MNRAS 446, 2659–2672 (2015)

doi:10.1093/mnras/stu2298

Supernova dust formation and the grain growth in the early universe: thecritical metallicity for low-mass star formation
Gen Chiaki,1‹ Stefania Marassi,2 Takaya Nozawa,3 Naoki Yoshida,1,4
Raffaella Schneider,2 Kazuyuki Omukai,5 Marco Limongi2,4,6 and Alessandro Chieffi7
1 Department of Physics, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
2 INAF/Osservatorio Astronomico di Roma, via Frascati 33, I-00040 Roma, Italy
3 National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan

277-8583, Japan
5 Astronomical Institute, Tohoku University, 6-3 Aramaki, Aoba, Sendai 980-8578, Japan
6 European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85748 Garching bei Munchen, Germany
7 INAF/IASF, Via Fosso del Cavaliere 100, I-00133 Roma, Italy

Accepted 2014 October 24. Received 2014 October 23; in original form 2014 July 17

ABSTRACT

We investigate the condition for the formation of low-mass second-generation stars in theearly Universe. It has been proposed that gas cooling by dust thermal emission can trigger
fragmentation of a low-metallicity star-forming gas cloud. In order to determine the criticalcondition in which dust cooling induces the formation of low-mass stars, we follow the thermal
evolution of a collapsing cloud by a one-zone semi-analytic collapse model. Earlier studies
assume the dust amount in the local Universe, where all refractory elements are depleted on to
grains, and/or assume the constant dust amount during gas collapse. In this paper, we employ
the models of dust formation and destruction in early supernovae to derive the realistic dust
compositions and size distributions for multiple species as the initial conditions of our collapse
calculations. We also follow accretion of heavy elements in the gas phase on to dust grains, i.e.
grain growth, during gas contraction. We find that grain growth well alters the fragmentation
property of the clouds. The critical conditions can be written by the gas metallicity Zcr andthe initial depletion efficiency fdep,0 of gas-phase metal on to grains, or dust-to-metal mass
ratio, as (Zcr /10−5.5 Z ) = (fdep,0 /0.18)−0.44 with small scatters in the range of Zcr = [0.06–
3.2] × 10−5 Z . We also show that the initial dust composition and size distribution areimportant to determine Zcr .
Key words: stars: formation – stars: low-mass – stars: Population II – ISM: abundances – dust,
extinction – galaxies: evolution.

1 I N T RO D U C T I O N
The first stars (Population III or Pop III stars) formed in metal-free
gas are thought to be predominantly massive with several tens tothousand solar masses (Bromm et al. 2001; Abel, Bryan & Norman
2002; Omukai & Palla 2003; Yoshida et al. 2006; Hosokawa et al.
2011; Susa, Hasegawa & Tominaga 2014; Hirano et al. 2014). This
is in stark contrast with the typical mass of the Galactic stars whichis less than the solar mass (Kroupa 2002). Therefore, how andwhen the transition of the typical stellar mass occurred is one of thecritical issues for understanding the star-forming history throughout
the cosmic time. The long-lived low-mass stars discovered in the
Galactic halo may be the fossils of the first low-mass stars.

 E-mail: gen.chiaki@utap.phys.s.u-tokyo.ac.jp

Recent studies propose that the formation of the first low-massstars is driven by the fine-structure line cooling by carbon and oxygen (Bromm et al. 2001; Bromm & Loeb 2003; Santoro & Shull
2006; Frebel, Johnson & Bromm 2007; Ji, Frebel & Bromm 2014).
However, the gas density at which the fine-structure line cooling cancause fragmentation of star-forming clouds is nH ∼ 104 –105 cm−3 ,where the typical mass of the fragments (comparable to the Jeansmass) is expected to be as large as ∼100 M . It appears that the efficient cooling by fine-structure lines at the low densities cannot be
the main mechanism to form Jeans-unstable fragments with small
masses. Hence, in order to produce low-mass stars in this scenario,
protostars must be formed and ejected from such massive clumps
by N-body interactions before they grow to high masses (e.g. Ji
et al. 2014). Although not entirely impossible, the scenario seems
to require rather specific conditions. Intriguingly, other studies argue that the presence of dust grains is essential for the formation

C 2014 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society

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4 Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba

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MNRAS 446, 2659–2672 (2015)

expanding ejecta and the destruction of dust by the reverse shocks,
but present different compositions, size distributions, and massesof dust. Thus, the fragmentation condition can be different for thetwo dust models. In this paper, we explore the thermal evolutionof low-metallicity star-forming clouds by employing these two setsof supernova dust models and by taking into account the growth ofmultiple grain species with size distributions.
In Section 2, we describe our one-zone model of the cloud
evolution and the calculation of grain growth. In Section 3, we discuss the effect of grain growth on the thermal evolution of clouds.
Then, we present the critical abundances of heavy elements for theformation of low-mass fragments. Concluding remarks and discussion are given in Section 4.
2 NUMERICAL METHOD
2.1 Collapse model and gas-phase chemistry
We follow the evolution of the cloud temperature T along with theincreasing density calculated by a one-zone semi-analytic collapse
model of Omukai (2000). In this paper, the cloud density is described as the hydrogen number density nH = XH ρ/mH , where XH
is the hydrogen mass fraction, ρ is the total (gas and dust) massdensity, and mH is the mass of a hydrogen atom. We include in thismodel non-equilibrium gas chemistry of eight species of primordial elements H+ , e− , H, H− , H2 , D+ , D, and HD, and 19 species
made of heavy elements C+ , C, CH, CH2 , CO+ , CO, CO2 , O+ , O,
OH+ , OH, H2 O+ , H2 O, H3 O+ , O+
2 , O2 , Si, SiO, and SiO2 . We solve
the reduced chemical networks of Omukai, Hosokawa & Yoshida(2010) supplemented with the silicon chemistry. For species containing silicon, oxidation of Si and SiO is considered as the major
reactions. Table 1 and Fig. 1 present the gas-phase chemical reactions of silicon and the rate coefficients used in this work. We also
Table 1. Silicon chemical reactions.
No.

Reaction

Rate coef.

Ref.

Si1
Si2
Si3

Si + OH → SiO + H
Si + O2 → SiO + O
SiO + OH → SiO2 + H

3 × 10−11
1.3 × 10−11 exp (−111/T)
2 × 10−13

H80
LG90
H80

Note: The units of temperature T and rate coefficients are K and cm3 s−1 ,respectively. Ref.: H80, Hartquist, Dalgarno & Oppenheimer (1980);
LG90, Langer & Glassgold (1990).

Figure 1. Chemical network of silicon (white arrows) which are included in
our one-zone calculations. We also consider accretion of gas-phase species
on to grains (indicated by black arrows). The reaction tabs above the arrows
are identical with the reaction numbers in Table 1. Circles and squares depict
gas-phase (g) and solid-phase (s) species, respectively.

