EThe Astrophysical Journal, 598:785–803, 2003 December 1# 2003. The American Astronomical Society. All rights reserved. Printed in U.S.A.

DUST IN THE EARLY UNIVERSE: DUST FORMATION IN THE EJECTA OF
POPULATION III SUPERNOVAE
Takaya Nozawa and Takashi Kozasa
Division of Earth and Planetary Sciences, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan;
nozawa@ep.sci.hokudai.ac.jp, kozasa@ep.sci.hokudai.ac.jp

andHideyuki Umeda, Keiichi Maeda, and Ken’ichi Nomoto
Department of Astronomy, School of Science, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan;
umeda@astron.s.u-tokyo.ac.jp, maeda@astron.s.u-tokyo.ac.jp, nomoto@astron.s.u-tokyo.ac.jp
Received 2003 June 28; accepted 2003 August 11

ABSTRACTDust grains play a crucial role in the formation and evolution history of stars and galaxies in the early
universe. We investigate the formation of dust grains in the ejecta of Population III supernovae, including
pair-instability supernovae, which are expected to occur in the early universe, applying a theory of non–steady state nucleation and grain growth. Dust formation calculations are performed for core-collapse
supernovae with progenitor mass Mpr ranging from 13 to 30 M and for pair-instability supernovae with
Mpr ¼ 170 and 200 M. In the calculations, the time evolution of gas temperature in the ejecta, whichstrongly aﬀects the number density and size of newly formed grains, is calculated by solving the radiativetransfer equation, taking account of the energy deposition of radioactive elements. Two extreme cases are
considered for the elemental composition in the ejecta, unmixed and uniformly mixed cases within the He
core, and formation of CO and SiO molecules is assumed to be complete. The results of calculations forcore-collapse supernovae and pair-instability supernovae are summarized as follows: in the unmixed ejecta, avariety of grain species condense, reﬂecting the diﬀerence of the elemental composition at the formation site
in the ejecta; otherwise only oxide grains condense in the uniformly mixed ejecta. The average size of newly
formed grains spans a range of 3 orders of magnitude, depending on the grain species and the formation
condition, and the maximum radius is limited to less than 1 lm, which does not depend on the progenitor
mass. The size distribution function of each grain species is approximately lognormal, except for Mg silicates,
MgO, Si, and FeS in the unmixed case and Al2O3 in both cases. The size distribution function summed up
over all grain species is approximated by a power-law formula whose index is 3.5 for the larger radius and2.5 for the smaller one; the radius at the crossover point ranges from 0.004 to 0.1 lm, depending on themodel of supernovae. The fraction of mass locked into dust grains increases with increasing the progenitor
mass: 2%–5% of the progenitor mass for core-collapse supernovae and 15%–30% for pair-instability supernovae whose progenitor mass ranges from 140 to 260 M. Thus, if very massive stars populated the ﬁrst
generation of stars (Population III stars), a large amount of dust grains would be produced in the early
universe. We also discuss the dependence of the explosion energy and the amount of 56Ni in the ejecta, as wellas the eﬃciency of formation of CO and SiO molecules, on the formation of dust grains in the ejecta ofsupernovae.
Subject headings: dust, extinction — early universe — supernovae: general
On-line material: color ﬁgures

the most iron-poor star so far discovered, HE 01075240(Christlieb et al. 2002), has raised important questions
about the formation of such a metal-poor (½Fe=H ¼
5:3  0:2) but low-mass (0.8 M) star. Umeda &Nomoto (2003a) showed that this star can be the secondgeneration star formed from a gas cloud enriched by a
Population III supernova that produced very little Fe but
enough C and O for eﬃcient gas cooling. Schneider et al.
(2003b) also considered this star as the second generation
but argued that the dust grains produced by Population III
objects could play an important role in forming low-massstars from a gas cloud as metal-poor as metallicity Z ¼
105:1 Z . Furthermore, dust grains residing in interstellar
space in galaxies and in the intergalactic medium cause
obscuration and reddening of starlight, and the thermalradiation from the dust grains distorts the cosmic background radiation (see Hauser & Dwek 2001 for details).

1. INTRODUCTION

The recent observations of the reddening of backgroundquasars and the damped Ly systems in the spectra of distant quasars have conﬁrmed the presence of dust grains in
the high-redshift universe (e.g., Pettini et al. 1994, 1997; Pei& Fall 1995). Dust grains play a critical role in the formation and evolution history of stars and galaxies in the early
universe: dust grains absorb stellar light and reemit it bythermal radiation, which controls the energy balance ininterstellar space and the evolution of gas clouds. Also, thesurface of dust grains is an eﬃcient site for the formation of
H2 molecules, which act as an eﬀective coolant at the time of
the formation of stars from interstellar clouds, enhance thestar formation rate (SFR), and strongly aﬀect the initial
mass function (IMF) in the metal-poor universe (Hirashita,
Hunt, & Ferrara 2002; Hirashita & Ferrara 2002). In fact,
785

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NOZAWA ET AL.

How much dust grains absorb stellar light and reemit it bythermal radiation heavily depends on their chemical composition, size, and abundance. Therefore, the investigation ofdust grains in the early universe is inevitable not only toreveal the structure and evolution of the early universe, butalso to deduce the SFR and the IMF during the evolution of
the universe from the observations of the cosmic microwave
background (CMB) and the cosmic infrared background(CIB), which could be one of the main subjects of theplanned Atacama Large Millimeter Array (ALMA) andthe James Webb Space Telescope (JWST, formerly the
Next-Generation Space Telescope; Loeb & Haiman 1997).
In cosmic environments, dust grains form in a cooling gas
outﬂowing from star to interstellar space, such as in thestellar winds from asymptotic giant branch (AGB) stars andin the ejecta of supernovae. The major source of dust grains
in our Galaxy is considered to be AGB stars evolving fromstars with a main-sequence mass of 8 M. However, theage of AGB stars is too old to contribute to dust grains in
the early universe. Thus, supernovae evolving from stars
whose main-sequence mass is larger than 8 M are considered to be the major source of dust grains in the early
universe. Dust formation in the ejecta of supernovae has
been suggested from the isotopic anomalies in meteorites,
and the microscopic analysis of dust grains extracted frommeteorites has identiﬁed SiC, graphite, and Si3N4 grains as
the supernovae condensates (SUNOCONs) from their isotopic composition (see Zinner 1998 for details). SN 1987A is
the ﬁrst supernova in which the in situ dust formation in the
expanding ejecta was observed (Lucy et al. 1989; Whitelock
et al. 1989; Meikle et al. 1993; Wooden et al. 1993; Colgan
et al. 1994), and the model of formation of dust grains in theejecta was investigated by Kozasa, Hasegawa, & Nomoto(1989a, 1989b, 1991). Also, dust formation was conﬁrmed
in the ejecta of SN 1999em from the appearance of blueshifted line emissions, as observed in SN 1987A (Elmhamdi
et al. 2003).
What kind of and how many dust grains condense in theejecta are still in debate. The observations of young supernova remnants (SNRs) with the Infrared Space Observatory(ISO) revealed the thermal radiation from dust grains condensed in the ejecta of Cas A (Lagage et al. 1996; Arendt,
Dwek, & Moseley 1999), and the observed thermal radiation is well ﬁtted by a mixture of Al2O3, MgSiO3, and SiO2
grains (Douvion, Lagage, & Pantin 2001b); however, thespectroscopic observations at 2.4–45 lm suggested that thepristine dust in Cas A is a peculiar class of silicate material(Arendt et al. 1999). On the other hand, no thermal radiation originating from dust grains formed in the ejecta wasobserved toward the Tycho, Kepler, and Crab SNRs by
ISO (Douvion et al. 2001a), despite the fact that the opticalobservations have suggested the presence of dust grains in
the Crab SNR (Fesen & Blair 1990; Hester et al. 1990). The
estimated amount of hot dust observed in the Cas A SNR
by ISO (Arendt et al. 1999) is 10–100 times smaller than thevalue previously estimated from IRAS observations (e.g.,
Mezger et al. 1986; Dwek et al. 1987). However, the recent
submillimeter observations by SCUBA have revealed theexistence of cold dust in the young SNR of Cas A, leading
to the conclusion that the amount of dust condensed in theejecta is estimated to reach a few solar masses (Dunne et al.
2003). These observations strongly support the hypothesisthat Type II supernovae could be the major source of dust
grains in the early universe.

Vol. 598

Of course, the chemical composition, size, and amount of
dust grains in the early universe are determined by the balance between production in the ejecta of supernovae anddestruction by interstellar shock, as well as reverse shock penetrating into the ejecta (see, e.g., Dwek 1998 for details), andthe investigation of dust formation in the ejecta of Population
III supernovae is the ﬁrst step to revealing the nature ofdust grains in the early universe. Todini & Ferrara (2001)
performed dust formation calculations in primordial corecollapse supernovae (CCSNe) with progenitor mass Mpr
ranging from 12 to 35 M, adopting the models of supernovae by Woosley & Weaver (1995) and applying the methodof dust formation calculation for SN 1987A by Kozasa et al.
(1989a, 1991) as a template. In the calculations, they considered a one-zone model for elemental composition and density
in the ejecta, assuming the uniform mixing of elements within
the He core. Taking account of the formation and destruction of CO and SiO molecules, they showed that carbon
grains condense ﬁrst and that Al2O3, Mg silicates, and Fe3O4
grains then condense in this order as the gas cools down.
Also, they showed that the size of newly formed grains is relatively small (300 Å in radius) and is sensitive to the value of
adiabatic index  and that the total mass of dust produced isabout 0.08–0.3 M per supernova.
The recent theoretical investigations of fragmentation ofgas clouds in the metal-free early universe have claimed thatstars more massive than some tens to 100 M populate theﬁrst generation (Nakamura & Umemura 2001; Bromm,
Coppi, & Larson 2002). Such metal-free stars as massive as
140 M  Mpr  260 M may evolve stably to end theirlives as pair-instability supernovae (PISNe; e.g., Umeda &Nomoto 2002; Heger & Woosley 2002). Also Umeda &Nomoto (2002) have suggested that hypernovae (HNe),
whose explosion energy is more than several to 10 times thatof ordinary CCSNe, are necessary to reproduce theenhancement of Zn and iron-peak elements observed inmetal-poor stars. Therefore, in this paper, we explore dust
formation in the ejecta of Population III supernovae,
including HNe and PISNe, to investigate the dependence ofthe progenitor mass and the explosion energy on the formation of dust grains in the ejecta. Our main aim is to clarify
the dependence of the yield of newly formed grains and theirsize on the progenitor mass, focusing on the ordinary
CCSNe and PISNe. We also investigate the eﬀects of the
explosion energy and the mass of 56Ni in the ejecta, as wellas the eﬃciency of formation of CO and SiO molecules, on
the formation of dust grains.
As discussed by Kozasa et al. (1989a), the elemental composition in the ejecta controls what kind of grain species
condenses, and the temporal evolution of gas density andtemperature in the ejecta strongly aﬀects the size of newly
formed grains in the ejecta. Thus, in the dust formation
calculations we employ the hydrodynamic models andelemental compositions of Population III supernovae calculated by Umeda & Nomoto (2002). The time evolution ofgas temperature is calculated by the multifrequency radiative transfer code, taking account of the energy deposition
from radioactive elements (Iwamoto et al. 2000). Two
extreme cases for the elemental composition in the ejecta areconsidered: the original onion-like structure (hereafterreferred to as the unmixed case) and the uniformly mixed
case within the He core. A theory of non–steady state nucleation and grain growth is applied for the calculation of dust
formation, under the assumption that the formation of CO

