Planetary and Space Science 133 (2016) 36–46

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Properties of interstellar dust responsible for extinction laws withunusually low total-to-selective extinction ratios of RV ¼ 1  2
Takaya Nozawa
National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan

art ic l e i nf o

a b s t r a c t

Article history:
Received 16 December 2015
Accepted 18 August 2016
Available online 21 August 2016

It is well known that the extinction properties along lines of sight to Type Ia supernovae (SNe Ia) aredescribed by steep extinction curves with unusually low total-to-selective extinction ratios of
RV = 1.0 − 2.0 . In order to reveal the properties of interstellar dust that causes such peculiar extinction
laws, we perform the ﬁtting calculations to the measured extinction curves by applying a two-component dust model composed of graphite and silicate. As for the size distribution of grains, we consider two
function forms of the power-law and lognormal distributions. We ﬁnd that the steep extinction curves
derived from the one-parameter formula by Cardelli et al. (1989) with RV ¼ 2.0, 1.5, and 1.0 can be reasonably explained even by the simple power-law dust model that has a ﬁxed power index of −3.5 withthe maximum cut-off radii of amax ≃ 0.13 μm , 0.094 mm, and 0.057 μm , respectively. These maximum cutoff radii are smaller than amax ≃ 0.24 μm considered to be valid in the Milky Way, clearly demonstratingthat the interstellar dust responsible for steep extinction curves is highly biased to smaller sizes. We
show that the lognomal size distribution can also lead to good ﬁts to the extinction curves with
RV = 1.0–3.1 by taking the appropriate combinations of the relevant parameters. We discuss that theextinction data at ultraviolet wavelengths are essential for constraining the composition and size distribution of interstellar dust.
& 2016 Elsevier Ltd. All rights reserved.

Keywords:
Dust
Extinction curves
Galaxy evolution
Interstellar medium
Milky Way
Type Ia supernovae

1. Introduction
The reddening of Type Ia supernovae (SNe Ia) by dust grains is
one of the largest uncertainties that limit the current precision of
the cosmological parameters (e.g., Nordin et al., 2008). The reddening laws along lines of sight to SNe Ia are generally measured
through the total-to-selective extinction ratio RV ≡ AV /(AB − AV ),where AV and AB are the extinction in V and B bands, respectively.
From the analyses of numerous samples of SNe Ia, many studies
suggest that, to minimize the residual on the Hubble diagram, RV
toward SNe Ia must be in a range of RV = 1.0–2.0 (e.g., Conley et al.,2007; Sullivan et al., 2010; Lampeitl et al., 2010), which is considerably lower than the range RV = 2.2–5.5 measured in the Milky
Way (MW). The origin of such unusually low RV thus has been an
important issue to be resolved for the applicability of SNe Ia as thecosmic standard candles.
The multi-band observations of individual SNe Ia also seem tocommonly show the values as low as RV ≤ ∼ 2.0 (Howell, 2011, andreference therein), although there is one study suggesting thatweakly reddened SNe Ia have the values similar to the average
RV ¼ 3.1 in the MW (Folatelli et al., 2010). For these nearby SNe Ia,
E-mail address: takaya.nozawa@nao.ac.jp
http://dx.doi.org/10.1016/j.pss.2016.08.006
0032-0633/& 2016 Elsevier Ltd. All rights reserved.

not only RV but also the extinction curves, namely the wavelengthdependence of extinction, can be obtained from optical to nearinfrared photometries and spectra. The derived extinction curves
are much steeper than the MW average extinction curve and arenicely ﬁtted with the one-parameter formula given in Cardelli
et al. (1989) by taking RV ≃ 1.0–2.0. Such non-standard extinction
laws indicate that the properties of interstellar dust in host galaxies of SNe Ia are different from those in the MW or that otherenvironmental effects associated with SNe Ia affect the apparent
shapes of extinction curves.
One of the possible processes that have been suggested as
provoking the steep extinction curves is the multiple scattering by
dust grains surrounding SNe Ia; it has been shown that multiple
scattering of photons in a circumstellar dust shell with a visual
optical depth of τ ≃ 1.0 can substantially steepen the extinction
curve (Wang, 2005; Goobar, 2008; Amanullah and Goobar, 2011,
but see also Nagao et al. 2016). However, if there exists such amoderately optically thick dust shell, we also expect thermal
emission from the circumstellar dust that is heated by the SN radiation (so-called infrared light echo). Johansson et al. (2013) observed three nearby SNe Ia with Herschel but did not detect any
far-infrared emission. Johansson et al. (2014) also reported thenon-detection with Spitzer at 3.6 mm and 4.5 mm for several SNe Ia

T. Nozawa / Planetary and Space Science 133 (2016) 36–46

and placed the mass limit of ≤ ∼ 10−5M⊙ on the amount of thecircumstellar dust. Furthermore, by comparing an infrared lightecho model and near-infrared observations of SNe Ia samples,
Maeda et al. (2015) put the upper limits of τB ≤ ∼ 0.1 on the B-band
optical depths of circumstellar dust shells. These works point out
that there would not be massive dust shells around SNe Ia so thatthe multiple scattering by local dust might not be a valid explanation of the unusual extinction laws.
Recently, Type Ia SN 2014J was discovered in the starburst galaxy M82 at a distance of ≃3.5 Mpc (Dalcanton et al., 2009), whichis the nearest among SNe Ia reported in the last thirty years. This
SN is highly reddened and thus offers the best opportunity to
study the extinction property on its sightline. The extensive observations have revealed that the extinction curve derived for SN
2014J is highly steep with a very low value of RV ≃ 1.5 (e.g., Goobar
et al., 2014; Amanullah et al., 2014; Foley et al., 2014; Marion et al.,2015). From a simultaneous ﬁt to the extinction and polarization
data, Hoang (2016) claim that both interstellar and circumstellar
dust are responsible for the anomalous extinction property toward
SN 2014J. However, the time invariability of the color (Brown et al.,2015) and polarization (Kawabata et al., 2014; Patat et al., 2015)
indicates that its peculiar extinction is mainly of interstellar-dust
origin. This seems to be also supported by non-detection of infrared excess, which otherwise would be caused by the circumstellar dust shell around SN 2014J (Johansson et al., 2014; Telesco
et al., 2015).
These studies above provide an increasing number of evidence
that the unusual extinction curves toward SNe Ia are likely to beoriginated by interstellar dust in their host galaxies. Hence, theextinction curves measured for SNe Ia can be powerful tools forprobing the properties of interstellar dust in external galaxies. In
general, a low RV is interpreted as corresponding to a smaller
average radius of dust than that in the MW. Gao et al. (2015)
searched for a physical model of relevant dust grains via the ﬁtting
to the measured color-excess curve of SN 2014J and found that
their size distribution is biased to small radii relative to that in the
MW. However, it has not been systematically investigated how
small the interstellar dust should be, to produce the steep extinction curves appeared for SNe Ia. As stated above, the extinction
laws toward SNe Ia are well described by the extinction curves
derived from the empirical formula by Cardelli et al. (1989) (which
we, hereafter, refer to as the CCM curves). Therefore, in this paper,
we aim to comprehensively understand what composition andsize distribution of interstellar dust can reproduce the CCM curves
with exceptionally low values of RV ≤ 2.0.
The paper is organized as follows. In Section 2, the procedure of
the ﬁtting calculations and the model of dust used for deriving theextinction curves are described. The results of the calculations arepresented in Section 3 and discussed in Section 4. In Section 5, ourmain conclusions are remarked. Throughout this paper, dust grains
are assumed to be spherical.

37

Table 1
Reference wavelengths adopted in this study.

