The Astrophysical Journal Letters, 765:L3 (5pp), 2013 March 1

doi:10.1088/2041-8205/765/1/L3

C 2013. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

GROWTH OF DUST GRAINS IN A LOW-METALLICITY GAS AND ITSEFFECT ON THE CLOUD FRAGMENTATIONGen Chiaki1 , Takaya Nozawa2 , and Naoki Yoshida1,2

1 Department of Physics, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
2 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
Received 2012 December 28; accepted 2013 January 25; published 2013 February 11

ABSTRACTIn a low-metallicity gas, rapid cooling by dust thermal emission is considered to induce cloud fragmentation and playa vital role in the formation of low-mass stars (1 M ) in metal-poor environments. We investigate how the growthof dust grains through accretion of heavy elements in the gas phase onto grain surfaces alters the thermal evolution
and fragmentation properties of a collapsing gas cloud. We directly calculate grain growth and dust emission
cooling in a self-consistent manner. We show that MgSiO3 grains grow sufficiently at gas densities nH = 1010 , 1012 ,and 1014 cm−3 for metallicities Z = 10−4 , 10−5 , and 10−6 Z , respectively, where the cooling of the collapsinggas cloud is enhanced. The condition for efficient dust cooling is insensitive to the initial condensation factor of
pre-existing grains within the realistic range of 0.001–0.1, but sensitive to metallicity. The critical metallicity is
Zcrit ∼ 10−5.5 Z for the initial grain radius rMgSiO3 ,0  0.01 μm and Zcrit ∼ 10−4.5 Z for rMgSiO3 ,0  0.1 μm.
The formation of a recently discovered low-mass star with extremely low metallicity (4.5 × 10−5 Z ) could havebeen triggered by grain growth.
Key words: dust, extinction – galaxies: evolution – ISM: abundances – stars: formation – stars: low-mass – stars:
Population II

Letter, we investigate the effects of grain growth on the thermal
evolution of star-forming clouds by calculating grain growth, theformation of hydrogen molecules on the grain surfaces, and dust
cooling self-consistently. Then, by investigating possible rangeof metallicities and initial condensation efficiency factors, we
can determine the critical metallicity, the minimal metallicity
required for the formation of low-mass fragments.

1. INTRODUCTION
Dust grains in the early universe are considered to be crucial
for low-mass star formation in the low-metallicity gas.3 The
efficient cooling by dust thermal emission can trigger the
fragmentation of the gas at densities nH ∼ 1013 –1015 cm−3 andgas temperature Tg ∼ 1000 K, where the corresponding Jeansmass is less than one solar mass (Schneider et al. 2003; Omukai
et al. 2005). In the early universe, dust grains are considered to
be predominantly supplied by core-collapse supernovae; grains
are formed in the supernova ejecta (Todini & Ferrara 2001;
Nozawa et al. 2003) but part of them is destroyed by the reverse
shock penetrating into the ejecta (Nozawa et al. 2007; Bianchi& Schneider 2007). Thus, the depletion factor of dust, fdep
(the mass ratio of dust to metals), is determined by the balance
between the formation and destruction of grains.
Schneider et al. (2012) followed the thermal evolution ofgas clouds adopting the depletion factors for their models of
dust formation and destruction in supernovae. For metallicity
Z, the absolute amount of dust is related to the dust-to-gas
mass ratio, D = fdep Z. Schneider et al. (2012) showed thatthe condition of cloud fragmentation owing to rapid dust
cooling is described as the minimal dust-to-gas mass ratio,
Dcrit , under the assumption that the dust-to-gas mass ratio isconstant in the collapsing cloud. However, Nozawa et al. (2012)
recently argued that the dust abundance can be increased by
the growth of dust grains through accretion of heavy elementsin the gas phase onto grain surfaces. They found that thecondensation efficiencies (the number fraction of an element
depleted onto grains) of Si- and Fe-grains increase up to unity at
gas densities nH = 1010 –1014 cm−3 for elemental abundances[Si, Fe/H] = −5 to −3. However, they did not follow thethermal evolution of the gas clouds to determine whether theconditions for fragmentation are met in the clouds. In this3

