The Astrophysical Journal, 863:100 (13pp), 2018 August 10

https://doi.org/10.3847/1538-4357/aad0a2© 2018. The American Astronomical Society. All rights reserved.

Abundances of Ordinary Chondrites in Thermally Evolving Planetesimals
1

Shigeru Wakita1,2, Yasuhiro Hasegawa3, and Takaya Nozawa4

Center for Computational Astrophysics, National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan; shigeru@elsi.jp
2
Earth-Life Science Institute, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8550, Japan3
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
4
Division of Theoretical Astronomy, National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan
Received 2018 April 24; revised 2018 June 26; accepted 2018 June 28; published 2018 August 14

Abstract
Chondrites are some of the most primitive objects in the solar system, and they maintain a record of the degree ofthermal metamorphism experienced in their parent bodies. This thermal history can be classiﬁed by the petrologic
type. We investigate the thermal evolution of planetesimals to account for the current abundances (known as thefall statistics) of petrologic types 3–6 of ordinary chondrites. We carry out a number of numerical calculations inwhich formation times and sizes of planetesimals are taken as parameters. We ﬁnd that planetesimals that form
within 2.0 Myr after the formation of Ca-Al-rich inclusions (CAIs) can contain all petrologic types of ordinarychondrites. Our results also indicate that plausible scenarios of planetesimal formation, which are consistent withthe fall statistics, are that planetesimals with radii larger than 60 km start to form around 2.0 Myr after CAIs and/orthat ones with radii less than 50 km should be formed within 1.5 Myr after CAIs. Thus, thermal modeling ofplanetesimals is important for revealing the occurrence and amount of metamorphosed chondrites and forproviding invaluable insights into planetesimal formation.
Key words: meteorites, meteors, meteoroids – planets and satellites: formation
the observations. Investigating planetesimals is therefore notstraightforward even for nearby, young stars.
The solar system stands out in exploring planetesimalformation because we can obtain key information fromasteroids and meteorites. It is reasonable to consider that
asteroids are the remnants of planetesimals that survived anumber of constructive and destructive events occurring over
the age of the solar system (e.g., Bottke et al. 2005a, 2005b;
Morbidelli et al. 2009; Zheng et al. 2017; Tsirvoulis et al.
2018). In other words, asteroids are unique objects that containa wealth of information about planetesimals. There arecompleted and ongoing sample return missions for investigating the composition of asteroids: Hayabusa, OSIRIS-REx, andHayabusa 2. For instance, the Hayabusa revealed that thechemical and isotopic compositions of particles on the asteroid
Itokawa are similar to those of meteorites falling onto the Earth(e.g., Nakamura et al. 2011; Yoshikawa et al. 2015). Asteroids
can also be used as a probe to specify the size distribution ofplanetesimals. Indeed, theoretical studies suggest that primordial planetesimals would be larger than 100 km in diameter inorder to explain the size distribution of current main belt
asteroids (e.g., Bottke et al. 2005a; Morbidelli et al. 2009;
Tsirvoulis et al. 2018). Thus, asteroids give a number ofinvaluable insights about planetesimals. It is however importantto realize that these insights are derived from the current statusof asteroids, and hence one has to take into account the full
history of a number of events that occurred for asteroids.
Meteorites act as important and complimentary objects forunderstanding planetesimal formation (e.g., McSween 1999).
This is because they literally contain the fossil record of how
the solar system formed. For example, chondrites are known as
the most primitive meteorites and contain the ﬁrst condensed
materials in the solar system, called Ca-Al-rich inclusions(CAIs; e.g., MacPherson 2005). It is well known that theage of the solar system is measured from CAIs, which is4567.30±0.16 million years (e.g., MacPherson et al. 1995;

