MNRAS 432, 637–652 (2013)

doi:10.1093/mnras/stt506

Advance Access publication 2013 April 17

What determines the grain size distribution in galaxies?
Ryosuke S. Asano,1‹† Tsutomu T. Takeuchi,1 Hiroyuki Hirashita2
and Takaya Nozawa3
1 Department of Particle and Astrophysical Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan
2 Institute of Astronomy and Astrophysics, Academia Sinica, PO Box 23-141, Taipei 10617, Taiwan
3 Kavli Institute for the Physics and Mathematics of the Universe, Todai Institutes for Advanced Study, the University of Tokyo, Kashiwa, Chiba 277-8583,

Japan

Accepted 2013 March 18. Received 2013 March 18; in original form 2012 December 28

Dust in galaxies forms and evolves by various processes, and these dust processes change thegrain size distribution and amount of dust in the interstellar medium (ISM). We construct a dust
evolution model taking into account the grain size distribution, and investigate what kind of dust
processes determine the grain size distribution at each stage of galaxy evolution. In addition to
the dust production by Type II supernovae (SNe II) and asymptotic giant branch (AGB) stars,
we consider three processes in the ISM: (i) dust destruction by SN shocks, (ii) metal accretionon to the surface of pre-existing grains in the cold neutral medium (CNM; called grain growth)and (iii) grain–grain collisions (shattering and coagulation) in the warm neutral medium and
CNM. We found that the grain size distribution in galaxies is controlled by stellar sources inthe early stage of galaxy evolution, and that afterwards the main processes that govern thesize distribution changes to those in the ISM, and this change occurs at earlier stage of galaxy
evolution for a shorter star formation time-scale (for star formation time-scales = 0.5, 5 and50 Gyr, the change occurs about galactic age t ∼ 0.6, 2 and 5 Gyr, respectively). If we only take
into account the processes which directly affect the total dust mass (dust production by SNe II
and AGB stars, dust destruction by SN shocks and grain growth), the grain size distribution isbiased to large grains (a ∼ 0.2–0.5 μm, where a is the grain radius). Therefore, shattering is
crucial to produce small (a  0.01 μm) grains. Since shattering produces a large abundanceof small grains (consequently, the surface-to-volume ratio of grains increases), it enhances theefficiency of grain growth, contributing to the significant increase of the total dust mass. Grain
growth creates a large bump in the grain size distribution around a ∼ 0.01 μm. Coagulation
occurs effectively after the number of small grains is enhanced by shattering, and the grain size
distribution is deformed to have a bump at a ∼ 0.03–0.05 μm at t ∼ 10 Gyr. We conclude thatthe evolutions of the total dust mass and the grain size distribution in galaxies are closely related
to each other, and the grain size distribution changes considerably through the galaxy evolution
because the dominant dust processes which regulate the grain size distribution change.
Key words: stars: formation – ISM: clouds – dust, extinction – galaxies: evolution – galaxies:
general – galaxies: ISM.

1 I N T RO D U C T I O NDust is one of the most important factors for the understanding ofgalaxy evolution. Since hydrogen molecules are efficiently formedon the surface of dust grains, the molecular formation rate is muchlarger than the case without dust. Such an enrichment of molecularabundance by dust realizes a favourable condition for star forma-

 E-mail: asano.ryosuke@g.mbox.nagoya-u.ac.jp† Fellow of the Japan Society for the Promotion of Science (JSPS).

tion (e.g. Hirashita & Ferrara 2002). Dust grains also absorb stellar
light mainly at ultraviolet and optical wavelengths and re-emit in
the infrared. Consequently, dust affects the spectral energy distribution (SED) of galaxies (e.g. Takagi, Vansevičius & Arimoto 2003).
Furthermore, the formation rate of hydrogen molecules on the grainsurface and the mass absorption coefficient of radiation depend
strongly on the grain size distribution (e.g. Hirashita & Ferrara
2002; Takeuchi et al. 2003).
Dust grains form by condensation of elements heavier than helium (i.e. metals). Metals are mainly supplied from asymptotic giant branch (AGB) stars and supernovae (SNe), and part of themC 2013 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society

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ABSTRACT638

R. S. Asano et al.
cally coupled with magnetized interstellar turbulence. Shattering
is indeed a promising mechanism of small-grain production (e.g.
Hirashita 2010). Shattering also occurs in SN shocks (Jones et al.
1996). In dense regions, coagulation can occur, so that the grain size
distribution shifts to larger sizes (e.g. Hirashita & Yan 2009; Ormelet al. 2009). The various dust processes above in galaxies occur ontime-scales dependent on the metallicity, the total dust amount, thegrain size distribution and so on. Hence, it is crucial to consider all
dust processes in a unified framework to understand the evolutionof both the total dust amount and the grain size distribution.
There have been a number of studies on the evolution of the grain
size distribution in galaxies. Liffman & Clayton (1989) discussed
the evolution of grain size distribution considering dust destruction
by SN shocks and grain growth. However, they did not consider
shattering and coagulation by grain–grain collisions. O’Donnell &Mathis (1997) suggested a dust evolution model in a multiphase ISM[warm neutral medium (WNM) and cold neutral medium (CNM)],and also considered the collisional processes of dust grains. However, they did not consider the size distribution of grains released
by stars in order to simplify their model. Hirashita et al. (2010) discussed the grain size distribution in young starburst galaxies. They
assumed that SNe II are the source of dust in these galaxies andfocused on the production of small grains by shattering. Yamasawa
et al. (2011) constructed a dust evolution model taking into accountdust formation and destruction by SNe II along with the formationand evolution of galaxies. However, since they focused on galaxies
in the high-z Universe, they did not consider dust formation in AGB
stars, grain growth, shattering and coagulation.
In this work, we construct a dust evolution model taking into
account the dust formation by SNe II and AGB stars, dust destruction by SN shocks, grain growth and shattering and coagulation, toinvestigate what kind of dust processes determine the grain size distribution at each stage of galaxy evolution. In our model, we do notconsider mass exchange among various ISM phases in detail (e.g.
Ikeuchi & Tomita 1983), but our results contain the contributions ofdust processes in the two ISM phases, WNM (∼6000 K, 0.3 cm−3 )and CNM (∼100 K, 30 cm−3 ) by assuming these mass fractions in
the ISM to be constant.
This paper is organized as follows. In Section 2 we introduce
the dust evolution model based on chemical evolution of galaxies.
In Section 3 we examine the contribution of each dust process to
the grain size distribution. Section 4 is devoted to the discussion onwhat kind of dust processes regulate the grain size distribution ingalaxies. We conclude this work in Section 5. Throughout this paper
the solar metallicity is set to be Z = 0.02 (Anders & Grevesse1989).
2 G A L A X Y E VO L U T I O N M O D E L
In this section, we introduce our dust evolution model in a galaxy.
First, we show the basic equations of the chemical evolution model.
We then describe the dust evolutions based on the chemical evolution model, involving dust production by SNe II and AGB stars,
dust destruction by SN shocks, grain growth and shattering andcoagulation by grain–grain collisions.
Some grain processing mechanisms work in a different way in adifferent ISM phase (O’Donnell & Mathis 1997). In this work, while
we use a one-zone model to examine the representative propertiesof a galaxy, we consider the effects of the dust processes in WNM
and CNM by introducing the mass fractions of WNM and CNM,
ηWNM and ηCNM . Considering temperatures less than 104 K, we findthat an equilibrium state of two thermally stable phases (WNM and

