M on.N ot.R .A stron.Soc.000,000{000 (0000)
P rinted 23 July 2013
(M N LA TEX style le v1.4)
E ects of C lum ping on the O bserved P roperties of D usty
G alaxies
S.Bianchi1?,A .Ferrara2,J.I.D avies1,P.B.A lton1
1 D epartm
arXiv:astro-ph/9909395v1 23 Sep 1999
2O
ent of P hysics and A stronom y, C ardi U niversity, P .O . B ox 913, C ardi , C F2 3Y B , U K
sservatorio A stro sico di A rcetri, Largo E . Ferm i 5, 50125 Firenze, Italy
23 July 2013
A B ST R A C T
W e present M onte C arlo radiative transfer sim ulations for spiral galaxies m odelled
as a stellar disk and a two-phase clum py dust distribution. W e divide the volum e
occupied by the dust into a three-dim ensionalgrid and assign each cella clum p or
sm ooth m edium status.C elldim ension,clum p dust m ass and spatialdistribution are
derived from the observed propertiesofG iantM olecularC louds and m oleculargasin
the G alaxy.W e produce m odels for severalvalues ofthe opticaldepth and fraction
ofthe interstellarm edium residing in clum ps.A s a generalresult,clum py m odels are
lessopaquethan thecorresponding hom ogeneousm odels.Fortheadopted param eters,
the increase in the fraction ofenergy that escapes the disk is m oderate,resulting in
surface brightnesspro lesthatare lessthan one m agnitude brighterthan those ofthe
hom ogeneousm odels.T he e ectsofclum ping are largerforedge-on view softhe disk.
T hisisin contrastw ith previousprelim inary resultsforclum ping in the literature.W e
show how di erences arise from the di erent param etrisation and clum p distribution
adopted.W e also considerm odelsin w hich a fraction ofthe stellarradiation isem itted
w ithin the clum ps.In this case,galaxies are less transparent than in the case w hen
only dust is clum ped.T he opacity can be even higher than in the hom ogeneous case,
depending on the fraction ofem bedded stellarem ission.W e pointoutthe im plications
ofthe results for the determ ination ofthe opacity and dust m ass ofspiralgalaxies.
K ey w ords: dust,extinction { ISM :cloud { galaxies:spiral{ m ethods:num erical{
radiative transfer { scattering
1
IN T R O D U C T IO N
A n understanding ofthe e ects ofdust extinction in spiral
galaxies is crucial,both for deriving the intrinsic properties
ofthe galactic radiation eld and interpreting observations
ofthe distant universe in the background.
W hilein earlierstudiestheopacity ofa galactic disk w as
often treated in a sim plistic w ay,causing m isinterpretations
and m utually exclusive results (D isney,D avies & Phillipps
1989), a num ber of m ore realistic m odels has been developed in the lastyears;the radiative transferhave been m odelled for appropriate galactic geom etries,including m ultiple
scattering,via analyticalapproxim ations(B yun,Freem an &
K yla s 1994; Silva et al. 1998) or M onte C arlo techniques
(B ianchi, Ferrara & G iovanardi 1996, hereafter B FG ; D e
Jong 1996;W ood 1997).R ecently,the B FG code has been
exploited to produce data for a conspicuous set of m odels
(Ferrara et al.1999,hereafter T he A tlas).
T he m ajority ofthe radiative transfer m odels for spiral
? E m ailaddress:Sim one.B ianchi@ astro.cf.ac.uk
c 0000 R A S
galaxies deal w ith sm ooth distributions of dust and stars.
Instead,the interstellar m edium is observed to have a com plex structure. It is di cult to solve the radiative transfer problem for a clum py dust distribution.T herefore m ost
m odels have been im plem ented for sim ple geom etries.A nalyticalsolutions for a clum py plane paralleldust layer have
been provided by B oisse (1990) for a tw o-phase m edium
and by H obson & Scheuer (1993) for a generic N -phase
m edium ,under the assum ption ofisotropic scattering.W itt
& G ordon (1996)and W olf,Fischer& Pfau (1998)perform ed
M onte C arlo sim ulationsforthe radiative transferthrough a
spherical,tw o-phase dusty m edium ,illum inated by a central
source.T hey divided the sphere in a cubic lattice,assigning
each cella low orhigh density statusaccording to chosen values for the lling factor and density ratio ofsm ooth/clum p
m edium .Sphericalclum ps inside a sphericaldust distribution have been studied by V arosi& D w ek (1999),allow ing
radiation to com e from a centralpointsource,a uniform distribution of em itters,or a uniform distribution of external
sources;analyticalapproxim ations are tested against M onte
C arlo sim ulations.
2
S.Bianchietal.
W e sum m arise here a few ofthe properties derived for
clum py distributions of dust. A s a general rule, a clum py
m edium has a higher transparency w ith respect to a hom ogeneous one w ith the sam e m ass of dust. T he e ect of
scattering is reduced,because light entering a dense clum p
has a low probability ofescaping:from the point ofview of
the radiative transfer,a dense clum p therefore behaves sim ilarly to a dustgrain w ith sm aller albedo than the one characterising single grains in a sm ooth m edium .For a clum py
distribution the am ount ofenergy absorbed varies less w ith
the w avelength of radiation than for a hom ogeneous distribution:this concurs,together w ith geom etric e ects (see
T he A tlas),in an apparentextinction,orm ore appropriately
attenuation,curve atter than the actualone.
H ow ever, the details of radiative transfer in a clum py
m edium depend not only on the param eters used for the
clum ping itself, but also on the distribution of dust w ith
respect to the stars. In this paper w e present a study of
the radiative transfer through a clum py dust distribution
for geom etries typicalofa spiralgalaxy.A lthough the ISM
m ay have a w ide range ofdensities (H artm ann 1994;H eiles
1998),w e adopta sim ple tw o-phase schem e for the dustdistribution:a sm ooth m edium associated w ith the low -density
di use atom ic gasand clum psassociated to high density G iantM olecularC louds(G M C s).T he distribution ofclum psis
derived from observationsofthem oleculargasin theG alaxy.