σi aPhotometric ﬁlters

(μm)

1/λ i
(μm  1)0.125
0.15280.19280.22240.23590.26
0.2704
0.33550.35310.365
0.39210.44
0.4627
0.55
0.614
0.66

8.000
6.545
5.187
4.496
4.239
3.846
3.698
2.981
2.832
2.740
2.550
2.273
2.161
1.818
1.629
1.515

0.32*
0.19*
0.15*
0.2*
0.2
0.12*
0.12
0.065*
0.065
0.022*
0.022
0.02*
0.02
—
0.02
0.02*

0.7467
0.81

1.339
1.235

0.02
0.027

0.8887
1.25
1.65
2.16
3.4

1.125
0.800
0.606
0.463
0.294

0.027*
0.03*
0.034*
0.04*
0.06*

—
FUV (GALEX)
uvw2 (Swift)
F218W (HST), NUV (GALEX)c
F225W (HST)
uvw1 (Swift)
F275W (HST)F336W (HST)u bandU bandF390W (HST)
B bandg bandV bandr bandRc bandi bandIc bandz bandJ bandH bandKs band
L bandWavelength λi

Referencesb(1)(2)(3), (1)
(3)
(2)
(3)
(3)
(4)
(3)
(4)
(4)
(4)
(4)
(5)
(5)
(5)

a
The uncertainties of extinction (A λ /AV )CCM obtained from the CCM formula atithe reference wavelengths λi. The values marked by asterisks are taken from Cardelli et al. (1989), while the others are deduced from those at the adjacentreference wavelengths.
b
References for the wavelengths: (1) Morrissey et al. (2005), (2) Poole et al.
(2008), (3) Dressel (2016), (4) 2.5 m reference in Doi et al. (2010), and (5) 2MASSbands in Skrutskie et al. (2006).
c
For the GALEX NUV band, the effective wavelength is 0.2271 mm.

(Cardelli et al., 1989). Given that the extinction curves are usually
extracted on the basis of photometric observations, it is practicalto consider the extinction at the speciﬁc wavelengths where theobservational data are available. In this study, we take, as the reference wavelengths, the effective wavelengths of widely used
optical to near-infrared photometric ﬁlters and wide-band UV
photometric ﬁlters onboard representative satellites such as Habble Space Telescope (HST), GALEX, and Swift. Table 1 presents thereference wavelengths λi adopted in this study. At these wavelengths, we calculate the extinction values A λi /AV normalized tothat in V band for different RV by means of the CCM formula anduse them as the data of extinction curves to be ﬁtted.
2.2. Model of interstellar dust
For the model of interstellar dust, we adopt a two-component
model consisting of graphite and silicate. As for the size distribution of dust, we consider two simple function forms: one is thepower-law size distribution given as

2. Fitting model of extinction curves

nj (a) = Cjf j (a) = Cja−qj,

2.1. Data on extinction curves

where nj(a)da is the number density of grain species j (j denotes
graphite or silicate) with radii between a and a + da . The normalization factor Cj is related to the speciﬁc mass of dust grains,mj, asThe main aim of this study is to explore how a variety of theCCM curves described by different RV can translate to the properties of interstellar dust. In order to do this, we perform thecalculations of ﬁtting to the CCM curves by applying a simple dust
model. However, we do not try to ﬁt the whole shape of the CCM
curve by ﬁnely spacing wavelengths from 0.125 mm to 3.5 mm
under consideration. Indeed, the continuous CCM curves havebeen derived by interpolating the data of extinction obtained fromultraviolet (UV) spectra and optical to near-infrared photometries⎛ 4πρ
jCj = mj ⎜⎜
⎝ 3
with

∫a

amax, jmin, j⎞−1 m
j
a3f j (a)da⎟⎟ =
,
Xj
⎠

(1)

(2)

ρj being the material density ( ρgra = 2.24 g cm−3 andρsil = 3.3 g cm−3), amax, j the maximum cut-off radius, and amin, j
the minimum cut-off radius of grain species j. It is well38

T. Nozawa / Planetary and Space Science 133 (2016) 36–46

known that, for this grain composition and size distributionfunction, the average extinction curve in the MW is nicely
reproduced by taking qgra = qsil = 3.5, amax,gra = amax,sil ≃ 0.25 μm ,amin,gra = amin,sil ≃ 0.005 μm , which has been referred to as the
MRN dust model (Mathis et al., 1977; Draine and Lee, 1984). Notethat the power-law distribution is originated from the collisional
fragmentation of dust grains (Biermann and Harwit, 1980), whichis considered as one of the main physical processes that modulate
the size distribution of interstellar dust (e.g., Hirashita and Nozawa, 2013; Asano et al., 2013).
On the other hand, the size distributions of dust produced instellar sources would not necessarily follow the power-law distribution; it has been suggested that the size distributions of dust
ejected from core-collapse supernovae (CCSNe) and asymptotic
giant branch (AGB) stars are lognormal-like with peaks around
0.1–1.0 mm (e.g., Nozawa et al., 2007; Yasuda and Kozasa, 2012).
Hence, as the other size distribution function, we adopt the lognormal form ofnj (a) = Cjf j (a) =⎡ (ln a − ln a )2 ⎤0, j ⎥,
exp⎢ −⎢⎣⎥⎦
2π aγj
2γj2
Cj

(3)

where a0, j and γj are the characteristic grain radius and the standard deviation of the lognormal distribution, respectively. The
normalization factor Cj in Eq. (3) is also determined from Eq. (2),for which we ﬁx amax, j = 10 μm and amin, j = 5 × 10−4 μm .
There are some elaborate interstellar dust models that consider
various grain components and more complicated functional forms
for grain size distributions (e.g., Weingartner and Draine, 2001;
Zubko et al., 2004; Jones et al., 2013). These dust models are tailored to produce excellent ﬁts to the average MW extinction curve
and to meet the constraints on elemental abundances in the interstellar medium. However, the information on elemental abundances is poorly available in general, especially for host galaxies of
SNe Ia. Thus, we do not take into account the abundance constraints in the present analysis and address only the reproductionof wavelength dependence of extinction. Furthermore, our goal is
to grasp the systematic behavior of how the properties of dust vary
according to the extinction curves, not to seek a unique set of thegrain composition and size distribution that gives the best ﬁt toeach extinction curve. For this purpose, the simple dust models as
proposed here are favorable, and they may also be useful to illustrate the applicability of the power-law and lognormal size
distributions.
Under the assumption that both graphite and silicate grains are
uniformly distributed in interstellar space, the extinction curve
Aλ /AV is calculated, for the dust model described above, as

Table 2
Dust models with power-law size distributions and their parameter sets.
Dust
model

Npara Free parameters

Model 1

2

amin, jAλ=,a
AV
∑ Kj ∫ max, j πa2f (a)Q Vext
, j (a )dajamin, jjModel 2

where

Kgra =⎛ X ⎞= fgs ⎜⎜ sil ⎟⎟ andCsil⎝ Xgra ⎠Cgra

Ksil = 1

2

q = qgra = qsil ,q, fgsamax,gra = amax,sil = 0.25 μm
Model 33

amax,gra , amax,sil , fgs

qgra = qsil = 3.5

Model 4

3q, amax, , fgs

q = qgra = qsil , amax = amax,gra = amax,sil

Model 55qgra , qsil , amax,gra ,–amax,sil , fgsa
For all of the models here, the minimum cut-off radii are ﬁxed asamin,gra = amin,sil = 0.005 μm .

