The Astrophysical Journal, 823:104 (10pp), 2016 June 1

doi:10.3847/0004-637X/823/2/104© 2016. The American Astronomical Society. All rights reserved.

EXTINCTION LAWS TOWARD STELLAR SOURCES WITHIN A DUSTY CIRCUMSTELLAR MEDIUM
AND IMPLICATIONS FOR TYPE IA SUPERNOVAE
Takashi Nagao1, Keiichi Maeda1,2, and Takaya Nozawa3
1

Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan; nagao@kusastro.kyoto-u.ac.jp
2
Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan
3
National Astronomical Observatory of Japan, 2-21-1, Osawa, Mitaka, Tokyo 181-8588, Japan
Received 2016 January 30; accepted 2016 March 24; published 2016 May 26

ABSTRACT
Many astronomical objects are surrounded by dusty environments. In such dusty objects, multiple scattering
processes of photons by circumstellar (CS) dust grains can effectively alter extinction properties. In this paper, we
systematically investigate the effects of multiple scattering on extinction laws for steady-emission sources
surrounded by the dusty CS medium using a radiation transfer simulation based on the Monte Carlo technique. In
particular, we focus on whether and how the extinction properties are affected by properties of CS dust grains byadopting various dust grain models. We conﬁrm that behaviors of the (effective) extinction laws are highly
dependent on the properties of CS grains, especially the total-to-selective extinction ratio RV, which characterizes
the extinction law and can be either increased or decreased and compared with the case without multiple scattering.
We ﬁnd that the criterion for this behavior is given by a ratio of albedos in the B and V bands. We also ﬁnd thateither small silicate grains or polycyclic aromatic hydrocarbons are necessary for realizing a low value of RV as
often measured toward SNe Ia if the multiple scattering by CS dust is responsible for their non-standard extinction
laws. Using the derived relations between the properties of dust grains and the resulting effective extinction laws,
we propose that the extinction laws toward dusty objects could be used to constrain the properties of dust grains in
CS environments.
Key words: circumstellar matter – dust, extinction – radiative transfer – scattering – stars: mass-loss –supernovae: general
Type Ia supernovae (SNe Ia). It has been shown that theextinction laws toward SNe Ia deviate from the standard
extinction laws derived for the Milky Way, Large Magellanic
Clouds (LMCs) or Small Magellanic Clouds (SMCs), and show
the total-to-selective extinction ratio, RV, as small as ∼2 (e.g.,
Nobili & Goobar 2008; Folatelli et al. 2010). One possible
scenario for explaining this non-standard nature of theextinction law involves the multiple scattering by CS dust
grains around SNe Ia (Wang 2005; Goobar 2008). Goobar(2008) showed that the non-standard extinction with a low
value of RV can be achieved by the multiple scattering, if CS
dust grains have the properties of the MW or LMC dust. He
argued that the multiple scattering causes photons with shorter
wavelengths to be more efﬁciently scattered and absorbed thanthose with longer wavelengths, leading to the steepening of theeffective extinction law as compared with the case of the single
scattering case. He also pointed out that this scenario requires
dust grains with a large albedo, mentioning that the SMC-type
dust having a small albedo in the optical to near-IR
wavelengths is not suitable for this scenario. However, it was
not demonstrated in detail how sensitively the effective
extinction laws depend on properties of CS dust and what
properties of dust are necessary for realizing unusually low RVthrough the multiple scattering.
In this paper we investigate what is a main factor in
determining the characteristic properties of the effective
extinction laws toward a steady-emission source surroundedby CS dust. In doing this, we adopt various dust models andsystematically compute resulting effective extinction laws, byperforming Monte Carlo radiation transfer simulations. We
especially focus on RV, which represents a slope of theextinction curve in the optical regime. In Section 2 we ﬁrst

1. INTRODUCTION
Objects surrounded by the dusty circumstellar (CS) media
are often found in the universe, including some classes ofsupernovae (SNe) (Fox et al. 2011), asymptotic giant branchstars, planetary nebulae, and protostars. Understanding theproperties of the CS dust grains is highly important as they are
directly linked to the yet unresolved host star’s evolutionary
status. Indeed, different classes of objects are expected to havedifferent components of CS dust grains depending on the gas
composition and chemistry responsible to the creation of thosedust grains. We also need to understand the relation between
the properties of CS dust grains and the extinction laws theycause in order to extract intrinsic (dust-free) information onthese objects.
The properties of dust grains are linked to extinction. It is
straightforward to analytically derive the extinction laws
caused by interstellar (IS) dust under the widely adopted
assumption that photons scattered off by the dust grains do notcome back to a line of sight (hereafter a “single scattering”case). This is a reasonable assumption if the dust grains arelocated far away from the source, i.e., IS dust grains. However,in a case of extinction caused by CS dust grains that are
distributed in the vicinity of the light source, we have to deal
with a situation in which photons originally emitted toward a
direction out of the line of sight can ﬁnally reach an observer
after multiple scattering of photons within CS environments(hereafter a “multiple scattering” case).
Multiple scattering should be in general taken into account instudying objects within dusty CS environments, or inversely instudying the CS environments using the extinction laws. As an
example, an increasing attention has been directed to themultiple scattering process within CS environments around1