O nly a few w orksinclude clum ping in a radiative transfer m odelappropriate for galactic geom etries.K uchinskiet
al.(1998) use prelim inary resultsfrom a M onte C arlo m odel
(based on W itt & G ordon 1996 form alism ) to derive opacitiesofedge-on galaxiesfrom theircolourgradients.A lthough
a few aspects ofthe inclusion ofclum ping are presented in
thatpaper,theauthorsdefera detailed discussion to a forthcom ing paper.C lum ping has been also included by Sauty,
G erin & C asoli (1998) in their m odel of the FIR and C +
em ission from the spiral galaxy N G C 6946; how ever, it is
di cult to single-out the e ect ofclum ping from the results
show n.
Since starform ation occursin the dense phase ofthe interstellar m edium ,itis justi ed to assum e that a fraction of
the stellar em ission occurs w ithin clum ps ofdust.T his m ay
change the e ects of clum ping. In their m odel for a centrally illum inated dust distribution,V arosi& D w ek (1997)
nd thatthe absorption e ciency m ay increase,ratherthan
decrease, w hen clum ps have a fractal distribution:being a
fractaldistribution a m ore connected set w ith respect to a
uniform random distribution ofsphericalclum ps,the fractal
cloud can behave like a shield in front of the source. Em bedded stars are included in the radiative transfer m odelof
Sauty et al.(1998) and Silva et al.(1998).In this paper w e
w illalso consider the e ectsofa clum py distribution for the
radiation sources.
T he paper is organised as follow s. In x 2 w e present a
detailed description ofthe param eters used to describe the
stellar em ission and the clum py dust distribution,referring
to B FG and T he A tlas for the generaltreatm ent ofthe radiative transfer; x 3 is dedicated to the e ect of clum ping
for radiation com ing from a sm ooth stellar distribution;x 4
show s the results w hen the possibility ofsources em bedded
in dust clum ps is taken into account.Finally,x 5 contains a
sum m ary and a discussion of the application ofthe results
obtained in this w ork. T hroughout the rest of this paper
w e w ill use the notation C M to refer to m odels including
clum ping and H M for hom ogeneous m odels.
2
T H E M O D EL
T he results are obtained using an adapted version of the
M onte C arlo code forthe radiative transferin dusty galaxies
described in B FG .T he originalcode has been sim pli ed:a
single H enyey-G reenstein scattering phase function (H enyey
& G reenstein 1941;see also B FG ) is used,w ith em pirically
derived values for the albedo ! and of the asym m etry param eter g. W e use here the values ! = 0:6 and g = 0:6y,
appropriate for radiation in the V -band (G ordon, C alzetti
& W itt1997).T he polarisation partofthe radiative transfer
code has been om itted.T he sam e approach has been used
for T he A tlas.
In thispaperw e restrictourselvesto a single stellarspatialdistribution,a three-dim ensionaldisk w ith exponential
behaviour in both horizontal and vertical directions. T he
horizontal and vertical scale lengths have been chosen to
m atch the observed values for the old disk population of
the G alaxy, ? = 3 kpc (K ent,D am e & Fazio 1991;Fux &
M artinet 1994) and ? = 0:26 kpc,respectively.N ote that
in B FG and in T he A tlas w e used ? = 4 kpc (B ahcall &
Soneira 1990): w hile in the previous m odels only the ratio
betw een the scale lengths w as relevant,for the m odelofthis
paper the absolute values m ust be speci ed,since the physicaldim ensions of the clum ps are required (see later).T he
ratio betw een the horizontaland verticalscale lengths used
in this paper is the sam e as in B FG and the A tlas.
Som e speci ed fraction of the total dust m ass is distributed in a sm ooth exponentialdisk,sim ilar to the stellar
one,w ith scale lengths d = ? and d = 0:4 ? (B FG and
A tlas),w hile the rem aining m ass is distributed in clum ps.
T he totalm ass of dust (sm ooth m edium + clum ps) is
de ned in term s of the opticaldepth of the m odel V ,i.e.
the V -band face-on opticaldepth through the centre ofthe
disk for a H M .For the exponentialdisk,the m ass is easily
com puted using the form ula
M
dust
=
V
=
6:8
2
d
8 a
3 QV
106
V
1
[M
(n + 1)e
n
];
w here a = 0:1 m and = 3g cm 3 are the grain radius and
density, Q V = 1:5 the extinction e ciency in the V -band
(H ildebrand 1983) and n = 4:6 is the radial truncation of
the disk (in scale length units;see later).In the follow ing,
w hen a C M is said to have been produced for a value of V ,
itm eansthatthe C M hasthe sam e dustm ass asa H M w ith
a face-on V -band optical depth V . T herefore, only in the
case of a H M V is a real optical depth, w hile for C M s is
m ainly an indicator of the m ass of dust in the m odel, the
e ective opacity depending in a com plex w ay on V and the
y T he values for ! and g are m easured in G alactic re ection nebulae,by com paring the intensity ofscattered light w ith radiative
transfer m odels. Since a clum py m edium has a low er e ective
albedo,neglecting clum ping in the m odelm ay bias the m easure
of! tow ards low ervalues,especially foroptically thick clouds and
stronger forw ard scattering (W itt & G ordon 1996).
c 0000 R A S,M N R A S 000,000{000
C lum py dustdisks
3
T able 1.Param eters ofthe clum py distributions.
fc
V
0.1
0.5
1.0
2.0
5.0
10.0
0.25
s
V
0.075
0.375
0.75
1.5
3.75
7.5
0.50
Nc
255
1277
2554
5108
12769
25539
s
V
0.05
0.25
0.5
1.0
2.5
5.0
0.75
Nc
511
2554
5108
10215
25539
51077
s
V
0.025
0.125
0.25
0.5
1.25
2.5
Nc
M du st
106 M
784
3831
7662
15476
38308
76616
0.7
3.4
6.8
13.6
34.0
68.0
For each value of the fraction of gas in clum ps fc and of the optical depth V ,the
opticaldepth ofthe sm ooth m edium Vs and the totalnum berofclum psN c are listed.