2. NUMERICAL METHOD
We explore the thermal evolution of a collapsing gas cloudwith metallicities of Z = 10−6 –10−3 Z . The evolutions of density ρ and gas temperature Tg of the cloud core are followed bya one-zone model. Our calculations include chemical reactionsof gas species and radiative transfer for line emissions from thegas, and continuum emissions from the gas and dust (see Chiaki
et al. 2013 for details).
In our calculations where the growth of dust grains is
included, the mass density of dust ρd increases with time. Thus,
we calculate the continuum optical depth as follows;τcont = (κg ρg + κd ρd )lBE ,(1)

where κg and κd denote the Planck mean opacities of gas
and dust, respectively,4 ρg denotes the mass densities of gas(ρ = ρg + ρd ), and lBE is the Bonnor–Ebert length as the typicalsize of the cloud. Planck mean opacities of dust κd are calculatedusing optical constants from references tabulated in Nozawa
et al. (2008) and shown in Figure 1 for grain species consideredin this study. We then obtain the escape fraction of continuum−2
emission as βcont = min{1, τcont} (Omukai 2000).
4

We should note that here the definition of dust Planck mean opacity κd isdifferent from that of κgr in Omukai (2000) in that the former is defined as theabsorption cross section per unit dust mass, while the latter is defined as theabsorption cross section per unit gas and dust mass.

In this Letter, we use the term low-mass for a mass less than 1 M .

1

The Astrophysical Journal Letters, 765:L3 (5pp), 2013 March 1

Chiaki, Nozawa, & Yoshida

104
3

fij

2

10

ri [µm]

10

101
0

10
10

-1

102M

10M M
10-1M

i=MgSiO3, j=Si

-2

103

MgSiO3
Mg2SiO4
Fe3O4
Fe
Si

-3

10-4
10

10-2
3
MJeans=10 M

10-2
10

-1

-5

101

102

10 M

Tg[K]

2 -1

Planck mean opacity [cm g ]10log(Z/Z )=-4log(Z/Z )=-5log(Z/Z )=-6

100

2

10

103

w/o grain growth
w/ grain growth

Dust temperature [K]1

10

Figure 1. Planck mean opacities per unit dust mass for MgSiO3 (solid),
Mg2 SiO4 (dashed), Fe3 O4 (dotted), Fe (dot-dashed), and Si (dot-dot-dashed)
as a function of dust temperature for a grain radius 0.01 μm. The values areinsensitive to the grain radius except for Fe and Fe3 O4 .



4π 3
a
3 i,0



kTg
2π mj

gas

108

1010

1012

1014

1016

nH [cm ]Figure 2. Condensation efficiency (top), the radius of MgSiO3 grains (middle),
and gas temperature (bottom) as a function of hydrogen number density ingas clouds with metallicities 10−4 Z (solid), 10−5 Z (dotted), and 10−6 Z(dot-dashed) for the initial condensation factor 0.1 and the initial grain radius
0.01 μm. With (red) and without (blue) grain growth. The filled circles markthe states where the fragmentation conditions are met (see the text).

We set the initial density and temperature of a gas cloud to be
nH,0 = 3000 cm−3 and Tg,0 = 200 K, respectively.5 The initial
number fractions of chemical species relative to hydrogen nuclei
are Ae = 10−5 , AD = 2.87 × 10−5 , AH2 = 10−3 , AHD = 10−6 ,irrespective of the metallicity Z. The number abundance of anelement j is given as Aj = Aj, (Z/Z ), where the solar number
abundances of heavy elements, Aj, , are taken from Caffau et al.
(2011a).
The average radius of an atom per key element in the dust
phase is taken from Nozawa et al. (2003), and the sticking
coefficient is assumed to be unity. For the scaled solar elemental
abundances (AMg /ASi > 1), the key element of MgSiO3 is
Si, whose initial condensation factor is taken as fMgSiO3 ,Si,0 =0.001, 0.01, and 0.1. As for the initial radius of pre-existinggrains, we consider four cases of rMgSiO3 ,0 = 0.001, 0.01, 0.1,
and 1 μm. Dust temperature is calculated from the balance
equation between heating due to collisions with the gas andcooling by dust thermal emission. Also, the formation rateof hydrogen molecules on grain surfaces is calculated byconsidering impact of hydrogen atoms onto grain surfaces as
in Schneider et al. (2006).

