eqn. 3, with a FWHM that increases with
galactocentric distance. Following the molecular gas, the dis­
tribution of clumps is more concentrated on the Galactic
plane than the smooth dust. At any galactocentric distance,
the probability of finding clumps for jzj ? 2:5fid is nil. The
distribution has been rescaled assuming 8.5 kpc as the Sun
distance from the Galactic centre: the position of the Galac­
tic molecular ring is thus between 1.5 and 3.5 radial scale
lengths from the centre. Outside 13 kpc (ú 4.5 ff ?) the
molecular gas is not detected (see also Heyer et al. 1998). In
the model, both the dust and the stellar disks are truncated
at this distance from the centre.
Each clump has been given a typical GMC mass of 10 5
M fi (Blitz 1993). The total number of clouds in the model is
then directly derived from the mass of gas in clumps. Assum­
ing that the dust has a uniform density inside each cloud,
the absorption coefficient in the clump can be easily com­
puted from the clump gas mass and dimensions, using the
adopted dust properties and gas­to­dust ratio: each clump
has an absorption coefficient in the V­band of 110 kpc \Gamma1 ,
corresponding to an optical depth ÜV ú 4. Since the effects
of a clumpy structure are stronger when individual clumps
are optically thick (Boiss'e 1990; Boiss'e & Thoraval 1996) our
simulations provide an upper limit to the effects of clumping
in a spiral galaxy.
For all the cells that have been assigned a clump status,
the clump absorption coefficient is summed to the absorp­
tion coefficient of the smooth medium. The optical depth
along any photon path is computed integrating the absorp­
tion coefficient array along the specified direction. Apart
from this, the Monte Carlo code follows the same scheme
as described in BFG. A list of the parameters defining the
clumpy dust distribution is given in Table 1.
To reduce the computational time and the dimension of
the array involved, we use only one octant of the galaxy and
assign values in the other octant according to the model
symmetries: the dust absorption coefficient is stored for
383x383x12 cells. Because of this, clumps are distributed
randomly only in one octant. This does not affect the sta­
tistical properties of the model, as long as the number of
clumps is large and the position of emission of photons is
random. Nevertheless, we have tested the validity of this
simplification comparing the results to a case with clumps
distributed randomly all over the space, obtaining the same
results. Finally, simulated images of 201x201 pixels are pro­
duced for several inclinations (BFG).
In the following sections, we present results and plots
for a few representative cases of optical depth and incli­
nation. The whole dataset of this paper is available at
http://www.arcetri.astro.it/~ sbianchi/clumping.html
3 RESULTS
As already described in the introduction, the main effect of
clumping is that of an overall increase in the transparency
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Table 2. Fraction of energy absorbed in each model.
f emb 0.00 0.15 0.50
fc 0.00 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
Ü V
0.1 0.02 0.01 0.01 0.01 0.09 0.08 0.08 0.25 0.25 0.25
0.5 0.07 0.06 0.05 0.04 0.13 0.12 0.11 0.29 0.28 0.28
1.0 0.12 0.11 0.09 0.07 0.17 0.16 0.15 0.32 0.31 0.30
2.0 0.19 0.18 0.15 0.12 0.24 0.22 0.19 0.38 0.36 0.35
5.0 0.32 0.31 0.28 0.23 0.37 0.34 0.30 0.49 0.47 0.44
10.0 0.44 0.43 0.39 0.34 0.48 0.45 0.40 0.59 0.57 0.54
For each value of Ü V , the first column gives the fraction for the HM (fc = 0), then each group
of three columns gives the values for CMs with a specific fraction of the total energy emitted within
clumps (f emb ; f emb = 0 refers to the case with a homogeneous stellar distribution) and for the three
values of the fraction of gas in clumps fc .
of the model. In Table 2 we show the fraction of the total
energy that is absorbed in each model, as a function of ÜV
(that defines the mass of dust) and fc . Obviously, for the
same value of the optical depth, more energy is absorbed in
a HM (defined in the table by fc=0.0). The results presented
for the HM are the same as for the spiral galaxy disk in The
Atlas. When the clumping is introduced, CMs with larger
fc are increasingly more transparent.