No. 2, 2003

DUST FORMATION IN POPULATION III SUPERNOVAE

and SiO molecules is complete. The formation of all possible
condensates is taken into account simultaneously, to clarify
what kind of grain species really condenses.
In x 2 we present the models of supernovae used in the calculations, and the method of dust formation calculation is
described in x 3. The results of calculations for ordinary
CCSNe and PISNe are presented and discussed in x 4 andsummarized in x 5. The eﬀects of the explosion energy andthe amount of 56Ni in the ejecta, as well as the eﬃciency offormation of CO and SiO molecules, on dust formation in
the ejecta of supernovae are presented in Appendices A andB, respectively.

787

the ejecta on dust formation. The elemental composition andtemporal evolution of gas density and temperature in theejecta, which are essential to investigating dust formation,
are summarized in the following subsections.
2.1. Elemental Composition
The elemental composition in the ejecta determines whatkind of grain species condenses. No dust grain condenses in
the hydrogen envelope of Population III supernovae,
because the hydrogen envelope is metal-free. Even if heavy
elements intrude into the hydrogen envelope during the evolution of the progenitor, the low density and the high expansion velocity prevent dust grains from condensing in thehydrogen envelope, as discussed by Kozasa et al. (1989a,
1991). Thus, the dust formation region in the ejecta ofPopulation III supernovae is conﬁned within the He core,
where heavy elements exist.
Figure 1 shows the elemental composition within theunmixed He core at day 600 after the explosion, taking
account of the decay of radioactive elements. Figure 1a isfor a model of CCSNe (C20) and Figure 1b for PISNe(P170). The ejecta is divided into ﬁve regimes, according to
the elemental composition of interest to dust formation: forexample, in the ejecta of model C20, an Fe-Si-S layer(2:45 M  Mr  2:52 M ), a Si-S-Fe layer (2:52 M 
Mr  2:95 M ), an O-Si-Mg layer (2:95 M  Mr 
3:14 M ), an O-Mg-Si layer (3:14 M  Mr  4:92 M ),and a He layer (4:92 M  Mr  5:79 M ).
The He layer is carbon-rich (C=O > 1) for CCSNe, andits mass irregularly varies with the progenitor mass inthe models of CCSNe used in this paper. The mass of theoxygen-rich layer (the O-Si-Mg and O-Mg-Si layers)
increases with increasing progenitor mass, and the mass of
the inner Si-S-Fe and Fe-Si-S layers depends on the value ofthe mass cut. On the other hand, in the ejecta of PISNe, theHe layer is very thin for model P170, and the mass increases
with increasing progenitor mass.
The mixing of elements in the ejecta of SN 1987A wasconﬁrmed from the early emergence of X-rays and -rays(e.g., Dotani et al. 1987; Itoh et al. 1987; Kumagai et al.
1988), and also hydrodynamic simulations (Hachisu et al.
1990; see Arnett et al. 1989 for a review and references) andlaboratory experiments (Drake et al. 2002) clearly demonstrate that the mixing of elements is caused by the RayleighTaylor instability at each interface of layers with diﬀerent

2. MODEL OF THE POPULATION III SUPERNOVAE

The nature of Population III supernovae has been
extensively investigated to decipher the chemical evolutionof the early universe compared with the observed elemental
abundance of metal-poor stars (Umeda & Nomoto 2002,
2003b). Umeda & Nomoto (2002) have suggested that, in
addition to ordinary CCSNe with explosion energies of1051 ergs, HNe, whose explosion energies are larger than
ð5 50Þ  1051 ergs (e.g., Nomoto et al. 2001, 2003 forreviews), are necessary to reproduce the enhancement of Zn
and iron-peak elements observed in metal-poor stars. Also,in the metal-free universe very massive stars, with mass
ranging from 140 to 260 M, evolve stably and explode as
PISNe, in which a star is completely disrupted by a nuclearpowered explosion (Umeda & Nomoto 2002; Heger &Woosley 2002).
We apply the hydrodynamic models and the results ofnucleosynthesis calculations of Population III supernovae by
Umeda & Nomoto (2002) for the dust formation
calculations. The details of the models used in the present calculations are summarized in Table 1, where the labels C, P,
and H in the model names represent ordinary CCSNe,
PISNe, and HNe, respectively, and the numerical value
denotes the mass of the progenitor in units of solar mass. The
mass cut is the mass coordinate of the boundary between theejecta and the remnant, and its value speciﬁes the mass of
56Ni in the ejecta; Mð56 NiÞ ¼ 0:07 M is taken as a typicalvalue for ordinary CCSNe to reproduce the observed
behavior of the early light curve. The models of HNe (H20A,
H20B, H30A, and H30B) are adopted to investigate thedependence of the explosion energy and the mass of 56Ni in

TABLE 1Models of Supernovae Used in the Calculations

Model

Progenitor Mass
(M)

Explosion Energy E51(1051 ergs)

Mass Cut(M/Mr)He Core Mass(M)

M(56Ni)(M)

C13 ..................
C20 ..................
C25 ..................
C30 ..................
P170.................
P200.................
H25A ...............
H25B ...............
H30A ...............
H30B ...............

13
20
25
30
170
200
25
25
30
30

1
1
1
1
20
28
10
10
30
30

1.50
2.45
2.54
2.54
0
0
3.29
2.47
5.31
4.33

2.05
5.79
8.01
10.7
82.4
117.3
8.01
8.01
10.9
10.9

0.07
0.07
0.07
0.07
3.56
7.25
0.07
0.7
0.07
0.7

Note.—In the model names, the labels C, P, and H represent ordinary CCSNe, PISNe, and HNe,
respectively, and the numerical value denotes the mass of the progenitor in units of solar mass.

788

NOZAWA ET AL.

Vol. 598

like structure, and the other is the mixed case, for which theelements are uniformly mixed within the He core.
2.2. The Time Evolution of Gas Density andTemperature in the Ejecta
The time evolution of gas density and temperaturestrongly aﬀects the number density and size of newly formed
grains in the ejecta. As the explosion shock propagates
through the star, the reverse shock generated at the interfaceof layers with diﬀerent chemical compositions changes thedensity structure. Although Deneault et al. (2003) show thatthe reverse shock generated at the He-H interface introduces
a very nonhomologous readjustment after t ¼ 106 s fromthe explosion, our hydrodynamic calculations including theinteraction with the reverse shock have shown that the interaction makes the expansion ﬁnally homologous in less than
1 day after the explosion (see Shigeyama & Nomoto 1990).
Thus, the time evolution of gas density at a given mass
coordinate Mr is calculated by 3tðMr ; tÞ ¼ ðMr ; t0 Þ
;
ð1Þ
t0

Fig. 1.—Number abundance of elements relevant to dust formation
within the unmixed He core at 600 days after the explosion, taking into
account the decay of radioactive elements, for (a) C20 and (b) P170. [Seethe electronic edition of the Journal for a color version of this ﬁgure.]elemental compositions. The isotopic signatures in the presolar carbon and SiC grains identiﬁed as SUNOCONs havesuggested extensive and microscopic mixing of elements inthe diﬀerent layers in the explosion (Amari & Zinner 1997;
Travaglio et al. 1999). Thus, as an ideal case, the uniform
mixing of elements within the He core was considered toinvestigate dust formation in the ejecta of supernovae(Kozasa et al. 1989a; Todini & Ferrara 2001; Schneider,
Ferrara, & Salvaterra 2003a). On the other hand, Douvion
et al. (2001b) have suggested that the mixing in the ejecta of
the Cas A SNR is knotty rather than microscopic. A simple
calculation of molecular diﬀusion has shown that mixing at
the atomic level is impossible before the condensation ofdust grains in the ejecta (Deneault, Clayton, & Heger 2003).
It is still debated at this moment whether the mixing ismacroscopic at the knotty level or microscopic at the atomic
level and how large the mixing is that is extended into theejecta. Therefore, in the following calculation of dust
formation we consider two extreme cases for the mixing ofelements; one is the unmixed case, with the original onion-

where the reference time t0 is 1 day after the explosion. The
distribution of gas density within the He core at day 600
after the explosion is shown in Figure 2, where the mass
coordinate is normalized to the He core mass and the histogram presentation of the density structure reﬂects the size ofthe mesh used in the calculations; Figure 2a is for CCSNe(C20 and C25) and HNe (H25) and Figure 2b for PISNe(P170 and P200). The gas density within the He cores of
CCSNe with the same explosion energy is of almost thesame order of magnitude and does not depend so much on
the progenitor mass, apart from the detailed structurecaused by the propagation of the reverse shock originating
at each interface of layers. The gas density in the ejecta of
H25 is an order of magnitude lower than that of CCSNe,
reﬂecting the explosion energy of 10 times that of CCSNe.
Also, the gas density in the ejecta of P200 is a little bit lower
than that in the ejecta of P170, but the gas density within the
He core is 1014 to 1013 g cm3 at day 600 after the explosion and is of almost the same order of the magnitude as thegas density in the ejecta of CCSNe, except for the region
Mr  40 M.
The gas temperature in the ejecta is determined by thedetailed process of the degradation of -rays and X-rays
deposited from the radioactive elements. In the previous
calculations (Kozasa et al. 1989a, 1991; Todini & Ferrara
2001), the time evolution of gas temperature wasapproximated by 3ð1Þt;ð2Þ
T ð Mr ; t Þ ¼ T ð Mr ; t 0 Þ
t0with a parameter , and the reference temperature
T ðMr ; t0 Þ was determined by the observations of SN
1987A. However, this method is not applicable for Population III supernovae with no observational data. Therefore,
the time evolution of gas temperature is calculated by using
the multifrequency radiative transfer code together with theenergy equation, taking account of the deposition of energy
from radioactive elements (see Iwamoto et al. 2000 for
details).