T he nalcolum n gives the totaldust m ass ofthe m odel.
other param eters for the clum py dust distribution.M odels
have been produced for opticaldepths V = 0.1,0.5,1,2,5,
10.
U sing a G alactic gas-to-dust ratio of150 (D evereux &
Young 1990;Sodroskiet al.1994),the gas m ass can be derived from the dustm ass.A fraction fc ofthe totalgasm ass
is then attributed to the m olecular com ponent.In this paper w e explore three di erent values, fc = 0.25, 0.5, 0.75.
T hese values are representative ofactualratios observed in
late type galaxies (from Scd to Sb, respectively; Young &
Scoville 1991).A llofthe m olecular gas (and the associated
m ass ofdust) is supposed to be distributed in clum ps.O ur
choice is appropriate,m ost ofthe m olecular gas ( 90% in
m ass) in the G alaxy being in the form of G M C s (C om bes
1991).
A sforthe clum py structure,w e use the sam e tw o-phase
form alism as in W itt & G ordon (1996).T he space occupied
by dustisdivided into cubic cellsofthe dim ension ofG alactic G M C s.B litz (1991) quotes a typicaldiam eter of45 pc:
a cubic cellof36 pc w illtherefore have the sam e volum e as
a m olecular cloud.
First,each cellis assigned a value corresponding to the
localabsorption coe cient(i.e.the inverse ofthe lightm ean
free path through dust) for the sm ooth distribution ofdust.
If V is the opticaldepth ofthe m odel,a fraction (1-fc) of
the dust m ass is distributed in the sm ooth m edium .T hus,
the sm ooth m edium opacity is de ned by a face-on optical
depth Vs = V
(1: fc).T he localabsorption coe cient
for the sm ooth m edium is then com puted from the assum ed
double-exponentialdistribution,using Vs .
Second,w e distribute the clum ps.T he position ofeach
clum p is derived from the distribution of m olecular gas in
the G alaxy,using the M onte C arlo m ethod.W e have used
the C O observationsofthe rstG alactic quadrantdescribed
by C lem ens,Saunders & Scoville (1988).T he radialdensity
distribution has been derived from the plot of the m ass of
m olecularhydrogen vs.galactocentric radiusin theirFig.11;
w e have adopted a gaussian verticaldistribution,as param eterised in their Eqn.3,w ith a FW H M that increases w ith
galactocentric distance.Follow ing them oleculargas,thedistribution of clum ps is m ore concentrated on the G alactic
plane than the sm ooth dust.A tany galactocentric distance,
the probability of nding clum ps for jzj> 2:5 d is nil.T he
distribution has been rescaled assum ing 8.5 kpc as the Sun
distance from the G alactic centre:the position ofthe G alactic m olecular ring is thus betw een 1.5 and 3.5 radial scale
lengths from the centre. O utside 13 kpc ( 4.5 ? ) the
c 0000 R A S,M N R A S 000,000{000
m olecular gas is notdetected (see also H eyeretal.1998).In
the m odel,both the dustand the stellar disks are truncated
at this distance from the centre.
Each clum p has been given a typicalG M C m ass of105
M (B litz 1993).T he totalnum berofcloudsin the m odelis
then directly derived from them assofgasin clum ps.A ssum ing that the dust has a uniform density inside each cloud,
the absorption coe cient in the clum p can be easily com puted from the clum p gas m ass and dim ensions,using the
adopted dust properties and gas-to-dust ratio: each clum p
has an absorption coe cient in the V -band of 110 kpc 1 ,
corresponding to an opticaldepth V
4.Since the e ects
ofa clum py structure are stronger w hen individualclum ps
areoptically thick (B oisse 1990;B oisse & T horaval1996)our
sim ulationsprovide an upperlim itto the e ectsofclum ping
in a spiralgalaxy.
Forallthe cellsthathave been assigned a clum p status,
the clum p absorption coe cient is sum m ed to the absorption coe cient of the sm ooth m edium . T he optical depth
along any photon path is com puted integrating the absorption coe cient array along the speci ed direction. A part
from this, the M onte C arlo code follow s the sam e schem e
as described in B FG .A list ofthe param eters de ning the
clum py dust distribution is given in Table 1.
To reduce the com putationaltim e and the dim ension of
the array involved,w e use only one octantofthe galaxy and
assign values in the other octant according to the m odel
sym m etries: the dust absorption coe cient is stored for
383x383x12 cells. B ecause of this, clum ps are distributed
random ly only in one octant.T his does not a ect the statistical properties of the m odel, as long as the num ber of
clum ps is large and the position of em ission of photons is
random . N evertheless, w e have tested the validity of this
sim pli cation com paring the results to a case w ith clum ps
distributed random ly allover the space,obtaining the sam e
results.Finally,sim ulated im ages of201x201 pixels are produced for severalinclinations (B FG ).
In the follow ing sections, w e present results and plots
for a few representative cases of optical depth and inclination. T he w hole dataset of this paper is available at
http://w w w .arcetri.astro.it/~ sbianchi/clum ping.htm l
3
R E SU LT S
A s already described in the introduction,the m ain e ect of
clum ping is that of an overallincrease in the transparency
4
S.Bianchietal.
