Modelling the dust emission from dense interstellar clouds: disentangling the effects of radiative transfer and dust properties

Abstract

With Planck and Herschel, we now have the spectral coverage and angular
resolution required to observe dense and cold molecular clouds. As these clouds
are optically thick at short wavelength but optically thin at long wavelength,
it is tricky to conclude anything about dust properties without a proper
treatment of the radiative transfer (RT). Our aim is to disentangle the effects
of RT and of dust properties on the variations in the dust emission to provide
observers with keys to analyse the emission arising from dense clouds. We model
cylindrical clouds, illuminated by the ISRF, and carry out full RT
calculations. Dust temperatures are solved using DustEM for amorphous carbons
and silicates, grains coated with carbon mantles, and mixed aggregates of
carbon and silicate. We allow variations of the grain optical properties with
wavelength and temperature. We determine observed colour temperatures, T, and
emissivity spectral indices, beta, by fitting the dust emission with modified
blackbodies, to compare our models with observations. RT effects can neither
explain the low T nor the increased submm emissivity measured at the centre of
dense clouds, nor the observed beta-T anti-correlation. Adding noise to the
modelled data, we show that it is not likely to be the unique explanation for
the beta-T anti-correlation observed in starless clouds. It may be explained by
intrinsic variations in the grain optical properties with temperature. As for
the increased submm emissivity and the low T, they have to originate in
variations in the grain optical properties, probably caused by their growth to
form porous aggregates. We find it difficult to track back the nature of grains
from the spectral variations in their emission. Finally, the column density is
underestimated when determined with blackbody fitting because of the
discrepancy between T and the true dust temperature at the cloud centre.

Full text

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Astronomy & Astrophysics manuscript no. paper
February 28, 2012
c ? ESO 2012
Modelling the dust emission from dense interstellar clouds:
disentangling the effects of radiative transfer and dust properties
N. Ysard1, M. Juvela1, K. Demyk2, V. Guillet3, A. Abergel3, J.-P. Bernard2, J. Malinen1, C. M´ eny2, L. Montier2, D.
Paradis2, I. Ristorcelli2, and L. Verstraete3
1Department of Physics, PO Box 64, FI-00014 University of Helsinki, Finland, e-mail: nathalie.ysard@ias.u-psud.fr
2IRAP, CNRS (UMR5277), Universit´ e Paul Sabatier, 9 avenue du Colonel Roche, BP 44346, F-31028 Toulouse cedex 4, France
3IAS, CNRS (UMR8617), Universit´ e Paris-Sud 11, Bˆ atiment 121, F-91400 Orsay, France
Preprint online version: February 28, 2012
ABSTRACT
Context. Dust emission is increasingly used as a tracer of the mass in the interstellar medium. With the combination of Planck
and Herschel observatories, we now have both the spectral coverage and the angular resolution required to observe dense and cold
molecular clouds. However, as these clouds are optically thick at short wavelength but optically thin at long wavelength, it is tricky to
conclude anything about dust properties without a proper treatment of the radiative transfer.
Aims. Our aim is to disentangle the effects of radiative transfer and of dust properties on the variations in the dust emission at long
wavelength. This enables us to provide observers with keys to analyse the dust emission arising from dense clouds.
Methods. We model cylindrical clouds with visual extinctions between 1 and 20 magnitudes, illuminated by the standard interstellar
radiation field, and carry out full radiative transfer calculations using a Monte-Carlo code. Dust temperatures are solved using the
DustEM code for amorphous carbons and silicates representative of dust at high Galactic latitude (DHGL), carbon and silicate grains
coated with carbon mantles, and mixed aggregates of carbon and silicate. We also allow variations of the optical properties of the
grains with wavelength and temperature. We determine observed colour temperatures, Tcolour, and emissivity spectral indices, βcolour,
by fitting the dust emission with modified blackbodies using a standard χ2fitting method, in order to compare our models with
observational results.
Results. Radiative transfer effects can neither explain the low Tcolournor the increased submillimetre emissivity measured at the centre
of dense clouds, nor the observed βcolour− Tcolouranti-correlation for the models considered. Adding realistic noise to the modelled
data, we show that it is not likely to be the unique explanation for the βcolour− Tcolouranti-correlation observed in starless clouds.
