Mon. Not. R. Astron. Soc. 404, 1437–1448 (2010)

doi:10.1111/j.1365-2966.2010.16354.x

Effects of grain shattering by turbulence on extinction curves in starburst
galaxies
Hiroyuki Hirashita,1 Takaya Nozawa,2 Huirong Yan3 and Takashi Kozasa4
1 Institute of Astronomy and Astrophysics, Academia Sinica, PO Box 23-141, Taipei 10617, Taiwan
2 Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa 277-8568, Japan
3 University of Arizona, LPL, Steward Observatory and Physics Department, 933 N Cherry Avenue, Tucson, AZ 85721, USA
4 Department of Cosmosciences, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan

Accepted 2010 January 13. Received 2010 January 10; in original form 2009 October 11

ABSTRACTDust grains can be efficiently accelerated and shattered in a warm ionized medium (WIM)
because of the turbulent motion. This effect is enhanced in starburst galaxies, where gas is
ionized and turbulence is sustained by massive stars. Moreover, dust production by Type IIsupernovae (SNe II) can be efficient in starburst galaxies. In this paper, we examine the effectof shattering in a WIM on the dust grains produced by SNe II. We find that, although thegrains ejected from SNe II are expected to be biased to large sizes (a  0.1 μm, where a is
the grain radius), because of the shock destruction in supernova remnants the shattering in a
WIM is efficient enough in ∼5 Myr to produce small grains if the metallicity is nearly solar
or more. The production of small grains by shattering steepens the extinction curve. Thus, thesteepening of extinction curves by shattering should always be taken into account for systems
in which the metallicity is solar and the starburst age is typically greater than 5 Myr. These
conditions may be satisfied not only in nearby starburst galaxies but also in high-redshift(z > 5) quasars.
Key words: turbulence – supernovae: general – dust, extinction – H II regions – galaxies:
evolution – galaxies: starburst.

1 I N T RO D U C T I O NType II supernovae (SNe II) are considered to be a source of grain
production (e.g. Kozasa, Hasegawa & Nomoto 1989; Todini &Ferrara 2001; Nozawa et al. 2003). The significance of SNe II in thisrespect is enhanced if the cosmic age is young enough (typicallyat redshift z > 5) that low-mass stars, namely asymptotic giant
branch (AGB) stars and Type Ia supernovae, cannot contribute significantly to the dust formation (Dwek, Galliano & Jones 2007; but
see Valiante et al. 2009), or if the current starburst activity is strong
enough. For some nearby blue compact dwarf galaxies (BCDs), thelatter condition may be satisfied (Hirashita & Hunt 2004; Takeuchi
et al. 2005). Thus, the dust production by SNe II is tested for high-z
objects and nearby starburst galaxies.
In galaxies where active star formation (starburst) is occurring,
dust grains are produced in and ejected from stars and then processedin the interstellar medium (ISM). In such star-forming galaxies, it isexpected that the supernova (SN) rate is enhanced, which leads to an
efficient destruction of large grains with a  0.1 μm, where a is thegrain radius, by SN shocks (e.g. Jones et al. 1994; Jones, Tielens & E-mail: hirashita@asiaa.sinica.edu.twC 2010 The Authors. Journal compilation 
C 2010 RAS

Hollenbach 1996). Furthermore, large amounts of ionizing photons
are supplied and H II regions develop. Indeed, giant H II regions withsizes of 100 pc are found in nearby galaxies (Kennicutt 1984).
Such large H II regions are also theoretically expected to occur ina starburst region (Hirashita & Hunt 2004). Moreover, their size islarger than the typical expansion radius of SN shells when the dust
condensed in SNe II is finally supplied to the ISM (hereafter N07
Nozawa et al. 2007). Thus, it is reasonable to assume that the dust
ejected from SNe II is supplied to the ionized regions in starburst
galaxies.
Hirashita & Yan (2009, hereafter HY09) show that shattering
occurs efficiently in a warm ionized medium (WIM), where grainsare efficiently accelerated by magnetohydrodynamic (MHD) turbulence (Yan & Lazarian 2003; Yan, Lazarian & Draine 2004). The
same mechanism is also expected to work in the ionized regions of
starburst galaxies. Indeed, turbulence is ubiquitous in the ISM, andthe collective effects of OB stellar winds and supernovae (SNe) canplay an important role in sustaining turbulence (e.g. Elmegreen &Scalo 2004). Therefore, it is probable that the grains ejected fromSNe II into ionized regions are efficiently shattered by turbulence.
Because it has been suggested that large grains with a  0.1 μm
are injected into the ISM from SNe II selectively owing to thedestruction of small grains in hot plasma caused by the reverse and1438

