The Astrophysical Journal, 770:27 (13pp), 2013 June 10

doi:10.1088/0004-637X/770/1/27


C 2013. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

PROPERTIES OF DUST GRAINS PROBED WITH EXTINCTION CURVES
Takaya Nozawa1 and Masataka Fukugita1,2,3

1 Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa 277-8583, Japan
2 Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan
3 Institute for Advanced Study, Princeton, NJ 08540, USA

Received 2013 January 15; accepted 2013 April 23; published 2013 May 21

ABSTRACT
Modern data of the extinction curve from the ultraviolet to the near-infrared are revisited to study properties ofdust grains in the Milky Way (MW) and the Small Magellanic Cloud (SMC). We confirm that the graphite–silicate
mixture of grains yields the observed extinction curve with the simple power-law distribution of the grain size
but with a cutoff at some maximal size: the parameters are tightly constrained to be q = 3.5 ± 0.2 for the size
distribution a −q and the maximum radius amax = 0.24 ± 0.05 μm, for both MW and SMC. The abundance ofgrains, and hence the elemental abundance, is constrained from the reddening versus hydrogen column density,E(B − V )/NH . If we take the solar elemental abundance as the standard for the MW, >56% of carbon should
be in graphite dust, while it is <40% in the SMC using its available abundance estimate. This disparity and therelative abundance of C to Si explain the difference of the two curves. We find that 50%–60% of carbon may
not necessarily be in graphite but in the amorphous or glassy phase. Iron may also be in the metallic phase or up
to ∼80% in magnetite rather than in silicates, so that the Mg/Fe ratio in astronomical olivine is arbitrary. With
these substitutions, the parameters of the grain size remain unchanged. The mass density of dust grains relative+70
to hydrogen is ρdust /ρH = 1/(120+10−16 ) for the MW and 1/(760−90 ) for the SMC under the elemental abundance
constraints. We underline the importance of the wavelength dependence of the extinction curve in the near-infraredin constructing the dust model: if Aλ ∝ λ−γ with γ  1.6, the power-law grain-size model fails, whereas it works
if γ  1.8–2.0.
Key words: dust, extinction – Galaxy: abundances – ISM: abundances – ISM: general
Online-only material: color figures
These arguments indicate that the size distribution of grains
may be key to understanding the formation of grains andsmall astronomical objects. The size distribution of grains
has been studied in the literature (e.g., Draine & Lee 1984,
hereafter DL84; Kim et al. 1994; Weingartner & Draine 2001,
hereafter WD01; Clayton et al. 2003b; Zubko et al. 2004). Thesestudies stress accurately reproducing the extinction curve thatis progressively more accurately measured and is extended to
the NIR region, to find the detailed size distribution of grains.
A significant variation has become apparent in the extinction
curve depending on the line of sight (Mathis & Cardelli 1992;
Clayton et al. 2000). The variation typically amounts to RV =2.2–5.5, assuming the one-parameter formula of Cardelli et al.
(1989, hereafter CCM: their formula is referred to as the CCM
curve), who showed that the family of extinction curves iswell described by a formula with the single free parameter,
RV = AV /E(B − V ), where AV is the extinction in the V band
and E(B − V ) is the reddening. It should be asked how thevariation of the extinction curve translates to the properties ofdust grains. Another important problem is determining othergrain constituents beyond silicate and graphite.
Further to the shape of the extinction curve, the amounts ofextinction, usually represented with E(B − V ) per hydrogen,gives an important constraint on the abundance of dust grains.
An important question is whether the grain parameters derived
from the extinction curve are consistent with those from theamount of extinction, and whether they are consistent with theelemental abundance.
It has been known that the extinction curve for the Small
Magellanic Cloud (SMC) is markedly different from that for
the Milky Way (MW) in that it lacks the feature at 2175 Å andit shows a significantly steeper rise to the far-UV side beyond
2000 Å. What would cause this difference is the problem to be

1. INTRODUCTION
Submicrometer-size grains cause attenuation of light fromultraviolet (UV) to near-infrared (NIR) wavelengths. Larger
grains may form small astronomical objects. Thirty years
ago, Mathis et al. (1977, hereinafter MRN) showed that dust
composed of a mixture of silicate and carbonaceous grainsaccounts for the extinction curve from the UV to optical
wavelengths, from which they derived the size distribution ofgrains for a = 0.005–0.25 μm. Only with nine bins of thehistogram, they advocated that the size distribution is consistent
with the power law a −q with the index q from 3.3 to 3.6.4
This dust model is consistent with the fact that Mg, Si, and Fe
are highly depleted in diffuse interstellar matter. These threeelements are also major constituents of astronomical silicate
with the corroborating fact that the abundance of dust needed toaccount for the observed extinction is on the order of magnitudeof their cosmic abundance, and hence the three elements could
dominantly be locked in dust.
Independently, the argument appeared that small astronomical bodies in frequent collisions would obey a power-law distribution in size (Dohnanyi 1969; Hellyer 1970; Biermann &Harwit 1980; Dorschner 1982). It was argued that the powerlaw derived by collisional equilibrium typically has q = 3.5(Dohnanyi 1969; Pan & Sari 2005), which agrees with the power
favored by interstellar dust from extinction studies. For larger
sizes, the distribution of small astronomical objects (1–10 km)
is not in conflict with this power, while the available information
is limited to derive it more accurately.
4

For reference we attempt to fit the size distribution derived by MRN (fivedata points for silicate, and six for graphite) by the power law: it results in
q = 3.8 ± 0.7 for graphite and q = 3.5 ± 0.6 for silicate, where the error
stands for the dispersion.

1

The Astrophysical Journal, 770:27 (13pp), 2013 June 10

Nozawa & Fukugita

Table 1
Interstellar Extinction Data for the Milky Way
Reference Wavelength

1/λ

(μm)

(μm−1 )

Lower

Upper

8.00
6.25
4.60
2.78
2.27
1.82
0.80
0.61
0.46

2.573
2.167
2.712
1.449
1.252
1
0.211
0.127
0.077

3.894
3.003
3.625
1.62
1.331
1
0.278
0.166
0.101

0.125
0.16
0.2175
0.36
0.44
0.55
1.25
1.65
2.17

(far-UV rise)(far-UV dip)
(UV bump)(U band)(B band)(V band)(J band)(H band)(K band)

1.25
1.65
2.17

(J band)(H band)(K band)(Aλ /AV )obs

Ranges from the CCM formula for NIR
0.80
0.61
0.46

0.267
0.171
0.110

0.299
0.191
0.123

Notes. The allowed ranges of the interstellar extinction (Aλ /AV )obs at thereference wavelengths constructed from the data of extinction curves of FM07.
The ranges of the NIR extinction from the CCM formula is for RV = 2.75–3.60.
In optical regions, the ranges virtually agree with the 1σ ranges of FM07.

resolved. The SMC-type extinction curve is also indicated forinterstellar clouds such as Mg ii absorbers (York et al. 2006); it
is shown that the majority, typically ∼70%, of the cloud obeys
the SMC-type extinction law.
In this paper, we revisit gross but generic features of dust
grains using the modern data of extinction curves from 1150 Å
to 2.2 μm derived for the MW and, in the other extreme, theSMC, along with modern optical data of the relevant dielectricmaterial. We examine whether some departure from power-law
size distributions, other than cutoffs in the size, is compellingin reproducing the observed extinction curve. We are alsointerested in how dust in the SMC should differ from thatin the MW in its properties. Furthermore, we would like to
see whether the graphite–silicate model is unique and what isthe possible range of dust to hydrogen mass ratio. We assume
spherical grains and their size distribution obeys a power law
allowing for a truncation at some maximum size. The powerlaw distribution, at least, can be understood with simple physics
for the evolution of grains. We do not treat polarization andinfrared emission from dust since either requires knowledge
other than the property of dust grains and so, in turn, requires
extra assumptions that would introduce further uncertainties.
In Section 2, we review the data of extinction curves used in
the present study, and we note a problem in the average NIR
extinction curve. After defining the model of interstellar dust
used in this paper in Section 3, we search, in Section 4, forthe dust model that could reproduce the observation, present
parameters of the model and ask various possibilities of thefavorable composition of grain species from the extinction data
for the MW. We carry out a similar analysis for SMC dust
in Section 5. Section 6 gives a summary of this study withdiscussion.