Instead, it may be explained by intrinsic variations in the grain optical properties with temperature. As for the increased submillimetre
emissivity and the low Tcolour, they have to originate in variations in the grain optical properties, probably caused by their growth to
form porous aggregates. We find it difficult to track back the nature of the grains from the spectral variations in their emission. This
difficulty comes from radiative transfer effects for λ ? 300 µm, and from the mixture of different grain populations otherwise. Finally,
the column density is underestimated when determined with modified blackbody fitting because of the discrepancy between Tcolour
and the ”true” dust temperature in the innermost layers of the clouds.
Key words. Radiative transfer – ISM: general – ISM: clouds – Dust, extinction – Submillimeter: ISM – Infrared: ISM
1. Introduction
Even if the main lines are understood, a number of questions re-
main regarding the processes that form a star from a cloud of gas
and dust. Recent observations indicate that the Galactic interstel-
lar medium (ISM) is organised as a network of elongated fila-
ments (Miville-Deschˆ enes et al. 2010; Andr´ e et al. 2010; Juvela
et al. 2010) and that pre-stellar cores seem to form preferentially
inside the densest of these filaments (Hill et al. 2011; Nguyen
Luong et al. 2011). However, the details of the formation of the
filaments or of their later fragmentation into dense molecular
clumps are still poorly known.
The physical, chemical, and dynamical state of the filaments
is expected to influence the star formation efficiency and the
mass of the new-born stars. It is thus crucial to characterize
the properties of the gas and in particular, in the context of the
Planck and Herschel missions, of the dust grains that compose
Galactic interstellar filaments. Indeed the combination of these
two observatories offers both the spectral coverage and the angu-
lar resolution necessary to enable the detection and the charac-
terisation of dense, molecular, and potentially pre-stellar clouds,
through the emission of the cold dust they contain. A lot can be
learnt about the cold clumps from this emission as the properties
of the grains are known to vary strongly from the diffuse ISM to
the centre of dense molecular clouds (Tab. 1). For instance, the
temperature of the grains decreases from the diffuse to the dense
medium (Schnee et al. 2010), while their submillimetre emis-
sivity is increased (Cambr´ esy et al. 2001; Stepnik et al. 2003;
Ridderstad et al. 2006; Kiss et al. 2006; Lehtinen et al. 2007;
Schnee et al. 2008; Juvela et al. 2011; Planck Collaboration
2011d). There are also observational proofs that the small grains
disappear towards dense molecular clouds, as the deficit of emis-
sion in the mid-IR (Bernard et al. 1999; Stepnik et al. 2003;
Flagey et al. 2009). Besides, the ratio of total to selective ex-
tinction RV increases from a standard value of 3.1 in the dif-
fuse medium to up to ∼ 5.5 towards dense clouds (Fitzpatrick &
Massa 1988; Mathis & Whiffen 1989; Cardelli & Clayton 1991;
Campeggio et al. 2007), which can be explained by the disap-
pearance of the small grains (Kim et al. 1994). This indicates
that the grains grow from the diffuse medium to the centre of
dense molecular clouds. It also appears to be increasingly clear,
1
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N. Ysard et al.: Modelling the dust emission from dense interstellar clouds
though still debated, that dust emission at long wavelength is
more complex than the usual modified blackbody at tempera-
ture T, with a constant emissivity spectral index β ∼ 2 (Tab.
1). Observations towards the diffuse gas as well as the dense
medium show an anti-correlation between β and T (Dupac et al.
2003; D´ esert et al. 2008; Paradis et al. 2010; Veneziani et al.
