Earth Planets Space, 65, 183–192, 2013

Synthesized grain size distribution in the interstellar medium
Hiroyuki Hirashita1 and Takaya Nozawa2
1 Institute of Astronomy and Astrophysics, Academia Sinica, P.O. Box 23-141, Taipei 10617, Taiwan
2 Kavli Institute for the Physics and Mathematics of the Universe, Todai Institutes for Advanced Study, the University of Tokyo,

Kashiwa, Chiba 277-8583, Japan(Received November 7, 2011; Revised March 15, 2012; Accepted March 16, 2012; Online published March 12, 2013)

We examine a synthetic way of constructing the grain size distribution in the interstellar medium (ISM). First,
we formulate a synthetic grain size distribution composed of three grain size distributions processed with thefollowing mechanisms that govern the grain size distribution in the Milky Way: (i) grain growth by accretion andcoagulation in dense clouds, (ii) supernova shock destruction by sputtering in diffuse ISM, and (iii) shattering
driven by turbulence in diffuse ISM. Then, we examine if the observational grain size distribution in the Milky
Way (called MRN) is successfully synthesized or not. We find that the three components actually synthesize the
MRN grain size distribution in the sense that the deficiency of small grains by (i) and (ii) is compensated by
the production of small grains by (iii). The fraction of each contribution to the total grain processing of (i), (ii),and (iii) (i.e., the relative importance of the three contributions to all grain processing mechanisms) is 30–50%,
20–40%, and 10–40%, respectively. We also show that the Milky Way extinction curve is reproduced with thesynthetic grain size distributions.
Key words: Cosmic dust, interstellar medium, grain size distribution, extinction, Milky Way.

1.

Introduction

Dust grains are important in some physical processes in
the interstellar medium (ISM). For example, they dominate the absorption and scattering of the stellar light, affecting the radiative transfer in the ISM. The extinction(absorption + scattering) by dust in the ISM as a function of
wavelength is called the extinction curve (Wickramasinghe,
1967; Hoyle and Wickramasinghe, 1991; Draine, 2003 forreview). Extinction curves are important not only in basic radiative processes in the ISM, but also in interpreting observational data: part of stellar light in a galaxy is
scattered, or absorbed, by dust grains within the galaxy
in a wavelength-dependent way according to the extinction curve. Therefore, to derive the intrinsic stellar spectral energy distribution of a galaxy, we always have to correct for dust extinction by considering the extinction curve(Calzetti, 2001).
Extinction curves generally reflect the grain composition
and the grain size distribution. Mathis et al. (1977, hereafter MRN) show that a mixture of silicate and graphite
dust, as originally proposed by Hoyle and Wickramasinghe(1969), with a grain size distribution (number of grains per
grain radius) proportional to a −3.5 , where a is the grain radius (a ∼ 0.001–0.25 µm), reproduces the Milky Way extinction curve. Pei (1992) shows that the extinction curvesin the Magellanic Clouds are also explained by the same
power-law grain size distribution (i.e., ∝ a −3.5 ) with different abundance ratios between the silicate and graphite. Kim
c The Society of Geomagnetism and Earth, Planetary and Space SciCopyright ences (SGEPSS); The Seismological Society of Japan; The Volcanological Societyof Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB.

doi:10.5047/eps.2012.03.003

et al. (1994) and Weingartner and Draine (2001) have applied a more detailed fit to the Milky Way extinction curvein order to obtain the grain size distribution. Although theirgrain size distributions deviate from the MRN size distribution, the overall trend from small to large grain sizes
roughly follows a power law with an index near to −3.5.
Therefore, the MRN grain size distribution is still valid as afirst approximation of the interstellar grain size distribution
in the Milky Way.
What regulates or determines the grain size distribution?
There are some possible processes that actively and rapidly
modify the grain size distribution. Hellyer (1970) showsthat the collisional fragmentation of dust grains finally leads
to a power law grain size distribution similar to the MRN
size distribution (see also Bishop and Searle, 1983). In fact,
Hirashita and Yan (2009) show that such a fragmentationand disruption process (or shattering) can be driven efficiently by turbulence in the diffuse ISM. However, theyalso show that grain velocities are strongly dependent on
grain size; as a result, the grain size distribution does notconverge to a simple power-law after shattering. Moreover,dust grains are also processed by other mechanisms. Various authors show that the increase of dust mass in the Milky
Way ISM is mainly governed by grain growth through theaccretion of gas phase metals onto the grains (elements
comprised of dust grains are referred to as “metals”) (Dwek,
1998; Inoue, 2003; Zhukovska et al., 2008; Draine, 2009;
Inoue, 2011; Asano et al., 2013). In the dense ISM, coagulation also occurs, making the grain sizes larger (e.g.,
Hirashita and Yan, 2009). In the diffuse ISM phase, interstellar shocks associated with supernova (SN) remnants(simply called SN shocks in this paper) destroy dust grains,
especially small ones, by sputtering (e.g., McKee, 1989).