No. 2, 2003

DUST FORMATION IN POPULATION III SUPERNOVAE

Fig. 2.—Distribution of gas density within the He core at 600 days after
the explosion for (a) C20, C25, and H25 and (b) P170 and P200. Note thatthe mass coordinate is normalized to the He core mass. [See the electronic
edition of the Journal for a color version of this ﬁgure.]

As an example, Figure 3 shows the time evolution of gas
temperature at a location in the oxygen-rich layer withalmost the same elemental composition in the ejecta of
CCSNe, PISNe, and HNe: Figure 3a for the unmixed ejectaof C20 (at Mr ¼ 3:7 M), C25 (at Mr ¼ 4:6 M), P170 (atMr ¼ 56:5 M), and P200 (at Mr ¼ 64:9 M) and Figure 3b
for the mixed ejecta of C25 (at Mr ¼ 4:6 M) and P200
(at Mr ¼ 64:9 M) and the unmixed ejecta of H30A (atMr ¼ 6:8 M) and H30B (at Mr ¼ 6:8 M). It should bepointed out here that, as can be seen from Figure 3, the temporal evolution of gas temperature in the ejecta is approximately determined only by the explosion energy and theamount of 56Ni in the ejecta and does not heavily depend on
the progenitor mass in the mass range considered in thispaper. Also, the gas temperature in the mixed ejecta isalmost the same as that in the unmixed ejecta, except for thegas temperature around the day 200 after the explosion. The
gas temperature in the ejecta of H30, with its large explosion
energy, quickly decreases in comparison with that in
CCSNe, because X-rays and -rays escape easily as a result

789

Fig. 3.—Time evolution of gas temperature at a location in the oxygenrich layer with almost the same elemental composition in the ejecta of
CCSNe, PISNe, and HNe. (a) At Mr ¼ 3:7, 4.6, 56.5, and 64.9 M for C20,
C25, P170, and P200, respectively, in the unmixed ejecta. (b) At Mr ¼ 4:6
and 64.9 M for C25 and P200, respectively, in the mixed ejecta andMr ¼ 6:8 M for H30A and H30B in the unmixed ejecta. [See the electronic
edition of the Journal for a color version of this ﬁgure.]of the low gas density in the ejecta. The eﬀect of Mð56 NiÞ inthe ejecta on the gas temperature is clearly shown by the difference in the gas temperature in the unmixed ejecta
between H30A and H30B. The gas temperature in the ejecta
of PISNe with a large explosion energies is much higher
than that in CCSNe, because the progenitors are massive
and a large amount of 56Ni resides in the ejecta.
From Figures 2 and 3, we can easily estimate the condensation time and the behavior of the average radius of a grain
species formed in the ejecta of Population III supernovae; if
the grains condense around a gas temperature of 1500 K in
the oxygen-rich layer, the condensation time tc after the
explosion would be around day 400 for CCSNe, day 600 for
PISNe, and day 200 for HNe with the same Mð56 NiÞ as
CCSNe. The average radius of the condensate would not bemuch diﬀerent, since the average size is roughly proportional to 1=3 / ð1=tc Þ, so long as the abundance of the790

NOZAWA ET AL.

elements available for dust formation is the same, which is
conﬁrmed by the results of the dust formation calculations
given in x 4 and Appendix A. Note that the radiationtransfer calculation does not include the radiative cooling ofmolecules, such as CO and SiO, that act as eﬀective coolants
of gas in the ejecta, as demonstrated by Liu & Dalgarno(1995) for SN 1987A. The eﬀect of radiative cooling ofmolecules would make the condensation time of dust grains
earlier and the average radius larger than those presented in
x 4. However, the average size and the number density of
newly formed grains could not be much diﬀerent, so long as
the condensation time is on the order of 100 days.
3. FORMULATION AND CALCULATION OF
DUST FORMATION

3.1. Formulation of Nucleation and Grain Growth
In astrophysical environments, dust grains condense via
formation of condensation nuclei and their growth throughthe collisions of gaseous species in a cooling gas outﬂowing
from a star into interstellar space.
The nucleation rate is usually formulated under the condition of the presence of monomer molecule whose chemicalcomposition is the same as the condensate. However, forformation of compound grains of astrophysical interest,
such as silicates, no gaseous monomer molecule exists, and
it is impossible to derive rigorously the nucleation rate without specifying the chemical pathways and reaction constants. Nevertheless, even if no information on chemical
pathways and reaction constants is available for the formation of compound grains, as is discussed by Yamamoto
et al. (2001), the nucleation rate can be evaluated by applying the concept of the key species, introduced by Kozasa &Hasegawa (1987), when the net reaction rate is much larger
than the decay rate. Under the assumption that the key
species, deﬁned as the gaseous species with the least collision
frequency among the reactants, controls the kinetics ofnucleation and grain growth, the steady state nucleation
rate of the jth grain species, Jjs ðtÞ, is given by
"
#

  
l3j
2j 1=2 T 1=2
4s
Jj ðtÞ ¼ sj jj c1j exp ;Td
27 ðln Sj Þ2
m1j
ð3Þ
where sj is the sticking probability of the key species for thejth grain species, j is the volume of the condensate per molecule of the key species, and j is the surface energy. We
assume sj ¼ 1 in the calculations (see Hasegawa & Kozasa
1988; Kozasa et al. 1996; Chigai, Yamamoto, & Kozasa
1999 for the derivation of steady state nucleation rate). The
concentration and the mass of the key species for the jthgrain species are denoted by c1j and m1j , respectively. Td isthe temperature of the condensation nuclei, and T is the gas
temperature (see Kozasa et al. 1996). The factor j is a function of the partial gas pressures of reactants and products
except for the key species (Yamamoto et al. 2001), and weset j ¼ 1 in the calculations, partly because the appearanceof the factor is somehow related to the detailed chemical
reaction mechanism at the condensation and partly because
the factor can be negligible under the condition that the
number of molecules of the key species contained in the condensation nuclei is much larger than unity. The quantity lj ,representing the energy barrier for nucleation, is deﬁned byVol. 598

lj ¼ 4a20j j =kTd , with the hypothetical radius of conden1=3,sate per molecule of the key species a0j ¼ 3j =4
where k is Boltzmann’s constant. We assume that the temperature of the condensation nuclei is the same as the gas
temperature in what follows. The supersaturation ratio Sj ,taking into account the chemical reaction at the condensation of the jth grain species, is calculated byln Sj ¼ 

DGj0 Xþ
ij ln Pij ;
kT
i

ð4Þ

where DGj0 is the Gibbs free energy of formation of the jthgrain species from the reactants per molecule of the key species and Pij are the partial gas pressures of the reactant andproduct gas species. The stoichiometric coeﬃcients ij of thereactants and products are normalized to the key species; ij
is positive for a reactant and negative for a product (seeKozasa & Hasegawa 1987; Hasegawa & Kozasa 1988).
In the ejecta of supernovae, where the gas cooling time
and/or the collision time controlling the process of nucleation and grain growth are comparable to the dynamicaltimescale, the application of the steady state nucleation rate
is questionable. In this paper, following Gail, Keller, &Sedlmayr (1984), we employ the non–steady state
nucleation rate Jj ðtÞ evaluated by the equation@ Jj ð t Þ
1 
Jj ðtÞ  Jjs ðtÞ ;
ð5Þ
¼
@t j
;j jwith the relaxation time toward the steady state ; j deﬁnedbyðln Sj Þ2
kT 1=2
c1j ðtÞ
1; j ¼ 4a20j sj
2m1jlj
ðln Sj Þ2
ð6Þ
lj

2
1
and j ¼ coll;j rc; j =a0j , where coll;j is the collision time of
the key species and rc; j ¼ 2a0j lj =3 ln Sj is the critical radius
for the jth grain species.
Given the nucleation rate Jj ðtÞ at a time t, the process of
nucleation and grain growth is described by the two basic
equations in a frame comoving with gas. One is the equationof continuity for the key species, given byZ t
c1j ðtÞ
Jj ðt0 Þ 4 3
¼ 1  Y1j ¼
r ðt; t0 Þ dt0 ; ð7Þ
1
~c1j ðtÞ
c1j ðt0 Þ 3j j
te ~1¼ coll;jwhere te is the equilibrium time, deﬁned as the time at which
the supersaturation ratio Sj reaches unity without depletionof the key species, ~c1j is the nominal concentration of the
key species expected without depletion of the key species
due to nucleation and grain growth, Y1j ¼ c1j =~c1j represents
the degree of depletion of the key species due to nucleationand grain growth, and rj ðt; t0 Þ is the radius of a grain
nucleated at t0 and measured at t. Another is the equation ofgrain growth, given by
drj
kT 1=21
1¼ sj jc1j ðtÞ ¼ a0j coll;jðt Þ :
ð8Þ
2m1j
3
dt
By diﬀerentiating equation (7) with t subsequently, theintegral equation is reduced to the simultaneous ordinary