2010; Planck Collaboration 2011d,c). This behaviour can be
explained by models of the physics of the emission of amor-
phous solids (Meny et al. 2007; Paradis et al. 2011), and is also
corroborated by laboratory measurements on interstellar dust
analogues of amorphous carbons (Mennella et al. 1998) and
amorphous silicates (Agladze et al. 1996; Boudet et al. 2005;
Coupeaud et al. 2011). Malinen et al. (2011) also showed that a
β−T anti-correlation can be produced by the presence of a warm
source inside the clouds. Besides, Shetty et al. (2009) found that
it naturally results from noise uncertainties and suggested that
the anti-correlation may also be produced by temperature varia-
tions along the line-of-sight. Furthermore, laboratory measure-
ments and models also predict spectral variations in the dust
emissivity (Boudet et al. 2005; Meny et al. 2007; Mennella et al.
1998; Coupeaud et al. 2011), which have already been observed
in the Galactic ISM (Paradis et al. 2009; Planck Collaboration
2011d).
Thanks to the new generation of instruments available, and
in particular to the combination of Planck and Herschel data, we
are now able to observe dense and cold molecular clouds, and to
test detailed models of grain emission and growth. However, in
such dense media, where the illuminated edge is much warmer
than the shielded centre, detailed calculations of the radiative
transfer are compulsory. The dust emission spectrum arising
from dense regions has already been studied by various authors.
For instance, Fischera & Dopita (2008) and Fischera (2011) in-
vestigatedthelinkbetweenthedustemissionspectruminspheri-
cal,passivelyheatedcoresandtheirenvironment(radiationfield,
externalpressure).Theyshowedthattheradiativetransfereffects
lead to an overestimate of the sizes of the cores and to an un-
derestimate of their masses, agreeing on this matter with Evans
et al. (2001); Stamatellos & Whitworth (2003); Stamatellos et al.
(2004) and Malinen et al. (2011). These authors used various
dust models and cloud geometries, spheres or cylinders, sym-
metric or not, considered them as embedded or directly submit-
ted to the insterstellar radiation field, but all concluded that the
radiative transfer has a very strong impact on the observed dust
emission. They showed that the shape of the emission profiles
differs from the shape of the column density profiles, that the
apparent temperature diverges from the dust physical tempera-
ture, and also that the sizes and the masses of the clouds cannot
be accurately estimated by fitting the dust emission with a modi-
fied blackbody (Stamatellos et al. 2010; Fischera 2011; Malinen
et al. 2011). All these results emphasized the importance of a
proper treatment of the radiative transfer if one wants to trace
the variations in the dust properties from the diffuse to the dense
medium, or at least of the use of sensible corrections otherwise.
Therefore, the first question to answer concerns our ability
to disentangle the effects of radiative transfer, i.e. the mixing of
different dust temperatures along one line-of-sight, and the ef-
fects of the variations in the grain properties on the dust emis-
sion coming from dense clouds. Can we link the ”true” dust tem-
perature at the centre of the clouds to the average temperature
we measure? Then, how should the grains grow to have the low
temperaturesandtheincreasedsubmillimetreemissivitytheyex-
hibit in dense clouds? Does the observed β − T anti-correlation
come from intrinsic properties of the grains or is it only an ef-
fect of radiative transfer and/or noise in the data? What intrinsic
relation would be needed to reproduce the observed β − T anti-
correlation, when the effects of radiative transfer are considered?
What is the impact of the variations in the grain properties and of
the radiative transfer on the quantities that are usually measured:
column density, dust opacity? These are the main questions we
try to answer in this paper. Our aim is to give keys for the ob-
servers to be able to analyse quantitatively observations of dense
starless clouds.
The paper is organised as follows. In Section 2 we describe
the dust emission model and the radiative transfer code we use
throughout the paper. In Section 3 we detail the characteristics of
the dust emission from dense interstellar filaments when assum-
ing that the grains have the same optical properties as in the high
Galactic latitude diffuse ISM (Compi` egne et al. 2011). We ex-
plore the effects of radiative transfer and of noise on the relation
between the emissivity spectral index and the colour tempera-
ture. In Section 4 we present the effects of grain growth on the
emerging dust emission. We consider two scenarios of growth,
accretion and aggregation, and investigate the relation between
the colour temperature and the grain physical temperature. In
Section 5 we show the impact of intrinsic variations in the dust
optical properties with wavelength and then with temperature. In
Section 6 we test the accuracy of the estimate of the hydrogen
column density from fitting of the dust emission with a modified
blackbodyandhowthevariationsinthedustemissivityfromdif-
fuse to dense regions can be used to probe grain growth. Finally,
we present in Section 7 our conclusions.
