Gravitational collapse and the thermal evolution of low-metallicity gas clouds in the early Universe

0.57 MB·

by  Gen Chiaki

Mon.Not.R.Astron.Soc.000,1–??(2015) Printed5January2016 (MNLATEXstylefilev2.2)
Gravitational collapse and the thermal evolution of
low-metallicity gas clouds in the early Universe
Gen Chiaki,1⋆ Naoki Yoshida1,2 and Shingo Hirano1
6
1 1Department of Physics, Graduate School of Science, Universityof Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
0 2Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for Advanced Study,
2 The University of Tokyo, Kashiwa, Chiba 277-8583, Japan

n
a
J

3

] ABSTRACT
A We study gravitational collapse of low-metallicity gas clouds and the formation
G of protostars by three-dimensional hydrodynamic simulations. Grain growth, non-
equilibrium chemistry, molecular cooling, and chemical heating are solved in a self-
.
h consistent manner for the first time. We employ the realistic initial conditions for the
p abundances of metal and dust, and the dust size distribution obtained from recent
-
o Population III supernova calculations. We also introduce the state-of-the-art parti-
r cle splitting method based on the Voronoi tessellation and achieve an extremely high
t mass resolution of ∼ 10−5 M (10 earth masses) in the central region. We follow
s J
a the thermal evolution of severalearly clouds with various metallicities. We show that
[ the condition for cloud fragmentation depends not only on the gas metallicity but

