Dust Formation in Stellar Winds of Very Massive
Population III Stars

National Astronomical Observatory of Japan, Japan, E-mail: takaya.nozawa@nao.ac.jp

Takashi Kozasa
Department of Cosmosciences, Graduate School of Science, Hokkaido University, Japan

Sung-Chul Yoon
Department of Physics and Astronomy, Seoul National University, Korea

Keiichi Maeda
Department of Astronomy, Kyoto University, Japan

Norbert Langer
Argelander-Institut für Astronomie der Universität Bonn, Germany

Ken’ichi Nomoto
Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo,JapanThe origin of dust grains in the early universe is one of the most challenging subjects in astrophysics. Recent study of the evolution of massive Population III stars has suggested that themetal-enriched mass-loss winds from very massive Population III red-supergiants with the initial
masses higher than MZAMS ≃ 250 M⊙ could be potential formation sites of dust grains. In thisproceedings, in order to explore the possibility of dust formation in such stellar winds, we present
the analytical formulae that describe the formation condition of dust grains in steady stellar winds.
We find that, in a stellar wind of a Population III red-supergiant with MZAMS = 500 M⊙ , the formation of carbon grains is possible if the mass-loss rate and wind velocity satisfy the condition(Ṁ/10−5 M⊙ yr−1 )(vw /10 km s−1 )−2 & 0.06. We note that a series of formulae given in thisstudy would be useful in evaluating the feasibility of dust formation in steady mass-loss winds ofvarious types of stellar populations.
The Life Cycle of Dust in the Universe: Observations, Theory, and Laboratory Experiments
18-22 November, 2013
Taipei, Taiwan∗ Speaker.

c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence.

http://pos.sissa.it/PoS(LCDU 2013)023

Takaya Nozawa∗

Dust Formation in Stellar Winds of Very Massive Population III Stars

Takaya Nozawa

1. Introduction

2. The Model
Recently, Nozawa & Kozasa [9] formulated the non-steady-state formation of small clusters
and the growth of grains accompanied by chemical reactions. By examining the results of dust formation calculations for various gas densities and cooling rates, they found that the condensation efficiency and average radius of newly formed grains are universally described by a non-dimensional
quantity Λon defined as the ratio of the supersaturation timescale τsat to the collision timescale τcollat the onset time ton of dust formation. They also showed that the formation of C and MgSiO3
grains is realized when Λon > 1. Hence, the formation of dust in mass-loss winds can be evaluatedby deriving Λon for given wind properties such as the mass-loss rate and wind velocity.
In general, the mass-loss rate and wind velocity of massive stars are not well known; the massloss may be caused pulsationally and dynamically as a result of multiple effects of centrifugal
force, magnetic fields, turbulence, and outburst in stellar surface. Thus, it might be hard to predict
accurately the total amount and size distribution of dust formed in the course of the stellar evolution.
In this study, instead of modeling the detailed mass-loss history, we aim at presenting the ranges ofthe mass-loss rate and wind velocity with which the formation of dust is feasible in stellar winds.
In the following, we describe the model of density, temperature, and chemical composition of theoutflowing gas, and demonstrate how Λon is expressed by the mass-loss rate and wind velocity.
2PoS(LCDU 2013)023

The first cosmic dust is considered to be produced in the metal-rich ejecta of supernovae (SNe)that occur from the first stellar generation, so-called Population III (Pop III) stars. Numericalsimulations of the formation of metal-free stars have shown that the mass of Pop III stars would bemuch higher than those of the local Pop I/II stars [1, 2], supporting that SNe arising from massivestars would play dominant roles in dust enrichment history of the early universe. However, a typicalmass of Pop III stars is a matter of controversy; the predicted mass-scale ranges from a few tens M⊙[3, 4] up to more than a few hundreds M⊙ [5, 6]. If Pop III stars have formed with the initial masses
above MZAMS ≃ 250 M⊙ , they are likely to finally collapse into black holes and fail to disperse alarge amount of heavy elements at the end of their lives [7, 8].
On the other hand, such very massive Pop III stars could chemically enrich the surrounding
medium during their evolutionary stages. Yoon et al. [8] extensively investigated the evolution ofmassive Pop III stars for the mass range of 10–1000 M⊙ , and found that non-rotating stars withMZAMS & 250 M⊙ experience convective dredge-up of a significant amount of carbon and oxygenfrom the helium-burning core into the hydrogen envelope at the red-supergiant (RSG) phase. Thus,
if the mass loss is driven efficiently during the RSG phase, the CNO-enriched stellar winds canpollute the ambient medium and can also offer formation sites of the first dust grains in the universe.
In this proceedings, with the aim of assessing the possibility of dust formation in RSG winds ofvery massive Pop III stars, we present the analytical formulae describing the formation conditionof carbon grains in steady stellar winds. The formula is based on the scaling relation of dust
formation given in Nozawa & Kozasa [9] and would be very useful in evaluating the feasibility of
dust formation in mass-loss winds of any kinds of stellar populations.