183

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H. HIRASHITA AND T. NOZAWA: GRAIN SIZE DISTRIBUTION

Shattering also occurs in SN shocks (Jones et al., 1996).
Modeling the evolution of the grain size distribution in
the ISM is a challenging problem because a variety ofprocesses are concerned as mentioned above. Those processes are also related to the multi-phase nature of the ISM.
Liffman and Clayton (1989) calculate the evolution of grain
size distributions by taking into account grain growth andshock destruction. However, their method could not treat
disruptive and coagulative processes (i.e., shattering and coagulation). O’Donnell and Mathis (1997) also model theevolution of grain size distribution in a multi-phase ISM,
taking into account shattering and coagulation, in addition
to the processes considered in Liffman and Clayton (1989).
They use the extinction curve and the depletion of gas-phasemetals as quantities to be compared with observations. Although their models are broadly successful, the fit to the ultraviolet extinction curve is poor, which they attribute to theerrors caused by their adopted optical constants. They also
show that inclusion of molecular clouds in addition to diffuse ISM phases improves the fit to the observed depletion,
but they did not explicitly show the effects of molecular
clouds on the extinction curve. Yamasawa et al. (2011) haverecently calculated the evolution of the grain size distribution in the early stage of galaxy evolution by considering
the ejection of dust from SNe and subsequent destruction in
SN shocks. Since they focus on the early stage, they did notinclude other processes such as grain growth and disruption (shattering), which are important in solar-metallicity
environments such as in the Milky Way (Hirashita and Yan,
2009).
Comparing theoretical grain size distributions with observations is not a trivial procedure. In a line of sight, wealways observe a mixture of grains processed in various
ISM phases. Therefore, a “synthetic” grain size distribution, which is made by summing typical grain size distributions in individual ISM phases with certain weights, isto be compared with observations. In this paper, we first
formulate a synthetic way of reproducing the grain size distribution. Then, we carry out a fitting of synthetic grain size
distributions to the observational grain size distribution, inorder to obtain the relative importance of individual grain
processing mechanisms. We do not model the multi-phase
ISM in detail, but our fitting contains the information on
the weights (i.e., relative importance) of different grain processing mechanisms, which depend on the ISM phase.
This paper is organized as follows. In Section 2, we
explain our synthetic method to reconstruct the grain size
distribution in the ISM. In Section 3, we fit our synthetic
models to the observational grain size distribution in the
Milky Way, and examine if the fitting is successful or not.
In Section 4, after we discuss our results, we calculate theextinction curves to examine if our synthesized grain size
distributions are consistent with the observed extinction ornot. In Section 5, we give our conclusions.

size distributions are explained in Subsection 2.1.
First, the ISM is divided into two parts: one is the part
where the grain processing is occurring (called the “grainprocessing region”), and the other is the area where thegrains already processed in the various grain-processing regions are well mixed (called the “mixing region”). The
mass fractions of the former and the latter regions are, respectively, f proc and 1− f proc . It is reasonable to assume thatthe grain size distribution in the mixing region should be themean grain size distribution in the ISM (Subsection 2.2).
We assume that all grains are spherical with material
density s; thus, the grain mass m is expressed as m =4
πa 3 s. Although coagulation may produce porous grains
3(e.g., Ormel et al., 2009), we neglect the effects of porosityand assume all grains to be compact. Two grain species aretreated in this paper; silicate and graphite. To avoid complexity arising from compound species, we treat these two
species separately. This separate treatment is also practical in this paper as we (and other authors usually) assume
that the observed extinction curve can be fitted with the two
species (Subsection 4.4). Before being processed, the grain
size distribution is assumed to be MRN: a power-law function with power index −r (r = 3.5), and upper and lower
bounds for the grain radii (whose values are determined below) amin and amax , respectively:n MRN (a) =(4 − r )ρda −r4  4−r4−r
π s amax − amin n H
3(1)

for amin ≤ a ≤ amax . If a < amin or a > amax , n MRN (a) =0. The grain size distribution is defined so that n MRN (a) dais the number of grains whose sizes are between a anda + da per hydrogen nucleus. The dust mass density, ρd ,is related to the metallicity Z (the mass fraction of elements
heavier than helium in the ISM) and the hydrogen number
density n H as (Hirashita and Kuo, 2011)  
XmXZ
ρd =(1 − ξ )
nH,
(2)
fX
Z
H 