T able 2.Fraction ofenergy absorbed in each m odel.
fem b
fc
0.00
0.25
0.00
0.50
0.75
0.25
0.15
0.50
0.75
0.25
0.50
0.50
0.75
0.02
0.07
0.12
0.19
0.32
0.44
0.01
0.06
0.11
0.18
0.31
0.43
0.01
0.05
0.09
0.15
0.28
0.39
0.01
0.04
0.07
0.12
0.23
0.34
0.09
0.13
0.17
0.24
0.37
0.48
0.08
0.12
0.16
0.22
0.34
0.45
0.08
0.11
0.15
0.19
0.30
0.40
0.25
0.29
0.32
0.38
0.49
0.59
0.25
0.28
0.31
0.36
0.47
0.57
0.25
0.28
0.30
0.35
0.44
0.54
V
0.1
0.5
1.0
2.0
5.0
10.0
For each value of V , the rst colum n gives the fraction for the H M (fc = 0), then each group
ofthree colum ns gives the values for C M s w ith a speci c fraction ofthe totalenergy em itted w ithin
clum ps (fem b ;fem b = 0 refers to the case w ith a hom ogeneous stellar distribution) and for the three
values ofthe fraction ofgas in clum ps fc .
of the m odel.In Table 2 w e show the fraction of the total
energy that is absorbed in each m odel,as a function of V
(that de nes the m ass of dust) and fc . O bviously,for the
sam e value ofthe opticaldepth,m ore energy is absorbed in
a H M (de ned in the table by fc = 0.0).T he resultspresented
for the H M are the sam e as for the spiralgalaxy disk in T he
A tlas. W hen the clum ping is introduced, C M s w ith larger
fc are increasingly m ore transparent.
W ithin the sam e m odel,the e ects ofclum ping depend
also on the disk inclination.T his is clearly show n in Fig.1,
w herew e haveplotted theattenuation (i.e.theratio betw een
the observed ux and the intrinsic, unextinguished, total
one) as a function of the optical depth, for the H M s and
the C M s for three values of the inclination angles, i= 20 ,
66 , 90 . A s seen before, C M s have the sam e attenuation
as H M s w ith a sm aller V ,for any inclination.T he largest
di erences betw een the tw o cases are found in m ore face-on
cases,and for high values offc.For instance,a face-on C M
fc= 0.75 and opticaldepth V = 10 hasthesam e attenuation
as an H M w ith V
4, w hile for an inclination of 20 , it
correspondsto a H M w ith V
5.Forthe sam e valuesof V
and fc ,the sm ooth m edium has an opticaldepth Vs = 2.5.
T he di erent behaviour of C M s w ith inclination can
also be seen analysing the disk m ajor axes pro les.In Fig.2
and 3 w e show the m ajor axispro les for tw o representative
cases oflow ( V = 1) and high ( V = 10) opticaldepth,for
inclinationsofi= 20 ,and 90 .In each leftpanelw eplotw ith
solid lines the pro les for C M s:the brighter pro le alw ays
refers to the case w ith higher fraction of dust in clum ps,
fc = 0:75,w hile the fainter is for fc = 0:25.A s a com parison,w e have also plotted a series ofpro lesforH M s(dotted
lines);thethreebrighterpro leshavethesam e opticaldepth
s
V asthe sm ooth m edium in each ofthe C M ,from fc = 0:75
(upper pro le) to fc = 0:25.T he faintest ofthe H M is characterised by the opticaldepth V .T he di erences betw een
each pro le and the pro le for the H M w ith opticaldepth
V are show n in the rightpanels,forthe chosen inclinations.
A s expected,a C M characterised by a value of V and
fc has a m ajor axis pro le at an interm ediate brightness
betw een the H M of optical depth V and the H M of opticaldepth Vs ,i.e.the hom ogeneous sm ooth com ponent.T he
di erences betw een H M and C M are not large,and alw ays
sm aller than 1 m ag.Foroptically thin cases,m odels are virtually indistinguishable atlow inclinations (asin the V = 1
pro lesofFig.2).Foroptically thick cases(e.g.them odelsin
Fig.3),the pro lesforC M are closerto those forthe sm ooth
F igure 1.A ttenuation as a function ofthe opticaldepth V ,for
the H M and C M s w ith fc = 0.25,0.50,0.75.D ata are plotted for
three inclinations,i= 20 ,66 and 90 .
m edium only for R < ? ,w here the num ber ofclum ps,and
therefore their lling factor,is sm all.B etw een 1 and 3 ? ,
w here the m olecular distribution peaks (the G alactic ring),
the pro le iscloser to thatforthe H M w ith the sam e optical
depth.O n the contrary,pro lesfor edge-on cases are alw ays
brighter than the corresponding H M .
T hese results are in contrast w ith those presented by
K uchinski et al. (1998). T hey derived opacities of highly
inclined galaxies, by com paring optical and near-infrared
colour gradients w ith M onte C arlo radiative transfer sim ulations,including scattering and clum py dust distribution.
T hey found thatthe derived opticaldepthsare insensitive to
the dusthaving eitherhom ogeneous orclum py distribution.
T his is explained by the signi cant num berofclum ps intersected by any line ofsight through a nearly edge-on galaxy.
O n this basis they argue that clum ping e ects m ay be m ore
im portant for low er inclinations, w here som e lines of sight
c 0000 R A S,M N R A S 000,000{000
C lum py dustdisks
5
F igure 2. M ajor axis pro les (left panel) for m odels w ith V = 1.0 and inclination i= 20 (face-on), 90 (edge-on). Solid lines refer to
C M s, for fc = 0.25, 0.5, 0.75. T he brighter pro les corresponds to the case w ith higher fraction of gas in clum ps, fc = 0.75, w hile the
dim m er to the case w ith fc= 0.25.H M s are presented for com parison (dotted lines).T he brighter pro le corresponds to a H M w ith the
sam e opticaldepth as the hom ogeneous dust ( Vs ) for the case fc = 0.75,follow ed by the analogous pro les for fc = 0.5,1.0.T he dim m er
pro le refers to a H M w ith the opticaldepth V .T he right panelpresents the di erence in m agnitude betw een each m odeland the H M
w ith the opticaldepth V .A llcurves have been sm oothed w ith a box of10 pixels.