Figure 1. (a) Interstellar extinction curves from the 328 stars in the Milky Way
from FM07. Our 1σ (thick solid bars) estimates are denoted at our reference
wavelengths. The plus symbol shows the V band used as the normalization. (b)The expanded figure for the extinction in NIR. The bars are 1σ . The extinction
curves of the CCM formula with RV = 2.75 (the upper dashed curve in (a) andthe lower dashed curve in (b)) and RV = 3.60 (the lower dashed curve in (a) andthe upper dashed in (b)), and the model of WD01 (RV = 3.1, the solid curves)
are shown.
(A color version of this figure is available in the online journal.)

ranges from 1150 Å to 2.2 μm. We are primarily interested in
the global average of the extinction curves, so we consider aset of the curves at several specific reference wavelengths, atwhich we derive the allowed ranges of extinction. We takenine reference wavelengths in total with the V band to givethe normalization. We consider wavelengths corresponding to
UBJHK and three wavelengths, λ = 0.2175 μm, 0.16 μm, and0.125 μm, in UV, which characterize the hump, the bottom in
the UV region, and an arbitrary chosen wavelength in the riseof the Galactic extinction curve toward shorter wavelengths,
respectively. The UBVJHK passbands are those FM07 used toderive the extinction curve in the optical and NIR regions fromthe observation (they have not used R and I). We take the 1σ
allowed ranges corresponding to 68% (224 curves) of the 328
curves at each reference point. Our wavelength mesh is toocoarse to study the 2175 Å feature, so it is considered separately.
We do not treat diffuse interstellar bands which are not apparent
in the FM07 extinction curve. Possible line features beyond the
K band are not treated.
The 1σ ranges of Aλ /AV are shown in Table 1 and in Figure 1:(a) for the entire wavelength range considered and (b) for NIR

2. EXTINCTION CURVES
2.1. Milky Way Extinction
We take the extinction curves derived from UV to NIR for the
328 stars in Fitzpatrick & Massa (2007, hereafter FM07) withthe aid of stellar atmosphere, replacing the traditional method
using reddened-comparison pairs of stars. The extinction curve2The Astrophysical Journal, 770:27 (13pp), 2013 June 10

Nozawa & Fukugita
Table 2Extinction Data in the Small Magellanic Cloudin an expanded scale.5 Figure 1(a) shows that the CCM curvewith RV = 2.75–3.60 and the dust-model calculation of WD01
(which gives RV = 3.1) are both consistent with the data ofFM07 in the optical and the UV regions. In the NIR region,
however, the expanded figure, Figure 1(b), indicates that bothcurves are off from the FM07 data by ≈1σ or more. This arises
from the fact that CCM adopted the NIR extinction that follows
the power law Aλ /AV ∝ λ−γ with the index γ = 1.61 (Rieke& Lebofsky 1985), whereas the FM07 data are consistent witha steeper index, γ = 1.84. The steeper index (Martin & Whittet
1990) has gained more support in a recent work: for example,
Fitzpatrick & Massa (2009) reported the index 1.78–2.0 and that
it varies according to the line of sight (see also Fritz et al. 2011
for a summary of the recent work). The slope changes between
IC and J.
We take the FM07 data as our prime choice, while tentatively
retaining the possibility that the NIR power index is moderate,
as with CCM and WD01. We also consider this possibility andstudy the implication on the dust model. We take the range of
the CCM curves with RV = 2.75–3.60, which correspond to 1σof the extinction data of FM07 in the UV and optical range, as
seen in Figure 1(a). Our 1σ range adopted for NIR covers thevariation of the index reported by Fitzpatrick & Massa (2009),
while CCM and WD01 are beyond 1σ . The difference in the
NIR slope leads to a significant difference in the conclusion
concerning the model.
In addition, we consider the absolute amount of extinction, orreddening. Bohlin et al. (1978) obtained for 75 stars NH /E(B −V ) = 5.8 × 1021 cm−2 mag−1 , which is widely adopted in
the literature, and claimed that the data for different lines ofsight rarely fall beyond the lines 1.5 or 1/1.5 times the value
indicated in this expression. Actually, we see in their figure (their
Figure 2(b)) that about 85% of stars are located between the twolines. (Most of the deviants are toward the smaller NH side.)
The NH value includes a 25% contribution from H2 molecules.
Unfortunately, they have not given the error or the dispersion of
the fit, so we have re-fitted their data to obtain the 1σ error. We
have also tried to include the recent enlarged data set compiled
by Gudennavar et al. (2012). After the selection similar to thatin Bohlin et al. (1978), we obtain, using 174 data,
NH /E(B − V ) = (5.7 ± 1.7) × 1021 cm−2 mag−1 ,Reference Wavelength

1/λ

(μm)

(μm−1 )

Lower

Upper

8.00
6.25
4.60
2.78
2.27
1.82
0.80
0.61
0.46

4.839
3.813
2.772
1.521
1.303
1
0.108
0.000
0.000

6.752
5.005
3.946
1.892
1.488
1
0.324
0.184
0.060

0.125
0.16
0.2175
0.36
0.44
0.55
1.25
1.65
2.17

(far-UV rise)(far-UV dip)
(UV Bump)(U band)(B band)(V band)(J band)(H band)(K band)(Aλ /AV )obs

214, AzV 398, and AzV 456 using the traditional reddenedcomparison pair of stars. The more recent result of Cartledge
et al. (2005) is consistent with G03.
As known, the bump at 2175 Å is not apparent except forthe sightline toward AzV 456, which is located in a “wing”
region of quiescent star formation, unlike the other four thatpass through the star-forming bar of the SMC. We include theextinction curve toward AzV 456 in our consideration.
For our measure, we define, because of the paucity of data,
the “1σ ranges” of the extinction to be the maximum and theminimum of the five curves at each reference wavelength. If thelower value becomes negative, it is set to zero. The ranges ofextinction are given in Table 2 and plotted in Figure 2.
The recent analysis for the column density of neutral hydrogen NH i is by Welty et al. (2012). The abundance of H2
molecules is estimated to be 2% of hydrogen, which is compared to 25% for the MW (Bohlin et al. 1978), giving an example of molecular formation depending on environment. Theirestimate is consistent with that of Tumlinson et al. (2002),6
2NH2 /NH = 1+0.5−0.2 %. Welty et al. (2012) gave
22
−2
NH /E(B − V ) = 2.3+2.8
mag−1 .
−1.3 × 10 cm(2)This is larger than the MW value given in Equation (1) by afactor of 4.0 ± 2.4. Russell & Dopita (1992) give a summaryof the elemental abundance for the SMC, which indicates thatits metallicity is 5.6 times smaller than solar. We note thatthe abundance in SMC is poorly known. The inferred dust
abundance per hydrogen of the SMC, relative to the MW, isconsistent with, or somewhat less suppressed compared with,
the inferred heavy element abundance.