also on the collapse timescale. In many cases, the cloud fragmentation is prevented
1
v by the chemical heating owing to molecular hydrogen formation even though dust
0 cooling becomes effective. Meanwhile, in several cases, efficient OH and H2O cool-
8 ing promotes the cloud elongation, and then cloud “filamentation” is driven by dust
2 thermal emission as a precursor of eventual fragmentation. While the filament frag-
0 mentation is driven by rapid gas cooling with &10−5 Z , fragmentation occurs in a
J
0 different manner by the self-gravity of a circumstellar disk with . 10−5 Z . We use
. J
1 a semi-analytic model to estimate the number fraction of the clouds which undergo
0 the filament fragmentationto be a severalpercents with 10−5–10−4 Z . Overall,our
J
6 simulationsshowaviableformationpathoftherecentlydiscoveredGalacticlow-mass
1 stars with extremely small metallicities.
:
v
Key words: dust, extinction — galaxies: evolution — ISM: abundances — stars:
i
X formation — stars: low-mass — stars: Population II
r
a
1 INTRODUCTION formed from a metal-free, primordial gas haveawide range
of masses (Hirano et al. 2014; Susa et al. 2014). Such a no-
The origin of Galactic extremely metal-poor stars remains
tionissupportedbyobservationsofmetal-poorstarswhose
largely unknown. Their small metal content suggests that
peculiar abundance patterns are consistent with the nucle-
they were born very early, while their small masses pose
osyntheticmodelsofmassiveprimordialstars(Caffau et al.
interesting questions on star formation in a metal-poor en-
2011b;Keller et al.2014).Interestingly,therecentlydiscov-
vironment. For example, the star SDSS J102915+172927
ered star SDSS J001820.5 −093939.2 shows characteristic
with the mass 0.8 M has an extremely low metallicity
of 4.5×10−5 Z (CaJffau et al. 2011b, 2012), and thus is signatures of the metal yield of a very massive primordial
J star, suggesting a broad mass distribution of the Pop III
thought to be one of the second generation stars; it was
stars (Aoki et al. 2014).
probably born in a gas cloud that had been enriched with
A trace amount of heavy elements in a star-forming
heavy elements synthesized by the first generation of stars
gas plays an important role in the formation of the low-
(Pop III stars).
mass stars with extremely low metallicities (Omukai 2000;
Recenttheoreticalstudiesbasedoncosmologicalhydro-
Bromm & Loeb 2003; Schneideret al. 2003). Efficient gas
dynamic simulations suggest that the first generation stars
cooling owing to metal and dust can trigger thecloud frag-
mentation.Ingeneral,gascoolingpromotestheclouddefor-
⋆ E-mail:[email protected] mation into sheets or filaments, whereas heating processes
(cid:13)c 2015RAS
2 G. Chiaki et al.
stabilizethegasagainstasphericalperturbations.Thelinear phase metals onto dust grains, can enhance the dust cool-
analyses by Lai (2000) and Hanawa & Matsumoto (2000) ing efficiency in a collapsing cloud even with an extremely
demonstrate that a contracting cloud gets highly elongated low metallicity to be close to that of the local interstellar
when the specific heat ratio γ = dlnT/dlnρ+1, an indi- medium (Nozawa et al. 2012;Chiaki et al. 2013).
cator of the effective gas equation of state, is less than 1.1. Thereareotherimportantprocessesinalow-metallicity
Inacooling gas withγ <1,significantlyelongated filamen- gas that have often been overlooked in previousstudies. O-
tary structures tend to beformed. When thedensity of the andC-bearingmoleculessuchasOHandH2Oactasimpor-
filament increases up to a value where gas cooling becomes tantcoolants atintermediatedensitiesnH ∼104–108 cm−3.
insufficient (γ > 1), multiple cores are formed and begin Although OH and H2O cooling is less effective in a gas
contractingseparately.Therefore,thecharacteristicmassof with the “normal” elemental composition (Omukai 2000),
the fragments is set by the Jeans mass at the density and ourpreviouswork(Chiaki et al.2015,hereafterC15)reveals
temperaturewherethecoolingbecomesinefficient.Analytic thatthemolecularcoolingisefficientinsomecaseswhenthe
modelssuggestthatlow-masscorescanbeformedbytransi- realistic initial condition (composition) is adopted. There
tionlinecoolingofCi,Cii,andOi(Bromm & Loeb2003; is yet another important heating process. When hydrogen
Frebel et al. 2005; Santoro & Shull 2006; Ji et al. 2014). molecules are formed, the binding energy is released to be
Dust thermal emission is another important process that converted into the gas kinetic energy. Tsuribe & Omukai
determine the thermal evolution of a low-metallicity gas. (2008) argue that the gas heating can stabilize the gas to
A semi-analytic approach reveals that dust cooling be- preventfragmentation usingsimulations of apolytropicgas
comes important at densities nH & 1012 cm−3, where the mimicking thethermal evolution of low-metallicity clouds.
corresponding Jeans mass is ∼ 0.1 M (Omukai 2000; The metal and dust properties in the early Universe
J
Schneideret al. 2003). can be determined theoretically under the assumption that
Three-dimensional hydrodynamic simulations have the formation sites of low-metallicity stars are in clouds
been also performed to study the conditions for gas cloud partly polluted with metals and dust by Pop III super-
fragmentation and for the formation of low-mass stars. novae in the same or nearby halos (Ritteret al. 2012;
Dopckeet al. (2011, 2013) follow the dynamical evolution Smith et al.2015).Thus,theabundancesofheavyelements
of a turbulent gas with metallicities below 10−4 Z . They and dust species, and the dust size distribution are ex-
J
showthatseveraltensofsinkparticlesareformedinthegas pected to be set by the nucleosynthetic process of the su-
andthatthemassdistributionofthesinkparticlesbecomes pernova explosion (Umeda & Nomoto 2002) and the dust
from flat to peaky distribution with increasing metallicity. formation and destruction history during propagation of
Safranek-Shraderet al. (2014a,b) perform simulations with blast waves (Todini & Ferrara 2001; Nozawa et al. 2003;
a proper cosmological set-up to investigate star formation Schneideret al. 2006).
in dense clumps in early galaxies with gas metallicities of Inthepresentpaper,weperformhydrodynamicsimula-
10−4–10−2 Z . They find that the gas fragments into sink tionsofcollapsinggascloudswithvariousmetallicities10−6–
J
particles,whosemassdistributionisconsistentwiththeob- 10−3 Z . We study in detail the critical conditions for gas
J
servationally derived stellar masses of the ultra faint dwarf fragmentationandfortheformationoflow-massprotostars.
spheroidalgalaxies.Morerecently,Smith et al.(2015)study Ourthree-dimensionalsimulationsfollow, forthefirsttime,
theeffectsofmetalpollutionandturbulencedrivenbyaPop all the necessary thermal processes including dust thermal