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condense into dust grains (e.g. Mathis 1990). Dust grains are not
only supplied by stars but are also destroyed by SN shocks in theinterstellar medium (ISM; e.g. Jones, Tielens & Hollenbach 1996;
Nozawa, Kozasa & Habe 2006; Zhukovska, Gail & Trieloff 2008).
Furthermore, it is thought that metal accretion on to the surfaceof grains in the ISM (referred to as ‘grain growth’ in this paper)
is an important process for explaining the amount of dust in the
Milky Way (e.g. Draine 2009; Pipino et al. 2011). To the presentday, there have been a lot of studies that investigate the evolutionof the total dust mass in galaxies by taking into account these processes (e.g. Dwek & Scalo 1980; Dwek 1998; Hirashita 1999a,b;
Inoue 2003, 2011; Calura, Pipino & Matteucci 2008; Zhukovska
et al. 2008; Pipino et al. 2011; Asano et al. 2013). They assumed a
representative grain size, but the efficiencies of dust destruction andgrain growth depend on the grain size distribution. Thus, we should
consider the evolution of the grain size distribution to understandthe total dust mass precisely.
The grain size distribution is derived from observed extinction
curves (which mainly depend on the grain size distribution and thegrain species). According to Mathis, Rumpl & Nordsieck (1977),
if spherical grains are assumed, the extinction curve in the Milky
Way is reproduced by f (a)da ∝ a−3.5 da [0.005 < a < 0.25 μm;
this grain size distribution is referred to as the Mathis–Rumpl–
Nordsieck (MRN) distribution], where a is the grain radius andf (a)da is the number density of grains in size interval [a, a + da](see Kim, Martin & Hendry 1994; Weingartner & Draine 2001,
for more detailed fitting to the Milky Way extinction curve). The
situation seems to be very different for distant galaxies. Recently,
Gallerani et al. (2010) discussed the extinction curves of sevenquasars at high redshift (3.9 ≤ z ≤ 6.4). They showed that theseextinction curves tend to be flat at wavelengths <0.2 μm in thequasar’s rest frame. The difference between extinction curves in
distant and nearby objects may indicate that different processes
dominate the dust evolution at different epochs.
In young galaxies, Type II SNe (SNe II) are thought to be thedominant sources of dust because of short lifetime of their progenitors. However, Valiante et al. (2009) suggested that AGB stars arealso important sources of dust production even at galactic age lessthan 1 Gyr. In addition, grain growth is expected to be the dominant
process to increase dust mass in galaxies if the metallicity becomeslarger than a certain value (Inoue 2011; Asano et al. 2013). Furthermore, if the metallicity reaches a subsolar value, grain–grain
collisions in the ISM (shattering and coagulation) become efficientenough to change the grain size distribution significantly (e.g. Hirashita & Yan 2009). We call all processes affecting the grain size
distribution ‘dust processes’.
These dust processes affect the different sizes of grains in galaxies. Nozawa et al. (2007) showed that SNe II supply relatively large
grains (a  0.01 μm) into the ISM because small grains are destroyed by reverse shocks before they are ejected into the ISM (seealso Bianchi & Schneider 2007; Silvia, Smith & Shull 2012). The
size distribution of grains produced by AGB stars is thought to bebiased to large (∼0.1 μm) sizes (e.g. Groenewegen 1997; Winters
et al. 1997; Yasuda & Kozasa 2012). Furthermore, the smaller grainsin the ISM are more easily destroyed by interstellar shocks driven
by SNe (e.g. Nozawa et al. 2006). If grain growth occurs, since thetime-scale of this process is proportional to the volume-to-surface
ratio of a grain, smaller grains grow more efficiently (e.g. Hirashita& Kuo 2011). After the dust grains are released into the diffuse ISM,
shattering can also occur. Yan, Lazarian & Draine (2004) showedthat large grains (a  0.1 μm) acquire larger velocity dispersions
than the shattering threshold velocity if the grains are dynami-

Evolution of the grain size distribution

2.1 Chemical evolution model
In this subsection, we describe our model of chemical evolution
in a galaxy. From the above assumptions, the equations of time
evolution of the total stellar mass, M∗ , the ISM mass, MISM , and themass of a metal species X, MX , in the galaxy are expressed asdM∗ (t)= SFR(t) − R(t),dt(1)

dMISM (t)= −SFR(t) + R(t),dt(2)

dMX (t)= −ZX (t)SFR(t) + YX (t),(3)
dtwhere t is the galaxy age, SFR(t) is the star formation rate, ZX =MX /MISM , and R(t) and YX (t) are the masses of the total baryons andtotal metal species X released by stars in a unit time, respectively.
In this paper, we consider two dust species, carbonaceous dust andsilicate dust, and we adopt two key elements of dust species (X =C for carbonaceous dust and X = Si for silicate dust) in calculating
dust evolution (see Section 2.2 for details). We adopt M∗ (0) = 0,
MISM (0) = Mtot and MX (0) = 0 as initial conditions.
n
In our work, we adopt the Schmidt law for the SFR: SFR ∝ MISM
(Schmidt 1959), and the index n is thought to be 1–2 observationally(e.g. Kennicutt 1998). We here adopt n = 1:
SFR(t) =MISM (t)
,
τSF

(4)

where the star formation time-scale τ SF is a constant. For comparison, the case with n = 1.5 is also shown in Appendix A1. As longas we adopt the same star formation time-scale at t = 0, there islittle difference between the two cases with n = 1 and 1.5.
R(t) and YX (t) are written as
 100 M
(5)
R(t) =[m − ω(m, Z(t − τm ))] φ(m)SFR(t) dm,
mcut (t)YX (t) = 100 M
mcut (t)mX (m, Z(t − τm ))φ(m)SFR(t) dm,(6)

where φ(m) is the stellar initial mass function, τ m is the lifetime of astar with mass m at the zero-age main sequence, Z is the metallicity(= X MX /MISM ) and ω(m, Z) and mX (m, Z) represent the mass of
remnant stars (white dwarfs, neutron stars or black holes) and themass of metal species X ejected by a star of mass m and metallicity Z,
respectively. For the lifetime of stars, we adopt the formula derived
by Raiteri, Villata & Navarro (1996), and the formula is obtainedby the fitting to the stellar models of the Padova group (Bertelli
et al. 1994). Since its metallicity dependence is weak, we always
adopt the stellar lifetime for solar metallicity as a representativevalue. The lower bound of the integration, mcut (t) is the mass of astar with τ m = t. We adopt the Salpeter IMF for stellar mass range
0.1–100M (Salpeter 1955):
φ(m) ∝ m−q ,(7)where q is set to be 2.35, and the normalization is determined by
 100 Mmφ(m) dm = 1.
(8)
0.1 M
To check the variation of the results with q, we examine the casewith q = 1.35 (a top heavy IMF) in Appendix A2. For q = 1.35, theprocesses in the ISM occur at earlier phases of galaxy evolution thanfor q = 2.35, because a larger amount of dust is supplied by stars.
However, we find that the sequence of the dominant dust processes
along the age does not change so the following discussions are notaltered significantly by the change of q. Thus, we only consider q =2.35 in the following discussion.
To calculate equations (5) and (6), we quote the remnant and metal
mass data of stars with mass m and metallicity Z from some previous
works. We assume that the mass ranges of AGB stars and SNe II are
1–8 and 8–40 M , respectively, and that all stars with initial masses
more than 40 M evolve into black holes without ejecting any gas,
metals or dust (Heger et al. 2003). The data for AGB stars withmass 1–6 M and metallicity Z = (0.005, 0.2, 0.4, 1.0) Z is takenfrom Karakas (2010) and the data for SNe II with mass 13–40 M
and metallicity Z = (0.0, 0.05, 0.2, 1.0) Z is from Kobayashi et al.
(2006). We interpolate and extrapolate the data for all values ofmass and metallicity (also for the dust data in Sections 2.2.1 and2.2.2).

2.2 Dust evolution
For dust evolution, we consider dust production by SNe II and AGB
stars, dust destruction by SN shocks in the ISM, grain growth in the
CNM and shattering and coagulation by grain–grain collisions in
the WNM and CNM. In this work, as mentioned in Section 1, we
assume a two-phase ISM (WNM and CNM) to calculate the dust
evolution (see also Section 2.2.5).
We neglect the contribution of Type Ia SNe (SNe Ia) to the production of metals and dust, and the destruction of dust. Nozawa
et al. (2011) showed that SNe Ia release little dust into the ISM.
Furthermore, dust destruction by SNe Ia is expected to be insignificant to the total dust budget in galaxies (less than 1/10 of thecontribution of SNe II; Calura et al. 2008). As for metals, although
Nomoto et al. (1997) showed that the contribution of SNe Ia to thesilicon and carbon enrichment in the ISM can be comparable to thatof SNe II, the ratio between SN Ia rate and SN II rate is unknown(Nomoto et al. 1997 suggested that it is about 0.1 taking into accounta chemical evolution model). Thus, to simplify the discussion, we
neglect the contribution from SNe Ia, keeping in mind a possible
underproduction of metallicity.
The dust production data we adopt contain a lot of dust species(C, Si, SiO2 , SiC, Fe, FeS, Al2 O3 , MgO, MgSiO3 , FeSiO3 , Mg2 SiO4

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CNM) is established in the ISM (e.g. Wolfire et al. 2003). Thus,
we calculate dust evolution taking into account a two-phase neutral
ISM. We also assume that the galaxy is a closed-box; that is, thetotal baryon mass Mtot (the sum of the stellar mass and the ISM
mass in the galaxy) is constant. Since Mtot is just a scale factor in
our work, the total dust mass just scales with Mtot . Throughout thispaper Mtot is set to be 1010 M .
Inflow and outflow are not considered in our model for simplicity.
Since inflowing gas is considered to be not only metal poor but also
dust poor, the abundance of both metals and dust is diluted with thesame (or similar) fraction by inflow. This effect is degenerate witha slower chemical enrichment, under a longer τ SF , where τ SF is thestar formation time-scale. As for outflow, since ISM components(namely gas, metals and dust) are blown out of a galaxy, the total
gas mass in a galaxy decreases. In this case, star formation rate
decreases at earlier phase of galaxy evolution; that is, the effect ofoutflow is degenerate with a shorter τ SF . Thus, we just absorb theeffects of inflow and outflow into τ SF .