F igure 3.Sam e as Fig.2,but for m odels w ith V = 10.0.To avoid overlap,a constant value of2 m agnitudes has been subtracted from
the surface brightness for the edge-on m ajor axis pro le presented in the left panel.
c 0000 R A S,M N R A S 000,000{000
6
S.Bianchietal.
F igure 4.M ajor axis pro les (left panel) for C M s w ith constant lling factor allover the dust distribution and a ratio of100 betw een
densities in clum ps and in the nearby sm ooth m edium (See text for details).M odels have V = 10 and i= 20 and 90 .Solid lines refer
to fc = 0.25,0.5,1.0 (B righter pro le).W ith a dashed line the case fc= 0.95 is plotted,w hile the dotted line refers to the H M w ith the
sam e V .To avoid overlaps,a constant value of0.75 m agnitudes has been subtracted from the surface brightness for the edge-on m ajor
axis pro le.T he right panelpresents the di erence in m agnitude betw een each m odeland the H M w ith the opticaldepth V .A llcurves
have been sm oothed w ith a box of10 pixels.
pass across clum ps, w hile others do not. H ow ever, the description of the clum py structure of dust in K uchinski et
al.(1998) is di erent from the one adopted here.Follow ing
W itt & G ordon (1996),they assign a constant lling factor
(ff= 0.15) for the high density cells allover the galaxy,and
assum e a ratio of100 betw een densitiesin clum psand in the
nearby sm ooth m edium .Since the sm ooth m edium has an
exponentialdistribution,clum ps close to the galactic centre
have a higher optical depth than those in the outer disk.
C lum ps in our sim ulations, instead, are distributed into a
ring-like structure and have the sam e opticaldepth.W e feel
these assum ptions are m ore realistic and are supported by
the available observationaldata.
To clarify the reasonsforthe discrepancy,w e have m odi ed the code to dealw ith the K uchinskiet al.(1998) form alism .C lum ps are now characterised by a constant lling
factor allover the dust distribution and by a value for the
localratio ofdensitiesofthe tw o dustphases.Forthese tests
w e have used the value 100 for the density ratio.T he geom etry ofthe galaxy and the cells dim ensions have been kept
asin x2.In Fig.4 w e show the m ajor axispro lesin the case
V = 10,for the tw o inclination i = 20 and 90 .T he solid
lines refer to C M s w ith the usualvalues fc= 0.25,0.50,0.75.
In these m odels fc can be directly converted into the lling
factor ff using the form ula
ff =
fc
fc + k(1
;
fc)
w here k = 100 is the density ratio. T he fc values of this
paper thus correspond to quite low global lling factors,
ff= 0.003, 0.01, 0.03, respectively. T he dashed line is for
ff= 0.15, as in K uchinski et al. (1998). For the assum ed
density ratio,this value of the lling factor corresponds to
fc
0:95, i.e. 95% of the totalm ass of dust in the m odel
is distributed in clum ps.Indeed,for this sim ulation setup,
C M s have pro les closer to the H M w hen view ed at larger
inclinations.
A sfor the pro les in Fig.2 and Fig.3,an increase offc
in the range 0.25 - 0.75 produces an increase in the global
transparency ofthe dustdistribution,thatisshow n in Fig.4
by the brighterpro les(especially in the face-on case) w hen
a larger fraction ofgas is distributed in clum ps.T he e ects
ofclum ping are instead reduced forfc = 0.95 (ff= 0.15).T his
is consistent w ith the sim ulations of radiative transfer in a
clum py m edia presented by V arosi & D w ek (1999). T hey
have studied the case of a clum py dust sphere under different radiation elds:a centralsource,a hom ogeneous distribution ofisotropic em itters and a uniform external eld.
W hen alltheotherparam etersde ning thedustdistribution
are xed,the fraction ofabsorbed photons as a function of
the clum ps lling factor show s a m inim um ,for any of the
radiation elds. T he exact value of ff for w hich the dust
distribution has a m axim um transparency depends on the
details ofthe m odel.In our sim ulation,for the value ofthe
clum p/sm ooth m edium density ratio adopted,clum ping has
the m ajor e ect in the reduction ofthe fraction ofabsorbed
energy for 0:03 ff < 0:15 (or0.75 fc < 0.95).In the C M
for fc= 0.95, 43% of the radiation is absorbed, alm ost the
sam e quantity as for the fc = 0.25 case (and for the H M ).A s
c 0000 R A S,M N R A S 000,000{000
C lum py dustdisks
7
F igure 5. E dge-on (i = 90 ) m inor axis pro les (left panel) for m odels w ith fc = 0.25, 0.5, 0.75 (solid lines). A s in Fig. 2, brighter
pro les corresponds to higher values offc .D otted lines refers to H M s as described for Fig.2.T he right panelpresents the di erence in
m agnitude betw een each m odeland the H M w ith the opticaldepth V .
already stated,the m odelfor fc = 0.75 is m ore transparent,
w ith 38% of the totalradiation absorbed.W ith respect to
the m ain m odelsofthispaper,the constant lling factordistributionsare slightly m ore opaque,forthe sam e value offc.
In the V = 10 m odelw ith fc= 0.75 ofTable 2,for instance,
the fraction ofabsorbed energy is 34% .
T he di erences in behaviour betw een the m ain C M of
this paper and the one w ith constant lling factor arise because oftw o reasons:(i) the di erent num ber ofclum ps for
a given value offc and (ii) the di erent geom etricaldistribution. In the m ain C M clum ps have all the sam e m ass,
and therefore the sam e density,for a given celldim ension.