III supernova explosion into the neighboring halo. The gas emission, gas heating by H2 formation, and OH and H2O
polluted to metallicity of 2×10−5 Z fragments by dust cooling. The metal abundances, initial grain condensation
J
thermal emission. Their study explicitly includes the pro- efficiency, and the dust size distribution are calculated in
cess of thedispersion of metals and dust. the stellar evolution and supernova explosion models of a
These previous studies assume the metal abundances, Pop III star. We calculate thechemical reactions and grain
grain condensation efficiency,and thedust size distribution growth in a direct, self-consistent manner in order to com-
to be those in the Galactic interstellar medium despite the pute the cooling and heating rates. Recently, Hirano et al.
factthatthemetal-poorstarsobservedsofarshowpeculiar (2014) find that the different collapse timescales of the pri-
abundance patterns. For example, the abundance ratio of mordialstar-formingcloudgivesthewidemassdistribution
light elementssuch as carbon and oxygen relative toiron is of Pop III stars, ∼ 10–1000 MJ. In this paper, we employ
enhanced with respect to the solar value (Sudaet al. 2008; four gas clouds with different collapse timescales to exam-
Tominaga et al. 2014). Because the efficiency of metal line ine the effect of the cloud variation on the low-metal star
cooling is determined by the abundance of respective ele- formation.
ments,itisimportanttoadoptrealisticvaluesbasedon,for
example, the nucleosynthesis calculations of Pop III super-
novae.
The same is truefor thecondensation efficiency,or the 2 NUMERICAL SIMULATIONS
massfraction ofheavyelementscondensedintodustgrains.
2.1 Chemistry and cooling
It is known that a significant fraction of heavy elements
are locked into dust grains in the local interstellar medium We use the parallel N-body/smoothed particle hydrody-
(Pollack et al. 1994), but recent spectroscopic observations namics (SPH) code gadget-2 (Springel 2005) with non-
of damped Lyman-α systems with metallicity ∼ 10−3 Z equilibriumchemistryandradiativecooling.Wesolvechem-
J
suggest that the condensation efficiency is smaller at red- ical networks of 54 reactions for 27 gas-phase species: H+,
shifts z ∼ 6 (Schadyet al. 2010; Zafar et al. 2011). It is e−, H, H−, H2, D+, D, HD, C+, C, CH, CH2, CO+, CO,
also known that grain growth, i.e., accretion of the gas- CO2, O+, O, OH+, OH, H2O+, H2O, H3O+, O+2, O2, Si,
(cid:13)c 2015RAS,MNRAS000,1–??
Low-metallicity gas collapse 3
SiO,andSiO2.ThechemicalreactionratesaregiveninC15 in which the daughter particles are distributed based on
forSi-bearingspecies,andinOmukaiet al.(2010)forother the Voronoi diagram tessellated by parent particles. With
species.Wesolvethechemicalreactionsimplicitlytoobtain thismethod,thedensity structuresof thecloud is well pre-
the abundances of the gas-phase species in each fluid ele- served.Intheentirecourseofoursimulations,aJeansmass
ment at each time step. We then calculate the associated is required to be resolved by more than 1000Mmin, where
cooling and heating rates. Mmin =Nngbmp is the minimal resolvable mass of the par-
We implement radiative cooling by C+, C, and O, and entSPHparticleswithmassmp,andNngb isthenumberof
byH2andHDmolecules.Wecalculatethelevelpopulations the neighbor particles. We set Nngb =64±8, and thus the
for each species in a time-dependent manner. Gas opacity Jeans mass is always resolved by ∼105 particles.
is explicitly calculated and the cooling rate for each emis-
sion line is reduced by a factor determined by the local ve-
locity gradient in the optically thick regime (the so-called 2.4 Later accretion phase
Sobolev approximation). Wecalculate thevelocity gradient
As the density in the central core increases, the dynami-
in the three directions x, y, and z as in Hirano & Yoshida
cal time decreases and the necessary integration time step
(2013). We also include metal molecular line cooling, us-
gets progressively shorter. Sink particle techniques are of-
ing the formulation and the cooling tables presented by
ten employed to save computational time, where the gas
Neufeld & Kaufman (1993) and Neufeld et al. (1995) for
inside a pre-determined accretion radius is replaced with
H2O and Omukaiet al. (2010) for CO and OH. We ob- a sink particle (Dopckeet al. 2013; Safranek-Shrader et al.
tain thecooling rate Λ (m) of molecules m=CO, OH,and
i 2014b; Smith et al. 2015). The evolution of a circumstellar
H2O with thecolumn densityN˜(m)=n(m)|dvi/dri|in the disk, through which the gas is accreted, is not followed ac-
three directions i = x, y, and z, and take the average as
curatelywith sinkparticletechniques.Itisalso knownthat
Λ(m)=[Λ (m)+Λ (m)+Λ (m)]/3.
x y z theseparationoffragmentsdependsontheaccretionradius
Hydrogen molecules are formed via endothermic reac-
(Machida & Doi2013;Greif et al.2012).Insteadofemploy-
tionswithE =4.48eVperformedmolecule.Thethree-body
ing a sink particle technique, we resort to following the gas
reactions particularly enhance the temperature at densities
dynamics in and around the proto-stellar core by setting
108–1011 cm−3.Tsuribe & Omukai(2008) findthat thegas
the specific heat ratio to be γ = 1.4 after the density of
heatingresultingfrom therapidhydrogenmolecularforma- fluid elements exceeds nH = 1016 cm−3, where the gas be-
tion mitigates the cloud deformation and makes the cloud
comesopticallythickeveninaprimordialgas.Althoughwe
core rounder. We consider the heating mechanism accord-
are not able to follow the thermal evolution accurately be-
ing to the formulation of Hollenbach & McKee (1979) and yond nH = 1016 cm−3, the treatment allows us to examine
Omukai(2000).
gas fragmentation owing todustthermalemission, which is
expected to beeffective at nH &1012–1015 cm−3.
2.2 Grain growth
2.5 Initial conditions
The dust species considered here are metallic silicon (Si),
metalliciron(Fe),forsterite(Mg2SiO4),enstatite(MgSiO3), 2.5.1 Cloud models
amorphous carbon (C), silica (SiO2), magnesia (MgO),
We use two different sets of initial conditions; one spher-
troilite (FeS), and alumina (Al2O3). We follow the evolu-
ical cloud as a controlled simulation, and three clouds
tion of the size distributions and abundances of the grain
hosted by small dark matter halos selected from cosmolog-
species by essentially treating the grain growth as chemical
ical simulations. The spherical cloud has a uniform den-
reactions. Weconsider 13 reactions as in C15.Weintegrate
thegrowthrate(Equation(9)inC15)toderivethegrainra- sity nH,ini = 0.1 cm−3 and temperature Tini = 300 K.
dius and condensation efficiency at each time for each fluid The cloud radius and mass are Rini = 551 pc and Mini =