2. Models and methods
2.1. Dust emission model
We use the dust model, DustEM1, described in Compi` egne et al.
(2011). DustEM is a versatile numerical tool that computes dust
emission and extinction as a function of the grain size distribu-
tion and optical properties. Our standard dust model is defined as
the DustEM model for dust in the diffuse ISM at high Galactic
latitude (DHGL according to the nomenclature of Compi` egne
et al. (2011)). Three dust populations are taken into account:
interstellar polycyclic aromatic hydrocarbons (hereafter PAHs),
amorphous carbons (small, hereafter SamC, and large, LamC),
and the so-called astronomical silicates (hereafter aSil). The op-
tical properties of the SamC and LamC grains come from the
laboratory measurements of Colangeli et al. (1995), who mea-
sured the opacity of a sample of sp2-rich hydrogenated amor-
phous carbon for λ ? 1900 µm (Compi` egne et al. 2011). For
longer wavelengths, the absorption efficiency is extrapolated us-
ing a single emissivity spectral index β = 1.55, equal to its aver-
age value for 800 ? λ ? 1900 µm. The optical properties of the
aSil population were built to reproduce the observations of the
diffuse ISM (Draine & Lee 1984) and have a single emissivity
spectral index equal to 2.11 in the far-IR and in the submillime-
tre2. Then, for the small grains (PAHs & SamC), we use log-
normal size distributions with central radii a0and widths σ. For
the larger grains, we use a power-law distribution aα, starting at
a = 4 nm, with an exponential cut-off e−[(a−at)/ac]γfor a ≥ at
(Weingartner & Draine 2001; Compi` egne et al. 2011). The dust
model abundances and the parameters of the size distributions
1Available at htpp://www.ias.u-psud.fr/DUSTEM.
2We do not use the update of the optical properties made by Draine
(2003b), who added a flattening of the emissivity spectral index in the
submillimetre (β = 2.11 → 1.7). Spectral variations in β are addressed
separately in Section 5.1.
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N. Ysard et al.: Modelling the dust emission from dense interstellar clouds
Table 1. Characteristics of the clouds used to compare the models of dense molecular clouds described in the paper. The first column
gives the name of the observed cloud, the second column the value of the visual extinction at the centre of the cloud, the third column
the visual extinction around the cloud (extinction of the incident radiation field), the fourth column the lowest colour temperature
measured in the cloud, the fifth column indicates if the submillimetre emission of big grains is enhanced towards the cloud centre,
the sixth column shows if an anti-correlation between the emissivity spectral index and the colour temperature is observed, and the
last column indicates the corresponding references.
Object
Filament in Taurus
Filament in Musca
Planck - G126.6+24.5
Lynds 1780
IC5146 (core # 1)
Central AV
≤ 10
∼ 8
∼ 3.5
4
∼ 21
External AV
∼ 0.4 − 0.5
∼ 0.3 − 0.4
? 1
∼ 0.25
?
Tcolour(K)
12.1+0.2
∼ 12.5
11.6 ± 0.8
14.9 ± 0.4
10.2 ± 0.1
Submm
×3.4+0.3
?
?
∼ ×1.5
Yes
β − T
Yes
Yes
Yes
?
?
References
Stepnik et al. (2003); Planck Collaboration (2011d)
Juvela et al. (2011)
Planck Collaboration (2011b)
Ridderstad et al. (2006); Ridderstad & Juvela (2010)
Kramer et al. (2003)
−0.1
−0.7
Table 2. Dust model abundances and size distribution parame-
ters (see Section 2.1 for details). ρ is the grain mass density, Y
is the mass abundance per H, κ250 µmis the opacity at 250 µm,
and β is the intrinsic opacity spectral index for λ > 100 µm for
each dust population. For aggregates, the percentages give their
degree of porosity (see Section 4.2 for details).