1. Introduction
Planetesimals are some of the central objects that play crucial
roles in planet formation (e.g., Pollack et al. 1996; Ida &Lin 2004; Mordasini et al. 2009). It has been suggested for along time that planetary cores were built from planetesimalcollisions and resulting mergers in protoplanetary disks (e.g.,
Wetherill & Stewart 1989; Kokubo & Ida 1998; Morishima
et al. 2010; Raymond et al. 2014). The presence ofplanetesimals in planet-forming regions can also affect theorbital evolution of protoplanets there, which is referred to asplanetesimal-driven migration (e.g., Levison et al. 2010;
Kominami et al. 2016). More recently, planetesimals havebeen regarded as the main contributors to metal enrichment ofJovian planets both in the solar system and extrasolar planetarysystems (e.g., Saumon & Guillot 2004; Miller & Fortney 2011;
Mordasini et al. 2016). These are all regulated by the dynamics
of planetesimals, and hence are determined by the properties ofplanetesimals such as their abundances and masses. Thus, it isof fundamental importance to identify and characterize whenand how planetesimal formation takes place in protoplanetarydisks for fully exploring the formation and migration historiesof planets.
Understanding of planetesimal formation is still far fromcomplete despite recent progress (e.g., Johansen et al. 2014).
There are two reasons for this. One is that, theoretically, noforce (i.e., surface adhesion force or gravity) is effective
for kilometer-sized bodies. Many theoretical studies thereby
attempt to establish high density regions of solids in protoplanetary disks that eventually become gravitationally unstable
and lead to direct and instant formation of planetesimals due toself-gravity (e.g., Cuzzi et al. 2001; Johansen et al. 2007; Kretke& Lin 2007). The other is that, observationally, planetesimals areinvisible at almost all wavelengths in the nearby star-forming
regions. This arises from the fact that emission from gas anddust and their scattered light become generally dominant in
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The Astrophysical Journal, 863:100 (13pp), 2018 August 10

Wakita, Hasegawa, & Nozawa

Amelin et al. 2010; Connelly et al. 2012; Davis & McKeegan2014). Also, the detailed chemical and isotopic analyses of
meteorites suggest that planetesimals should have formed
soon after CAI formation and continued for about a few
million years (Davis et al. 2014; Gail et al. 2014; Blackburn
et al. 2017). Moreover, meteorites contain another primitive
component called chondrules that are considered to haveformed within 5 Myr after CAI formation (e.g., Connelly et al.
2012; Bollard et al. 2015). The formation mechanisms ofchondrules are currently under active investigation, one ofwhich is highly relevant to planetesimal collisions (e.g.,
Johnson et al. 2015; Hasegawa et al. 2016a, 2016b; Matsumoto
et al. 2017; Wakita et al. 2017a).
Another remarkable feature of meteorites is that more than
40,000 of them have been found. Interestingly, 90% of theanalyzed meteorites are ordinary chondrites (Meteoritical
Bulletin websitehttp://www.lpi.usra.edu/meteor/metbull.php;
Grady et al. 2014). Thanks to such abundant samples, statisticalstudies have become possible, including classiﬁcation. In
general, meteorites are grouped by their chemical composition
and texture (e.g., Van Schmus & Wood 1967; Krot et al. 2005;
Scott & Krot 2005; Weisberg et al. 2006). We focus here on the
petrologic type that links the degree of thermal metamorphism.
The degree of metamorphism is considered to reﬂect the
maximum temperatures that the parent bodies of meteorites
experienced (e.g., Dodd 1981; Scott & Krot 2005; Husset al. 2006). Type 3 chondrites are the least metamorphosed
ones and would not pass through temperatures higher than∼600 °C. Type 6 chondrites are the most highly metamorphosed, possibly in the environment above ∼800 °C, andtype 4 and 5 chondrites would be between them (Keil 2000;
Scott & Krot 2005; Wlotzka 2005; Huss et al. 2006). Since theexperienced temperatures depend on the position insideplanetesimals, previous studies make use of thermal evolution
models of planetesimals (e.g., McSween et al. 2002; Henke
et al. 2013; Gail et al. 2014). Attempts to reproduce the fall
statistics of ordinary chondrites (Ghosh & McSween 2000;
McSween et al. 2002) have not worked well under theassumption that parent bodies of meteorites are constituted ofsingle-sized planetesimals.
It is, however, highly possible that ordinary chondrites
originate from a variety of planetesimals (e.g., Monnereau
et al. 2013; Vernazza et al. 2015; Blackburn et al. 2017).
Furthermore, the internal temperature of planetesimals is
sensitive not only to the location within the planetesimals,
but also to their formation times and initial sizes (e.g., Gailet al. 2014; Wakita et al. 2014; Lichtenberg et al. 2016; Ricard
et al. 2017). In this paper, therefore, we extensively investigate
the thermal evolution of planetesimals with a wide range ofmodel parameters such as formation times and sizes ofplanetesimals. By comparing our numerical results with thefall statistics of ordinary chondrites in each petrologic type, wewill specify the most plausible values of the formation times
and size distributions of planetesimals.
The plan of this paper is as follows. In Section 2, we describe
how we calculate the thermal evolution of planetesimals withmodel parameters. Our numerical results are shown in
Section 3, where we discuss the dependencies of the model
parameters on the results. In Section 4, we compare our resultswith the fall statistics of ordinary chondrites and discuss
implications for planetesimal formation. Our conclusions are
given in Section 5.