Within the same model, the effects of clumping depend
also on the disk inclination. This is clearly shown in Fig. 1,
where we have plotted the attenuation (i.e. the ratio between
the observed flux and the intrinsic, unextinguished, total
one) as a function of the optical depth, for the HMs and
the CMs for three values of the inclination angles, i=20 ffi ,
66 ffi , 90 ffi . As seen before, CMs have the same attenuation
as HMs with a smaller ÜV , for any inclination. The largest
differences between the two cases are found in more face­on
cases, and for high values of fc . For instance, a face­on CM
fc=0.75 and optical depth ÜV = 10 has the same attenuation
as an HM with ÜV ú 4, while for an inclination of 20 ffi , it
corresponds to a HM with ÜV ú 5. For the same values of ÜV
and fc , the smooth medium has an optical depth Ü s
V =2.5.
The different behaviour of CMs with inclination can
also be seen analysing the disk major axes profiles. In Fig. 2
and 3 we show the major axis profiles for two representative
cases of low (ÜV = 1) and high (ÜV = 10) optical depth, for
inclinations of i=20 ffi , and 90 ffi . In each left panel we plot with
solid lines the profiles for CMs: the brighter profile always
refers to the case with higher fraction of dust in clumps,
fc = 0:75, while the fainter is for fc = 0:25. As a compari­
son, we have also plotted a series of profiles for HMs (dotted
lines); the three brighter profiles have the same optical depth
Ü s
V as the smooth medium in each of the CM, from fc = 0:75
(upper profile) to fc = 0:25. The faintest of the HM is char­
acterised by the optical depth ÜV . The differences between
each profile and the profile for the HM with optical depth
ÜV are shown in the right panels, for the chosen inclinations.
As expected, a CM characterised by a value of ÜV and
fc has a major axis profile at an intermediate brightness
between the HM of optical depth ÜV and the HM of opti­
cal depth Ü s
V , i.e. the homogeneous smooth component. The
differences between HM and CM are not large, and always
smaller than 1 mag. For optically thin cases, models are vir­
tually indistinguishable at low inclinations (as in the ÜV = 1
profiles of Fig. 2). For optically thick cases (e.g. the models in
Fig. 3), the profiles for CM are closer to those for the smooth
Figure 1. Attenuation as a function of the optical depth Ü V , for
the HM and CMs with fc=0.25, 0.50, 0.75. Data are plotted for
three inclinations, i=20 ffi , 66 ffi and 90 ffi .
medium only for R ! ff ? , where the number of clumps, and
therefore their filling factor, is small. Between 1 and 3 ff ? ,
where the molecular distribution peaks (the Galactic ring),
the profile is closer to that for the HM with the same optical
depth. On the contrary, profiles for edge­on cases are always
brighter than the corresponding HM.
These results are in contrast with those presented by
Kuchinski et al. (1998). They derived opacities of highly
inclined galaxies, by comparing optical and near­infrared
colour gradients with Monte Carlo radiative transfer sim­
ulations, including scattering and clumpy dust distribution.
They found that the derived optical depths are insensitive to
the dust having either homogeneous or clumpy distribution.
This is explained by the significant number of clumps inter­
sected by any line of sight through a nearly edge­on galaxy.
On this basis they argue that clumping effects may be more
important for lower inclinations, where some lines of sight
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Clumpy dust disks 5
Figure 2. Major axis profiles (left panel) for models with Ü V = 1.0 and inclination i=20 ffi (face­on), 90 ffi (edge­on). Solid lines refer to
CMs, for fc= 0.25, 0.5, 0.75. The brighter profiles corresponds to the case with higher fraction of gas in clumps, fc= 0.75, while the
dimmer to the case with fc= 0.25. HMs are presented for comparison (dotted lines). The brighter profile corresponds to a HM with the
same optical depth as the homogeneous dust (Ü s
V ) for the case fc=0.75, followed by the analogous profiles for fc= 0.5, 1.0. The dimmer
profile refers to a HM with the optical depth Ü V . The right panel presents the difference in magnitude between each model and the HM
with the optical depth Ü V . All curves have been smoothed with a box of 10 pixels.