H. Hirashita et al.

forward shocks in supernova remnants (SNRs) (N07),1 shattering ina WIM is important for the production of small grains. If a significant
number of small grains are produced, optical–ultraviolet (UV) grain
opacity is enhanced, and the slope of the extinction curve becomessteep. In particular, the extinction curves of young starburst galaxies
are used to constrain the composition and size distribution of grains
formed in SNe II (Maiolino et al. 2004b; Hirashita et al. 2005).
Thus, it is important to quantify the effect of grain shattering on theextinction curve.
The production of small grains in starburst environments is also
worth investigating in the observational context. The modificationof the grain-size distribution should have an impact on dust extinction and emission properties (e.g. Dopita et al. 2005). By studying
the spectral energy distributions (SEDs) of the dust and stars ofsome actively star-forming dwarf galaxies, Galliano et al. (2005)
showed that the grain-size distribution is biased to small sizes (∼ a
few nanometres). The extinction curves of starburst galaxies in general show a significant reddening in the optical–UV range (Calzetti2001), indicating that there should be some contribution from small
grains. Galliano et al. (2005) also suggest that their results are consistent with the shattering and erosion of ISM grains by SN shocks(Jones et al. 1994, 1996; see also Borkowski & Dwek 1995). Bothshock and turbulence are efficient drivers of the relative motion
between grains, as the grain acceleration occurs in a way strongly
dependent on the grain size (Shull 1977; McKee et al. 1987; Joneset al. 1996; Yan et al. 2004).2 At present, however, it is not known
which of the two drivers is more important. Thus, in this paper, asa first necessary step to gaining a full understanding of the possible
mechanisms of small-grain production, we focus on turbulence asa possible driver of grain shattering.
The paper is organized as follows. We explain the method usedin Section 2, and describe some basic results of our calculationsin Section 3. We discuss the results and note some observational
implications in Section 4. Finally, Section 5 gives the conclusion.
2 METHOD
We consider a young starburst, where dust is predominantly supplied
by SNe II. We then calculate the modification of the grain-size
distribution by shattering in the WIM. Finally, we examine whether
or not the shattering effect is apparent in the extinction curve. In
this section, we first explain the initial grain-size distribution in
the ejecta from SNe II (Sections 2.1 and 2.2). Next we review thetreatment of shattering that was adopted from Jones et al. (1994,
1996) in HY09 (Section 2.3). We use the grain velocities calculated
by a MHD turbulence model to obtain the relative grain velocities
in shattering (Section 2.4). We also describe the method used to
calculate the extinction curve (Section 2.5). Throughout this paper,
grains are assumed to be spherical with radius a.
2.1 Initial grain-size distribution
The size distribution of grains ejected from SNe II into the ISM(WIM) is adopted from N07. This size distribution is used as theinitial condition for the calculation of shattering in the WIM. N07
have shown that the size distribution of dust formed in the ejecta is

1 We call this destruction process ‘destruction in SNRs’.
2 Galactic-scale bulk motions such as collective outflow from stellar feed-backs are not efficient at producing the relative motions among grains.

strongly modified by sputtering in ionized gas heated by the reverseand forward shocks. Not only N07 but also Bianchi & Schneider(2007) treat the effect of destruction by shocks in SNe II. N07
consider some aspects that were neglected by Bianchi & Schneider(2007): they solve the motion of dust grains by taking into accountthe gas drag, and treat the destruction of dust in the radiative phaseas well as in the non-radiative phase of SNRs. Thus, we adopt N07’s
results for the size distribution of grains ejected from SNe II into theISM, although the qualitative behaviour of our results are the sameeven if we adopt the size distribution of Bianchi & Schneider (2007).
We also note that both N07 and Bianchi & Schneider (2007) neglect
the effects of dust electrical charge and the effects of magnetic fields
on grain kinematics. These physical processes, which should bequantified in future work, are discussed further in Section 4.4.
Here we briefly summarize the calculation of N07. N07 started
from the grain-size distribution calculated by Nozawa et al. (2003),
who treated dust nucleation and growth in SNe II based on the SN
model of Umeda & Nomoto (2002). Then, N07 took into account the
dust destruction by kinetic and thermal sputtering in hot gas swept
up by the reverse and forward shocks after the interaction of the
SN ejecta with the ambient ISM. Thus, the grain-size distribution
calculated by N07 is regarded as being equal to that injected into
the ISM.
N07 treated two extreme cases for the mixing of elements in a
SN II: the unmixed case, in which the original onion-like structureof elements is preserved; and the mixed case, in which the elements
are uniformly mixed within the helium core. In this paper, we adopt
the unmixed case, as the extinction features of carbon and silicon,
which are the major grain components in the unmixed case, areconsistent with observations (Hirashita et al. 2005; Kawara et al.
2010). Even if the mixed case is adopted, the grain-size distribution
is similarly biased to large grain sizes, so that the behaviour of theextinction curve calculated later is expected to be similar.
As a representative progenitor mass, we adopt 20M , following
N07. The grain species formed are C, Si, SiO2 , Fe, FeS, Al2 O3 ,MgO, MgSiO3 and Mg2 SiO4 . According to their calculation, small
grains with a  0.02 μm are trapped in the shocked region because
the deceleration rate of a grain by the gas drag is inversely proportional to its size (e.g. Nozawa, Kozasa & Habe 2006). Thus,
these small grains are efficiently destroyed by thermal sputtering if the ambient hydrogen number density, nH , is larger than
0.1 cm−3 . Moreover, the destruction efficiency depends sensitively
on nH . With nH as large as 10 cm−3 , only a few per cent of grains
survive, and the grain-size distribution is strongly biased to large(a  0.1 μm) radii. It is interesting to note that Slavin, Jones &Tielens (2004) derived similar grain radii for the dynamical decoupling of grains from the interstellar gas, although they include
magnetic fields in their calculation (see Section 4.4). The following
results on the effects of grain shattering by turbulence in a WIM
are not greatly affected by the details in the hydrodynamical treatment of SNe as long as the initial size distribution is biased to large(a  0.1 μm) grains.
2.2 Normalization of the grain-size distribution
The total grain mass density integrated for all the size range, ρ dust ,is normalized to the gas mass density, 1.4nH mH , to obtain the dustto-gas ratio, D ≡ ρdust /(1.4nH mH ), where mH is the mass of thehydrogen atom and the factor 1.4 is the correction for species otherthan hydrogen. Because dust grains are composed of metals, it
is useful to label the dust abundance in terms of metallicity. We
parametrize the dust abundance by the oxygen abundance, because
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Effects of grain shattering on extinction curves
Table 1. Characteristic quantities for SN II dust production for various ambient densities nH .
nH (cm−3 )md (M )

D0

0.1
1
10

0.35
0.14
0.033

2.2 × 10−3
8.7 × 10−4
2.0 × 10−4Note. md is the dust mass ejected per SN II (a progenitor
mass of 20M is assumed), and D0 is the dust-to-gas
ratio at the solar metallicity (oxygen abundance) ZO .