2. Methods
2.1. Thermal Evolution Models of Planetesimals
We adopt thermal evolution models of planetesimals thatwere constructed in our previous work (Wakita et al. 2014,2017b). We assume that planetesimals are spherical and have aspatially uniform chemical composition. It is also assumed that,once planetesimals form with a radius of Rpl at a given time(t = tpl), they do not experience any further growth anddestructive processes afterward. Then, thermal evolutionof a planetesimal is numerically calculated by solving theequation of heat conduction,rc¶T (r , t )1 ¶ ⎛ 2 ¶T (r , t ) ⎞⎜r K⎟ + A exp ( - lt ) ,= 2¶tr ¶r ⎝¶r ⎠(1 )where t is the time measured since the formation of theplanetesimal, r is the distance from the center of the planetesimal,
T(r, t) is the temperature of materials located at r in theplanetesimal at a time t, A is the radiogenic heat generation rate
per unit volume, and λ is the decay constant of the radionuclides.
In our simulations, decay heat arising from the short-lived
radionuclide, 26Al is taken as the heating source of planetesimals(e.g., Miyamoto et al. 1982). The initial temperature ofplanetesimals is set at T(r, t = tpl)=−123 °C (150 K), which is
low enough not to affect the results of our calculations. We
assume that physical quantities, such as ρ, c, and K, are constantfor simplicity; ρ=3300 kg m−3, c=910 J kg−1 K−1, and K=
2 J s−1 m−1 K−1 are the adopted values of the bulk density,
speciﬁc heat, and thermal conductivity, respectively (Yomogida &Matsui 1983; Opeil et al. 2010).
If planetesimals contain volatiles such as water, CO2, andorganics, then different values should be adopted for the above
three quantities. Furthermore, if this were the case, thetemperature distribution of planetesimals would be regulated
not only by thermal conductivity, but also by transport of suchvolatiles. However, these complexities should be taken into
account in exploring thermal evolution of volatile-rich bodies
such as parent bodies of carbonaceous chondrites (e.g., Grimm& McSween 1989; Travis & Schubert 2005; Bland & Travis2017). In this paper, we focus on parent bodies of ordinarychondrites that accreted in very volatile-poor environments,and most of them are (highly) thermally metamorphosed.
Therefore, the impact of volatiles can be negligible, except forthe ones that contain a small amount of volatiles and/or
experience the least metamorphism. In this sense, the physicalquantities used here are reasonable, and considering heat
conduction is valid for thermal evolution of planetesimals in
the context of this paper.
2.2. Key Model Parameters
The main purpose of this study is to carry out a parameter
study to determine the fundamental quantities of planetesimals.
In this paper, we introduce ﬁve parameters (see Table 1): theradius of planetesimals (Rpl), the power-law index (α) for thesize distribution of planetesimals, the initial time (tint) at whichplanetesimal formation begins, the ﬁnal time (tﬁn) whenplanetesimal formation ends, and the formation timescale (τf)
of planetesimals. In the following, we describe what values are
considered for these parameters.
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Wakita, Hasegawa, & Nozawa

mass of planetesimals under consideration. Here, we summarize the total mass of planetesimals for three cases.

