The Astrophysical Journal Letters, 756:L35 (5pp), 2012 September 10

doi:10.1088/2041-8205/756/2/L35

C 2012. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

CAN THE GROWTH OF DUST GRAINS IN LOW-METALLICITY STAR-FORMING
CLOUDS AFFECT THE FORMATION OF METAL-POOR LOW-MASS STARS?
Takaya Nozawa1 , Takashi Kozasa2 , and Ken’ichi Nomoto1

1 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; takaya.nozawa@ipmu.jp
2 Department of Cosmosciences, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan

Received 2012 June 21; accepted 2012 July 31; published 2012 August 23

ABSTRACTThe discovery of a low-mass star with such a low metallicity as 4.5 × 10−5 Z reveals the critical role of dustin the formation of extremely metal-poor stars. In this Letter, we explore the effect of the growth of dust grains
through accretion of gaseous refractory elements in very low metallicity pre-stellar cores on cloud fragmentationinduced by dust emission cooling. Employing a simple model of grain growth in a gravitationally collapsing gas, weshow that Fe and Si grains can grow efficiently at hydrogen densities of 1010 –1014 cm−3 in the clouds with metalabundances of −5  [Fe, Si/H]  −3. The critical metal number abundances, above which the grain growth could
induce the fragmentation of the gas clouds, are estimated to be Acrit  10−9 –10−8 , unless the initial grain radius is
too large (1 μm) or the sticking probability is too small (0.01). We find that even if the initial dust-to-gas mass
ratio is well below the minimum value required for the dust-induced fragmentation, the grain growth increases thedust mass high enough to cause the gas to fragment into sub-solar mass clumps. We suggest that as long as thecritical metal abundance is satisfied, grain growth could play an important role in the formation of low-mass stars
with metallicity as low as 10−5 Z .
Key words: dust, extinction – ISM: clouds – stars: formation – stars: low-mass – stars: Population II – supernovae:
generalOnline-only material: color figures

if the majority of grains formed in the SN ejecta are destroyed by
the reverse shock (see also Schneider et al. 2012b). However,it could be possible that the accretion of gaseous refractoryelements released by dust grains in the shocked gas onto thesurfaces of the SN dust surviving in star-forming clouds changes
the mass and size distribution of the dust, and thus affects thethermal evolution of the collapsing cores.3
In this Letter, we investigate the feasibility of grain growth
in low-metallicity star-forming clouds to explore whether thegrain growth can facilitate the formation of metal-poor lowmass stars. In Section 2, we describe the model of grain
growth in collapsing dense clouds, and present the results ofthe calculations in Section 3. In Section 4, we estimate thecritical metal abundances above which the grain growth could
encourage the gas fragmentation into sub-solar mass clumps,
and discuss the corresponding dust-to-gas mass ratio and total
metallicity. Our conclusion is given in Section 5.

1. INTRODUCTION
Dust grains in the early universe are considered to beimportant agents in triggering the formation of low-mass stars inmetal-poor environments (Omukai 2000; Schneider et al. 2003);
the cooling of the gas through the thermal emission of dust
makes collapsing dense cores gravitationally unstable, leading
to fragmentation into multiple sub-solar mass clumps at gas
densities of 1012 –1014 cm−3 (Tsuribe & Omukai 2006; Dopcke
et al. 2011). This scenario has been recently supported by thediscovery of a Galactic low-mass star, SDSS J102915+172927(Caffau et al. 2011), whose metal content is too low (Z 
4.5 × 10−5 Z ) to induce the fragmentation of star-forming
clouds by metal-line cooling (see Klessen et al. 2012; Schneider
et al. 2012b for details).
The condition that allows dust-induced fragmentation to berealized depends on the amount of dust grains as well as their size
distribution in pre-stellar clouds (Omukai et al. 2005; Schneider
et al. 2006, 2012a). Schneider et al. (2012a) found that theformation condition of the low-mass fragments obtained fromnumerical simulations is fully described in terms of the productof the dust-to-gas mass ratio D and geometrical cross section
per unit dust mass S as follows:
SD > 1.4 × 10−3 cm2 g−1