oxygen is one of the main metals produced by SNe II and oxygen
emission lines in the optical are often used to estimate the gasphase metal abundance. The oxygen mass produced by a SN II of
a 20M progenitor is mO = 1.58 M (Umeda & Nomoto 2002).
The dust mass after the destruction in SNRs (md ) is listed for eachambient density in Table 1. The solar oxygen abundance is assumed
to be ZO = 9.7 × 10−3 in mass ratio (Anders & Grevesse 1989).
(The oxygen abundance is denoted as Z O and is simply called themetallicity in this paper.) Therefore, in the solar metallicity case, we
assume that the dust-to-gas ratio is D0 = ZO md /mO , which is also
listed in Table 1. The dust-to-gas ratio is assumed to be proportional
to the metallicity: D = (ZO /ZO )D0 . This is equivalent to theassumption that both dust and oxygen are predominantly supplied
from SNe II.
2.3 Shattering
Dust grains are subject to shattering if the relative velocity between
grains is larger than 2.7 and 1.2 km s−1 for silicate and graphite,
respectively (Jones et al. 1996). As shown by Yan et al. (2004),
these velocities can be achieved in a magnetized and turbulent ISM.
In particular, grains are efficiently accelerated in a WIM, because
damping is weak (Yan et al. 2004). The parameters adopted for
MHD turbulence and the grain velocities obtained are describedin Section 2.4. The time evolution of the grain-size distribution by
shattering is calculated by adopting the formulation of Jones et al.
(1994, 1996). We briefly review the calculation method. The details
are described in HY09.
We solve the shattering equation discretized for the grain size.
Although nine grain species are treated here (Section 2.1), the material properties needed for the calculation of shattering are notnecessarily available for all species. Thus, we divide the grains into
two groups, namely carbonaceous dust and all the other species ofdust, and apply the relevant material quantities of graphite and silicate, respectively. In fact, as shown later, the mass and opacity of
the latter group are dominated by Si. The validity of this approximation, in which all species other than carbonaceous dust are treated assilicate (called the one-species method), is examined in comparison
with another extreme approximation (individual-species method) in
Appendix A. Because of the lack of experimental data for Si, we
assume that Si (and also the other ‘silicate’ species) can be treatedas silicate in shattering because of a similar hardness.3 Because
3 Among the materials whose shattering properties are available in table 1of Jones et al. (1996) (i.e. silicate, SiC, ice, iron and diamond), silicate is
expected to have the nearest atomic binding energy to Si. As shown by SerraDı́az-Cano & Jones (2008), who derived the previously unknown shattering
properties of hydrogenated amorphous carbon, it is not impossible to estimate relevant quantities, but some guiding quantities from other materials
are still necessary even in their method. Thus, here we simply adopt thematerial quantities of silicate for Si.

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the cratering volume in a grain–grain collision is approximately
proportional to 1/Pcr , where Pcr is the critical shock pressure forshattering (Jones et al. 1996), the shattering time-scale is roughly
proportional to Pcr . Thus, if Pcr is obtained for appropriate materials(especially Si) in some future experiment, our results can easily be
scaled. The material properties of silicate and graphite are takenfrom Jones et al. (1996) as summarized in HY09.
The number density of grains whose radii are between a anda + da is denoted by n(a) da, where the entire range of a is fromamin to amax . To ensure the conservation of the total mass of grains,
it is convenient to consider the distribution function of grain massinstead of that of grain size. We denote the number density ofgrains whose masses are between m and m + dm as ñ(m) dm. The
two distribution functions are related as n(a) da = ñ(m) dm andm = (4π/3)a 3 ρgr , where ρ gr is the grain material density (3.3 and2.2 g cm−3 for silicate and graphite, respectively).
For numerical calculation, we consider N discrete bins for thegrain radius. The grain radius in the ith (i = 1, . . . , N ) bin is(b)(b)
and ai(b) , where ai(b) = ai−1
δ, a0(b) = amin , and aN(b) =between ai−1
amax [i.e. log δ specifies the width of a logarithmic bin: log δ =(1/N ) log(amax /amin )]. We represent the grain radius and mass in(b)+ ai(b) )/2 and mi ≡ (4π/3)ai3 ρgr . The
the ith bin as ai ≡ (ai−1boundary of the mass bin is defined as m(b)≡ (4π/3)[ai(b) ]3 ρgr .
iGiven amin , amax and N, all bins can be set. A grain in the ith
bin is called ‘grain i’. We adopt amin = 3 × 10−4 μm (3 Å) andamax = 3 μm to cover the entire grain size range in N07, andN = 32. We have confirmed that the results are not altered even if
N is doubled.
The mass density of grains contained in the ith bin, ρ̃i , is defined
as


(b)
(1)
ρ̃i ≡ mi ñ(mi ) m(b)i − mi−1 .
Note that ρ̃i = ρi δi in the expression of Jones et al. (1994, 1996).
The time evolution of ρ̃i by shattering can be written as


N 
N
N


dρ̃i
kj
= −mi ρ̃i
αki ρ̃k +αkj ρ̃k ρ̃j mshat (i) ,(2)
dt shatk=1
j =1 k=1⎧⎨ σki vki
mi mk
αki =
⎩0
kj