639

640

R. S. Asano et al.
Table 1. Parameters for each dust species.
SpeciesX

gX

mX (amu)s (g cm−3 )bv shat (km s−1 )γ (erg cm−2 )c

E (dyn cm−2 )cνcGraphite
Silicate

CSi

1.0
0.166a

12
28.1

2.26
3.3

1.2
2.7

75
25

1.0 × 1011
5.4 × 1011

0.32
0.17

Note. X is the key element of dust species, gX is the mass fraction of the key element X in the grains, mX is the atomic
mass of X, s is the bulk density of dust grains, v shat is the shattering threshold velocity, γ is the surface energy perunit area of grains, E is Young’s modulus and ν is Poisson’s ratio.
a We assume Mg Fe SiO for the composition of silicate (Draine & Lee 1984).
1.1 0.94
b Draine & Lee (1984).
c Chokshi, Tielens & Hollenbach (1993).

m(a) =

4πa 3
s,
3

(9)

where s is the bulk density of dust grains. In our model, we consider
that the minimum and maximum radii of grains, amin and amax ,are 0.0003 and 8 μm, respectively. Although the minimum size ofgrains is poorly known, even if amin = 0.001 μm, the evolution ofboth the total dust mass and the grain size distribution does notchange significantly (Hirashita 2012).

2.2.1 Dust production by AGB stars
The size distribution of grains produced by AGB stars is not wellknown. Winters et al. (1997) suggested that the size distribution is
log-normal with a peak at ∼0.1 μm based on the fitting to observed
SEDs. Yasuda & Kozasa (2012) have recently calculated the size
distribution of SiC produced by C-rich AGB stars by performing
dust formation calculation coupled with a hydrodynamical model.
They showed that the mass distribution, a4 f (a), is close to lognormal with a peak at 0.2–0.3 μm, where the grain size distribution
f (a) is defined so that f (a) d a is the number density of dust grains
with radii in the range [a, a + da] (The size distribution multiplied by
a4 means the mass distribution per logarithmic grain radius). Hence,both theory and observations suggest that AGB stars preferentially
produce large grains (a  0.1 μm). In this paper, we simply assume
that the mass distribution, a4 f (a), of each species produced by AGB
stars is log-normal with a peak at 0.1 μm with standard deviation
σ = 0.47, so that the shape of the mass distribution in fig. 7 inYasuda & Kozasa (2012) is reproduced. We normalize f (a) by ∞4π 3
md (m) =a sf (a)da,(10)
30
where md (m) is the dust mass released by a star with mass m.
The size distributions of all species are assumed to be the samefor simplicity. Dust mass data for AGB stars with mass 1–7 M
and metallicity Z = (5.0 × 10−2 , 0.1, 0.2, 0.4, 0.75, 1.0) Z is takenfrom Zhukovska et al. (2008). The size distributions of dust species
other than carbonaceous dust are summed to compose the grain
size distribution of silicate (the same procedure is also applied in
Section 2.2.2). We define f X (a) as the size distribution of dust
species, where X represents the key element of dust species (X = C
for carbonaceous dust and X = Si for silicate dust).
2.2.2 Dust production by SNe II
Some fraction of dust grains in galaxies are produced in the ejectaof SNe II (e.g. Matsuura et al. 2011). After a SN explosion, reverse
shock occurs because of interactions between the ISM surrounding
the SN and its ejecta, and dust grains are destroyed by sputteringin the shock (e.g. Bianchi & Schneider 2007; Nozawa et al. 2007).
Nozawa et al. (2007) calculated the total mass and size distributionof dust grains ejected by SNe II considering the dust destruction in
the radiative and non-radiative phases of SN remnants. We adopt
the data for dust mass and size distribution derived by Nozawa et al.
(2007) for SNe II with mass 13–30 M .1 They also considered twocases for mixing in the helium core: unmixed and mixed models.
Hirashita et al. (2005) showed that the data from the unmixed model
are in better agreement with observations than that of the mixed
model. Hence, we adopt the unmixed model. Nozawa et al. (2007)
showed that the size distribution of grains supplied by SNe II isbiased to large (∼0.1 μm) grains due to the destruction of small
grains by the reverse shock.
The amount and size distribution of grains injected by SNe II
depend on the density of the surrounding ISM because the dust
destruction efficiency of the reverse shock is higher in the denser
ISM. However, the trend that smaller grains are more easily destroyed does not change. In this paper, the hydrogen number densityof the ISM surrounding the SNe II, nSN , is set to be 1.0 cm−3 as afiducial value, but the cases with nSN = 0.1 and 10.0 cm−3 are also
examined.
1 Although Nozawa et al. (2007) investigated only the dust formation in
SNe II evolving from zero-metallicity stars, the grain species formed in theejecta of SNe II and their size distribution are insensitive to the metallicity
of progenitor stars (e.g. Todini & Ferrara 2001; Kozasa et al. 2009). In
addition, the destruction efficiency of dust by the reverse shocks is almost
independent of metallicity in the ISM; its difference between Z = 0 and Z =Z is less than 15 per cent (see Nozawa et al. 2007).

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and Fe2 SiO4 ; Nozawa et al. 2007; Zhukovska et al. 2008). However,the physical properties of grain species other than carbonaceousand silicate grains are not fully known. Hence, we categorize all
grain species other than carbonaceous dust as silicate and calculate
their growth, shattering and coagulation by adopting the physicalparameters of silicate grains. In particular, after grain growth andcoagulation occur, the dust species categorized as silicate dust do not
evolve separately and our simplification can avoid the complexity
arising from the compound species. In fact, the mass of dust grainsejected by SNe is dominated by Si grains, which would grow into
silicate grains in the oxygen-rich environments such as molecular
clouds. For carbonaceous dust, we adopt material properties ofgraphite. The adopted parameters of these two grain species areshown in Table 1 and are the same as in Hirashita & Yan (2009)
and Kuo & Hirashita (2012). Although we calculate silicate andcarbonaceous dust separately, we are interested in how the overallgrain size distribution is affected by each dust process. Therefore,
we focus on the total grain size distribution.
In this work, we assume that grains are spherical. Thus, the massof a grain with radius a is

Evolution of the grain size distribution

641

2.2.4 Grain growth

Dust grains in the ISM are destroyed or become smaller by sputtering due to the passage of interstellar shocks driven by SNe. Since
the destruction changes and depends on the grain size, it is important to consider dust destruction taking into account the grain size
distribution.
To calculate this destruction process, we adopt the formulae in
Yamasawa et al. (2011), which we show here briefly. The number
density of dust grains with radii in the range [a, a + da] after thepassage of a SN shock, fX (a)da, is given by ∞ξX (a, a ) dafX (a ) da ,(11)fX (a) da =Here, we formulate the growth process of grains taking into accountthe grain size distribution. In the ISM, particularly in dense andcold regions, metals accrete on to the surface of grains efficiently(e.g. Liffman & Clayton 1989; Inoue 2003; Draine 2009). Recently,
various studies have shown the importance of grain growth for dust
enrichment in galaxies (e.g. Zhukovska et al. 2008; Michałowski
et al. 2010; Hirashita & Kuo 2011; Pipino et al. 2011; Valiante
et al. 2011). Hirashita & Kuo (2011) showed quantitatively thatthe grain size distribution has a very important consequence forgrain growth. Here, we follow the formulation by Hirashita & Kuo(2011) and consider only grain growth of refractory dust (silicate
and carbonaceous dust in this paper). Although volatile grains suchas water ice also exists in clouds in reality, they evaporate quickly
when the clouds disappear or the gas temperature rises.
For grain growth, the following equation holds:awhere ξ X (a, a )da is the number fraction of grains that are eroded
from the initial radii [a , a + da ] to radii [a, a + da] by sputteringin the SN shock and has been obtained using the models by Nozawa
et al. (2006). Note that if a > a , ξ X (a, a ) = 0. Thus, the change in
the number density of grains with radii [a, a + da], dNd, X (a), after
the passage of a single SN shock is expressed as ∞ξX (a, a ) dafX (a ) da − fX (a) da.
(12)
dNd,X (a) =0Accordingly, the change of mass density, dMd, X (a), is
4 3
πa sdNd,X (a)
3
 ∞
4πa 3
ξX (a, a ) dafX (a ) da − Md,X (a) da, (13)=30dMd,X (a) =where Md,X (a) da = 43 πa 3 sfX (a) da = Md,X (a) is the total dust
mass with radii [a, a + da] before the dust destruction. The dust
destruction efficiency ξ X depends on the hydrogen number densityof the ISM, nSN , such that dust grains are destroyed more efficiently
in denser regions. As mentioned in Section 2.2.2, nSN = 1.0 cm−3
as a fiducial value in this paper.
The equation for the time evolution of Md, X (a, t) for dust destruction by SN shocks in the ISM is expressed asMsweptdMd,X (a, t)=−γSN (t) Md,X (a, t) − m(a)dt
MISM (t) ∞×ξX (a, a ) dafX (a , t) da ,(14)0where γ SN (t) is the SN rate and Mswept is the ISM mass swept up by
a SN shock. To express the dependence on the galaxy age, we write
Md, X (a) and f X (a) as Md, X (a, t) and f X (a, t), respectively.
The SN rate, γ SN (t), is expressed as
 40 M
φ(m)SFR(t − τm )dm,γSN (t) =(15)
8 M
where we assume that the range of integration in equation (15) is
the mass range where SNe occur (Heger et al. 2003) (if t − τ m < 0,
SFR(t − τ m ) = 0).
The ISM mass swept up by a SN shock, Mswept , depends on thedensity and metallicity of the ISM. In our model, we adopt thefollowing formula used in Yamasawa et al. (2011):