1/2
nj (t),

106-3

In this study, we primarily consider the growth of MgSiO3
grains; Mg-silicate is one of the major grain species ejected bycore-collapse supernovae (Bianchi & Schneider 2007; Nozawa
et al. 2007). We also investigate the growth of Mg2 SiO4 grains
and its effects on the thermal evolution of the gas clouds (seeSection 4 below). We assume that the growth of these multielement grains is regulated by the least abundant gaseous species
among reactants, which we call the key element hereafter (seeHirashita & Kuo 2011).
Using the concept of the key element, we calculate grain
growth by following the prescription in Nozawa et al. (2012).
On the assumption that pre-existing dust grains are sphericalwith a single initial radius ri,0 , the evolution of grain radius, ri ,is described by
dri
= si
dt

104

(2)

where si denotes the sticking probability of the gaseous atoms
incident onto grain surfaces, ai,0 are the average radius of amonomer molecule in the dust phase per key element, and mj
gas
and nj are the mass and the number density of key element
j, respectively. In Equation (2), the evaporation term can beneglected. When the cloud fragmentation is triggered throughefficient cooling by dust, the temperature of dust increases only
up to several hundred kelvin, well below the sublimation temperature of MgSiO3 (∼1000 K under the condition consideredhere). Therefore, dust evaporation does not affect the fragmentation condition in the situations we investigate.
The time evolution of condensation efficiency fij , definedas the number fraction of the key element j locked in grain
species i, is described by fij (t) = fij,0 [ri (t)/ri,0 ]3 , where fij,0represents the initial condensation factor. Then, the mass densityof the grains is given by ρd,i (t) = ρXH Aj fij μi , where XH is
the mass fraction of hydrogen nuclei (set 0.76 here), and μi is
the molecular weight of grains per key element.

3. RESULTS
Figure 2 shows the evolutions of the condensation efficiency
fMgSiO3 ,Si (t), the radius rMgSiO3 (t), and the gas temperature
Tg for the initial condensation factor fMgSiO3 ,Si,0 = 0.1 andthe initial grain radius rMgSiO3 ,0 = 0.01 μm in gas clouds
5

The initial condition of the gas cloud is taken from the values when asupernova shell becomes gravitationally unstable in our one-dimensional
calculations of a supernova remnant (Chiaki et al. 2013). We have confirmedthat the evolutions of collapsing gas clouds at nH > 108 cm−3 are insensitive
to these initial values.

2The Astrophysical Journal Letters, 765:L3 (5pp), 2013 March 1

Metallicity log(Z/Z )-2

Chiaki, Nozawa, & Yoshida-2

rMgSiO ,0=0.01 µm
3

3

-3

-3

-4

-4

-5

-5

-6

-6
-3

10

-2

10

fMgSiO ,Si,0=0.1

-1

-3

10

10

Initial condensation factor

-2

10

-1

10

0

10

Initial grain radius [µm]Figure 3. Fragmentation properties of the collapsing gas clouds for each model on Z–fMgSiO3 ,Si,0 (left) and Z–rMgSiO3 ,0 (right) planes. For models marked by double
circles, the fragmentation conditions are met both with and without grain growth. For models marked by filled circles, the conditions are met only with grain growth. For
models marked with open circles, the conditions are not met both with and without grain growth. In the left panel, the dashed line shows the metallicity corresponding
to the critical dust-to-gas mass ratio given by Schneider et al. (2012; see the text) and the solid line shows the critical metallicities obtained by detailed calculations.