silicate grains, and qgra , qsil , amax,gra , amax,sil , amin,gra , and amin,sil,which regulate the size distribution of each grain species. As discussed in Nozawa and Fukugita (2013), amin, j cannot be constrained
very much unless the extinction data at wavelengths shorter than≃0.1 mm are used. Therefore, we ﬁx amin, j = 0.005 μm for bothgraphite and silicate. For this power-law distribution, we consider
ﬁve dust models, depending on the combinations of the freeparameters and ﬁxed parameters, as summarized in Table 2. For
example, the dust model in which graphite and silicate have the
same
size
distribution
withandq = qgra = qsil = 3.5
amax = amax,gra = amax,sil is referred to as Model 1. On the other
hand, the dust model in which all of fgs , qgra , qsil , amax,gra , and amax,sil
are treated as free parameters is referred to as Model 5. Hereafter,
we mainly show the results of the ﬁtting calculations for Model 1.
This simple dust model reduces the number of parameters to only
two ( fgs and amax ) and can be regarded as a natural extention of
the MRN dust model.
For the lognormal size distribution, the adjustable parameters
are fgs , a0,gra , a0,sil , γgra , and γsil . For this size distribution, two dust
models are considered, according to the numbers of the freeparameters. The details of the dust models for the lognormal size
distribution are given in Table 3 and are described in Section 3.2.
2.3. Fitting procedure
We carry out the ﬁtting through comparison between the extinction data and the calculated extinction ycal, i = (A λi /AV )cal at eachreference wavelength λi. The goodness of the ﬁtting is evaluated by
χ2, which might be, for instance, given as

χ02 =

(4)

qgra = qsil = 3.5,

amax , fgs

amax = amax,gra = amax,sila∑j Kj ∫ max, j πa2f j (a)Q λext
, j (a )daConstraintsa(

Ndata y−y〈σ 〉2
∑ CCM, i 2 cal, iNdata − Npara i
σi

2

),

(6)

where yCCM, i = (A λi /AV )CCM is the extinction derived from the CCM
formula at λi, Ndata is the number of the data to be ﬁtted (i.e., thenumber of the reference wavelengths under consideration except
for V band), Npara is the number of free parameters, si are weights,(5)with the mass ratio fgs = mgra /msil of graphite to silicate. The extinction coefﬁcients Q λext
, j (a ) are calculated with Mie scattering from
dielectric constants for graphite and astronomical silicate (which
we refer to simply as silicate in what follows) in Draine and Lee(1984) and Draine (2003). In computing Q λext
, j for graphite, the 1/3–
2/3 approximation is employed to take the anisotropy into consideration (Draine and Malhotra, 1993).
For the power-law size distribution, there are seven adjustableparameters: fgs , which prescribes the mixture of graphite andTable 3
Dust models with lognormal size distributions and their parameter sets.
Constraintsa

Dust model

NparaFree parameters

Model 1s3

a0,gra , a0,sil , fgs

γgra = γsil = 0.5

Model 5s

5a0,gra , a0,sil , γgra , γsil , fgs—

a
For the models with lognormal size distributions, the maximum and miniamax,gra = amax,sil = 10 μm andmum cut-off radii are ﬁxed to beamin,gra = amin,sil = 5 × 10−4 μm , respectively.

T. Nozawa / Planetary and Space Science 133 (2016) 36–46
N

and 〈σ 〉2 = [(1/Ndata ) ∑i data (1/σi2)]−1 is the adjustment coefﬁcient foralleviating the large external errors arising from si2. As for si,
which are generally assigned as the uncertainties of observational
data (e.g., Gao et al., 2015), we take the standard deviation of Aλ /AVgiven in Table 2 of Cardelli et al. (1989). For si not given in Cardelli
et al. (1989), we infer their values from those at the closest reference wavelengths. The adopted values of si are provided inTable 1.
It should be noted that a set of the best-ﬁt parameters thatminimizes χ02 in Eq. (6) does not necessarily produce apparently
good ﬁts to the entire range of extinction curves; since si areconsiderably large at UV wavelengths (see Table 1), the data of UV
extinction are unimportantly treated in the ﬁtting calculations,
resulting in poor ﬁts at UV wavelengths. Therefore, as in someprevious studies (e.g., Weingartner and Draine, 2001), we adopt an
identical weight σi = 1, independent of λi, and assess the dispersion by

χ12 =

Ndata

1
∑
Ndata − Npara i(yCCM, i2)− ycal, i .

(7)In the case that the extinction data at all the reference wavelengths covering UV to near-infrared are exploited, Ndata = 22. In
Section 4.2, we will also show the results of the ﬁtting calculationsin the cases that the extinction data at UV wavelengths are nottaken into account.

3. Results of ﬁtting calculations
3.1. Extinction curves from dust models with power-law size
distributions
We ﬁrst consider the average MW extinction curve withRVCCM = 3.1 to check if our ﬁtting method yields the plausible results. As mentioned above, this extinction curve can be wellmatched by the MRN dust model that is represented by the powerlaw size distribution with q¼ 3.5 and amax = 0.25 μm (Mathis et al.,
1977; Draine and Lee, 1984; Nozawa and Fukugita, 2013). Thus, ourﬁtting calculations should lead to the similar values for q and amax .
The results of the ﬁtting calculations are given in Table 4, whichsummarizes a set of parameters that give the best ﬁt for each dust
model with a power-law size distribution. For Model 1, whichassumes q ¼3.5 and amax = amax,gra = amax,sil , the best ﬁt is achieved
with amax = 0.243 μm and fgs = 0.57. On the other hand, if we ﬁx as
amax = 0.25 μm (Model 2), the power index q¼ 3.53 offers the best
ﬁt with fgs = 0.60. Hence, our simplest dust models (Models 1 and
2) present the results consistent with the MRN dust model, demonstrating that our ﬁtting method works well.
The extinction curves derived from these best-ﬁt parameters
are shown in Fig. 1(a). We can see that the extinction curves obtained from Model 1 and Model 2 are fully overlapped and reproduce the whole shape of the CCM curve with RVCCM = 3.1. In
Fig. 1(a), we also depict the extinction curve from Model 5. Since
Model 5 treats the ﬁve quantities ( qgra , amax,gra , qsil , amax,sil , and fgs )as free parameters, the best-ﬁt values result in the least dispersion
among the dust models considered for the power-law size distribution (see Table 4). However, as is observed from Fig. 1(a), thematch between the resulting extinction curve and the extinction
data is not largely superior to that for Model 1. Given that Model 1(and Model 2) gives a satisfactory ﬁt despite its small number offree parameters, such a simplest dust model should be regarded as
being instructive for understanding the properties of dust incomparison with the MRN model.
Next we search for the dust models that can account for very
steep extinction curves with RVCCM ≤ 2.0. Figs. 1(b), (c), and39

Table 4
A set of the best-ﬁt parameters obtained for dust models with power-law size
distributions.
Dust modela

qgra

amax,gra

qsil

(μm)

amax,sil

fgs

χ1

RVmod

(μm)

RVCCM = 3.1
Model 1Model 2Model 3Model 4Model 5

3.50
3.53
3.50
3.58
3.78

0.243
0.250
0.119
0.274
0.471

3.50
3.53
3.50
3.58
3.42

0.243
0.250
0.544
0.274
0.290

0.57
0.60
0.25
0.60
0.44

0.04640.04300.04620.04130.0358

3.38
3.46
3.44
3.56
3.46

RVCCM = 2.0
Model 1Model 2Model 3Model 4Model 5

3.50
4.05
3.50
3.84
4.04

0.134
0.250
0.0386
0.174
0.164

3.50
4.05
3.50
3.84
3.76

0.134
0.250
0.270
0.174
0.230

0.46
0.50
0.16
0.49
0.35

0.09320.08930.05500.05400.0368

2.16
2.66
2.41
2.41
2.34

RVCCM = 1.5
Model 1Model 2Model 3Model 4Model 5

3.50
4.40
3.50
3.67
4.10

0.09440.250
0.08840.102
0.0903

3.50
4.40
3.50
3.67
3.85

0.09440.250
0.0735
0.102
0.200

0.49
0.47
0.64
0.48
0.31

0.07070.231
0.05350.06430.0465

1.75
2.34
1.72
1.89
1.74

RVCCM = 1.0
Model 1Model 2Model 3Model 4Model 5

3.50
5.08
3.50
3.28
3.86

0.05720.250
0.06270.05460.0600

3.50
5.08
3.50
3.28
3.77

0.05720.250
0.08440.0546
0.128

0.60
0.42
0.42
0.63
0.30

0.223
0.663
0.171
0.218
0.158

1.49
2.10
1.39
1.47
1.34

a
The free parameters and constraints adopted in the power-law dust models
are described in Table 2.