−0.289
Z
Mswept= 1535n−0.202
.
(16)
+
0.039
SN
M
Z

∂
∂fX (a, t)+[fX (a, t)ȧ] = 0,
∂t
∂a

(17)

where ȧ ≡ da/dt is the growth rate of the grain radius.2
From equation (9),dm(a)= 4πa 2 s.
da

(18)

Also, from Hirashita & Kuo (2011), the rate of mass increase of agrain with radius a is expressed asdm(a)= gX−1 mX αR,
dt

(19)

where gX is the mass fraction of the key species X in the grains,
mX is the atomic mass of X, α is the sticking coefficient of the key
species and R is the collision rate of X to a grain with radius a,defined as follows (Evans 1994):
kTgas 1/2,
(20)
R = 4πa 2 nX (t)2πmX
where nX (t) is the number density of X in the gas phase in the CNM,
k is the Boltzmann constant and Tgas is the gas temperature in the
CNM. We apply gX = 1.0 and 0.166 for carbonaceous and silicate
dust, respectively (Table 1), and Tgas = 100 K.
Next, we consider nX (t), which is evaluated by
Md,X (t)ρ eff MX (t)1 − gX,(21)nX (t) = ISM
mX MISM (t)MX (t)
eff
is the average mass density of the ISM in which grain
where ρISM
growth occurs. As grain growth occurs, the number of gaseous
metals decreases. Thus, nX is a decreasing function of time if only
eff=
grain growth is concerned. The mass density is estimated as ρISM
μmH nH,CNM , where μ is the mean atomic weight, assumed to be
1.33 (the mass ratio of hydrogen to helium is 3:1). In addition, mH
and nH, CNM are the mass of a hydrogen atom and the hydrogen
number density in the CNM, respectively, and we apply nCNM =30 cm−3 . Hence, from the above four equations (equations 18–21),
we obtain
 


eff
αρISM
MX (t)
kTgas 1/2
Md,X (t)da=ȧ ≡1 − gX. (22)
dt
gX s MISM (t) 2πmX
MX (t)

2 Note that equation (17) is valid for the case where only grain growth isconsidered, i.e. without sputtering, shattering and coagulation.

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2.2.3 Dust destruction by SN shocks in the ISM

642

R. S. Asano et al.
it is difficult to separate them. To compare our work with previous studies (Hirashita et al. 2010; Kuo & Hirashita 2012), we only
consider shattering in turbulence.

2.2.5 Shattering

2.2.6 Coagulation

Turbulence occurs in the ISM ubiquitously, and it is confirmed that
turbulence is maintained by thermal conduction from simulations(e.g. Koyama & Inutsuka 2002). In a turbulent medium, dust grains
are accelerated by turbulence (e.g. McKee & Ostriker 2007), andthey collide with each other and shattering can occur (e.g. Yan et al.
2004; Hirashita & Yan 2009; Hirashita et al. 2010). Hirashita &Yan (2009) suggested that the grain size distribution in the ISMchanges significantly by shattering due to collisions between dust
grains accelerated by magnetohydrodynamic turbulence (Yan et al.
2004). In our model, to calculate shattering process, we adopt thegrain velocity calculated by Yan et al. (2004), and the shattering
equation and parameters used by Hirashita & Yan (2009), whose
formulation is based on Jones et al. (1996).
We outline the treatment of shattering. We define ρ X (a, t) da =m(a)f X (a, t) da as the mass of grains with radii [a, a + da] in aunit volume (refer to as ‘mass density’ in this paper). Considering
shattering in the collision between two grains with radii a1 and a2(called grains 1 and 2, respectively), the time evolution of ρ X (a,t)da for shattering is expressed asdρX (a, t)da
= −m(a)ρX (a, t) dadtshat amax×α[m(a), m(a1 )]ρX (a1 , t) da1 da1

In low-temperature and high-density regions of the ISM, it is expected that coagulation by grain–grain collisions occurs. Indeed,
Stepnik et al. (2003) observed dense filaments and showed that theratio of the intensity in the filaments, I60 µ m /I100 µ m , is smaller thanthat in the diffuse ISM. They concluded that this trend resulted fromthe decrease of small grains due to coagulation. For coagulation,
we adopt the formulation, the velocity of grains and the parameters
used by Hirashita & Yan (2009).
The time evolution of ρ X (a, t) da for coagulation is expressed asfollows:dρX (a, t)da
= −m(a)ρX (a, t) da
dt
coag

amin

+

 amax  amax
amin

amin

α[m(a1 ), m(a2 )]ρX (a1 , t)× da1 ρX (a2 , t) da2 m1,2
shat (a) da1 da2andα[m(a1 ), m(a2 )] =0
σ1,2 v1,2m(a1 )m(a2 )(v1,2 ≤ vshat ),(v1,2 > vshat ),

(23)

(24)

where m1,2
shat (a) is the total mass of shattered fragments of grain 1
within size bin [a, a + da] by a collision between grains 1 and 2, anddepends on the relative velocity of the grains. The size distributionof shattered fragments is proportional to a−3.3 (e.g. Jones et al.
1996).3 The cross-section of the collision between grains 1 and 2 isassumed to be σ1,2 = π(a1 + a2 )2 . The shattering threshold, v shat , isassumed to be 1.2 and 2.7 km s−1 for carbonaceous dust and silicate
dust, respectively (Jones et al. 1996). We adopt the same treatment
for the relative velocity as Jones et al. (1994) and Hirashita & Yan(2009): each time-step is divided into four small time-steps, andwe consider shattering under the following four relative velocities
in each small time-step: (i) front collision (v 1, 2 = v 1 + v 2 ); (ii)back-end collision (v 1, 2 = |v 1 − v 2 |); (iii) side collision v 1, 2 = v 1and (iv) v 1, 2 = v 2 . Here, v 1 and v 2 are the velocities of the grainwith radius a1 and a2 , respectively.
Shattering can occur not only in turbulence but also in SN shocks(e.g. Jones et al. 1996). However, both of these shattering mechanisms have similar consequences on the grain size distribution, so
3 The method of calculation of the maximum and minimum size of fragments
is described in detail in section 2.3 in Hirashita & Yan (2009).

×+ amaxamin

α[m(a1 ), m(a)]ρX (a1 , t) da1 da1

 amax  amax
amin

amin

α[m(a1 ), m(a2 )]ρX (a1 , t) da1

× ρX (a2 , t) da2 m1,2
coag (a) da1 da2andα[m(a1 ), m(a2 )] =



0

1,2
(v1,2 ≥ vcoag),

βσ1,2 v1,2m(a1 )m(a2 )

1,2(v1,2 < vcoag
),

(25)

where β is the sticking coefficient of dust grains, and m1,2
coag (a) =m(a1 ) if the mass range of m(a1 ) + m(a2 ) is within [m(a), m(a) +dm(a)]; otherwise m1,2
coag (a) = 0.
We assume that coagulation occurs if the relative velocity is less
1,2
. In our model, it is calculated
than the coagulation threshold vcoag
in the same way as Hirashita & Yan (2009):
 3
1/2
a1 + a23
γ 5/6
1,2
,
(27)
vcoag = 21.4
5/6
3
(a1 + a2 )E 1/3 R1,2 s 1/2
where R1, 2 ≡ a1 a2 /(a1 + a2 ), γ is the surface energy per unit areaand 1/E = [(1 − ν 1 )2 /E1 + (1 − ν 2 )2 /E2 ], where ν 1 and E1 are
Poisson’s ratio and Young’s modulus of grain 1. The parameters weused are shown in Table 1. Here, we assume β = 1 for simplicity.
The treatment of the relative velocity is the same as for shattering.

2.2.7 Formulation of the grain-size-dependent evolutionof dust mass
Here, using the dust processes introduced above, we show the equation for the dust mass evolution in a galaxy at each grain radius bin,so that we can finally obtain the evolution of grain size distribution.
Defining Md (a, t) ≡ m(a)f (a, t)a as the mass density of grains
with radii [a, a + a]4 and a galactic age t, it is formulated asdMd (a, t)=
dt
4 In this section, we use the symbol ‘’ to emphasize that it stands not for
infinitesimal but a certain small amount.

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We assume α = 1 for simplicity, which means that when the key
species collide with a dust grain, it definitely sticks. In our study,
we calculate the grain growth using equations (17) and (22).