The Astrophysical Journal, 823:104 (10pp), 2016 June 1

Nagao, Maeda, & Nozawa

where Il0 is an incident intensity and Il1 is an intensity at theanalytically derive expected relations between an effective
extinction law and properties of dust grains under the planeparallel approximation for the distribution of dust grains. This
provides a useful guide for further investigation of a case of CS
dust surrounding the source. In Section 3 we describe ourmethods for the radiation transfer simulations within the dusty
CS environments. In Section 4 we present the results of oursimulations. Discussion on relations between dust properties(grain species and size) and values of RV is given in Section 5.
In Section 6 we discuss implications for the extinction law
toward SNe Ia as an application of the present results, while weemphasize that the general properties we derive in this paper
can be applied to any objects surrounded by dusty CS
environments. In Section 7 we provide our conclusions.

emergence through the plane-parallel cloud. Here, text (l ) is an
optical depth of the cloud along the line of sight at awavelength λ. A quantity to characterize the extinction, k (l ),is determined by the following equation:w (l ) =RVm =2.1. Values of RV Resulting from a Semi-inﬁnite
Plane-parallel Dusty CloudIn this subsection we derive a value of RV resulting fromextinction in a semi-inﬁnite plane-parallel dusty cloud within
which dust grains are uniformly distributed. It is assumed thatthe incident radiation is orthogonal to the plane. Here, twocases are considered. The ﬁrst is a case in which photons thatare scattered off do not come back to a line of sight by anotherscattering; that is, a case where “multiple scattering” processes
are negligible (the “single scattering” case). The second is acase in which this “single scattering” assumption does not hold(the “multiple scattering” case).
In the single scattering case, a value of RV (hereafter RVs ) is
determined only by the ratio of the mass extinction coefﬁcientof an mixture of dust grains in the B band to the V band(kext (B) kext (V ))4 as follows:(1 )Here, kext (B) and kext (V ) include contributions from different
dust species, taking into account the size distribution.
In the multiple scattering case, we have to consider photons
that are scattered multiple times and ﬁnally reach an observer.
In this subsection, we assume that a dust grain scatters off
photons isotropically, in order to separate effects of anisotropic
scattering by dust grains as discussed in Section 4.2. Taking themultiple scattering into account under the plane-parallel
approximation, the radiation transfer can be expressed asfollows (Chandrasekhar 1960);
Il1 = Il0 exp [ - ∣k (l)∣text (l)] ,1.
k (B) kext (B)-1
k (V ) kext (V )(4 )If the emission source is embedded in the dusty environment(e.g., CS dust), the effect of the multiple scattering must always
be considered. On the other hand, if the scattering clouds arelocated far away from the source, the importance of themultiple scattering is dependent on a ratio of size of eachscattering cloud orthogonal to the line of sight (L) to the mean
free path in scattering (r0 ~ 1 (kscat r0 ), where kscat is a mass
scattering coefﬁcient and r0 is a typical mass density in thecloud). If L < r0 for all of the clouds through which theincident radiation propagates, the effects of the multiple
scattering should be unimportant. This case applies toextinction by IS dust, and thus RVs can be used (Spitzer 1998).
If L > r0 in a speciﬁc cloud, the multiple scattering must be
taken into account. An exception for the latter is the case where
the length scale of scattered echo (r0) is spatially resolved,
because in the scale smaller than r0, the scattering cloud can be
regarded as the single scattering component.
From Equations (3) and (4) it is clear that the value of RVm
depends not only on the ratio of kext (B) to kext (V ), but also on
the absolute values of w (B) and w (V ), or equivalently on theratio of w (B) to w (V ) and the value of w (B). It should be notedthat the multiple scattering can be important even when the
optical depth along a line of sight is smaller than unity, as longas the optical depth of a direction orthogonal to the line of sight
is not negligibly small (i.e., L > r0 ). In plane-parallel dusty
clouds, the optical depth of a direction orthogonal to a line ofsight is assumed to be inﬁnity. Therefore, RV can be determinedfrom Equation (4), regardless of the optical depth along the lineof sight.
Equation (4) and Figure 1 show that RVm is lower than RVs ,when k (B) k (V ) > 1, namely, when w (B) < w (V ). However,when w (B) > w (V ), RVm should be higher than RVs . Namely,
whether RVm becomes higher or lower than RVs depends strongly
on the dust properties related to the behavior of the albedo as afunction of wavelength. As we show below, different dust
models predict different ratios of the albedos in the B and V
bands.
Figure 2 shows the values of RVm and DRV1 (=RVm - RVs ,difference between RVm and RVs ) for various ratios of the mass
extinction coefﬁcient in the B band to that in the V band(kext (B) kext (V )) and for various ratios of the albedo in the Bband to that in the V band (w (B) w (V )). For the purpose ofillustration, two cases for w (B) = 0.6 and w (B) = 0.7 areshown in the ﬁgure. Figure 2 shows how strongly RVm depends
on parameters of dust models. RVm is lower whenkext (B) kext (V ) is higher and/or w (B) w (V ) is lower, if