where m X is the atomic mass of the key element X (X = Sifor silicate and C for graphite), f X is the mass fraction of X
in the dust, ξ is the fraction of element X in the gas phase(i.e., the fraction 1 − ξ is in the dust phase), and (X/H)
is the solar abundance relative to hydrogen in the number
density. The metallicity is assumed to be solar (Z = Z ).
We fix the maximum grain radius as amax = 0.25 µm(MRN). Although the lower bound of the grain size ispoorly determined from the extinction curve (Weingartnerand Draine, 2001), we assume that amin = 0.3 nm, since alarge number of very small grains are indeed necessary to
explain the mid-infrared excess of the dust emission in the
Milky Way (Draine and Li, 2001). For the other parameters,
we follow Hirashita (2012). We assume that 0.75 of Si iscondensed into silicate (i.e., ξ = 0.25) while 0.85 (i.e.,
2. Synthetic Grain Size Distribution
ξ = 0.15) of C is included into graphite. Those values
As explained in Introduction, we “synthesize” the obser- are roughly consistent with the observed depletion (e.g.,vational grain size distribution (here, the MRN size distri- Savage and Sembach, 1996), and reproduce the Milky Way
bution) by summing some representative grain size distri- extinction curve (Subsection 4.4). We adopt the following
butions in various ISM phases. These representative grain abundances for Si and C: (Si/H) = 3.55 × 10−5 andH. HIRASHITA AND T. NOZAWA: GRAIN SIZE DISTRIBUTION(C/H) = 3.63 × 10−4 . We assume that Si occupies amass fraction of 0.166 ( f X = 0.166) in silicate, while C
is the only element composing graphite ( f X = 1). We
adopt s = 3.3 and 2.26 g cm−3 for silicate and graphite,
respectively.
By using the MRN size distribution as the initial condition, we calculate the evolution of grain size distribution by the various processes treated in Subsection 2.1.
In the numerical calculation, the grains outside the radius
range between amin and amax are removed from the calculation (the removed mass fraction is < 1%). The processes considered are (i) “grain growth”—grain growth byaccretion and coagulation in a dense medium, (ii) “shock
destruction”—destruction by sputtering in SN shocks, and(iii) “grain disruption”—grain disruption by shattering ininterstellar turbulence. Shattering in SN shocks (Jones etal., 1996) could be included as a separate component, but inour framework, it is not possible to separately constrain thecontributions from the two shattering mechanisms because
both shattering mechanisms (turbulence and SN shocks) selectively destroy grains with a  0.03 µm and increasesmaller grains, predicting similar grain size distributions.
Thus, we simply assume that the size distribution of shattered grains, whatever the shattering mechanism may be, isrepresented by the one adopted in Subsection 2.1.3.
Although our fitting procedures are based on grain size
distributions, we should keep in mind that observational
constraints on the grain size distribution are mainly obtained by extinction curves. Weingartner and Draine (2001)
performed a detailed fit to the Milky Way extinction curve.
However, the grain size distributions derived by thembroadly follow an MRN-like power law, although there arebumps and dips at some sizes. We will examine the consistency with the extinction curve later in Subsection 4.4.
2.1 Processes considered
2.1.1 Grain growth Grain growth occurs in thedense ISM, especially in molecular clouds, through theaccretion of metals (called accretion) and the sticking ofgrains (called coagulation). The change of grain size distribution by grain growth has been considered in our previous
paper (Hirashita, 2012). The grain size distribution aftergrain growth is denoted as n grow (a, tgrow ), where tgrow is theduration of grain growth. We adopt tgrow = 10 and 30 Myr
based on a typical lifetime of molecular clouds (e.g., Lada etal., 2010). The metallicity in the Milky Way is high enoughto allow the complete depletion of grain-composing materials into dust grains in ∼10 Myr. Thus, the total massesof silicate and graphite become 1.33 (= 1/0.75) and 1.18(= 1/0.85) times as large as the initial values, respectively(recall that the dust mass becomes 1/(1 − ξ ) times as much
if all the dust grains accrete all the gas-phase metals). The
difference in the grain size distribution between tgrow = 10
and 30 Myr is predominantly caused by coagulation rather
than accretion.
2.1.2 Shock destruction We calculate the change ofgrain size distribution by SN shock destruction in a medium
swept by an SN shock, following Nozawa et al. (2006).
All SN explosions are represented by an explosion of a star
which has a mass of 20 M at the zero-age main sequence,and the SN explosion energy is assumed to be 1051 erg. For

185

the ISM, we adopt a hydrogen number density of 0.3 cm−3
(since the destruction is predominant in the diffuse ISM;
McKee, 1989), and solar metallicity. The calculation ofgrain destruction is performed until the shock velocity is
decelerated down to 100 km s−1 (8 × 104 yr after the explosion). We apply the material properties of Mg2 SiO4 andcarbonaceous dust in Nozawa et al. (2006) for silicate andgraphite, respectively. We denote the grain size distribution
after shock destruction by n shock (a). The destroyed massfractions of silicate and graphite are 0.38 and 0.27, respectively.
2.1.3 Disruption Grain motions driven by interstellar
turbulence lead to grain disruption (shattering) in the diffuse
ISM (Yan et al., 2004; Hirashita and Yan, 2009). Amongthe various ISM phases, dust grains can acquire the largest
velocity dispersion in a warm ionized medium (WIM). We
recalculated the results of earlier workers based on our assumed initial conditions. We adopt the same grain velocity dispersions and hydrogen number density (n H = 0.1
cm−3 ) in the WIM as adopted in Hirashita and Yan (2009).
The fragments are assumed to follow a power-law size distribution with a power index of −3.3 (Jones et al., 1996;
Hirashita and Yan, 2009). We denote the grain size distribution after disruption as n disr (a, tdisr ), where tdisr is theduration of shattering in the WIM. The lifetime of WIM is
estimated to be a few Myr from the recombination timescale
and the lifetime of ionizing stars (Hirashita and Yan, 2009).
Thus, we adopt tdisr = 3 and 10 Myr for our calculation incausing moderate and significant disruption.
2.2 Synthesizing the grain size distribution
At the beginning of this section, we introduced the massfraction ( f proc ) of ISM hosting grains which are now being
processed (“grain-processing region”). The mean grain size
distribution over all the grain-processing region, n synt (a),can be synthesized with the processed grain size distributions, n grow (a, tgrow ) (grain size distribution after grain
growth with a growth duration of tgrow ), n shock (a) (grain
size distribution after shock destruction), and n disr (a, tdisr )(grain size distribution after disruption with a shattering duration of tdisr ):n synt (a) = f grow n grow (a, tgrow ) + f disr n disr (a, tdisr )+ f shock n shock (a),(3)

where f grow , f disr and f shock are the mass fractions of
medium hosting, respectively, grain growth, disruption, andgrain destruction in the grain-processing region. We calln synt (a) the “synthetic grain size distribution”. If both
species are spatially well mixed, they would have common
values for f grow , f shock , and f disr .
The mean grain size distribution in the ISM is denoted asn mean (a) and expressed asn mean (a) = (1 − f proc )n mean (a) + f proc n synt (a),(4)

since it is assumed that the grain size distribution in the mixing region has already become the mean grain size distribution. By assumption, the mean size distribution is MRN:n mean (a) = n MRN (a). This condition is equivalent to
n synt (a) = n MRN (a).

