MNRAS 439, 3121–3127 (2014)

doi:10.1093/mnras/stu178

Advance Access publication 2014 February 20

Dust grain growth and the formation of the extremely primitive star
SDSS J102915+172927
Gen Chiaki,1‹ Raffaella Schneider,2 Takaya Nozawa,3 Kazuyuki Omukai,4Marco Limongi,2,3 Naoki Yoshida1,3 and Alessandro Chieffi5
1 Department of Physics, Graduate School of Science, The 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 Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for Advanced Study, The University of Tokyo,

Kashiwa, Chiba 277-8583, Japan
5 INAF/IASF, Via Fosso del Cavaliere 100, I-00133 Roma, Italy

Accepted 2014 January 22. Received 2014 January 22; in original form 2013 October 11

ABSTRACTDust grains in low-metallicity star-forming regions may be responsible for the formation of thefirst low-mass stars. The minimal conditions to activate dust-induced fragmentation require
the gas to be pre-enriched above a critical dust-to-gas mass ratio Dcr = [2.6–6.3] × 10−9 . The
recently discovered Galactic halo star SDSS J102915+172927 has a stellar mass of 0.8 M anda metallicity of Z ∼ 4.5 × 10−5 Z and represents an optimal candidate for the dust-induced
low-mass star formation. Indeed, the critical dust-to-gas mass ratio can be overcome provided
that at least 0.4 M of dust condenses in Pop III supernova ejecta, allowing for moderate
destruction by the reverse shock. Here, we show that grain growth during the collapse of theparent gas cloud is sufficiently rapid to activate dust cooling and fragmentation into low-massstars, even if dust formation in the first supernovae is less efficient or strong dust destruction
does occur. We find that carbon grains do not experience grain growth because at densities
below nH ∼ 106 cm−3 carbon atoms are locked into CO molecules. Silicates and magnetite
grains accrete gas-phase species in the density range 109 < nH < 1012 cm−3 , until their gasphase abundance drops to zero, reaching condensation efficiencies ≈1. The corresponding
increase in the dust-to-gas mass ratio allows dust-induced cooling and fragmentation to be
activated at 1012 < nH < 1014 cm−3 , before the collapsing cloud becomes optically thick tocontinuum radiation.
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 physical conditions which enable the first low-mass and longlived stars to form in the Universe is still a subject of debate(Karlsson, Bromm & Bland-Hawthorn 2013). Recent numericalsimulations of primordial star formation suggest that the first stars
could have a wide variety of masses (for a recent comprehensive
review, see Greif et al. 2012; Bromm 2013; Hirano et al. 2014).
Indeed, no evidence for the existence of primordial low-mass stars
has been found in surveys of the Galactic halo and in nearby dwarfs(Beers & Christlieb 2005).

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

Once the first supernovae (SNe) explode and start to seed thegas with metals and dust grains, the physical properties of starforming regions change (Bromm, Yoshida & Hernquist 2003;
Chiaki, Yoshida & Kitayama 2013a). Low-metallicity gas canachieve larger cooling rates through additional molecular species(OH, CO, H2 O), fine-structure line cooling (mostly O I and C II),and thermal emission from dust grains (Omukai 2000; Bromm et al.
2001; Schneider et al. 2002; Omukai et al. 2005). The relative importance of these coolants depends on the density (time) regime
during the collapse and on the initial metallicity and dust content of
the collapsing core.
O I and C II fine-structure line cooling dominates the thermal
evolution at nH < 104 cm−3 until the transition from non local thermal equilibrium to local thermal equilibrium occurs and the coolingefficiency decreases. Bromm & Loeb (2003) and Frebel, Johnson

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

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4 Astronomical Institute, Tohoku University, Sendai 980-8578, Japan

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G. Chiaki et al.