In the constant lling factor C M the density ofa clum p depends on the localdensity ofthe sm ooth m edium (it is 100
tim es for the m odel analysed here). B ecause of the exponentialdistribution ofthe sm ooth m edium ,clum psatlarger
distance from the centre have sm aller density (and m ass)
than those in the inner part of the galaxy. C onsequently,
w hen the fraction of ISM m ass that is locked in clum ps is
xed,the m ain C M w ill have a sm aller num ber of clum ps
of higher m ass than the one w ith a constant lling factor.
For the case V = 10 w ith fc = 0.75 analysed before there are
roughly 4 tim es m ore clum ps in the latter than in the form er.T his sm aller num ber ofclum ps is then distributed on
preferential places in the m ain C M , m ore concentrated on
the galactic plane and in the m olecular ring. W hen seen
face-on the large num ber ofclum ps in the ring position attenuates a lot of the radiation com ing from the half of the
galaxy below the plane,acting alm ostasa screen.O nly outside the ring the lling factor ofclum ps is sm all,and in this
regions the face-on pro le is closer to that for a H M w ith
the optical depth of the sm ooth m edium . In the edge-on
c 0000 R A S,M N R A S 000,000{000
case,instead,it is the light em itted outside the ring,attenuated by the sm ooth dust distribution,that dom inates the
radiative transfer, thus resulting in a m ajor axis pro le of
interm ediate brightness betw een those ofthe H M s w ith V
and Vs .
In Fig.5 w e plot the edge-on m inor axis pro les for the
m ain m odelw ith opticaldepths V = 1 and 10.G eom etricaland opticalparam eters ofthe dust distribution in spiral
galaxies are m ainly retrieved by tting radiative transfer
m odels to im ages of edge-on galaxies (X ilouris et al.1997;
X ilouris et al. 1998; X ilouris et al. 1999). A t this inclination,in fact,w e can have separate view s of the radial and
vertical dust distribution. Furtherm ore, dust extinction is
increased as a result ofthe projection.T he pro les ofFig.5
show a behaviour analogous to the edge-on m ajor axis proles in Fig.2 and 3,w ith a reduced extinction that results
in shallow er extinction lanes on the galactic plane.O n the
contrary,edge-on m inor axis pro les for C M s w ith constant
lling factor are virtually indistinguishable from the H M .
4
E M B E D D E D SO U R C E S
So far clum ping has been introduced only in the description
of the dust distribution. Since w e use a clum py structure
to sim ulate G M C s, it is logical to assum e that part of the
stellar em ission com es directly from inside the clum ps,the
stars being form ed in the higher density phases ofthe ISM .
In thissection w e study the e ectsofclum ping both fordust
and stars.
R adiation from stars em bedded in G M C s is sim ulated
assum ing thata fraction fem b ofthe totalenergy em itted by
stars com es from inside the clum ps of dust.W e have used
8
S.Bianchietal.
F igure 6.Sam e as Fig.1,but for m odels w ith fem b = 0.15 and 0.50.
the values 0.15 and 0.50.T he actualvalue ofthis param eter
is di cult to be derived from observations, since it w ould
require sensitive indicators of the (optically) obscured star
form ation rates.T he above range is an educated guess that
com esfrom theestim ateoftheaverage fraction ofstellarlifetim e spent inside dark clouds. W ood & C hurchw ell(1989)
and C hurchw ell(1991),from their study ofU C H II regions,
concluded that O stars are em bedded in their natalm olecular clouds on average about 15% of the m ain sequence of
an O 6 star. H ow ever, this m ight be a low er lim it and the
fraction oftotalem bedded star form ation could be as high
as 50% w hen the contribution from low m ass stars is taken
into account.W ithin the M onte C arlo code,a fraction fem b
of the photons is em itted inside the cells. T he position of
the em ission inside the cellis random ly distributed w ithin
itsboundaries.T hisallow s m ore radiation to escape clum ps,
w ith respectto thecase w herethephotonsareem itted in the
centre ofthe cell.O nce the photon is em itted,the radiative
transfer is carried out through the clum py dust distribution
as described in x 2.
T hefraction ofabsorbed energy in C M sw ith em bedding
is show n in Table 2,for the tw o values offem b.In general,a
larger fraction ofenergy is absorbed in these C M s w ith respect to the case w hen only dust has a clum py distribution.
T he fraction ofabsorbed energy m ay be even larger than for
a H M .W hen the degree ofclum piness increases (larger fc )
the e ect of em bedding is m itigated and m ore energy can
escape.A s an exam ple,for fem b = 0.15,optically thick C M s
w ith fc
0:5 have an overall opacity sim ilar or sm aller
than for the H M . T he absorption in the case fem b = 0.5 is
larger and although it decreases w ith increasing fc as for
the previous case,the fraction ofabsorbed energy is alw ays
higher than in the H M . T he large fraction of energy ab-
sorbed in the optically thin cases, especially for fem b= 0.5
m ight seem surprising. T his is because in our form alism ,
for optically thin cases (sm allm ass ofdust),the em bedded
em ission com es from relatively few opaque clum ps. For a
cubic cellofopticaldepth 4 through its side and w ith a hom ogeneous distribution of internalem itters,nearly 50% of
the radiation is absorbed w ithin the cell.For instance,in a
m odel w ith V = 0.1 and fem b= 0.5, half of the radiation is
em itted w ithin clum ps,and half of it is absorbed,thus resulting in a fraction ofabsorbed energy 0.25,as in Table 2.