In this section we consider an effective extinction law towarda steady light source through a plane-parallel dusty cloud. Thissubject can be examined in an analytic way. While thissituation is not readily applicable to the case of CS dust, theresult in this section will provide a useful guide to understanding relations between properties of dust grains and a
resulting effective extinction law in the case of CS dust.

1.
kext (B)-1kext (V )(3 )where w (l ) is an albedo at a wavelength λ. We can then obtain
RV in the “multiple scattering” case (hereafter RVm ) as follows:

2. EXTINCTION THROUGH A PLANE-PARALLEL
DUSTY CLOUD

RVs =2 k (l ),⎡ 1 + k (l ) ⎤ln ⎢⎥⎣ 1 - k (l ) ⎦

(2 )

4

The subscript “ext” is used for values related to “extinction,” i.e., a sum of
the contributions from the absorption and scattering.

2The Astrophysical Journal, 823:104 (10pp), 2016 June 1

Nagao, Maeda, & Nozawa

used to describe the interstellar extinction within the MW(MW1 and MW2) and within the LMC (LMC1 and LMC2),
but when the multiple scattering is important, they show the
opposite behavior in RVm . From Table 1, k (B) k (V ) is higher inorder of LMC2, MW2, LMC1, and MW1 (see also Figure 1).
According to the above discussion, the value of RV is foundto highly depend on the properties of dust (kext (B) kext (V ),w (B) w (V ) and w (B)) in the case of extinction through theplane-parallel dusty cloud. In the following sections we study
extinction caused by the CS dust grains around a steadyemission source. In this case, we have to deal with extinction
taking into account the multiple scattering process, even if the
optical depth is low. Unlike the plane-parallel case, theextinction by the CS dust is not described analytically, thus
we calculate the value of RVm using the Monte Carlo radiationtransfer method.

Figure 1. Values of k (B ) k (V ) (see the vertical color bar on the right side for ascale) as a function of w (B ) and w (B ) w (V ) (Equation (3)). The properties ofthe MW1, LMC1, MW2, and LMC2 dust models (see Section 2.2) areindicated by black dots. The line of k (B ) k (V ) = 1 provides a criterion for
either RVm < RVs or RVm > RVs . The MW1 and LMC1 dust models have theproperty k (B ) k (V ) > 1, thus RVm > RVs while the MW2 and LMC2 dust
models have the opposite property.

3. METHODS
We simulate the radiation transfer of photons within the CS
dust shell around a steadily emitting source, following theprescriptions used by Goobar (2008). The contribution of reemitted photons following the absorption of a photon within
the CS dust is negligible in the B and V bands, given that thetemperature of CS dust is at most ∼2000 K.
We assume that the CS dust grains are distributed uniformly
within a sphere of a radius RCS, where RCS is measured fromthe central emitting source. Let n 0 (B) and n 0 (V ) be thenumbers of photons in the B band and V band wavelengths,
respectively, which are originally emitted from the central
source. The propagation of these photons within the CS
environment is calculated by three-dimensional Monte Carloradiation transfer simulations until they reach RCS and escape
the CS environment or until they are absorbed by the CS dust.
For each photon path, an optical depth for each interaction of aphoton packet is determined following a probability distribution exp (-text ). The distance of the ﬂight is then obtained fromthe equation text (l ) = ò kext (l ) rdl , where kext is the mass
extinction coefﬁcient of a dust model including both theabsorption and scattering, and ρ is the mass density of the CS
dust and is taken to be constant. At each interaction the fate of
the photon, absorbed or scattered, is determined following thevalue of albedo w (l ), through the Monte Carlo random number
generation. If the photon is determined to be absorbed, thecalculation of this photon packet is ﬁnished. If the scattering
takes place, a new direction of the photon is speciﬁed following
a scattering angle distribution. In Section 4.3 the scattering
distribution is assumed to be isotropic and in Sections 4.1 and
4.2 it is assumed to be given by the following Henyey–
Greenstein approximation (Henyey & Greenstein 1941):

w (B) is ﬁxed. On the other hand, given kext (B) kext (V ) andw (B) w (V ), ∣DRV1 ∣ is higher for higher w (B), which means thatas the albedo in the B band becomes higher, the effect of themultiple scattering becomes more signiﬁcant.