Tgas
103 K

− 12 2. MODEL OF GRAIN GROWTH IN METAL-POOR
STAR-FORMING CLOUDS
We consider the growth of dust grains in collapsing clouds
that have been enriched with metals and dust grains produced
by very early generation of SNe. Dust formation calculations by
Nozawa et al. (2003) showed that various grain species condensein the unmixed ejecta of Population III SNe, and that Fe, Si, andC grains have relatively large average radii (0.01 μm). Basedon their dust models, Nozawa et al. (2007) investigated theevolution of dust in the hot gas swept up by the SN shocks and

− 12
cH
, (1)
1012 cm−3

where Tgas is the temperature of the gas and cH is the hydrogen
number density. Treating self-consistently the dust formation in
the ejecta of supernovae (SNe) and the subsequent destructionof the dust by the reverse shocks, Schneider et al. (2012a) argued
that the condition could not be satisfied in the collapsing starforming clouds enriched with metals and dust from the first SNe

3

Hirashita & Omukai (2009) examined the coagulation of dust in collapsing
clouds with a variety of metallicity. They found that the dust coagulation canproceed even at metallicity as low as 10−6 Z but it does not have any impact
on the thermal evolution of the star-forming clouds.

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Nozawa, Kozasa, & Nomoto

Table 1
Numerical Values Used in the Calculations
Numerical Values
s=1
Tgas = 103 K
μ = 2.18
aFe,0 = 1.441 Å
aSi,0 = 1.684 Å
mFe = 56mH
mSi = 28mHAFe, = 3.26 × 10−5ASi, = 3.58 × 10−5
ρFe = 7.90 g cm−3
ρSi = 2.32 g cm−3

Explanation of Symbols
Sticking probability
Gas temperature
Mean molecular weight
Radius of an Fe atom in the solid phasea
Radius of an Si atom in the solid phasea
Mass of an Fe atom
Mass of an Si atom
Solar abundance of Fe relative to Hb
Solar abundance of Si relative to Hb
Bulk density of Fec
Bulk density of Sic

Notes.
a Nozawa et al. (2003).
b Anders & Grevesse (1989).
c Nozawa et al. (2006).

Figure 1. Time evolutions of the condensation efficiency fFe (t) (red) andgrain radius rFe (t) (blue) through the growth of Fe grains with the initial dust
abundance fFe,0 = 0.1 and the initial grain radius rFe,0 = 0.01 μm as a functionof hydrogen number density cH (t). Dot-dashed, solid, and dashed lines depict
the results for the collapsing clouds with the Fe abundances of [Fe/H] = −3,−4, and −5, respectively.
(A color version of this figure is available in the online journal.)

found that most of such large Fe, Si, and C grains can survive thedestruction by the reverse shock to be predominantly injected
into the early interstellar medium.
Being motivated by these studies, and to simplify the chemistry, we consider that Fe and Si grains composed of only one
element grow through accretion of Fe and Si atoms in the gas
phase, respectively.4 We assume that dust grains are spheres,
all of which have a single initial radius, although they might be
expected to have the distribution of grain sizes. Suppose thatthe number density of a given refractory element i at a time t0
is ci,0 = ci (t0 ) = Ai cH,0 , where Ai is the number abundanceof the element i relative to hydrogen. In order to specify the
number abundance of pre-existing seed grains, we introduce aparameter fi,0 (>0), which is defined as the number fraction of
the elements i originally locked in dust grains (i.e., condensation
efficiency at t0 ). Then, the number density of the dust whose ini3
tial radius is ri,0 is described as ndusti,0 = fi,0 ci,0 (ai,0 /ri,0 ) , where
ai,0 is the hypothetical radius of an atom in the dust phase.
The time evolution of the number density ci (t) of an element
i in the gas clouds collapsing with the timescale of free fall isgiven by


t −2
,
(2)
ci (t) = ci,0 1 − ff
2τ0

In the dense clouds where almost all gaseous atoms areneutral, the growth rate of grain radius is given by