Downloaded from http://mnras.oxfordjournals.org/ at National Astronomical Observatory of Japan on December 11, 2014of the first low-mass stars; heat transfer via collisions from gas
to dust effectively cools the gas and induces the gravitational instability, to trigger the core fragmentation into small clumps. Thisoccurs at very high gas densities of nH ∼ 1012 –1014 cm−3 , where thetypical mass scale of fragments is in the range of ∼0.01–0.1 M
(e.g. Schneider et al. 2003, 2006). Assuming a certain grain size
distribution and the dust-to-metal mass ratio (∼0.5) of the local
interstellar medium, Omukai et al. (2005) derive the critical metallicity of Zcr ∼ 10−5.5 Z , above which the dust cooling can trigger
the gas fragmentation.
It is important to note that the conditions for fragmentation of lowmetallicity gas clouds depend on the properties of dust grains, suchas the size distribution and the composition, which are expected to bedifferent in the early Universe. For example, recent observations ofdamped Lyman α systems with metallicity Z ∼ 10−3 Z reveal thatthe mass fraction of dust relative to metal is smaller than the presentday value (Molaro et al. 2000; De Cia et al. 2013). In the local
Universe, dust grains are thought to be formed in the stellar wind of
asymptotic giant branch stars as well as in the ejecta of supernovae.
Subsequent accretion of heavy elements on to dust in molecular
clouds also contributes the enhancement of dust mass fraction. The
prompt formation path of dust in the early Universe is limited to the
Pop III supernovae whose progenitors have short lifetimes (Todini &Ferrara 2001; Nozawa et al. 2003). Newly formed grains in the
supernova ejecta are, however, destroyed by sputtering after thereverse shocks penetrate into the ejecta (Bianchi & Schneider 2007;
Nozawa et al. 2007, hereafter N07; Silvia, Smith & Shull 2010,2012). Metals locked up in dust grains are partly returned into thegas phase. Therefore, the mass ratio of grains to gas-phase metals
can be significantly small. It is thus important to study the role ofdust grains in the first galaxies by employing realistic dust formation
models.
Schneider et al. (2012a, hereafter S12) and Schneider et al.
(2012b) investigate the fragmentation properties of gas clouds preenriched with the grains that are produced by Pop III supernovae.
They show that the initial dust-to-gas mass ratio is a key quantity forthe fragmentation condition. The dust-induced fragmentation is not
triggered if this ratio is significantly reduced by the destruction ofdust in the supernova ejecta. The conclusion is drawn under the assumption that the size distribution and dust-to-gas mass ratio neverchange in the course of cloud collapse. Nozawa, Kozasa & Nomoto(2012) point out that the growth of dust grains due to the accretionof heavy elements can take place and increase the dust-to-gas mass
ratio even in very low metallicity gas clouds.
Chiaki, Nozawa & Yoshida (2013) show that the cloud fragmentation can be triggered by the grain growth even if the initial fractionof heavy element condensed into grains is as small as 0.001. Whenthe grain growth is considered, the fragmentation conditions relyon the gas metallicity and the initial grain size rather than only on
the initial dust-to-gas mass ratio. The estimated critical metallicity is Zcr ∼ 10−4.5 Z for the initial grain radii r0 ≥ 0.1 µm, andZcr ∼ 10−5.5 Z for r0 ≤ 0.01 µm. Chiaki et al. (2013), however,assume a single size and a single component of dust (MgSiO3 , enstatite) which may be too simple as a realistic dust model in theearly Universe. Clearly, further studies of the thermal evolutionof the collapsing clouds based on more realistic dust models arenecessary to clarify the formation condition of the first low-massstars.
There are two leading studies that independently predict the size
distribution and the total amount of dust ejected from Pop III supernovae with a wide range of progenitor masses: the models of
N07 and S12. Both of the studies treat the formation of dust in theSN dust and grain growth: the critical metallicity

2661

solve the cooling rates by emission lines of atoms/ions (C I, C II,
and O I), and by molecules (H2 , HD, CO, OH, and H2 O). Whenthe cloud becomes optically thick, the cooling rate from each line
is reduced by the corresponding escape fraction. To this end, we
determine the column density assuming the size of the cloud to begiven by the Jeans length.
We set the initial hydrogen number density and temperature
nH,0 = 0.1 cm−3 and T0 = 300 K, respectively.1 The initial number abundances of H+ and H2 relative to hydrogen nuclei arey0 (H+ ) = 10−4 and y0 (H2 ) = 10−6 . The total number abundancesof deuterium and helium are 3.0 × 10−5 and 0.083 (corresponding
to the mass fraction YHe = 0.25), respectively. We assume that elements C, O, and Si are initially in the form of C+ ions, neutral O
atoms and Si atoms, respectively.
We consider the clouds to be enriched by Pop III supernovae. The
supernova models give the mass yield Mj of heavy element j andthe total metal yield Mmetal = Mj . Pop III supernovae are generally characterized by metallicities in the range 1–30 Z (Umeda& Nomoto 2002; Limongi & Chieffi 2012). Heavy elements are
diluted within the expanding ejecta and eventually mixed with theambient primordial gas. Thus, the formation site of the secondgeneration stars, polluted with the metal and dust, has typically alower metallicity than the solar value. Setting the cloud metallicity
Z as an effective dilution factor, we obtain the number abundanceof heavy element j relative to hydrogen nuclei in the cloud as
Aj =

Mj
Z
,
μj XH Mmetal(1)

where μj is the molecular weight of element j, and XH = 1 − YHe is
the hydrogen mass fraction.
Both N07 and S12 employ the core-collapse supernova (CCSN)
models with progenitor masses Mpr = 13, 20, 25, and 30 M (hereafter called M13, M20, M25, and M30 models, respectively). N07
further explore the pair-instability supernova (PISN) models with
Mpr = 170 (M170) and 200 M (M200). S12 adjust the mass cut,for each SN model, in order to fit the abundance pattern of the most
1 Hereafter, the subscript ‘0’ is referred to as the initial values.

primitive star SDSS J102915+172927 (Caffau et al. 2011). Such a
procedure, as well as the properties of each progenitor mass and explosion details, has been extensively explained in Limongi & Chieffi(2012). Fig. 2 shows the abundances of the major heavy elementsfor various progenitor masses. For all the progenitor models, theabundances deviate from the solar values by ∼0.5 dex. While S12model can predict the abundances less sensitive to the progenitor
mass, N07 model presents the various patterns. N07M13 models
predict C > O, and heavier elements such as Si and Fe increase
with the increasing progenitor mass.
2.2 Dust models
2.2.1 Dust properties
We consider 10 dust species: silicon (Si), iron (Fe), forsterite(Mg2 SiO4 ), enstatite (MgSiO3 ), magnetite (Fe3 O4 ), amorphous
carbon (C), silica (SiO2 ), magnesia (MgO), troilite (FeS), and alumina (Al2 O3 ). Note that, depending on the employed supernova
models, some of the species are not efficiently formed in the supernova ejecta, or are fully destroyed by the reverse shocks.
We quantify the amount of dust grains by the condensation efficiency fij , which is defined as the number fraction of nuclei of
element j locked in dust species i.2 By using the condensation efficiency, the mass density of dust species i in the cloud is written asρ i = fij Aj nH μij mH , where μij is the molecular weight of the grain
species i per nucleus of element j (see Table 2).
In this study, we introduce the differential
 size distribution function ϕ i (r) of grain species i normalized as ϕ i (r)dr = 1. Then, thenumber of dust particles per unit volume isni =(4π/3)siρir 3 ϕi (r)dr,(2)

where si is the bulk density of an individual dust particle derived
from the values of μij and aij,0 in Table 2. We can calculate thenumber density of grains with radii between r and r + dr as ni ϕ i (r)dr.
2 In this paper, the subscripts i and j denote grain species and heavy elements,
respectively.

MNRAS 446, 2659–2672 (2015)

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Figure 2. Number abundance Aj of heavy element j for the Pop III supernova models of N07 (left) and S12 (right). We plot the values derived by equation (1)with Z = Z . For each supernova model, red pentagons, orange diamonds, yellow opposite triangles, green triangles, blue circles, and purple squares show
the abundances for supernovae with progenitor masses 13, 20, 25, 30, 170, and 200 M , respectively.

2662

G. Chiaki et al.
Table 2. Grain species considered in the calculations.
Key species

Chemical reaction

Si(s)
Fe(s)
Mg2 SiO4 (s)

Si(g)
Fe(g)
Mg(g)
SiO(g)
Mg(g), SiO(g)
Fe(g)
C(g)
SiO2 (g)Mg(g)Fe(g), S(g)Al(g)Si(g)→ Si(s)Fe(g)→ Fe(s)2Mg(g) + SiO(g) + 3H2 O(g) → Mg2 SiO4 (s) + 3H2 (g)2Mg(g) + SiO(g) + 3H2 O(g) → Mg2 SiO4 (s) + 3H2 (g)Mg(g) + SiO(g) + 2H2 O(g) → MgSiO3 (s) + 2H2 (g)
3Fe(g) + 4H2 O(g) → Fe3 O4 (s) + 4H2 (g)C(g)→ C(s)SiO2 (g)→ SiO2 (s)Mg(g) + H2 O(g) → MgO(s) + H2 (g)Fe(g) + S(g) → FeS(s)2Al(g) + 3H2 O(g) → Al2 O3 (s) + 3H2 (g)MgSiO3 (s)Fe3 O4 (s)C(s)
SiO2 (s)MgO(s)
FeS(s)Al2 O3 (s)μijaij,0 (Å)

28.0
56.0
70.0
140.0
100.0
77.3
12.0
60.0
40.0
88.0
51.0

1.684
1.411
2.055
2.589
2.319
1.805
1.281
2.080
1.646
1.932
1.718

Note: The subscripts ‘(s)’ and ‘(g)’ are attached to solid- and gas-phase species, respectively. The values aretaken from Nozawa et al. (2003). μij and aij,0 are the molecular weight and the hypothetical radius of a monomer
molecule of grain species i per nuclei of the key element j.