(d) show the results of the ﬁtting to the CCM curves withRVCCM = 2.0, 1.5, and 1.0, respectively. For Model 1 with q ¼3.5 andamax,gra = amax,sil, the least dispersions are achieved withamax = 0.134 μm , 0.0944 μm , and 0.0572 μm for RVCCM = 2.0, 1.5, and1.0. For these cases, the mass ratio of graphite to silicate is in arange of fgs = 0.45–0.6. The dispersions are considerably small for
RVCCM = 2.0 and 1.5, demonstrating that the ﬁts are adequate. The ﬁt
to the RVCCM = 1.0 curve is not good very much at optical to nearinfrared wavelengths, but the calculated extinction curve canroughly reproduce the entire wavelength-dependence of
extinction.
In Figs. 1(b)–(d), we also plot the extinction curves calculated
from the best-ﬁt combination of fgas and q for Model 2. When the
maximum cut-off radii are ﬁxed as amax = 0.25 μm , the optimum
values of q are 4.05, 4.40, and 5.08, respectively, for RVCCM = 2.0, 1.5,
and 1.0. This implies that as the extinction curves become steeper,
the steeper size distributions are required. However, as is obvious
from Figs. 1(c) and (d), the best-ﬁt parameters for Model 2 lead toa poor match to the CCM curves with RVCCM = 1.5 and 1.0. Therefore, only enhancement in q, with a ﬁxed amax , is not sufﬁcient for
describing the highly steep extinction curves with RVCCM ≤ 1.5.
Nevertheless, for RVCCM = 2.0, the size distribution from Model
2 with q ¼4.05 and amax = 0.25 μm yields a better ﬁt than that for
Model 1 with q¼ 3.5 and amax = 0.134 μm .
These results indicate that the steeper extinction curves represented by the values as low as RVCCM = 1.0–2.0 can be explainedby the dust model whose power-law size distribution is skewed tosmaller radii than the MRN model. More speciﬁcally, if
amax = 0.25 μm is held, the power index is needed to be increased
from q ≃ 3.5 for RVCCM = 3.1 up to q≃ 4.0 for RVCCM = 2.0. On theother hand, for RVCCM = 1.0–2.0 , a better ﬁt can be generally obtained through reducing the maximum cut-off radius by a factor of
2–5 in comparison to the MRN model, with a ﬁxed index of q ¼3.5
and a mass ratio of fgs = 0.45–0.6. Given the peculiarity of the40

T. Nozawa / Planetary and Space Science 133 (2016) 36–46

Fig. 1. Extinction curves calculated from the best-ﬁt parameters in Model 1 (thick solid), Model 2 (dashed), and Model 5 (thin solid) with power-law size distributions for (a)RVCCM = 3.1 (b) 2.0, (c) 1.5, and (d) 1.0. The ﬁlled circles are the extinction data at the reference wavelengths, derived from the CCM formula for each RVCCM value. For the panels(b)–(d), the extinction curve derived from Model 1 for RVCCM = 3.1 is shown by the dot-dashed line for reference.

extinction curves as never appeared in the MW, the maximum cutoff radii of amax ≃ 0.06–0.13 μm are not too small and thus may not
be unrealistic. The important consequence of the present analysis
is that the remarkably tilted extinction curves measured for SNe Ia
can be described in the context of the simple dust model with thepower-law size distribution.
3.2. Extinction curves from dust models with lognormal size
distributions
In this section, we examine whether the dust models withlognormal size distributions can lead to the good ﬁt to the CCM
curves with RVCCM = 1.0–3.1. As mentioned in Section 2.2, the lognormal-like size distributions have been suggested as those of dust
supplied from CCSNe and AGB stars. Hirashita et al. (2015) adopted
the standard deviation of γj = 0.49 for the lognormal size distribution of dust from these stellar sources. Following this study,
we ﬁrst consider the case with γj = 0.5, treating a0,gra , a0,sil , and fgs
as parameters (referred to as Model 1s). We note that γj = 0.5
presents a well peaked distribution, so such lognormal distributions would be useful to seek the characteristic grain radius responsible for the measured extinction curves.
The best-ﬁt parameters from the ﬁtting calculations for lognormal grain size distributions are given in Table 5. For a ﬁxed
value of γj = 0.5, the best-ﬁts are obtained with a0, j ≃ 0.03 μm (forRVCCM = 3.1) down to a0, j ≃ 0.01 μm (for RVCCM = 1.0), indicating thatthe characteristic grain radius decreases as the extinction curve
becomes steeper. However, as shown by dashed lines in Fig. 2, theTable 5A set of the best-ﬁt parameters obtained for dust models with lognormal size
distributions.
Dust modela

γgra

a0,graγsil(μm)

RVCCM = 3.1
Model 1s
Model 5s

0.50
1.32

0.0358RVCCM = 2.0
Model 1s
Model 5s

0.50
1.18

0.0238RVCCM = 1.5
Model 1s
Model 5s

0.50
0.86

0.0186RVCCM = 1.0
Model 1s
Model 5s

0.50
0.50

a0,sil0.50
1.34

0.02530.50
1.25

0.02010.50
1.06

0.01852.84 × 10−3
0.01270.01140.50
0.71

0.01725.05 × 10−4

5.60 × 10−4

fgs

χ1

RVcal

0.91
0.55

0.139
0.0675

2.81
3.56

0.78
0.42

0.208
0.0608

1.87
2.48

0.69
0.38

0.220
0.0890

1.61
1.95

0.54
0.39

0.239
0.232

1.49
1.50

(μm)

5.12 × 10−4

5.00 × 10−4

1.48 × 10−3

8.63 × 10−3a
The free parameters and constraints adopted in the lognormal dust models
are described in Table 3.

matches are poor, especially for RVCCM = 3.1, 2.0, and 1.5. Hence, thelognormal distribution with the standard deviation as low as
γj = 0.5 is not likely to be suitable for managing these steep CCM
curves.
On the other hand, if we treat all of the ﬁve quantities ( fgs , a0,gra ,a0,sil , γgra , and γsil ) involved in the lognormal distribution as freeparameters (referred to as Model 5s), the nice ﬁts are obtained

T. Nozawa / Planetary and Space Science 133 (2016) 36–46

41

Fig. 2. Extinction curves calculated from the best-ﬁt parameters in Model 1s (dashed lines) and Model 5s (solid lines) with lognormal size distributions for (a) RVCCM = 3.1(b) 2.0, (c) 1.5, and (d) 1.0. The ﬁlled circles are the extinction data at the reference wavelengths, derived from the CCM formula for each RVCCM value. For the panels (b)–(d),the extinction curve derived from Model 1 with the power-law size distribution for RVCCM = 3.1 is shown by the dot-dashed line for reference.