1
kTgas 2 gas
dri
4π 3
1 Tdust= siaci (t) 1 −
, (5)
dt
3 i,0
2π mi
Si Tgaswhere si is the sticking probability of the gaseous element i
incident onto grains, k is the Boltzmann constant, and mi is themass of the element i. The gas temperature Tgas is assumed tobe constant during the evolution of clouds in this study. The
supersaturation ratio Si is a function of the dust temperature
Tdust . Since Tdust /Tgas  1 and Si  1 under the condition
considered here (Dopcke et al. 2011), Equation (5) is reduced
to


t −2
Yi (t)dXi= gg 1 − ff
,
(6)
dt
2τ0
τi,0
3
(kTgas /2π mi )1/2 Ai cH,0 /3ri,0 .
by introducing (τi,0 )−1 = si 4π ai,0Then, integration of Equation (6) leads to the ratio of grainradius to the initial one Xi (t):gg

where t is the elapsed time from t0 , τ0ff = (3π/32GμmH cH,0 )1/2
is the free-fall time at the density cH,0 with the gravitational
constant G, the mean molecular weight μ, and the mass of ahydrogen atom mH . Once grain growth is activated, the gaseousgas
atoms are consumed, and the number density ci (t) of anelement i in the gas phase at the time t can be written asgas

ci (t) = ci (t){1 − fi,0 [ri (t)/ri,0 ]3 },Xi (t) = 1 +u0

Y (u )du ,(1 − u )2(7)

with u = t/2τ0ff .
In principle, by solving Equations (4) and (7) for a given set offi,0 , ri,0 , and Ai , we can calculate the time evolution of fi (t) andri (t) = ri,0 Xi (t). The values of the other parameters necessary
for the calculations are summarized in Table 1. As shown later,
for the metal abundances considered in this Letter, the grain
growth operates at high gas densities of cH  1010 cm−3 , where
the gas temperature is expected to be in the range of 500–2000 K(Dopcke et al. 2011). Thus, the calculations are started from
cH,0 = 108 cm−3 , with Tgas = 103 K. Note that the results of thecalculations are not sensitive to Tgas as long as the above rangeof Tgas is considered.

(3)

with ri (t) being the radius of the ith grain species at t.
Equation (3) is reduced tofi (t) = 1 − Yi (t) = fi,0 Xi3 (t),

2τ0ff
gg
τi,0

(4)

where fi (t) is the condensation efficiency at t, Yi (t) =gasci (t)/ci (t) represents the depletion of the gaseous atoms due
to grain growth, and Xi (t) = ri (t)/ri,0 .

3. RESULTS OF CALCULATIONS OF GRAIN GROWTH

4

The growth of C grains may not be expected in dense clouds. This isbecause at the high gas densities considered in this Letter (cH  108 cm−3 ), all
C atoms can be bounded in CO molecules in gas clouds with the metallicity
scaled by the solar abundance (Omukai et al. 2005).

Figure 1 depicts the growth of Fe grains with fFe,0 = 0.1
and rFe,0 = 0.01 μm; the time evolutions of the condensation
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The Astrophysical Journal Letters, 756:L35 (5pp), 2012 September 10

Nozawa, Kozasa, & Nomoto

Figure 2. Time evolutions of the condensation efficiency fSi (t) (red) and grainradius rSi (t) (blue) through the growth of Si grains of rSi,0 = 0.01 μm in thecollapsing clouds with the Si abundance of [Si/H] = −4. Solid, dashed, and dotdashed lines represent the results for the initial dust abundances of fSi,0 = 0.1,
0.01, and 0.001, respectively.
(A color version of this figure is available in the online journal.)