Figure 3. Same as Fig. 2, but for models with Ü V = 10.0. To avoid overlap, a constant value of 2 magnitudes has been subtracted from
the surface brightness for the edge­on major axis profile presented in the left panel.
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6 S. Bianchi et al.
Figure 4. Major axis profiles (left panel) for CMs with constant filling factor all over the dust distribution and a ratio of 100 between
densities in clumps and in the nearby smooth medium (See text for details). Models have Ü V =10 and i=20 ffi and 90 ffi . Solid lines refer
to fc= 0.25, 0.5, 1.0 (Brighter profile). With a dashed line the case fc= 0.95 is plotted, while the dotted line refers to the HM with the
same Ü V . To avoid overlaps, a constant value of 0.75 magnitudes has been subtracted from the surface brightness for the edge­on major
axis profile. The right panel presents the difference in magnitude between each model and the HM with the optical depth Ü V . All curves
have been smoothed with a box of 10 pixels.
pass across clumps, while others do not. However, the de­
scription of the clumpy structure of dust in Kuchinski et
al. (1998) is different from the one adopted here. Following
Witt & Gordon (1996), they assign a constant filling factor
(ff=0.15) for the high density cells all over the galaxy, and
assume a ratio of 100 between densities in clumps and in the
nearby smooth medium. Since the smooth medium has an
exponential distribution, clumps close to the galactic centre
have a higher optical depth than those in the outer disk.
Clumps in our simulations, instead, are distributed into a
ring­like structure and have the same optical depth. We feel
these assumptions are more realistic and are supported by
the available observational data.
To clarify the reasons for the discrepancy, we have mod­
ified the code to deal with the Kuchinski et al. (1998) for­
malism. Clumps are now characterised by a constant filling
factor all over the dust distribution and by a value for the
local ratio of densities of the two dust phases. For these tests
we have used the value 100 for the density ratio. The geom­
etry of the galaxy and the cells dimensions have been kept
as in x2. In Fig. 4 we show the major axis profiles in the case
ÜV =10, for the two inclination i = 20 ffi and 90 ffi . The solid
lines refer to CMs with the usual values fc=0.25, 0.50, 0.75.
In these models fc can be directly converted into the filling
factor ff using the formula
ff = fc
fc + k(1 \Gamma fc) ;
where k = 100 is the density ratio. The fc values of this
paper thus correspond to quite low global filling factors,
ff=0.003, 0.01, 0.03, respectively. The dashed line is for
ff=0.15, as in Kuchinski et al. (1998). For the assumed
density ratio, this value of the filling factor corresponds to
fc ú 0:95, i.e. 95% of the total mass of dust in the model
is distributed in clumps. Indeed, for this simulation setup,
CMs have profiles closer to the HM when viewed at larger
inclinations.
As for the profiles in Fig. 2 and Fig. 3, an increase of fc
in the range 0.25 ­ 0.75 produces an increase in the global
transparency of the dust distribution, that is shown in Fig. 4
by the brighter profiles (especially in the face­on case) when
a larger fraction of gas is distributed in clumps. The effects
of clumping are instead reduced for fc=0.95 (ff=0.15). This
is consistent with the simulations of radiative transfer in a
clumpy media presented by V'arosi & Dwek (1999). They
have studied the case of a clumpy dust sphere under dif­
ferent radiation fields: a central source, a homogeneous dis­
tribution of isotropic emitters and a uniform external field.
When all the other parameters defining the dust distribution
are fixed, the fraction of absorbed photons as a function of
the clumps filling factor shows a minimum, for any of the
radiation fields. The exact value of ff for which the dust
distribution has a maximum transparency depends on the
details of the model. In our simulation, for the value of the
clump/smooth medium density ratio adopted, clumping has
the major effect in the reduction of the fraction of absorbed
energy for 0:03 Ÿ ff ! 0:15 (or 0.75Ÿ fc !0.95). In the CM
for fc=0.95, 43% of the radiation is absorbed, almost the
same quantity as for the fc=0.25 case (and for the HM). As
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Clumpy dust disks 7
Figure 5. Edge­on (i = 90 ffi ) minor axis profiles (left panel) for models with fc= 0.25, 0.5, 0.75 (solid lines). As in Fig. 2, brighter
profiles corresponds to higher values of fc . Dotted lines refers to HMs as described for Fig. 2. The right panel presents the difference in
magnitude between each model and the HM with the optical depth Ü V .
already stated, the model for fc=0.75 is more transparent,
with 38% of the total radiation absorbed. With respect to
the main models of this paper, the constant filling factor dis­
tributions are slightly more opaque, for the same value of fc .