with metallicities 10−6 –10−4 Z . For the models with grain
growth, the condensation efficiency (mass density of grains)
increases at nH = 1010 , 1012 , and 1014 cm−3 for Z = 10−4 ,10−5 , and 10−6 Z , respectively. It eventually converges tounity regardless of metallicities. This is because the timescalecharacterizing grain growth eventually becomes smaller than
the timescale characterizing the free-fall collapse (see Nozawa
et al. 2012). The grain radius also reaches to a single value
rMgSiO3 ,0 (1/fMgSiO3 ,Si,0 )1/3 . Since the increase in grain massdensity causes enhanced dust cooling, the models with grain
growth lead to lower gas temperatures than the models inwhich the condensation efficiency is constant (i.e., without grain
growth).
The circles in Figure 2 depict the phase (on each trajectory)
where the cloud is expected to fragment (see Schneider &Omukai 2010 for the detailed description of the criterion forfragmentation). Interestingly, for Z = 10−5 Z , the trajectory
reaches this phase with grain growth, whereas, without grain
growth, the trajectory does not. We therefore conclude that grain
growth can reduce the metallicity (Si abundance) above which
low-mass fragments can form.

where ρmetal is the mass density of metal in both gas and dust
phase and μMgSiO3 = 100. With the solar metallicity Z =0.0153 (Caffau et al. 2011b), we obtain fdep = 0.16fMgSiO3 ,Si,0 .
In Figure 3, we plot Z = Dcrit /(0.16fMgSiO3 ,Si,0 ) by the dashedline. Our results are consistent with the critical dust-to-gas mass
ratio found by Schneider et al. (2012).
Then, we discuss how the critical condition is changed if we
consider grain growth. The solid line in Figure 3 shows thecritical line above which the fragmentation condition is met
with grain growth. Comparison of the two critical lines (solidand dashed lines) shows that lower metallicity is generally
allowed for low-mass fragment to form when we consider
grain growth. Also, the solid line in Figure 3 shows that thecritical metallicity is insensitive to the initial condensation
factor. This is because even if the initial condensation factor
is small (fMgSiO3 ,Si,0 ∼ 0.001), the grains grow up to unity
before the dust emission cooling becomes effective unless themetallicity is too small.
Instead of the condensation efficiency (dust-to-gas mass
ratio), the critical condition is determined by the metallicity anddust size. The estimated critical metallicity is Zcrit  10−5.5 Z
for the initial radius of MgSiO3 grains rMgSiO3 ,0  0.01 μm, andZcrit ∼ 10−4.5 Z for fMgSiO3 ,Si,0  0.1 μm. For a larger initial
radius, the gas density at which grains grow is higher because thetotal surface area of dust particles, which determines the growthrate, is proportional to ri−1 (for a fixed mass density of dust).
The right panel of Figure 3 shows that for rMgSiO3 ,0 = 0.001 andZ = 10−3 Z , the fragmentation conditions are not met without
grain growth, since heating due to H2 formation compensates
for cooling by dust emission.

3.1. Critical Metal Abundance
Figure 3 shows fragmentation properties for each model weinvestigate. The double and filled circles depict the models forwhich the fragmentation conditions are met when we consider
grain growth, and double circles depict the models for which theconditions are met without grain growth. From these results, we
can discuss the critical condition for the formation of low-massfragments.
First, we make the comparison of our results without grain
growth with previous works. As the minimal condition offragmentation, Schneider et al. (2012) found the critical dustto-gas mass ratio Dcrit = 4.4 × 10−9 (= fdep Z), above which
the dust-induced fragmentation occurs. In the dust model theyused, the grains have size distribution functions with the peak at
0.01 μm. Thus, we can compare their result with our models
for rMgSiO3 ,0 = 0.01 μm. To apply this, we should convert
fMgSiO3 ,Si,0 into fdep asfdep =

ρd,MgSiO3
XH ASi μMgSiO3
fMgSiO3 ,Si,0 ,=ρmetalZ

3.2. Application to SDSS J102915+172927
We perform the calculations of grain growth and evolutionof collapsing gas, adopting the abundance of heavy elements
for the star SDSS J102915+172927: AC = 1.58 × 10−8 ,AO = 2.51 × 10−8 , AMg = 8.91 × 10−10 , ASi = 1.78 × 10−9 ,and AFe = 3.39 × 10−10 , corresponding to the metallicity
4.5 × 10−5 Z . Note that the abundance of carbon is an upper
limit because no strong carbon enhancement is evident in thestellar spectrum. Also, they estimate the oxygen abundance,
assuming the typical excess of the α-element oxygen (Caffau
et al. 2011a). Given the abundance of the gas, we aim atconstraining the initial condensation factor and the initial grain(3)3