Figure 3. Critical abundances of Fe (upper panel) and Si (lower panel), for which
the grain growth can lock up a fraction fi,∗ = 0.8 (dot-dashed), 0.5 (solid), and0.25 (dashed) of Fe and Si atoms in dust grains at the hydrogen number densitycH,∗ = 1012 cm−3 . The horizontal axis is the initial dust abundance fi,0 , andthe initial grain radius is set to be ri,0 = 0.01 μm.
(A color version of this figure is available in the online journal.)

efficiency fFe (t) and grain radius rFe (t) versus hydrogen number
density cH (t) for [Fe/H] = −5, −4, and −3, which correspond to
AFe = 3.26×10−10 , 3.26×10−9 , and 3.26×10−8 , respectively,
with the solar abundance by Anders & Grevesse (1989). We cansee that the grain growth is activated efficiently even in the gas
clouds with [Fe/H] = −5, and the grain radius finally reaches
a constant value rFe,0 (1/fFe,0 )1/3 by consuming all gaseous Fe
atoms. However, the gas density at which a considerable fraction(fFe ∼ 0.5) of Fe atoms is locked up in dust grains is higher
for lower Fe abundance; cH  1010 , 1012 , and 1014 cm−3 for[Fe/H] = −3, −4, and −5, respectively. Also, the gas density at
which fFe reaches ∼0.5 is two orders of magnitude higher (lower)
for rFe,0 = 0.1 (0.001) μm than that for rFe,0 = 0.01 μm, though
it is not presented in the figure. These behaviors of grain growthin the collapsing gas clouds can be seen from Equation (9) (seebelow); for given values of fi,0 and fi,∗ , cH (t) ∝ (ri,0 /Ai )2 if
cH (t)/cH,0  1. Hence, for a fixed fi,0 , one order of magnitude
higher ri,0 or one order of magnitude lower Ai is compensatedby two orders of magnitude higher cH .
Figure 2 shows the time evolutions of the fSi (t) and rSi (t) of Si
grains with rSi,0 = 0.01 μm for different initial dust abundancesof fSi,0 = 0.1, 0.01, and 0.001. Here, the total abundance of
Si atoms is set to be [Si/H] = −4 (ASi = 3.58 × 10−9 ). For
fi,0 = 0.1, the growth of Si grains proceeds somewhat earlier
than that of Fe grains for [Fe/H] = −4, and the condensation
efficiency increases to fSi = 0.5 at cH  1011 cm−3 . In cases oflower fSi,0 , higher gas densities are needed to achieve some levelof the condensation efficiency, although the final grain radii arelarger when all Si atoms are tied up in dust grains.

fi,∗ of the element i should be locked up in dust grains at
a given hydrogen number density cH,∗ . Noting that Yi (t) =1 − fi,0 Xi3 (t) = 1 − fi (t), Equation (7) can be rewritten as
Xi,∗

Ki (fi,0 , fi,∗ ) =
1

2τ ff
= gg0
τi,0

dXi
1 − fi,0 Xi3
u∗02τ0ff u∗du=,gg
τi,0 1 − u∗
(1 − u )2(8)

where Xi,∗ = (fi,∗ /fi,0 )1/3 and u∗ = 1 − (cH,0 /cH,∗ )1/2 . From
this equation, we can derive the critical metal abundances above
which the grain growth could facilitate the fragmentation of theclouds as follows:−9Ai,crit = (1.0–2.5) × 10


Ki

ri,0
0.01 μm1012 cm−3
cH,∗

 12
,

(9)
using cH,0 /cH,∗  1. The numerical factor 2.5 (1.0) corresponds
to Fe (Si) grains. The function Ki increases with increasing fi,∗and/or decreasing fi,0 ; for 0.2  fi,∗  0.8, Ki = 0.3–2.4(5.2–15) at fi,0 = 0.1 (fi,0 = 0.001).
Figure 3 presents the critical abundances of Fe and Si in
the form of [X/H] versus fi,0 for ri,0 = 0.01 μm; the dotdashed, solid, and dashed lines give the abundances necessary
for the condensation efficiency fi,∗ to reach 0.8, 0.5, and 0.25,
respectively, at cH,∗ = 1012 cm−3 . As expected, higher metalabundances are needed to attain higher fi,∗ and/or for lower
fi,0 . For the case of fi,∗ = 0.5, the critical abundances ofFe and Si spans the ranges of −4.11  [Fe/H]  −3.19
and −4.54  [Si/H]  −3.62, respectively, for the range of
0.1  fi,0  0.001. It would be interesting to mention that theabove range of [Si/H] covers the abundance of Si observed for
SDSS J102915+172927 ([Si/H] = −4.27; Caffau et al. 2011).
This could suggest that the growth of Si grains might haveworked in the parent cloud of this star.