In the ÜV =10 model with fc=0.75 of Table 2, for instance,
the fraction of absorbed energy is 34%.
The differences in behaviour between the main CM of
this paper and the one with constant filling factor arise be­
cause of two reasons: (i) the different number of clumps for
a given value of fc and (ii) the different geometrical distri­
bution. In the main CM clumps have all the same mass,
and therefore the same density, for a given cell dimension.
In the constant filling factor CM the density of a clump de­
pends on the local density of the smooth medium (it is 100
times for the model analysed here). Because of the expo­
nential distribution of the smooth medium, clumps at larger
distance from the centre have smaller density (and mass)
than those in the inner part of the galaxy. Consequently,
when the fraction of ISM mass that is locked in clumps is
fixed, the main CM will have a smaller number of clumps
of higher mass than the one with a constant filling factor.
For the case ÜV =10 with fc=0.75 analysed before there are
roughly 4 times more clumps in the latter than in the for­
mer. This smaller number of clumps is then distributed on
preferential places in the main CM, more concentrated on
the galactic plane and in the molecular ring. When seen
face­on the large number of clumps in the ring position at­
tenuates a lot of the radiation coming from the half of the
galaxy below the plane, acting almost as a screen. Only out­
side the ring the filling factor of clumps is small, and in this
regions the face­on profile is closer to that for a HM with
the optical depth of the smooth medium. In the edge­on
case, instead, it is the light emitted outside the ring, atten­
uated by the smooth dust distribution, that dominates the
radiative transfer, thus resulting in a major axis profile of
intermediate brightness between those of the HMs with ÜV
and Ü s
V .
In Fig. 5 we plot the edge­on minor axis profiles for the
main model with optical depths ÜV = 1 and 10. Geometri­
cal and optical parameters of the dust distribution in spiral
galaxies are mainly retrieved by fitting radiative transfer
models to images of edge­on galaxies (Xilouris et al. 1997;
Xilouris et al. 1998; Xilouris et al. 1999). At this inclina­
tion, in fact, we can have separate views of the radial and
vertical dust distribution. Furthermore, dust extinction is
increased as a result of the projection. The profiles of Fig. 5
show a behaviour analogous to the edge­on major axis pro­
files in Fig. 2 and 3, with a reduced extinction that results
in shallower extinction lanes on the galactic plane. On the
contrary, edge­on minor axis profiles for CMs with constant
filling factor are virtually indistinguishable from the HM.
4 EMBEDDED SOURCES
So far clumping has been introduced only in the description
of the dust distribution. Since we use a clumpy structure
to simulate GMCs, it is logical to assume that part of the
stellar emission comes directly from inside the clumps, the
stars being formed in the higher density phases of the ISM.
In this section we study the effects of clumping both for dust
and stars.
Radiation from stars embedded in GMCs is simulated
assuming that a fraction femb of the total energy emitted by
stars comes from inside the clumps of dust. We have used
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8 S. Bianchi et al.
Figure 6. Same as Fig. 1, but for models with f emb =0.15 and 0.50.
the values 0.15 and 0.50. The actual value of this parameter
is difficult to be derived from observations, since it would
require sensitive indicators of the (optically) obscured star
formation rates. The above range is an educated guess that
comes from the estimate of the average fraction of stellar life­
time spent inside dark clouds. Wood & Churchwell (1989)
and Churchwell (1991), from their study of UCHII regions,
concluded that O stars are embedded in their natal molec­
ular clouds on average about 15% of the main sequence of
an O6 star. However, this might be a lower limit and the
fraction of total embedded star formation could be as high
as 50% when the contribution from low mass stars is taken
into account. Within the Monte Carlo code, a fraction femb
of the photons is emitted inside the cells. The position of
the emission inside the cell is randomly distributed within
its boundaries. This allows more radiation to escape clumps,
with respect to the case where the photons are emitted in the
centre of the cell. Once the photon is emitted, the radiative
transfer is carried out through the clumpy dust distribution
as described in x 2.