(5)

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H. HIRASHITA AND T. NOZAWA: GRAIN SIZE DISTRIBUTION

In the Milky Way ISM, since the grain mass is roughly
in equilibrium between the growth in clouds and the destruction by SN shocks (Inoue, 2011), we apply f grow R1 =f shock R2 , where R1 is the fraction of dust mass growth inclouds (0.33 and 0.18 for silicate and graphite, respectively;
Subsection 2.1.1), and R2 is the destroyed fraction of dust
in a SN blast (0.38 and 0.27 for silicate and graphite, respectively; Subsection 2.1.2). Thus, we put a constraint,
f shock = (R1 /R2 ) f grow .

(6)

We approximately adopt R1 /R2 = 0.8 as a mean value
between silicate and graphite. As mentioned above, if thetwo species (silicate and graphite) are spatially well-mixed,
both species would have common values for f grow , f shock ,and f disr . Thus, we adopt a single value for R1 /R2 .
Using the above constraints, Eq. (3) is reduced ton synt (a) = f grow n g,s (a) + f disr n disr (a, tdisr ),(7)

where n g,s (a) ≡ n grow (a, tgrow ) + (R1 /R2 )n shock (a). Thus,
we treat f grow and f disr as free parameters. We define thesum of all the fractions as
f tot ≡ f grow + f shock + f disrR1f grow + f disr .
= 1+
R2

(8)

If the grain size distribution is predominantly modified by
the three processes considered in this paper, we expect thatf tot = 1. The deviation of f tot from 1 is an indicator of
goodness of our assumption that the grain size distribution
is modified by the three processes.
2.3 Best fitting parameters
We search for a set of parameters, ( f grow , f disr ), which
minimizes the square of the difference:
δ2 =

N
1 
[log n synt (ai ) − log n MRN (ai )]2 ,
N i=1

(9)

where ai is the grain size sampled by logarithmic bins (i.e.,
log ai+1 − log ai is the same for any i), and N is the numberof the sampled grain radii (N = 512 in our model, but theresults are insensitive to N ).
The individual components of the processed grain size
distributions [n grow (a, tgrow ), n shock (a), and n disr (a, tdisr )] as
well as n g,s (a) for silicate and graphite are shown in Fig. 1.
The MRN size distribution, which should be fitted, is alsopresented. Our fitting procedure is first applied separately
for silicate and graphite, although we discuss a possibility
that both species have common values for ( f grow , f disr ) laterin Section 4.

3.

Results

In Table 1, we show the best-fitting values of f grow andf disr . We examine tgrow = 30 and 10 Myr, and tdisr = 3
and 10 Myr as mentioned in Subsection 2.1. We observethat f grow = 0.16–0.55 and f disr = 0.06–0.57 fit the MRN
grain size distribution. The sum of all the fractions, f tot(Eq. (8)) is unity with the maximum difference of 15% (seethe column of f tot in Table 1).

In order to show how the synthetic grain size distributions
reproduce the MRN size distribution, we present Fig. 2,
where we only show Models A and D for the smallest andthe largest residuals δ 2 . We observe that the best-fitting results are fairly consistent with the MRN size distribution. In
particular, the enhanced and depleted abundances of small
grains at a  0.001 µm in n disr and n g,s , respectively, cancel out very well, especially in Model A. In Model D, thesynthesized size distribution slightly fails to fit the MRN
around a ∼ 0.001–0.002 µm because both n g,s and n disr
(which are used for the fitting) show an excess around thisgrain radius range; thus, the excess around these sizes inthe synthesized grain size distribution inevitably remains inModel D. However, the Milky Way extinction curve is reproduced even by Model D within a difference of ∼10%(see Subsection 4.4).
In order to see the details of the fitting, we show theratio between the synthesized grain size distribution and
the MRN distribution in Fig. 3. There is a general trendof excess around a ∼ 0.001–0.003 µm, which is due tograin growth (see Fig. 1). The excess is stronger in Models
B and D than in Models A and C, which is why the fit isworse in Models B and D than Models A and C (Table 1).
Since the bump comes from grain growth, the fit tends tosuppress f grow in the presence of a strong bump. As a result,f tot is smaller in Models B and D than Models A and C(Table 1). We also observe in Fig. 3 that the bump appears at
different grain radii between Models A/C and B/D becauseof the difference in the duration of grain growth. Thisbump may disappear if coagulation is more efficient thanassumed here: more efficient coagulation may be realized
if tgrow
30 Myr and/or coagulation also occurs in denser
regions (Subsection 4.2).
Figure 3 also indicates that the synthetic grain size distributions tend to be deficient at the largest grain sizes(a  0.1 µm). This is because shattering tends to process
large grains into small sizes (see Fig. 1). The deficiencyof large grains may be overcome if we include the supplyof large grains by stellar sources of efficient coagulation as
discussed in Subsection 4.2. Because of significant grain
growth in Models A and C, the deficiency of large grains isrecovered by grain growth at a ∼ 0.01–0.03 µm for silicate.
In Models A and C of graphite, shattering causes a dip feature around a ∼ 0.03 µm as seen in Fig. 1, which also appears in Fig. 3. In the WIM, where shattering is assumed tooccur in this paper, grains with a  a few×10−2 µm are accelerated up to velocities larger than the shattering thresholdby turbulence. Shattering efficiently destroys small grains
because of their large surface-to-volume ratios. Thus, theshattering efficiency is the largest for the smallest grainsthat attain a velocity above the shattering threshold. This is
the reason why the grains around a ∼ 0.03 µm are particularly destroyed by shattering. This dip feature would besmoothed out in reality since the grain velocity driven by
turbulence has a dependence on grain charge, gas density,
magnetic field, etc., all of which have a wide range within agalaxy.