where S is the total cross-section of dust grains per unit dust
mass and the expression holds in the regime where dust coolingis effective; hence, grain temperatures are much lower than gas
temperature. An extensive parameter-space exploration shows that
Dcr ∼ 4.4 × 10−9 can be considered as a good representative value
for the minimal dust enrichment required to activate dust-induced
fragmentation (Schneider et al. 2012a).
The recently discovered Galactic halo star SDSS
J102915+172927 represents a perfect candidate for dustinduced fragmentation (Klessen, Glover & Clark 2012; Schneider
et al. 2012b). It has a stellar mass of <0.8 M and a metallicity
of only Zobs = 4.5 × 10−5 Z , being the most chemically pristine
star ever observed in the Universe (Caffau et al. 2011). Schneider
et al. (2012b) investigate the origin of SDSS J102915+172927 byreconstructing the physical conditions of the parent cloud out of
which the star had formed. They assume pre-enrichment of the gas
with metals and dust released by primordial core-collapse SNe,allowing for the partial destruction of grains by the reverse shock.
To simulate the physical conditions of the parent cloud of SDSS
J102915+172927 at the onset of collapse, the resulting metal anddust yields are characterized by the initial depletion factor fdep,0 ,defined to be the ratio of dust mass relative to the total mass ofheavy elements and are rescaled to the total metallicity of SDSS
J102915+172927 yielding an initial dust-to-gas mass ratio of
D0 = fdep,0 Zobs . Given these initial conditions, Schneider et al.
(2012b) follow the thermal evolution of the collapsing clouds andfind dust-induced fragmentation to be effective only in four out of
the eight models considered. Not surprisingly, since both fdep,0 andD0 are assumed to be constant during the course of the collapse,
the successful models conform to the condition that fdep,0 > 0.01,
that is D0 > Dcr .
In a recent paper, Nozawa, Kozasa & Nomoto (2012) show that as
the density in the collapsing cloud increases, dust grains can grow inmass by accreting metal species. Grain growth during the collapse isvery efficient and the dust-to-gas mass ratio increases even for verysmall initial gas metallicity and depletion factor. Earlier, in Chiaki,
Nozawa & Yoshida (2013b), we have studied the impact of grain
growth on the thermal evolution of collapsing clouds. It is shownthat even for small values of the initial depletion factor, fdep,0 ≈10−3 , and for initial metallicities in the range 10−5.5 –10−4.5 Z ,grain growth enables us to activate dust cooling at densities
MNRAS 439, 3121–3127 (2014)nH ∼ 1012 –1014 cm−3 , thereby inducing fragmentation into lowmass clumps. However, the model is based on a few simplifying
assumptions, such as the use of an average grain size as a proxy
to the full grain size distribution and of a single dust species todominate the total dust mass.
The aim of the present study is to investigate the implications
that the process of grain growth may have on the formation of alow-mass star with a metallicity of Z = 4.5 × 10−5 Z such as
SDSS J102915+172927. Starting from the same SN progenitor
models and initial conditions as Schneider et al. (2012b), we take
into account the modifications induced by grain growth on the size
distribution of the grains to check whether conditions favourable
to fragmentation can be met. A more systematic investigation of
the parameter space, including different SN models and initial gas
metallicities, will be presented in a forthcoming paper (Chiaki et al.,in preparation).
2 METHOD
We base our calculation on a one-zone semi-analytical model thatenables us to follow the thermal evolution of the collapsing cloud
and the process of grain growth in a self-consistent way. Here, we
describe the new features of the model required to implement theprocess of grain growth. For each dust species initially present in thecollapsing cloud, we calculate the time evolution of condensation
efficiency, fij , defined as the number fraction of nuclei of element
j condensed into grains of species i. We assume that (i) grains
are spherical particles and (ii) the rate is controlled by a single
chemical species, referred to as the key species, which corresponds
to the reactant with the least collisional frequency on the target.
Hence, the growth rate of a dust species i can be expressed as