The fraction of absorbed energy in CMs with embedding
is shown in Table 2, for the two values of femb . In general, a
larger fraction of energy is absorbed in these CMs with re­
spect to the case when only dust has a clumpy distribution.
The fraction of absorbed energy may be even larger than for
a HM. When the degree of clumpiness increases (larger fc)
the effect of embedding is mitigated and more energy can
escape. As an example, for femb=0.15, optically thick CMs
with fc – 0:5 have an overall opacity similar or smaller
than for the HM. The absorption in the case femb=0.5 is
larger and although it decreases with increasing fc as for
the previous case, the fraction of absorbed energy is always
higher than in the HM. The large fraction of energy ab­
sorbed in the optically thin cases, especially for femb=0.5
might seem surprising. This is because in our formalism,
for optically thin cases (small mass of dust), the embedded
emission comes from relatively few opaque clumps. For a
cubic cell of optical depth 4 through its side and with a ho­
mogeneous distribution of internal emitters, nearly 50% of
the radiation is absorbed within the cell. For instance, in a
model with ÜV =0.1 and femb=0.5, half of the radiation is
emitted within clumps, and half of it is absorbed, thus re­
sulting in a fraction of absorbed energy 0.25, as in Table 2.
The global increase in extinction can also be seen in
the attenuation plots of Fig. 6. Models with clumping dis­
tributions for both stars and dust have a transparency lower
than those with a clumpy dust distribution only, at any in­
clination. As already shown before, for femb = 0:50, all the
CMs are more opaque than the corresponding HM with the
same optical depth. For the more face­on inclinations, the
behaviour of the attenuation is less dependent on ÜV with re­
spect to the HM. This is because the projected optical depth
of the smooth medium is lower, and the clumps dominate
the emission (and absorption).
Face­on images, especially in the case for a higher frac­
tion of embedded emission, clearly show a ring structure, due
to the photons being emitted in clumps distributed accord­
ingly to the ring­like distribution of molecular gas. When the
models are seen edge­on, the ring is smoothed out by the pro­
jection and it is clearly visible only for optically thin cases
and femb=0.50. In Fig. 7 we plot the major axis profiles for
the edge­on cases with ÜV = 1 and 10. For femb=0.15 and
for optically thick models with femb=0.50, the major axis
profiles are similar to those of HM with different effective
opacities. It is interesting to note that the major axis pro­
files can be brighter than those of the HM with optical depth
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Clumpy dust disks 9
Figure 7. Edge­on major axis profiles (left panels) for models with embedded sources. Profiles are plotted for optical depths Ü V =1, 10
and for the three values of fc (solid lines). As in Fig. 2, brighter profiles corresponds to higher values of fc . Dotted lines refers to HM as
described for Fig. 2. To avoid overlap, a constant value of 1 magnitude has been added to the surface brightness for the Ü V =10 profile.
The right panel presents the difference in magnitude between each model and the HM with the optical depth Ü V . All curves have been
smoothed with a box of 10 pixels.
ÜV , even when the global opacities are larger. For example,
in an edge­on model with femb=0.50, fc=0.75 and ÜV = 10
only 20% of the radiation escapes, while for the correspond­
ing HM the fraction that escapes is 25%. The profile along
the major axis is brighter, being very similar to that of the
HM with ÜV =7.5. The brighter profile can be accounted for
by the embedded emission, that is more concentrated in the
plane than the smooth one. While the CM is generally dim­
mer than the HM, the major axis emission is higher: embed­
ding concurs therefore with dust clumping in reducing the
contrast of the dust edge­on absorption lane. This is clearly
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10 S. Bianchi et al.
Figure 8. Edge­on minor axis profiles (left panels) for models with embedded sources. Profiles are plotted for optical depths Ü V =1, 10
and for the three values of fc (solid lines). As in Fig. 2, brighter profiles corresponds to higher values of fc . Dotted lines refers to HM as
described for Fig. 2. The right panel presents the difference in magnitude between each model and the HM with the optical depth Ü V .
seen in the edge­on minor axis profiles in Fig. 8, especially
for high optical depth and femb=0.50.