Small grains (DHGL)
σ
a0
(nm)
0.400.64
0.35 2.00
Big grains (DHGL)
α
ac,at
(nm)
-2.8150.0
-3.4 200.0
Accreted grains
α
ac,at
-2.8 150.0
-3.4 200.0
Aggregates
α
ac,at
-2.4234.0
-2.4242.0
-2.4256.0
-2.4 276.0
ρ
Y
κ250 µm
(cm2/g)
0.001
0.002
β
(g/cm3)
2.24
1.81
(Mdust/MH)
7.80×10−4
1.65×10−4
PAH
SamC 1.55
ρ
Y
κ250 µm
β
LamC
aSil
1.81
3.5
1.45×10−3
7.8×10−3
0.014
0.034
1.55
2.11
ρ
Y
κ250 µm
0.015
0.066
β
accC
accSi
1.81
3.22
1.6×10−3
8.6×10−3
1.55
1.52
ρ
Y
κ250 µm
0.111
0.140
0.208
0.331
β
0%
10%
25%
40%
2.87
2.59
2.16
1.72
1.02×10−2
1.02×10−2
1.02×10−2
1.02×10−2
1.33
1.32
1.30
1.27
are given in Tab. 2. We also consider the change of the grain opti-
cal properties towards dense molecular clouds, usually assumed
to be caused by their growth by accretion and/or aggregation.
The populations of evolved dust are described in Section 4 and
Tab. 2.
2.2. Radiative transfer model
We use the Monte-Carlo radiative transfer model described in
Juvela & Padoan (2003) and Juvela (2005), CRT3(Continuum
Radiative Transfer). Regarding the cloud geometry, until now,
two types of clouds could be used to perform the calculations:
spheres, divided into discrete cells with constant density, which
are concentric shells, or 3D clouds, divided into cubic cells also
with constant density. The second case is time consuming as no
assumption is made regarding symmetries. Observations of the
Galactic ISM show that interstellar matter is mostly distributed
along elongated filaments both in the dense and in the diffuse
3Available at http://wiki.helsinki.fi/display/˜mjuvela@
helsinki.fi/CRT.
medium (Abergel et al. 2010; Andr´ e et al. 2010; Juvela et al.
2010, 2011; Miville-Deschˆ enes et al. 2010; Arzoumanian et al.
2011). To account for this particular geometry, we implemented
a third type of cloud in CRT: finite circular cylinders. Our cylin-
ders are divided into layers perpendicular to the axis of symme-
try and each layer is divided into concentric rings. Each con-
centric ring is thus a cell of the modelled cloud, except for the
central cell of each layer which is a disk. One discrete cell has a
constant density but density can vary from one cell to the other
in both directions, that is to say as a function of the height along
the axis of symmetry and as a function of the radial position.
CRT consists of two independent parts. First the radiation
field is estimated at each position inside the cloud using Monte-
Carlo methods as described in Juvela & Padoan (2003). In the
case of cylinders, we consider incoming and outcoming pho-
tons over the entire surface of the cloud, including the two caps.
Second, using the radiation field and the selected dust model,
dust temperature and emission are calculated. CRT allows to
add different populations of grains, with distinct spatial distribu-
tions. Dust temperature distributions are solved using DustEM
(Compi` egne et al. 2011). The non-isotropic and multiple scat-
tering events are taken into account, as well as the re-emission
by dust grains by iterating CRT (Juvela & Padoan 2003; Juvela
2005).
2.3. Definition of the test cloud
In CRT, a cloud is defined by its hydrogen density distribution,
ρ(r). Cylindrical isothermal self-gravitating clouds have distri-
butions varying as ρ(r) ∝ r−4in the outer regions (Ostriker
1964). However, interstellar filamentary clouds usually exhibit
much gentler profiles from ρ(r) ∝ r−1.5to r−2.5, associated to a
flat distribution in the centre (Alves et al. 1998; Stepnik et al.
2003; Arzoumanian et al. 2011). Fiege & Pudritz (2000) showed
that such profiles are representative of magnetized filaments
where the magnetic field influence dominates over gravity in the
outer regions. For isothermal filaments, they found ρ(r) ∝ r−1.8
to r−2and for logatropic filaments ρ(r) ∝ r−1to r−1.8. Such gen-
tle profiles may also be explained by non-isothermal, collaps-
ing clouds, accreting the surrounding diffuse gas (Nakamura &
Umemura 1999).