(1)where the error stands for the dispersion of the fit. This is ingood agreement with the original Bohlin et al. (1978) result,including the size of the dispersion we re-estimated from theirdata (±1.7).

3. DUST MODEL2.2. Extinction in the Small Magellanic CloudWith spherical particles uniformly distributed in interstellar
space the total extinction, Aλ , along the line of sight l, is amax,jAλ = 1.086dl
π a 2 Qext
(3)λ,j (a)nj (a)da,

Only a handful of sightlines are studied for the extinction
curve for the SMC. Gordon et al. (2003, hereafter G03)
presented the curve toward five stars, AzV 18, AzV 23, AzV

jamin,j5

If we derive the extinction for RC and IC from FM07 in the same manner as
the other passbands, the 1σ ranges are 0.722–0.778 and 0.492–0.587,
respectively. We note that these data are not directly constrained byobservations. In fact, direct observational determinations for the extinction for
RC and IC are scant. Winkler (1997) estimated it assuming the fiducial colors
for the estimated types of stars. His values overlap with those from FM07 at
1σ only marginally. The extinction in the IC band discussed in Draine (2003a)
extends from −1.5 to +2.7 σ for RV = 3.1. The range derived above mayunderestimate the error. We do not use the extinction for RC and IC as theconstraint: if we would take their 1.5σ error range for FM07, all constraints
discussed in this work will be unchanged.

where Qext
λ,j (a) is the extinction efficiency defined as the ratioof the extinction cross section σλ,j (a) at wavelengths λ to thegeometric cross section π a 2 for grain species j and is calculatedusing Mie scattering with the laboratory optical data of dielectric
6

The estimate of Cartledge et al. (2005) for H2 for some of the five stars
wildly varies from 5% to 50%. We quote that the cosmic global abundance of
2N (H2 )/N (H) is inferred to be 0.30 (Fukugita 2011).

3

The Astrophysical Journal, 770:27 (13pp), 2013 June 10

Nozawa & Fukugita

and corundum (Al2 O3 ; Toon et al. 1976). We do not consider
enstatite (MgSiO3 ). One may expect that the optical propertyof MgSiO3 is not much different from that of olivine, andits contribution may not disturb much the calculation withastronomical silicate.8 We ignore SiO2 , since its Q factor issmall. Inclusion of SiO2 only increases the amount of Si locked
in dust grains. The Ca- and Ti-bearing grains may constitute
some components of dust, but they are not considered becauseof their small cosmic abundance.
We do not consider polycyclic aromatic hydrocarbons (PAHs)
separately. The bump at 2175 Å in the Galactic extinction curve
can be accounted for either with or without PAH insofar as small
graphite grains (molecules with the effective radius of 200 Å)
are included (Stecher & Donn 1965; Joblin et al. 1992; Clayton
et al. 2003a).
Dust grains dominated by a single size do not give theobservationally obtained extinction curve. Some distribution
over the size is necessary. A typical distribution that is known
to work is the power law, which we also use herenj (a) = nH Kj (a/a0 )−qj ,where nH is the hydrogen number density, Kj is the fraction of
species j, and a0 is a size of the normalization. We limit therange to amin,j  a  amax,j . We assume the same grain-size
distribution independent of grain species.
We take qj , amax,j , and the condensed fraction fi,j , i.e., thefraction of element i contained in grain species j, as parameters,
and find a set of the parameters that satisfy 1σ of extinction dataat all nine reference wavelengths. We also study the dependence
on the parameter amin,j , but the fit varies little insofar as thisminimum size is smaller than 0.005 μm. The minimum size
amin affects the extinction curve in the way that increasing
it diminishes the rise in the far-UV for wavelengths roughly
shorter than O(2π ) × amin . With our choice of amin = 50 Å
it affects little the curve for λ > 1150–1250 Å we consider.
The increasing variation of amin also affects the 2175 Å bumpcaused by graphite, reducing the hump, but the resulting changeof the extinction curve from the decreasing variation of aminfrom 0.005 μm to 0 is small, typically <1/3 the 1σ error range.
The parameter amin is not well determined if it is left as a freeparameter, unless the extinction data of shorter wavelengths areused. Thus we fix amin at 0.005 μm.9 The change of lower cutoff
affects little the determination of the other parameters, unlessq  4, which is deep in the region not allowed in our study.
It increases the total amount of mass, say, by 10% if amin is
reduced from 0.005 μm to 0.001 μm. A possible increase of thelower cutoff for graphite is considered for SMC dust separately.
The extinction curve we refer to is Aλ /AV , normalized in the V
band

Aλ
j Kj Aλ,j
=
,
(5)
AV
j Kj AV ,jFigure 2. (a) The extinction curves for the SMC derived from five stars by G03,
with 1σ (thick solid bars) significance denoted at the reference wavelengths.
The plus symbol corresponds to the V band. (b) The expanded figure for the
NIR extinction. The extinction curve given by WD01 (for AzV 398) is shownby the solid line. The extinction curve for the MW, taken from Figure 3 (ourmodel), is also added (thin dotted curves) for comparison.
(A color version of this figure is available in the online journal.)

constants: nj (a)da is the number density of the grain of species
j with radius between a and a + da.
The “standard composition” of dust is graphite and silicate.
Following DL84, we take “astronomical silicate” (we simply call it silicate unless otherwise stated) with the composition MgFeSiO4 . We assume the simple approximation that
dust grains are bare refractory particles without substructure.
For graphite, we calculate Qext
λ,gra using the usually adopted(1/3) − (2/3) approximation for the dielectric constant components perpendicular and parallel to the basal plane (e.g., DL84)
to represent anisotropy.
In order to see their possible importance we also consider
other refractory components and carbonaceous materials, which
are likely to condense into grains, insofar as their opticalproperties are known. We consider 10 grain species, graphite(gra; optical data from Draine 2003b), glassy carbon (glC;
Edoh 1983), amorphous carbon (amC; Zubko et al. 1996),
silicon carbide (SiC; Choyke & Palik 1985), astronomicalsilicate (asil; Draine 2003b) of the chemical composition ofolivine (Mgx Fe1−x SiO4 ) with x ≈ 1, forsterite (Mg2 SiO4 ;Semenov et al. 2003), pure iron (Fe, Semenov et al. 2003),
magnetite (Fe3 O4 ; Triaud),7 troilite (FeS; Semenov et al. 2003),
7(4)

where Aλ,j is the component contribution to Equation (3).
8

The imaginary part of the presently available refractive index of enstatite(Dorschner et al. 1995) deviates largely from that of forsterite in the UV
region, while the real part differs little. The Q factor of enstatite differs little
from that of forsterite. We do not treat enstatite separately in this paper.
9 Considering a possible importance of small grains beyond the power law in
the IR emission, we examined the possibility that some non-negligible
amounts of grains are distributed below amin . The presence of such grains, say,at 0.001 or 0.005 μm, modifies most characteristically the UV slope, making it
steeper at wavelengths shorter than 2000 Å and the hump at 2175 Å larger.
Insofar as these extra components are less than 15% in mass, our resulting
extinction curves are not changed beyond our reference errors.

A. Triaud, http://www.astro.uni-jena.de/Laboratory/OCDB/oxsul.html

4

The Astrophysical Journal, 770:27 (13pp), 2013 June 10

We also calculateE(B − V )= 1.086KjNHj amax,j

Nozawa & Fukugita
Table 3Extinction Data Aλ /AV for the Milky Wayda(σB,j − σV ,j )(a/a0 )−qj .