1/2
 
4π 3
kT
dr
ai,0
= αi
ni1 (t),
(2)
dt i
3
2πmi1
where α i is the sticking probability of a gas particle on to grains ofspecies i, ai,0 is the radius of a single monomer of species i in thecondensed phase, and mi1 and ni1 are the mass and number densityof the key species for grains of species i.1 Note that the growth rateis independent of the grain radius. In this paper, we consider size
distribution function
ϕ i (r) for grains of species i. The function is
normalized as ϕi (r) dr = 1, and thus the number density of dust
i with radii between r and r + dr is ni ϕi (r) dr, where ni is the total
number density of dust grains i. Given the size distribution at thetime t, we can calculate the condensation efficiency of grains. Condensation efficiency is obtained by integrating the size distributionas 3
r ϕi (r) dr,(3)
fij (t) = fij ,0  3
r ϕi,0 (r) dr
where the subscript ‘0’ indicates the initial values.
As the collapse proceeds, equation (2) allows us to computethe time-dependent grain size distribution, dust mass, and depletion
factor. In addition, we need to compute the condensation efficiencies
fij , defined as the number fraction of element j condensed into
dust species i. We solve the non-equilibrium chemistry of eightprimordial elements H+ , e− , H, H− , H2 , D+ , D, and HD, and 19
heavy elements C+ , C, CH, CH2 , CO+ , CO, CO2 , O+ , O, OH+ ,1 The subscript i1 denotes the key chemical species for grains of species i(e.g. Mg atoms or SiO molecules for Mg2 SiO4 grains). We use a different
subscript j to indicate elements (e.g. Mg, Si, and so on).

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& Bromm (2007) have discussed the critical O and C abundances
required to overcome the compressional heating rate and fragment
the gas, [C/H]cr = −3.5 and [O/H]cr = −3.05. Given the typicalgas temperatures and densities where line-induced fragmentation
takes place, the associated Jeans-unstable fragment masses are still
relatively large, ≥10 M (Schneider et al. 2006; Safranek-Shrader,
Milosavljevic & Bromm 2014).
A pervasive fragmentation mode that allows the formation ofsolar or sub-solar mass fragments is activated at higher gas densitiesnH ∼ 1012 –1014 cm−3 where dust grains are collisionally excited andemit continuum radiation, thereby decreasing the gas temperature,T, until it becomes coupled with the dust temperature, or the gas
becomes optically thick (Schneider et al. 2002). The efficiency ofthis physical process depends on the dust-to-gas mass ratio and on
the total cross-section of the grains, with smaller grains providing
a larger contribution to cooling (Schneider et al. 2006). Hence,
the minimal conditions for dust-induced fragmentation have been
expressed in terms of a minimal critical dust-to-gas mass ratio,−1/2 
−1/2
T
nH
−3
2 −1
, (1)
SDcr = 1.4 × 10 cm g
103 K
1012 cm−3

Grain growth and the origin of SDSS J102915

allowing for the partial destruction of the newly formed dust by
the reverse shock. As is the same as Schneider et al. (2012b), foreach progenitor mass, we consider a model where no dust destruction is assumed to take place (norev) and three additional models
with increasing dust destruction efficiency by the reverse shock(rev1, rev2, and rev3 models which correspond to circumstellar
medium gas densities of 10−25 , 10−24 , and 10−23 g cm−3 , respectively). The initial depletion factors for each of these eight models,fdep,0 , are 0.26, 0.045, 0.015, and 0.0057 for norev, rev1, rev2, andrev3 models with Mpr = 20 M , respectively, and 0.076, 0.0082,
0.0025, and 0.0010 for norev, rev1, rev2, and rev3 models with
Mpr = 35 M , respectively.
3 R E S U LT S
In the left-hand panel of Fig. 1, we show the predicted temperature
evolution of collapsing clouds for all the initial conditions (progenitor masses and fdep,0 ) considered in this paper. The points mark thestate where fragmentation conditions are met (Schneider & Omukai2010). Molecular cooling leads to a first temperature dip at intermediate densities n ≈ 107 cm−3 where gravitational fragmentation
is expected to form relatively massive fragments, with tens of solar masses. At higher densities, n > 1011 cm−3 , grain growth has alarge effect on the thermal evolution of the clouds: all the models
show that enhanced dust cooling leads to a rapid decrease in gas
temperature and fragmentation into sub-solar mass fragments, independently of fdep,0 . This is different from what was originally foundby Schneider et al. (2012b) where dust cooling was unable to activate fragmentation for rev2 and rev3 models with Mpr = 20 M ,and rev1, rev2, and rev3 models with Mpr = 35 M , which did
not satisfy the condition of equation (1). To better clarify the impactof grain growth, in the right-hand panel of Fig. 1, we compare the

Figure 1. Left-hand panel: the temperature evolution for the norev (solid), rev1 (dotted), rev2 (dashed), and rev3 (dot–dashed) models with grain growthfor progenitor mass Mpr = 20 M (top) and 35 M (bottom) as a function of the cloud density. Right-hand panel: comparison between the temperature
evolution with (red) and without (blue) grain growth. We plot the results for norev (solid) and rev3 (dot–dashed) models. Triangles identify the states where
the fragmentation conditions are met.