Dapp & Basu (2009) and Arzoumanian et al. (2011) showed
that the column density profiles of interstellar filaments can be
fitted with the following radial hydrogen density distribution:
ρ(r) =
ρC
1 + (r/H0)α,
(1)
where ρC is the central density, kept constant along the axis
of symmetry, and H0is the internal flat radius. We define the
3

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N. Ysard et al.: Modelling the dust emission from dense interstellar clouds
(ρC,H0)-parameters for the clouds to be stable with their mass
per unit length equal to the critical value defined by Ostriker
(1964), Mcrit = 2c2
The sound speed, cS ∼ 0.2 km/s, is assumed to be constant and
is computed for a gas temperature of Tgas = 12 K. The steep-
ness of the density profile, α, usually lies between 1.5 ? α ? 2.5
(Dapp & Basu 2009; Arzoumanian et al. 2011). For our study
we assume that α = 2. The mass per unit length for the profile
defined by Eq. 1 is:
?R
M = πρCH2
H0
S/G, where G is the gravitational constant.
M =
0
2πρ(r)rdr(2)
0ln
1 +
?R
?2,
(3)
where R is the outer radius of the cloud. Assuming that M =
Mcrit, the (ρC,H0)-parameters can be solved numerically:
1 +
We set R = 1 pc to be able to compare our results with those
of Arzoumanian et al. (2011) who used R = 1.5 pc and those
of Juvela et al. (in prep) who chose R = 0.5 pc. The clouds
are divided into 217 cells, consisting in 31 rings and 7 layers
(see Section 2.2). We compute the (ρC,H0)-parameters to model
clouds with visual extinction at the centre from AV = 1 to 20
(Tab. 3). The emission maps produced have 41 × 41 pixels with
a single pixel size of 0.025 pc, equivalent to 0.85’ at a distance
of 100 pc or 5” at 1 kpc. Finally, when not specified, these ide-
alized filaments are illuminated by the isotropic interstellar stan-
dard radiation field, ISRF, as defined by Mathis et al. (1983). We
also study an alternative geometry for the clouds: Bonnor-Ebert
spheres (Bonnor 1956; Ebert 1955). The results are presented
in Appendix B and allow us to test the influence of the cloud
geometry on the results.
πρCH2
0ln
?R
H0
?2=2c2
S
G.
(4)
2.4. Dust colour temperature and emissivity spectral index
Thermal emission from large dust grains dominates the observed
emission in the far-IR and in the submillimetre. These grains are
in thermal equilibrium with the interstellar radiation field and
the observers often assume that their spectral energy distribution
(SED)iswellapproximatedbyasinglegrainsizewithaconstant
temperature and abundance along the line of sight:
?ν
where Sν is the brightness, Bν is the Planck function, Tcolour
is the dust colour temperature, ?0is the dust emissivity at fre-
quency ν0(or the optical depth per unit column density τ0/NH
in cm2/H), and βcolouris the emissivity spectral index. In order
to compare our model with observations, we fit the model cen-
tral pixel, which is also the brightest and the coldest pixel, with
this modified blackbody where the factor NH?o, the spectral in-
dex βcolourand the colour temperature Tcolourare free parame-
ters. Three cases are considered. Except where otherwise stated,
either we fit the entire SED from 100 µm to 3000 µm (∼ 100
GHz), or perform the fit using the Planck-HFI4(High Frequency
Instrument) filters at 857, 545, 353, and 143 GHz (350, 550,
Sν= NH?0
ν0
?βcolour
Bν(Tcolour),
(5)
4We exclude the two HFI channels which are contaminated by CO
emission, 100 and 217 GHz, as in Planck Collaboration (2011d).
Fig.2. Colour temperature for the central brightest pixels (for
SEDs fitted between 100 µm and 3000 µm) of the idealized fil-
aments as a function of Aext
(and the correspondingG0factor). The blue line shows the cylin-
der with AV= 1 at the centre, green shows 5, red 10, light blue
15, and magenta 20. The true grain temperatures, Tdust, can be
seen in Fig. 8.