Wavelength

amin,jAλ /AV(μm)(6)

0.35310.365
0.44
0.46270.55
0.61400.66
0.74670.81
0.8887
1.25
1.65
2.17
3.35
3.55
4.44
4.60

The amount of reddening per hydrogen leads to the constraint
on the abundance of dust that can be compared with otherelemental abundance estimates. Assuming that the abundance
does not vary from place to place, we take as standard for
the MW the solar elemental abundance estimated by Grevesse& Sauval (1998, hereafter GS98).10 Asplund et al. (2009),
using three-dimensional (3D) calculation, claimed that the solar
abundance is generally lower than that of GS98 by as muchas 30%, especially for C, O, and a few others. A comparable
3D calculation (Caffau et al. 2011), however, has given anabundance that is lower than GS98 only by 10%.
The elemental abundance is generally not tightly converged
among the authors. For instance, for the carbon abundance,
which is of one of our major concerns, the GS98 value
C/H = (3.3 ± 0.5) × 10−4 may be compared with 3.62 × 10−4of Anders & Grevesse (1989), 2.69 × 10−4 of Asplund et al.
(2009), 3.16 × 10−4 of Caffau et al. (2011), (1.9–2.9) ×10−4 of
Cardelli et al. (1996), 2.14 × 10−4 of Nieva & Przybilla (2012),
2.45×10−4 of Lodders (2010), and so forth. The iron abundance,
for which GS98 give log(Fe/H) + 12 = 7.50 ± 0.05, too, varies
between 7.45 and 7.66. Keeping these uncertainties in mind,
we take as our default GS98, which leads to a satisfactory solar
structure.
For the SMC we take the composition given by Russell &Dopita (1992), with which the total metallicity is 1/5.6 times
solar. The abundance of refractory elements, Mg, Si, andFe, is smaller than their MW values by factors 1/3.3–1/4.6.
The abundance of O and C in the SMC seems more strongly
depressed by 1/(6.2–6.3) times, compared with the solar abundance. Taken literally, this leads to the significant result thatthe ratio of carbonaceous material to silicate in the SMC is
50% lower than the corresponding value for the MW. We must
remember, however, that the elemental abundance for SMC isprobably more uncertain than for MW.

(u)
(U)
(B)
(g)
(V)
(r)
(Rc )
(i)
(Ic )
(z)
(J)
(H)
(K)
(WISE1)(IRAC1)(IRAC2)(WISE2)

Our Work

CCM (RV = 3.1)

WD01 (RV = 3.1)

1.633
1.583
1.306
1.232
1.000
0.866
0.785
0.660
0.582
0.498
0.267
0.153
0.0836
0.03480.03130.02130.0201

1.584
1.557
1.325
1.243
1.000
0.884
0.812
0.678
0.583
0.489
0.282
0.180
0.116
0.05770.05250.03670.0346

1.660
1.608
1.318
1.246
1.000
0.859
0.774
0.646
0.571
0.493
0.287
0.185
0.117
0.05130.04630.02970.0280Note. Our model extinction is compared with those of CCM and WD01.

rise in the far-UV and have a larger scattering efficiency in the
NIR bands, and we suppose that the mixture of these two species
would give the correct extinction curve.
4.2. Graphite–Silicate Dust Model
The two-component dust model, consisting of graphite andastronomical silicate, has been widely taken for interstellar dust
since MRN. Mg, Si, and Fe are similar in the cosmic number
abundance (within 20%). If we assume that Si is all condensed
into astronomical silicate, fSi,asil = 1, then fMg,asil = 0.93 andfFe,asil = 1, if Fe:Mg = 1:1 (see Draine 2003a) with the GS98
abundance. Here, iron is slightly (10%) deficient and extra Si
and Mg may condense into forsterite. Oxygen locked in silicate
is fO,asil = 0.26. Carbon is also depleted in interstellar matter.
Following Sofia et al. (2011), it is approximately 60%–70%,
i.e.,
fC,gra = 0.6 − 0.7(7)

4. RESULTS FOR MW DUST

if all condensed carbon is in graphite. This is higher than theirearlier estimate, 30%–40%.11 The depletion given by Cardelli
et al. (1996) is also consistent with Equation (7) if the GS98
abundance is adopted.
The model attains good fits from far-UV to NIR within 1σ ofthe data with

4.1. Single Grain Species
We first study the extinction curve with a single grain species.
We consider the allowed region in the q and amax plane, withwhich the dust model satisfies the 1σ ranges of the FM07
extinction curve. We find that there is no overlap among thethree regions derived from the UV group (0.125 μm, 0.16 μm,
and 0.2175 μm), the UBV group, and the NIR group (J, H, andK) for any species of grains we considered. The regions required
for the UV and UBV groups are always disjoint. It often happensthat no consistent parameters exist for J, H, and K, the NIR
passbands alone. We cannot make a dust model which explains
the extinction curve over the wide range of wavelength with onlya single grain species. For instance, astronomical silicate gives
too steep a rise in the far-UV if amax is chosen to account forthe optical extinction curve, and the predicted NIR extinction is
too small by a factor of 3–4. We expect that these problems areoffset by introducing carbonaceous grains, which give a milder

q = 3.5 ± 0.2 and amax = 0.24+0.10−0.05 .

(8)

The range of fC,gra /fFe,asil is 0.25–2.23, allowing for fFe,asil < 1.
Figure 3 presents examples of the resulting extinction curvefrom our model at the central values of Equation (8) with
fC,gra /fFe,asil = 0.7. We give in Table 3 numerical values ofAλ /AV for selected wavelengths (and compare them with themodels of WD01 and CCM). We underline the discrepancy
between our model and WD01 (and also CCM) increasing fromthe J band longward. It becomes 30% for the K band. We draw
several curves in addition that are somewhat away from the bestfit value (say, about 1.5σ of the observation) to indicate how the

10 We remark that GS98 agree with the earlier table of Anders & Grevesse(1989) up to oxygen, for which the GS98 value is lower by 0.1 dex.

11 Sofia et al. (2004) used only weak absorption lines of carbon, while their

updated analysis (2011) includes strong absorption lines.

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Figure 4. Allowed parameter regions for graphite–silicate models in q and amax
plane for the 1σ range in the UV group (blue), the UBV group (green), and the
NIR (JHK) group (red). We take fC,gra /fFe,asil = 0.7 in (a) and 0.2 in (b), thelatter of which is a model out of 1σ . Thin dotted curves show the parameters
for the 1σ range in the NIR (JHK) group when the NIR wavelength dependent
power is γ = 1.6 as with CCM or WD01.
(A color version of this figure is available in the online journal.)