No. 2, 2003

DUST FORMATION IN POPULATION III SUPERNOVAE

equationsði Þ

dKj
Jj ðtÞ 4 iði1Þ drj¼
r þ iKj~c1j ðtÞ 3j c; j
dt
dt

ð9Þ

(for i ¼ 1 3) andð0Þ

dKj
Jj ðtÞ 4
¼:~c1j ðtÞ 3j
dt

ð10Þ

In principle, by solving equations (5), (8), (9), and (10),coupled with the temporal evolution of gas density ðtÞ andtemperature T ðtÞ with the number abundances of reactants,
the number density ngr; j ðtÞ and volume equivalent averageradius rgr; j ðtÞ of the jth grain species are calculated byngr; jð0Þ
¼ a30j Kj ðtÞ
~c1j

ð11Þ

and

791

Deneault, & Meyer (2001), and they have shown that carbon grains can condense even in the region of C=O < 1.
Also, the formation eﬃciency of SiO molecules is expected
to aﬀect the abundance and the size of Si- and/or Mgbearing grains. The eﬀect of the formation eﬃciency of COand SiO molecules on dust formation in the ejecta is
discussed in Appendix B.
The grain species, the chemical reactions at the condensations, and the basic data necessary for dust formation calculations are in Table 2. The Gibbs free energy for formationof a condensate from the reactants is approximated by
DGj0 =kT ¼ A=T þ B, and the numerical values of A and B
are evaluated by least-squares ﬁtting of the thermodynamic
data (Chase et al. 1985). It should be recognized here thatwhat kind of grain species really condenses in a gas with agiven elemental composition is determined by the competitive process of formation of each grain species, because formation of a grain species depletes the gaseous atoms andmolecules available for the condensation of other grains. So
in the calculations, the formation of all possible condensatesin Table 2 is taken into account simultaneously.

ð3Þ

r3gr; j ¼

K j ðt Þ
ð0Þ;

K j ðt Þ

ð12Þ

respectively. Furthermore, the size distribution function
fj ðrÞ of the newly formed jth grain species at a time t iscalculated byð0Þ

fj ðrÞdr ¼ a30j ~c1jdKj ðt0 Þ 0
dt ;
dt0

ð13Þ

since the grains with radii between r and r þ dr are nucleatedin the time interval between t0 and t0 þ dt0 .
3.2. Calculation of Dust Formation in theEjecta of Supernovae
The formation of dust grains in the ejecta of supernovae
is calculated by solving equations (5), (8), (9), and (10) withthe temporal evolution of gas density and temperature,
given the grain species, the chemical reaction at the condensation, and the abundance of the reactants. The formationof CO and SiO molecules prior to the formation of dust
grains was observed in the ejecta of SN 1987A (e.g., Bouchet& Danziger 1993). The formation of CO and SiO molecules
plays a crucial role in the formation of dust grains, because
CO molecules lock the oxygen atoms available for theformation of oxide grains and SiO molecules are considered
to be the starting molecules to form silicate grains.
As discussed by Liu & Dalgarno (1994, 1996), in theejecta of supernovae, these molecules are destroyed by
impact with energetic electrons and charge transfer
reactions with the ionized inert gaseous atoms, which are
created by the decay of radioactive elements whose abundances depend on the elemental composition, as well as on
the degree of the mixing in the ejecta. In this paper, to simplify the calculation of dust formation, we assume that theformation of CO and SiO molecules is complete, that is, thatno carbon-bearing grain condenses in the region of
C=O < 1 and no Si-bearing grain, except for oxide grains,condenses in the region of Si=O < 1. The eﬀect of incomplete formation of CO molecules has been extensively investigated by Clayton, Liu, & Dalgarno (1999) and Clayton,

4. THE RESULTS OF CALCULATIONSAND DISCUSSIONSIn this section, the results of the calculations of dust
formation in the ejecta of ordinary CCSNe and PISNe
are presented and discussed. The behaviors of nucleationand growth of dust grains are depicted in x 4.1. The
detailed results on the condensation times, average radii,
and size distribution functions of each grain species
formed in the unmixed and mixed ejecta of CCSNe andPISNe are presented and discussed in xx 4.2 and 4.3,
respectively. The dependence of the total mass of newly
formed grains and of the mass yield of each grain specieson the progenitor mass are given in xx 4.4 and 4.5,
respectively. The eﬀects of the explosion energy, as wellas the amount of 56Ni in the unmixed ejecta, on dust formation are investigated in Appendix A, and the eﬀect of
the formation eﬃciency of CO and SiO molecules isdiscussed in Appendix B.
4.1. The Behavior of Nucleation and Grain Growth
Figure 4 depicts the behaviors of nucleation and grain
growth as a function of time after the explosion at a locationof Mr ¼ 3:5 M in the O-Mg-Si layer, with the elemental
abundances relative to oxygen Si=O ¼ 2:97  102 ,Mg=O ¼ 8:25  102 , and Al=O ¼ 9:38  104 ; Figure 4a
is for the behaviors of nucleation rate Jj ðtÞ and the depletionof the key species Y1j , and Figure 4b is for the number density ngr; j and the average radius rgr; j . At this location, theexpected condensates are Al2O3 and Si- and/or Mg-bearing
grains. The key species is Al for Al2O3 and Mg or SiO for
Mg- and/or Si-bearing grains, depending on what kind ofgrain species really condenses.
As the gas cools down with time, it can be clearlyshown from Figure 4 that the nucleation rate rapidly
increases with increasing supersaturation ratio and thendecreases with the depletion of the key species due tograin growth; thus, the nucleation rate has a maximum
at a given time. This is a typical behavior of nucleationand grain growth in a cooling gas. The time at which thenucleation rate reaches the maximum is deﬁned as the

792

NOZAWA ET AL.

Vol. 598

TABLE 2Grain Species Considered in the Calculations

GrainsKey Species

Chemical Reactions

A/104
(K)

Fe(s) .........................
FeS(s).......................
Si(s) ..........................
Ti(s) .........................
V(s) ..........................
Cr(s) .........................
Co(s) ........................
Ni(s) .........................
Cu(s) ........................
C(s) ..........................
SiC(s) .......................
TiC(s).......................
Al2O3 (s)...................
MgSiO3 (s) ...............
Mg2SiO4 (s) ..............

Fe(g)
Fe(g), S(g)
Si(g)
Ti(g)
V(g)
Cr(g)
Co(g)
Ni(g)
Cu(g)
C(g)
Si(g), C(g)Ti(g), C(g)Al(g)Mg(g), SiO(g)Mg(g)
SiO(g)
SiO(g)
Mg(g)
Fe(g)
Fe(g)

Fe(g)!Fe(s)Fe(g) + S(g)!FeS(s)Si(g)!Si(s)Ti(g)!Ti(s)V(g)!V(s)Cr(g)!Cr(s)Co(g)!Co(s)Ni(g)!Ni(s)Cu(g)!Cu(s)C(g)!C(s)Si(g) + C(g)!SiC(s)Ti(g) + C(g)!TiC(s)2Al(g) + 3O(g)!Al2O3 (s)Mg(g) + SiO(g) + 2O(g)!MgSiO3 (s)2Mg(g) + SiO(g) + 3O(g)!Mg2SiO4 (s)2Mg(g) + SiO(g) + 3O(g)!Mg2SiO4 (s)SiO(g) + O(g)!SiO2 (s)Mg(g) + O(g)!MgO(s)3Fe(g) + 4O(g)!Fe3O4 (s)Fe(g) + O(g)!FeO(s)4.84180
9.31326
5.36975
5.58902
6.15394
4.67733
5.03880
5.09130
3.97955
8.64726
14.8934
16.4696
18.4788
25.012918.6200
37.2400
12.6028
11.9237
13.2889
11.1290

SiO2 (s) .....................
MgO(s) ....................
Fe3O4 (s) ..................
FeO(s) ......................

Bj(ergs cm2)

References for j

a0j
(Å)

16.5566
30.777117.4349
16.607117.870216.7596
16.8372
17.1559
14.9083
19.042237.3825
37.230145.3543
72.001552.4336104.872
38.150733.159339.1687
31.9850

1800
380
800
1510
1697
1880
1936
1924
1300
1400
1800
1242
690
400
436
436
605
1100
400
580

1
2
3
3
3
3
3
3
3
4
5
6
7
8
8
8
7
7
7
7

1.411
1.932
1.684
1.615
1.490
1.421
1.383
1.377
1.412
1.281
1.702
1.689
1.718
2.319
2.055
2.589
2.080
1.646
1.805
1.682

Notes.—The key species is deﬁned as the gaseous species with the least collisional frequency among the reactants. The subscripts ‘‘ (s) ’’ and ‘‘ (g) ’’ denotecondensate and gas species, respectively. The Gibbs free energy DGj0 for formation of the condensate from the reactants per molecule of the key species isapproximated by DGj0 =kT ¼ A=T þ B, and the numerical values A and B are evaluated by least-squares ﬁtting of the thermodynamic data (Chase et al.
1985) in the range of temperatures of interest; j is the surface energy of the condensate, and a0j is the hypothetical radius of the condensate per molecule ofthe key species, whose values are calculated from the molar volumes tabulated by Robie & Waldbaum 1968.
References.—(1) Elliott, Gleiser, & Ramakrishna 1963; (2) Kozasa & Hasegawa 1988; (3) Elliott & Gleiser 1960; (4) Tabak et al. 1975; (5) Kozasa et al.
1996; (6) Rhee 1970; (7) Overbury, Bertrand, & Somorjai 1975; (8) Boni & Derge 1956.

condensation time. At this location, Al2O3 grains condense ﬁrst at day 400, then Mg2SiO4 grains condense,
consuming the key species SiO, at day 421, and ﬁnallythe remaining Mg atoms are locked into MgO grains at
day 439. Also, it should be noted that each grain grows
to the ﬁnal radius in a very short time interval: less than
20 days after the condensation time.
Furthermore, we can see from Figure 4 that a smaller
abundance of the key species leads to a larger number density of condensation nuclei, with a broad peak of the nucleation rate, and results in a small average radius. Thus, theabundance of the key species is very sensitive to the number
density and average radius of newly formed grains, which iswell reﬂected by the size distribution function of grains
given in Figure 5, in Figure 5a at Mr ¼ 3:5 M and in Figure
5b at Mr ¼ 4:0 M, where the abundances of Si, Mg, and Al
are more than an order of magnitude smaller than at
Mr ¼ 3:5 M: Si=O ¼ 1:50  104 , Mg=O ¼ 6:40  103 ,and Al=O ¼ 3:95  105 . At Mr ¼ 4:0 M, the smaller
abundance of the key species SiO for the formation of
Mg2SiO4 grains results in a larger number of condensation
nuclei with a very broad peak of the nucleation rate, corresponding to a widespread size distribution, which is true forthe formation of Al2O3 grains. On the other hand, the size
distribution function of MgO grains at Mr ¼ 4:0 M isalmost the same as that at Mr ¼ 3:5 M, because the abundance of Mg, being the key species for the formation of
MgO and remaining after the formation of Mg2SiO4, is onlya factor of 4 smaller at Mr ¼ 4 M than at Mr ¼ 3:5 M.
Note that the size distribution function of a grain species
formed at a location is approximately lognormal so long as
the average size is larger than 0.01 lm.