Table 1
Summary of Parameters in This Study
Parameters of Planetesimals
Radius (km)Power-law index for size distribution
Initial time of formation (Myr)Final time of formation (Myr)Timescale for the formation rate (Myr)

Symbol
Rpl
α
tint
tﬁn
τf

(i) The simplest case considered in this paper is thatplanetesimals form at a single epoch (tpl = tint = tﬁn) andhave a single size (Rpl). In this case, the total mass ofplanetesimals (M1) is given as

Value
1–500
0.0, 2.8, 3.5, 4.5
0.1, 1.0, 1.5, 2.0, 2.5, 3.0
tint, 5.0 or 7.0
1.0, 4.0a

M1 = N1

Note.
a
The formation rate of planetesimals is assumed to be constant (t f = ¥) for
all of the simulations, except in Section 3.4.4.

(3 )where N1 is the number of planetesimals formed.
(ii) A more realistic situation is that planetesimals form
during some period (tint  tpl  tﬁn), but all of them havethe same size (Rpl). In this case, the total mass ofplanetesimals (M2(t)) is written as

The typical size of planetesimals and their size distribution
are hardly constrained. Accordingly, we take the broad range
from Rpl=1 km to Rpl=500 km. We assume that the size
distribution can be described as a power law in terms of theirradius. Then, the number (npl) of planetesimals between Rpl and
Rpl+dRpl can be written as
npl = n 0 (R pl R pl,0)-a ,4pr 3
R pl ,3t

M2 (t ) = å f (t ¢)t ¢= tint

4pr 3
R pl .
3(4 )When t>tﬁn, the summation is carried out up to t=tﬁn.
(iii) The most realistic case in this paper is that planetesimals
form over certain times (tint  tpl  tﬁn), and they have asize distribution. Here, we assume that the power-law
index (α) and its size range (Rpl,min and Rpl,max) do notchange with time. Then, the total mass of planetesimal(M3(t)) is given as(2 )for Rpl,min  Rpl  Rpl,max , where n0 is the normalization
constant, and we set Rpl,0=Rpl,min. For the power-law index(α), we adopt four values: α=0, 2.8, 3.5, and 4.5. In our
ﬁducial model, we choose α=2.8, following Johansen et al.
(2015) and Simon et al. (2016), who propose that it would
represent the primordial distribution of planetesimals.
The formation time of planetesimals is also uncertain since
their formation mechanisms are still unclear. It nonethelessmight be reasonable to consider that planetesimal formation
would take place when the (gaseous) solar nebula was present.
Then, we assume that formation time (tpl) is in the range of
0.1 Myrtpl7.0 Myr. In this parameter study, we consider
that the onset time (tint) of planetesimal formation is between
0.1 Myr and 3.0 Myr. For the termination time (tﬁn) of
planetesimals, we consider two cases. For one case, we
assume tpl=tint=tﬁn to focus on the single generation ofplanetesimals (see Sections 3.1 and 3.2). For the other case, inwhich multiple generations of planetesimals are examined,tﬁn=5.0 Myr or 7.0 Myr (see Sections 3.3 and 3.4). Note thatin the following sections, the time is all measured since theformation of CAIs: CAIs are regarded as the ﬁrst condensatesin the solar system that formed 4567 Myr ago with the initialratio of 26Al/27Al=5.25×10−5 (e.g., Connelly et al. 2012).
Given that the abundance of 26Al decreases exponentially witha half-life of 0.72 Myr, the earlier formed planetesimals have ahigher abundance of the heating source.
The formation rate of planetesimals is another important
quantity that can affect global thermal histories of forming
planetesimals. In this paper, we simply assume that theformation rate (f) of planetesimals is constant with time formost cases. We will also consider the case that the formation
rate of planetesimals decreases exponentially with time.
Namely, it is given as f=f0exp(−t/τf), where f0 is the
hypothetical formation rate of planetesimals in number att=0, and τf is the formation timescale (see Section 3.4.4).