if vki > vshat ,(3)otherwise,

where mshat (i) is the total mass of the shattered fragments of a grain
k that enters the ith bin in the collision between grains k and j , σki
and vki are, respectively, the grain–grain collision cross-section andthe relative collision speed between grains k and i, and v shat is thevelocity threshold for shattering to occur. For the cross-section, weapply σki = π(ak + ai )2 .
The grain velocities given by Yan et al. (2004) are typical velocity
dispersions. In order to take into account the directional information,
we follow the method in Jones et al. (1994): we divide each timestep into four small steps, and we apply vik = vi + vk , |vi − vk |, vi
and vk in each small step, where vi and vk are the velocities ofgrains i and k, respectively (see Section 2.4). Note that the mass
kj
of the shattered fragment mshat (i) depends on vkj , as described in
Jones et al. (1996). Briefly, the total fragment mass is determined by
the shocked mass in the collision, and the fragments are distributedwith a grain-size distribution ∝ a −αf with αf = 3.3 unless otherwise
stated (see HY09 for the size range of the fragments).
For the shattering duration, several mega-years may be appropriate, as this is a typical lifetime of ionizing stars with mass  20 M
(lifetime <10 Myr) (Bressan et al. 1993; Inoue, Hirashita &
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Kamaya 2000). HY09 also indicate that the small (a  0.01 μm)
grains depleted by coagulation in dense clouds are recovered if shattering in the WIM lasts for 3–5 Myr, which justifies the necessityof shattering in the WIM for mega-years. We also examine a longer
time-scale, 10 Myr, to investigate a starburst environment in which
intense star formation occurs continuously. Such a situation may
be realized in extragalactic giant ionized regions (Hirashita & Hunt
2004; Hunt & Hirashita 2009). In summary, we examine t = 3, 5
and 10 Myr, where t is the elapsed time of shattering.
2.4 Grain velocity
The grain velocity as a function of grain radius a in the presenceof interstellar MHD turbulence is calculated using the method described in Yan et al. (2004). They considered the grain acceleration
by hydrodrag and gyroresonance and calculated the grain velocities
achieved in various phases of the ISM. Among the ISM phases, we
focus on the WIM to investigate the possibility of efficient shatteringin actively star-forming environments.
We adopt three cases for the hydrogen number density of theWIM (nH = 0.1, 1 and 10 cm−3 ), as N07 applied these densities forthe ambient medium. For the WIM, a density of nH ∼ 0.1–1 cm−3
is usually considered (McKee & Ostriker 1977), but we also examine a density as high as nH ∼ 10 cm−3 for young H II regions
around massive stars, as observed in starburst environments (Hunt& Hirashita 2009). Embedded starburst regions may also have suchdense ionized regions. We adopt a gas temperature T = 8000 K, anelectron number density ne = nH , an Alfvén speed VA = 20 km s−1
and an injection scale of the turbulence L = 100 pc, following Yan
et al. (2004). The effect of the injection scale is minor comparedwith that of the sound and Alfvén velocities. Because both the soundspeed and the Alfvén speed are fixed, the plasma β is constant inall cases. The grain charge is assumed to be the same as that in
Yan et al. (2004), who calculated it by assuming a typical Galactic condition. Because we expect a higher value for the interstellar
radiation field and a higher electron density for starburst environments, the absolute values for the grain charge can be larger thanthose assumed here. For grains with a  0.1 μm, where most of
the grain mass is contained in our cases, the grain velocity is governed by the gyroresonance. The acceleration rate of gyroresonanceincreases with the grain charge, but the acceleration duration, thegaseous drag time, decreases with the grain charge (Yan & Lazarian2003). As a result, the acceleration efficiency of gyroresonance isinsensitive to the grain charge.
In Fig. 1, we show the grain velocities. In general, larger grains
tend to acquire larger velocities because they are coupled withlarger-scale motions. For small grains, the motion is governedby the gaseous drag, which has a linear dependence on the graincharge (Yan et al. 2004). This is the reason for the complex (nonmonotonic) behaviour of the grain velocity as a function of a forsmall grains (< 0.1 μm). We also observe that the grain velocity
is not very sensitive to nH for large (a  0.1 μm) grains, whose
shattering is important in this paper.

2.5 Extinction curves
Extinction curves have been an effective tool with which to examine
dust properties (e.g. Mathis 1990). For the calculation of extinction
curves, we adopt the same optical constants as in Hirashita et al.
(2008) for the grain species formed in SNe II (C, Si, SiO2 , Fe,
FeS, Al2 O3 , MgO, MgSiO3 and Mg2 SiO4 ). The grain properties
of individual species and the references for the optical constants
are listed in table 1 of Hirashita et al. (2008). Using these opticalconstants, we calculate the absorption and scattering cross-sectionsof homogeneous spherical grains with various sizes based on the
Mie theory (Bohren & Huffman 1983). Then, the grain extinction
coefficient as a function of wavelength is obtained by weighting thecross-sections with the grain-size distribution. The total extinction
as a function of wavelength λ, denoted as Aλ , is calculated bysumming the contributions from all species.
As stated in Section 2.3, we divide the grain species into twogroups in the calculation of shattering: carbonaceous dust and silicate, which in fact contains all species other than carbonaceousdust. In the calculation of the extinction curves, the size distributionof the silicate species is redistributed to each component (Si, SiO2 ,Fe, FeS, Al2 O3 , MgO, MgSiO3 and Mg2 SiO4 ) in proportion to thegrain volume (i.e. the total mass of each component divided by its
material density) with a fixed shape of the grain-size distribution. In
fact, Si is dominant in the extinction curve, so the uncertainty resulting from the above rough treatment does not affect our conclusion(Appendix A).

Figure 1. Grain velocities v calculated from the turbulence model as a function of grain radius a. Two grain species, (a) silicate and (b) carbonaceous dust, areshown. The solid, dotted and dashed lines indicate the velocities with hydrogen number densities of nH = 0.1, 1 and 10 cm−3 , respectively.
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Figure 2. Grain-size distributions per hydrogen atom. The solid and dotted lines show the results at t = 5 Myr for metallicities of 1ZO and 0.1ZO ,respectively. The hydrogen number density nH is assumed to be 0.1 cm−3 . The dashed line presents the initial grain-size distribution before shattering. Two grain species, (a) silicate and (b) carbonaceous dust, are shown. The case with 0.1ZO is multiplied by 10 for display purposes to offset the
10-times smaller dust abundance. The arrow at a = 0.03 μm is a rough representative size of the grains contributing to the steepening of the UV extinction
curve.

Figure 3. As Fig. 2, but for nH = 1 cm−3 .