(solid lines in Fig. 2); in the case of Model 5s, a0, j ≤ ∼ 3 × 10−3 μm
and γj ≃ 1.0 are necessary for producing the good ﬁts to the extinction curves with RVCCM = 3.1, 2.0, and 1.5. Note that the size
distribution obtained from such a small a0, j and a large γj can beviewed as an exponential-like distribution rather than a lognormal
distribution with a sharp peak (see Fig. 3). For RVCCM = 1.0, a peaked
lognormal distribution seems to yield the entire agreement withthe extinction data, although the match is not necessarily sufﬁcient in optical and near-infrared regions.
The inspection of Fig. 3 allows us to point out that the size
distributions offering the best-ﬁts for Model 5s have well similarslopes to the corresponding power-law distributions at the radii
between ≃0.01 mm and ≃0.2 mm. Then, we see if this similarity also
holds with regard to the average radii given as1⎛ amax, j amn(a)da ⎞ m⎜ ∫amin, j⎟m〈aave, j〉 = ⎜⎟a⎜ ∫ max, j n(a)da ⎟⎝ amin, j⎠

the average radius when the properties of dust are discussed in thecontext of extinction curves. Indeed, the radii of grains whichmainly contribute to extinction are different at different wavelengths, so the wavelength-dependence of extinction could not be
described only with a single characteristic grain radius.
We have revealed that the lognormal size distribution can leadto good ﬁts to the CCM curves with RVCCM = 1.0–3.1 by adopting theappropriate a0, j and γj. However, the required values of a0, j is below
0.01 mm, which is much smaller than the typical radii (0.1–1.0 mm)
of dust expected from CCSNe and AGB stars. In fact, dust grainsfrom CCSNe could cause the ﬂat extinction curves because of theirrelatively large radii (Hirashita et al., 2008). We have also noticed
that the average radius could not be a good quantity to characterize the size distribution of dust that reproduces the extinction
curves.
3.3. Allowed ranges of amax , q and fgs for power-law size distribution(m = 1, 2, 3).
(8)

Table 6 presents the average radii from the best-ﬁt parameters formsome of the dust models in this study. We can see that 〈aave,j〉 from
Models 5 and 5s do not coincide at all, despite the fact that theirsize distributions highly resemble ata ≃ 0.01–0.2 μm . Form
RVCCM = 3.1, 2.0, and 1.5, 〈aave,j〉 from Model 5s are always smaller
than those from Model 5. This is because amin, j (and a0, j ) in Model
5s is much smaller than amin, j = 0.005 μm in Model 5. Meanwhile,CCMm= 1.0 due to the lack of grains〈aave,j〉 is higher in Model 5s for RV
smaller than ≃0.01 mm. Hence, it would not be proper to invoke

In Section 3.1, we have shown that the CCM extinction curves
with RVCCM = 1.0–3.1 can be reasonably ﬁtted by the simplest
power-law dust model with q ¼3.5 through taking an appropriate
set of amax and fgs . We found that, for Model 1, the optimum
maximum cut-off radius amax decreases from amax = 0.24 μm for
RV ¼ 3.1 down to amax = 0.057 μm for RV ¼1.0, with a range of
fgs = 0.45–0.6. However, the best-ﬁt value is not a unique solution;
there should be other combinations of amax and fgs that still yieldreasonable ﬁts to the extinction data. Therefore, given that there
are some uncertainties on the data of extinction curves, it should

42

T. Nozawa / Planetary and Space Science 133 (2016) 36–46

Fig. 3. Grain size distributions that lead to the best-ﬁt to the extinction curves with (a) RVCCM = 3.1 (b) 2.0, (c) 1.5, and (d) 1.0. In each panel, the solid lines show the powerlaw size distributions obtained from Model 5, while the dashed lines indicate the results from lognormal size distributions from Model 5s. Graphite and silicate are drawn in
red and blue, respectively. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

Table 6m
Average radii 〈aave,j〉 of graphite and silicate grains in units of μm for Models 1, 2, 4,

and 5 in Table 2 and for Model 5s in Table 3.
Dust model

Graphite

Silicate

m¼ 1

m¼ 2

m¼ 3

m ¼1

m¼ 2

m¼ 3

RVCCM = 3.1
Model 1Model 2Model 4Model 5Model 5s

0.00830.00820.00820.00780.00220.01030.01020.01000.00930.00410.01550.01520.01500.01390.00870.00830.00820.00820.00850.00220.01030.01020.01000.01090.00430.01550.01520.01500.01710.0095RVCCM = 2.0
Model 1Model 2Model 4Model 5Model 5s

0.00830.00740.00770.00740.00190.01000.00850.00900.00840.00310.01380.01110.01190.01100.00560.00830.00740.00770.00780.00200.01000.00850.00900.00930.00340.01380.01110.01190.01290.0066

Model 1Model 2Model 4Model 5Model 5s

0.00820.00710.00790.00740.00420.00980.00780.00930.00820.00600.01280.00940.01200.00990.00870.00820.00710.00790.00770.00300.00980.00780.00930.00900.00490.01280.00940.01200.01210.0084RVCCM = 1.0
Model 1Model 2Model 4Model 5Model 5s

0.00810.00660.00850.00760.01290.00940.00700.01000.00860.01460.01140.00770.01220.01020.01660.00810.00660.00850.00780.01110.00940.00700.01000.00910.01430.114
0.00770.01220.01190.0184RVCCM = 1.5

be inspected what extent of the change in amax and fgs is allowable.
In order to quantify the allowed ranges of amax and fgs , weintroduce the average extinction uncertainty σ̃ , deﬁned asσ˜ = ∑ σi/Ndata , where si are the uncertainties of extinction data at
the reference wavelengths λi. Then, if the dispersion χ1 calculated
for a given combination of amax and fgs is smaller than σ̃ , weconsider it as reproducing the extinction curves within the 1s
errors. Using the values of si in Table 1 leads to the average uncertainty of σ̃ = 0.115. It should be kept in mind that some of si in
Table 1 are inferred from those at the closest reference wavelengths, so the absolute value of σ̃ is quite arbitrary. Nevertheless,the introduction of such a criterion for the dispersion would give ameaningful indication for the allowed range of amax and fgs .
Fig. 4 shows the contours within which the combinations ofamax and fgs satisfy the 1s condition χ1 ≤ σ˜ for RVCCM = 3.1, 2.0, and1.5. For RVCCM = 1.0, there is no combination of amax and fgs within
the 1s error. Thus, we plot the 2s range (that is, combinations ofamax and fgs with χ1 ≤ 2σ˜ ) for RVCCM = 1.0 (we also show the 2s
contours for RVCCM = 2.0 and 1.5). We observe that, for each RVCCM , ahigher fgs is needed for a lower amax to produce ﬁts within the 1s
errors for RVCCM = 1.5–3.1. For RVCCM = 3.1, the 1s ranges of amax andfgs are 0.199 mm ≤amax ≤ 0.294 μm and 0.31 ≤ fgs ≤ 1.07, whereas
the allowed ranges of amax lie just around its optimum values for
RVCCM ≤ 2.0, keeping the trend that amax decreases with decreasing
RVCCM. On the other hand, fgs is relatively uniform with the values of≃0.4–0.6, regardless of RVCCM . This means that the fraction of

T. Nozawa / Planetary and Space Science 133 (2016) 36–46

43

4. Discussions
4.1. Dependence of RV on amax

Fig. 4. Contour plots showing the allowed regions of amax and fgs with which
Model 1 can yield ﬁts to the CCM extinction curves within the 1s errors (solid lines)
for RVCCM = 1.5, 2.0, and 3.1. For RVCCM = 1.0 , 1.5, and 2.0, the contours within the 2s
errors are also drawn by the dashed lines. The best-ﬁt combination of amax and fgs
is marked by the ﬁlled star for each RVCCM . Note that the size of 2s contour for
RVCCM = 1.0 is comparable with that of the star symbol.