2.2.2 H2 formation on grains
In the presence of dust grains, H2 molecules are efficiently formedon grain surfaces. Since H2 molecules are major coolants in a lowmetallicity gas, the thermal evolution of the collapsing gas is significantly affected (especially in the early stages of collapse) by theamount of dust. The formation rate of hydrogen molecules per grainof species i with radius r is
1n(HI )vH π r 2 H2 SH ,(3)2
where n(x) is the number density of gas-phase chemical species x,
v H = (8kT/π mH )1/2 is the average velocity of hydrogen atoms,
H2 is the efficiency of H2 recombination on grain surfaces, andSH is the sticking efficiency of hydrogen atoms impacting grainsurfaces (Cazaux & Tielens 2002). The values of H2 and SH are
functions of both gas and grain temperatures (see Schneider et al.
2006, for detailed formulation). We calculate RH2 ,i (r) for each dust
species and each grain radius. Then, the formation rate of hydrogenmolecules on grain surfaces per unit volume is described as
dnH2 
=(4)
RH2 ,i (r)ni ϕi (r)dr.

dt on grainsi
RH2 ,i (r) =equation of a dust grain, Gi (r) = Li (r), we can obtain dust temperature for each dust species and size. Then, the cooling rate of
the gas owing to dust per unit volume is calculated by summing up
equation (6) over all sizes and species:(7)
Li (r)ni ϕi (r)dr.
d =i

2.2.4 Continuum opacity
The Planck-mean absorption coefficients of the grain species aretaken from the references in table 1 of Nozawa et al. (2008). The
continuum optical depth is
 

(8)
Qνi (r) π r 2 ni ϕi (r)dr lsh ,τcont = κg ρg +iwhere κ g is the Planckopacity of gas taken from Mayer & Duschl(2005), and ρ g = ρ − i ρ i is the mass density of gas-phase species.
We calculate the shielding length lsh as in Omukai (2000). Then,the escape fraction of continuum emission is calculated as βcont =−2} (Omukai 2000).
min{1, τcont2.2.5 Grain growth

2.2.3 Dust temperature and cooling
The temperature Ti (r) of grain species i with radius r can be derived
from the balance between the heating by collisions with the gas particles (mostly hydrogen and helium atoms and electrons) and thecooling by thermal emission of dust. We ignore the heat exchange
among dust particles because the grain–grain collision rate is muchsmaller than the gas–grain collision rate in the low metallicity environments considered in this paper. The heating rate of a dust grainwith radius r owing to collisions with the gas particles is(5)
Gi (r) = π r 2 nvg [2kT − 2kTi (r)],√where nvg = [n(HI ) + n(H2 )/ 2 + n(HeI )/2](8kT /π mH )1/2 isthe average velocity of gas particles. The cooling rate of a grainthrough thermal radiation is(6)
Li (r) = 4σB Ti4 (r) Qνi (r) π r 2 βcont ,where σ B is the Stephan–Boltzmann coefficient, Qνi (r) is the
Planck-mean of the absorption coefficient, and β cont is the continuum escape fraction (see the next section). From the energy balance
MNRAS 446, 2659–2672 (2015)

We consider that the grain growth by the accretion of the gas species
proceeds via the reactions in Table 2. Here, we assume that theheavy element species with the smallest time-scale of collisions
with grains control the kinetics of grain growth (hereafter, referred
to as key species). In this model, silicon, silicates (forsterite andenstatite), and silica grains grow through the accretion of Si atoms,
SiO molecules, and SiO2 molecules, respectively, as indicated byfilled arrows in Fig. 1. The flow of silicon from the gas phase to thesolid phase is shown in Fig. 1, along with the gas phase reactions.
The growth rate of grain radius is given by
dr
dt

= αi
i

4π 3
a
3 ij ,0kT
2π mi11/2

ni1 (t),(9)where α i denotes the sticking probability of the gaseous species
incident on to surfaces of grains i, and aij,0 denotes the hypotheticalradius of a monomer molecule of grain species i in the dust phase,and mi1 and ni1 , respectively, denote the mass and the number densityof the key species i1. We here assume the sticking probability α i = 1(see Section 4 for the justification of the select of this value and for

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GrainsSN dust and grain growth: the critical metallicity

2663

Figure 4. Same as Fig. 3 but for models by S12.

the effect of smaller α i on the results). Note that the growth rate
does not depend on the grain radius r. Therefore, the size distribution
ϕ i (r, t) at a time t is equivalent to the initial one ϕ i,0 (r) shifted totlarger radii by r = 0 (dr/dt)i dt. Then, the time evolution of thecondensation efficiency is given as
 3
r ϕi (r, t)dr.
(10)
fij (t) = fij ,0  3
r ϕi,0 (r)dr
The density n(xj ) of the gas-phase species xj containing the element
j is determined by the condensation of the elementj on to all the
relevant grain species i as n(xj , t + t) = n(xj , t) − i {fij (t + t)− fij (t)}Aj nH as well as the relevant gas-phase chemistry.

the supernovae. N07 investigate the reverse shock models when thenumber densities of the ambient gas is namb = 0.1, 1, and 10 cm−3 .
We call these models n0.1, n1, and n10, respectively. S12 investigate
for the mass densities of the ambient gas ρ amb = 10−25 (rev1), 10−24(rev2), and 10−23 g cm−3 (rev3).
These models give the mass Mi of grain species i surviving against
the destruction by the reverse shocks. Figs 3 and 4 show the ratio
fdep,i of dust mass to the total metal mass for N07 and S12 models,
respectively. Table 3 shows the values for carbon and silicate. One
can see that the dust mass decreases with the increasing ambient gas
density. We can also calculate the initial condensation efficiency ofthe collapsing cloud as

2.3 Supernova dust models

fij ,0 =N07 and S12 calculate the formation of grains in the supernova
ejecta for different progenitor masses, predicting the compositionand size distribution of newly-formed grains, hereafter called n0
and norev models for N07 and S12, respectively. For each dust
formation model, they also study dust destruction by reverse shocks
which occurs after the grain formation. The strength of the reverse
shock is parametrized with the density of the ambient gas aroundThe efficiency of gas cooling is determined by the dust composition and size distribution. N07 and S12 obtain different dust
compositions. First, the mass fraction of carbon grains is generally
larger for S12 model than N07 model. Carbon grains are formedby condensation of carbon atoms that are not oxidized to form
CO molecules in the supernova ejecta. S12 consider the molecular
destruction by the collision with high-energy electrons from 56 Co

Mi /μij.
Mj /μj(11)

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Figure 3. Mass fraction fdep,i of grain species i relative to the total metal mass for supernova model of N07. Each histogram represents the mass fraction for
n0, n0.1, n1, and n10 model from top to bottom. The first histogram represents the depletion factor fdep whereas the other histograms show the contribution ofeach dust species.