t

M3 (t ) = å f (t ¢)t ¢= tint

R pl,maxòRdR pl n pl (R pl )pl,min4pr 3
R pl .
3(5 )In the next section, we present the results for these threecases.
3. Results
3.1. Thermal Evolution of a Single-sized Planetesimal Formed
at a Single EpochIn this section, we demonstrate the fundamental properties ofthermal evolution of a single-sized planetesimal. The results are
obtained, using the following set of parameters: Rpl=100 km,
and tpl=tint=tﬁn=2.0 Myr or 2.5 Myr, that is, a single
generation of planetesimals is considered.
Figure 1 shows the results for the cases of tpl=2.0 Myr andtpl=2.5 Myr on the left and the right panels, respectively. We
ﬁrst discuss the case of tpl=2.0 Myr. The top-left panel
depicts temperature evolutions at the positions of r=0, 80,
and 90 km. Initially, the temperature increases with time, as aresult of the continuous energy input from radionuclides 26Al.
The central region in the planetesimal (r=0 km) achieves thehighest temperature, which exceeds 1000 °C at t=5.5 Myr.
On the other hand, the temperature at r=80 km (r=90 km)
reaches 920 °C (680 °C) at t=5.5 Myr (4.2 Myr). And then it
starts to decrease because the heating rate due to decay of 26Al
becomes lower than the cooling rate at the position. This meansthat each position of the planetesimal has a peak temperature(Tpeak(r)) at t=tpeak(r). The bottom-left panel of Figure 1
shows the temperature evolution at all the positions of
the planetesimal. We can see that the temperatures atr65 km follow the same evolution and can increase to
more than 1000 °C.
For the case of tpl=2.5 Myr (see right panels of Figure 1),
Tpeak(r) at the center is 600 °C, demonstrating that the

2.3. Normalization Factors for Mass Fractions
As discussed below, we compute and examine massfractions of planetesimals that are normalized by the total
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Figure 1. Temperature evolutions of single-sized planetesimals (Rpl=100 km) formed at (a) tpl=2.0 Myr and (b) tpl=2.5 Myr. Top panels show temperature
evolutions at positions of r=0 km (red solid lines), r=80 km (black dashed lines), and r=90 km (blue dotted lines), respectively. Bottom panels showtemperatures at each position of the planetesimal. Each black line is drawn every 100 °C.

We then introduce the mass fraction ( f (t, Tth)) ofplanetesimals that is determined by a threshold temperature(Tth). When the single generation of planetesimals with thesingle size is considered, a fraction of mass that experiences thetemperatures equal to and higher than Tth can be computed asplanetesimal formed later has the lower Tpeak at the same
position than the earlier formed one. This indicates that thethermal evolution is signiﬁcantly affected by the formation timeof planetesimals.
3.2. The Maximum Temperature and theCorresponding Mass Fractionf (t , Tth) =As discussed above, the peak temperatures inside planetesimals that are heated by radioactive nuclei are different at
different locations. For the analysis that will be developed forunderstanding the origin of the petrologic type of ordinarychondrites, we manipulate the computed values of thetemperature (T(r, t)) as follows.
In this work, we are interested in the maximum temperaturesthat planetesimals experience through thermal evolution. Given
that the peak temperatures (Tpeak(r)) at position r are achievedat different time (t = tpeak(r)), the maximum temperature(Tmax (r, t)) is described as⎧T (r , t ) at t < tpeak (r )Tmax (r , t ) = ⎨⎩Tpeak (r ) at t  tpeak (r ) ,N1 4pr 3
r (Tmax (t )  Tth) ,
M1 3

(7 )

where t is explicitly labeled both in f and Tmax for clearly
representing that both quantities are functions of time.
The bottom panels of Figure 2 show the time evolutions off (t, Tth) for different Tth. Our results show that f (t, Tth) withhigher Tth rises up as time goes on. This is a direct reﬂection of
the fact that the temperature at each position increases with time(see the middle panels of Figure 2). Then, any values of f (t, Tth)
become almost constant after t;6 Myr for tpl=2.0 Myr. Thisis because, at this time, almost all positions already pass throughTpeak, and Tmax no longer changes. Also, our results clearly showthat a mass fraction of the planetesimals that undergo a hightemperature is reduced for ones formed at later times (see thebottom-right panel of Figure 2): f (t, Tth = 600°C)=0.3 andf (t, Tth = 800°C)=0 at t7 Myr for the planetesimal formed
at tpl=2.5 Myr. On the contrary, the planetesimal formed at
tpl=2.0 Myr has f (t, Tth = 600°C)=0.76 and f (t, Tth =800°C)=0.65 at t7 Myr.