2.3×106 M , corresponding to the half Jeans length and
element. We then calculate the reaction rate of hydrogen J
the Jeans mass, respectively, for the uniform cloud with
molecularformationongrainsurfaces,radiativecoolingeffi-
ciency,andcontinuumopacityforeverygrainspeciesandra- density nH,ini/fenh, where fenh = 1.8 is the enhancement
factor (Matsumoto & Hanawa 2003). We perturb the ini-
dius. Theoptical depthto continuumemission is estimated
tial densities so that the root mean square of the density
asτcont =κρλJ,whereκisthetotalabsorptioncrosssection
is 10% of the mean density. We impose solid rotation such
ofgas anddustperunitfluidmass, ρisthetotal densityof
thattherotationenergyrelativetothegravitationalenergy
gas and dust,and λJ is thelocal Jeans length. βini =Ω2iniRi3ni/3GMini is 10−3, corresponding to the angu-
larvelocityΩini =1.4×10−17 s−1.Theinitialnumberfrac-
tions of deuterium and helium relative to hydrogen nuclei
2.3 Particle splitting
are5.3×10−5 and0.079,respectively.Thenumberfractions
Weneedtofollow thegascloudcollapsetoveryhighdensi- ofelectronandhydrogenmoleculesrelativetohydrogennu-
ties.TheJeanslengthofthecentral,densestpartdecreases clei are y(e) = 10−4 and y(H2) = 10−6, respectively. Both
eventually down to ∼ 0.1 AU when a protostellar core is deuteriumandheliumareinitiallyassumedtobeinneutral
formed. To save the computational cost, we use the parti- atoms. The spherical cloud calculations are dubbed “UNI”
cle splitting technique,when the resolution is about to vio- hereafter.
late the Jeans criterion (Truelove et al. 1997, 1998). Dense We also perform collapse simulations for gas clouds
gas particles are replaced with less massive daughter parti- hosted by minihalos chosen from the cosmological simula-
cles.InChiaki & Yoshida(2015),wepresentanovelmethod tionsof Hirano et al. (2014).From thelargesimulation box
(cid:13)c 2015RAS,MNRAS000,1–??
4 G. Chiaki et al.
Table 1.Abundances ofheavyelements
X C O Mg Al Si S Fe
AX 1.08×10−4 1.19×10−3 4.19×10−5 8.29×10−7 6.67×10−5 3.01×10−5 6.76×10−6
[X/Fe] 0.22 1.01 0.77 0.14 0.99 1.01 0.00
Note—Abundances ofheavyelementsreleasedbyaPopIIIsupernovawithprogenitor mass30M inanambientgaswithdensity
J
namb=1cm−3 (Nozawaetal.2007).Weshowthenumberabundances relativetohydrogennucleiwithmetallicityZ=ZJ.The
valuesarereducedbythefactorZ/Z formetallicityZ.Inthesecondrow,therelativeabundance
J
[X/Fe]=log(AX/AFe)−log(AX,J/AFe,J)isshown.Thesolarabundance AX,J istaken fromCaffauetal.(2011a).
Table 2.Propertiesofgrains
i Silicon Iron Forsterite Enstatite Carbon Silica Magnesia Troilite Alumina Total
fdep,i,0 [×10−3] 29.14 1.93 0.77 <0.01 0.62 5.27 1.34 0.45 <0.01 39.53
rgrow [×10−2 µm] 38.67 28.30 1.20 1.38 1.97 5.76 5.31 3.53 0.11
i,0
Note—Initialdepletionfactorfdep,i,0 relativetototal massofmetalanddustandthecharacteristicradiusrig,r0ow =hr3ii/hr2ii of
grainspecies i.
of ∼ 1 Mpc on a side, we cut out the region centered at ThePopIIIsupernovamodelgivesustherelativeabun-
a minihalo with radii ∼ 1 kpc so that sound waves can- dances of metal and dust. Their absolute values are deter-
not cross the region during the cloud collapse. We select mined for a given metallicity, or the total mass fraction of
threeminihalosdubbed“MH1”,“MH2”,and“MH3”,which heavyelementtothegas.Weexaminefourcaseswithmetal-
cover the different types of the early clouds. In the simula- licities 10−6, 10−5, 10−4, and 10−3 times the solar value
tionsperformed byHirano et al.(2014)forthethreeclouds Z = 0.02. Hereafter we call the models as Z6, Z5, Z4,
J
without metal, the masses of the hosted Pop III stars are andZ3,respectively.WeassumethatinitiallyallofCnuclei
283.9, 751.3, and 60.5 M , respectively. The thermal evo- are in the form of C+, and O and Si are in neutral atoms.
J
lution differs from cloud to cloud, and probably affects the Theinitialmassfractionf ofgrainspeciesirelativeto
dep,i,0
resultingfragmentationpropertiesasdiscussedinAppendix metal and the characteristic radius of grains (see C15) are
C of Hirano et al. (2014). shown in Table 2.
2.5.2 Metal and dust properties
3 CHEMO-THERMAL EVOLUTION OF
In our simulations, metals and dust are uniformly added SPHERICAL CLOUDS
to the simulation gas particles. The metal abundance,
Particle splitting allows us to follow the nonlinear gravita-
dust condensation efficiency, and dust size distribution are
tional evoltution from a diffuse interstellar cloud to proto-
taken from a model of nucleosynthesis and grain forma-
stellar core(s) over 20 orders of magnitude in density. The
tion/destruction in a Pop III supernova. For the first time,
we adopt such the realistic initial conditions to the three- mass resolution Mmin is initially 320 MJ and eventually
reaches 3.2×10−5 M (10 Earth masses) in the central
dimensional simulations of thePop II star formation, while J
densest region with & 1016 cm−3 after splitting particles
previous works often assume the solar abundance pattern.
seven times.
We employ a Pop III supernova model with the progenitor
mass 30 M and the ambient gas density n = 1 cm−3 Figure1showsthetemperatureevolutionasafunction
J amb of density of the core in UNI simulations for Z3–6.1 Figure
taken from Nozawa et al. (2007) as a characteristic model.
2 shows the evolution of the gas cooling and heating rates,
Tables 1 and 2 show themetal and dust properties, respec-
and the abundances of Si-bearing species and grains as a
tively.
function of thecentral density for UNIZ5.
We briefly summarize the difference between the Pop
Adiabaticcompressionalheatingisdominantforallthe
III supernova model and the present-day model. First, the
casesatdensities.1cm−3.Afterthat,molecularhydrogen
number fractions of the light elements relative to iron are
coolingbecomesimportantuntilthedensityreachesthecrit-
largerinthelattercase(secondrowofTable1).Inparticu-
lar,thePopIIImodelpredictsalargefractionof[O/Fe]≃1. icalvalueforH2 molecules, nH ∼103 cm−3,wherethelevel
population is set by the LTE one. For Z4–6, the tempera-
We thus expect that the radiative cooling by OH and H2O
ture increases only slowly by the balance between the gas
molecules is more efficient than in the case of solar abun-
dance. Second, the total condensation efficiency of metal is
only 4%,which is muchsmaller than∼50% in thepresent-
1 Hereafter, we measure the quantities such as the tempera-
day.Althoughthegrain growth can enhancethefinaldust-
ture,density,chemicalabundance,andcoolingrateinthecentral
to-gas mass ratio, the efficiency of H2 formation on grains region by mass-weighting over the gas particles with densities
and dust cooling rate are still smaller than in the present- nH > nH,peak/3, where nH,peak is the largest density in each
day case. snapshot.
(cid:13)c 2015RAS,MNRAS000,1–??
Low-metallicity gas collapse 5
] UNI Z6 4 UNI Z5 adi
K