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G. Chiaki et al.
Table 3. Initial values for supernova models and the critical conditions without grain growth.
Mpr

rev.

fdep,C,0SC,0fdep,Sil,0

SSil,0fdep,0D0

Dcr,ng[Zcr,ng ]N07M13n0
N07M13n0.1
N07M13n1
N07M13n10
N07M20n0
N07M20n0.1
N07M20n1
N07M20n10
N07M2513n0
N07M25n0.1
N07M25n1
N07M25n10
N07M30n0
N07M30n0.1
N07M30n1
N07M30n10
N07M170n0
N07M170n0.1
N07M170n1
N07M170n10
N07M200n0
N07M200n0.1
N07M200n1
N07M200n10

13

n0
n0.1
n1
n10
n0
n0.1
n1
n10
n0
n0.1
n1
n10
n0
n0.1
n1
n10
n0
n0.1
n1
n10
n0
n0.1
n1
n10

0.101
0.096
0.049
0.011
0.023
0.020
0.013
0.002
0.058
0.053
0.041
0.016
0.004
0.003
<0.001<0.001
0.003
0.002<0.001<0.001
0.009
0.007< 0.001<0.001

2.65
2.64
2.50
3.88
4.51
3.46
1.96
2.24
2.42
1.91
1.12
0.83
6.85
7.14
10.27
7.52
4.66
4.88
10.03
12.31
5.39
5.20
15.37
18.02

0.024
0.016
<0.001<0.001
0.049
0.028
0.002
<0.001
0.060
0.048
0.017
<0.001
0.068
0.031<0.001<0.001
0.072
0.016<0.001
0.000
0.052
0.007
<0.001
0.000

3.83
2.77
3.76
7.77
4.22
2.96
3.61
4.87
3.53
1.42
1.35
3.05
3.99
4.04
8.57
7.17
6.62
4.76
1.27
–
9.29
4.94
1.60
–

0.305
0.233
0.093
0.020
0.220
0.137
0.055
0.013
0.240
0.195
0.118
0.036
0.184
0.113
0.040
0.015
0.315
0.138
0.013
<0.0010.359
0.091<0.001<0.001

6.106
4.659
1.851
0.392
4.402
2.750
1.110
0.259
4.797
3.907
2.351
0.726
3.671
2.265
0.791
0.292
6.297
2.761
0.266
0.001
7.189
1.824
0.020
<0.001

3.060
2.335
2.330
1.966
2.777
3.462
5.563
8.199
3.810
3.907
4.692
4.578
2.916
3.590
6.280
11.6393.156
4.377
16.796
1.488
3.603
2.892
4.935
<0.001

− 5.3
− 5.3
− 4.9
− 4.3
− 5.2
− 4.9
− 4.3
− 3.5
− 5.1
− 5.0
− 4.7
− 4.2
− 5.1
− 4.8
− 4.1
− 3.4
− 5.3
− 4.8
− 3.2
> − 2.0
− 5.3
− 4.8
− 2.6
> − 2.0

norev
rev1
rev2
rev3
norev
rev1
rev2
rev3
norev
rev1
rev2
rev3
norev
rev1
rev2
rev3

0.249
0.185
0.101
0.044
0.095
0.030
0.013
0.005
0.162
0.134
0.090
0.047
0.161
0.144
0.097
0.052

4.59
3.35
3.22
3.28
37.56
32.77
27.85
23.72
11.51
3.49
2.32
2.04
1.44
1.12
1.15
1.18

0.033
0.003
0.001
<0.001
0.179
0.030
0.010
0.004
0.068
0.010
0.003
0.001
0.088
0.013
0.004
0.002

45.76
53.73
52.36
47.14
14.68
18.12
18.22
17.00
29.42
34.48
34.39
33.17
26.22
31.72
32.36
28.84

0.413
0.197
0.106
0.047
0.369
0.081
0.030
0.011
0.315
0.155
0.097
0.050
0.324
0.164
0.104
0.055

8.260
3.947
2.117
0.943
7.384
1.618
0.591
0.219
6.293
3.106
1.934
1.000
6.485
3.285
2.071
1.094

0.826
1.571
1.336
1.494
0.738
0.811
0.937
1.098
0.997
2.467
2.435
2.512
1.629
2.609
2.607
2.182

− 6.0
− 5.4
− 5.2
− 4.8
− 6.0
− 5.3
− 4.8
− 4.3
− 5.8
− 5.1
− 4.9
− 4.6
− 5.6
− 5.1
− 4.9
− 4.7

S12M13norev
S12M13rev1
S12M13rev2
S12M13rev3
S12M20norev
S12M20rev1
S12M20rev2
S12M20rev3
S12M25norev
S12M25rev1
S12M25rev2
S12M25rev3
S12M30norev
S12M30rev1
S12M30rev2
S12M30rev3

20

25

30

170

200

13

20

25

30

Note: Progenitor mass Mpr is in the unit of M . The third column ‘rev.’ refers to the reverse shock model (see text).
fdep,i,0 is the initial mass fraction of grain species i relative to the total metal mass and Si,0 is the initial geometricalcross-section of grain i per unit dust mass (×104 cm2 g−1 ). We write fdep,i,0 and Si,0 in bold if either carbon or silicate
dust is the dominant coolant for each model. For silicate grains (i = Sil), we show the values for the dominant species:
MgSiO3 for S12M13 and S12M25 models, and Mg2 SiO4 for the other models. D0 is the dust-to-gas mass ratio for our
Z = 10−5 Z calculations (× 10−8 ). Dcr,ng and Zcr,ng are the critical dust-to-gas mass ratio and metallicity which we
determine by one-zone calculations without grain growth ([Zcr,ng ] = log (Zcr,ng /Z )). Since we set metallicities varying
every 0.1 dex, the value [Zcr,ng ] = −5.9 indicates, for example, that fragmentation condition is met at [Zcr,ng ] ≥ −5.9,
but not for [Zcr,ng ] ≤ −6.0.

(Todini & Ferrara 2001). N07 do not include the dissociation of themolecules. Although carbon grains are formed in the outer layerswith C > O in the ejecta, most of the carbon nuclei are in the inner
layers, where oxygen nuclei are also abundant. Thus, the formationof carbon grains is mitigated in N07 model.
Secondly, magnetite grains are produced only in S12 models.
This results from the different ejecta models. Our N07 model istaken from their unmixed ejecta model, where the ejecta is considered to remain stratified, i.e. keeping the original onion-likestructure of composition of heavy elements during the explosions.
While the oxygen-rich layer is originally in the outer region, Fe is
MNRAS 446, 2659–2672 (2015)

in the innermost layer. In this case, Fe3 O4 grains are not formed
because of the assumed inefficient mixing. On the other hand, S12assume the fully mixed ejecta, where the composition of heavy
elements is uniform, and magnetite grains can be formed.
N07 and S12 models also produce different size distribution functions. In Fig. 5, we compare the initial size distributions of Mg2 SiO4
grains for the two extreme cases: without dust destruction (n0 andnorev) and with the largest efficiency of the dust destruction among
their reverse shock models (n10 and rev3). The size distributionof newly formed grains is mainly regulated by the number density of condensable gas species, i.e. elemental composition at the

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Model

SN dust and grain growth: the critical metallicity

2665

3 R E S U LT S
3.1 Thermal evolution of gas clouds

formation site of the grains. For the dust model by S12, where theuniform elemental composition in the ejecta is assumed, the size
distribution of newly formed grains is confined to a narrow range ofgrain radius (see the red solid curve in Fig. 5). On the other hand, inthe unmixed ejecta applied by N07, dust grains form with different
characteristic radii in the layers with the different elemental compositions. Thus, the resulting size distribution, which is made of thecontributions of grains formed in each formation region, spreads ina wider range of radius than S12 model (green solid curve).
The successive process of dust destruction also affects the size
distribution of surviving grains. S12 assume that the grains aretrapped in the high-temperature region defined by the reverse andforward shocks, where the grains undergo sputtering by the impactof high-energy ions. The grain size continues to decrease until theejecta cools down sufficiently, and the size distribution of surviving
dust has a flat tail at the smaller radii (see the red dot–dashed curvein Fig. 5). N07 consider the relative motion between the expanding gas and dust grains. The erupted dust grains are decelerated
by the drag force from the gas. Smaller grains are more tightly
coupled with the gas so that they remain in the shocked hot ejecta,
continuing to be destroyed. Some of the grain species are totally destroyed and returned to the gas phase. Since larger grains are more
likely to escape from the ejecta because of their larger inertia, theysurvive without being significantly destroyed. As a result, the size
distribution is truncated at a given radius (green dot–dashed curve).
These different dust properties can affect the dust amount above
which dust cooling activates the gas fragmentation: for S12 model,
which predicts the smaller grain radii than N07, the efficiency ofgas cooling by grains is expected to be larger because the total
grain cross-section is larger with the fixed dust mass. In this paper,
we define the critical condition for N07 and S12 supernova dust
models separately. We consider the dust models where grains are
not destroyed by reverse shocks (n0 and norev), and also three
dust models where the reverse shock destruction is calculated for
different ambient gas densities for each progenitor mass. In order
to see the effect of grain growth, we study both cases with andwithout grain growth for each supernova dust model. We determine
the critical dust amount by varying the gas metallicity in the rangeof Z = 10−7 –10−2 Z , where Z = 0.02.