4. DISCUSSION: CRITICAL METAL ABUNDANCESAs shown in the last section, grain growth can be activatedeven in metal-poor star-forming clouds whose metallicity is
only [Fe, Si/H]  −5. However, in order that the grain
growth affects the thermal evolution of collapsing cores, it
must become effective before the cloud density increases to
cH = 1012 –1014 cm−3 , where the optical depth becomes highenough to suppress dust emission cooling, and as a result gas
fragmentation is expected to occur (e.g., Schneider et al. 2012a).
The metal abundance above which grain growth becomes
important can be estimated by requiring that a certain fraction
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The Astrophysical Journal Letters, 756:L35 (5pp), 2012 September 10

Nozawa, Kozasa, & Nomotoin Figure 4 are independent of the initial grain radius ri,0 since−1
Si ∝ ri,0and Di ∝ Ai,crit ∝ ri,0 .
Finally, we relate the critical abundance to the total
metallicity Z. Equation (9) suggests that the critical metal abundances are generally in the range of Acrit  10−9 –10−8 , depending on fi,0 , fi,∗ , ri,0 , and grain species. By representing the mass
ratio of refractory elements condensible into dust grains to thetotal heavy elements as R, the metallicity Z can be related to
Acrit as RZ  Acrit (μref /μ) with μref being the mean atomic
mass of refractory elements. Then, we have
−6 0.2Z ,(11)Z  (5–50) × 10
R
where we use μref /μ = 20 and Z = 0.02. Equation (11)implies that, if the grain growth does work efficiently, it candrive the gas fragmentation of low-mass clumps in the starforming clouds enriched with metallicity ∼10−5 Z . In otherwords, as long as the abundance of a given refractory element
satisfies the critical abundance in Equation (9), the formationof hyper-metal-poor low-mass stars with metallicity lower thanZ  4.5 × 10−5 Z observed in SDSS J102915+172927 could
be possible.

Figure 4. Products of dust-to-gas mass ratio D and geometrical cross section
per unit dust mass S for Fe (blue) and Si (red) as a function of the initialdust abundance fi,0 . The dashed lines depict (SD)i,crit corresponding to thecritical abundances given by the solid lines in Figure 3, whereas the solid lines
depict the resulting dust-to-gas mass ratio after the grain growth (SD)i,∗ . The
horizontal dot-dashed line indicates the minimum value above which the dust
emission cooling causes the gas to fragment into low-mass clumps (Schneider
et al. 2012a).
(A color version of this figure is available in the online journal.)

5. CONCLUDING REMARKS

Here, we present how the fragmentation condition induced bygrain growth depends on unknown parameters such as ri,0 and si .
The time duration Δt ff for which the gas density increases from
cH,0 to cH,∗ by free fall is given by Δt ff = 2τ0ff (1 − cH,0 /cH,∗ ).
On the other hand, the time duration Δt gg for which thecondensation efficiency fi,0 at cH,0 increases up to fi,∗ atcH,∗ through grain growth is derived from Equation (8) asgg(Δt gg )−1 = (Ki τ0 )−1 +(2τ0ff )−1 . Since Δt ff /Δt gg  1 is required
for the fragmentation, we can obtain the condition
0.01 μm
Ai
1.0
2.5 × 10−9
ri,0
 c
 12  1.0 H,∗× 1.0,
1012 cm−3
Ki(1.0–2.5)