4.2. Dust Formation in the Unmixed Ejectaof CCSNe and PISNe
In the unmixed ejecta of supernovae, a variety of grain
species condense in each layer, corresponding to the diﬀerence in the elemental composition. Figures 6a and 6b show
the condensation times of dust grains formed in theunmixed ejecta of C20 and P170, respectively. At ﬁrst, carbon grains condense in the He layer, which is followed by
the condensation of Al2O3 and Mg silicates (Mg2SiO4 andMgSiO3) in the oxygen-rich layer, MgO in the O-Mg-Si
layer, SiO2 in the O-Si-Mg layer, Si and FeS in the Si-S-Fe
layer, and Fe in the innermost Fe-Si-S layer in this sequence.
Note that in the O-Si-Mg layer, SiO2 grains condense from
SiO molecules left over after the formation of Mg silicate
grains; MgO grains condense from Mg atoms remaining in
the O-Mg-Si layer. The major grain species are Mg2SiO4
and MgO in the O-Mg-Si layer and MgSiO3 and SiO2 in the
O-Si-Mg layer. In the ejecta of C20, Si grains condense in
the innermost region of the He layer because a signiﬁcantamount of Si atoms are present around the interface
between the He layer and the oxygen-rich layer (see Fig. 1a),but no SiC grains condense in this region. Anyway, in theunmixed ejecta of C20 and P170, the grain species condensed in the ejecta are the same. Also, metallic Cr and Ni
grains do not condense signiﬁcantly in the Fe-Si-S layer ofboth ejecta because of low number density, as well as thehigh energy barrier for nucleation.
The condensation time generally depends not only on thetemporal evolution of gas temperature, but also on theconcentration of the key species at the formation site. On
the other hand, given the time evolution of gas temperature,

No. 2, 2003

DUST FORMATION IN POPULATION III SUPERNOVAE

Fig. 4.—Behaviors of nucleation and growth of each grain species condensed at Mr ¼ 3:5 M in the O-Mg-Si layer of the unmixed ejecta for C20.
(a) Time evolution of the nucleation rate Jj ðtÞ normalized to ~c1j ðtÞ (solidline) and the depletion of key species Y1j (dashed line). (b) Time evolution of
the number density ngr;j normalized to ~c1j ðtÞ (solid line) and the averagegrain radius rgr;j (dot-dashed line). The number abundances of dust-forming
elements relative to oxygen at this location are Si=O ¼ 2:97  102 ,Mg=O ¼ 8:25  102 , and Al=O ¼ 9:38  104 . [See the electronic editionof the Journal for a color version of this ﬁgure.]

the condensation temperature, deﬁned as the gas temperature at the condensation time, depends only on the concentration of the key species; the condensation temperature
decreases with decreasing the concentration. In the unmixed
ejecta of CCSNe and PISNe, the condensation temperaturedoes not heavily depend on the progenitor mass: 1900 K
for carbon, 1600–1700 K for Al2O3, 1400–1500 K for Mg
silicates, 1350–1450 K for MgO, 1300–1400 K for SiO2,
1100–1200 K for Si, 1000–1100 K for FeS, and 800–850 K
for Fe. The reason is as follows: as described in x 2, the concentration of the key species is a few times larger in theejecta of PISNe than in that of CCSNe, and the elemental
composition in each layer and the gas density at a given time
are not signiﬁcantly diﬀerent between CCSNe and PISNe.
On the other hand, the gas temperature in the ejecta of
PISNe is higher than that of CCSNe. Thus, as can be seenfrom Figure 6, the condensation time of each grain species

793

Fig. 5.—Size distribution function of each grain species formed at a location in the O-Mg-Si layer of the unmixed ejecta for C20 at (a) Mr ¼ 3:5 M
and (b) Mr ¼ 4 M, with Si=O ¼ 1:50  104 , Mg=O ¼ 6:40  103 , and
Al=O ¼ 3:95  105 . [See the electronic edition of the Journal for a color
version of this ﬁgure.]

in the ejecta of PISNe is delayed about 150 days, in comparison with that of CCSNe. Anyway, dust grains condense in
the ejecta of ordinary CCSNe around day 300–600 after the
explosion and around day 500–800 in the ejecta of PISNe,
which is almost independent of the progenitor mass.
Figures 7a and 7b show the average radius of each grain
species in the ejecta of C20 and P170, respectively. The average radius of each grain species well reﬂects the concentration of the key species at the condensation time and heavily
depends on the elemental composition and the gas density
at the formation site in the ejecta. In the ejecta of C20, theaverage radii of Fe and Si condensed in the innermost region
are relatively large: about 0.2 lm for Fe and 0.4 lm for Si.
The range of the average radius spans about an order ofmagnitude for carbon, Al2O3, SiO2, and FeS, whose maximum radii are 0.4, 0.004, 0.1, and 0.06 lm, respectively. The
range spans more than 2 orders of magnitude for MgO,
Mg2SiO4, and MgSiO3, with maximum radii of 0.2, 0.1, and
0.5 lm, respectively. In the ejecta of P170, the averageradius of each grain species is a little smaller than that in the

794

NOZAWA ET AL.

Fig. 6.—Condensation times of dust grains formed in the unmixed ejectafor (a) C20 and (b) P170. [See the electronic edition of the Journal for a color
version of this ﬁgure.]ejecta of C20, but the diﬀerence is not signiﬁcant, except for
Fe, FeS, and Si condensed in the Fe-Si-S layer, where thegas density is substantially lower than that in the ejecta of
C20 (see Fig. 2). The average radii of dust grains formed in
the unmixed ejecta are limited to less than 1 lm.
The size distribution functions of newly formed grains in
the ejecta of C20 and P170 are given in Figures 8a and 8b,
respectively. As is shown in x 4.1, the size distribution function at a location is approximately lognormal for a grain
species with an average radius larger than 0.01 lm. However, the size distribution function of a grain species
summed up over the formation region does not keep theoriginal shape at the formation site, reﬂecting the diﬀerencein the elemental composition and gas density in the formation region. Generally, for a grain species whose formation
region is widespread in the ejecta and whose average radiusspans more than an order of magnitude, the size distributionfunction completely deviates from the original lognormal
shape, which is true for MgO, Mg2SiO4, MgSiO3, and FeS
grains. Although the average size of carbon grains ranges

Vol. 598

Fig. 7.—Average radii of dust grains formed in the unmixed ejecta for(a) C20 and (b) P170. [See the electronic edition of the Journal for a color
version of this ﬁgure.]from 0.01 to 0.4 lm in the ejecta of C20, the size distributionfunction appears lognormal because the large grains formedin the innermost region of the He layer do not contribute to
the size distribution function, as a result of their small numbers relative to the small grains condensed in the outer He
layer. The size distribution functions of SiO2 and Fe in theejecta of C20 are approximately lognormal. Si grains show abimodal distribution function with large grains condensed
inside the Si-S-Fe layer and smaller ones condensed aroundthe interface between the He and O-Mg-Si layers. In theejecta of P170, the behavior of the summed-up size distribution function of each grain species is the same as that in theejecta of C20. However, in the ejecta of P170, the bimodalityof Si stems from the wide span of average radii arising fromthe density variation within the Fe-Si-S layer.
The size distribution function summed up over all grain
species formed within the He core is shown by the thickcurve in Figure 8. The straight lines represent power-law
formulae with indexes of  ¼ 3:5 and 2.5. As can be seenfrom Figure 8, in both cases the size distribution summed
up over all grain species is well ﬁtted with a power-law

No. 2, 2003

DUST FORMATION IN POPULATION III SUPERNOVAE

Fig. 8.—Size distribution function of each grain species summed up overthe formation region in the unmixed ejecta for (a) C20 and (b) P170. The
thick curve represents the size distribution function summed up over all
grain species, and the straight lines indicate power-law formulae withindexes of  ¼ 2:5 and 3.5. [See the electronic edition of the Journal for acolor version of this ﬁgure.]

formula whose index is 3.5 for the larger size and 2.5 forsmaller one, which is diﬀerent from the conventional MRN
size distribution function (Mathis, Rumpl, & Nordsieck1977) used in the astrophysical literature. The radius at thecrossover point is 0.06 lm for C20 and 0.02 lm for P170.
Although in this subsection only one model each for CCSNe
and PISNe is presented, the behaviors of dust formationand the average size, as well as the size distribution function
described above, are almost the same for the other models,being independent of the progenitor mass.
4.3. Dust Formation in the Uniformly Mixed Ejecta of
CCSNe and PISNe
This subsection presents the results of the dust formation
calculations in the ejecta with the elemental composition
uniformly mixed within the He core, as an extreme case. In
the calculations we assume that the density structure in theejecta is not aﬀected by mixing, since the numerical simulation suggested that the Rayleigh-Taylor instability produces