Dust Formation in Stellar Winds of Very Massive Population III StarsTakaya Nozawa

2.1 Hydrodynamic Model of the Circumstellar Gas
We consider a simple model where the circumstellar gas is a spherically symmetric outflow
with a constant wind velocity. In this case, the density profile of the outflowing gas is given byṀṀ=
ρ (r) =
24π r vw 4π R2star vw

r
Rstar

−2,

(2.1)

T (r) = Tstar

r
Rstar

− α,

(2.2)

where Tstar is the effective temperature of the star, and α ≃ 1 for adiabatic expansion, and α = 0.5
for the case of radiative equilibrium.
2.2 Chemical Composition of the Circumstellar Gas
As mentioned in the Introduction, very massive Pop III stars with MZAMS & 250 M⊙ could
undergo extensive dredge-up during helium-core burning, which conveys a considerable amount ofcarbon and oxygen to the hydrogen envelope [8]. For the model of a 500 M⊙ RSG without rotation,
the average number fractions of carbon and oxygen in the outer envelope are AC = 3.11 × 10−3 and
AO = 1.75 × 10−3 , respectively [8]. Thus, the gas outflowing from very massive RSGs is expected
to be carbon-rich with a high C/O ratio (AC /AO = 1.78 for the 500 M⊙ RSG model).
In a carbon-rich cooling gas, all oxygen atoms react with carbon atoms to form CO molecules,
and carbon atoms and/or carbon-bearing molecules left after the CO formation are available forthe formation of carbon clusters and grains. Figure 1 shows the result of the chemical equilibrium
calculation along the gas flow [10], as an example, for the 500 M⊙ Pop III RSG (see below for theproperties of this star). It can be seen that CO molecule is the stable gas species in the temperature
range considered here, and most of the carbon unbound in CO molecules exists as atomic carbon
at T > 1750 K and as C2 H at 1400 K < T < 1700 K. This indicates that the formation of carbon
grains is likely to proceed through successive attachment of carbon atoms (or C2 H molecules) at
high temperatures.
2.3 Derivation of the non-dimensional quantity Λon
For the wind model given above, we here provide a non-dimensional quantity Λon = τsat /τcoll ,considering that carbon grains condense via the successive attachment of carbon atoms.
The timescale of collisions of carbon atoms is given by−1= π a20
τcoll8kT
π mC

 21ñC ,(2.3)

where a0 is the radius of the condensate per atom, k is the Boltzmann constant, mC is the mass of acarbon atom, and ñC is the nominal number density of free carbon atoms unbound in CO molecules(see Section 2.3 in Nozawa & Kozasa [9] for the definition of the nominal number density). By
3PoS(LCDU 2013)023

where Ṁ is the mass-loss rate, vw is the wind velocity, Rstar is the stellar radius, and r is the distancefrom the center of the star. The temperature distribution of the gas is assumed to beDust Formation in Stellar Winds of Very Massive Population III Stars