Figure 6. Temperature evolution as a function of the hydrogen number
density of the cloud centres for our N07M30n1 model with metallicities
Z = 10−6 Z (dot–dashed), 10−5 Z (solid), and 10−4 Z (dashed), respectively. Red thick and blue thin curves depict the cases with and without
grain growth, respectively. Triangles are plotted at the states where thefragmentation condition (see text) is met.

MNRAS 446, 2659–2672 (2015)

Downloaded from http://mnras.oxfordjournals.org/ at National Astronomical Observatory of Japan on December 11, 2014

Figure 5. Initial size distribution of Mg2 SiO4 grains for the N07 (green)
and S12 (red) models without the effect of dust destruction (solid) and withthe largest efficiency of dust destruction among the three reverse shockmodels (dot–dashed).

We highlight and discuss the results of some specific cases in thissection. Fig. 6 shows the thermal evolution of the cloud centre
for our N07M30n1 model with metallicities Z = 10−6 , 10−5 , and10−4 Z . Around nH ∼ 104 cm−3 , H2 cooling becomes efficient fora higher metallicity because H2 molecules are formed on grain surfaces more efficiently. If the gas temperature drops below ∼100 K,
HD cooling becomes dominant (Omukai et al. 2005; Hirano et al.
2014), which can be seen at nH ∼ 104 –106 cm−3 with metallicities
Z > 10−5 Z . Then, OH (Z ≥ 10−5 Z ) becomes a major coolant
at nH ∼ 105 –108 cm−3 . For Z = 10−5 and 10−4 Z , the gas temperature increases by the heating owing to the exothermic reactionof the formation of H2 molecules via rapid three-body reactions at
nH = 109 –1012 cm−3 . If the amount of dust is sufficiently large, dust
cooling becomes effective at nH = 1011 –1012 cm−3 .
We examine the condition for the formation of low-mass fragments by radiative cooling on the basis of the analysis by Schneider& Omukai (2010). First, the gas becomes gravitationally unstableand deformed due to rapid cooling when the specific heat ratio, γ ,drops below unity. When the gas cooling is inefficient (γ  1) andthe gas is likely to collapse in approximately a spherical manner,
yielding fragments whose mass is about the Jeans mass corresponding to the gas density and temperature at the fragmentation (e.g.
Larson 1985; Inutsuka & Miyama 1997). Then, the fragmentation
condition for a gas cloud is defined as the following set of three criteria: (1) the gas cloud undergoes rapid radiative cooling (γ < 0.8), butthen (2) gas cooling becomes soon inefficient (γ > 0.97). Since weare interested in the cases where low-mass fragments are formed,
we add another criterion to our fragmentation condition: (3) the
cloud Jeans mass is less than 0.8 M when both (1) and (2) aresatisfied.

2666

G. Chiaki et al.

In some models, we find O I cooling is efficient enough to trigger
fragmentation, i.e. the criteria (1) and (2) given above are met, at
nH ∼ 102 –103 cm−3 for metallicities Z ≥ 10−4 Z . However, since
the mass of the clump is ∼100 M in this regime, the criterion (3)is not satisfied (see Section 4 for the further discussion of this). It
is clearly shown in Fig. 6 that dust cooling is effective at higher
densities, nH = 1012 –1015 cm−3 , where the Jeans mass is∼0.01 M . In this regime, all of the three criteria are met, suggesting that dust cooling can drive the fragmentation of the gas into
small mass clumps.
Let us discuss the effect of grain growth in detail. Fig. 6 showsthat, for Z = 10−5 Z (solid curves), the fragmentation condition is
met with grain growth (red) while dust cooling is inefficient without
grain growth (blue). For Z = 10−4 Z (dashed), the fragmentation
condition is met even for the case without grain growth. Overall,
the metallicity, or the initial amount of dust required for the gas
fragmentation, is reduced by the effect of grain growth.
Fig. 7 shows the size distribution of grains for N07M30n1 modelwith Z = 10−5 Z before (nH = 0.1 cm−3 ) and after (1016 cm−3 )grain growth. We see that the radii of grains increase especiallyfor Mg2 SiO4 (red solid curve). If linearly plotting the figure, thedistribution functions just shift from left to right because the increment of grain radii is independent from their initial radii as wehave mentioned above. Fig. 8 shows the condensation efficiencies ofgrain species i (top) and the number fractions of Mg atoms, and SiO
and SiO2 molecules relative to total Mg and Si nuclei, respectively
MNRAS 446, 2659–2672 (2015)

Figure 8. Top: number fraction of nuclei which are concentrated into grains
i relative to total nuclei of element j in both the gas- and solid-phases(condensation efficiency fij ). Bottom: number fraction of nuclei which are in
the form of SiO molecules (blue dashed), and SiO2 molecules (green dotted)
relative to total Si nuclei. Red solid curves depict the number fraction of Mg
in the gas phase to total Mg nuclei. This figure shows the evolution for thesame case as Fig. 7.

(bottom). The condensation efficiencies of all species but carbon(black dot-dot–dashed) and silicon (blue dotted) increase by accreting the gas-phase species. Almost all Si atoms are oxidized into
SiO molecules at nH < 108 cm−3 , where silicate grains (Mg2 SiO4
and MgSiO3 ) grow most rapidly by accretion of Mg atoms and
SiO molecules at nH ∼ 1011 –1013 cm−3 . Further, MgO grains growalmost at the same time as silicate until gas-phase magnesium is
exhausted. Some fractions of SiO molecules survive without being
incorporated into dust grains, because AMg < 2ASi . SiO2 molecules
fully condense into SiO2 grains at nH  1015 cm−3 . Even if SiO2
molecules are totally depleted, SiO molecules are not further oxidized because the major oxidizer, OH, is already depleted into H2 O.
At nH ∼ 1015 cm−3 , FeS grains begin to grow. The time when grains
grow depends mostly on the abundances of gas-phase elements aswell as the monomer radius aij,0 . Carbon grains do not grow because C atoms have been already depleted into CO molecules at
nH = 106 cm−3 . As a result of grain growth, at nH = 1012 cm−3 , thetotal depletion factor (total mass fraction of dust relative to metal)
increases from 0.040 to 0.045 and 0.14 for Z = 10−5 , and 10−4 Z ,respectively.
Fig. 9 shows the contribution of each grain species to gas coolingas a function of gas density for N07M30n1 model. Without grain
growth, Si and SiO2 grains make a major contribution to gas cooling.
On the other hand, the cooling rate of Mg2 SiO4 grains is enhancedby accretion of heavy elements, and eventually becomes larger thanDownloaded from http://mnras.oxfordjournals.org/ at National Astronomical Observatory of Japan on December 11, 2014

Figure 7. Size distribution function ni ϕ i /nH of dust species i before(nH = 0.1 cm−3 ) and after (nH = 1016 cm−3 ) grain growth for our
N07M30n1 model with Z = 10−5 Z . Size of all species but for C and
Si grains increases by accretion of heavy elements. C and Si grains do not
grow because the key species, C and Si atoms, are depleted on to CO andSiO molecules, respectively.

SN dust and grain growth: the critical metallicity

the compressional heating rate. This means that the small initialdust amount suffices for cloud fragmentation by the effect of grain
growth.
3.2 Conditions for cloud fragmentation
3.2.1 Critical dust-to-gas mass ratio and metallicity
In this section, we discuss the results for all our dust models, anddetermine the conditions for the cloud fragmentation. First of all, we
find that dust cooling triggers the gas fragmentation for the models
where the fragmentation condition is satisfied. This indicates thatthe dust-to-gas mass ratio D is the key quantity to determine thefragmentation properties of the gas as suggested by S12. We firstshow the initial value D0 with Z = 10−5 Z in Table 3. The dust-togas mass ratio for arbitrary metallicity Z can be obtained simply by(Z/10−5 Z )D0 . Then, varying the metallicity by 0.1 dex, we define
the critical dust-to-gas mass ratios Dcr,ng and Dcr,gg above which
dust cooling triggers the fragmentation into low-mass gas clumps,
based on our one-zone calculations without and with grain growth
as Tables 3 and 4 show, respectively. For almost all the models,the critical dust-to-gas mass ratio is reduced by the effect of grain
growth. For example, for N07M30n1 model, Dcr,ng = 6.3 × 10−8
without grain growth, while it decreases to Dcr,gg = 0.63 × 10−8
with grain growth. The initial dust-to-gas mass ratio D0 for this caseis 0.79 × 10−8 with Z = 10−5 Z , which is lower than Dcr,ng but
higher than Dcr,gg . For all our supernova models, the critical dust-