MNRAS 439, 3121–3127 (2014)

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OH, H2 O+ , H2 O, H3 O+ , O+
2 , O2 , Si, SiO, and SiO2 . Silicon atoms
and SiO molecules are oxidized mainly by OH molecules (Langer& Glassgold 1990).
We adopt the same initial conditions as Schneider et al. (2012b)
to understand how grain growth may affect the resulting cloudfragmentation properties. In these models, dust species included
are forsterite (Mg2 SiO4 ), enstatite (MgSiO3 ), magnetite (Fe3 O4 ),amorphous carbon (C), silica (SiO2 ), and alumina (Al2 O3 ). Then,
we assume that silicate (Mg2 SiO4 and MgSiO3 ) and silica grains
grow when SiO and SiO2 molecules stick to the grains, respectively.
Dust temperature, dust continuum opacity, and cooling rate are calculated at each time using the time-dependent grain size distributionfunction on the basis of formulae of Schneider et al. (2006). The
size bins have an initial width of 0.1 dex. The results are not significantly different from the ones obtained when the initial size bins
are 0.01 dex width.
Additional features of the model, other than the part of grain
growth, are the same as in the model used by Schneider et al.
(2012b). The collapsing gas cloud has a total metallicity ofZobs = 4.5 × 10−5 Z , equal to the observed metallicity of SDSS
J102915+172927. The initial metal and dust contents are taken fromtwo different Population III SN models with progenitor masses andexplosion energies selected to minimize the scatter between thetheoretical yields and the observed ones. Despite its very low ironcontent, SDSS J102915+172927 shows an abundance pattern consistent with typical Galactic halo signatures and the best agreement
is found for 20 and 35 M progenitors with mass cuts, explosion energies, and ejected 56 Ni mass in the range of what is typically foundfor ordinary core-collapse SNe (Limongi & Chieffi 2012). Using
these SN models as input for the dust nucleation model (Bianchi& Schneider 2007), they compute the mass, composition, and size
distribution of dust grains released in the interstellar medium, after

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G. Chiaki et al.

thermal evolution of the norev and rev3 models with and without
grain growth.2 At densities n < 1011 cm−3 , the thermal tracks arestill hardly affected by grain growth because dust cooling itself is
not effective in this regime. The norev tracks appear to be only
moderately affected because almost all refractory elements are already depleted (0.93 of Mg, 1.0 of Fe for Mpr = 20 M ) if they
are not destroyed by the effects of reverse shocks. Conversely, theevolution of rev3 models, which suffers the strongest dust destruction by the reverse shock and have very low initial depletion factors,
show that grain growth leads to efficient cooling and fragmentation
at n > 1011 cm−3 , where the tracks strongly deviate from the cases
without grain growth.
We first discuss the results for rev3 model with Mpr = 35 M .
In Fig. 2, for each dust species we show the predicted grain size
distribution functions, ni ϕ i (r)/nH , in unit of cm−1 before (upper
panel at nH = 0.1 cm−3 ) and after (bottom panel at nH = 1016 cm−3 )grain growth. We can see that the radii of all grain species, except
for carbon dust, increase. The size distribution of carbon grainsis nearly unaffected because C atoms are already depleted into CO
molecules early at densities below nH ∼ 106 cm−3 . Since the growthrate, (dr/dt)i , is independent of grain radius (equation 2), the net
effect is an overall shift of the size distribution function to larger
radii. For the same model, we also show in Fig. 3 the evolution of thecondensation efficiencies and the fractional abundance of gas-phaseelements as a function of the cloud central density. As expected, thecondensation efficiencies of carbon atoms remains constant andalmost equal to its initial value. The first grain species to growat nH ∼ 109 –1012 cm−3 are silicate (Mg2 SiO4 and MgSiO3 , solid
2 Our calculations without grain growth are the same as the calculations
performed by Schneider et al. (2012b).