T he global increase in extinction can also be seen in
the attenuation plots of Fig.6.M odels w ith clum ping distributionsforboth starsand dusthave a transparency low er
than those w ith a clum py dust distribution only,at any inclination.A s already show n before,for fem b = 0:50,allthe
C M s are m ore opaque than the corresponding H M w ith the
sam e optical depth.For the m ore face-on inclinations, the
behaviouroftheattenuation islessdependenton V w ith respectto the H M .T hisisbecause the projected opticaldepth
of the sm ooth m edium is low er, and the clum ps dom inate
the em ission (and absorption).
Face-on im ages,especially in the case for a higher fraction ofem bedded em ission,clearly show a ring structure,due
to the photons being em itted in clum ps distributed accordingly to the ring-likedistribution ofm oleculargas.W hen the
m odelsareseen edge-on,thering issm oothed outby theprojection and it is clearly visible only for optically thin cases
and fem b = 0.50.In Fig.7 w e plot the m ajor axis pro les for
the edge-on cases w ith V = 1 and 10.For fem b = 0.15 and
for optically thick m odels w ith fem b= 0.50, the m ajor axis
pro les are sim ilar to those of H M w ith di erent e ective
opacities.It is interesting to note that the m ajor axis prolescan be brighterthan those ofthe H M w ith opticaldepth
c 0000 R A S,M N R A S 000,000{000
C lum py dustdisks
9
F igure 7.E dge-on m ajor axis pro les (left panels) for m odels w ith em bedded sources.P ro les are plotted for opticaldepths V = 1,10
and for the three values offc (solid lines).A s in Fig.2,brighter pro les corresponds to higher values offc .D otted lines refers to H M as
described for Fig.2.To avoid overlap,a constant value of1 m agnitude has been added to the surface brightness for the V = 10 pro le.
T he right panelpresents the di erence in m agnitude betw een each m odeland the H M w ith the opticaldepth V .A llcurves have been
sm oothed w ith a box of10 pixels.
V ,even w hen the globalopacities are larger.For exam ple,
in an edge-on m odelw ith fem b= 0.50,fc = 0.75 and V = 10
only 20% ofthe radiation escapes,w hile for the corresponding H M the fraction that escapes is 25% .T he pro le along
the m ajor axis is brighter,being very sim ilar to that ofthe
H M w ith V = 7.5.T he brighter pro le can be accounted for
c 0000 R A S,M N R A S 000,000{000
by the em bedded em ission,thatis m ore concentrated in the
plane than the sm ooth one.W hile the C M is generally dim m erthan the H M ,the m ajoraxisem ission ishigher:em bedding concurs therefore w ith dust clum ping in reducing the
contrast ofthe dust edge-on absorption lane.T his is clearly
10
S.Bianchietal.
F igure 8.E dge-on m inor axis pro les (left panels) for m odels w ith em bedded sources.P ro les are plotted for opticaldepths V = 1,10
and for the three values offc (solid lines).A s in Fig.2,brighter pro les corresponds to higher values offc .D otted lines refers to H M as
described for Fig.2.T he right panelpresents the di erence in m agnitude betw een each m odeland the H M w ith the opticaldepth V .
seen in the edge-on m inor axis pro les in Fig.8,especially
for high opticaldepth and fem b= 0.50.
5
SU M M A R Y A N D D ISC U SSIO N
W e have presented in this paper the results ofM onte C arlo
radiative transfer sim ulations through a tw o-phase clum py
dust distribution, for geom etries typical of a spiral galaxy
disk.T he space occupied by the dust distribution has been
divided into a three-dim ensionalgrid and each cellhasbeen
assigned a clum p or sm ooth m edium status.T he dim ension
ofthe cells and the dust m ass ofa clum p have been derived
from the properties of G alactic G M C . C lum ps have been
random ly distributed according to the ring-like distribution
ofm olecular gas observed in our G alaxy.W e have explored
c 0000 R A S,M N R A S 000,000{000
C lum py dustdisks
severalvaluesforthe opticaldepth V (i.e.the opticaldepth
ofa hom ogeneousdustdistribution w ith thesam e m ass)and
for the fraction oftotalgas residing in clum ps.
T he m ain conclusions are the follow ing. A s predicted
by sim ple argum ents and previous studies,a m odelw ith a
clum py dustdistribution (C M ) su ers a reduced extinction.
For the param eters adopted in this paper, w hich em ulate
realdust distributions in spiraldisks,the reduction of the
fraction of absorbed energy is m oderate, resulting in surface brightness pro les that are alw ays less than one m agnitude brighter than the corresponding hom ogeneous m odel
(H M ). T he m ajor di erences betw een H M s and C M s are
found for edge-on inclinations.T his is in contrast to the results presented by K uchinski et al. (1998). U sing a m odel
w ith spatially constant clum p lling factor and w ith alm ost
all (95% ) of the galactic dust locked in clum ps, they nd
that the sm allest di erences occur in edge-on cases.To ascertain the reasons for thisdiscrepancy,w e have reproduced
their experim ent and con rm ed their results.W e have concluded that the disagreem ent depends on the di erent param eters and distribution adopted for the clum ps. T his is
unfortunate, how ever, as it indicates a strong dependence
of the observed brightness pro les on the detailed internal
and spatial distribution properties of clum ps w hich m akes
the interpretation ofthe data very di cult.
Since star form ation occurs in high density clouds, it
is logicalto assum e that part ofthe stellar em ission occurs
w ithin clum ps.W e have therefore produced m odels w here a
fraction ofthe photons is em itted in the high density cells.
W hen thisclum py stellardistribution isconsidered,C M sare
less transparent than for a clum py dust distribution only.
D epending on the fraction ofgas in clum ps and on the fraction ofem bedded stellarem ission,galaxiescan be even m ore
opaque than predicted by H M s w ith the sam e m ass.