(6 )from the temperature evolution as given in Figure 1. The time
evolutions of Tmax(r, t) are shown in the top and middle panelsof Figure 2. As is obvious from Equation (6), Tmax becomes
constant after t=tpeak=5.5 Myr (4.2 Myr) at r=80 km(r=90 km) for tpl=2.0 Myr.
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Figure 2. Maximum temperatures of single-sized planetesimals (Rpl=100 km) formed at (a) tpl=2.0 Myr and (b) tpl=2.5 Myr. Top panels show time evolutionsof Tmax(r, t) at positions of r=0 km (red solid lines), r=80 km (black dashed lines), and r=90 km (blue dotted lines), respectively. Middle panels show Tmax(r, t)evolutions at each position of planetesimals. Bottom panels show the mass fraction of the planetesimals, f (t, Tth), that experienced the maximum temperatures higher
than Tth, deﬁned in Equation (7).

We ﬁrst see the effect of planetesimal formation time onf (t, Tth). Figure 3 plots f (t, Tth) at t=7.0 Myr≡tms as afunction of tpl(tint  t  tﬁn). We conﬁrm that the mass fraction
becomes constant at tms tﬁn. We ﬁnd that as the formation
time shifts later, f (tms, Tth) becomes lower. For instance, theplanetesimals formed after tpl=2.3 Myr cannot reach 800 °C.
Also, f (t = tms, Tth = 0 °C)=0 at tpl > 4.3 Myr, indicating thatthe planetesimals formed later than tpl=4.3 Myr cannot beheated above 0 °C even at the center.
We then examine the behavior of time-integrated massfractions of planetesimals formed at multiple epochs. The
fraction of mass F(t, Tth) that experienced Tmax Tth for all ofThis clearly demonstrates that a mass fraction of planetesimals that undergo the temperature above 600 °C strongly
depends on the formation time of planetesimals.
3.3. Multiple Generations of Planetesimals with a Single Size
We have so far considered a single generation of planetesimals, that is, tpl=tint=tﬁn. In general, however, planetesimals
are expected to form over some period. Here, we present theresults from multiple generations of planetesimals (tint  tpl  tﬁn)with single sizes. As an example, we adopt the following set
of model parameters: Rpl=100 km, tint=0.1 Myr, and tﬁn=
7.0 Myr with a constant formation rate.

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Figure 3. Mass fraction f (t, Tth), deﬁned in Equation (7), that experienced
Tmax(t) Tth and is measured at tms=7.0 Myr for planetesimals that haveradii of Rpl=100 km and form at tpl=0.1 Myr to 7.0 Myr.

Figure 4. Time evolution of mass fraction F(t, Tth) deﬁned in Equation (8) for
planetesimals with Rpl=100 km that formed at tpl=0.1 Myr to 7.0 Myr.

the formed planetesimals is given as
F (t , Tth) =t
14pr
å f (t ¢) 3 r 3 (Tmax (t ¢)  Tth).
M2 (t ) t ¢= tint

(8 )Figure 4 shows the time evolution of F(t, Tth). It can be seenthat F(t, Tth) for Tth 800 °C increases until t;1.5 Myr. This
stems from the fact that it takes about 1 Myr for earlier formedplanetesimals to reach Tpeak(r). On the other hand, F(t, Tth) forTth 800 °C starts to decreases after t=1.5 Myr because asmall, or no, part of later formed planetesimals can reachtemperatures higher than 800 °C (also see Figure 3). Our results
also show some spiky structures at the early times. These are
numerical noises, but do not affect our conclusions that arederived from the later stage of the evolutions.
In this study, we examine the abundance of ordinarychondrites moderately metamorphosed in thermally evolving
planetesimals. In order to perform direct comparison among
our results, we deﬁne the ratio of F(t, Tth) as follows,
Fratio (t , Tth) ºFigure 5. Time evolution of fraction ratio Fratio(t, Tth) deﬁned in Equation (9)for planetesimals of Rpl=100 km that formed at tpl=0.1 Myr to 7.0 Myr.