2.2. Cases of Typical Dust Models
In this subsection we apply the general discussion in theformer subsection to widely used dust models. For demonstration we consider four dust models, which enables us to provide
a fair comparison to the results by Goobar (2008) and obtainfurther insight; the MW dust and LMC dust models with apower-law size distribution (hereafter MW1 and LMC1,
respectively), and the MW dust and LMC dust models withthe size distributions proposed by Weingartner & Draine (2001;
hereafter MW2 and LMC2). MW2 and LMC2 are the dust
models used by Goobar (2008). These dust models areconstructed as a mixture of astronomical silicate and graphite
grains of various sizes (and also polycyclic aromatic hydrocarbons (PAHs) for MW2 and LMC2), assuming that dust
grains have a spherical shape. We consider that the power-law
size distribution (for MW1 and LMC1) has an index of −3.5
and that the maximum and minimum grain radii are 0.25 μm
and 0.005 μm, respectively (Mathis et al. 1977). The mass ratio
between astronomical silicate and graphite is 0.642:0.358 for
the MW1 dust model and 0.843:0.157 for the LMC1 dust
model. These values are determined so that the dust models
well reproduce the observed extinction curves in the MW andthe LMC.5 Table 1 summarizes the detailed characteristics ofthese dust models.
The values of RVs and RVm in these models, which are
expressed as RVs (pp) and RVm (pp)6, respectively, are shown in
Table 1. In the LMC1 and MW1 models w (B) < w (V ) (seeFigure 1) and therefore RVm is higher than RVs . On the other
hand, RVm is lower than RVs in LMC2 and MW2 models. It
should be emphasized that these models are both extensivelyds
1 - g2=,d [cos q ][1 + g2 - 2g cos q ]3 2

(5 )where g = ácos (q ) ñ (see Table 1) and σ is a scattering crosssection toward a scattering angle θ. The scattering is isotropic
when g = 0, while it is perfect forward scattering when g = 1.
We denote the numbers of photons in the B and V bands
escaping the system without absorption as n1 (B) and n1 (V ),5

The mass ratios between silicate and graphite given by Draine & Lee (1984)and Pei (1992) are 0.64:0.36 and 0.88:0.12, respectively, for the MW and LMC
dust models. The difference between their values and ours mainly stems fromthe fact that we adopt the updated extinction curves (Gordon et al. 2003;
Whittet 2003) to construct these dust models.
6
The notation “pp” is for plane parallel.

3

The Astrophysical Journal, 823:104 (10pp), 2016 June 1

Nagao, Maeda, & Nozawa

Figure 2. Values of RVm and DRV1 (=RVm - RVs ) under the plane-parallel approximation with isotropic scattering. The color and the black contour lines show the values(RVm and DRV1 ) as a function of kext (B ) kext (V ) and w (B ) w (V ) when w (B ) = 0.6 and 0.7. (a) RVm for w (B ) = 0.6 . (b) DRV1 for w (B ) = 0.6 . (c) RVm for w (B ) = 0.7.
(d) DRV1 for w (B ) = 0.7. The red lines show values of RVs . The criterion for RVm < RVs (RVm > RVs ) is given by w (B ) w (V ) < 1 (w (B ) w (V ) > 1), and RVm is smaller
for smaller w (B ) w (V ) and/or larger kext (B ) kext (V ).

value of DRV1 = RVm - RVs depends strongly on dust models. In
the following subsections we discuss how the detailed properties of the dust models affect the value of RVm .

respectively. Finally, RVm can be calculated as follows:
RVm =
=

AV
E (B - V )log (n1 (V ) n 0 (V )).
log (n1 (B) n 0 (B)) - log (n1 (V ) n 0 (V ))4.2. Dependence of RVm on g = ácos (q ) ñ(6 )Here we perform the same calculations as in Section 4.1, butchanged the value of g artiﬁcially to investigate how the valueof RVm depends on the scattering angular distribution. Figure 4
shows values of RVm as a function of E (B - V ) calculatedusing a dust model based on LMC2 but with various values ofg. To simplify the analysis we used the same value for g in boththe B and V bands, ignoring the wavelength-dependence. RVm issmallest when the scattering is isotropic (g = 0) and largest
when it is described as a perfect forward scattering (g = 1).
When g = 1, the value of RVm can be evaluated by theformula for the single scattering (Equation (1)), but using kabs
instead of kext ;7