MNRAS 439, 3121–3127 (2014)

Figure 3. Top panel: the evolution of the condensation efficiency as afunction of the cloud central density. Bottom panel: the relative abundanceof gas-phase elements. Red solid curve represents the number fraction ofgas-phase magnesium atom relative to total Mg nuclei. Green dashed anddot–dashed curves depict the number fraction of SiO and SiO2 molecules
relative to total Si nuclei, respectively.

and dashed lines in the upper panel, respectively) grains whose key
species is the SiO molecule. As silicate grains grow, the abundanceof SiO molecules decreases until its abundance eventually drops tozero (see the dashed line in the bottom panel). The growth of silica
grains is controlled by SiO2 molecules whose formation rate grows
with density. Though the abundance of SiO2 molecules remains
nearly constant because the abundance of SiO2 grains are small at
nH ∼ 109 –1013 cm−3 , it eventually drops to zero (dot–dashed linein the bottom panel). The condensation efficiency of silicon atoms
on to silica grains grows by more than four orders of magnitudes.
Finally, although with different evolutionary properties, Fe and Al
condensation efficiencies reach unity, meaning that all the available
Al and Fe grains are condensed into solid grains.
We consider that grain growth is significant when the collapse
has reached the density (time) threshold, ngrow,i , at which the condensation efficiency becomes 0.5. From the results of calculations,
we find a fitting function of ngrow,i as

−2 

Aj
fij ,0 −0.8
ngrow,i = 1.0 × 1012 cm−3
7.1 × 10−10
0.1
 grow 2 

−6 
ri,0
ai,0
mi1
×
,
0.01 μm
mH
1Å
grow

grow

1/3

where ri,0 is the mass-weighted average radius, ri,0 = r 3 i,0 .
Hence, the condition for efficient grain growth depends on specificproperties of the grain species as well as on the abundance of the
key species. As an example, grains with larger ai,0 , such as silicates,
grow earlier in the evolution (at lower densities), whereas in rev3

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Figure 2. Size distribution functions, ni ϕ i (r)/nH , of different dust species
before (top) and after (bottom) grain growth for the rev3 model with
Mpr = 35 M .

Grain growth and the origin of SDSS J102915

3125

model with Mpr = 35 M , aluminium abundance is so small that thegrowth of Al2 O3 grains is inefficient at all but the largest densities.
While the growth rate of each specific grain depends also on
the initial composition of gas-phase elements, we find that grain
growth is able to activate dust cooling and fragmentation for all
the explored initial conditions (two SN progenitor models, a total
gas metallicity set to reproduce the observed surface metallicity of
SDSS J102915+172927, and 0.001 < fdep,0 < 0.26). In Fig. 4, weshow the cooling rate for rev3 models with Mpr = 35 M , eachline representing the contribution of a single dust species. Left-handpanel shows the result without grain growth. In this case, total dust
cooling does not exceed gas compressional heating. This means thatthe gas does not meet the condition for fragmentation. With grain
growth (right-hand panel), dust cooling is increased and dominates
over compressional heating rate.
The dominant coolants are silicate and magnetite grains. Even
if only MgSiO3 grains are present, their effect on the cooling ratewould be enough to overcome compressional heating and enablegas fragmentation. This conclusion is independent of the specificreverse shock model (the value of fdep,0 ) as silicates and magnetite grains always represent the dominant dust species. Compressional heating is smaller for the model with grain growth because the rate is proportional to temperature which is reduced byenhanced dust cooling. Cooling rates increase with density for
109 cm−3 < nH < 1012 cm−3 as a consequence of the combined
effects of grain growth and the increasing collisional rate between
dust and gas particles. Above nH ∼ 1012 cm−3 , the cooling rates
start to decline as the gas temperature approaches the dust temperature. For silicates and magnetite grains, this decrement is partly
compensated by grain growth. At nH ∼ 1014 cm−3 , the gas cloud
starts to be optically thick to continuum opacity.

4 DISCUSSION
The results discussed above have been obtained under the assumptions of a sticking probability of α i = 1 for all dust species. In addition, there may be physical mechanisms that limit grain growth,such as dust evaporation, and dust coagulation due to grain–grain
collisions. However, none of these affects the gas fragmentation
properties, at least in the models that we have investigated, as we
explain in what follows.