H. HIRASHITA AND T. NOZAWA: GRAIN SIZE DISTRIBUTION

187

Fig. 1. Individual components for synthesized grain size distributions. The thin solid, dashed, and dot-dashed lines represent individual components
processed by disruption (shattering) for 3 Myr in Panels (a) and (b), and for 10 Myr in Panels (c) and (d), growth for 10 Myr in Panels (a) and (b) andfor 30 Myr in Panels (c) and (d), and shock, respectively. The thick solid line shows n g,s (a) = n grow (a, tgrow ) + 0.8n shock (a). The dotted line shows
the MRN size distribution adopted in this paper. Panels (a) and (c) present silicate while Panels (b) and (d) show graphite.

4.

Discussion

4.1 Derived parameters
The obtained values of the parameters f grow and f disrreflect the fraction of individual grain processing mechanisms. In other words, these two quantities show the relative importance of grain growth and disruption. Note thatthe efficiency of shock destruction is automatically constrained by the balance with the mass growth by grain
growth (Eq. (6)). Table 1 shows that the best-fitting parameters are not very sensitive to tgrow (duration of grain growth)
but that they are sensitive to tdisr . For larger tdisr , only asmaller f disr is necessary because the grain size distribution
is more modified. As expected, f disr tdisr is less sensitive totdisr ; note that f disr tdisr is the mean duration of disruption perprocessed grain.

Seeing all the models, we find f grow ∼ 0.2–0.6 andf disr ∼ 0.06–0.6 (or f disr tdisr ∼ 0.6–1.7 Myr). As mentioned in Subsection 2.2, if silicate and graphite are well
mixed in the ISM, they are expected to have the common
values for f grow , f shock , and f disr . In this sense, Models
C and D work better than Models A and B. In summary,
20–60% of processing occurs in dense clouds (i.e., grain
growth), while a processed dust grain experiences disruption for ∼1 Myr on average (or disruption accounts for 6–
60% of processing). From the equilibrium constraint of thetotal dust mass (Eq. (6)), the fraction of shock destruction
to all the processing is f shock = 0.8 f grow ∼ 0.1–0.4.
The sum of all the fractions, f tot (Eq. (8)), is unity, witha maximum deviation of 15%. In other words, we cannot
reject other processing mechanisms, which could contribute

188

H. HIRASHITA AND T. NOZAWA: GRAIN SIZE DISTRIBUTION
Table 1. Models.
Name
Aspecies

tgrow
(Myr)
30

silicategraphiteBsilicate
10
graphiteCsilicate
30
graphiteDsilicate
10
graphite
Note: f shock = 0.8 f grow from Eq. (6).

tdisr
(Myr)
3
3
10
10

f growf disr

0.26
0.43
0.16
0.40
0.42
0.55
0.40
0.50

0.46
0.24
0.57
0.20
0.16
0.073
0.15
0.057

δ2
(10−3 )
3.7
4.2
16
13
6.4
7.3
33
15

f tot
0.93
1.01
0.86
0.92
0.92
1.06
0.88
0.96

Fig. 2. Best-fitting synthetic grain size distributions to the MRN size distribution. Panels (a) and (b) show silicate and graphite, respectively. We only
show two models (A and D; solid and dot-dashed lines, respectively) for the smallest and largest residuals (δ 2 ) among the four models. The dottedline shows the MRN size distribution.

to the grain processing with 15%.
We have shown that the grain size distributions after (i)grain growth, (ii) shock destruction, and (iii) grain disruption can synthesize the MRN size distribution. It is also
likely that we can state the opposite; that is, to realize the
MRN size distribution, those three processes are crucial.
Without (i), the grain mass just decreases; without (ii), thegrain mass just increases; without (iii), there is no mechanism that produces the large abundance of small grains.
4.2 Stellar sources?
In this paper, we have not considered dust supply fromstars, because the dust mass in the Milky Way is governed by the equilibrium between grain growth in molecular clouds and grain destruction by SN shocks (e.g. Inoue,2011). Dust grains supplied from stars may be biased
to large sizes. Production of large grains from asymptotic giant branch (AGB) stars is indicated observationally(Groenewegen, 1997; Gauger et al., 1998; Höfner, 2008;
Mattsson and Höfner, 2011). The dust ejected from SNe
is also biased to large grain sizes, because small grains are
selectively destroyed by the shocked region within the SNe(Bianchi and Schneider, 2007; Nozawa et al., 2007). Coagulation associated with star formation is also a source oflarge grains, if coagulated grains in circumstellar environments are somehow ejected into the ISM. For this possibility, Hirashita and Omukai (2009) have shown that dust
grains can grow up to micron sizes by coagulation in starformation (see also Ormel et al., 2009). As mentioned in
Section 3, efficient coagulation may also solve the bumpproblem around a ∼ 0.001–0.003 µm. These possible
sources of large grains may be worth including in dust evolution models in the future.
4.3 Fitting under other constraints
In Subsection 2.2, we adopted the balance between the
dust mass growth by accretion and the dust mass loss byshock destruction (Eq. (6)) as a constraint. Although thisconstraint is reasonable for the dust content in the Milky
Way (e.g., Inoue, 2011), it may be useful to apply otherconstraints without using Eq. (6), to see how the best-fitting
parameters have been controlled by Eq. (6).
First we try to fit the MRN size distribution with thethree components under the condition that the sum of all theH. HIRASHITA AND T. NOZAWA: GRAIN SIZE DISTRIBUTION