Two million photon packets per band are used in thesecalculations.
4. RESULTS
4.1. Dependence of RVm on E (B - V )The Monte Carlo calculations with various amounts of dust
are performed. We adopt four different models for theproperties of the CS dust grains MW1, LMC1, MW2, andLMC2 (see Table 1). The scattering angle of a photon adopted
in the simulations is computed using the Henyey–Greenstein
Approximation.
Figure 3 shows the values of RVs and RVm for various values ofE (B - V ). For 0.2 < E (B - V ) < 2.0 , RVm of each model
does not depend strongly on E (B - V ), as shown in Figure 3,
conﬁrming the result by Goobar (2008). Furthermore, thesevalues of RVm are different from the values of RVs of thecorresponding models. Figure 3 shows RVm (MW1) ∼3.9,
RVm (LMC1) ∼3.3, RVm (MW2) ∼2.5, RVm (LMC2) ∼1.6. For theMW2 and LMC2 dust models that Goobar (2008) used, ourresults are consistent with their results. We also ﬁnd that theRVm (g = 1) =1.
kabs (B)-1kabs (V )(7 )Namely, the scattering does not contribute to the extinction law
when g = 1. In this case, due to the perfect forward scattering,
the scattering is considered to have null effect on the7
Hereafter, RVm (g = 1) is used for a case of perfect forward scattering. RVm isused for a case of isotropic scattering unless otherwise mentioned.

4The Astrophysical Journal, 823:104 (10pp), 2016 June 1

Nagao, Maeda, & Nozawa
Table 1Dust Parameters

Filterλ(μm)

kabs (cm2 g−1)

kscat (cm2 g−1)

Albedog

RVs

RVm (pp)

DRV1

MW1

B
V

0.44
0.55

1.839E+04
1.473E+04

2.253E+04
1.745E+04

0.55060.54230.45730.4187

3.680

3.759

7.682E-02

LMC1

B
V

0.44
0.55

9.744E+03
7.735E+03

1.990E+04
1.470E+04

0.67130.65520.50510.4737

3.112

3.320

2.092E-01

MW2

B
V

0.44
0.55

1.191E+04
8.551E+03

2.240E+04
1.764E+04

0.65290.67350.5654
0.5382

3.225

2.973

−2.520E-01

LMC2

B
V

0.44
0.55

7.542E+03
4.666E+03

1.900E+04
1.503E+04

0.71590.76310.61530.6059

2.876

2.262

−6.139E-01

Dust Model

Note. The values of MW2 and LMC2 are from Table 1 in (Goobar 2008) (The original data is in Weingartner & Draine 2001 and Draine 2003.)

w (B) w (V ), and ω(B)), as is the case for the plane-parallel
cloud.
To investigate how these three parameters affect RVm , weparameterize kext (B) kext (V ), w (B) w (V ), and w (B). Oncethe behaviors are clariﬁed through our analysis, these “input”
dust parameters can be compared with any speciﬁc physicaldust models. The amount of CS dust grains is set so thatE (B - V ) is between 0.2 and 2.0.
Figure 5 shows the values of RVm and DRV1 as a function of
kext (B) kext (V ) and w (B) w (V ). We also ﬁx w (B) to be 0.6 or
0.7, which almost cover the range expected for physical dust
models (Table 1). In these calculations, the scattering ofphotons is assumed to be isotropic (g = 0). We ﬁnd that thevalue of RVm strongly depends on the detailed properties(kext (B) kext (V ), w (B) w (V ), and w (B)) of dust grains. We
also ﬁnd that the dependence of RVm on the detailed propertiesof dust is qualitatively similar to the case of the plane-parallel
dusty cloud. RVm is lower when kext (B) kext (V ) is higher, andw (B) w (V ) is lower if w (B) is ﬁxed. DRV1 is lower for a higherw (B) when kext (B) kext (V ) and w (B) w (V ) are ﬁxed.

attenuation. This is different from a case with g < 1, where themultiple scattering brings photons originally emitted toward a
direction out of the line of sight to the observer’s direction, andthus effectively changes the path lengths depending on analbedo.
Equation (7) can be rewritten in a similar form toEquation (4) (that assumes isotropic scattering) as follows:
RVm (g = 1) =1.
⎛ 1 - w (B) ⎞ kext (B)-1⎜⎟⎝ 1 - w (V ) ⎠ kext (V )(8 )From Equations (1) and (8) it is clear that RVm (g = 1) is lower
than RVs when w (B) < w (V ). We note that the criterion for
RVm < RVs is indeed the same for the cases with g = 1 andg = 0, indicating that this is a general criterion independent
from the scattering angular distribution.
Equation(7)
providesthefollowingvalues:RVm (g = 1; MW1) = 4.025,
RVm (g = 1; LMC1) = 3.850 ,RVm (g = 1; MW2) = 2.546,
RVm (g = 1; LMC2) =and
1.622 . For the MW2 and LMC2 dust models, these values
are close to the values of RVm with g adopted from the dust
models under consideration (i.e., RVm in the former subsection).
From this analysis, we clarify that the result by Goobar (2008)
can be understood as a limiting case of the general descriptionof the radiation transfer within the dusty CS environments,
which is affected both by the scattering angular distribution andby the ratio of the albedos (w (B) w (V )). The dust models
Goobar (2008) adopted have relatively large values of g (close
to the forward scattering) and thus RVm approaches the above
value described mainly by the pure absorptive component(Equation (7)). Their dust models have w (B) < w (V ), leading
to RVm < RVs (Equation (8)).