We first consider the effect of a smaller sticking probability.
In fact, the exact value of the sticking probability for each dust
species is unknown. A more careful treatment might be requiredfor grain species on to which more than one gas-phase element
is accreted such as silicates. Let us assume that two elements, 1
and 2, are accreted on to a grain species i, with corresponding
growth rates (dr/dt)i1 and (dr/dt)i2 , respectively. If element 1 isthe key species, then by definition, (dr/dt)i1 < (dr/dt)i2 and thegrowth rate of ith grains is set to be (dr/dt)i = (dr/dt)i1 . As long as(dr/dt)i1 < (dr/dt)i2 , the sticking probability would not be affectedby the species which collides with the second most frequent collision
rate. However, when (dr/dt)i1 ∼ (dr/dt)i2 , element 2 collides lessfrequently and the sticking probability might be further reduced. We
also run models assuming that α i = 0.1 for all dust species. In thisconservative limit, dust formation in the SN ejecta is also affected
and fdep,0 are further reduced with respect to the reference model.
However, we find grain growth to be effective and the fragmentation
condition to be verified for all progenitor and reverse shock models.
The effects of dust coagulation in star-forming clouds with different initial metallicities have been discussed by Hirashita &Omukai (2009). They show that as long as the grain velocities
are governed by thermal motions, coagulation occurs at densities
nH > 107 (Z/Z )−2 (T/100 K)−1 cm−3 but it does not have significant effects on the thermal evolution of clouds. In addition,
grain growth increases the average grain mass, thereby decreasing
their thermal velocity which eventually drops below the coagulation threshold velocity. Dust evaporation may be important if thedust temperature becomes comparable to the sublimation temperature. To check the importance of this effect, we have conservatively
assumed that grains whose temperature is above the sublimation
temperature are immediately returned to the gas phase. The sublimation temperatures as a function of density are taken from Pollack
et al. (1994). We find that dust evaporation can be safely ignored
because the dust temperature always remains well below the sublimation temperature for all the models which we investigate here.

5 CRITICAL INITIAL CONDITIONS FORT H E F O R M AT I O N O F S D S S J 1 0 2 9 1 5 +1 7 2 9 2 7
The results of the present study may have important implications for the formation of the first low-mass stars in the Universe.
MNRAS 439, 3121–3127 (2014)

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Figure 4. Compressional heating (red) and dust cooling rates as a function of gas density for rev3 models with Mpr = 35 M with (left) and without (right)
grain growth. Each line represents the contribution of a single dust species and the light grey solid line shows the total cooling rate. The abrupt change of slope
at nH 1014 cm−3 is where to continuum opacity becomes effective.

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G. Chiaki et al.
Table 1. Critical abundances.
Model

cool
rMgSiO(µm)
3cool (µm)
rFe
3 O4

fMgSiO3 ,Mg

fFe3 O4 ,Fe[Mg/H]cr[Fe/H]cr

20

norev
rev1
rev2
rev3

8.39e−03
1.57e−022.31e−02
3.27e−02

2.38e−02
2.44e−02
2.55e−02
2.78e−02

0.20
0.35
0.37
0.40

1.00
0.43
0.19
0.07

−5.02
−4.99
−4.84
−4.73

−4.92
−4.53
−4.16
−3.70

cool
rMgSiO(µm)
3cool (µm)
rFe
3 O4

fMgSiO3 ,Si

fFe3 O4 ,Fe[Si/H]cr[Fe/H]cr

1.45e−02
1.89e−02
2.51e−02
3.24e−02

7.27e−03
1.01e−02
1.19e−02
1.43e−02

0.60
0.44
0.32
0.19

1.00
0.55
0.28
0.12

−5.24
−4.99
−4.72
−4.39

−5.43
−5.03
−4.66
−4.21

35

norev
rev1
rev2
rev3

Note. We calculate the critical abundances of Si or Mg, and Fe (seventh and eighth columns, respectively) byusing values obtained from our calculations at nH = 1012 cm−3 (third to sixth columns).