3 R E S U LT S
3.1 Grain-size distribution after shattering
The grain-size distributions after shattering are shown in Figs 2–4
for nH = 0.1, 1 and 10 cm−3 , respectively. The grain-size distribution is shown by n(a)/nH . We adopt t = 5 Myr as a typical
time-scale on which the WIM is sustained by the radiation frommassive stars (Section 2.3). Two cases for the metallicity, ZO = 0.1
and 1 ZO , are examined.
We observe that shattering affects the grain-size distribution for
all densities. In particular, the abundance of small grains with a 
0.1 μm significantly increases after the shattering of a small portionof larger grains. If the metallicity is 1ZO , a continuous powerlaw-like size distribution is realized for a  0.1 μm. Although thedust abundance is lower for higher nH (Table 1), because of more
efficient shock destruction in the SN remnant before the ejection
to the ISM (N07), the grain–grain collision rate is enhanced inenvironments with higher nH .
The increase of grains with a  0.1 μm could have a strong
effect on the UV and optical extinction curves. This point is quantitatively addressed in Section 3.2. Large grains with a > 0.1 μm are
marginally affected by shattering; that is, the shattering of a small
fraction of large grains can produce a large number of small grains.
In the case of HY09, on the other hand, grains with a > 0.1 μm are
more shattered because abundant small grains in the MRN (Mathis,
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H. Hirashita et al.

Figure 4. As Fig. 2, but for nH = 10 cm−3 .

Rumpl & Nordsieck 1977) grain-size distribution, which they assumed as the initial condition, enhance the grain–grain collisionrate.
3.2 Extinction curves
The extinction curve of grains ejected from SNe II tends to be flatbecause small grains are efficiently destroyed in SNRs before theycan escape into the ISM (Hirashita et al. 2008). Here we investigate
whether or not the increase of small grains by shattering effectively
steepens the extinction curves.
In Fig. 5, we show the time variation of extinction curves fornH = 0.1, 1 and 10 cm−3 with ZO = 1 ZO . We normalize theextinction to AV (i.e. at λ = 0.55 μm). As stated by Hirashita
et al. (2008), the initial extinction curve is steeper for lower nH ,Figure 5. Extinction curves normalized to the V-band extinction. The solid, dotted and dot–dashed lines indicate the results at t = 3, 5 and 10 Myr, respectively.
Panels (a), (b) and (c) present the results for nH = 0.1, 1 and 10 cm−3 , respectively, with ZO = 1 ZO . Panel (d) shows the result for nH = 1 cm−3 with
ZO = 0.1 ZO .
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Effects of grain shattering on extinction curves

Figure 6. Contributions from Si, C and the other grain species (dotted,
dashed, and dot–dashed lines, respectively) for the case of Fig. 5(b) (nH =1 cm−3 and ZO = 1 ZO ) at t = 10 Myr. The solid line shows the total
extinction.

as more small grains survive after the shock destruction in SNRs.
We also observe that the extinction curve becomes steeper as thegrains are shattered for a longer time because of the production ofsmall grains. Indeed, at t = 5 Myr, Aλ /AV at λ ∼ 0.2 μm increases
by more than 20 per cent for nH = 1 cm−3 . The variation of theslope by shattering is more pronounced for larger nH , as the originalextinction curve is flatter.
The extinction curves are dominated by Si and C, which survive after the shock destruction in SNRs because of their relatively
large sizes (N07). Therefore, the steepening of the extinction curve
results mainly from the production of small Si and C grains by
shattering. The contributions from Si, C and the other species areshown in Fig. 6. The ‘bump’ features at 1/λ ∼ 4 and 7μm−1
originate from the absorption by C and Si, respectively. Such features tend to be prominent for smaller grains, because as grains
become larger the extinction cross-sections are determined moreby the geometrical ones, not by the grain properties (Bohren &Huffman 1983). Thus, not only the steep slope but also various features in the extinction curve become apparent as grains are subject toshattering.
We also examine the dependence on the dust abundance (metallicity). In Fig. 5(d), we show the evolution of the extinction curvefor ZO = 0.1 ZO . The effect of shattering is significantly reduced
compared with the case of solar metallicity. If the grain velocity as afunction of grain size is fixed, a constant ZO t gives the same result.
Thus, if Z O is 10 times lower, a time 10 times longer is required forthe same shattering effect to appear. The scaling with ZO t is useful
if one would like to know the results for other time-scales and/or
metallicities.
4 DISCUSSION
4.1 Steepening of the UV extinction curve
In the above section, we showed that the abundance of small(a < 0.1 μm) grains increases through the shattering of large(a  0.1 μm) grains. Consequently, the slope of the UV extinction curve becomes steep after shattering. Here, we discuss thisphenomenon in terms of the grain-size distribution.
The contribution from grains in a logarithmic size range[ln a, ln a + d ln a] to the extinction can be written as

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dκext ≡ πa 2 Qλ (a)n(a)a d ln a, where Qλ (a) is the extinction crosssection normalized to the geometrical cross-section (πa 2 ). If thesize distribution is approximated by a power law (n ∝ a −p ) over acertain size range, dκext /d ln a ∝ a 3−p Qλ (a). Because Qλ (a) ∼ 1
for 2πa  λ and Qλ (a) ∝ a for 2πa  λ (e.g. Bohren &Huffman 1983), we obtain dκext /d ln a ∝ a 3−p for 2πa  λ anddκext /d ln a ∝ a 4−p for 2πa  λ. Thus, if p < 3, the largest grains
make the largest contribution to the extinction. In order for small
grains to make a significant contribution to the extinction, p ≥ 3should be satisfied. If 3 < p < 4, the largest contribution to theextinction comes from the grains with 2πa ∼ λ. In other words,
the UV (λ ∼ 0.2 μm) extinction curve is steepened significantly
if grains with a ∼ 0.03 μm are produced and p  3 is satisfied
around this grain size.
From Figs 2–4, we observe that a large number of grains witha ∼ 0.03 μm are produced, and the slope around this grain radius
is p  3 for the solar metallicity cases. Indeed, the UV slope ofthe extinction curve is steepened for the solar metallicity cases, as
shown in Fig. 5.
In this paper, the shattered fragments are distributed with a size
distribution with exponent αf = 3.3 (Section 2.3). Jones et al.
(1996), from a discussion on the cratering flow, argue that a valueof αf slightly larger than 3 is robust. Even if αf = 2.5 is assumed as
an extreme case, the difference in the extinction curve is less than
10 per cent at λ = 0.1 μm and smaller at longer wavelengths (seeAppendix B for details).