5. DUST PROPERTIES AND VALUES OF RVm
In this section we investigate how the value of RVm depends
on dust properties such as the composition and size bychanging the relative mass ratio of silicate and graphite as wellas the power-law index, maximum grain radius, and minimum
grain radius in the power-law size distribution (i.e., variants of
MW1 and LMC1). The values of kext (B) kext (V ) andw (B) w (V ) in the various dust models are computed andcompared with the values of RVm under the plane-parallel
approximation with w (B) = 0.6 and isotropic scattering. This
provides a useful estimate for the following reasons: (1) thegeneral behaviors for the plane-parallel dusty cloud and for the
CS dust are similar (Sections 2.1 and 4.3); (2) the criterion for
RVm < RVs or RVm > RVs (i.e., a ratio of w (B) w (V )) is
independent from g; and (3) for the parameter range expected
for the typical dust models, the effect of g is not large(Figure 4). However, we caution that for a detailed analysis,
especially for dust models with extreme parameters, one has totake into account the effects of the dust spatial distribution andthe scattering angular distribution.
Figure 6 presents the dependencies of kext (B) kext (V ) andw (B) w (V ) on the mass ratio between silicate and graphite,and on the power-law index in the size distribution. Here, themaximum and minimum grain radii are ﬁxed to be 0.25 and0.005 μm. RVs and RVm become lower for a steeper size

4.3. Dependence of RVm on kext (B) kext (V ),w (B) w (V ), and w (B)In this subsection we examine the dependence of RVm on
kext (B) kext (V ), w (B) w (V ), and w (B). At ﬁrst look, there are
four parameters (kscat (B), kabs (B), kscat (V ), and kabs (V )) that
describe the system and determine a value of RVm , except for g
discussed in Section 4.2. However, once the model is speciﬁedby E (B - V ), i.e., optical depth, the absolute value of mass
absorption coefﬁcient is already absorbed to this parameter.
Thus, one can determine the values of RVm from only
three parameters in the dust properties (kext (B) kext (V ),5The Astrophysical Journal, 823:104 (10pp), 2016 June 1

Nagao, Maeda, & Nozawa

Figure 3. RVs and RVm as a function of E (B - V ) calculated adopting four dust models: MW1 (orange circles), LMC1 (cyan squares), MW2 (green asterisks), andLMC2 (violet crosses). The values of RVs for the same dust models are shown by the dashed lines (with the same colors as the corresponding multiple scattering cases).
The MW2 and LMC2 dust models result in RVm < RVs , while it is not the case for the MW1 and LMC1 dust models.

Figure 4. RVm as a function of E (B - V ) for various values of g based on the LMC2 model. In these simulations, g is artiﬁcially changed from the scattering angular
distribution in the original LMC2 dust model. The scattering angular distribution has the effect such that smaller g leads to lower RVm in this case.

distribution and/or for a larger fraction of silicate grains. For
example, when the power-law index is −3.5, RVs ~ 5 andRVm ~ 4 for the carbon-rich dust model and RVs ~ 2.5 andRVm ~ 2.5 for the silicate-rich dust model, under the planeparallel approximation with w (B) = 0.6 and isotropic
scattering.

The cases with various maximum grain radii, while the otherparameters (the grain size distribution, the mass ratio between
silicate and graphite and the minimum grain radius) are ﬁxed atthe values adopted in the MW1 model, are shown in Figure 7.
The maximum grain radius was changed from 5.0 ´ 10-3 m m
to 1.0 ´ 101 m m. RVs and RVm take the minimum values when
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Nagao, Maeda, & Nozawa

Figure 5. Values of RVm and DRV1 (=RVm - RVs ) for the case of the CS dust with isotropic scattering. The color and the black contour lines show the values as a functionof kext (B ) kext (V ) and w (B ) w (V ) when w (B ) = 0.6 and 0.7. (a) RVm for w (B ) = 0.6 . (b) DRV1 for w (B ) = 0.6 . (c) RVm for w (B ) = 0.7. (d) DRV1 for w (B ) = 0.7.
The red lines show values of RVs . The general behavior is similar to the case of the plane-parallel dusty cloud (Figure 2).

Figure 6. Values of kext (B ) kext (V ) and w (B ) w (V ) in the dust models with various mass ratios of astronomical silicate and graphite, and for various size
distributions (assumed to be a power-law function in the dust grain radius). The violet crosses, green squares, and cyan circles represent the dust grains where thepower-law index in the size distribution is −3.0, −3.5, and −4.0, respectively. For a given power-law index, kext (B ) kext (V ) is larger (to the right) when silicate dust
is more abundant. The dashed lines represent contour lines of RVm in the plane-parallel dusty cloud case (Equation (4)) for w (B ) = 0.6 . The red lines show values of
RVs . Both RVm and RVs become smaller for a dust model with a steeper size distribution and/or a larger abundance of silicate.