A smaller dust-to-gas mass ratio may be sufficient to activate dustinduced fragmentation by the accretion of heavy elements on to
grains during cloud collapse. In particular, our analysis shows thatthe dust species that dominate cooling are silicates and magnetite.
Grain growth do not affect carbon grains, even if these are initially present, because carbon atoms are rapidly locked into CO
molecules during the first stages of the collapse. Hence, the conditions for low-mass star formation may rely on the initial abundanceof other refractory elements, such as key elements of silicates andmagnetite: Mg and Fe for our Mpr = 20 M models, and Si and Fe
for our Mpr = 35 M models.
A conservative lower limit to the initial abundance of refractory elements can be obtained as follows. The critical condition
expressed by equation (1) depends on the product of the dust-to-gas
mass ratio and the grain cross-section per unit mass of dust. If weassume a single dominant grain species i, the critical condition aftergrain growth can be written as
SD = XH Aj fij μi

πr 2 i
≥ SDcr ,(4π/3)si r 3 i

(4)where si is the material density of grains i. Here, we define another characteristic size for dust cooling as ricool = r 3 i /r 2 i . Fromequation (4), the critical abundance reads−1/2 −1/2TnH
4 si ricool,
Aj ,cr = 1.4 × 10−33
12−3
3XH fij μi 10 K
10 cm
for (i, j) = (MgSiO3 , Si or Mg) and (Fe3 O4 , Fe), respectively. Table 1 shows the critical abundances that correspond to the valuesof ricool and fij at nH = 1012 cm−3 , obtained for each of the SN
model that we have explored. Given the large variation of condensation efficiencies experienced by the key elements in the different models, we can take as indicative critical abundances the
maximum and minimum values that we find for each refractoryelement: [Mg/H]cr = [−5.02: −4.73], [Si/H]cr = [−5.24: −4.73],
and [Fe/H]cr = [−5.43: −3.70], where we have normalized to solar
abundances.3
The initial abundances of Mg, Si, and Fe predicted by the initial
SN progenitors are [Mg/H] = −4.02 and [Fe/H] = −4.55 for the
3 We have considered A−5−5Fe, = 3.31 × 10 , ASi, = 3.31 × 10 , and−5AMg, = 3.47 × 10 (Caffau et al. 2011). Also, the mean molecularweights per unit key elements are μMgSiO3 = 100 and μFe3 O4 = 77.3. The
material densities are sMgSiO3 = 3.21 g cm−3 and sFe3 O4 = 5.25 g cm−3

(Nozawa et al. 2003).

MNRAS 439, 3121–3127 (2014)

Mpr = 20 M models and [Si/H] = −4.39 and [Fe/H] = −4.71
for the Mpr = 35 M models with Z = 4.5 × 10−5 Z . Hence, forall our models, the Si or Mg abundances are sufficient or marginallysufficient (for rev3 model with Mpr = 35 M ) to enable fragmentation even if there were (hypothetically) only MgSiO3 grainsin the clouds. Conversely, in all models but the rev1 model with
Mpr = 35 M , the Fe abundance is not sufficient to enable fragmentation if only Fe3 O4 grains are present in the clouds. These
conclusions are consistent with the results obtained by the full calculations shown in Fig. 1 and in the right-hand panel of Fig. 4.
Recently, Ji, Frebel & Bromm (2014) have compared criticalsilicate abundances for dust cooling to chemical abundances ofmetal-poor stars, with [Fe/H] < −4. Assuming silicon-based dustwith different grain size distributions, they conclude that, out of thenine stars considered, two to four stars show silicate abundances
below the critical ones, pointing to a different formation pathway.
Interestingly enough, all the stars that cannot form through siliconbased dust satisfy the criterion for O I and C II fine-structure line
cooling (Frebel et al. 2007).
We conclude that low-mass star formation induced by dust cooling may be more common than previously thought based on therequired initial dust-to-gas mass ratio of very low-metallicity starforming clouds. Although the first SNe may already provide enoughdust grains, our study – which is motivated by the observed properties of SDSS J102915+172927– suggests that even in the extreme
conditions where a large fraction of the newly formed dust is destroyed by the reverse shock of the SN, grain growth during thecollapse is able to replenish the required amount of dust provided
that a minimum abundance of refractory elements (mostly Si, Fe,
and Mg) is initially present in the gas phase.

AC K N OW L E D G E M E N T S
We thank Simone Bianchi for his kind collaboration. 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 (23540324,
25287040, 22684004, and 23224004). The research leading to theseresults has received funding from the European Research Council under the European Union’s Seventh Framework Programme(FP/2007-2013) / ERC Grant Agreement no. 306476.

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Mpr (M )Grain growth and the origin of SDSS J102915
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