Takaya Nozawa

introducing a parameter f = AC /AO , which is the number ratio of carbon atoms to oxygen atoms,
the nominal number density of carbon atoms is written as α2
Ṁ
1ACT
1−ñC =,
(2.4)
µ mHfTstar
4π R2star vw
where µ is the mean molecular weight, and mH is the mass of hydrogen. The supersaturation ratio
S is described byñC kT
γñC kT∆g̊+ ln= − δ + ln,(2.5)
ln S = −kT
pv
Tpv
where pv is the vapor pressure of the condensate, and the Gibbs free energy for the formation ofbulk condensate from the vapor is approximated by ∆g̊/kT = −γ /T + δ with the numerical values
γ and δ derived by least-squares fittings of the thermodynamics data [11] in the temperature rangeof T = 200–2500 K. Then, the timescale on which the supersaturation ratio increases is given as α1 d ln S α vw
γT2−1τsat =(2.6)=− −1 .
dt
Rstar Tstar
T α
From Equations (2.3) and (2.6), the non-dimensional quantity Λon is written as−1
1 
1 
AC π a20
Ton α
8kTon 22Ṁ1
τsatγ,(2.7)=− −11−
Λon =τcoll t=ton α µ mH
fπ mC
4π Rstar v2w
Tstar
Ton α
where Ton is the gas temperature at t = ton (that is, condensation temperature of dust). In general,α = 0.5 is used for the temperature profile of the outflowing gas [e.g., 12]. In this case,−2 
 1Lstar − 21vw
ṀAC
Λon ≃ 150 1 −,(2.8)
f
10−310−5 M⊙ yr−1105 L⊙
10 km s−14 ), a = 1.28 Å , and γ = 8.64 × 104 K
where Lstar is the luminosity of the star (= 4π R2star σ Tstar
0[13], and we adopt µ = 1.3 and Ton = 1800 K. Equation (2.8) demonstrates that, for α = 0.5, theformation process of dust (represented by Λon ) in stellar winds is sensitive to the mass-loss rate Ṁ
and wind velocity vw but does not depend explicitly on Tstar .

4PoS(LCDU 2013)023

Figure 1: Number fractions of carbon-bearing gas species as a function of gas temperature obtained fromthe chemical equilibrium calculation for the circumstellar envelope model of the 500 M⊙ Pop III RSG.

Dust Formation in Stellar Winds of Very Massive Population III Stars

Takaya Nozawa

3. Formation Condition of Dust in a RSG Wind of a Pop III 500 M⊙ StarAs shown in Nozawa & Kozasa [9], the formation of carbon grains is possible when Λon > 1.
Thus, from Equation (2.8), the formation condition for carbon grains in a steady RSG wind of a
Pop III 500 M⊙ star ( f = 1.78, AC = 3.11 × 10−3 , Lstar = 107.2 L⊙ ) is given as follows:

φ=Ṁ
−510 M⊙ yr−1

vw
10 km s−1

−2& 0.06.

(3.1)

This equation indicates that, for a constant wind velocity of 20 km s−1 , carbon grains can condense
if Ṁ & 2.5 × 10−6 M⊙ yr−1 .
Nozawa & Kozasa [9] also presented the approximation formula describing the average grainradius aave,∞ and condensation efficiency fcon,∞ as a function of Λon ; the approximation formulae
for carbon grains are given by
aave,∞− 1 = −1.21 + 0.854 log Λon(3.2)
loga∗andlog fcon,∞ = 4.15 [tanh (1.91 log Λon + 0.088) − 1] ,(3.3)

where a∗ = 5.946 Å . The solid lines in Figure 2 shows aave,∞ and fcon,∞ obtained from the above
analytic formulae as a function of φ for the 500 M⊙ Pop III RSG model. The condensation efficiency rapidly increases from φ ≃ 0.06, reaching unity at φ ≃ 1. The dashed lines in Figure 25PoS(LCDU 2013)023

Figure 2: Final average radius aave,∞ and condensation efficiency fcon,∞ of carbon grains that can form in a
RSG wind of a Pop III star with MZAMS = 500 M⊙ as a function of φ defined in Equation (3.1). The solidlines are the results obtained analytically from Equations (2.8), (3.2), and (3.3), while the dashed lines are
the results of detailed numerical simulations of dust formation. The vertical dotted line marks φ = 0.06,
above which the formation of dust is found to be possible [see Equation (3.1)].

Dust Formation in Stellar Winds of Very Massive Population III Stars

Takaya Nozawa

Acknowledgments
This research has been supported by World 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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6PoS(LCDU 2013)023

plot the results of the numerical simulations of non-steady-state dust formation, which are in good
agreement with the analytical ones. In fact, there are small differences in aave,∞ at φ & 10 and infcon,∞ at φ . 0.2, both of which mainly come from the simple assumption of Ton = 1800 K in thepresent analysis; in reality, Ton is higher than 1800 K for a higher Λon , and vice versa.
In this study, we have put the constraints on the mass-loss rate and wind velocity with whichcarbon grains can condense in RSG winds of very massive Pop III stars. Here, it should be emphasized that our analysis can be also applicable for dust formation in mass-loss winds of any types ofstellar populations, as long as the mass-loss wind is steady. Therefore, a series of formulae givenin this work would be powerful tools in assessing the possibility of dust formation in stellar winds.