Figure 3. (a) Typical extinction curves from the graphite–silicate models, and(b) the expanded figure of (a) for NIR wavelengths. The error bars stand for theobserved 1σ ranges. The thick solid curve is with q = 3.5, amax = 0.24 μm,
and fC,gra /fFe,asil = 0.7, taken as our fiducial. The other curves denoted by thincurves are examples where one of the three parameters is shifted to demonstrate
the response to the parameter. The legend shows the parameters when changed
from the fiducial choice.
(A color version of this figure is available in the online journal.)

to hydrogen. The allowed region extends in a belt, running frombottom left to top right, representing fC,gra /fFe,asil = const. We
also indicate the region allowed from E(B − V )/NH , which islocated near the center of the belt (lightly shaded). This leads tothe elemental abundance carried by dust grains,

curve shifts with the variation of q, amax , and fC,gra /fFe,asil .
The mass density ratio of graphite to astronomical silicate isgiven by ρgra /ρasil = 0.73fC,gra /fFe,asil . We note that our curve
is in close match with the WD01 curve (RV = 3.1) from the
U to IC passbands.12 A significant departure starts from theJ band longward. The role of graphite is, in addition to producing
the 2175 Å bump, to increase the extinction in NIR, which is too
small with silicate alone when normalized in the optical region.
Graphite makes the rise in the far-UV more moderate.
In Figure 4, we show in q − amax plane the region where
the model gives the 1σ ranges of the extinction for the choicesof fC,gra /fFe,asil = 0.7 and 0.2. The overlapping region is seen
for UV, UBV, and NIR for fC,gra /fFe,asil = 0.7, validating thegraphite–silicate model. Such allowed regions disappear for
fC,gra /fFe,asil  0.25.
Figure 5 shows the abundances of carbon and silicon in
grains necessary to account for the extinction curve relativelog (C/H) + 12 = 8.4 ± 0.3 and log (Si/H) + 12 = 7.6 ± 0.4.
(9)
We also indicate the abundance of GS98 for Si (7.55; horizontal
dashed line) and C (8.52; vertical horizontal line) with theneighboring shade showing various abundance estimates.
The abundances resulted from the interstellar extinction are
consistent with other estimates. If we take the GS98 value forthe total abundance of C and Si including the gas phase, thefigure, showing log (C/H) + 12  8.27, means
fC,gra  0.56,(10)

for fSi,grain = 1 assumed. This carbon fraction in graphite isconsistent with the depletion estimated for interstellar matter
quoted in Equation (7).13

12 Our curve lies at the 1σ edges if we take the R and I data constructed
C
Cfrom the FM07 curves. The constraints on the q − Amax plane are unchanged if
we take the 1.5 σ of RC and IC . Note that we do not use the RC and IC

13 The carbon abundance required from the extinction is still consistent withthe lower abundance of Asplund et al. (2009), but then the depletion must be as
high as >80%.

extinction data of FM07, which are not observationally constrained.

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Figure 5. Abundance of C and Si in dust grains relative to hydrogen for the MWto satisfy the 1σ range of the extinction curve. The range within the oblique
belt is allowed from the extinction curve, and the further restricted range in the
middle is from E(B − V )/NH . The two horizontal and vertical dashed lines
show the total abundance of Si and C from GS98, with the shaded regions
corresponding to the range of various estimates.

Figure 6. Allowed regions of q and amax with which the model satisfies the 1σ
ranges of the observed extinction for the MW, marginalized over the graphiteto-silicate ratios fC,gra /fFe,asil  0.25. The three contours inlaid in the allowed
region indicate those with fC,gra /fFe,asil = 1.0, 0.7, and 0.4. The constraints
from the reddening E(B − V )/NH are shown for fC,gra = 0.4, 0.7, and 1.0(with fFe,asil = 1.0), drawn by thin (brown, green, and blue) contours.

A similar consideration is made for the 1σ ranges from theCCM formula, where the NIR extinction takes a smaller powerindex of the wavelength dependence. The range of q and amax
required for the NIR extinction curve is shown in Figure 4 withdotted curves (those for UV and UBV remain unchanged). We
see that the region allowed simultaneously for the three groupsof color bands does not exist. The region for NIR is always
disjoint from those for UBV and UV. No graphite–silicate model
is consistent with the observed extinction, if the NIR extinction
power is ≈λ−1.6 , for grains with the power-law size distribution.
The steeper wavelength dependence for the NIR power thus
looks more easily accommodated from the model point of view.
We remark that WD01 tweaked significantly the grain-size distribution from the power law, separately for graphite and silicate,adding different components so that the graphite–silicate model
becomes consistent with the CCM-like extinction. This is also
true for the size distributions obtained by other authors (listed
earlier) when the CCM-like extinction is reproduced with thegraphite–silicate model. We note that the solution of WD01
takes the Si, Mg, and Fe cosmic abundance larger by 30%–50%
than the GS98 cosmic value.
We note that the allowed regions in Figure 4 above standfor the ranges consistent with the variation of dust properties
along lines of sight in the MW. They are well converged to thenarrow regions of q and amax indicated in Equation (8), despite
an apparently significant variation of extinction curves. The sizeof grains is similar; only a small variation of the size parameter
could cause the difference in the extinction curve.
We summarize in Figure 6 the region allowed for amax and q,
marginalizing over the ratio of condensed fractions of graphite
to silicate for fC,gra /fFe,asil  0.25. The curves are inlaid for
fC,gra /fFe,asil = 0.4, 0.7, and 1. The 1σ allowed range is realizedwith 3.2  q  3.7 and 0.19 μm  amax  0.34 μm. With
increasing fC,gra , amax becomes smaller. The other set of curves
is the constraint from E(B − V )/NH for fC,gra = 0.4, 0.7, and 1with fFe,asil = 1. Overlaps are seen between the two curves for
fC,gra /fFe,asil = 0.7 and 1, but not for 0.4.
For our model that satisfies the 1σ constraints of both
extinction curve and the E(B − V ) size we have the massdensity of dust relative to that of the hydrogen gas, ρdust /ρH =1/(122+48−75 ). The allowed range would be tightly constrained if

the abundance is fixed to GS98 and fFe,asil = 1 is assumedρdust /ρH = 1 122+9
(11)
−16 .
The mass extinction constant Kext,λ = Aλ /Σdust for the V bandis
Kext,V = (3.7 ± 0.5) × 104 mag cm2 g−1 ,(12)which is compared to 2.8 × 104 mag cm2 g−1 of WD01. (Ifthe abundance constraints are removed, Equation (12) becomes(3.6 ± 1.0) × 104 mag cm2 g−1 .) The ratio of the mass densityof graphite to silicate is ρgra /ρasil = 0.51+0.22−0.10 , or in terms ofthe number of C and Fe atoms in grains, NC /NFe = 7.3+3.2−1.4 ,
corresponding to Equation (11).
4.3. Inclusion of Other Carbonaceous Grains
Various populations of carbonaceous grains may constitutecosmic dust in addition to graphite. We explore the possible significance of other carbonaceous species, such as glassy carbon
or amorphous carbon. We find, however, that the two-component
dust model, composed of glassy or amorphous carbon and astronomical silicate, does not give the extinction curve that lies in the1σ ranges of observation for any fC,glC /fFe,asil and fC,amC /fFe,asil
ratios. These carbonaceous grains do not give the proper UVbump at 2175 Å. Glassy carbon shows a broad hump at around2000 Å, and amorphous carbon only has a broad maximum ataround 2000 Å. We consider in the following the model in whichgraphite is partly replaced with glassy or amorphous carbon.
Figure 7 shows a few example extinction curves for threecomponent models, compared with our fiducial graphite–silicate
model, with q = 3.5 and amax = 0.24 μm, fixed at the same
parameters. While some of the resulting curves displayed do
not satisfy the 1σ range, as we replace graphite with thesecomponents excessively and/or we do not tweak the otherparameters, we maintain this set of parameters to see how
the extinction curve is modified with the inclusion of othercarbonaceous material.
Varying the (q, amax ) parameters, we examine to what extent
graphite can be replaced with glassy or amorphous carbon in thegraphite–silicate model to maintain the extinction curve within
the 1σ range. Figure 8 presents the maximum fraction of C
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Figure 9. Allowed regions of q and amax with which the model satisfies the1σ extinction ranges for the MW when glassy carbon, amorphous carbon, or
silicon carbide is included in the graphite–silicate model. The parameters are
marginalized over fC,grain /fFe,asil . The regions that satisfy the observed sizeof reddening E(B − V ) are also shown (dashed curves) for fC,grain = 0.7 andfFe,asil = 1.0. The blue, red, and green lines are, respectively, for the cases where
glassy carbon, amorphous carbon, and SiC are added to the graphite–silicate
model, which is represented by the black curves for comparison. For theE(B −V )/NH constraint, it is taken that fC,glC /fC,grain = 0.3, fC,amC /fC,grain =0.3, and fC,SiC /fC,grain = 0.1.