4.2 Grain properties in starburst environments
From the results above, the presence of small grains in starburst
environments is generally predicted, although SNe II tend to ejectlarge grains because of the shock destruction in SNRs. For example,
BCDs (or H II galaxies) in the nearby Universe host large ionizedregions, and the age of the current star-formation episode is a few
Myr to 20 Myr (e.g. Hirashita & Hunt 2004; Takeuchi et al. 2005).
These ages are just in the range where shattering could modify
the grain-size distribution and extinction curve, although we shouldtake into account the low metallicity in BCDs. Some BCDs show an
excess of near-infrared emission (e.g. Hunt, Vanzi & Thuan 2001),
which can be attributed to the emission from transiently heated verysmall grains (Aannestad & Kenyon 1979; Sellgren 1984; Draine &Anderson 1985). Galliano et al. (2005) carried out a comprehensive
analysis of the SEDs of dust and stars in some dwarf galaxies (dwarf
irregular galaxies and BCDs), and showed that the grain size isbiased to small grains with radii of a few nanometres. Because theirsample galaxies have metallicities larger than 1/10 Z , shattering in
the WIM can work as a production source of nanometre-sized grainson time-scales of a few mega-years, and thus can be considered as
an origin of small grains in these galaxies.
It is natural to expect that a similar condition (i.e. turbulencein a WIM sustained for more than a few mega-years) is generally
realized in starburst galaxies. Although it is hard to compare theextinction curve with the observed wavelength dependence of thedust attenuation because of the effects of radiative transfer (Calzetti
2001; Inoue 2005), shattering may be crucial to reproducing the reddening observed in starburst galaxies. Therefore, shattering should
be considered as a source of small grains, which contribute to thereddening. Alternatively, dust produced by AGB stars in the underlying old population (older than several ×108 yr; Valiante et al.
2009) could contribute to the steepening if they produce small
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grains produced in AGB stars are large (a ∼ 0.1 μm) (Groenewegen1997; Gauger et al. 1999).
Efficient shattering also occurs in the ISM by means of the passage of SN shocks. Jones et al. (1996) show that a large fractionof large grains with a > 0.1 μm is redistributed into smaller grains
by a single passage of a shock with a velocity of ∼100 km s−1 .
In their calculation, large grains take a longer time before they
are dynamically coupled with gas and are subject to more collisions with dust. Jones et al. (1996) consider the MRN distribution as the initial grain-size distribution, which enhances theshattering efficiency compared with our case, because of the enhanced collision with the abundant small grains. Below, we estimate the time-scale on which shattering in SN shocks destroys
large grains based on Jones et al. (1996), although the timescale obtained might be an underestimate for the grains producedby SNe II, because of the enhanced collision rate in the MRN
distribution.
The time-scale on which shattering in SN shocks effectively
destroys large grains can be estimated in a way similar to in
McKee (1989). A single SN can sweep Msw ∼ 104 M of gas(i.e. Msw vs2 /2 ∼ ESN , with shock velocity vs ∼ 100 km s−1 andthe energy given to the gas by a SN equal to ESN ∼ 1051 erg).
Then, the gas mass swept by SN shocks with vs  100 km s−1
per unit time can be estimated as Msw γ , where γ is the SN rate.
Thus, the time-scale on which the entire gas mass M g is affectedby shattering by SN shocks is estimated as τsw ∼ Mg /(Msw γ ).
Because γ /ψ ∼ 10−2 M−1
 for a Salpeter initial mass function(Salpeter 1955) (ψ is the star-formation rate), the above time-scaleis estimated as τsw ∼ 10−2 Mg /ψ. This estimate indicates that the
shattering time-scale by SN shocks is about 0.01 times the gas consumption time-scale by star formation. In starburst environments,
Mg /ψ ∼ 108 –109 yr may be reasonable (Young et al. 1986), andshattering in SN shocks occurs in 1–10 Myr, which is comparable tothe time-scale investigated in this paper. Therefore, both shatteringin turbulence and that in SN shocks can affect the grain-size distribution. A detailed calculation of shattering in SN shocks of grains
produced by SNe II is required before we judge which of these two
shattering mechanisms is dominant.
It might also be useful to discuss our results in terms of the extinction curves of the Large and Small Magellanic Clouds (LMCand SMC), both of which have developed H II regions such as 30
Doradus. Indeed, Bernard et al. (2008) indicate that the 70-μm excess around 30 Doradus can be explained by an enhancement of theabundance of very small grains, possibly by the destruction of large
grains. Bot et al. (2004) find this excess in the SMC. Paradis et al.
(2009) show that the very small grain abundance is substantially enhanced around 30 Doradus using an SED model of dust emission.
However, the extinction curves in these galaxies are much steeper
than our results (Aλ /AV  2.9 and 3.2 at λ  0.2 μm for the
LMC and the SMC, respectively; Pei 1992). Because these galaxies have less intense star formation than BCDs, non-starbursting
components are likely to contribute significantly to the extinction
curve. The steep extinction curves of the LMC and the SMC indicate that we should consider not only the dust production/shatteringin star-forming regions but also various other mechanisms that act
as efficient production sources of small grains. For example, shattering in a warm neutral medium works on a time-scale of 100
Myr (HY09). ISM phase exchange, which occurs on a time-scale of
50–100 Myr, also affects the evolution of the grain-size distribution(O’Donnell & Mathis 1997). Such longer-time-scale mechanisms
could also have affected the extinction curves (grain-size distributions) of those galaxies. The current paper, which focuses on a

Figure 7. The extinction curves in Fig. 5(b) plotted only in the UV range(ZO = 1 ZO and nH = 1 cm−3 ). The shaded area shows the observed
extinction curve for SDSS J1048+4637 (z = 6.2) by Maiolino et al. (2004b),
including the uncertainty.

short-time-scale (<10 Myr) grain processing, is a starting point for
the inclusion of other physical processes in future work.