Vthe extinction applied to the ISRF
850, and 2100 µm respectively), and the IRIS5100 µm filter as
in Planck Collaboration (2011d), or we use the Herschel PACS6
and SPIRE7filters at 100, 160, 250, 350, and 500 µm. Except
in Section 3.2 where we consider the effects of noise, we do
standard weighted least square fits, where all bands are given
an equal weight, in order to determine Tcolourand βcolour. The
results do not depend on the absolute value adopted for the un-
certainty (0.01% here), but only on the relative weighting of the
data points. The values of Tcolourand βcolourare also completely
dependent on the wavelength range/instrumental filters used in
the fitting procedure. This dependence can be seen in Fig. 1 for
example and comes from the different spectral distribution of the
filters regarding the position of the peak wavelength of the big
grains’ emission.
3. Dust as in the diffuse high galactic latitude ISM
We first assume that in the ten dense idealized cylindrical fil-
aments defined by Eq. 1 with the parameters listed in Tab.
3, dust has the same properties as in the diffuse ISM, DHGL
(Compi` egne et al. 2011). The dust populations present in our
modelled cylindrical clouds, with constant abundances, are thus
PAHs, SamC, LamC, and aSil, described in Tab. 2.
3.1. Effect of the radiative transfer
As shown in Fig. 1c, a cloud with a visual extinction of AV= 1 at
the centre has a dust colour temperature larger than 17.5 K, irre-
spective of wether the dust SED is fitted between 100 and 3000
µm or in the Planck HFI and IRIS 100 µm bands. When AV= 10
or 20, the colour temperature decreases and can be as low as 14.8
or 14 K. Indeed, in a cloud with high visual extinction, the radia-
5IRIS is the new generation of IRAS (InfraRed Astronomical
Satellite) images described in Miville-Deschˆ enes & Lagache (2005).
6Photodetector Array Camera and Spectrometer.
7Spectral and Photometric Imaging Receiver.
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N. Ysard et al.: Modelling the dust emission from dense interstellar clouds
Table 3. Parameters defining the gas distribution of the modelled clouds. Central density (ρCin H/cm3) and internal flat radius (H0
in pc) for the clouds with visual extinction at the centre AV= 1 to 20, and for all the dust populations considered in the paper (see
Tab. 2).
DHGL (Dust at High Galactic Latitude)
5 7.5
61000 155800
0.0240.014
Accreted grains
3900099800
0.0310.018
Aggregates 0%
23600 61000
0.0420.024
Aggregates 10%
2180056500
0.044 0.025
Aggregates 25%
18894 48968
0.0480.027
Aggregates 40%
1740044600
0.0500.029
Av
12 3.5 1012.5
496000
0.0073
15 17.5 20
ρC(H/cm3)
H0(pc)
1190
0.30
6900
0.088
26300
0.039
300600
0.0097
742500
0.0059
1.05 × 106
0.0048
1.43 × 106
0.0041
ρC
H0
740
0.44
4400
0.12
16700
0.051
196000
0.012
322000
0.0093
485000
0.0074
682000
0.0061
921600
0.0052
ρC
H0
400
0.91
2500
0.17
10000
0.070
120000
0.016
199000
0.012
305000
0.0096
426000
0.0080
576500
0.0067
ρC
H0
369
0.96
2308
0.18
9230
0.074
110000
0.017
184000
0.013
280000
0.010
395000
0.0083
535000
0.0070
ρC
H0
340
0.81
2137
0.19
8000
0.081
95650
0.019
106500
0.014
245000
0.011
345600
0.0090
470000
0.0076
ρC
H0
280
0.66
1800
0.22
7250
0.086
88500
0.019
146300
0.015
225000
0.011
315000
0.0094
425000
0.0080
tion field at short wavelengths (visible/UV), mainly responsible
for the grain heating, is more extinguished in the inner layers of
the cloud. However, these colour temperatures are still higher by
several Kelvins than what is observed in dense molecular clouds
(see Tab. 1). These dense clouds are often embedded in large HI
or molecular complexes leading to a lower and reddened inci-
dent radiation field at the surface of the cloud and thus to lower
colour temperatures. Using the same clouds with central AV= 1
to 20, we illuminate them with an extinguished isotropic ISRF8.