respectively. In other words, more than 60%–70% or 40%–50%
of C atoms must be in graphite. With this inclusion, theparameters (q, amax ) are shifted only a little; see Figure 9. Ourrepresentative parameters q = 3.5 and amax = 0.24 μm still
remain to be in the allowed solution.
SiC is another candidate material that can be in dust grains.
The upper limit of carbon contained in SiC is approximately
15% to reproduce the extinction curve (see Figure 8). Thisimplies that SiC could be a major component of Si-bearing
grains. The abundance of SiC, however, has been tightly limited
to <4% of silicon from the lack of 11.3 μm feature in theextinction curve (Whittet et al. 1990; Chiar & Tielens 2006).
The SiC component can be neglected in the discussion of theextinction curve.
In conclusion, the inclusion of amorphous or glassy carbon
has little effect on the agreement of the graphite–silicate modelwith observations. The inclusion, however, does not make theoverlap of the two constraints easier. The dust to gas ratio
is unchanged from the graphite–silicate model. With SiC theoverlap of the two constraints becomes marginal, and the dust
to gas ratio becomes ρdust /ρH = 1/(122+60−16 ).

Figure 7. (a) Extinction curves from the graphite–glassy (or amorphous)
carbon–silicate models with q = 3.5 and amax = 0.24 μm, and (b) the expanded
figure of (a) for NIR, as compared with our fiducial graphite–silicate model(thick black curve). fC,grain /fFe,asil = 0.7 is taken for all cases. The 1σ ranges
from the observed extinction curves in FM07 are indicated with error bars.

4.4. The 2175 Å FeatureWith small graphite, the feature at 2175 Å is generated. It is
known that this feature is observationally fit well with the Drude
formula, including a smooth background,

1
c2
Aλ
=
c0 + c1 λ−1 +2 + 1. (13)
2
2
AV
RV
γ + λ λ−2 − λ−2Figure 8. Allowed fractions of C atoms in the glassy phase (fC,glC /fC,grain ,blue), amorphous phase (fC,amC /fC,grain , red), and silicon carbide(fC,SiC /fC,grain , green). The abscissa is fC,grain /fFe,asil , where fC,grain =fC,glC + fC,gra or fC,grain = fC,amC + fC,gra or fC,grain = fC,SiC + fC,gra . The
ordinate is the maximally allowed fraction of the individual component.

10

In Figure 10, we give the central wavelength x0 = λ−10 and thewidth of the profile γ1 for 328 extinction curves of FM07, which
are summarized as

atoms in glassy (fC,glC ) or amorphous carbon (fC,amC ) relative
to the entire condensed component of carbon fC,grain against
fC,grain /fFe,asil , where fC,grain = fC,glC + fC,gra or fC,grain =fC,amC + fC,gra . We see that up to ≈30%–40% or ≈50%–60%
of carbon in grains can be in the glassy or amorphous phase,

λ0 = 2178 ± 10 Å,

γ1 = 0.95 ± 0.1 μm−1 .

(14)

We fit the feature from our graphite–silicate model, which reads
λ0 = 2186 ± 12 Å,
8γ1 = 1.21 ± 0.08 μm−1 ,(15)The Astrophysical Journal, 770:27 (13pp), 2013 June 10

Nozawa & Fukugita

Figure 10. Central wavelength and the width parameter that describe the 2175 Å
feature of the MW extinction. The observed data from FM07 (squares) arecompared to our model with the parameters that gives a good fit (within 1σof the full extinction curve) to the global extinction curve. The horizontal andvertical lines indicate 1σ of the data.
(A color version of this figure is available in the online journal.)

indicating that the model width is broader by 30% (see also
Draine & Malhotra 1993), while the central wavelength agrees
with the observation. We also show models with amorphousor glassy carbon included with its fraction and size parametersin the range allowed from the entire extinction curve. Their
inclusions make the width slightly (≈10%) smaller, but not
sufficient to give the observed width. This seems to be a problem
intrinsic to the optical data we adopted for graphite. Moredetailed treatments may be needed for small graphite or PAH.
4.5. Inclusion of Fe, Fe3 O4 , FeS, and Al2 O3
Figure 11. (a) Extinction curves for the graphite–silicate model where ironbearing silicate is replaced with metallic Fe, Fe3 O4 , or FeS, while thecorresponding silicate portions are replaced with forsterite (Mg2 SiO4 ). We takethe grain-size parameters, q = 3.5 and amax = 0.24μm. (b) The expanded
figure for the NIR. We take fC,grain /fFe,grain = 0.7.
(A color version of this figure is available in the online journal.)

We consider how much iron can be incorporated ingeneric iron-bearing grains rather than in astronomical silicate.
Figure 11 shows the extinction curve for the model with our typical parameters q = 3.5 and amax = 0.24 μm. Figure 12 exhibits
the upper limits of the fraction of Fe in the metallic phase (fFe,Fe ,solid line) and in magnetite, Fe3 O4 (fFe,Fe3 O4 , dashed line), forthe model that satisfies the 1σ ranges of the extinction curve. The
rest of the Fe atoms in interstellar space are assumed to be condensed in astronomical silicate, i.e., fFe,grain = fFe,Fe + fFe,asil =1 or fFe,grain = fFe,Fe3 O4 + fFe,asil = 1, and Mg not includedin astronomical silicate is assumed to be in forsterite, so thatj =grain fMg,j = 1. Here, the abscissa is chosen to be the C/Fe
ratio of the condensation.
There are parameters (fC,gra /fFe,grain = 0.15–1.43) for which
Fe may stay entirely in the metallic phase rather than in
astronomical silicate, yet the model gives the correct extinction
curve. A similar parameter range, however, is very small with
the Fe3 O4 –graphite–forsterite model. Practically, the maximum
allowed abundance of Fe3 O4 is about 80%. The NIR extinctionof Fe3 O4 is large by more than a factor of two compared withthe observed extinction, whereas Fe grains give the extinctionin NIR only by ≈30% larger, and, therefore, the abundanceof magnetite is limited from the extinction curve in NIR morestrongly. The 1σ allowed (q, amax ) parameters with these models
are shown in Figure 13.
The combination of metallic Fe and forsterite works virtuallyin the same way as astronomical silicate, which means that the
Mg/Fe ratio in astronomical silicate is arbitrary. The situation
is somewhat different if iron is in magnetite. The inclusion of
Fe3 O4 disturbs the allowed range in the (q, amax ) plane by an

Figure 12. Maximum fractions of Fe atoms allowed in the metallic(fFe,Fe /fFe,grain , red solid) or the magnetite phase, Fe3 O4 (fFe,Fe3 O4 /fFe,grain ,blue dashed), with which the model satisfies the 1σ extinction range for the
MW. The maximum fraction is also drawn for FeS (fFe,FeS /fFe,grain , green
dotted). The abscissa is fC,gra /fFe,grain , where fFe,grain = fFe,Fe + fFe,asil ,fFe,grain = fFe,Fe3 O4 + fFe,asil , or fFe,grain = fFe,FeS + fFe,asil , and the ordinate is the fraction of each species added to the model.
(A color version of this figure is available in the online journal.)