795

Fig. 9.—Condensation times of dust grains formed in the mixed ejectafor (a) C25 and (b) P200. [See the electronic edition of the Journal for a color
version of this ﬁgure.]

the clumpy structure, but the gas density does not changemuch on average (Hachisu et al. 1990).
Under the assumption that the formation of CO
molecules is complete, only oxide grains condense in theuniformly mixed ejecta, because the ejecta is oxygen-rich.
Figure 9 shows the condensation time of each grain species
formed in the ejecta, Figure 9a for C25 and Figure 9b for
P200. In both models, Al2O3, Mg silicates (Mg2SiO4 andMgSiO3), SiO2, and Fe3O4 grains condense in that order.
Unlike in the unmixed ejecta, no MgO grain condenses,
because Si > Mg in the uniformly mixed ejecta, and all ironatoms are locked into Fe3O4 grains. Except for the innerand outermost regions of P200, the condensation temperature of each dust grain is almost the same within the He
core: 1550–1600 K for Al2O3, 1450 K for Mg silicates, 1350
K for SiO2, and 1300–1350 K for Fe3O4.
In the ejecta of C25, the gas temperature decreases more
rapidly in the outer region than in the inner region ofMr  6:8 M, where the gas temperature at a time t isalmost the same. Thus, well reﬂecting the temperature structure, in the ejecta of C25, Al2O3 grains start to condense in

796

NOZAWA ET AL.

the outer edge of the He core at day 330 after the explosion,
Mg silicate grains at day 350, SiO2 grains at day 364, andFe3O4 at day 370. In the region of Mr  6:8 M, each grain
species condenses at almost the same time: Al2O3 aroundday 385, Mg silicates around day 400, SiO2 around day 415,
and Fe3O4 around day 420. Unlike the formation of dust
grains in the unmixed ejecta, all dust grains condense within
100 days after the onset of dust formation. On the other
hand, in the ejecta of P200, the condensation time of eachgrain species increases with a decrease in the mass coordinate, because the gas density gradually decreases withdecreasing Mr : Al2O3 grains from day 450 to day 560, Mg
silicate grains from day 450 to day 570, SiO2 grains fromday 465 to day 583, and Fe3O4 grains from day 470 to day
590. In the ejecta of P200, the condensation times of dust
grains are about 150 days later than those in the ejecta of
C25, since the gas temperature at a given time is higher thanthat of C25.
As shown in Figures 10a for C25 and 10b for P200, therange of average radius of each grain species spans less thanan order of magnitude throughout the He core. Given theVol. 598

same elemental composition within the He core, the averageradius depends not only on the density structure of the gas,
but also on the time evolution of the gas temperature; in theouter region of the He core, the gas density is lower than in
the inner region, but the gas temperature decreases faster inthe outer region, which compensates for the decrease in gas
density at the condensation time. In the inner region of
P200, the gas temperature is almost the same, being independent of Mr , and the variation of the gas density is lessthan an order of magnitude. In the mixed ejecta of C25, theaverage radii of SiO2 and Mg silicates are a few hundredthsof a micron, with a maximum of 0.1 lm for SiO2, 0.05 lm
for Mg2SiO4, and 0.07 lm for MgSiO3. The average radii of
Fe3O4 and Al2O3 are small: several tens of angstroms for
Fe3O4 and several angstroms for Al2O3, consistent with theresults of Todini & Ferrara (2001). Despite the fact that thegas temperature in the mixed ejecta of P200 is higher thanthat in C25, the average radius of each grain species formedin the region of Mr > 40 M is almost the same as that in
the ejecta of C25, because the abundance of the key speciesof dust grains is a few times larger in P200 than in C25. The
low density in the region of Mr < 40 M of P200 (see Fig.
2b) results in a smaller average radius for each grain species.
Figure 11 shows the size distribution function of each
dust grain formed in the mixed ejecta, Figure 11a for C25
and Figure 11b for P200. Except for Al2O3 grains with avery small average radius, in both models the size distribution functions of Mg2SiO4, MgSiO3, SiO2, and Fe3O4 grains
tend to be approximately lognormal because of the narrowsize range of each grain species. The size distribution function summed up over all grain species is also ﬁtted with apower-law formula, whose index is 3.5 for radii larger
than 0.004 lm in P200 and 2.5 for radii smaller than 0.02
lm in C25. The deviation from the power-law formula isdistinctive for the larger size in model C25, because only
SiO2 grains with a lognormal distribution function contribute to the larger size. However, this is an untypical case forthe behavior of the summed-up size distribution function.
Here it should be addressed that, irrespective of the progenitor mass, the size distribution function summed up over all
grain species newly formed in the ejecta with and without
mixing is approximated by power-law distribution whose
index is 3.5 for larger radii and 2.5 for smaller one,
although the radius at the crossover point depends on themodel of supernovae, with a range from 0.004 to 0.1 lm.
4.4. The Amount of Freshly Formed Dust Grains

Fig. 10.—Average radii of dust grains formed in the mixed ejecta for(a) C25 and (b) P200. [See the electronic edition of the Journal for a color
version of this ﬁgure.]

Figure 12 shows the total mass of dust grains produced in
the ejecta versus the progenitor mass, Figure 12a for CCSNeand Figure 12b for PISNe, where triangles denote the resultsof calculations for the unmixed ejecta and squares those forthe mixed ejecta. In Figure 12a, as a reference, the results ofcalculations in the unmixed ejecta of HNe are also plotted
as crosses for H25A and H30A and as circles for H25B and
H30B. The straight lines in Figures 12a and 12b are for the
linear least-squares ﬁts to the calculated mass for ordinary
CCSNe and linearly connect the two data points for PISNe,
respectively: the solid line for the unmixed ejecta and thedashed line for the mixed ejecta.
The total mass of newly formed grains increases withincreasing progenitor mass in both the unmixed and mixed
ejecta of CCSNe and PISNe. In the unmixed ejecta of ordinary CCSNe, the total dust mass is 0.57 M for C20 andNo. 2, 2003

DUST FORMATION IN POPULATION III SUPERNOVAE

Fig. 11.—Same as Fig. 8, but in the uniformly mixed ejecta for (a) C25
and (b) P200. [See the electronic edition of the Journal for a color version ofthis ﬁgure.]

1.32 M for C30. It is generally expected that the dust massis larger in the mixed ejecta than in the unmixed ejecta,
because many more oxygen atoms are locked into dust
grains in the mixed ejecta. However, in the mixed ejecta of
C13, the total mass of dust grains is 0.22 M, a little smaller
than the dust mass 0.23 M in the unmixed ejecta. The reason is that in the mixed ejecta of C13, the formation of CO
molecules consuming the oxygen atoms available for oxide
grains limits the amount of oxide grains; in this case, a small
amount of Si and FeS grains condense instead of SiO2 andFe3O4 grains. Also in model C25, the dust mass in the mixed
ejecta is not much diﬀerent from that in the unmixed ejecta,
because in that model the comparatively massive He layer
somehow depresses the increase in the amount of massavailable for the formation of oxide grains within the oxygen-rich layer.
In the unmixed ejecta of HNe with the sameM ð56 NiÞ ¼ 0:07 M as CCSNe, H25A and H30A produce
almost the same dust mass as C25 and C30, respectively. On
the other hand, with a deep mass cut corresponding toM ð56 NiÞ ¼ 0:7 M, H25B and H30B produce about 0.7 M
more dust than C25 and C30, respectively. In the ejecta of

797

Fig. 12.—Total mass of dust grains formed in the unmixed (triangles)
and mixed (squares) ejecta of CCSNe and PISNe vs. progenitor mass Mpr
for (a) CCSNe and (b) PISNe. Lines indicate the least-squares ﬁts to thecalculated mass in the unmixed (solid line) and mixed (dashed line) ejecta of
CCSNe in (a) and connect the calculated masses for PISNe in (b). Also, in(a) the total mass of dust grains produced in the unmixed ejecta of HNe isplotted as crosses (H25A and H30A) and circles (H25B and H30B). [See theelectronic edition of the Journal for a color version of this ﬁgure.]

the ordinary CCSNe, the ratio of dust mass to progenitor
mass is 0.02–0.05, which increases with increasing progenitor mass. Although the total mass of newly formed grains
depends on the detailed elemental composition in the ejecta,
as well as the position of mass cut, in general the total mass
produced in the mixed ejecta becomes larger than that in theunmixed ejecta with increasing Mpr.
PISNe, for which the progenitors are very massive andare completely disrupted, produce much more dust than
CCSNe. For model P200, the total mass of newly formed
grains is 40 M in the unmixed ejecta and 60 M in themixed ejecta. The mass ratio of newly formed grains to
the progenitor mass for PISNe reaches 0.15–0.3 at least, andthe ratio increases with increasing progenitor mass. Therefore, in the early universe, much more dust could be produced and injected into interstellar space if very massive
stars populated the ﬁrst generation of stars.

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Vol. 598

4.5. The Mass Yield of Each Grain Species
Figures 13 and 14 present the mass yield of each grain species versus the progenitor mass Mpr in the unmixed ejecta
and mixed ejecta, respectively, Figures 13a and 14a for
CCSNe and Figures 13b and 14b for PISNe. The smooth
curves in Figures 13a and 14a are for the least-squares spline
ﬁts to the calculated values for CCSNe, and those in Figures
13b and 14b represent, in loglinear scale, the straight lines
connecting the two calculated values in linear scale for
PISNe. Thus, for PISNe the mass yield for a given
progenitor mass should be taken as a nominal value.
In the unmixed ejecta of CCSNe (Fig. 13a), with increasing Mpr the mass yield of carbon grains, apart from a small
ﬂuctuation, somehow increases up to around Mpr ¼ 25 M
and then decreases, while the yields of other grain species,
except for Fe, increase. The grain species contributing to thetotal dust mass heavily depend on the progenitor mass forFig. 14aFig. 13aFig. 13bFig. 13.—Mass yield of each dust grain formed in the unmixed ejecta for(a) CCSNe and (b) PISNe. In (a) the curves indicate the least-squares spline
ﬁts to the calculated yields, and in (b) they represent the straight lines
connecting the calculated yields in linear scale. [See the electronic edition of
the Journal for a color version of this ﬁgure.]