We have investigated the growth of dust grains in metalpoor protostellar clouds. Our simple model shows that grain
growth can operate efficiently even in collapsing dense coreswith metal abundances as low as [Fe, Si/H]  −5. We alsopresent the critical metal abundances above which grain growth
could affect the fragmentation process of collapsing gas clouds.
This abundance is estimated to be Acrit  10−9 –10−8 , whichsuggests that the formation of low-mass stars with metallicityof ∼10−5 Z can be possible. We conclude that even if the initialdust-to-gas mass ratio does not satisfy the condition required forthe dust-induced fragmentation, the grain growth can increase
the dust-to-gas mass ratio high enough to facilitate the formationof metal-poor low-mass stars.
Our results suggest that if grain growth is considered, theformation of low-mass protostars can occur not only at very
low metallicity but also at higher metallicity. The final mass of a
newly born star is determined by the accretion of the surrounding
gas onto the protostars (McKee & Ostriker 2007 and references
therein), and its mass accretion rate would be regulated by themass of the central protostar induced by the grain growth. Thus,
the grain growth in collapsing clouds might be a fundamental
physical process to control the stellar initial mass function in
the present universe.
It should be mentioned that we have considered only thegrowth of single-component Fe and Si grains with a single
initial radius. However, it might be possible that Si and Fe atoms
condense as silicates or oxides in an oxygen-rich gas. Since thegrowth of such compound grains with no monomer moleculehas been usually treated by considering Si or Fe element as a
key element (e.g., Zhukovska et al. 2008), the mass and radiusof dust given in this Letter are considered to be lower limits.
On the other hand, the timescale of grain growth is sensitive
to the initial grain radius (Hirashita & Kuo 2011). Thus, theeffect of the initial size distribution as well as the growth ofcompound grains should be explored. Furthermore, we haveassumed the sticking probability of si = 1 and a constant gas
temperature Tgas = 103 K during the collapse of the clouds. In
particular, too low sticking probabilities (si  0.01) may prevent s 
i(10)

where the numerical factor 1.0 (2.5) is for Fe (Si) grains.
This inequality demonstrates that the fragmentation condition isachieved more easily for initial grain radii smaller than 0.01 μm.
In contrast, if ri,0 > 1.0 μm or si < 0.01, grain growth can nolonger induce fragmentation for Ai  2.5 × 10−7 ([Fe, Si/H] −2). We also note that a smaller fi,0 producing a larger Ki acts
against gas fragmentation by grain growth.
Next we consider the dust-to-gas mass ratio to see whether
the condition for the dust-induced fragmentation given in Equation (1) can be met. In the context of this Letter, we suppose that the product of S and D without grain growth isgiven as (SD)i,crit = 3fi,0 Ai,crit mi /4ρi ri,0 μmH , adopting thecritical metal abundance evaluated in Equation (9), whereas
the product resulting from the grain growth is given by(SD)i,∗ = (SD)i,crit (fi,∗ /fi,0 )2/3 . Figure 4 shows the dependence of (SD)i,crit and (SD)i,∗ on fi,0 , adopting fi,∗ = 0.5. We
can see that (SD)i,crit is well below the minimum value required
for the dust-induced fragmentation (dot-dashed line in Figure 4),
whereas (SD)i,∗ exceeds this value. This indicates that even if
the destruction by the SN reverse shock results in a lower SD
than the criterion for the dust-induced fragmentation, the grain
growth can enhance SD in the clouds and can enable the gas
to fragment into sub-solar mass clumps. Note that the results
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The Astrophysical Journal Letters, 756:L35 (5pp), 2012 September 10

Nozawa, Kozasa, & Nomoto

grain growth from becoming efficient for metal abundances of
Ai  2.5 × 10−7 . We note that our conclusions obtained withthe simple model should be confirmed by more sophisticatedsimulations of the thermal evolution of star-forming clouds
involving grain growth.

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We thank Hiroyuki Hirashita for useful comments. We are
grateful to the anonymous referee for critical comments that
improved the manuscript. This research has been supported byWorld Premier International Research Center Initiative (WPI
Initiative), MEXT, Japan, and by the Grant-in-Aid for Scientific
Research of the Japan Society for the Promotion of Science(22684004, 23224004).
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