189

Fig. 3. The ratio of the synthetic grain size distribution to the MRN size distribution. Panels (a) and (b) show silicate and graphite, respectively. The
solid, dotted, dashed, and dot-dashed lines represent Models A, B, C, and D, respectively.
Table 2. Models with f tot = 1.
Name

species

tgrow
(Myr)
30

A

tdisr
(Myr)
3

silicategraphiteBsilicate
10
graphiteCsilicate
30
graphiteDsilicate
10
graphite
Note: f shock = 1 − f grow − f shock .

3
10
10

f growf disr

0.19
0.32
0.065
0.19
0.34
0.53
0.22
0.33

0.45
0.24
0.43
0.22
0.16
0.074
0.14
0.060

δ2
(10−3 )
2.1
3.5
6.8
6.2
4.5
7.8
20
10

f tot
1
1
1
1
1
1
1
1

Table 3. Models with three free parameters.
NamespeciesAsilicategraphitesilicategraphitesilicategraphitesilicategraphite

B
C
D

tgrow
(Myr)
30

tdisr
(Myr)
3

10

3

30

10

10

10

fractions is unity: f grow + f shock + f disr = 1. The results withthis fitting are shown in Table 2. The best-fitting values off disr vary from those in Table 1 within a difference of 10%
except for Model B of silicate (25% less). However, thebest-fitting values of f grow is broadly 1/2–2/3 of those in
Table 1. As a result, Eq. (6) is not satisfied, and the total
dust mass decreases.
Next, we perform fitting to the MRN size distribution
with the parameters f grow , f shock , and f disr free. The results are shown in Table 3. Again, the values of f disr differ
by only 10% except for Model B of silicate (18% less).
However, f grow is only ∼1/3–2/3 of the values in Table 1,
and f shock is made large to compensate for the decreasedf grow

f shockf disr

0.13
0.20
0.038
0.11
0.27
0.42
0.16
0.24

0.62
0.76
0.83
0.92
0.80
0.69
1
0.99

0.45
0.25
0.47
0.23
0.16
0.074
0.15
0.061

δ2
(10−3 )
0.98
1.8
2.4
2.1
3.0
6.4
10
5.8

f tot
1.2
1.2
1.3
1.3
1.2
1.2
1.3
1.3

f grow . This means that the fitting is practically dominated by
the balance between the decreased small grains in shock destruction and the increased small grains in disruption (shattering). Because of the dominance of f shock , Eq. (6) is notsatisfied, and the total dust mass decreases.
4.4 Extinction curve
The MRN grain size distribution is originally derived
from the Milky Way extinction curve. Therefore, in orderto check if our fitting by synthetic grain size distributions is
successful or not, it is useful to calculate extinction curves.
Extinction curves are calculated by using the same optical properties of silicate and graphite as those in Hirashita
and Yan (2009). The grain extinction cross-section as a

190

H. HIRASHITA AND T. NOZAWA: GRAIN SIZE DISTRIBUTION

Fig. 5. Extinction curves (extinction per hydrogen nucleus as a functionof wavelength) calculated for Models A–D (upper panel). The ratio tothe extinction curve for the MRN size distribution is also shown (lower
panel). The solid, thick dotted, dashed, and dot-dashed lines represent
Models A, B, C, and D, respectively. The thin dotted line shows theextinction curve for the MRN size distribution. The points show theobserved Milky Way extinction curve taken from Pei (1992).

Fig. 4. Upper panel: Extinction curves (extinction per hydrogen nucleus
as a function of wavelength) calculated for the components used for thefitting to the grain size distribution. These components are shown in
Fig. 1. The thin solid, dashed, and dot-dashed lines represent individualcomponents processed for the following processes: disruption (shattering) for 3 Myr in Panel (a) and for 10 Myr in Panel (b), growth for 10
Myr in Panel (a) and for 30 Myr in Panel (b), and shock, respectively.
The thick solid line shows the extinction curve for n g,s (a)/1.8 (divided
by 1.8 because the component “g,s” contains a grain mass 1.8 times asmuch as the MRN). The dotted line presents the extinction curve for the
MRN size distribution. The points show the observed Milky Way extinction curve taken from Pei (1992). Lower panel: Ratio of extinction
curves to the extinction curve of the MRN size distribution. The line
species in the lower panel correspond to those in the upper panel.

function of wavelength and grain size is derived from theMie theory, and is weighted for the grain size distribution
per hydrogen nucleus to obtain the extinction curve per unithydrogen nucleus (denoted as Aλ /NH ). The abundances ofsilicate and graphite relative to hydrogen nuclei are alreadyinherent in the models through the abundances of Si and C
and ξ (Section 2).
First, we show the extinction curves of the individualcomponents, which are used to fit the MRN size distribution, in Fig. 4. Grain growth does not make the extinction
curve flatter in spite of the increase of the mean grain size.
The reason is already explained in Hirashita (2012): accretion predominantly occurs at the smallest sizes. Since theextinction at short wavelengths is more sensitive to the in-Fig. 6. Same as Fig. 5 but we use the mean values between silicate andgraphite for f grow and f disr .

crease of the mass of small grains than that at long wavelengths, the extinction curve becomes rather steeper. Although coagulation flattens the extinction curve, the flattening due to coagulation does not overwhelm the steepening
due to the above effect of accretion.
Shock destruction makes the extinction curve flatter because small grains are more easily destroyed than large
grains. Grain disruption steepens the extinction curve because of the production of a large number of small grains.
The 0.22 µm bump created by small graphite grains in thismodel becomes also prominent by grain disruption. We
also show the extinction curve for the grain size distribution n g,s (a) (The component n g,s has a total dust mass 1.8
times as large as the initial value. To see the difference inthe extinction curve, it would be helpful to make a comparison under the same dust mass, so n g,s /1.8 is compared with
the MRN in Fig. 4.)