Evolution of the grain size distribution−Md (a, t)
Mswept+ Yd (a, t) −γSN (t)MISM (t)MISM (t)× Md (a, t) − m(a) ∞3.1 Without the effects of grain–grain collisions
3.1.1 Stellar processesξ (a, a )af (a , t)da0
+ ηCNM m(a)a+ ηCNMdMd (a, t)∂[f (a, t)]+ ηWNM∂tdtshat,WNMdMd (a, t)dMd (a, t)+ ηWNMdt
dt
shat,CNM
coag,WNM

dMd (a, t),dt
coag,CNM+ ηCNM(28)

 100 M
mcut (t)md (m, Z(t − τm ), a)φ(m)SFR(t − τm )dm,
(29)

where md (m, Z, a) is the total mass of grains with radii [a, a +a] released by stars with mass m and metallicity Z.

First, we consider the stellar processes including dust ejection fromstars (SNe II and AGB stars) and dust reduction via astration. Fig. 1
shows the result. The size distribution is expressed by multiplying
a4 to show the mass distribution in logarithmic grain radius bin. We
adopt τ SF = 5 Gyr and nSN = 1.0 cm−3 . We also show the caseswith SNe II only. As mentioned in Section 2, since Mtot is just a
scale factor, the shape of the size distribution does not change withMtot . From Fig. 1, throughout any galactic age, we can observe thatthe grain size distribution has a peak at around 0.5 μm, and that
only a small amount of grains with a < 0.01 μm can be formed
by stars. In Fig. 2, we show the grain size distribution for othervalues of nSN : 0.1 cm−3 in the left-hand panel and 10.0 cm−3 in theright-hand panel, respectively. From Fig. 2, we find that a larger
amount of dust grains with radii less than ∼0.1 μm are destroyed
by reverse shocks in the case of higher nSN , and a smaller amount of
dust is supplied to the ISM. However, even if nSN changes, the trendthat a small amount of dust grains with radii less than 0.01 μm aresupplied to the ISM does not change. Thus, stars are the sources ofdust grains with large radii (≥0.05 μm).
From Figs 1 and 2, we observe that dust from SNe II always
dominates the grain size distribution, while the contribution of dust
from AGB stars is seen only around 0.1 μm at a galactic age t =10 Gyr. From our calculation, for the case with nSN = 1.0 cm−3 ,the dust mass ratios produced by AGB stars and SNe II are 1.6 ×
10−3 , 0.16 and 0.37 at t = 0.1, 1.0 and 10 Gyr, respectively. On theother hand, Valiante et al. (2009) suggested that the contribution of
AGB stars to the total dust mass in galaxies approaches or exceedsthat of SNe II at t ∼ 1 Gyr. This difference mainly results fromthe dust mass data adopted. We adopt the data of Nozawa et al.
(2007), whereas Valiante et al. (2009) adopted those of Bianchi &Schneider (2007). The dust mass of Nozawa et al. (2007) is larger
than that of Bianchi & Schneider (2007) because of the differencein the treatment of the dust condensation and the destruction by

3 M O D E L R E S U LT S
In this paper, as mentioned above, we consider the effects of dust
formation by SNe II and AGB stars, dust destruction by SN shocks
in the ISM, grain growth, shattering and coagulation on the evolution of grain size distribution in galaxies. Among these processes, dust formation by SNe II and AGB stars, dust destruction and grain growth directly increase or decrease the total dust
mass, while shattering and coagulation modify only the grain size
distribution. The evolution of the total dust mass in galaxies is often modelled by taking into account the former four contributions(dust formation by SNe II and AGB stars, dust destruction andgrain growth; e.g. Dwek & Scalo 1980; Dwek 1998; Calura et al.
2008; Zhukovska et al. 2008; Inoue 2011; Pipino et al. 2011; Asanoet al. 2013). They calculated the dust evolution by assuming a representative grain size, but the dust destruction and grain growthdepend on the grain size distribution. Thus, it is unknown whether
these four contributions can reproduce the grain size distribution
in nearby galaxies even though they can explain the evolution of
the total dust mass. In Section 3.1, we first investigate the contributions of the processes that directly affect the total dust mass,and then in Section 3.2, we examine the effects of shattering andcoagulation.

Figure 1. Grain size distribution taking into account the dust production
by AGB stars and SNe II and dust reduction through astration (solid lines).
Red, blue and purple lines represent the cases at t = 0.1, 1.0 and 10 Gyr,respectively, with τ SF = 5 Gyr and nSN = 1.0 cm−3 . Dashed lines are caseswith dust production by SNe II only and dust reduction through astration,
the same colour corresponding to the same age. Note that the red dashedline overlaps with the red solid line.

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where ηWNM and ηCNM are the mass fraction of WNM and CNM in
the ISM, respectively. From top to bottom, the terms on the righthand side describe reduction of dust due to astration, ejection of dust
from stellar sources, dust destruction by SN shocks, grain growthin the CNM, shattering in the WNM and CNM and coagulation in
the WNM and CNM. To calculate the dust processes which occur
in each ISM phase, as mentioned in Section 1, we assume (1) that
ηWNM and ηCNM are constants and (2) that there are two stablephases, WNM and CNM, in the ISM (namely, the sum of ηWNM andηCNM is unity).
The total mass of grains with radii [a, a + a] ejected by stars
per unit time, Yd (a, t), is expressed asYd (a, t) =

643

644

R. S. Asano et al.

reverse shocks. However, even if the contribution of AGB stars islarger, the result that only a small amount of grains with 0.01 μm
are produced by stars does not change.
From the right-hand panel in Fig. 2, we find that the contributionof dust from AGB stars is relatively large for nSN = 10.0 cm−3 at t =10 Gyr. At t = 10 Gyr, the dust mass ratios produced by AGB stars
and SNe II are 0.16, 0.37 and 1.39 for the cases with nSN = 0.1, 1.0
and 10.0 cm−3 , respectively. This is because a larger amount of dust
grains condensed in the ejecta of SNe II are destroyed by reverse
shocks for higher nSN .

3.1.2 Dust destruction by SN shocks in the ISM
In Fig. 3, we show the evolution of the grain size distribution taking
into account dust destruction by SN shocks in the ISM in addition
to the dust production by SNe II and AGB stars. We also present thecases without the dust destruction (i.e. the same as the solid lines ofFig. 1). The values of τ SF and nSN are set to the same values as inFig. 1. At t  1.0 Gyr, the grain size distributions with and without
the dust destruction by SN shocks in the ISM are very similar toeach other.
Now we estimate the dust destruction time-scale. First, we introduce the sweeping time-scale, τ sweep , at which SN shocks sweep
the whole ISM, asτsweep ≡MISM.
Mswept γSN

From equation (15), if C is defined as
 40 Mφ(m) dm,
C≡
8 M

(30)

(31)

equation (15), with equation (4), can be approximated as
γSN

C

MISM
,
τSF

(32)

where C is about 1.5 × 10−2 from our calculation. Thus, if nSN =1.0 cm−3 , τ sweep ∼ 2–4 × 10−2 τ SF . Next, we approximate the dust
destruction rate by introducing the dust destruction time-scale, τ SN ,as
dMd 
Md
∼−
.
(33)
dt 
τ
SN

SN

The right-hand hand side of equation (14) can be approximated−1
Md (1 − ξ ), where ξ is a typical value of ξX (a, a ); then,as −τsweep
equation (14) reduces toτSN ∼ (1 − ξ )−1 τsweep .

Figure 3. Grain size distribution taking into account the dust destruction
by SN shocks in the ISM in addition to the processes in Fig. 1 (dotted lines).
The values of τ SF and nSN are the same as in Fig. 1. Solid lines are the sameas in Fig. 1, the same colour corresponding to the same age. Note that the
red dotted line overlaps with the red solid line.

(34)

Since the overall efficiency of dust destruction, (1 − ξ ), is ∼0.3 fornSN = 1.0 cm−3 (Nozawa et al. 2006), we obtain τ SN ∼ 0.1τ SF . Thus,
the difference between the cases with and without dust destruction
cannot be seen at t = 0.1 Gyr in Fig. 3, where τ SN ∼ 0.1τ SF ∼0.5 Gyr.
We find that dust grains with a  0.1 μm are destroyed effectively at 10 Gyr >τ SN (compare the solid and dotted lines in Fig. 3).
Since the decreasing rate of grain radius by sputtering does not depends on the grain radius, smaller grains are effectively destroyedin SN shocks, and the amount of smaller grains decreases (Nozawa
et al. 2006).

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Figure 2. Same as in Fig. 1, but we adopt different values of nSN : 0.1 cm−3 in the left-hand panel and 10.0 cm−3 in the right-hand panel. Note that the reddashed lines overlap with the red solid lines.