graphite mass of the total dust mass is in a narrow range of
fgs /(1 + fgs ) ≃ 0.3–0.4 .
Next we consider the variation of the power-law index q. As
demonstrated in Section 3.1, Model 2, in which amax is ﬁxed as
0.25 mm, cannot reproduce the CCM curves with RVCCM= 1.5 and 1.0;
in these cases, the dispersion χ1 is larger than 2σ̃ = 0.23 even for
their best-ﬁt combinations of q and fgs (see Table 4). This implies
that the change in amax must be essential for the reproduction of
the steep extinction curves with RVCCM ≤ 1.5. Fig. 5 presents the 1σ(2σ ) ranges of combination of amax and q for RVCCM = 3.1, 2.0, and 1.5(RVCCM = 1.0) under the assumption that graphite and silicate havethe same size distribution (i.e., amax = amax,gra = amax,sil andq = qgra = qsil , which corresponds to Model 4). The contours show
that, for a given value of RVCCM , a higher q is needed for a higher
amax .1 However, the power index does not increase with goingfrom RVCCM = 2.0 down to 1.0; q is conﬁned within the range ofq = 3–4 for any value of RVCCM considered in this work, while there
is a clear tendency that amax decreases with decreasing RVCCM. This
indicates that the reduction in amax is preferable to the enhancement in q for describing the highly steep extinction curves. We
also note that, for the cases of RVCCM = 2.0, 1.5, and 1.0, the best-ﬁt
combinations of amax and q in Model 4 lead to the similar average3
radii of 〈aave,j〉 ≃ 0.012 μm . This is an additional indication that the
average radius is not a good measure of size distributions thatfeature the extinction curves.
In summary, the variation of the extinction curves is mostly
described by the change in amax in the context of the power-law
size distribution. For RVCCM = 1.0–3.1 , the range of q = 3–4 reasonably ﬁts to the CCM curves. The mass ratio of graphite to silicate is constrained to be 0.4 ≤ fgs ≤ 0.6, which interestingly well

We have demonstrated that the two-component model ofgraphite and silicate with power-law size distributions is a competent dust model that can explain the systematic behaviors of
extinction curves for a wide variety of RVCCM. However, the RVmod
calculated from our dust models does not match the RVCCM that isreferred to in deriving the data of the extinction curve; for all thepower-law dust models considered in this study, RVmod is higher
than RVCCM as seen from Table 4. This indicates that our dust model
can account for the overall shape of extinction curves but cannot
accurately reproduce the extinction ratio between speciﬁcwavelengths.
Fig. 6 depicts the RVmod calculated from Model 1 as a function of
amax for fgs = 0.4 , 0.6, and 1.0. We can see that the dependence of

RVmod on fgs is weak, although there appears slight difference at
amax = 0.1–0.4 μm . More importantly, as amax increases, RVmod decreases at amax ≤ 0.06 μm and increases at amax ≥ 0.07 μm , having
the minimum value of RVmod ≃ 1.46 at amax ≃ 0.065 μm for any valueof fgs . We have conﬁrmed that this dependence of RVmod on amax is
not sensitive to the minimum cut-off radius amin as long as
amin ≤ 0.01 μm . Thus, if the measured values of RV < 1.4 are real,
they could not be interpreted by the simple dust model consideredhere. The unfeasibility of such low RV, as well as the discrepancy
between RVmod and RVCCM , implies that more sophisticated dust
models which adopt more ﬁne-tuned size distribution function
and involve additional components other than graphite and silicate are necessary for consistently explaining the shape of extinction curve and the value of RV.
We also note that the measured value of RV is unlikely to be agood probe for constraining the properties of interstellar dust. For
example, as seen in Fig. 6, there are two possible values of amax
( amax ≃ 0.026 μm and 0.11 mm) that realize RV ¼2.0. This suggeststhat the properties of dust cannot be necessarily determined uniquely only from RV being the ratio of extinction in V and B bands,
unless RV is an extremely low or high value. In order to extract the
information on the properties of interstellar dust, the extinction

agrees with the range 0.3 ≤ fgs ≤ 0.7 estimated from the analysis of
the average extinction curve and abundance constraints in the
MW (Nozawa and Fukugita, 2013). This suggests that the composition of dust toward lines of sight with RVCCM ≤ 2.0 would not belargely different from that in the MW.
1

Nozawa and Fukugita (2013) showed that, to reproduce the average MW
extinction curve, the ranges of q = 3.5 ± 0.2 and amax = 0.24 ± 0.05 μm are demanded, which are narrower than that given in Fig. 5. They took into account theconstraints of elemental abundances, and adopted the deﬁnition different from thatin this study for the 1σ error of the extinction curve.

Fig. 5. Contour plots showing the allowed regions of amax and q with which Model
4 can yield ﬁts to the CCM extinction curves within the 1s errors for RVCCM = 1.5, 2.0,
and 3.1 (thick solid lines). For RVCCM = 1.0 , the contour within the 2s errors is drawn
by thin solid line. The best-ﬁt combination of amax and q is marked by the ﬁlled starfor each RVCCM . The dotted curves indicate the trajectories of speciﬁc average radii
3
3
〈aave
j〉 calculated from Eq. (8) adopting m ¼3: 〈aavej〉 = 0.01, 0.012, 0.015, and0.02 mm from top to bottom.

44

T. Nozawa / Planetary and Space Science 133 (2016) 36–46

Table 7
A set of the best-ﬁt parameters obtained for some dust models that do not consider
the UV extinction data.
Dust modela

qgra

amax,gra

qsil

(μm)

RVmod onFig. 6. Dependence of
amax calculated from Model 1 with q¼ 3.5 andamax = amax,gra = amax,sil for fgs = 0.4 , 0.6, and 1.0. The minimum cut-off radii are
ﬁxed to amin = 0.005 μm . The two dashed horizontal lines represent RV = 3.1 and
1.0, and the dotted line exhibits RV = 2.0.

data over a broad range of wavelengths are essential, which isdemonstrated in more details in the next subsection.
4.2. Necessity of UV extinction data
So far, we have performed the ﬁtting to the extinction data at
all the reference wavelengths covering UV to near-infrared. However, the data at UV wavelengths cannot be always acquired, andin most cases, we have to derive the extinction law by relying only
on the data at optical to near-infrared wavelengths. In addition,
the extinction curves in external galaxies such as Large and Small
Magellanic Clouds do not show a remarkable 2175 Å bump (Gordon et al., 2003) and cannot be described as the CCM curves. Thus,
it would be interesting to see how the results of ﬁtting can bechanged in the cases that the UV extinction data are not taken into
account, especially in the absence of data around the 2175 Å bump.
From this motivation, we do the ﬁtting calculations for the data
sets that do not include the wavelengths (1/λ = 3.0–5.2 μm−1)
around the 2175 Å bump (but still include the two far-UV wavelengths at 1/λ= 6.5 and 8.0 mm  1). For purposes of illustration, we
here focus on the dust models with power-law size distributions.
As has been done before, we evaluate the goodness of ﬁtting with
Eq. (7), for which Ndata = 17 with the ﬁve data around the UV
bump being excluded.
Table 7 presents the best-ﬁt parameters obtained from theﬁtting calculations that do not take account of the data around the
UV bump for some dust models considered in this study. The
corresponding extinction curves are shown in Fig. 7(a) and (b) for
RVCCM = 2.0 and 1.5, respectively. We can see that, when we assume
q ¼3.5 and amax,gra = amax,sil (referred to as Model 1nb), the best-ﬁt
combinations of amax and fgs , even if the UV-bump data are notconsidered, are chosen so that the resulting extinction curves havea prominent 2175 Å bump like those from Model 1, which do notneglect the UV-bump data. This seems that the data near 2175 Å
may be insigniﬁcant in extracting the properties of dust from themeasured extinction curves. However, this is not true; when weparameterize all of the ﬁve quantities ( qgra , amax,gra , qsil , amax,sil , andfgs ) relevant to the power-law size distribution (referred to as
Model 5nb), the best-ﬁt models yield a much weaker 2175 Å bumpfor both RVCCM = 2.0 and 1.5. In particular, the result without theUV-bump data for RVCCM = 2.0 prefers a highly silicate-dominateddust composition ( fgs = 0.18, compared to the nominal range of

fgs = 0.4–0.6), indicating that the data around the 2175 Å bump areamax,sil

fgs

χ1

RVcal

(μm)