The Astrophysical Journal Letters, 765:L3 (5pp), 2013 March 1

Chiaki, Nozawa, & Yoshidamean opacities (Figure 1) and thus similar cooling efficiencies
through thermal emission.
On the other hand, Si grains have a mean opacity smaller
than MgSiO3 by orders of magnitude (Figure 1). Thus, even if
the condensation efficiency reaches unity, the gas cloud cannot
cool down enough to meet the fragmentation conditions evenfor metallicities Z  10−2 Z . The growth of Fe grains resultsin larger critical abundance than MgSiO3 because the species
also have smaller opacity than MgSiO3 . Hence, these grain
species are unlikely to affect heavily the evolution of clouds.
Indeed, Si and Fe grains could be rapidly oxidized into silicate
and magnetite in a gas cloud with oxygen-rich composition.
Supernova models (N. Tominaga et al. 2013, in preparation)and observations in the Galactic halo (Yong et al. 2012) havesuggested that the abundances of oxygen in metal-poor stars areusually AO /AX > 10 where X = Mg, Si, and Fe.
It is considered that C grains might have also large effects on
the evolution of collapsing clouds. Yet it is uncertain whether
C grains can grow sufficiently in the collapsing gas becausea large fraction of carbon atoms are expected to be locked
in CO molecules and are not available for the growth of C
grains (Nozawa et al. 2012). Even if CO molecules are accreted
onto dust nuclei, the CO ice mantle would be easily evaporated
because of its low sublimation temperature. Nevertheless, if
the initial condensation factor of C grains is high enough, thecooling of the gas by carbon dust thermal emission may havean impact on the cloud evolution.
We have examined the effect of growth of an individual dust
species on collapsing gas clouds separately. In a low-metallicity
gas, multiple dust species would grow simultaneously, and thus
their combined effects could further reduce the critical metallicity. Dust formation calculations in primordial supernovae (e.g.,
Todini & Ferrara 2001; Nozawa et al. 2003; Schneider et al.
2006; Bianchi & Schneider 2007) showed that various grains
species can be produced. It is important to study how the growthof multiple species of dust affects the thermal evolution of collapsing gas clouds based on the realistic supernova dust models.
Finally, we have set the sticking probability si = 1 in thecalculations. As seen from Equation (2), a lower sticking probability leads to a lower growth rate of dust. We perform anothercalculation, setting si = 0.1. As a result, the fragmentation
fails for Z = 10−5 Z and rMgSiO3 ,0 = 0.01 μm. However,even for si = 0.1, grains with rMgSiO3 ,0 = 0.001 μm rapidly
grow to cause the cloud fragmentation. We have considered dust
grains with a single initial size, but dust grains that were formed
and processed in supernovae may have size distribution. Since
smaller grains grow more rapidly, the lower limit of size distribution and the initial mass fraction of the small grains could be
crucial quantities for cloud fragmentation. We will investigate
these effects comprehensively in our future work.

Initial condensation factor

100
SDSS J102915+172927
10-1

-2

10

-3

10

10-3

10-2

10-1

100

Initial grain radius [µm]Figure 4. Fragmentation conditions of collapsing gas clouds on
fMgSiO3 ,Mg,0 –rMgSiO3 ,0 plane for the abundance of heavy elements of SDSS
J102915+172927. The symbols have the same meaning as in Figure 3.