O ne of the m ajor concerns about clum ping is that its
neglect w illproduce a signi cant underestim ate ofthe dust
m ass of a galaxy. X ilouris et al. (1999) analysed a sam ple
ofseven edge-on galaxies tting the surface brightness w ith
a radiative transfer m odel. T hey nd a m ean centralfaceon optical depth V = 0.5. C om paring the total dust m ass
of each galaxy w ith the m ass of gas, they derive a gas-todust m ass ratio of360 60.T he derived value is larger than
the G alactic value by m ore than a factor of tw o (Sodroski
et al. 1994), but closer than estim ates based on FIR dust
em ission observed by IR A S (D evereux & Young 1990).From
ourresultsin Fig.2 and Fig.5,m inorand m ajoraxispro les
for C M s w ith V = 1 and 0:50 < ff < 0:75 are sim ilar to
those for a H M w ith V = 0:5 (In the V = 1 panels, a
pro le for a H M w ith V = 0:5 is show n betw een the tw o
m entioned C M s w ith a dotted line).T herefore,a C M w ith
V = 1 can have an edge-on appearance very sim ilar to an
H M w ith V = 0:5. T he m ass of dust in the C M w ould
be tw ice the value derived for the H M and the gas-to-dust
m assratio w ould bereduced accordingly to 180,a valueclose
to the canonical.Sim ilar results can be obtained using the
V = 1 pro les for m odels w ith em bedded stellar em ission.
W e caution here that X ilouris et al. (1999) nd that dust
diskshave a larger radialscale lengthsthan the stellar w hile
in the m odels ofthis paper both disks have the sam e radial
scale lengths; this m ay a ect the num erical details of the
exercise carried out in this paragraph, but the qualitative
result holds.O n the contrary,a C M w ith a constant lling
c 0000 R A S,M N R A S 000,000{000
11
factor allover the galaxy w ould resem ble,w hen seen edgeon,a H M w ith the sam e opticaldepth,and there w ould be
no change in the m ass estim ate.
A cknow ledgem ents.W ew ish to thank D .G alliforinteresting
discussion aboutthe em bedding ofstars in m olecular clouds
and the referee,A .W itt,forusefulcom m entsthatim proved
the presentation ofthe data in the paper.
R EFER EN C ES
B ahcallJ.N .,Soneira R .M .,1980,A pJS,44,73
B ianchiS.,Ferrara A .,G iovanardiC .,1996,A pJ,465,127 (B FG )
B yun Y .I.,Freem an K .C .,K yla s N .D .,1994,A pJ,432,114
B litz L.,1991,in:T he P hysicsofStarForm ation and E arly Stellar
E volution,eds.C .J.Lada,N .D .K yla s,K luw er,D ordrecht,
p.3
B litz L.,1993,in:P rotostar and P lanets III,eds.E .H .Levy,J.I.
Lum ine,U niv.ofA rizona P ress,Tucson,p.125
B oisse P.,1990,A & A ,228,483.
B oisse P.,T horavalS.,1996, in:N ew E xtragalactic Perspectives
in the N ew South A frica,eds.D .L.B lock ,J.M .G reenberg,
K luw er,D ordrecht,p.187
C hurchw ell, 1991, in: T he P hysics of Star Form ation and E arly
StellarE volution,eds.C .J.Lada,N .D .K yla s,K luw er,D ordrecht,p.221
C lem ens D .P.,Sanders D .B .,Scoville N .Z.,1988,A pJ,327,129
C om bes F.,1991,A R A & A ,29,195
D e Jong R .S.,1996,A & A ,313,377
D evereux N .A .,Y oung J.S.,1990,A pJ,359,42
D isney M .,D avies J.,P hillipps S.,1989,M N R A S,239,939
Ferrara A ., B ianchi S., C im atti A ., G iovanardi C ., 1999, A pJS,
in press
Fux R .,M artinet L.,1994,A & A ,287,L21
G ordon K .D .,C alzettiD .,W itt A .N .,1997,A pJ,487,625
H artm ann D .,1994,P hD T hesis,Sterrew acht Leiden
H eiles C .,1998,1998,A pJ,498,689
H enyey L.G .,G reenstein J.L.,1941,A pJ,93,70
H eyer M .H .,B runt C .,SnellR .L.,H ow e J.E .,Schloerb F.P.,
C arpenter J.M .,1998,A pJS,115,241
H ildebrand R .H .,1983,Q JR A S,24,267
H obson M .P.,Scheuer P.A .G .,1993,M N R A S,264,145
K ent S.M .,D am e T .M .,Fazio G .,1991,A pJ,378,131
K uchinski L. E ., Terndrup D . M ., G ordon K . D ., W itt A . N .,
1998,A J,115,1438
Sauty S.,G erin M .,C asoliF.,1998,A & A ,339,19
Silva L.,G ranato G .L.,B ressan A .,D anese L.,1998,A pJ,509,
103
SodroskiT .J.et al.,1994,A pJ,428,638
V arosi F., D w ek E ., 1997, in: T he U ltraviolet U niverse at Low
and H igh R edshift,eds.W .H .W aller et al.,A IP,N ew Y ork,
p.370
V arosiF.,D w ek E .,1999,A pJ (1999),in press
W itt A .N .,G ordon K .D .,1996,A pJ,463,681
W olfS.,Fischer O .,P fau W .,1998,A & A ,340,103
W ood K .,1997,A pJ,477,L25
W ood D .O .S.,& C hurchw ell,E .1989,A pJ,340,265
X ilouris E . M ., K yla s N . D ., Papam astorakis J., Paleologou
E .V .,H aerendelG .,1997,A & A ,325,135
X ilouris E .M .,A lton P.B .,D avies J.I.,K yla s N .D .,Papam astorakis J.,Trew hella M .,1998,A & A ,331,894
X ilouris E .M .,B yun Y .I.,K yla s N .D .,Paleologou E .V .,Papam astorakis J.,1999,A & A ,344,868
Y oung J.S.,Scoville N .Z.,1991,A R A & A ,581,625