F (t , Tth - 100 C) - F (t , Tth)
, (9 )F (t , Tth = 0 C) - F (t , Tth = 1000 C)with Rpl50 km are different. Planetesimals of Rpl=50 km
have Fratio(Tth 600 °C) of 0.27. For Rpl=10 and 20 km,
Fratio(Tth 600 °C) are 0.21 and 0.25, respectively. This
indicates that smaller planetesimals produce lower Fratio(Tth600 °C) than larger ones, reﬂecting the fact that they cannot
reach higher temperatures than the larger ones due to more
effective cooling from their surfaces.
It should be stressed that the radius of the planetesimal (Rpl)
plays an important role in thermal evolution of planetesimals
and hence the resulting ratio (Fratio(Tth)). Also, the results are
not affected by planetesimals that form later than 5.0 Myr.

where Tth varies from 100 °C to 1000 °C.
Figure 5 shows the time evolutions of Fratio(t, Tth) for
different Tth. We ﬁnd that Fratio(t, Tth) becomes almost constantafter t=5.0 Myr. Planetesimals formed later than 5.0 Myr
cannot contribute to the change of Fratio(t, Tth), because they do
not experience the temperature higher than 100 °C. Accordingly, we focus on planetesimals that form earlier than 5.0 Myr
in the following sections, and denote Fratio(t = tms=5.0 Myr,Tth) as Fratio(Tth) for brevity.
We are now in a position to examine the effect of Rpl on
Fratio(Tth), which is shown in Figure 6. In this ﬁgure, we adopt
the following parameters: Rpl=5–500 km, tint=0.1 Myr,tﬁn=5.0 Myr, and the constant formation rate is considered.
We ﬁnd that Fratio(Tth) for larger planetesimals with Rpl100 km are very similar, and Fratio(Tth 600 °C) is 0.29. Thismeans that even if we consider the arbitrary size distribution forplanetesimals larger than 100 km, the results do not changevery much. On the other hand, Fratio(Tth) of planetesimals

3.4. Multiple Generations of Planetesimals
with the Size Distribution
In this section, we present the results that are obtained fromthe most realistic case where planetesimals form within thetime interval between t=tint and t=tﬁn and they have the size
distribution from Rpl,min to Rpl,max.
6

The Astrophysical Journal, 863:100 (13pp), 2018 August 10

Wakita, Hasegawa, & Nozawa

Figure 7. Cumulative representation of Fratio (Tth ) deﬁned in Equation (11) for
planetesimals with size ranges of (a) all (1–500 km), (b) small (1–50 km),(c) middle (60–90 km), and (d) large (100–500 km). The power-law index isα=2.8 and planetesimals form from 0.1 Myr to 5.0 Myr with a constantformation rate.

Figure 6. Cumulative representation of Fratio(Tth) deﬁned in Equation (9) for
planetesimals with the radius of 5, 10, 20, 30, 50, 70, 100, 200, 300, and500 km. The planetesimals form from 0.1 Myr to 5.0 Myr with a constantformation rate.

the ratio of all planetesimals combined from the other threeranges.
From this, we can conclude that the cumulative mass fraction
ratio of planetesimals that experience certain temperatures
depends on their size range. In particular, there is a signiﬁcantdifference between the smaller size range (Rpl50 km) andthe larger one (Rpl 60 km).