Fig. 14b
Fig. 14.—Same as Fig. 13, but in the mixed ejecta for (a) CCSNe and(b) PISNe. [See the electronic edition of the Journal for a color version of thisﬁgure.]

Mpr  28 M; in the ejecta with Mpr < 15 M, the major
grain species in mass are carbon, Fe, and Si, with an averageradius 0.1 lm. With increasing Mpr, the masses of theoxygen-rich layer and the Si-S-Fe layer increase. Therefore,
the yields of Si and Mg2SiO4 increase, and the mass of FeS
also increases, together with a decrease in the mass of Fe.
Around Mpr ¼ 20 M, Mg2SiO4, Si, and FeS grains tend to
dominate the mass abundance. Around Mpr ¼ 25 M, themajor grain species are Mg2SiO4, Si, and C. The mass abundances of FeS, SiO2, and MgO grains are larger than that ofcarbon grains for Mpr 28 M, and Mg2SiO4 and Si are
the major grain species, followed by MgO, SiO2, and FeS.
Although the masses of Al2O3 and MgSiO3 grains increase
with Mpr, their mass fractions are limited to less than 102.
The masses of Mg2SiO4 and MgO grains per supernova
are almost constant in the unmixed ejecta of PISNe, being
almost independent of the progenitor mass (see Fig. 13b):
M ðMg2 SiO4 Þ ’ 6 M and slightly decreases with increasing
Mpr, and MðMgOÞ ¼ 1:4 M. When the progenitor mass issmaller than 160 M, Mg2SiO4, Si, and SiO2 are the major

No. 2, 2003

DUST FORMATION IN POPULATION III SUPERNOVAE

grain species contributing to the total mass. In contrast to
the case in unmixed ejecta of CCSNe, the mass fraction ofcarbon grains is less than 5  102 , although the mass ofcarbon grains increases with increasing Mpr. It should be
noted here that for Mpr > 160 M, the most abundant grain
species in the unmixed ejecta of PISNe is Si, followed by
SiO2, because the masses of the Si-S-Fe and O-Si-Mg layers
increase with increasing Mpr. Also, the mass fractions of theiron-bearing grains (Fe and FeS) produced within the Si-SFe and Fe-Si-S layers are larger than those of the Mg2SiO4
grains for Mpr > 220 M.
In the mixed ejecta of CCSNe (see Fig. 14a), SiO2 and
Mg2SiO4 grains mainly contribute to the total dust mass for
Mpr  15 M. The mass of Fe3O4 for Mpr 20 M is 0.09
M, being almost independent of the progenitor mass. With
increasing Mpr, the mass of Mg2SiO4 increases and becomes
the same as that of SiO2 at Mpr ¼ 30 M. The dust mass ofmassive CCSNe with Mpr 25 M is dominated by SiO2
and Mg2SiO4 grains. Although the yield of Al2O3 increases
with increasing Mpr and tends to be the same mass as Fe3O4,
the mass fraction of Fe3O4 is limited to less than 0.1 for
Mpr 20 M, and that of Al2O3 is limited to less than 0.02
at most. In the mixed ejecta of PISNe (see Fig. 14b), SiO2 isthe most abundant grain, regardless of the progenitor mass.
As in the mixed ejecta of CCSNe, around the low-mass endof the progenitor mass of PISNe, SiO2 and Mg2SiO4 contribute to the total mass. As the progenitor mass increases,
the masses of MgSiO3 and Fe3O4 increase and approachalmost the same mass, while the amount of Mg2SiO4
decreases. The total mass of dust grains is dominated by
SiO2, MgSiO3, and Fe3O4, whose mass fractions are 0.7,
0.15, and 0.15, respectively.

5. SUMMARY

We investigate dust formation in the ejecta of Population
III core-collapse and pair-instability supernovae, employinghydrodynamic models and the results of nucleosynthesis
calculations by Umeda & Nomoto (2002). The time evolution of gas temperature in the ejecta is calculated by solving
the radiative transfer equation together with the energy
equation, taking into account the energy deposition fromradioactive elements. Two extreme cases are considered forthe elemental composition in the He-core: the unmixed caseand the uniformly mixed case. In order to clarify what kindof grain species condenses in the ejecta, the formation of all
possible condensates is calculated simultaneously by applying a theory of non–steady state nucleation and grain
growth, under the assumption of complete formation of COand SiO molecules. The results of the calculations are
summarized as follows:
1. The species of newly formed dust grains are aﬀected by
the elemental composition in the ejecta. In the unmixed
ejecta, a variety of grain species condense, corresponding to
the elemental composition at the formation site: C and Si in
the He layer; Al2O3, Mg2SiO4, and MgSiO3 in the oxygenrich layer (the O-Mg-Si and O-Si-Mg layers); MgO in the OMg-Si layer; SiO2 in the O-Si-Mg layer; Si and FeS in the SiS-Fe layer; and Fe in the Fe-Si-S layer. On the other hand,
only oxide grains, such as Al2O3, Mg2SiO4, MgSiO3, SiO2,
and Fe3O4, condense in the mixed ejecta. The main speciesof newly formed grains do not depend on the progenitor
mass.

799

2. The average size of newly formed dust grains strongly
depends on the concentration of the key species at the condensation time and at the formation site. In the unmixed
ejecta, the range of average radius of each grain species
spans a few orders of magnitude, depending on the grain
species and the formation region. On the other hand, in themixed ejecta, the average radius of each grain species is ofalmost the same order of magnitude in the entire region
within the He core, reﬂecting the uniform elemental composition. The average radius of each grain species in both theunmixed and the mixed ejecta does not strongly depend on
the progenitor mass and the explosion energy, but it isaﬀected by the amount of 56Ni in the ejecta. No dust grainscondense with an average radius larger than 1 lm in theejecta of Population III supernovae considered in thispaper.
3. The size distribution function of a grain species condensed at a location in the ejecta is almost lognormal so longas the average radius is larger than 0.01 lm. The size distribution function summed up within the He core is lognormal
for carbon, Fe, and SiO2 grains condensed in the narrow,conﬁned region of the unmixed ejecta; in the mixed ejecta,
except for Al2O3 grains, the summed-up size distributionfunction is lognormal. The size distribution function
summed up over all grain species is approximated by apower-law formula with an index of 3.5 for the larger radiiand 2.5 for the smaller ones for both the unmixed and themixed cases, being almost independent of the progenitor
mass; the radius at the crossover point ranges from 0.004 to
0.1 lm, depending on the model of supernovae.
4. The total mass of newly formed dust grains increases
with increasing progenitor mass and is generally muchlarger in the mixed case than in the unmixed case. The total
mass of dust grains is 2%–5% of the progenitor mass for
CCSNe with 13M  Mpr  40 M and 15%–30% of theprogenitor mass for PISNe with 140M  Mpr  260 M.1
Thus, if PISNe populated the early universe, a large amountof dust would have been injected into the primordial interstellar medium.
5. The mass yield of each grain species depends on theprogenitor mass. C, Fe, and Si are the major grain species in
the unmixed ejecta of CCSNe with Mpr < 15 M. With
increasing Mpr, Si and Mg2SiO4 grains become dominant,and Mg2SiO4 grains are the most abundant for Mpr 24
M. In the unmixed ejecta of PISNe, the most abundantgrain is Mg2SiO4 for Mpr < 160 M; for Mpr 160 M themost abundant grain is Si, followed by Mg2SiO4, SiO2, FeS,
and Fe. In the mixed ejecta of CCSNe, SiO2 and/or
Mg2SiO4 are the major grain species contributing to thetotal mass. Being independent of the progenitor mass, inthe mixed ejecta of PISNe, SiO2 is the most abundantspecies, followed by Mg2SiO4 for Mpr < 160 M and
MgSiO3 and Fe3O4 for Mpr 170 M.
In the calculations, we treat only the formation of homogeneous grains, taking into account the chemical reactions
at the condensation, but we do not consider the possibilityof heterogeneous nucleation on the surface of precondensed
grains. In fact, heterogeneous grains consisting of metal
carbide cores and graphite mantles have been observed in
1 After submitting this paper on 2003 June 28, we received a preprint
from A. Ferrara, who obtained results basically similar to ours for the massof the dust produced per PISN.

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NOZAWA ET AL.

the Murchison meteorite (Bernatowicz et al. 1991, 1996).
Also, we must consider the possibility of the formation ofcomposite grains, such as Fe-Ni-Cr alloys, and thus whether
the metal elements Ni, Co, and Cr, with relatively smallabundances, can be locked into dust grains in the ejecta ofsupernovae. Also, as is discussed in Appendix B, it is very
important to investigate the eﬀect of the formation eﬃciency of CO and SiO molecules on the formation of dust
grains, in order to clarify what kind of dust grains reallycondenses in the ejecta of supernovae. These subjects are left
for future work.
Anyway, dust grains in the early universe play a criticalrole in the formation and evolution history of stars and galaxies. The investigation of the nature of dust grains in theearly universe is essential, not only to investigating the evolution and structure of the early universe, but also to deducing the SFR and the IMF during the evolution of theuniverse from the relevant observations, because how muchVol. 598

dust grains absorb stellar light and reemit it by thermal radiation depends on the chemical composition, size distribution, and amount of dust grains residing in interstellar spacein galaxies and in the intergalactic medium. The evolutionof dust grains is determined by the balance between theirproduction in the ejecta of supernovae and their destruction
by interstellar shock, as well as by the reverse shock penetrating into the ejecta. The eﬃciency of destruction by shock
depends on the chemical composition and size of the dust
grains. Therefore, the results of the calculations presented
in this paper can be used as the basis to investigate theevolution of dust grains in the early universe.
The authors thank the anonymous referee for criticalcomments, which improved the manuscript. This work has
been supported in part by a Grant-in-Aid for ScientiﬁcResearch from the Japan Society for the Promotion of
Sciences (13640229).