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Nagao, Maeda, & Nozawa

Figure 7. The values of kext (B ) kext (V ) and w (B ) w (V ) in the dust models with various maximum grain radii, while the other parameters (the grain size distribution,
the mass ratio between silicate and graphite and the minimum grain radius) are ﬁxed at the values adopted in the MW1 model. The maximum grain radius is changed
from 5.0 ´ 10-3 m m to 1.0 ´ 101 m m. The dashed lines are the same as in Figure 6. The red lines show values of RVs . The maximum grain radius of (5–8)´ 10-2 μm
results in RVm (and RVs ) as low as ∼2.

Figure 8. The values of kext (B ) kext (V ) and w (B ) w (V ) in the dust models including PAHs, with various mass ratios of astronomical silicate and carbon grains. The
mass fraction of PAHs is varied in this ﬁgure. Violet-crosses show the case where PAHs are not included. The cyan asterisks, orange open squares, yellow squares,
and blue circles show the cases where the mass fraction of PAHs is taken as one-sixth, one-fourth, one-second, and the same amount, respectively, as compared to thecarbon grains. The dashed lines are the same as in Figure 6. The red lines show values of RVs . A dust model including a large amount of PAHs results in RVm as low as
2, while RVs is signiﬁcantly larger than RVm .

the maximum grain radius is set to be ~7.0 ´ 10-2 m m. Under
the plane-parallel approximation with w (B) = 0.6 and isotropic scattering, the corresponding minimum values of RVs and
RVm are ∼1.5 and ∼2, respectively, for a speciﬁc range of the

maximum grain radius between 5 ´ 10-2 m m and8 ´ 10-2 m m. We also performed the calculations withdifferent minimum grain radii, but we ﬁnd that RVm does notdepend strongly on the minimum grain radius.
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Nagao, Maeda, & Nozawa

Finally, we consider dust models including PAHs. As
demonstration, we convert some fraction of the carbon mass inthe dust model into PAHs, while the parameters in terms of size
distributions of silicate and graphite are the same as those in theMW1 dust model. The optical properties of PAHs are adopted
from Li & Draine (2001). We adopt 10 Å as a typical size of
PAH. Calculated values of kext (B) kext (V ) and w (B) w (V )for various mass ratios of astronomical silicate and graphite(and PAHs) are shown in Figure 8. For a large amount of PAH,
RVm can be much lower than RVs . For example, in the carbonrich dust model where one-third of the total mass is in PAHs,
RVs ~ 3 and RVm ~ 2 can be realized. In summary, RVm becomes
lower in the dust models that include more silicate grains or
smaller grains, and in these cases RVs becomes lower as well.
On the other hand, RVm can be lower without affecting RVs (∼3)
substantially in the dust models that include PAHs.

large fraction of PAHs, more than a half of graphite mass, areneeded to realize a situation that RVs ~ 3 and RVm < 2.
We should note that SNe Ia should be regarded as nonsteady sources. In this paper we provide analysis of the CS dust
multiple scattering model for SNe Ia using a steady source,
under the same assumption as in the previous study by Goobar(2008). In reality, time-dependent effects must be taken into
account, including the effect of time evolution of source ﬂux
and color (e.g., Amanullah & Goobar 2011; Krügel 2015, fortheoretical studies; Amanullah et al. 2015 for an observational
study). In fact, a value of RV is frequently derived around the Bband maximum. The echo should have a lower ﬂux and bluer
color than the corresponding steady source, because theobserved light echo at this moment is caused by the delayed
arrival of the radiation that was emitted earlier. Therefore, theeffects of the multiple scattering may be smaller than in a
steady source. Also, the strength of this effect may be different
for different SNe Ia, depending on the early-phase light curveand color evolutions which are more diverse than the maximum
phase (Cartier et al. 2011). However, in a case where thescattering is mainly forward scattering (see Section 4.2) or thetimescale of the echo is small, the effects of the source
variability can be ignored and the present results can apply. To
clarify how the effects of the multiple scattering contribute to
the value of RV toward SNe Ia requires radiation transfer
calculations for non-steady sources. We will present suchanalyses in our forthcoming paper (Nagao et al. 2016, in
preparation), considering the effect of the difference in dust
optical properties, taking into account the time evolution.