RVCCM = 3.1
Model 1nb
Model 1nu

3.50
3.50

0.259
0.266

3.50
3.50

0.259
0.266

0.50
0.41

0.02240.0234

3.41
3.42

RVCCM = 2.0
Model 1nb
Model 5nb
Model 1nu

3.50
1.50
3.50

0.128
0.0729
0.201

3.50
3.94
3.50

0.128
0.441
0.201

0.50
0.18
0.16

0.06050.02860.0354

2.12
2.11
2.30

RVCCM = 1.5
Model 1nb
Model 5nb
Model 1nu

3.50
2.00
3.50

0.08960.07410.0830

3.50
2.31
3.50

0.08960.03580.0830

0.55
0.63
4.00

0.05530.03220.0458

1.68
1.53
1.65

RVCCM = 1.0
Model 1nb
Model 1nu

3.50
3.50

0.0618
0.149

3.50
3.50

0.0618
0.149

0.53
0.00

0.215
0.115

1.46
1.18

a
Model 1nb and Model 1nu are the same as Model 1 in Table 2 but do not take
into account, respectively, the ﬁve data around the 2175 Å bump at
1/λ = 3.0–5.2 μm−1 and the seven UV data at 1/λ < 3.0 μm−1 for the ﬁtting
calculations. Model 5nb is the same dust model as Model 5 in Table 2 but does
not take the data around the UV bump in consideration.

essential for determining the mass fraction of graphite.
In addition, if we do not include the data at UV wavelengths
shorter than 0.3 mm, distinct extinction curves appear even in thecase of q ¼3.5 and amax,gra = amax,sil (referred to as Model 1nu forwhich Ndata = 15 without the seven data in UV regions). As seenfrom Fig. 7, the extinction curves from Model 1nu for RVCCM = 2.0
and 1.5 well reproduce the extinction data at 1/λ ≤ 3.0 μm−1.
However, at 1/λ ≥ 3.0 μm−1, these extinction curves do not resemble to their corresponding ones from Model 1; the UV extinction curve for RVCCM = 2.0 shows a less prominent 2175 Å bumpbecause of a smaller abundance of graphite (fgs = 0.16), whereas
the best ﬁt for RVCCM = 1.5 requires a very high abundance of graphite (fgs ≥ 4.0), resulting in an extremely conspicuous UV bump.
These analyses clearly demonstrate that, despite using thesame extinction data at optical to near-infrared wavelengths, thedifferent results come out depending on whether the UV extinction data are taken into account or not. Therefore, we conclude
that the extinction data at UV wavelengths are crucial for reliably
assessing the composition and size distribution of interstellar dust
from the measured extinction curves.

5. Conclusion
We have investigated the properties of interstellar dust responsible for the peculiar extinction laws with RV = 1.0–2.0 measured toward Type Ia supernovae (SNe Ia). We perform the ﬁtting
calculations to the measured extinction curves, for which weadopt the extinction values at the reference wavelengths derived
with the empirical one-parameter formula by Cardelli et al. (1989).
As a dust model for calculating the extinction curves, we consider
a two-component model composed of graphite and silicate withpower-law and lognormal size distributions.
We ﬁrst conﬁrm that our dust model can reproduce the extinction curves with RVCCM = 3.1 that is taken to be a typical value inthe Milky Way (MW). Then, we ﬁnd that even the simplest dust
model with the power index of q ¼3.5 can account for the entire
shapes of steep extinction curves described by RVCCM = 2.0, 1.5, and
1.0 with appropriate combinations of the maximum cut-off radiusamax and mass ratio of graphite to silicate fgs . In particular, amax is

T. Nozawa / Planetary and Space Science 133 (2016) 36–46

45

Fig. 7. Extinction curves for (a) RVCCM = 2.0 and (b) RVCCM = 1.5, calculated from the best-ﬁt combinations of amax and fgs for Model 1nb (solid) and Model 1nu (dashed) with
q¼ 3.5 and amax,gra = amax,sil . Model 1nb performs the ﬁtting to the data of the extinction curves in which the ﬁve data around the UV bump at 1/λ = 3.0–5.2 μm−1 are
excluded, while Model 1nu does not include the UV data at 1/λ < 3.0 μm−1 for the ﬁtting calculations. The dot-dashed line depicts the extinction curves from Model 5nb,
which uses the same extinction data as Model 1nb but parameterizes all of the ﬁve quantities relevant to the power-law distribution. The circles are the extinction data at thereference wavelengths, derived from the CCM formula for each RVCCM value, where the open symbols signify the data disregarded for Model 1nb and Model 5 nb.

found to be an important quantity to describe the variety of extinction curves, and decreases from ≃0.24 mm for RVCCM = 3.1 down

Acknowledgment

to ≃0.06 mm for RVCCM = 1.0. On the other hand, fgs takes a relatively

We thank the anonymous referees for their careful reading anduseful comments, which highly improved the paper. I also thankTakashi Kozasa for kindly providing me with the computational
resources to do the ﬁtting calculations in this study. This work wasachieved using the grant of Research Assembly supported by the
Research Coordination Committee, National Astronomical Observatory of Japan (NAOJ). This work is supported in part by the
JSPS Grant-in-Aid for Scientiﬁc Research (26400223).

narrow range of ≃0.4–0.6, indicating the mass ratio of graphite tosilicate is not changed dramatically for different RVCCM.
We have demonstrated that the lognormal grain size distribution can also work well in reproducing the CCM curves withRVCCM = 1.0–3.1 by taking the very small characteristic radius a0, j
and relatively large standard deviation γj of the distribution. Furthermore, from the comparison of the average grain radii between
the best-ﬁt power-law and lognormal size distributions, we suggest that the average radius is not a proper quantity as representing the properties of dust that explains the extinction
curves. Finally, we point out that the extinction data at a limited
range of wavelengths, such as a single value of RV, do not allow us
to uniquely determine the properties of dust, and that the extinction data over a wide range of wavelengths including the UVdata are essential for constraining the composition and size distribution of interstellar dust.
Our main conclusion is that, in order to explain the steep extinction curves suggested for SNe Ia, the size distribution of interstellar dust in their host galaxies is biased to small sizes, compared to that in the MW. Since SNe Ia are known to happen in any
types of galaxies, this implies that large grains (a ≥ ∼ 0.1 μm) arelacking in galaxies other than the MW. Why are the sizes of interstellar dust so small in external galaxies? This may presupposethat the evolution history of the MW is somehow different fromthe other galaxies so that the interstellar dust in the MW has the
exceptional properties. There is also a possibility that the measured extinction curves do not reﬂect pure extinction but are
distorted by other effects such as contamination of scattered lights
at shorter wavelengths. Nagao et al. (2016) showed that small silicate grains and/or polycyclic aromatic hydrocarbons (PAHs) areneeded for producing a low RV through multiple scattering. Hence,
this possibility may require that the host galaxies of SNe Ia containabundant small silicate grains and PAHs. Another possibility is thatsomething is amiss in the process of extracting the extinction
curves from the observations of SNe Ia. Given that SNe Ia are ideal
objects to measure the extinction curves in external galaxies, thesesubjects should be addressed from both observational and theoretical points of view.