/
e 2 chem
ur 3 Z5
at 1] 0 CIE H
er Z4 -s 2
p 1
m - -2
g
e HD
T g
[ Z3 er -4
g 2 /
o e
l t 4 dust
a
r

0 4 8 12 16 ng 2 H O
2
log [Central density / cm-3] oli
o 0
C
[

Figure1. ThermalevolutionofthecloudcoresoftheUNIsim- g -2
ulationswithmetallicities10−6 (Z6),10−5 (Z5), 10−4 (Z4), and lo adi OH
10−3 ZJ (Z3)fromtoptobottom. -4 OI
compressional heatingandH2 linecooling. ForZ3,thefine-
structure line cooling by O i becomes efficient. Thereafter, ]
e Si
OH molecules form via the reaction c -10
n
a SiO
O+H→OH+γ, (Reaction 30) d
Mg SiO
n 2 4
and then OH cooling becomes dominant at nH ∼ 104– u -12
108 cm−3 forZ3,and106–108 cm−3 forZ4.Thethree-body Ab
SiO
H2 formation reactions g [ 2 Mg
H+H+H→H2+H, (Reaction 5) lo -14
H+H+H2 →H2+H2 (Reaction 6)
0 4 8 12 16
operateat∼108cm−3,wheretheexothermicreactionsraise -3
log [Central density / cm ]
the gas temperature rapidly. Then the specific heat ratio
temporarilyexceedsthecriticalvalueof4/3forgravitational
collapse. Thenapressure-supportedcoresurroundedbyac- Figure2. Top:coolingfunctionsofprimordialspeciesandmetal
cretionshockisformedforZ3.Thegastemperaturecontin- forUNIZ5. Thelabel “adi”depicts theadiabatic compressional
uestoincreaseto∼1011 cm−3 whiletheH2Omoleculesact heating rate, and “chem” depicts the heating rate owing to H2
as a main coolant. The H2O molecules are formed via the molecule formation. Bottom: number abundances of Si atoms
reactions (blue solid), SiO molecules (green dashed), SiO2 molecules (red
dot-dashed), Mgatoms(purpledotted), andSinucleicondensed
OH+H2→H2O+H. (Reaction 24) intoMg2SiO4 grains(blackdot-dot-dashed)relativetohydrogen
nuclei.
ForZ5,H2Ocooling isadominantcooling processinanar-
row range of density around 1010 cm−3 as shown by the
cyan dashed curve in Figure 2. The gas fragments into two
clumps by efficient OH cooling for Z3 as we see in detail in
AppendixA1. Dustcoolingbecomesineffectivewhenthermalcoupling
Dust cooling becomes efficient at 1011–1013 cm−3 for between the gas and grains is established. The cloud core
Z3, and at slightly larger densities 1012–1014 cm−3 for Z4. becomes optically thick soon after this phase, at densities
The grain growth enhances the efficiency of the dust ther- 1×1013, 3×1014, 3×1015, and 1×1016 cm−3 for Z3, Z4,
malemissionforZ3–5.Forsteritegrainsgrowrapidlyviathe Z5, andZ6, respectively. Thegas pressure rapidly increases
reaction intheregionwhereradiativecoolingisinefficientduetothe
large opacity, and then a hydrostatic core bounded by ac-
2Mg(g)+SiO(g)+3H2O(g)→Mg2SiO4(s)+3H2(g), cretion shock is formed. Just inside the shock, the gas disk
where (g) and (s) denote the species in the gas- and solid- is formed, being supported by the rotation. As the temper-
phases, respectively. The magnesium atoms are eventually ature further increases toward the center, the central part
exhausted(seethepurpledottedlineinFigure2).ForZ6,al- becomes spherical because of the strong pressure support.
thoughthereislittleamountofdust,thetemperaturedrops Here, let us define a pressure-supported “protostar” as the
at 1014 cm−3 bycollision-induced continuum cooling. region bounded by a sufficiently spherical iso-density sur-
(cid:13)c 2015RAS,MNRAS000,1–??
6 G. Chiaki et al.
face.2 The cloud fragmentation again occurs for Z3 around ∼20–100 AU, appears. Such a peculiar structure is formed
the time of the first protostar formation. The detailed pro- typically when dust cooling operates, as we discuss later.
cess is described in AppendixA1. Hereafter,wedistinguishtheformerandlattertypesof
fragmentation as disk fragmentation and filament fragmen-
tation, respectively. Since the disk fragmentation has been
discussed in previous numerical studies (Clark et al. 2011;
4 CLOUD FRAGMENTATION
Greif et al. 2012), we in this paper focus on the filament
4.1 Thermal evolution and fragmentation of cloud fragmentation, and revisit the disk fragmentation briefly in
Section 4.5. The different points between the two types of
4.1.1 Overview of the global feature fragmentationareinmoredetaildiscussedinAppendixA3.
In order to see whether the gas clouds fragment or not, we
follow the evolution over several tens years after the first
4.2 Collapse timescale and the thermal evolution
protostar appears. Figure 3 shows the snapshots output at
the time t∗ after the first protostar formation. Contrary to As described in Section 4.1.1, the thermal evolution has
thepopular notion that cloud fragmentation conditions are an effect on the cloud fragmentation property even with a
largelydeterminedbythegasmetallicity,thefragmentation fixed metallicity. We find that the different thermal evolu-
properties are different even with the same metallicity (see tion among clouds can be explained by the variation of the
panels in each row for a given metallicity). For example, collapsetimescale.Inaslowlycollapsingcloud,theadiabatic
for MH3 Z4, fragmentation does not occur even 30 years compressional heating rate perunit volume
after the first protostar formation, while multiple clumps
d 1 p
are already formed before the first protostar is formed for Γ =−ρp = (1)
adi
MH2 Z4. Clearly, whether a cloud fragments or not is not dt(cid:18)ρ(cid:19) tcol
determined solely by thegas metallicity. is correspondingly small. Here, t is thecollapse timescale
col
In other words, the thermal evolution, which critically written as
affects the fragmentation property, is not uniquely deter-
ρ