5 SUMMARY AND DISCUSSION
We have presented in this paper the results of Monte Carlo
radiative transfer simulations through a two­phase clumpy
dust distribution, for geometries typical of a spiral galaxy
disk. The space occupied by the dust distribution has been
divided into a three­dimensional grid and each cell has been
assigned a clump or smooth medium status. The dimension
of the cells and the dust mass of a clump have been derived
from the properties of Galactic GMC. Clumps have been
randomly distributed according to the ring­like distribution
of molecular gas observed in our Galaxy. We have explored
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Clumpy dust disks 11
several values for the optical depth ÜV (i.e. the optical depth
of a homogeneous dust distribution with the same mass) and
for the fraction of total gas residing in clumps.
The main conclusions are the following. As predicted
by simple arguments and previous studies, a model with a
clumpy dust distribution (CM) suffers a reduced extinction.
For the parameters adopted in this paper, which emulate
real dust distributions in spiral disks, the reduction of the
fraction of absorbed energy is moderate, resulting in sur­
face brightness profiles that are always less than one mag­
nitude brighter than the corresponding homogeneous model
(HM). The major differences between HMs and CMs are
found for edge­on inclinations. This is in contrast to the re­
sults presented by Kuchinski et al. (1998). Using a model
with spatially constant clump filling factor and with almost
all (95%) of the galactic dust locked in clumps, they find
that the smallest differences occur in edge­on cases. To as­
certain the reasons for this discrepancy, we have reproduced
their experiment and confirmed their results. We have con­
cluded that the disagreement depends on the different pa­
rameters and distribution adopted for the clumps. This is
unfortunate, however, as it indicates a strong dependence
of the observed brightness profiles on the detailed internal
and spatial distribution properties of clumps which makes
the interpretation of the data very difficult.
Since star formation occurs in high density clouds, it
is logical to assume that part of the stellar emission occurs
within clumps. We have therefore produced models where a
fraction of the photons is emitted in the high density cells.
When this clumpy stellar distribution is considered, CMs are
less transparent than for a clumpy dust distribution only.
Depending on the fraction of gas in clumps and on the frac­
tion of embedded stellar emission, galaxies can be even more
opaque than predicted by HMs with the same mass.
One of the major concerns about clumping is that its
neglect will produce a significant underestimate of the dust
mass of a galaxy. Xilouris et al. (1999) analysed a sample
of seven edge­on galaxies fitting the surface brightness with
a radiative transfer model. They find a mean central face­
on optical depth ÜV =0.5. Comparing the total dust mass
of each galaxy with the mass of gas, they derive a gas­to­
dust mass ratio of 360\Sigma60. The derived value is larger than
the Galactic value by more than a factor of two (Sodroski
et al. 1994), but closer than estimates based on FIR dust
emission observed by IRAS (Devereux & Young 1990). From
our results in Fig. 2 and Fig. 5, minor and major axis profiles
for CMs with ÜV =1 and 0:50 ! ff ! 0:75 are similar to
those for a HM with ÜV = 0:5 (In the ÜV = 1 panels, a
profile for a HM with ÜV = 0:5 is shown between the two
mentioned CMs with a dotted line). Therefore, a CM with
ÜV = 1 can have an edge­on appearance very similar to an
HM with ÜV = 0:5. The mass of dust in the CM would
be twice the value derived for the HM and the gas­to­dust
mass ratio would be reduced accordingly to 180, a value close
to the canonical. Similar results can be obtained using the
ÜV = 1 profiles for models with embedded stellar emission.
We caution here that Xilouris et al. (1999) find that dust
disks have a larger radial scale lengths than the stellar while
in the models of this paper both disks have the same radial
scale lengths; this may affect the numerical details of the
exercise carried out in this paragraph, but the qualitative
result holds. On the contrary, a CM with a constant filling
factor all over the galaxy would resemble, when seen edge­
on, a HM with the same optical depth, and there would be
no change in the mass estimate.
Acknowledgements. We wish to thank D. Galli for interesting
discussion about the embedding of stars in molecular clouds
and the referee, A. Witt, for useful comments that improved
the presentation of the data in the paper.
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