We extinguish the radiation by Aext
G0= 1 to 0.005 respectively9. The results, shown for fits of the
dust SEDs between 100 and 3000 µm, are presented in Fig. 2. A
radiation field decreased by one magnitude of visual extinction,
or withG0= 0.3, leads to a decrease of the observed colour tem-
perature less than 2 K. This diminution becomes smaller when
the clouds central column density or visual extinction increases.
Indeed, most of the UV/visible photons, which disappear when
the radiation field is extinguished, have no influence in the in-
ner layers of the densest clouds. Consequently, the extinction of
the external radiation field alone cannot explain the colour tem-
peratures measured in the objects described in Tab. 1 (Bernard
et al. 1992; Stepnik et al. 2003). Similar results are obtained in
the case of clouds modelled as Bonnor-Ebert spheres instead of
cylinders (see Appendix B).
Fig. 1 also shows the βcolour−Tcolourrelation obtained by fit-
ting the dust SEDs between 100 and 3000 µm, and in the Planck
HFI and IRIS 100 µm bands. This relation exhibits a correlation
instead of the observed anti-correlation between the emissivity
spectral index and the colour temperature. Similar results were
already obtained by Malinen et al. (2011), who modelled turbu-
lent clouds containing gravitationally bound cores (see their Fig.
15a), and this correlation is also seen in the case of Bonnor-Ebert
spheres (see Appendix B). This can be explained by purely ra-
V
= 0 to 5, corresponding to
8As noted by Stamatellos & Whitworth (2003), the radiation field
illuminating embedded clouds is usually not isotropic but we keep this
assumption for the sake of simplicity.
9G0scales the radiation field intensity integrated between 6 and 13.6
eV. The Mathis radiation field, ISRF, G0= 1, corresponds to an inten-
sity of 1.6 × 10−3erg/s/cm2.
diative transfer effects for grains with constant emissivity spec-
tral indices. The clouds are optically thick at short wavelengths
and the UV/visible photons, which heat the dust grains, are ab-
sorbed efficiently in the outer regions. Thus the dust grains are
colder at the centre of the clouds than on the surface. As the
clouds are optically thin at long wavelengths, we observe the
emission of all the grains present along the line-of-sight. The
range of temperatures increases with the central density: the
dust temperature at the edge of the clouds, illuminated by the
ISRF, remains almost the same, whatever the central density is,
whereas it is strongly decreased at the centre. Consequently, the
resulting SED is broadened when the extinction of the clouds
increases, leading to a decrease of the emissivity spectral index
to account for the flatter shape, while the colour temperature de-
creases. This βcolour−Tcolourcorrelation, obtained using methods
including full radiative tranfer calculations, is in contradiction
with the results of Shetty et al. (2009) who found that the line-
of-sight temperature variations naturally lead to a βcolour−Tcolour
anti-correlation. This difference emphasizes the importance of
the use of full radiative transfer calculations when modelling the
dust emission from dense molecular clouds and is addressed in
details by Juvela & Ysard (2012). These authors showed that
the anti-correlation observed by Shetty et al. (2009) results from
the particular temperature structure of the objects they modelled,
which are not likely to be representative of externally heated in-
terstellar clouds.
3.2. Effect of noise on the β − T relation
Radiative transfer effects lead to a correlation between the dust
emissivity spectral index, βcolour, and the colour temperature,
Tcolour, whereas observations show an anti-correlation (see Tab.
1). Shetty et al. (2009) and Blain et al. (2003) showed that this
anti-correlation can be caused by intrinsic noise in the observa-
tions, which produces a degeneracy between Tcolourand βcolour.
Recently, this anti-correlation was observed in various interstel-
lar environments thanks to Planck and Herschel instruments,
for instance by Planck Collaboration (2011d,c), Paradis et al.
(2010), Rod´ on et al. (2010), Juvela et al. (2011) and Bracco et al.
5