to-gas mass ratio is reduced from Dcr,ng = [0.81 : 11.6] × 10−8 to
Dcr,gg = [0.07 : 3.82] × 10−8 , depending on the dust model as we
see in the next section. Note that we do not count n1 and n10
models for PISNe because the PISN models predict the much larger
amounts of heavy elements such as Si and Fe than those inferredfrom the Galactic metal-poor stars so far observed, and because theambient gas density of these progenitors is expected to be small
namb  0.1–1 cm−3 by the copious emission of ultraviolet photons
in their main sequence (Kitayama et al. 2004; Whalen, Abel &Norman 2004).
If the dust amount is constant during the collapse of the gas
clouds, the condition for gas fragmentation can be described simply by the initial dust-to-gas mass ratio (S12). We have shown thatgrain growth can alter this simple picture. If grains can completely
accrete the gas-phase refractory elements, the condition becomes
insensitive to the initial dust properties but sensitive to the gas
metallicity because the depletion efficiency of metals on to dust
grains converges to a certain value determined by the amountsof the refractory elements in this extreme case. To see how thefragmentation property depends on the metal and dust contents,
let us study the dependence of the critical metallicity on the dust
model. Although there are various quantities that characterize thedust for each supernova model, we here focus on the initial depletion factor. The metallicity Zcr above which gas fragmentation is−1. The last column intriggered is given by the relation Zcr = Dcr fdep,0Tables 3 and 4 shows the values Zcr,ng and Zcr,gg without and withgrain growth, respectively, and Table 3 shows fdep,0 . Without grain
growth, the critical metallicity is roughly inversely proportional−1with the averto the initial depletion factor as Zcr,ng = Dcr,ng fdep,0age critical dust-to-gas mass ratio Dcr,ng = [2.0 : 2.5] × 10−8 . Withgrain growth, the least-squares fitting to the results for both N07
and S12 models leads to the following relationship:
Zcr,gg10−5.5 Z

=fdep,00.18−0.44±0.21

.

(12)

The critical metallicity is still dependent on the initial depletion
factor, because the gas-phase heavy elements partly condense into
dust grains as we discuss below. Note that the uncertainty of thespectrum index stems from the dependence of the critical metallicity on the composition, size distribution, and initial condensation
efficiency of the dust models adopted in this paper as discussed in
the next subsection.
Our study has shown that the grain growth is important to alter
the fragmentation property for our supernova models. For the casewith the least depletion factor (0.011) among our supernova dust
models, Zcr is reduced by about a factor of 20. We also find that the
dust properties in the early star-forming regions are much different
from those of the local Universe, where all refractory elements are
depleted on to grains (Pollack et al. 1994).

3.2.2 Dependence of the critical metallicity on the initial dust
modelsWe note that there are appreciable scatters in the critical metallicity
from the average value of equation (12), depending on the initial dust
properties. Fig. 10 shows Zcr as a function of the initial depletion
factor for N07 and S12 models separately.
We begin with the cases without grain growth. Open symbols in
Fig. 10 present Zcr,ng , and the average values are drawn by the dotted lines. The critical metallicity roughly follows the relationship
MNRAS 446, 2659–2672 (2015)

Downloaded from http://mnras.oxfordjournals.org/ at National Astronomical Observatory of Japan on December 11, 2014

Figure 9. Gas cooling rate per unit gas mass owing to each dust species
as a function of gas density with metallicity Z = 10−5 Z for N07M30n1
models with (top) and without (bottom) grain growth. We also plot the total
dust cooling rate (thick grey) summed over all grain species and the gas
compressional heating rate (thick dark red).

2667

2668

G. Chiaki et al.
Table 4. Critical conditions with grain growth.
Mpr

rev.

rSil

grow

log(nH,Sil )growfdep,Sil,∗

SSil,∗

fdep,∗

Dcr,gg[Zcr,gg ]N07M13n0
N07M13n0.1
N07M13n1
N07M13n10
N07M20n0
N07M20n0.1
N07M20n1
N07M20n10
N07M2513n0
N07M25n0.1
N07M25n1
N07M25n10
N07M30n0
N07M30n0.1
N07M30n1
N07M30n10
N07M170n0
N07M170n0.1
N07M170n1
N07M170n10
N07M200n0
N07M200n0.1
N07M200n1
N07M200n10

13

n0
n0.1
n1
n10
n0
n0.1
n1
n10
n0
n0.1
n1
n10
n0
n0.1
n1
n10
n0
n0.1
n1
n10
n0
n0.1
n1
n10

0.00600.04440.03310.01470.00740.03420.03130.02280.00440.06150.07540.0511
0.02210.03390.01200.02460.00780.04390.1813–0.00450.04520.1440
–

no
no
15.2
15.9
∗
12.8
14.0
15.6
∗
∗
13.8
15.0
∗
12.7
13.2
15.9
∗
13.1
no
–
∗
13.8
no
–

0.024
0.017
0.001
<0.001
0.051
0.034
0.003
< 0.001
0.060
0.051
0.020
< 0.001
0.069
0.044
0.006
<0.001
0.072
0.029< 0.001
0.000
0.053
0.012
<0.001
0.000

3.83
2.78
3.95
8.45
5.22
3.12
4.00
5.25
4.24
1.44
1.39
2.93
3.99
4.01
8.80
5.43
6.62
4.07
1.17
–
9.68
4.19
1.48
–

0.351
0.270
0.128
0.043
0.224
0.146
0.057
0.013
0.241
0.200
0.121
0.036
0.184
0.128
0.045
0.015
0.317
0.154
0.013
<0.001
0.364
0.098
0.001
<0.001

0.769
1.473
0.928
0.312
2.777
2.184
1.397
0.820
3.810
3.103
2.351
1.448
2.916
2.265
0.628
0.734
3.156
2.194
2.662
1.488
3.603
1.824
0.031
<0.001

− 5.9
− 5.5
− 5.3
− 5.1
− 5.2
− 5.1
− 4.9
− 4.5
− 5.1
− 5.1
− 5.0
− 4.7
− 5.1
− 5.0
− 5.1
− 4.6
− 5.3
− 5.1
− 4.0
> − 2.0
− 5.3
− 5.0
− 4.8
− 3.2

norev
rev1
rev2
rev3
norev
rev1
rev2
rev3
norev
rev1
rev2
rev3
norev
rev1
rev2
rev3

0.00510.00390.00400.00470.01590.01150.01090.01180.00800.00610.00600.00610.00880.00650.00620.0071

10.4
11.2
11.6
12.4
∗
11.1
11.8
12.4
9.1
11.5
12.0
12.3
∗
11.3
12.0
12.5

0.200
0.203
0.178
0.069
0.179
0.141
0.107
0.064
0.086
0.093
0.074
0.049
0.088
0.068
0.045
0.024

25.28
14.62
10.90
10.79
14.68
11.56
9.24
7.43
27.19
17.77
13.58
11.21
26.22
19.56
16.09
13.24

0.602
0.476
0.333
0.164
0.370
0.242
0.172
0.105
0.337
0.259
0.181
0.105
0.325
0.240
0.157
0.084

0.826
0.497
0.336
0.188
0.466
0.204
0.118
0.069
0.997
0.982
0.770
0.501
1.629
1.039
0.825
0.548

− 6.0
− 5.9
− 5.8
− 5.7
− 6.2
− 5.9
− 5.7
− 5.5
− 5.8
− 5.5
− 5.4
− 5.3
− 5.6
− 5.5
− 5.4
− 5.3

S12M13norev
S12M13rev1
S12M13rev2
S12M13rev3
S12M20norev
S12M20rev1
S12M20rev2
S12M20rev3
S12M25norev
S12M25rev1
S12M25rev2
S12M25rev3
S12M30norev
S12M30rev1
S12M30rev2
S12M30rev3

20

25

30

170

200

13

20

25

30

grow

grow

Note: riis the characteristic size of grain species i for grain growth (µm; see text). nH,i is the density
where the condensation efficiency of grain species i becomes above 0.5 by grain growth for Z = 10−5 Z runsgrow
(cm−3 ). nH,i is not defined when the condensation efficiency is initially above 0.5 (‘∗’) or when the condensation
efficiency does not reach 0.5 until nH = 1016 cm−3 (‘no’). The subscript ‘∗’ denotes the value for Z = 10−5 Z
at density nH = 1012 cm−3 . The critical dust-to-gas mass ratio Dcr,gg and metallicity Zcr,gg are determined by our
one-zone calculations with grain growth.