4.3 Comparison with high-z data
At z > 5, it is usually assumed that the main production source ofdust is SNe II, whose progenitors have short lifetimes, as the cosmic
age is too young for low-mass stars to evolve (but see Valiante et al.
2009). Thus, the extinction curves at such high z are often used totest the theory of dust production in SNe II (Maiolino et al. 2004b;
Hirashita et al. 2005). As a representative case of observed high-z
extinction curve, we discuss the rest-frame UV extinction curve of
SDSS J1048+4637 (z = 6.2) obtained by Maiolino et al. (2004b).
In Fig. 7, we show the UV part of the extinction curves calculatedwith our models in comparison with the observed UV extinction
curve of SDSS J1048+4637. The extinction curves are normalized
to the value at λ = 0.3 μm. We show the result for nH = 1 cm−3 , but
the following discussions hold qualitatively also for other densities.
As discussed in Hirashita et al. (2008), the initial extinction curve
before shattering is too flat to explain the UV rise in the observed
extinction curve because small grains are selectively destroyed in
SNRs. However, after shattering, the extinction curve approaches
the observed curve because of the production of small grains. After
10 Myr of shattering, the observed extinction curve is reproduced.
Not only the slope but also the bump feature at 1/λ ∼ 4 μm−1 , whichbecomes prominent after shattering (Section 3.2), may account forthe behaviour of the observed extinction curve around 1/λ ∼ 3.5–
4 μm−1 .
In summary, if the metallicity is nearly solar and the age of thecurrent episode of starburst is greater than 5 Myr, we should take theeffect of shattering in turbulence into account when comparing theobserved extinction curve with the theoretical one, even at z > 5.
Because quasars tend to be found in evolved stellar systems whose
metallicity could be nearly solar (or more than solar; Juaerz et al.
2009), the UV rise of the extinction curve may be caused by the
production of small grains by shattering. The dependence of theextinction curve on age and metallicity may also be responsible forthe variation of the UV slope of the quasar spectra in the sample ofMaiolino et al. (2004a).

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4.4 Remarks on grain physics

AC K N OW L E D G M E N T S

Before concluding this paper, we mention some physical processes
to be considered in the future. In the calculation of the shock
destruction of grains in SNRs by N07, the effects of grain electrical charge and the effects of magnetic fields are ignored. As
shown in Jones et al. (1994, 1996) and more recently by Guillet,
Pineau Des Forêts & Jones (2007) and Guillet, Jones & Pineau Des
Forêts (2009), the dynamics of charged grains is critically modified
by magnetic fields. The gyration around the magnetic fields tends tostrengthen the coupling between gas and dust, and this effect could
suppress the ejection of large grains into the ISM. Thus, not only
small grains but also large grains with a  0.1 μm could be subject
to significant processing in the shock. Slavin et al. (2004) show thatthe presence of magnetic fields in shocks produces complexity in the
kinematics of large ( 0.1 μm) grains. Thus, it may be important
to trace the grain trajectory around the reverse and forward shocks.
The quantification of all these effects of magnetic fields is left for
future work.
The importance of shattering by turbulence for small-grain production in starburst galaxies should, however, still be consideredeven if we take into account the effect of the magnetic field in the
future, because it is still true that the shock destruction in SNRs suppresses the injection of small grains into the ISM. It should also be
kept in mind that at the smallest size ranges (a few Å), the treatmentof grains as a bulk solid may not be a good approximation. Becausesuch tiny grains do not affect the UV–optical extinction curve, as
discussed in Section 4.1, the results on the extinction curves are notaffected. Mid-infrared spectra of dust emission are more suitablefor constraining the abundance of such small grains (e.g. Mathis
1990).

We thank the referee, A. P. Jones, for useful comments that improved
this paper considerably. We thank T. T. Takeuchi and T. T. Ishii
for helpful discussions. HY is supported by a TAP fellowship in
Arizona. TN has been supported by the World Premier International
Research Center Initiative (WPI Initiative), MEXT, Japan, and by
a Grant-in-Aid for Scientific Research of the Japan Society for the
Promotion of Science (19740094, 20340038).

5 CONCLUSION
We have theoretically investigated the effect of shattering in a turbulent WIM on the grain-size distribution using the framework forshattering developed by Jones et al. (1994, 1996) and the calculationof interstellar MHD turbulence obtained by Yan et al. (2004). We
have focused on systems in which dust is predominantly producedby SNe II. Although SNe II tend to eject large (a  0.1 μm) grains
because of the shock destruction in SNRs (N07), shattering in the
WIM supplies small grains on a time-scale of several mega-years
in the solar-metallicity (i.e. Galactic dust-to-gas ratio) case. Consequently, the extinction curve is steepened and the features such as
the carbon bump around 1/λ ∼ 4 μm−1 and the Si bump around1/λ ∼ 7 μm−1 become apparent if the metallicity is solar and theduration of shattering is longer than ∼5 Myr. Therefore, when wetreat a system in which the metallicity is solar and the star formation
age is  5 Myr, we should take into account the effect of shattering in interstellar turbulence. In particular, the extinction curves ofhigh-z quasars, whose metallicity is typically (above) solar, may beaffected by shattering, and the UV rise of the extinction curve as
well as the bump feature at 1/λ ∼ 3.5–4 μm−1 can be attributed tothe small grains produced by shattering. If the metallicity is 1/10
solar, the extinction curve does not vary significantly on a timescale of 10 Myr because the frequency of grain–grain collisions is
reduced in proportion to the grain abundance. Thus, the steepening
mechanism of the extinction curve discussed in this paper is valid
for systems whose metallicities are significantly greater than 1/10
solar. We conclude that shattering in the WIM is generally of potential importance in starburst galaxies as a production mechanismof small grains.