9The Astrophysical Journal, 770:27 (13pp), 2013 June 10

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Figure 13. Allowed regions of q and amax with which the model satisfies the1σ extinction ranges for the MW, marginalized over fC,gra /fFe,grain . For theE(B − V )/NH constraint, we take fC,gra = 0.7 and fFe,grain = 1.0. The blue,
red, and green curves are, respectively, for the inclusion of metallic Fe, magnetite(fFe3 O4 /fFe,grain = 0.8), and FeS (fFeS /fFe,grain = 0.68). The black curve is
our fiducial graphite–silicate model.
(A color version of this figure is available in the online journal.)

appreciable amount, making amax smaller by 20%. Nevertheless,we see that the overlap of the two constraints is maintained.
The cosmic abundance of sulfur is about 0.7 times that of
Fe. We consider the case where all S is in troilite, FeS. The
rest of Fe is in astronomical silicate and the rest of Mg is in
forsterite. Troilite also gives large NIR extinction almost as
much as magnetite, but the maximal amount of FeS is allowed
because of the smaller cosmic abundance of sulfur. This casesatisfies the 1σ constraint with amax somewhat smaller than thegraphite–silicate model, as seen in Figure 13.
In conclusion, iron can be in a variety of grain species, including olivine, metallic phase, Fe3 O4 , or FeS without disturbing theextinction curve. The only condition is that Fe is not predominantly in magnetite, which produces too large an NIR extinction.
The grain-size parameters are nearly the same for all cases, up
to the result that the maximum size cutoff becomes 20%–30%
smaller if Fe3 O4 or FeS is the major component. All iron atoms
need not necessarily be locked in astronomical silicate, and theratio of Fe:Mg in astronomical silicate is arbitrary. Allowing forthe inclusion of a variety of iron material, the mass density of
dust differs little from the graphite–silicate model.
We can ignore the contribution from corundum (Al2 O3 ). In
addition to the small abundance of Al (1/13 of Mg), the Q factor
for corundum is small (Toon et al. 1976). Even if all Al atoms
are locked in corundum, they contribute little to modifying theextinction curve.
Table 4 summarizes our result for most grain species weconsidered in the present work. Limiting ourselves to at mostfour grain species, we consider what combination of species
would give the extinction curve consistent with the observation
at 1σ . Some of them were already discussed in the text above. We
take the species as valid when its fraction is more than 10%. We
here note one particular case that can reproduce the extinction
law of CCM: it is the combination of graphite, astronomicalsilicate, and magnetite. This is due to the particularly large Q
factor of magnetite in the NIR, which is excluded if we take
FM07.

Figure 14. (a) Extinction curves for the SMC from the graphite–silicate dust
models with q = 3.5 and amax = 0.24 μm, and (b) the expanded figureof (a) for the NIR. The solid, dashed, and dotted curves are, respectively,
for fC,gra /fFe,asil = 0, 0.1, and 0.4 with amin = 0.005 μm fixed as our
fiducial choice. The dot-dashed line is with the minimum cutoff increased toamin = 0.02 μm, while fC,gra /fFe,asil = 0.4. The error bars span the maximum
and minimum (taken as “1σ ”) of the observed SMC extinction curves of G03.
(A color version of this figure is available in the online journal.)

Table 4
Allowed Grain Species
Dust Grains(1) Graphite–astronomical silicate(2) Glassy carbon–astronomical silicate(3) Amorphous carbon–astronomical silicate(4) Graphite–glassy carbon–astronomical silicate(5) Graphite–amorphous carbon–astronomical silicate(6) Graphite–SiC–astronomical silicate(7) Graphite–Fe(8) Graphite–Fe3 O4(9) Graphite–Fe–astronomical silicate(10) Graphite–Fe3 O4 –astronomical silicate(11) Graphite–Fe–Mg2 SiO4(12) Graphite–Fe3 O4 –Mg2 SiO4(13) Graphite–Fe–astronomical silicate–Mg2 SiO4(14) Graphite–Fe3 O4 –astronomical silicate–Mg2 SiO4(15) Graphite–FeS–astronomical silicate–Mg2 SiO4

5. DUST IN THE SMC
Figure 14 shows examples of the model extinction curve for
the SMC with q = 3.5 and amax = 0.24 μm for fC,gra /fFe,asil =FM07 1σ

CCM 1σ

Yes
No
No
Yes
Yes
Yesa
Yes
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes

No
No
No
No
No
No
No
No
No
Yes
No
Yes
No
Yes
No

Notes. We count the species when its fraction is more than 10%.
a This model is ruled out if we consider the lack of 11.3 μm feature.

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0, 0.1, and 0.4. (The curve for fC,gra /fFe,asil = 0.4 appears
to be out of the 1σ range, but it satisfies 1σ if amax 
0.19 μm.) The model for fC,gra /fFe,asil  0.4 reproduces the
SMC extinction curve within 1σ when q and amax are adjusted.
With fC,gra /fFe,asil = 0.4 one sees in this figure the symptomthat a small 2175 Å bump starts appearing and the rise toward the
far-UV side becomes insufficient. We draw another curve (dotdashed curve) with the minimum grain-size cutoff increased
to amin = 0.02 μm, removing much of the small graphite
grains. This curve lies similar to the one suggested by Calzetti
et al. (1994) for star-forming galaxies. The bump at 2175 Å isreduced and becomes insignificant, but the removal of small
size grains makes the far-UV rise insufficient. Therefore, sucha case is excluded. Within the graphite–silicate model, the way
to make the predicted extinction consistent with the observation
is not to change the grain-size parameters but to decrease theabundance of graphite grains. The steeper rise toward the far-UV
is accounted for by the smaller silicate grains.
The abundance of Si and C contained in dust is shown in
Figure 15, where the abundance from Russell & Dopita (1992)
is also indicated with dashed lines. We have log(Si/H) + 12 =6.7–7.8 and log(C/H) + 12  7.9 from the extinction in the
SMC. When fSi,grain = 1, log(C/H) + 12  7.6 in agreementwith Equation (16) below. It is noted that the carbon abundancein graphite in the SMC is at least a factor of 1.5 times smaller
than the abundance of Russell & Dopita (if Si is all condensed
into dust; following the horizontal dashed line); the latter gives
7.73 for C and 7.03 for Si. The upper and lower curves
correspond to the 1σ error of NH /E(B − V ). For the SMC
ρgra /ρasil = 0.54fC,gra /fFe,asil  0.22.
Figure 16 is a summary of the region of q and amax , where
the graphite–silicate model gives the extinction curve within the1σ range marginalized over fC,gra /fFe,asil . The allowed grainsize distribution again lies in 3.3  q  3.8 and 0.19 μm
 amax  0.35 μm. It is interesting to observe that this range
for the SMC agrees with that for the MW (recapitulated in the
figure with the dotted curve) in spite of the significantly different
behavior of the two extinction curves, as was noted earlier in
Pei (1992).
The other curves in Figure 16 show the regions allowed by
NH /E(B − V ) for the 1σ error of Equation (2). The overlap of
the two constraints is seen for
fC,gra < 0.41.

Figure 15. Abundance of C and Si in dust grains relative to hydrogen for
the SMC to meet the extinction curve. The range left of the oblique line isallowed from the extinction curve, and that within the two curves runningnearly horizontal is from E(B − V )/NH . The two dashed lines (horizontal andvertical) show the total abundance of C and Si from Russell & Dopita (1992).
(A color version of this figure is available in the online journal.)