Evolution of the grain size distribution

645

In Fig. 4, we show the cases with nSN = 0.1 and 10 cm−3 . Comparing Figs 3 and 4, we find that a larger amount of dust grains aredestroyed for higher nSN . The destruction effect is more pronounced
at small sizes. Indeed, we observe that grains with a  1.0 μm are
effectively destroyed in the case with nSN = 10.0 cm−3 . Nevertheless, the result that smaller grains are effectively destroyed does notchange, and we find that dust grains with radii larger than 0.1 μm
mainly dominate the total dust amount in galaxies. Consequently,
if the dust destruction by sputtering in SN shocks is dominant, only
large (a  0.1 μm) grains can survive in the ISM.
3.1.3 Grain growthFig. 5 shows the evolution of the grain size distribution taking into
account the dust production from stellar sources, dust destruction

and grain growth. We adopt τ SF = 5 Gyr, nSN = 1.0 cm−3 and themass fraction for the CNM, ηCNM = 0.5. From Fig. 5, we observethat while the grain size distributions with and without grain growth
are almost the same at ages 0.1 and 1 Gyr, the difference is clear at
10 Gyr. The effect of grain growth is prominent around a ∼ 0.3 μm
at 10 Gyr, since the total surface area of grains is dominated bygrains with a ∼ 0.3 μm. The time-scale of grain growth is discussed
in detail in Section 4.
In Fig. 5, we also plot the slope of the MRN size distribution,f (a)da ∝ a−3.5 da (Mathis et al. 1977), which is thought to be thegrain size distribution in the Milky Way. From Fig. 5, it is clearthat the small grains with a  0.01 μm are too few to reproduce
the MRN size distribution. However, the existence of the 70 μm
excess is considered to be a proof of the existence of small grains(Bernard et al. 2008). Furthermore, Takeuchi et al. (2003, 2005)
argued by using their infrared SED model that small grains arenecessary to reproduce the near–mid infrared SEDs of star-forming
galaxies. Consequently, when we consider the case in which dust
production by SNe II and AGB stars, dust destruction and grain
growth take place, the grain size distribution is always dominatedby large grains, and we need to consider other processes to produce
small grains efficiently.
3.2 Grain–grain collision effects

Figure 5. Grain size distribution taking into account the dust production
from stellar sources, dust destruction and grain growth with τ SF = 5 Gyrand nSN = 1.0 cm−3 (long-dashed lines). We adopt ηCNM = 0.5. Dottedlines represent cases without grain growth (the same as in Fig. 3), the same
colour corresponding to the same age. Note that the red dotted line overlaps
with the red long-dashed line. Green solid line represents the slope of thepower-law grain size distribution with index −3.5 [f (a)da ∝ a−3.5 da (Mathis
et al. 1977)] which is thought to be the grain size distribution in the MilkyWay.

In the above we have investigated the dust processes which directly
affect the evolution of the total dust mass in galaxies: dust production by AGB stars and SNe II, dust destruction by SN shocks
and grain growth. As shown above, these processes cannot produce
small grains (a  0.01 μm) efficiently. Therefore, we now consider
the contributions of the grain–grain collisions, shattering and coagulation in turbulence, to the grain size distribution. If these processes
occur, although the total dust mass in galaxies does not change, thegrain size distribution does.

3.2.1 Shattering
In the left-hand panel of Fig. 6, we show the evolution of the grain
size distribution in the galaxy with and without shattering (all other
dust processes in Section 3.1 are included). The right-hand panel ofFig. 6 shows the time evolution of dust-to-gas mass ratio (Md /MISM )Downloaded from http://mnras.oxfordjournals.org/ at University of Tokyo Library on May 21, 2013

Figure 4. Same as in Fig. 3, but we adopt different values of nSN : 0.1 cm−3 in the left-hand panel and 10.0 cm−3 in the right-hand panel. Note that the red
dotted lines overlap with the red solid lines.

646

R. S. Asano et al.

for the cases with and without shattering, respectively. We also plot
the case without grain growth and shattering and the evolution of
metallicity in the same panel. The parameters τ SF and nSN are set
to be 5 Gyr and 1.0 cm−3 , respectively. We adopt ηWNM = ηCNM =0.5.
From the left-hand panel of Fig. 6, at the early stage of galaxy
evolution (0.1 Gyr) the size distributions with and without shattering
are similar with only a little difference at small sizes. At 1 Gyr, weobserve that the size distribution has a bump at a ∼ 0.001 μm inthe case with shattering. As time passes, the amount of large grainsdecreases, and as a result the size distribution is dominated by small
grains. This behaviour is substantially different from that of the casewithout shattering. We now discuss this behaviour in more detail.
As shown in equation (23), the efficiency of shattering is larger
for larger amount of grains (Hirashita 2010). At 0.1 Gyr, the efficiency of shattering is low because of the small dust abundance.
As a result, there is only a small difference between the cases withand without shattering. At t = 1 Gyr, since shattering occurs efficiently due to the increased amount of large grains, the amount ofsmall grains increases. At the same time, we observe that the grain
size distribution has little difference between the cases with andwithout shattering at a > 0.1 μm in the left-hand panel of Fig. 6.
This is because shattering of a tiny fraction of large grains can
produce a large amount of small grains (Hirashita et al. 2010). Furthermore, since the number of small grains increases, the small
grains dominate the total grain surface area. Consequently, grain
growth occurs at the smallest grain sizes (a  10−3 μm), forming abump at ∼10−3 –10−2 μm. At t = 10 Gyr, since the number of small
grains increases, large grains are shattered more efficiently by thefrequent collisions with the small grains. Consequently, comparing
the grain size distribution at 10 Gyr with that at 1 Gyr, the amount oflarge grains decreases significantly. Furthermore, because of grain
growth, the bump is shifted to a larger size at 10 Gyr than at 1 Gyr,and finally the size distribution has a large bump at a ∼ 0.01 μm at
10 Gyr.
Focusing on the grain size distribution at 10 Gyr, we find that if
shattering occurs, the amount of grains with a > 0.2 μm is morethan two orders of magnitude smaller than that of grains witha < 0.2 μm. Thus, the maximum size of grains in diffuse ISM isdetermined not by stardust but by the process of shattering.

In the right-hand panel of Fig. 6, we find that grain growth starts
to increase the total dust mass at around t = 1 Gyr as seen in the
rapid increase of dust-to-gas mass ratio, and grain growth becomes
more rapid in the case with shattering than in the case without
shattering because of the increased number of small grains. As
discussed in Kuo & Hirashita (2012), shattering contributes not only
to the evolution of the grain size distribution but also to the total
dust mass in galaxies indirectly through the enhanced grain growth.
Thus, shattering is a very important process in understanding theevolution of the size distribution and the amount of dust grains in
the ISM.

3.2.2 Coagulation
In Fig. 7, we show the evolution of the grain size distribution withand without coagulation (all the other dust processes are included)in the left-hand panel, and the time evolutions of dust-to-gas mass
ratio with and without coagulation and of metallicity in the righthand panel. The parameters adopted are the same as in Fig. 6.
From the left-hand panel of Fig. 7, we find that there is little difference between the cases with and without coagulation at 0.1 and
1.0 Gyr. Since larger grains are coupled with the larger-scale turbulence, they can obtain larger velocity dispersions. Thus, coagulation
mainly occurs by collisions between small grains whose velocity
dispersions are smaller than the coagulation threshold (equation
27). However, since the abundance of small grains is low, the contribution of coagulation is not seen at 0.1 and 1 Gyr before shattering becomes effective. After that, a large abundance of small
grains are produced by shattering so coagulation becomes effective.
Consequently, the bump at a ∼ 0.01 μm shifts to a larger size by
coagulation.
From the right-hand panel of Fig. 7, we find that the evolutionof the total dust mass does not change significantly by coagulation,
confirming the result obtained by Hirashita (2012). If coagulation
occurs, the number of small grains decreases; as a result, the surfaceto-volume ratio of grains decreases. This effect may suppress theincrease in dust mass due to grain growth. However, since grain
growth becomes inefficient to the dust evolution before coagulation becomes efficient (the details are shown in Section 4), theDownloaded from http://mnras.oxfordjournals.org/ at University of Tokyo Library on May 21, 2013

Figure 6. Left-hand panel: grain size distribution with (dot–dashed lines) and without (long-dashed lines) shattering (all other dust processes in Section 3.1
are included). Note that the red dot–dashed line overlaps with the red long-dashed line. Right-hand panel: time evolution of dust-to-gas mass ratio with (solidline) and without (dotted line) shattering. The case without grain growth and shattering (dashed line) is also plotted. Dot–dashed line represents the evolutionof metallicity. The parameters τ SF and nSN are set to be 5 Gyr and 1.0 cm−3 , respectively. We adopt ηWNM = ηCNM = 0.5.