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443

APPENDIX A: TEST FOR THE ‘ONE-SPECIES’METHODAs stated in Section 2.3, all grain species other than carbonaceous
grains are treated as a single species, formally termed ‘silicate’ incalculating the grain-size distribution. This approximation is called
the ‘one-species’ method, and it is exact if all the grain species
have the same shape of grain-size distribution. We expect that the
one-species method will give a reasonable answer, as Si is dominant among the ‘silicate’ category. Although the ‘silicate’ species
other than Si (we term these species non-Si grains) have minor
contributions in grain mass, some of them have a significant contribution to the number of small-sized grains, which affect the UV
slope of the extinction curve. Here we test the validity of the onespecies method in comparison with the ‘individual-species method’
as explained below.
The ‘individual-species’ method adopts the grain-size distribution of individual species, and the evolution of the grain-size distribution is calculated separately for individual species (note that the

Figure A1. Size distributions for (a) nH = 0.1 cm−3 and (b) nH = 1 cm−3 for the grains other than carbon (i.e. ‘silicate’). The solid and dotted lines show theresults with the individual-species method and with the one-species method, respectively. The dashed line represents the contribution from Si to the solid line.
The metallicity and the age are assumed to be 1ZO and 5 Myr, respectively.

Figure A2. Extinction curves normalized to the V-band extinction for the grain-size distributions presented in Fig. A1 (the grain-size distributions ofcarbonaceous grains are the same as those in Figs 2 and 3). Panels (a) and (b) present the cases for nH = 0.1 and 1 cm−3 , respectively. The solid and dottedlines show the results of the individual-species and one-species methods, respectively. The dashed lines represent the contributions from various species (C, Si
and others, as labelled) for the individual-species method.
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Effects of grain shattering on extinction curves
grain-size distribution summed over all species other than carbonaceous grains is adopted for ‘silicate’ in the one-species method). In
calculating the evolution of the grain-size distribution of a certain
species, the total mass density of the species relative to the gas
density is assumed to be the total dust-to-gas ratio (but the grainsize distribution after shattering is normalized again to recover thecorrect mass ratio of each species). This treatment maximizes the
production of small grains for non-Si species, which have smaller
sizes than Si, but minimizes the production of small Si grains. Thus,
this method is suitable for examining the maximum possible contribution from non-Si small grains to the UV extinction curve.
In Fig. A1, we compare the grain-size distributions predicted by
the one-species and individual-species methods for nH = 0.1 and1 cm−3 at 5 Myr. For nH = 10 cm−3 , the difference between thetwo methods is negligible because non-Si grains contribute little
to the total grain abundance. From the figure, we observe that thedifference is relatively large in the case of nH = 0.1 cm−3 . This isbecause the fraction of non-Si grains is larger for nH = 0.1 cm−3
than for nH = 1 cm−3 .
In Fig. A2, we show the extinction curves calculated by the two
methods. We observe that the extinction curves of the individualspecies method tend to be steeper than those of the one-species
method. As can be seen in the figure, the steeper slope comes fromthe contribution from the non-Si grains, indicated by ‘others’. In
the individual-species method, the size distribution of each non-Si
species, which has a larger fraction of small grains than that of Si, iscalculated separately, so that the production of small non-Si grainsis enhanced. We note that the ‘real’ grain-size distribution would
lie between the results of the two methods. This means that theapproximate treatment adopted in the text (i.e. one-species method)
is justified for nH  1 cm−3 . For nH  0.1 cm−3 , because thecontribution from non-Si species is significant, the error of the onespecies method is at most ∼10 per cent at λ ∼ 0.2 μm, and ∼40
per cent at λ ∼ 0.1 μm. In order to overcome this uncertainty, we
should develop a different scheme that could treat the collisions
between multiple species (in our case, nine species), which cannot
be treated in the current scheme in a reasonable computational
time.

1447

Figure B2. Extinction curves normalized to the V-band extinction for thegrain-size distributions presented in Fig. B1. The solid and dotted lines show
the results for αf = 3.3 and 2.5, respectively. The dashed line represents theinitial extinction curve before shattering.

A P P E N D I X B : F R AG M E N T- S I Z E
D I S T R I B U T I O N W I T H A S H A L L OW E R S L O P E
The size distribution of shattered fragments is assumed to be a
power law with an exponent of −αf . As discussed in the text, thesteepening of the extinction curve becomes prominent if the powerlaw exponent (p) of the grain-size distribution around a ∼ 0.03 μm
is steeper than ∼3 (Section 4.1). Jones et al. (1996) have shown thatthe size distribution after shattering is not sensitive to αf . They also
argue that αf slightly larger than 3 is robust against the change ofthe cratering flow parameters in shattering (αf = 3.3 is adopted in
the text). Nevertheless, it would be interesting to examine whetherp > 3 is realized even if we assume αf < 3.
Here we examine the smallest exponent adopted in Jones et al.
(1996), αf = 2.5, as an extreme case. The ambient hydrogen number
density is fixed at nH = 1 cm−3 . In Fig. B1, we show the result at
t = 5 Myr. As expected, the effect of αf is more prominent for

Figure B1. Grain-size distributions for nH = 1 cm−3 with a metallicity of 1ZO . The solid and dotted lines show the results at t = 5 Myr for αf = 3.3 and
2.5, respectively. The dashed line represents the initial grain-size distribution before shattering. Two grain species, (a) silicate and (b) carbonaceous dust, areshown. The arrow at a = 0.03μm indicates a rough representative size of the grains contributing to the steepening of the UV extinction curve.
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smaller grains, as shattering with large αf can supply small grains
more efficiently. However, we observe that the difference between
αf = 2.5 and 3.3 is small around a ∼ 0.03 μm, confirming theresult of Jones et al. (1996). The small difference comes from thefixed shattered mass in a collision; that is, the distribution of grain
fragments as a function of size has a minor effect compared withthe total mass of shattered fragments (shattering efficiency).

The extinction curves are shown in Fig. B2. We observe thatthe difference between the two curves with α = 2.5 and 3.3 isnegligibly small at λ ∼ 0.3 μm and is less than 10 per cent even at
λ ∼ 0.1 μm. The small difference is the natural consequence of thesmall variation of the grain-size distribution at a  0.03 μm.
This paper has been typeset from a TEX/LATEX file prepared by the author.

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