mined by the gas metallicity. The top panels of Figure 4 tcol = dρ/dt. (2)
shows the temperature evolutions as a function of the cen-
tral density of four clouds for Z5 and Z4. The evolutionary Therefore, the evolutionary track on the nH-T plane tends
tracksonthenH-T planevarysignificantlyevenwithafixed to belocated in a low-temperature region.
metallicity. Thevariation would generate thedifferent frag- We measure in our simulations that the time inter-
mentation property as discussed in more detail later. We val over which the peak density increases from nH,peak =
begin with discussing the physical processes that drive the 102 cm−3 to 1016 cm−3 is 3, 4, 11, and 19 Myr for UNI,
variation of the thermal evolution in Section 4.2. Then we MH1,MH2,andMH3Z4,respectively.ThecloudUNIcon-
discusstheeffectofthethermalevolutiononthecloudfrag- tracts most rapidly while the cloud MH3 contracts most
mentation in Section 4.3. Finally, in Section 4.4, we derive slowly among the four clouds. The evolutionary tracks in
criteria for cloud fragmentation. Figure 4 roughly trace the trend; the temperature of UNI
remainshighatalldensitiesbecauseofitsslowrotationand
hence the most rapid collapse. Contrastingly, the tempera-
4.1.2 Filament fragmentation vs. disk fragmentation tureofMH3remainslowbecausethelargecentrifugalforce
and the flat distribution of the dark matter slow the gas
Wenoteanimportantpointthatthecloudfragmentationis infall.
observedalsoat averysmall metallicity of10−6 Z ,where
J
dustcooling is notefficient.ForMH1Z6,theperturbations
growinthespiralarmsinacircumstellardiskwithasizeof 4.3 Important thermal processes
≃10AUaroundthecentralmostmassivecore.Suchastruc-
ture is reported also by Clark et al. (2011) and Greif et al. Figure 4 shows that different cooling and heating processes
(2012)inthecasewithoutmetalordust.Whereas,thesim- become important and characterize the various evolution-
ulationswithintermediatemetallcities10−5–10−3 Z show arytracks.Forexample,MH1andMH3Z5entertheregime
adifferenttypeoffragmentation.ForUNIZ3,MH1JZ4,and wheredustcoolingisefficientatnH ∼1012–1014 cm−3while
MH1 Z5, the “edamame”-like structure, where several pro- UNIandMH2donot.Inadditiontodustcooling,othertwo
tostellar cores are hosted in a long filament extending over cooling andheatingprocessesresponsible tothefragmenta-
tion can be identified: H2 formation heating and OH and
H2O cooling. We discuss the processes one-by-one in the
2 In practice, we calculate iso-density contours by dividing the following.
gas into 100 density bins equally separated with a logarithmic
scale from log(nH,peak)−3 to log(nH,peak). Fitting the distri-
bution of the gas particles in each density bin by an ellipsoid 4.3.1 Dust cooling
withthemajor-andminor-axesaandb,weobtaintheellipticity
Dust cooling is a crucial process to lower the gas tempera-
E =a/b−1 of each iso-density surface. We then define the pro-
ture,topromotecloudelongation,andtoinducetheforma-
tostellarsurfaceastheleastdens(mostdistantfromthecenter)
surfacethatsatisfiesE <E∗ withthecriticalellipticityE∗=0.3. tionoflow-massfragmentsathighdensities1012–1015cm−3,
Theprotostellarmassisonly20%smallerwithasmallerthresh- where the Jeans mass is small ∼ 0.1 MJ. Figure 4 clearly
oldofE∗=0.2. showstheeffectofdustcoolingonthefilamentformationfor
(cid:13)c 2015RAS,MNRAS000,1–??
Low-metallicity gas collapse 7
Figure3. Density-weighteddensityprojectionofthecloudsUNI,MH1,MH2,andMH3fromlefttorightwithmetallicities10−6,10−5,
10−4,and10−3Z frombottomtotopatthetimewherethesimulationsareterminated.Theplottedregionis100AU(upper8panels)
J
and20AU(lower8panels)onaside.ThecolorcontourdepictsthedensitynH=1010–1016 cm−3 (upper8panels)or1012–1018 cm−3
(lower 8 panels) from black to white. These snapshots are output at the time t∗ from the first protostar is formed as written in the
bottomofeachpane.Inthemodelswithoutthetime,thecentralblobdoesnotsatisfyourcriterionofaprotostar(seefootnote2)until
theendofthesimulation.
MH1 Z4 and Z5. The bottom panels present the cloud el- spectively.Figure3showsthatshortfilamentsareformedin
lipticity E = a/b−1, where a and b are the major- and theseclouds, but without fragmenting to multiple clumps.
minor-axes of the cloud. The gas becomes unstable ow-
We emphasize that the growth of dust grains further
ing to the dust cooling at densities nH ∼ 1012–1014 cm−3, enhances the cooling rate and thus promotes the gas elon-
and bar-mode perturbations grow. A filamentary structure
gation and fragmentation. In the Pop III supernova dust
is formed as clearly seen in Figure 3. When the ellipticity
model adopted here, Mg2SiO4 grains grow until gas-phase
becomes above 20–30, which is consistent with the critical Mgatomsareexhaustedat∼1014 cm−3.Figure5(a)com-
value defined by Tsuribe & Omukai (2006), the dense fila-
pares our simulations with (left) and without (right) grain
ment quickly fragments to yield several protostars.
growth for MH1 Z5 at the same time t∗ = 4.6 yr after the
ForUNIandMH2Z5,eventhoughthetemperaturede- firstprotostarformation.Whileseveralprotostellarcoresare
creases slightly by dust cooling at nH ∼1012–1014 cm−3, it born from a filament in the case with grain growth, only a
isinsufficienttoenhancetheellipticity.Thecloudellipticity single protostar is formed at the center of a slightly more
increases to only at most 11 and 12 for UNI and MH2, re- diffusefilament otherwise.
(cid:13)c 2015RAS,MNRAS000,1–??
8 G. Chiaki et al.
K] Z5 Z4