radius necessary for triggering the formation of the low-mass
star.
Figure 4 shows the results of calculations with the growthof MgSiO3 grains. In the present case, the key element is
Mg because of AMg /ASi < 1. Interestingly, for most of thecases considered here (all cases except for fMgSiO3 ,Mg,0 =0.01 and 0.001 with rMgSiO3 ,0 = 1 μm), the fragmentation
conditions are met. Thus, we argue that the parent cloud of
SDSS J102915+172927 could have been enriched with relatively small grains (rMgSiO3 ,0  0.1 μm) or otherwise with relatively high condensation factor of dust (fMgSiO3 ,Mg,0  0.1).
It is worth mentioning that the above results are approximately
consistent with those obtained with the scaled solar composition(Figure 3), although the abundance pattern is different. The
results for rMgSiO3 ,0 = 0.01 μm in Figure 4 lie between thosewith metallicity log(Z/Z ) = −5 and −4 in the left panel of
Figure 3, and the results for fMgSiO3 ,Mg,0 = 0.1 lie between thosewith metallicity log(Z/Z ) = −5 and −4 in the right panel of
Figure 3.
4. DISCUSSION
In our calculations for the scaled solar abundances, the least
abundant element, silicon, regulates the growth of MgSiO3
grains. Though we have assumed that Si atoms are directly
accreted onto the grains, a part of or all the Si atoms might bein the form of SiO molecules in dense clouds. We also examine
a model in which all Si atoms are in the form of SiO molecules
initially. As a result, the density at which grains grow rapidly
and the resulting critical metallicity are not changed from theresults of models in which we consider Si atoms as the key
element.
We also examine the growth of other grain species and theeffect of each species on the thermal evolution of clouds. We
find that, for the scaled solar composition (ASi /AFe = 1), Fe3 O4grains grow more slowly than MgSiO3 , and the gas density at
which the condensation efficiency of Fe3 O4 grains becomes 0.5
is higher than for MgSiO3 by a factor of ∼5. This is because the3volume of a monomer of Fe3 O4 grains (∝ aFe) is smaller
3 O4 ,0than MgSiO3 and thus the rate of grain growth is smaller (seeEquation (2)). This is also true for Mg2 SiO4 grains for which thevolume per key element, Mg atom, is smaller than for MgSiO3 .
However, for these grain species, the critical metallicity is thesame as for MgSiO3 grains since these grain species have similar

We thank K. Omukai, R. Schneider, N. Tominaga, andBeate Patzer for fruitful discussion. This work is supported inpart by the Grants-in-Aid for Young Scientists (S: 20674003,
A: 22684004) and for General Scientists (S: 23224004) by theJapan Society for the Promotion of Science. N.Y. acknowledges
financial support from World Premier International ResearchCenter Initiative (WPI Initiative), MEXT, Japan.
REFERENCES
Bianchi, S., & Schneider, R. 2007, MNRAS, 378, 973
Caffau, E., Bonifacio, P., François, P., et al. 2011a, Natur, 477, 67

4The Astrophysical Journal Letters, 765:L3 (5pp), 2013 March 1

Chiaki, Nozawa, & Yoshida

Caffau, E., Ludwig, H. G., Steffen, M., Freytag, B., & Bonifacio, P. 2011b, SoPh,
268, 255
Chiaki, G., Yoshida, N., & Kitayama, T. 2013, ApJ, 762, 50
Hirashita, H., & Kuo, T.-M. 2011, MNRAS, 416, 1340
Nozawa, T., Kozasa, T., Habe, A., et al. 2007, ApJ, 666, 955
Nozawa, T., Kozasa, T., & Nomoto, K. 2012, ApJL, 756, L35
Nozawa, T., Kozasa, T., Tominaga, N., et al. 2008, ApJ, 684, 1343
Nozawa, T., Kozasa, T., Umeda, H., Maeda, K., & Nomoto, K. 2003, ApJ, 598,
785
Omukai, K. 2000, ApJ, 534, 809

Omukai, K., Tsuribe, T., Schneider, R., & Ferrara, A. 2005, ApJ, 626, 627
Schneider, R., Ferrara, A., Salvaterra, R., Omukai, K., & Bromm, V. 2003, Natur,
422, 869
Schneider, R., & Omukai, K. 2010, MNRAS, 402, 429
Schneider, R., Omukai, K., Bianchi, S., & Valiante, R. 2012, MNRAS, 419,
1566
Schneider, R., Omukai, K., Inoue, A. K., & Ferrara, A. 2006, MNRAS, 369,
1437
Todini, P., & Ferrara, A. 2001, MNRAS, 325, 726
Yong, D., Carney, B. W., & Friel, E. D. 2012, AJ, 144, 95

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