Evolution of the grain size distribution

647

Figure 8. Grain size distribution with (ηWNM , ηCNM ) = (0.9, 0.1) (solid lines with filled circles in the left-hand panel) and (0.1, 0.9) (solid lines with filled
diamonds in the right-hand panel). The values of parameters τ SF and nSN are the same as in Fig. 7. Triple-dot–dashed lines are the case with ηWNM = ηCNM =0.5.

contribution of coagulation cannot be observed for the total dust
mass evolution. Consequently, the effect of coagulation on the evolution of the total dust mass in galaxies is negligible.
From the left-hand panel of Fig. 7, we find that the amount ofgrains with a > 0.2 μm does not change significantly by coagulation because coagulation cannot occur by collision between large
grains which have larger velocity dispersions than the coagulationthreshold. Thus, although the bump of the grain size distribution isshifted to a larger size by coagulation, coagulation does not affectthe maximum size determined by shattering (Section 3.2.1).
3.3 Parameter dependence
Shattering and coagulation occur differently in both ISM phases(WNM and CNM). Here, by adopting (ηWNM , ηCNM ) = (0.9, 0.1)
and (0.1, 0.9), we show the effect of ISM phases.
In Fig. 8, we show the evolution of the grain size distribution with(ηWNM , ηCNM ) = (0.9, 0.1) (left-hand panel) and (0.1, 0.9) (righthand panel). The case with ηWNM = ηCNM = 0.5 is shown forcomparison. At t = 0.1 Gyr, the grain size distributions are almost

the same in all the cases, since the dust process is dominated by
the production by stellar sources. At t = 1 Gyr, the amount of dust
grains with a < 0.01 μm is larger for the case with larger ηCNM
because grain growth is more efficient. At 10 Gyr, the differenceis clear at each grain size. For larger ηWNM , the decrements of theamount of dust grains with a > 0.2 μm is larger because shatteringin the WNM is more efficient. Furthermore, for larger ηCNM , thebump produced by grain growth around 0.01 μm shifts to a larger
size. Thus, we understand that the amount of dust grains with a >0.2 μm and the shift of the bump around 0.01 μm are dominatedby shattering in WNM and coagulation in CNM, respectively. In
addition, comparing the two panels in Fig. 8, we find that the dust
amount at a ∼ 0.1–0.2 μm tends to be smaller for a larger ηCNM .5 It
means that the amount of dust grains with a ∼ 0.1 μm is dominatednot by shattering in WNM but shattering in CNM. Hence, the grain
size distribution in galaxies is finally dominated by processes in

5 The mass ratio of grains with a ∼ 0.1–0.2 µm for the cases between(ηWNM , ηCNM ) = (0.5, 0.5) and (0.9, 0.1) is about 0.6 at t = 10 Gyr.

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Figure 7. Left-hand panel: grain size distribution with (triple-dot–dashed lines) and without (dot–dashed lines) coagulation (all the other dust processes areincluded). Note that the red and blue dot–dashed lines overlap with the red and blue triple-dot–dashed lines. Right-hand panel: time evolution of dust-to-gas
mass ratio with (solid line) and without (dotted line) coagulation. Dot–dashed line represents the evolution of metallicity. The values of parameters (τ SF , nSN ,ηWNM and ηCNM ) are the same as in Fig. 6.

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R. S. Asano et al.

WNM for large grains (>0.2 μm) and by processes in CNM forsmall grains (∼0.1 μm).
4 DISCUSSION

5 CONCLUSIONS
We constructed a dust evolution model taking into account thegrain size distribution in a galaxy, and investigated what kind of
dust processes dominate the grain size distribution at each stage ofgalaxy evolution. In this paper, we considered dust formation by
SNe II and AGB stars, dust destruction by SN shocks in the ISM,
grain growth in the CNM and grain–grain collisions (shattering andcoagulation) in the WNM and CNM.
We found that the grain size distribution in galaxies is dominated by large grains produced by stars in the early stage of galaxy
evolution, but as time passes the size distribution is controlled byprocesses in the ISM (grain growth, shattering and coagulation) andthe age at which these ISM processes enter depends on the star formation time-scale, as ∼1(τ SF /Gyr)1/2 Gyr. While dust production
by SNe II and AGB stars, dust destruction by SN shocks and grain
growth in the CNM directly affect the total dust mass evolution, we
found that the grains are predominantly large (a ∼ 0.2–0.5 μm) andonly a small amount of small grains (a < 0.01 μm) are produced bythese processes. If we take shattering and coagulation into account,the grain size distribution is modified significantly by these twoprocesses. In particular, shattering indirectly contributes to the large
increase of the total dust mass: After small grains (a  0.01 μm)
are produced by shattering, grain growth becomes more effective

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In Section 3, we showed the evolution of the grain size distribution
in galaxies for a variety of mixture of dust processes. We found thatthe grain size distribution is dominated by large grains produced bystars (SNe II and AGB stars) in the early stage of galaxy evolution,
but as the time passes the number of small grains increases due
to shattering, and the small grains grow to larger grains by grain
growth. After that, the size distribution shifts to larger sizes due tocoagulation. Thus, we conclude that, while the grain size distribution in galaxies is controlled by stellar processes in the early stageof galaxy evolution, the main driver to change the size distribution
is replaced with the processes in the ISM (shattering, coagulationand grain growth) at the later stage of galaxy evolution. These processes (shattering, coagulation and grain growth) have time-scales
dependent on the grain size distribution. In this section, by adopting
representative grain radii, 0.001, 0.01, 0.1 and 1.0 μm, we discuss
the evolution of the grain size distribution more quantitatively.
Fig. 9 shows the time evolution of the ratio between the size
distribution functions, f (a) and f (a)star , the latter being obtained byconsidering only the stellar processes (the first and second terms
in the right-hand side of equation 28). Panels (a), (b) and (c) are
the cases with (ηWNM , ηCNM ) = (0.5, 0.5), (0.9, 0.1) and (0.1, 0.9),
respectively, for τ SF = 5 Gyr and nSN = 1.0 cm−3 . From Fig. 9,
we find that behaviour of f (a)/f (a)star depends strongly on the grainradius. As mentioned above, this is because each dust process works
at different grain radii on different time-scales. First, we discuss theevolution of the grain size in panel (a). We find that f (a)/f (a)star at
a = 0.001 μm starts to deviate from unity at the earliest galactic age
among all four grain sizes. The process that causes this increase isshattering. These small grains are produced by shattering between
large grains produced by stars. Furthermore, as mentioned in Section 3.2.1, since shattering of a small number of large grains can
produce a large number of small grains, we cannot see the changeof f (a)/f (a)star for a = 0.1 and 1.0 μm.
At t ∼ a few hundreds of Myr, f (a)/f (a)star at a = 0.01 μm
increases. This increase is also due to shattering. The reason
why the effect of shattering appears at a = 0.01 μm later than at
a = 0.001 μm is that the size distribution of shattered fragments
is proportional to a−3.3 (see Section 2.2.5). In other words, theshattered fragments become dominant at smaller sizes on shortertime-scales than at larger sizes.
At t ∼ 1 Gyr, we find that the increase of f (a)/f (a)star at a =0.01 μm is accelerated. This indicates that another process becomesefficient, and it is grain growth. As seen from Fig. 6, as grain growth
becomes efficient around 1 Gyr, the amount of grains with less thana ∼ 0.01 μm increases significantly.
At t ∼ 2 Gyr, we find that f (a)/f (a)star decrease at all sizes. These
decreases are due to coagulation for small grains (a = 0.001 and0.01 μm) and shattering for large grains (a = 0.1 and 1.0 μm). As weshowed in Section 3.2.2, coagulation mainly occurs between small
grains. Thus, the coagulation effect cannot be seen at early phaseof galaxy evolution when the abundance of small (a  0.01 μm)
grains is small. Shattering can also occur effectively if there is a large
amount of small grains because of a high grain–grain collision ratewith small grains (cf. equation 24). In addition, the main reason why
the decrements of grains with a = 0.1 and 1.0 μm are different isshattering in different ISM phases. As shown in Section 3.3, grains
with a > 0.2 μm are mainly dominated by shattering in WNM,

while grains with a ∼ 0.1 μm are dominated by shattering in CNM.
In summary, at early phase of galaxy evolution (t  10 Myr), thesize distribution is dominated by dust grains produced by stars, aftert  100 Myr, the dust processes in the ISM begin to affect the size
distribution at small size and at t ∼ 2 Gyr (for τ SF = 5 Gyr), various
dust processes in the ISM affect all sizes of grains.
Panels (b) and (c) in Fig. 9 show the cases with (ηWNM , ηCNM ) =(0.9, 0.1) and (0.1, 0.9), respectively. Compared with panel (a), we
find that f (a)/f (a)star at a = 0.01 μm does not decrease at 10 Gyrin panel (b). This is because the time-scale of coagulation becomes
longer for smaller ηCNM . From panel (c) (ηWNM = 0.1, ηCNM = 0.9),
we find that the decrement at a = 1.0 μm is smaller than those in
the cases of panels (a) and (b). This is because the efficiency ofshattering in WNM is smaller for smaller ηWNM . However, from all
the three panels, we can observe that the timing at which f (a)/f (a)starat all sizes changes due to the dust processes in the ISM (in thiscase, it is about 2 Gyr) does not vary significantly by the change of(ηWNM , ηCNM ) for the same star formation time-scale.
In order to discuss the effect of τ SF on the size distribution, the results are shown for the same values of the parameters as in the panel(a) of Fig. 9, but for τ SF = 0.5 Gyr in panel (d) and τ SF = 50 Gyr inpanel (e). Compared with panel (a), we find that f (a)/f (a)star change
at earlier stages for shorter τ SF at all sizes. This is explained as follows. If τ SF is short, the amounts of dust and metals released bystars are large at early phases of galaxy evolution. The time-scales ofshattering and coagulation are inversely proportional to the dust-togas mass ratio (e.g. Hirashita & Omukai 2009; Hirashita 2010), andthe time-scale of grain growth is inversely proportional to metallicity (e.g. Asano et al. 2013). Thus, for shorter τ SF , dust processes in
the ISM (grain growth, shattering and coagulation) begin to affectthe size distribution at earlier stages of galaxy evolution (∼0.6, 2
and 5 Gyr for τ SF = 0.5, 5 and 50 Gyr, respectively). The timescale of the change of f (a)/f (a)star is roughly estimated to be ∼1(τ SF /Gyr)1/2 Gyr (Appendix B). We conclude that the grain size
distribution in galaxies changes drastically through the galaxy evolution because different dust processes operate on the grain size
distribution at different ages.