6. DISCUSSIONS
The analysis presented in this paper is generally applicable toa wide range of astronomical objects within dusty environments as long as the dusty region is relatively localized near thesource (e.g., CS dust environments). We plan to present several
applications in a forthcoming paper (T. Nagao et al. 2016, in
preparation), but in this paper we have focused on one speciﬁc
example: it is the non-standard extinction law toward SNe Ia.
SNe Ia show a non-standard extinction law, RV  2, in
contrast to the typical Galactic value of RV ~ 3 (e.g., Nobili &Goobar 2008; Folatelli et al. 2010). As one possible
interpretation, it has been claimed that multiple scattering of
SN photons by CS dust may lead to the non-standard extinction
law. Goobar (2008) presented this argument by calculating theextinction laws toward a steady source within a CS dusty
environment, adopting the LMC2 dust model using a Monte
Carlo simulation. We reproduce their result that multiple
scattering by the LMC2 dust model leads to low RVm . However,
we have shown that this result is strongly dependent on the dust
properties and that the multiple scattering can lead to either
higher or lower RVm than the single scattering case. Our analysis
found that RVm is lower for the CS dust with higher
kext (B) kext (V ) and lower w (B) w (V ).
From our analysis, we emphasize that the multiple scatteringmodel does not necessarily lead to RVm < RVs . If it is the mainprocess to explain the non-standard value of RV toward SNe Ia,
the properties of the CS dust must satisfy particular criteria.
Namely, this scenario requires the CS dust having highkext (B) kext (V ) and low w (B) w (V ). For example,w (B) w (V )  0.97 to realize a situation that RVs ~ 3 andRVm < 2 when w (B) = 0.7. The LMC2 and MW2 dust models(Weingartner & Draine 2001; Draine 2003) adopted by Goobar(2008) have these features, while the LMC1 and MW1 models
do not. Inversely, if one has evidence of a sufﬁcient amount of
CS dust around SNe Ia from arguments totally independent
from the extinction properties (e.g., Maeda et al. 2015), then
one could potentially distinguish the dust models, or generally
the nature of dust grains that SNe Ia progenitor produced before
the explosion. Again, this argument is not restricted to SNe Ia
and applies generally to any sources within dusty CS
environments. We have further analyzed the relation between
dust properties and resulting RVm . Thereby we have found that
small silicate or PAHs within the CS environments is necessary
to lead to low RVm . Especially in the carbon-rich dust model, a

7. CONCLUSIONS
Effects of the multiple scattering are important in considering extinction by dust, especially when the dust grains are
localized around/near the emission source. In a case of
extinction by interstellar dust, this effect is generally negligible
except for a case in which the size of (spatially unresolved)
scattering bodies, L, is larger than the mean free path inscattering, r0, in a system dominating the total extinction (e.g.,
extinction by a very thick molecular cloud). In a case of
circumstellar dust, we have to consider such effects even whenan optical depth of a scattering cloud is smaller than unity.
In this paper, we have systematically investigated the natureof extinction toward a steadily emitting source within dusty CS
environments, by performing Monte Carlo radiation transfer
simulations. Especially, we have clariﬁed how RV after themultiple scattering (i.e., as observed) is connected to the opticalproperties of dust (kext (B) kext (V ), w (B) w (V ), and w (B))
and the physical properties of dust (sizes and species
including PAHs).
Our ﬁndings are summarized as follows:
1. A value of RV after the multiple scattering (RVm ) for theextinction by CS dust is almost constant for
0.2 < E (B - V ) < 2.0 , conﬁrming the result by Goobar(2008).
2. RVm is generally dependent on the scattering angular
distribution. The expressions of RVm in the limiting cases
are derived for g = 0 (Equation (4)) under the planeparallel approximation, and for g = 1 (Equations (7)and (8).
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Nagao, Maeda, & Nozawa

3. A value of RVm depends strongly on properties of dust. RVm
is lower when kext (B) kext (V ) is higher and w (B) w (V )supported by the World Premier International Research Center
Initiative (WPI Initiative), MEXT, Japan.

is lower for given w (B).
4. The important optical property of dust as a criterion for
RVm < RVs (or RVm > RVs ) is the ratio of the albedos in the
B and V bands, but not the absolute values of the albedos.
This criterion applies irrespective of the scattering
angular distributions.
5. The ratio kext (B) kext (V ) is higher in the dust models
that include more silicate grains or more small-sizedgrains. On the other hand, w (B) w (V ) is lower in the
dust models that include PAHs.
6. To realize a situation in which RVs ~ 3 and RVm < 2 (as anapplication to the non-standard extinction toward SNeIa), the CS dust must include PAHs.

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Data analyses were carried out in part on the PC cluster at
the Center for Computational Astrophysics, National Astronomical Observatory of Japan. The authors thank Akihiro
Suzuki and participants in the “Workshop SN MAS 2016” forstimulating discussion. This work has been supported by theJapan Society for the Promotion of Science (JSPS), KAKENHI
grants 26800100 (K. Maeda) and 26400223 (T.Nozawa), andby JSPS Open Partnership Bilateral Joint Research Project
between Japan and Chile. The work by K. Maeda is partly

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