H. HIRASHITA AND T. NOZAWA: GRAIN SIZE DISTRIBUTIONIn Fig. 5, we show the extinction curves calculated forModels A–D. First of all, we confirm that the MRN size
distribution reproduces the observed extinction curve (somesmall deviations can be fitted further if we adopt a more detailed functional form of the grain size distribution, which isbeyond the scope of this paper; see Weingartner and Draine,
2001, for a detailed fitting). Comparing the extinction curvefor the MRN size distribution and those for Models A–D,
we observe that the extinction curve is reproduced within adifference of ∼10%. Models A and C are successful, while
Models B and D systematically underproduce the MRN extinction curve (although the difference is small). The underprediction by ∼10% in Models B and D occurs because f tot
is ∼0.9.
In the above, we adopted different values for f grow andf disr between silicate and graphite. As mentioned in Subsection 2.2, if both species are well mixed in the ISM, theywould have common values of these parameters. In Fig. 6,
we show the extinction curve by taking the average of thevalues for silicate and graphite (for example, f grow = 0.35
and f disr = 0.35 for Model A). We find that the difference between Figs. 5 and 6 is small. Therefore, the meanvalues work to reproduce the Milky Way extinction curves.
The mean values are in the range of f grow = 0.3–0.5 andf disr = 0.1–0.4. We conclude that the synthetic grain
size distributions with these parameter ranges reproduce theMilky Way extinction curve.

5.

Conclusion

In our previous papers (Nozawa et al., 2006; Hirashita
and Yan, 2009; Hirashita, 2012), we showed that dust grains
are quickly processed by shock destruction, disruption, andgrain growth. In this paper, we have examined if the MRN
grain size distribution, which is believed to represent thegrain size distribution in the Milky Way, can be reproducedby the processed grain size distributions. We “synthesized”
the grain size distribution by summing the processed grain
size distributions under the condition that the decrease of
dust mass by shock destruction is compensated by grain
growth. We have found that the synthetic grain size distribution can reproduce the MRN grain size distribution in thesense that the deficiency of small grains by grain growth andshock destruction can be compensated by the production ofsmall grains by disruption. The values of the fitting parameters indicate that, among the processed grains, 30–50% are
growing in a dense medium, 20–40% are being destroyedby shocks in a diffuse medium, and 10–40% are being shattered in a diffuse medium (the percentage shows the relative
importance of each process). The extinction curves calculated by the synthesized grain size distributions reproduce
the observed Milky Way extinction curve within a difference of ∼10%. This means that our approach of synthesizing the grain size distribution based on major processing
mechanisms (i.e., grain growth, shock destruction, and disruption) is promising as a general method to “reconstruct”
the extinction curve.
Acknowledgments. We are grateful to C. Wickramasinghe andan anonymous reviewer for helpful comments in their review ofthis paper. HH has been supported through NSC grant 99-2112-

191

M-001-006-MY3. TN has been supported by a World PremierInternational Research Center Initiative (WPI Initiative), MEXT,
Japan, and by the Grant-in-Aid for Scientific Research of theJapan Society for the Promotion of Science (20340038, 22684004,
23224004).