−1
Zcr,ng = Dcr,ng fdep,0
with the specific value of Dcr,ng for each progenitor mass. In more quantitative manner, we revisit the criterionof

SD > 1.4 × 10−3 cm2 g−1−1/2 
−1/2
T
nH
×
,
103 K
1012 cm−3
where S is the total geometrical cross-section of grains
per unit dust
mass (S12). We rewrite the left-hand side as SD = Z i fdep,i Si to
see the contributions of the dust composition and size distributionof different species. Here, fdep,i = ρ i /ρ metal = fij Aj nH μij mH /ρ metal
MNRAS 446, 2659–2672 (2015)is proportional to the mass fraction of the grain species i. Si is the
cross-section of species i per unit dust mass as 
π r2 i
3
  =
,
(13)
Si =
4si ricool(4π/3)si r 3 iwhere ricool = r 3 i /r 2 i is an average dust radius characterizing
dust cooling rate. This represents the contribution of the dust size
distribution to gas cooling.
Table 3 shows fdep,i and Si for the major species: carbon and silicate. For S12 model, the mass fraction of carbon grains is larger thanN07 model as shown in the fourth column of Table 3. Although themass fraction of silicate grains is similar between the two models,

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Model

SN dust and grain growth: the critical metallicity

2669

the contribution of this species to gas cooling is different because
the value SSil,0 for S12 model is larger than N07 model as a result of
the smaller characteristic size of silicate grains for the former case as
we have seen in Fig. 5. The contribution of magnetite grains, whose
cooling efficiency is also large, can reduce the critical metallicity
for S12 model. Therefore, the total dust cooling efficiency is larger
for S12 model and hence the lower initial dust amount suffices toactivate the gas fragmentation.
The critical dust-to-gas mass ratio Dcr,ng varies also with progenitor masses. For N07 model, this value is smallest for M13 models,followed by M25. This variation stems from the composition of thedominant species: carbon and silicate. For n0 cases, Dcr,ng is rather
insensitive to the progenitor mass because the sum of the contributions of carbon and silicate is similar to each other. On the other
hand, for n10 cases, the cooling efficiency is largely determinedby the mass fraction of carbon grains because silicate, which ismore efficiently destroyed by the reverse shock than carbon, can nolonger contribute to gas cooling (see Fig. 3). The fraction of carbon
grains is largest for M13, and thus Dcr,ng is smallest, followed by
M25. For S12 model, the critical dust amount is sensitive to thedust size distribution of only carbon grains because this species isthe dominant coolant. Dcr,ng decreases roughly with the increasing
SC,0 .
With grain growth, the critical metallicity is reduced, as seen by
the filled symbols in Fig. 10. The solid lines represent the results of
−0.51
−0.30
and fdep,0for N07 andthe least-squares fitting as Zcr,gg ∝ fdep,0S12, respectively, showing that the effect of grain growth is larger
grow
for the latter case. We compare the density nH,i at which the congrow
densation efficiency of grain i exceeds 0.5. Table 4 presents nH,Sil forsilicate grains, which grow most rapidly for most of our supernova
growmodels. For S12 model, nH,Sil is below a density nH ∼ 1012 cm−3
where dust cooling becomes efficient. On the other hand, for N07
model, grains can grow only after this threshold density. Thus, thecooling efficiency for N07 model is less modified by grain growththan S12 model especially for the cases with the dust destruction in
the supernovae.
The growth rate of grains is determined by their compositionand size. Since the initial dust composition is similar for N07 andS12 models, the growth rate is determined largely by the dust size
grow
distribution.3 The characteristic density nH,i where grains rapidly
grow decreases with decreasing dust size because the total crosssection of grains per unit dust mass is larger for smaller grains.
grow
Although the relation between nH,i and fij,0 is complicated, we finda fitting formula to the density where grains rapidly grow as
nH,i = 1.0 × 1012 cm−3
grow

grow

×

ri
0.01 µm

2

Aj
7.1 × 10−10
ai,0
1Å

−6

mi1
mH

−2

fij ,0
0.1

−0.8

,

which is valid in the range of fij,0  0.5. In the above equation,grow= r 3 1/3 is a measure of the average dust size which characrigrowis generally larger
terizes the growth rate. Table 4 shows that ri
for N07 model as we discuss in Section 2.3. Thus, the growth rateis smaller for this model.
The critical metallicity with grain growth depends also on theprogenitor mass. For N07M13 model, carbon grains grow because
3 In our previous paper (Chiaki et al. 2014), where we investigate the effect ofgrain growth on the formation of the specific star SDSS J102915+172927,
employing the part of S12 model, we conclude that the growth rate is almost
insensitive to the initial dust models. We in this paper survey a wider rangeof the initial conditions including N07 model, which predicts larger grain
radii than S12 model by about an order of magnitude. Thus, the dependenceof the growth rate on the initial dust size becomes apparent.

MNRAS 446, 2659–2672 (2015)

Downloaded from http://mnras.oxfordjournals.org/ at National Astronomical Observatory of Japan on December 11, 2014

Figure 10. Critical metallicity Zcr for N07 (left panel) and S12 (right) supernova dust models as a function of the initial dust-to-metal mass ratio (depletion
factor), fdep,0 . Four points with the same symbol (progenitor mass) correspond to our four reverse shock models: n0, n0.1, n1, and n10 from right to left. Open andfilled symbols depict the results obtained by our one-zone calculations with and without grain growth, respectively. Dotted and solid lines are fitted to these results−1−1
for each supernova model. The former are Zcr,ng = 3.4 × 10−8 fdep,0and Zcr,ng = 1.3 × 10−8 fdep,0, and the latter (Zcr,gg /10−5.5 Z ) = (fdep,0 /0.50)−0.51 and
−5.5
−0.30
Z ) = (fdep,0 /0.022)
for N07 and S12 models, respectively.
(Zcr,gg /10

2670

G. Chiaki et al.

4 CONCLUSION AND DISCUSSION
We have investigated the conditions for formation of the firstlow-mass stars in the early Universe in a low-metallicity gas byperforming one-zone collapse calculations including grain growth.
As the initial abundances of metal and dust and size distributionof grains, we have employed the model of dust formation and destruction in preceding Pop III supernovae presented by N07 andS12. Without grain growth, the critical metallicity is inversely pro−1
withportional to the initial depletion factor as Zcr,ng = Dcr,ng fdep,0
the critical dust-to-gas mass ratio Dcr,ng = [0.81 : 11.6] × 10−8 , depending on the supernova models. Dcr,ng is smaller for S12 model
than N07 because the mass fractions of carbon and magnetite grains
are larger for the former model, and silicon grains, which have lesscooling efficiency, account for a considerable mass fraction for N07
model. For S12 model, Dcr,ng is determined mainly by the sizeof carbon grains, which are dominant coolants for all progenitor
masses.
With grain growth, the critical dust-to-gas mass ratio is reduced down to Dcr,gg = [0.07 : 3.82] × 10−8 . The corresponding−0.44
metallicity is around Zcr,gg ∼ 10−5 Z , proportional to fdep,0(equation 12). This dependence on fdep,0 becomes milder than thecase without grain growth, but the dependence is not completely
washed out because of the incomplete accretion of the gas-phase
species on to grains. For S12 model, the accretion of magnesium on
to silicate grains rapidly occurs. Carbon atoms are depleted on CO
MNRAS 446, 2659–2672 (2015)