APPENDIX A
DUST FORMATION IN THE EJECTA OF HYPERNOVAE
HNe, with explosion energies larger than 1052 ergs, are proposed to reproduce the behavior of the observed early light curveof SN 1998bw (Iwamoto et al. 1998; Woosley, Eastman, & Schmidt 1999). Also, the analysis of the light curve of SN 1998bw(Nakamura et al. 2001) led to the conclusion that the mass of 56Ni in the ejecta is 0.4 M, which is larger than the typical valueof 0.07 M in the ejecta of ordinary CCSNe. Furthermore, Umeda & Nomoto (2002) have claimed that primordial HNe well
reproduce the observed elemental composition in metal-poor stars: [Zn/Fe] increasing with decreasing [Fe/H] for½Fe=H  2:5 (Primas et al. 2000). In this appendix, we perform calculations of dust formation in the ejecta of Population III
HNe to investigate the eﬀect of explosion energy, as well as the amount of 56Ni in the ejecta, on dust formation. We consider
two models: one is H30A, with Mpr ¼ 30 M, Eexp ¼ 3  1052 ergs, and M ð56 NiÞ ¼ 0:07 M, and the other is H30B, for which
Mpr and Eexp are the same as for H30A, but M ð56 NiÞ ¼ 0:7 M.
Figure 15a shows the condensation time of each dust grain in the unmixed ejecta of H30A, with an explosion energy 30 times
that of ordinary CCSNe but with M ð56 NiÞ the same. The grain species formed in the ejecta and their condensation sequences
are almost the same as in CCSNe and PISNe, except for Fe; the reason is that in this model, with an explosion energy of3  1052 ergs, the innermost Fe-Si-S layer is not ejected, because of a shallow mass cut caused by adjusting M ð56 NiÞ in theejecta to 0.07 M. The condensation time of each grain species is about 200 days earlier than in CCSNe, because the gas
temperature quickly decreases as a result of the high explosion velocity and low gas density in the ejecta. However, as can beseen from Figure 15b, the average radius of each dust grain formed in the ejecta is almost the same as that in the ejecta of
CCSNe, because the early condensation time compensates for the decrease in gas density at the condensation time.

Fig. 15.—(a) Condensation times and (b) average radii of dust grains formed in the unmixed ejecta of H30A. [See the electronic edition of the Journal for acolor version of this ﬁgure.]

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DUST FORMATION IN POPULATION III SUPERNOVAE

801

Fig. 16.—Same as Fig. 15, but for the unmixed ejecta of H30B. [See the electronic edition of the Journal for a color version of this ﬁgure.]With M ð56 NiÞ ¼ 0:7 M and a higher gas temperature at a given time than in H30A, as is shown in Figure 16a, thecondensation time of each grain species in the ejecta of H30B is about 150 days later than that of H30A. Therefore, the averageradii of dust grains formed in the ejecta are a factor of 4–10 smaller than those in H30A (see Fig. 16b), depending on theconcentration of the key species at the formation site. In contrast to the case in H30A, iron grains condense in the innermost
Fe-Si-S layer, reﬂecting the deep mass cut at Mr ¼ 4:33 M.
From the results of the calculations, we can conclude here that the explosion energy, as well as the amount of 56Ni in theejecta, strongly aﬀects the time evolution of the gas temperature. In the ejecta with a ﬁxed amount of 56Ni, a higher explosion
energy results in an early condensation time, because the gas temperature quickly decreases as a result of the low gas density.
However, the average radius of dust grains is not heavily aﬀected, because the early condensation time compensates for thedecrease in gas density at the formation site. On the other hand, the mass of 56Ni in the ejecta strongly aﬀects the average radiusof dust grains formed in the ejecta with the same explosion energy. With increasing M ð56 NiÞ, the more delayed condensation
time decreases the concentration of key species at the site of dust formation, which results in a smaller size of dust grains.
Finally, it should be pointed out here that the amount of 56Ni in the ejecta also aﬀects the total mass of dust grains formed in
the ejecta, as is depicted in Figure 13a; a large amount of 56Ni in the ejecta with a deep mass cut for a ﬁxed progenitor mass
results in a large mass in the ejecta, as well as a large total mass of dust grains, and vice versa.

APPENDIX B
THE EFFECT OF THE FORMATION EFFICIENCY OF CO AND SiO MOLECULES ON DUST FORMATIONIn the calculations of dust formation, we assumed the complete formation of CO and SiO molecules. Formation of CO andSiO molecules prior to dust formation was observed in SN 1987A (e.g., Bouchet & Danziger 1993). Furthermore, recent observations have suggested that the formation of CO molecules is a common phenomenon in the ejecta of CCSNe (e.g., Gerardy etal. 2002). Formation of CO and SiO molecules is usually considered to be complete and controls the chemistry in the astrophysical environments of interest to dust formation, such as in the circumstellar envelopes of AGB stars. However, in theejecta of supernovae, these molecules are destroyed by collisions with high-energy electrons, as well as with ionized inert gaseous atoms produced by the decay of radioactive elements, and the abundance of these molecules is determined by the balance
between formation and destruction (Liu & Dalgarno 1994, 1996). Thus, in the ejecta of supernovae, for example, it is expectedthat free carbon atoms are available for the formation of carbon-bearing dust grains even in the ejecta of C=O < 1. This aspect
has been extensively pursued by Clayton et al. (1999, 2001) in investigating the origin of presolar grains identiﬁed as
SUNOCONs. The abundances of CO and SiO molecules depend on the abundance of free electrons, as well as ionized inert
gas. Thus, in this appendix, to simplify the calculations, we investigate the eﬀect of the formation eﬃciency of CO and SiO
molecules on dust formation, by referring to the formation eﬃciency of CO and SiO molecules observed in SN 1987A as a template and introducing the parameters fC and fSi, which represent the mass fractions of C and Si atoms not locked into CO andSiO molecules, respectively.
In the ejecta of SN 1987A, the masses of CO and SiO molecules have been evaluated to be ð3 6Þ  103 and ð0:4 1Þ  103
M, respectively, during 300–600 days after the explosion (Liu & Dalgarno 1994, 1995), which correspond to fC ’ 0:99 andfSi ’ 0:9 in the oxygen-rich core, respectively. The mass fractions fC ¼ 0:99 and fSi ¼ 0:9 being ﬁxed, Figures 17a and 17b
show the mass of newly formed grain species versus fSi and fC, respectively, at the location of Mr ¼ 3:5 M in the O-Mg-Si
layer of C20. In the calculations, we consider the SiO molecule the key species for the formation of Mg silicate grains. Of
course, the formation of Al2O3 grains is not aﬀected by the formation eﬃciency of these molecules. The mass of C and Si grains

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NOZAWA ET AL.

Vol. 598

Fig. 17.—Mass of each grain species vs. mass fractions of carbon fC and silicon fSi not locked into CO and SiO molecules, at Mr ¼ 3:5 M in the O-Mg-Si
layer of the unmixed ejecta of C20. (a) Mass vs. fC when fSi is ﬁxed to 0.9. (b) Mass vs. fSi when fC is ﬁxed at 0.99. [See the electronic edition of the Journal for acolor version of this ﬁgure.]

increases with increasing f. As can be seen from Figure 17a, except for the formation of carbon grains, the formation eﬃciency
of CO molecules does not inﬂuence the formation of other grain species. Also, the formation of carbon grains does not aﬀect
the total mass of dust grains at this location, since O and Mg atoms are much more abundant than C atoms (see Fig. 1a). The
average size of carbon grains increases with increasing fC but is limited to less than 0.04 lm, even at fC ¼ 1. On the other hand,the formation eﬃciency of SiO molecules strongly aﬀects the mass abundance of Si- and/or Mg-bearing grains; withincreasing fSi, the decrease of SiO molecules not only results in the formation of Si grains, but also increases the mass of MgOgrains with the reduction of the mass of Mg2SiO4 grains, which is prominent for fSi 0:1. The average radii of Si and MgOgrains increase and reach 0.1 lm with increasing fSi; the average radius of Mg2SiO4 grain decreases. At this location, the formation eﬃciency of CO and SiO molecules does not aﬀect the total mass of dust grains, because the abundance of C atoms is 2
orders of magnitude lower than that of O atoms. However, if fC ¼ 0:99 in the outer region of the O-Mg-Si layer(4 M  Mr  5 M for C20), many more carbon dust grains could condense, and the average radius could reach to from
0.4 to 1 lm, depending on the abundance of carbon atoms at the formation site.
Finally, it should be addressed here that we cannot realize the formation of SiC grains with radii comparable to those of presolar SiC grains identiﬁed as SUNOCONs by changing the parameters fC and fSi. The formation of SiC grains competes withthe formation of carbon and silicon grains; carbon atoms available for formation of SiC grains are locked into carbon grains
prior to the formation of SiC grains, and the same is true for silicon atoms. This implies that the amount and size are very smalleven if SiC grains condense. Therefore, as proposed by Deneault et al. (2003), special conditions and/or chemical pathways
might have to be considered in order to realize the formation of SiC grains identiﬁed as SUNOCONs.
Anyway, the results of the calculations show that the reduction of the formation eﬃciency of CO and SiO molecules leads tothe formation of C and Si grains. Furthermore, the formation eﬃciency of SiO molecules aﬀects the abundance and size of
Si- and/or Mg-bearing grains. However, we must keep in mind that in the calculations the stability of C and Si grains is nottaken into account; in the environment with a lot of free oxygen atoms, C and Si grains could be easily oxidized if their sizes
were small, though Clayton et al. (2001) have claimed that carbon grains are stable against the oxidation in the oxygen-richenvironment. Oxidation of Si grains produces SiO2 grains and/or evaporates SiO molecules available for formation of Mg
silicate grains, and it may be possible that another chemical pathway leads to the formation of Mg silicate grains without SiO
molecules, resulting in more oxide grains even if the formation eﬃciency is very low. These aspects should be pursued to clarify
how much dust, as well as what kind of oxide grains, really forms in the ejecta of supernovae.

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