References
Asano, R. S., T. T. Takeuchi, H. Hirashita, and A. K. Inoue, Dust formation
history of galaxies: A critical role of metallicity for the dust mass growthby accreting materials in the interstellar medium, Earth Planets Space,
65, this issue, 213–222, 2013.
Bianchi, S. and R. Schneider, Dust formation and survival in supernova
ejecta, Mon. Not. R. Astron. Soc., 378, 973–982, 2007.
Bishop, J. E. L. and T. M. Searle, Power-law asymptotic mass distributions
for systems of accreting or fragmenting bodies, Mon. Not. R. Astron.
Soc., 203, 987–1009, 1983.
Calzetti, D., The dust opacity of star-forming galaxies, Publ. Astron. Soc.
Pac., 113, 1449–1485, 2001.
Draine, B. T., Interstellar dust grains, Ann. Rev. Astron. Astrophys., 41,
241–289, 2003.
Draine, B. T., Interstellar dust models and evolutionary implications, incosmic dust—near and far, in ASP Conference Series 414, edited by Th.
Henning, E. Grün, and J. Steinacker, 453–472, Astromomical Societyof the Pacific, San Francisco, 2009
Draine, B. T. and A. Li, Infrared emission from interstellar dust. I. Stochastic heating of small grains, Astrophys. J., 551, 807–824, 2001.
Dwek, E., The evolution of the elemental abundances in the gas and dust
phases of the galaxy, Astrophys. J., 501, 643–665, 1998.
Gauger, A., Y. Y. Balega, P. Irrgang, R. Osterbart, and G. Weigelt, Highresolution speckle masking interferometry and radiative transfer modeling of the oxygen-rich AGB star AFGL 2290, Astron. Astrophys., 346,
505–519, 1998.
Groenewegen, M. A. T., IRC +10 216 revisited. I. The circumstellar dust
shell, Astron. Astrophys., 317, 503–520, 1997.
Hellyer, B., The fragmentation of the asteroids, Mon. Not. R. Astron. Soc.,
148, 383–390, 1970.
Hirashita, H., Dust growth in the interstellar medium: How do accretionand coagulation interplay?, Mon. Not. R. Astron. Soc., 2012 (in press).
Hirashita, H. and T.-M. Kuo, Effects of grain size distribution on theinterstellar dust mass growth, Mon. Not. R. Astron. Soc., 416, 1795–
1801, 2011.
Hirashita, H. and K. Omukai, Dust coagulation in star formation withdifferent metallicities, Mon. Not. R. Astron. Soc., 399, 1795–1801, 2009.
Hirashita, H. and H. Yan, Shattering and coagulation of dust grains in interstellar turbulence, Mon. Not. R. Astron. Soc., 394, 1061–1074, 2009.
Höfner, S., Winds of M-type AGB stars driven by micron-sized grains,
Astron. Astrophys., 491, L1–L4, 2008.
Hoyle, F. and N. C. Wickramasinghe, Interstellar grains, Nature, 223, 450–
462, 1969.
Hoyle, F. and N. C. Wickramasinghe, The Theory of Cosmic Grains,
Kluwer Academic Publishers, Dordrecht, 1991.
Inoue, A. K., Evolution of dust-to-metal ratio in galaxies, Publ. Astron.
Soc. Jpn., 55, 901–909, 2003.
Inoue, A. K., The origin of dust in galaxies, Earth Planets Space, 63, 1027–
1039, 2011.
Jones, A., A. G. G. M. Tielens, and D. Hollenbach, Grain shattering inshocks: The interstellar grain size distribution, Astrophys. J., 469, 740–
764, 1996.
Kim, S.-H., P. G. Martin, and P. D. Hendry, The size distribution of interstellar dust particles as determined from extinction, Astrophys. J., 422,
164–175, 1994.
Lada, C. J., M. Lombardi, and J. F. Alves, On the star formation rates inmolecular clouds, Astrophys. J., 724, 687–693, 2010.
Liffman, K. and D. D. Clayton, Stochastic evolution of refractory interstellar dust during the chemical evolution of a two-phase interstellar
medium, Astrophys. J., 340, 853–868, 1989.
Mathis, J. S., W. Rumpl, and K. H. Nordsieck, The size distribution ofinterstellar grains, Astrophys. J., 217, 425–433, 1977 (MRN).
Mattsson, L. and S. Höfner, Dust-driven mass loss from carbon stars as afunction of stellar parameters. II. Effects of grain size on wind properties, Astron. Astrophys., 533, A42, 2011.
McKee, C. F., Dust destruction in the interstellar medium, in Interstellar
Dust, Proc. of IAU 135, edited by L. J. Allamandola and A. G. G. M.
Tielens, 431–443, Kluwer, Dordrecht, 1989.
Nozawa, T., T. Kozasa, and A. Habe, Dust destruction in the high-velocity

192

H. HIRASHITA AND T. NOZAWA: GRAIN SIZE DISTRIBUTION

shocks driven by supernovae in the early universe, Astrophys. J., 648,
435–451, 2006.
Nozawa, T., T. Kozasa, A. Habe, E. Dwek, H. Umeda, N. Tominaga,
K. Maeda, and K. Nomoto, Evolution of dust in primordial supernova
remnants: Can dust grains formed in the ejecta survive and be injected
into the early interstellar medium?, Astrophys. J., 666, 955–966, 2007.
O’Donnell, J. E. and J. S. Mathis, Dust grain size distributions and theabundance of refractory elements in the diffuse interstellar medium,
Astrophys. J., 479, 806–817, 1997.
Ormel, C. W., D. Paszun, C. Dominik, and A. G. G. M. Tielens, Dust coagulation and fragmentation in molecular clouds. I. How collisions between dust aggregates alter the dust size distribution, Astron. Astrophys.,502, 845–869, 2009.
Pei, Y. C., Interstellar dust from the Milky Way to the Magellanic Clouds,
Astrophys. J., 395, 130–139, 1992.
Savage, B. D. and K. R. Sembach, Interstellar abundances fromabsorption-line observations with the Hubble Space Telescope, Ann.
Rev. Astron. Astrophys., 34, 279–330, 1996.

Weingartner, J. C. and B. T. Draine, Dust grain-size distributions and extinction in the Milky Way, Large Magellanic Cloud, and Small Magellanic Cloud, Astrophys. J., 548, 296–309, 2001.
Wickramasinghe, N. C., Interstellar grains, in The International Astrophysics Series, Chapman & Hall, London, 1967.
Yamasawa, D., A. Habe, T. Kozasa, T. Nozawa, H. Hirashita, H. Umeda,
and K. Nomoto, The role of dust in the early universe. I. Protogalaxy
evolution, Astrophys. J., 735, 44, 2011.
Yan, H., A. Lazarian, and B. T. Draine, Dust dynamics in compressible
magnetohydrodynamic turbulence, Astrophys. J., 616, 895–911, 2004.
Zhukovska, S., H.-P. Gail, and M. Trieloff, Evolution of interstellar dust
and stardust in the solar neighbourhood, Astron. Astrophys., 479, 453–
480, 2008.
H. Hirashita (e-mail: hirashita@asiaa.sinica.edu.tw) and T. Nozawa