0.11 ± 0.11, or NC /NFe = 1.6 ± 1.6, in numbers of atoms. Thisratio is 4.5 times smaller than that for the MW. The abundance
itself indicates the ratio of C/Fe to be 1.4 times smaller, so theformation of carbonaceous grains is suppressed by ≈3 times
more in the SMC.
6. CONCLUSIONS AND DISCUSSION
We have confirmed that the graphite–silicate grain model
gives a satisfactory description of the extinction curves within
the simple power-law model of the size distribution. The grainsize distribution is tightly constrained to the index q = 3.5±0.2.
We showed that departures from a power law are not needed.
However, we need to cutoff the power law at some maximum
size, a = 0.2–0.3 μm. Grains may be a variety of carbonaceousand silicate materials. Their size parameters, however, vary
little and the same parameters also apply to both MW and
SMC extinction curves in spite of their apparently different
behavior. The difference between the two extinction curves canbe ascribed to the abundance of graphite relative to silicates, andhence to some lower efficiency (1/2) of graphite condensation,
beyond the lower ratio (1/1.5) of carbon to silicon in the
SMC indicated in abundance estimates currently available. We
also remarked on the somewhat dissatisfying description of the2175 Å feature.
While grains can be a variety of combinations that containsilicate and carbonaceous material, the presence of a significant abundance of graphite is important. It is interesting tonote that the resulting q = 3.5 is the power expected fromcollisional equilibrium for small grains. We have derived theelemental abundance of Si and C contained in grains from theextinction. For the MW it is consistent with solar. If we takethe widely adopted abundance of GS98, we infer the fractionof carbonaceous grains against total carbon, fC,gra = 0.6–0.7,
which agrees with the observational depression factor for carbon. This may be compared with the corresponding value for the
SMC, fC,gra  0.4, taking the elemental abundance of Russell& Dopita (1992). The required silicate is also consistent.
The upper cutoff is compelling for the grain-size distribution
to give the correct shape of the extinction curve. Extending it toa larger size would disrupt the agreement with observations(see Draine 2009): for example, the RV parameter becomes(16)

This allowed maximum fraction of graphite is smaller than theminimum fC,gra /fFe,asil favored for the MW (fC,gra  0.56),
suggesting that condensation of carbon into grains should be lessefficient in the SMC, as might be satisfied if, e.g., fC,gra ∝ NC .
More detailed arguments, however, depend on the accuracy of
the abundance estimates, in particular, for the SMC.
For the model that satisfies 1σ constraints of both extinction
curve and E(B − V ) size for the SMC, we find ρdust /ρH =1/760+150−660 , where the large error range comes from that ofE(B − V )/NH , and corresponds roughly to the upper and lower
curves of the allowed region in Figure 15. If the abundanceis constrained with Russell & Dopita’s value and Fe is allcondensed,
ρdust /ρH = 1/760+70
(17)
−90 .
We have
Kext = (2.2 ± 0.3) × 104 mag cm2 g−1 .

(18)

−1

(It is (2.1 ± 0.4) × 10 mag cm g if the abundance constraints
are removed.) The ratio of graphite to silicate is ρgra /ρasil =
4

2

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The Astrophysical Journal, 770:27 (13pp), 2013 June 10

Nozawa & Fukugita

the NIR wavelength dependence, such as γ = 1.8–2.1, however,the simple power-law model works for the observed extinction.
The NIR power γ = 1.6 can be consistent with the power-law
grain size only when the iron component of grains is largely inmagnetite, which has a large NIR scattering efficiency.
We have seen that the presence of graphite grains is uniquely
important, at least in the MW, although the condensation of C
atoms may not necessarily be all into graphite. Roughly half
the amount of carbon may be condensed into a glassy or an
amorphous phase, whereas some significant fraction of C atoms
must be in graphite. Iron may also remain in the metallic phase.
There is no need for all Fe atoms to be locked in astronomicalsilicate. The effective ratio of Mg to Fe in olivine seems arbitrary.
Fe3 O4 , however, cannot be predominant. Fe may also be introilite, as much as the sulfur abundance allows.
The size distribution is well converged to a narrow range,
regardless of whether other grain compositions are included.
The variation of extinction along lines of sight may be accounted
for by a small variation of the grain-size distribution withΔq ≈ 0.2 and/or Δamax /amax ≈ 0.3, and with changing thegraphite to silicate ratio as for the difference between the MW
and the SMC. It is noteworthy that the a −3.5 power as expectedin collisional equilibrium seems to be generic to dust grains.

Figure 16. Allowed regions of q and amax with which the model satisfies
the “1σ ” extinction ranges for the SMC. The graphite-to-silicate ratio ismarginalized over 0  fC,gra /fFe,asil  0.41. The inlaid curves indicate theregions for fC,gra = 0, 0.2, and 0.4. The regions for 0.4 is a point indicatedwith a small circle (the actual region is enlarged) near the bottom of the allowed
region. The constraints from the reddening E(B − V )/NH are shown withbrown, green, and blue contours for fC,gra = 0, 0.2, and 0.4 (fFe,asil = 1.0).
The thin dotted curve recapitulates the allowed region for MW dust in Figure 6
for comparison.

We thank Bruce Draine for many discussions and carefully
reading the manuscript, which have improved our work significantly. We also thank Kevin Bundy for many suggestions on the
manuscript. The work is supported in part by the Grants-in-Aid
for Scientific Research of the Japan Society for the Promotionof Science (22684004, 23224004, and 23540288). M.F. is supported by the Monell foundation and the W. M. Keck foundation
at Princeton.

intolerably large.14 While we are not able to find the reason
for the cutoff, we may see another argument that forces us toimpose the presence of a cutoff in the power law. The abundanceof grains is constrained at a submicrometer region by the amountof the observed optical extinction. The power law with q = 3.5
means that the integrated mass of grains is slowly increasing
with the upper cutoff mass mc as m0.17
c . If the cutoff were larger,
we would have too large a mass in grains to be accounted for asa product of stellar evolution. The dust mass density relative tohydrogen ρdust /ρH ≈ 0.008 means Ωdust ≈ 4.4 × 10−6 taking
the global hydrogen abundance for H i and H2 observations,ΩH = 5.4 × 10−4 . If dust is a product of stellar evolution overcosmic time, stars of the amount Ωstar  0.003 would produce
dust grains no more than Ωdust ≈ 1×10−5 (Fukugita 2011). This
amount is consistent with the power-law distribution constrained
from the extinction if the power law cutoff is at 0.25 μm. If the
maximum size cutoff were an order of magnitude larger, say,the dust abundance would be larger by ≈50%, more than thestar formation activity can account for.
Small astronomical objects may also obey the same powerlaw. Their normalization, however, should be smaller by a large
factor than that for the dust grains. The integrated mass densityof the core of planets for a > 100 km is estimated to be 1/300
that of dust grains (Fukugita 2011). The power law should
be broken by two orders of magnitude at a few tenths of a
micrometer. The addition of the planet core mass disturbs little
the estimate of the mass density borne by small objects.
We have emphasized that the behavior of the extinction curve
in NIR is important. If the power of the NIR extinction curve
λ−γ is as small as γ = 1.6, as was derived earlier and adopted
by CCM or WD01, one cannot reproduce the extinction curvefrom UV to NIR by the grain model with a simple power lawof the grain size. One needs to substantially adjust the size
distribution, by, e.g., adding extra components, as was done by
WD01, Zubko et al. (2004), and others. With a larger power of

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