References
Amanullah, R., et al., 2014. The peculiar extinction law of SN 2014J measured with
the Hubble Space Telescope. Astrophys. J. 788 (L21), 6pp.
Amanullah, R., Goobar, A., 2011. Perturbations of SNe Ia light curves, colors, andspectral features by circumstellar dust. Astrophys. J. 735 (20), 10pp.
Asano, R.S., Takeuchim, T.T., Hirashita, H., Nozawa, T., 2013. What determines thegrain size distribution in galaxies? Mon. Not. R. Astron. Soc. 432, 637–652.
Biermann, P., Harwit, M., 1980. On the origin of the grain-size spectrum of interstellar dust. Astrophys. J. 241, L105–L107.
Brown, P.J., et al., 2015. SWIFT ultraviolet observations of supernova 2014J in M82:
large extinction from interstellar dust. Astrophys. J. 805 (74), 13pp.
Cardelli, J.A., Clayton, G.C., Mathis, J.S., 1989. The relationship between infrared,
optical, and ultraviolet extinction. Astrophys. J. 345, 245–256.
Clayton, G.C., Wolff, M.J., Soﬁa, U., Gordon, K.D., Misselt, K.A., 2003. Dust grain size
distributions from MRN to MEM. Astrophys. J. 588, 871–880.
Conley, A., Carlberg, R.G., Guv, J., Howell, D.A., Jha, S., Ruess, A.G., Sullivan, M., Is
there evidence for a Hubble bubble? The nature of Type Ia supernova colors
and dust in external galaxies, Astrophys. J., 664, L13–L16.
Dalcanton, J.J., et al., 2009. The ACS nearby galaxies survey treasury. Astrophys J.
Suppl. 183, 67–108.
Doi, M., et al., 2010. Photometric response functions of the sloan digital sky Survey
imager. Astronom. J. 139, 1628–1648.
Draine, B.T., 2003. Scattering by interstellar dust grains. II. X-ray. Astrophys. J. 598,
1026–1037.
Draine, B.T., Lee, H.M., 1984. Optical properties of interstellar graphite and silicate
grains. Astrophys. J. 285, 89–108.
Draine, B.T., Malhotra, S., 1993. On graphite and the 2175 Å extinction proﬁle. Astrophys. J. 414, 632–645.
Dressel, L., 2016. Wide Field Camera 3 Instrument Handbook, Version 8.0, STScI,
Baltimore.
Folatelli, G., et al., 2010. The Carnegie supernova project: analysis of the ﬁrst sample
of low-redshift Type-Ia supernovae. Astronom J. 139, 120–144.
Foley, R.J., et al., 2014. Extensive HST ultraviolet spectra and multiwavelength observations of SN 2014J in M82 indicate reddening and circumstellar scatteringby typical dust, Type-Ia supernovae. Mon. Not. R. Astron. Soc. 443, 2887–2906.
Gao, J., Jiang, B.W., Li, A., Li, J., Wang, X., 2015. Physical dust models for the extinction toward supernova 2014J in M82. Astrophys. J. 807 (L26), 6pp.
Goobar, A., 2008. Low RV from circumstellar dust around supernovae. Astrophys. J.
686, L103–L106.

46

T. Nozawa / Planetary and Space Science 133 (2016) 36–46

Goobar, A., et al., 2014. The rise of SN 2014J in the nearby galaxy M82. Astrophys. J.
784 (L12), 6pp.
Gordon, K.D., Clayton, G.C., Misselt, K.A., Landolt, A.U., Wolff, M.J., 2003. A quantitative comparison of the Small Magellanic Cloud, Large Magellanic Cloud, andMilky Way ultraviolet to near-infrared extinction curves. Astrophys. J. 594,
279–293.
Hirashita, H., Nozawa, T., Takeuchi, T.T., Kozasa, T., 2008. Extinction curves ﬂattenedby reverse shocks in supernovae. Mon. Not. R. Astron. Soc. 384, 1725–1732.
Hirashita, H., Nozawa, T., 2013. Synthesized grain size distribution in the interstellar
medium. Earth Planets Space 65, 183–192.
Hirashita, H., Nozawa, T., Villaume, A., Srinivasan, S., 2015. Dust processing in elliptical galaxies. Mon. Not. R. Astron. Soc. 454, 1620–1633.
Hoang, T., 2016. Properties and alignment of interstellar dust grains toward Type Ia
supernovae with anomalous polarization curves. Mon. Not. R. Astron. Soc.,
Submitted for publication arXiv:1510.01822
Howell, D.A., 2011. Type Ia supernovae as stellar endpoints and cosmological tools.
Nat. Commun. 2 (350), 10pp.
Johansson, J., Amanullah, R., Goobar, A., 2013. Herschel limits on far-infraredemission from circumstellar dust around three nearby Type Ia supernovae.
Mon. Not. R. Astron. Soc. 431, L43–L47.
Johansson, J., et al., 2014. Spitzer observations of SN 2014J and properties of mid-IR
emission in Type Ia supernovae. Mon. Not. R. Astron. Soc., submitted arXiv:1411.
3332
Jones, A.P., Fanciullo, L., Köhler, M., Verstraete, L., Guillet, V., Bocchio, M., Ysard, N.,
The evolution of amorphous hydrocarbons in the ISM: dust modelling from a
new vantage point, Astronom. Astrophys. J., 558, A62 (22pp).
Kawabata, K.S., et al., 2014. Optical and near-infrared polarimetry of highly reddened Type Ia supernovae 2014J: peculiar properties of dust in M82. Astrophys.
J. 795 (L4), 5pp.
Lampeitl, H., et al., 2010. The effect of host galaxies on Type Ia supernovae in the
SDSS-II supernova survey. Astrophys. J. 722, 566–576.
Maeda, K., Nozawa, T., Nagao, T., Motohara, K., 2015. Constraining the amount ofcircumstellar matter and dust around Type Ia supernovae through near-infrared echoes. Mon. Not. R. Astron. Soc. 452, 3281–3292.
Marion, G.H., et al., 2015. Early observations and analysis of the Type Ia SN 2014J in
M82. Astrophys. J. 798 (39), 12pp.

Mathis, J.S., Rumpl, W., Nordsieck, K.H., 1977. The size distribution of interstellar
grains. Astrophys. J. 217, 425–433.
Morrissey, P., et al., 2005. The on-orbit performance of the Galaxy Evolution Explorer. Astrophys. J. 619, L7–L10.
Nagao, T., Maeda, K., Nozawa, T., 2016. Extinction laws toward stellar sources withina dusty circumstellar medium and implications for Type Ia supernovae. Astrophys. J. 823 (104), 10pp.
Nordin, J., Goobar, A., Jakob, J., 2008. Quantifying systematic uncertainties in supernova cosmology. J. Cosmol. Astropart. Phys. 02 (008), 21pp.
Nozawa, T., Fukugita, M., 2013. Properties of dust grains probed with extinction
curves. Astrophys. J. 770 (27), 13pp.
Nozawa, T., Kozasa, T., Habe, A., Dwek, E., Umeda, H., Tominaga, N., Maeda, K.,
Nomoto, K., 2007. Evolution of dust in primordial supernova remnants: candust grains formed in the ejecta survive and be injected into the early interstellar medium?. Astrophys. J. 666, 955–966.
Patat, F., et al., 2015. Properties of extragalactic dust inferred from linear polarimetry of Type Ia supernovae. Astron. Astrophys. J. 577 (A53), 10pp.
Poole, T.S., et al., 2008. Photometric calibration of the swift ultraviolet/opticaltelescope. Mon. Not. R. Astron. Soc. 383, 627–645.
Skrutskie, M.F., et al., 2006. The two micron all sky survey (2MASS). Astronom J.
131, 1163–1183.
Sullivan, M., et al., 2010. The dependence of type ia supernovae luminosities on
their host galaxies. Mon. Not. R. Astron. Soc. 406, 782–802.
Telesco, C.M., et al., 2015. Mid-IR spectra of Type Ia SN 2014J in M82 spanning theﬁrst 4 months. Astrophys. J. 798 (93), 9pp.
Wang, L., 2005. Dust around Type Ia supernovae. Astrophys. J. 635, L33–L36.
Weingartner, J.C., Draine, B.T., 2001. Dust grain-size distributions and extinction in
the milky way, large magellanic cloud, and small magellanic cloud. Astrophys. J.
548, 296–309.
Yasuda, Y., Kozasa, T., 2012. Formation of SiC grains in pulsation-enhanced dustdriven wind around carbon-rich asymptotic giant branch stars. Astrophys. J. 745(159), 14pp.
Zubko, V., Dwek, E., Arendt, R.G., 2004. Interstellar dust models consistent withextinction, emission, and abundance constraints. Astrophys. J. Suppl 152,
211–249.