molecules before accreted by carbon grains. For N07 model, silicate
grains do not completely accrete the gas-phase species because oflarger grain radii than S12 model.
It is conceivable that dust evaporation and coagulation by grain–
grain collision could reduce the cross-section of collisions between
dust grains and gas particles. These processes, however, have little effect on the thermal evolution of the clouds in the region of
our interest for the following reasons: first, silicate grains are sublimated at temperature ∼1100 K at high density (Pollack et al.
1994). Gas temperature rises up to such a value only when the
dust cooling is not efficient enough to induce cloud fragmentation. Next, grain–grain collision is effective only at densities
nH > 1016 (Z/10−5 Z )−2 (T/1000 K)−1 cm−3 (Hirashita & Omukai2009), where the gas is already optically thick. Therefore, we ignore these effects in this study. We should also mention that, if the
sticking coefficient α i is less than unity, the growth rate of grains is
reduced accordingly. This value is still uncertain but considered tobe 0.1–1 by various approaches (Leitch-Devlin & Williams 1985;
Grassi et al. 2011; Tachibana et al. 2011). We have explicitly tested
cases with α i = 0.1 and found that grain growth hardly affects thefragmentation condition: (Zcr /10−5.5 Z ) = (fdep,0 /3.5)−0.92 . In thiscase, the scaling between Zcr and Dcr is almost the same as expectedwithout grain growth. Let us finally remark that an experimental
measurement reveals that the large value of the sticking probability
is an order of unity, although this is for Fe grains (Tachibana et al.
2011).
As discussed above, the critical metallicity depends on the composition and size distribution of dust. However, it is not possible toobserve directly the properties of dust grains in the formation site oflong-lived low-metallicity stars discovered in the Galaxy. Instead,we define the domain of metal abundances where the formation of
low-mass stars is favoured. We focus here on carbon-normal stars,
which are characterized by [C/Fe] < 0.7 (Beers & Christlieb 2005;
Aoki et al. 2007). In such cases, we find that silicate grains are
the most important species for the gas cooling for the most cases
styled in bold in Tables 3 and 4. We define the range of the criticalconditions in terms of silicon abundance from equation (1), setting the metallicity Z = Zcr , which are shown by the green and red
shaded regions in Fig. 11, for N07 and S12 models, respectively.
Our analyses reveal that the dust-induced fragmentation is activatedin the region inside the colour-shaded regions. Interestingly, the carbon, oxygen and silicon abundances of the extremely metal-poor(EMP) stars which have so far been observed are located to the rightof the colour-shaded region. The formation of the most primitive
star ever observed, SDSS J102915+172927 with [Si/H] = −4.3
(Caffau et al. 2011), is favoured by about half of our supernova
models even if grain growth does not occur. Furthermore, most of
our models can explain the formation of this star by grain growth(Chiaki et al. 2014).
An empirical model for the formation of the first low-massstars posits that the fine-structure cooling by carbon and oxygen drives the gas fragmentation under the critical discriminant of
Dtrans = log (10[C/H] + 0.9 × 10[O/H] ) > −3.5 ± 0.2 (horizontal lines
in Fig. 10: e.g. Bromm et al. 2001; Frebel et al. 2005). However,this occurs only at the low densities nH = 102 –103 cm−3 , where
the Jeans mass is ∼ 100 M , which indicates that the low-massfragments could not be formed by the fine-structure cooling only
(Omukai et al. 2005; Schneider et al. 2006). Ji et al. (2014) determine the critical Si abundance above which dust cooling can induce
gas fragmentation by equating the gas compressional heating rateand dust cooling rate, assuming that dust consists only of Si-bearing
grains. They suggest that the formation of the low-mass stars b–e

Downloaded from http://mnras.oxfordjournals.org/ at National Astronomical Observatory of Japan on December 11, 2014

C > O. The condensation efficiency of carbon increases from theinitial value of fC, C, 0 = 0.04–0.38 up to fC,C,∗ = 0.13–0.55 atnH ∼ 1012 cm−3 with Z = 10−5 Z . Since the carbon abundanceis large relative to magnesium, log (AC /AMg ) = 1.24, the growth ofcarbon grains enhances gas cooling more efficiently than silicate.
The rate of heat transfer between gas and dust becomes proportional
to fdep,C,∗ SC, ∗ = [2.6–30.3] × 104 cm2 g−1 , which is larger than thevalues for silicate grains with the other progenitor masses. Therefore, Zcr,gg is smallest for M13. For N07M13n0 model, MgO grains,growwhich grow at gas density log(nH,MgO ) = 13.8, further enhance theefficiency of dust cooling. With the other progenitor masses forN07 model, silicate grains become the dominant species. For M20
and M30 models, silicate grow too slowly, at nH > 1015 cm−3 , toenhance the cooling efficiency for n10 models. In these cases, SiO2
grains become dominant species for gas cooling. For M25, the contribution of silicate grains to gas cooling becomes comparable tocarbon by grain growth for n1 and n10 models. Regardless of thedifferent composition of carbon to silicate, Zcr,gg is within ∼0.2 dex
from the value for M20 and M30 models.
For S12 model, the dominant species is changed from carbon
to silicate by grain growth for all progenitor masses. Magnetite
grains also contribute to gas cooling for M13 and M30 models
grow
via their growth at densities log(nH,Fe3 O4 ) = 11.8–12.7 and 12.7–
13.8 for the former and latter models, respectively. Fig. 10 showsthat the critical metallicity is small for M13 and M20 models. For
M13, both magnetite and silicate grow to become dominant. Although this occurs also for M30 models, the mass density of silicate
fdep,Sil,∗ is larger for M13 as shown in Table 4. In addition, boththe values fdep,Fe3 O4 ,∗ and SFe3 O4 ,∗ are larger for M13 than M30:
(fdep,Fe3 O4 ,∗ , SFe3 O4 ,∗ ) = (0.026–0.11, 18.0–40.9) for M13, while(0.004–0.070, 15.2–19.9) for M30, where the unit of SFe3 O4 ,∗ is
104 cm2 g−1 . For M20, the contribution of carbon grains to gas cooling is still large because of their small size, which further reduces
the value Zcr,gg .

SN dust and grain growth: the critical metallicity

2671

AC K N OW L E D G E M E N T S
Figure 11. Critical conditions for the formation of low-mass stars plotted on the Dtrans -[Si/H] plane as used by Ji et al. (2014). We show therange of the critical metallicity as the green- and red-coloured regions defined by our one-zone calculations with N07 and S12 models, respectively,
without (top panel) and with (bottom) grain growth. The subsets of the regions with darker colours show the range of C and O abundances examined
here. We also draw horizontal lines depicting the proposed discriminant,Dtrans = −3.5 ± 0.2, above which the fine-structure cooling activate thefragmentation (Frebel & Norris 2013). We also plot the abundances of EMP(black squares) and CEMP (blue circles) stars which have so far been observed. The symbols with a–e indicate the elemental abundances of therecently discovered primitive star SDSS J102915+172927 (a: Caffau et al.
2011), and most iron-poor CEMP stars HE0557−4840 (b: Norris et al.
2007), HE0107−5240 (c: Christlieb et al. 2004), HE 1327−2326 (d: Frebel
et al. 2008), and SMSS J0313−6708 (e: Keller et al. 2014). The other stars
are the carbon-normal main-sequence stars with the intermediate metallicities taken from the SAGA data base (Suda et al. 2008).

in Fig. 10 can not be explained by dust cooling scenario becauseof the lack of silicon abundance, but can by fine-structure coolingscenario because of more abundant C and O than their discriminant.
Although one-zone calculations reveal that subsolar-mass clumps
could not be formed in the fine-structure cooling model, Ji et al.
(2014) speculate that low-mass star formation could still be possible with the help of some dynamical effects. It is important tonote that the contribution of carbon grains, which can reduce thecritical Si abundances, is uncertain in their model. The mass ratio of this species is determined for our supernova dust models
as 0.003–0.65 and 0.26–0.96 for N07 and S12, respectively. The
contribution of the considerable amount of the carbon grains reduces the lower bound of our critical Si abundance down to thevalue lower than that of the stars b–e even without grain growth.
Grain growth can further reduce the critical Si abundance so thatWe thank Simone Bianchi for his kind contribution. GC is
supported by Research Fellowships of the Japan Society for the
Promotion of Science (JSPS) for Young Scientists. This work is
supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan and in part by Grant-in-Aid for
Scientific Research from the JSPS Promotion of Science (22684004,
23224004, 23540324, 25287040, 25287050, and 26400223). A partof calculations is performed with COMA at Center for Computational Sciences in University of Tsukuba. The research leading tothese results has received funding from the European ResearchCouncil under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement n. 306476. ML
acknowledges the following funding sources: PRIN INAF 2009
‘Supernova Variety and Nucleosynthesis Yields’, and PRIN MIUR
2010–2011, project ‘The Chemical and dynamical Evolution of the
Milky Way and Local Group Galaxies’, prot. 2010LY5N2T.

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