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649

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Figure 9. Time evolution of the ratio between the size distribution functions, f (a), and f (a)star , the latter being obtained by considering only stellar processes(the first and second terms in right-hand side of equation 28). Panel (a), (b) and (c) are the cases with (ηWNM , ηCNM ) = (0.5, 0.5), (0.9, 0.1) and (0.1, 0.9),
respectively, with τ SF = 5 Gyr and nSN = 1.0 cm−3 . Panel (d) and (e) are the case with τ SF = 0.5 and 50 Gyr, respectively, with (ηWNM , ηCNM ) = (0.5, 0.5).
Solid, dotted, dashed and dot–dashed lines represent the ratio of a = 0.001, 0.01, 0.1 and 1.0 µm, respectively.

because of the enhanced surface-to-volume ratio. Furthermore,grain growth produces a large bump in the grain size distribution
around a = 0.01 μm. The effects of shattering in WNM and CNM
on the size distribution appear at different grain radii: while grains
with a > 0.2 μm are mainly shattered in WNM, shattering in CNM
affects grains with a ∼ 0.1 μm. Furthermore, the effect of shatter-

ing, in particular shattering in WNM, is large enough to determine
the maximum size of grains in the ISM. Coagulation occurs effectively after the abundance of small grains is enhanced by shattering,and the grain size distribution is deformed to have a bump at alarger size (a ∼ 0.03–0.05 μm at t ∼ 10 Gyr) by coagulation. We
conclude that the evolution of both the total dust mass and the grain

650

R. S. Asano et al.

size distribution in galaxies are related strongly to each other andthe grain size distribution changes drastically through the galaxy
evolution.

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

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A P P E N D I X A : E X A M I N AT I O N O F
PA R A M E T E R D E P E N D E N C E
In this appendix, we show dust evolution models with parameters
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We thank the anonymous referee for many suggestions, which were
useful to improve the quality and the clarity of this paper. We are
grateful to Akio K. Inoue, Takashi Kozasa, Daisuke Yamasawa andAsao Habe for fruitful discussions, to Satoshi Okuzumi and Hiroshi
Kobayashi for helpful discussions on the process of grain–grain
collision, to Lars Mattsson for his comments which improve thepresentation of this paper and to Jennifer M. Stone for checking
the English. RSA acknowledges the hospitality of the members in
Institute of Astronomy and Astrophysics, Academia Sinica during his stay. RSA has been supported from the Grant-in-Aid for
JSPS Research under Grant No. 23-5514. RSA and TTT have been
also partially supported from the Grant-in-Aid for the Global COE
Program ‘Quest for Fundamental Principles in the Universe: From
Particles to the Solar System and the Cosmos’ from the Ministryof Education, Culture, Sports, Science and Technology (MEXT) of
Japan. TTT have been supported by the Grant-in-Aid for the Scientific Research Fund (TTT: 23340046, 24111707) commissioned by
the MEXT and by the Strategic Young Researcher Overseas Visits Program for Accelerating Brain Circulation commissioned by
the JSPS (R2405). HH is supported by NSC grant 99-2112-M-001006-MY3. TN has been supported by World Premier International
Research Center Initiative (WPI Initiative), MEXT, Japan, and by
the Grant-in-Aid for Scientific Research of the JSPS (22684004,
23224004).

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Evolution of the grain size distribution

651

A1 The Schmidt law index n = 1.5

A2 The index of the Salpeter IMF q = 1.35

In Fig. A1, we show star formation history (SFH) and the evolutionof the grain size distribution with the Schmidt law index n = 1.0
and 1.5. To compute the SFH and grain size distribution by using
star formation rate with the Schmidt index n = 1.5, the SFR with
the Schmidt law index n = 1.5 (SFR1.5 ) is expressed as

Fig. A2 shows the SFH and the evolution of the grain size distribution with the power-law index of the Salpeter IMF q = 1.35 and2.35 (fiducial value in this paper). We observe that SFRs are almost
the same, but the grain size distributions are different. If q is small,that is, a large number of SNe II are produced, the abundance of dust
and metals increase earlier than the case with large q. As a result,the dust amount of each size of grains [the values of a4 f (a)] withq = 1.35 is larger than the case with q = 2.35. The dust processes
in the ISM also become effective earlier because of the larger dust
abundance. However, we find that the trend of the evolution of thegrain size distribution (at early phases, stars are dominant sourcesof dust, as time passes, the processes in the ISM become important)does not change.

SFR1.5 (t) =

1.5
(t)
MISM
,
ν1.5

(A1)

where ν 1.5 is a constant. We define the value of ν 1.5 so that it satisfies
the following equation at t = 0:
MISM (t)= τSF .
SFR1.5 (t)

(A2)

0.5
.
ν1.5 = τSF Mtot

(A3)

From Fig. A1, we find that the results are not significantly different
between the cases with the n = 1.0 and 1.5.

APPENDIX B: TIME-SCALE OF THE CHANGE
O F f (a)/ f (a)star
In Section 4, we found that the time-scale of the change off (a)/f (a)star of all sizes of grains depends on star formation timescale, and the change are due to coagulation for small grains andFigure A1. Left-hand panel: the star formation history with n = 1.0 (solid line) and 1.5 (dotted line). Right-hand panel: grain size distribution with n = 1.0(triple-dot–dashed line) and 1.5 (solid line with triangles). We adopted τ SF = 5 Gyr, nSN = 1.0 cm−3 and ηWNM = ηCNM = 0.5 in these plots. Note that the
red and blue triple-dot–dashed lines overlap with the red and blue solid lines with triangles.

Figure A2. Left-hand panel: the star formation history with q = 2.35 (solid line) and 1.35 (dotted line). Right-hand panel: grain size distribution with q =2.35 (triple-dot–dashed lines) and 1.35 (solid lines with filled squares). We adopted τ SF = 5 Gyr, nSN = 1.0 cm−3 and ηWNM = ηCNM = 0.5 in these plots.

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shattering for large grains. Since both of shattering and coagulation are collisional processes, the time-scales scale with the grainabundance in the same way. In order to evaluate the dependence on
the star formation time-scale, we compare the contributions of stars
and shattering.
First, we consider the stellar contribution (equation 29). If D is
defined as ∞  100 Mmd (m, Z(t − τm ), a)φ(m) dm da,
(B1)
D≡
0

mcut (t)with equation (4), the stellar contribution can be approximated as
dMd 
MISM
D
.
(B2)

dt starτSF
Then, we consider the time-scale of shattering, τ shat . Since shattering is a collisional process, τ shat can be represented as1,
π a 2 vngrains(B3)

where a2  is the second moment of a grain size a, v is the relative
velocity of grains and ngrain is the number density of grains, which
is given byMd
4
π a 3 sngrain ∼ μnH,shat mH
,
3
MISM

In addition, by substituting equations (B3) and (B4) into equation(B5), we obtainτshat

4
π a 3 s
3

1/2

τ .
π a 2 vμmH nH,shat D SF

(B6)

To evaluate this value, we adopt s = 3.0 g cm−3 , v = 20 km s−1 andnH, shat = 0.3 cm−3 (WNM) as a representative value. Also, fromour calculation, D 10−3 , and a3 / a2  10−5 cm for dust grains
produced by stars. Then, we finally obtainτSF 1/2[Gyr].
(B7)
τshat ∼ 1
GyrThus, we conclude that the time-scale of shattering, that is, the
1/2time-scale of the change of f (a)/f (a)star , is proportional to τSF .

(B4)

where a3  is the third moment of a grain size, s is the bulk density of dust grains, nH, shat is the hydrogen number density in theThis paper has been typeset from a TEX/LATEX file prepared by the author.

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τshatregion where shattering occurs and mH is the mass of the hydrogenatom. We assume the contribution of shattering to the amount of
dust grains as Md /τshat , and comparing this equation with equation(B2), we obtain the relation between shattering time-scale and starformation time-scale:
Md 1
.
(B5)
τshat τSF
MISM D