/
e 3
r UNI
u UNI
t
a
r MH1
e MH1
p
m MH2
MH2
e
T 2 MH3
[
MH3
g
o
l
30
y
Fragmentation
t
ci 20
i MH1
t
p
MH1
i
l
l MH3
E
10
MH3
0
0 4 8 12 16 0 4 8 12 16
-3 -3
log [Central density / cm ] log [Central density / cm ]
Figure4. Temporalevolutionofthetemperature(top)andellipticity(bottom)asafunctionofthecentraldensityofcloudcoresUNI
(bluedot-dashed),MH1(greensolid),MH2(orangedashed),andMH3(reddotted)forZ5(left)andZ4(right).Weshowtheellipticities
onlyforMH1andMH3forclarity.Thegreyshadedregionindicatestheellipticitiesforwhichthecloudbreaksupintomultipleclumps.
4.3.2 H2 formation heating are exhausted by the formation process before the molecu-
lar formation via three-body reactions at nH ∼ 108 cm−3.
In many cases, dust cooling becomes efficient at high den- Intheirmodel,H2 formation heatingthereforedoesnotbe-
sities but the gas cloud does not fragment. We find that
come efficient and can not stabilize the gas. Meanwhile, in
chemical heating effectively makes the cloud stable against our simulations even with 10−4–10−3 Z , three-body reac-
J
deformation.Forexample,inMH3Z5,therapidgasheating tionsandtheassociatedgasheatingmakethecloudrounder.
by the formation of H2 molecules via three-body reactions In order to see the effect more clearly, we perform a
becomes efficient at nH ∼109–1012 cm−3.Thetemperature controlled simulation without the heating process. Figure 5
rapidly increases from 130 K at nH =5×108 cm−3 to 700 (b)showsthedensitydistributionsinthecentral100AUre-
Kat2×1011 cm−3,causingthespecificheatratiotobe1.7
gionofthecloudMH3Z4with(left)andwithout(right)H2
on average, which is sufficient to stabilize the gas against
formation heating. In the former case, the cloud core ellip-
elongation. Then the gas ellipticity rapidly decreases from
ticity E remains around 3 because of the effective chemical
12 to 2 as shown in thebottom panel of Figure 4. heating at ∼108 cm−3. In the latter case E increases up to
The timescale for which the ellipticity increase from 15byOHcoolingandthenfurtherincreasesto∼30bydust
such a small value to the critical ellipticity is longer than cooling without any stabilization. Density perturbations in
the dynamical timescale of the cloud. In the clouds UNI, the filament grow and yield several protostellar cores in a
MH1,andMH2withmetallicities10−4–10−3 ZJ andinthe thin filament.
cloud MH3 with 10−6–10−3 ZJ, the H2 formation heating
significantly suppresses thegas elongation before dustcool-
ing becomes effective to promote thegas elongation.
4.3.3 OH and H O cooling
2
Tsuribe & Omukai (2008) show that a gas cloud does
not fragment with the metallicity range of 10−5–10−4 ZJ, InthecloudMH1Z4,althoughH2formationheatingiseffec-
by an order of magnitude smaller than in our simulations. tive(seethegreensolidcurveinFigure4),athinfilamentis
In their present-daydust model predicts 30 times larger ef- formedandquicklyfragmentstoafewclumps.Wefindthat
ficiencyoftheH2 molecularformationongrainsthaninour radiative cooling owing to the transition lines of OH and
PopIIIsupernovadustmodel(seeTable2).Hydrogenatoms H2Omoleculesisimportantinthiscase.Figure6showsthe
(cid:13)c 2015RAS,MNRAS000,1–??
Low-metallicity gas collapse 9
4 MH1 Z4
adi
2 chem
]
1 0 CIE
- H
s 2

1
- -2
g

g
r HD
e -4

/

e
t 4
a
r dust
g
n 2
li H2O
o
o 0
C
OH
[ OI
g -2 adi
o
l
-4
-6
O H O
O 2
] 2
e
c
n -8
a
d
n OH
u
b -10
A
[

g
o
-12
l
Figure 5. We compare the cloud fragmentation in the regular 4 8 12 16
simulations (left) and controlled simulations (right) at the same -3
log [Central density / cm ]
epoch.Alltherelevantchemo-thermalprocessesareconsideredin
theformersimulations,while,inthelattersimulations,(a)grain
growth, (b) H2 formationheating, and (c) OH and H2O cooling Figure 6. Coolingandheating ratesofthedominantprocesses
arenotincluded.
(top)andtheabundancesoftheO-bearingspecies(bottom)asa
functionofthecentraldensityforMH1Z4.
contribution of various heating and cooling processes as a
function of density for the cloud. At nH =106–108 cm−3, a
formation ofalongfilament andsubsequentfragmentation.
large fraction of O atoms are converted into OH molecules
This is for the first time demonstrated by our simulations
(greendot-dashedcurve),andOHcoolingexceedstheadia-
whichexplicitlyincludethemetalmolecularcoolingandthe
baticcompressionalheatingrate(redsolid).EventhoughH2
formation heating is effective at nH ∼ 109–1011 cm−3 and Pop III supernovamodel with thelarge oxygen excess.
reducesthecloudellipticity,E evaluatedat1011 cm−3 isthe
largest among thefourclouds for Z4 thanksto efficient OH
andH2Ocooling.Thecloudisthensufficientlyelongatedto 4.4 Criteria for first low-mass star formation
trigger fragmentation. The ellipticity increases up to 33 in
the end (see Figure 4). Although H2O cooling is indeed ef- From the discussion in Sections 4.2–4.3, we can consider
fectiveforMH2Z4at106–108 cm−3(orangedashedcurvein that the three thermal processes, dust cooling, H2 for-
Figure4),itisnotsufficienttocompensatethestabilization mation heating, and OH and H2O cooling are important
effect byH2 formation. to determine the fragmentation properties of the clouds.
Again, in order to see clearly the effect of the OH and Therefore, the criteria for gas fragmentation can be trans-
H2O cooling, we perform a controlled simulation without lated into the condition where these processes become ef-
OH and H2O cooling for MH1 Z4. The result is shown in fective or not for a given combination of the metallicity
Figure 5 (c). Without the molecular cooling, even at den- Z and the collapse timescale t . Extending the approach
col
sities where dust cooling is efficient, a single protostar is ofSchneider& Omukai(2010),wedefinethefragmentation
formed.Clearly,theefficientOHandH2Ocoolingdrivesthe condition as
(cid:13)c 2015RAS,MNRAS000,1–??
10 G. Chiaki et al.
] Z4 UNI MH1MH2 MH3

K

/ 3
re log [f0] ]⊙ --33
ratu 0.50.0 / Z Filament Frag.
pe 1.0 ty
m OH ci --44
[ Te 2 H2O Dust etalli OH/H2O
g M
lo H2 form. [ --55 Dust cool.

g
0 4 8 12 16 lo Disk Frag. H
log [ Central density / cm-3 ] --66 2 form.
00 00..55 11
Figure 7. Temperature as a function of the density in a
cloud core obtained by the semi-analytic chemo-thermal evolu- log [ f ]
0
tionmodelofgascloudsforZ4withf0 (Equation(3))from1to
10 every 0.5 dex (thick curves) and every 0.1 dex (thin curves).
We also draw thick and colored segments in the regimes where
γ>1.1byH2formationheating(red),whereγ<0.5byOHand Figure8. Weplottheregionsofthecollapsetimescalef0andthe
metallicity Z favourable for the filament fragmentation (green-
H2O cooling(blue), andwhereγ<0.8bydustemission(green)
and blue- shaded regions) and for the disk fragmentation (grey-
oneachevolutionarytrack.
hatched region). Dustcoolingis efficientintheregionabove the
green dashed lines (labeled “Dust cool.”). H2 formation heating
(i) dust cooling is efficient at high densities ∼ 1012 cm−3 isefficientintheregionabovethereddotted lines(“H2 form.”).
to reducethespecific heat ratio to γ <0.8, and OHandH2Ocoolingisefficientintheregionsurroundedbyblue
(ii-a) H2 formation heating is not efficient at intermediate solidline(“OH/H2O”).Wedefinetheselinesbylinearinterpola-
(iid-ebn)siOtyH∼or10H82cOm−c3o,oloirng is efficient so that γ < 0.5 even tfWi0oen=aol1sf–ot1ho0veeervre-epsrulyoltt0s.f0o05fadtnhedxeZasenmodfiZ-tah/neZaJslyimt=iucl1ac0ate−lcd6u–cla1lot0ui−odn3sseUvwNeirtIyh(0bv.al1ureydiednxig-.
though H2 formation heating later increases γ above 1.1. amonds),MH1(greencircles),MH2(orangetriangles),andMH3
Thethresholdvaluesofγ(0.8,1.1,and0.5)usedintheabove (red squares). Open and close symbols indicate the models that
endwithandwithoutfragments,respectively.
are determined so that the criteria reproduce the results of
oursimulations.Inthissection,weaimatdefiningtheregion
of Z and t where thegas fragmentation is favored.
col 4.4.1 One-zone semi-analytic model
Thereremainsacomplexitythatthecollapse timescale
itself varies when the cloud collapse proceeds faster/slower It is costly to run additional three-dimensional simulations
owing to gas cooling/heating. We thus introduce a param- whichcoveralarge parameterregionsofZ andf0.Wethus
eter f0 as an indicator of the collapse timescale that char- resorttoutilizingasimplerapproachfocusingonthechemo-
acterizes each cloud. The parameter satisfies the equation thermalevolution ofacloud.Tothisaim,thesemi-analytic
as one-zonemodelofC15issuitable.Wemodifythecodesuch
that thedensity in thecloud center increases according to
tcol =f0tscol(γ). (3)
dρ ρ ρ
The factor ts (γ) accounts for the dependence on the spe- = = (6)
cific heat ractoiol γ, the indicator of the thermal evolution. dt tcol f0tscol(γ)
From theself-similar solution of a polytropic cloud without with various f0.3 For γ &4/3 we set the upper limit of tcol
rotation or dark matter halo, tscol(γ) can be written as as 5tff, and for γ <0.5 we use tscol(0.5)=0.32tff. For given
2 1/2 density and temperature, our one-zone code solves exactly
tscol(γ)=(cid:20)3π2Dcen(γ)(cid:21) tff, (4) thesamechemicalreactionsincludinggraingrowth andgas
heating/cooling processes as in our three-dimensional sim-
where ulations. The CMB temperature is set to be 50 K (redshift
Dcen(γ)=4πGρcen(−t)2 (5) e∼ve2r0y).0W.0e5pdeerxfoarmndthZe=cal1cu0−la6t–io1n0s−3wiZth vaevryerinyg0f.10 =de1x–1t0o
J
isthedimensionlessformofthedensityatthecenter(Yahil coverthe parameter range of early clouds.
1983; Hanawa & Matsumoto 2000; Lai 2000), and tff = Figure 7 shows the temperature evolution of the cloud
(3π/32Gρcen)1/2 isthefree-falltime.Dcen(γ)increaseswith
the increasing γ for γ > 0.8, and diverges as γ approaches
4/3. Hereafter, we utilize theparameter f0 torepresent the 3 C15investigatethecloudevolutiononlyinthecasewithtcol≃
bulk rate of the cloud collapse. 1tff,correspondingtof0∼3.
(cid:13)c 2015RAS,MNRAS000,1–??