Under this situation, the mass fraction of planetesimals thatexperience a threshold temperature (Tth) or higher is written as
F (t , Tth) =´R pl,max

òRt
1
f (t ¢)å
M3 (t ) t ¢= tintdR pl n pl (R pl )pl,min4pr 3r (Tmax (t ¢) > Tth, R pl ).
3(10)

3.4.2. Dependence on Onset of Planetesimal Formation

One of our aims in this paper is to constrain the formation
time of planetesimals. In previous sections, we have consideredthat planetesimals start to form at tint=0.1 Myr. In thissubsection, we see the dependence of the results on theformation timing (tint) of planetesimals.
As shown in Figure 3, the mass fraction with higher
temperatures decreases with increasing the formation time. If
the formation time is later than 2.0 Myr after CAI formation,
there is no chance to reach 1000 °C. Figure 8 shows Fratio (Tth )for the four size ranges with onset times of planetesimalformation, tint=0.1, 1.0, 1.5, 2.0, 2.5, and 3.0 Myr. When theonset time of planetesimal formation becomes later, Fratio (Tth 
600 C) decreases. In particular, Fratio (Tth  600 C) in thesmall size range (1–50 km) rapidly decreases as tint goes later(Figure 8(b)). This is because smaller planetesimals cannot
reach temperatures higher than 600 °C when they form
later (tpl=1.5 Myr for Rpl=10 km and tpl=2.5 Myr for
Rpl=50 km).
We should emphasize that the onset time of planetesimalformation should be earlier than 2.2 Myr (1.2 Myr) to containthe volume that experiences Tmax=800 °C for Rpl=50 km(Rpl=10 km). This is important to compare our results withfall statistics of ordinary chondrites (see Section 4).

Note that r is now a function of Rpl. Then, the ratio of F(t , Tth )can be given as
Fratio (t , Tth) ºF (t , Tth - 100 C) - F (t , Tth),F (t , Tth = 0 C) - F (t , Tth = 1000 C)(11)

for 100 °CTth1000 °C. Armed with this formulation,
we carry out a parameter study and examine how Fratio (t , Tth )at t=tms is inﬂuenced by model parameters of planetesimals.
For brevity, Fratio (t = tms=5.0 Myr, Tth) is denoted as
Fratio (Tth ). Unless otherwise mentioned, we take the powerlaw index α=2.8 and consider that planetesimals form fromtint=0.1 Myr up to tﬁn=5.0 Myr with a constant formation rate.
3.4.1. Dependence on Size Ranges of Planetesimals

Planetesimals are likely to form with some size range (sizedistribution) but not with a single size. Therefore, we consider
four size ranges; small (1–50 km in radius), middle (60–100 km
in radius), large (100–500 km in radius), and all (1–500 km in
radius).
Figure 7 shows the resulting values of Fratio (Tth ) for the four
size ranges. We ﬁnd that the results of the middle and large
size ranges ((c) and (d) in Figure 7) look similar. Both ofthem have Fratio (Tth  600 C)=0.29. On the other hand, thesmall size range (Figure 7(b)) takes a smaller value of
Fratio (Tth  600 C)=0.22. As we see in the previous section,
only a small portion of the planetesimal can reach higher
temperatures in small sized planetesimals. Figure 7(a) shows

3.4.3. Dependence on Size Distributions of Planetesimals

Next we see the effect of the size distribution ofplanetesimals. So far, we take α=2.8 as the power-law index
of the size distribution. This can be regarded as the initial size
distribution of planetesimals (Johansen et al. 2015; Simon
et al. 2016). Meanwhile, α=3.5 is known to be the size
7

The Astrophysical Journal, 863:100 (13pp), 2018 August 10

Wakita, Hasegawa, & Nozawa

Figure 8. Cumulative representation of Fratio (Tth ) deﬁned in Equation (11) for planetesimals with size ranges of (a) all (1–500 km), (b) small (1–50 km), (c) middle(60–90 km), and (d) large (100–500 km). The results are shown for the onset of planetesimal formation tint taken at 0.1, 1.0, 1.5, 2.0, 2.5, and 3.0 Myr. The power-law
index is α=2.8 and planetesimals form with a constant formation rate.

distribution to explain the current asteroid belt (Morbidelli
et al. 2009). We also consider α=0 and 4.5 as extreme cases.
The results for different size distributions with α=0, 3.5,
and 4.5 are shown in Figure 9. When we compare α=2.8(Figure 8(a)) and 3.5 (Figure 9(b)), we ﬁnd a moderate
difference between them: Fratio (Tth  600 C) at tint=2.0 Myr
are 0.21 and 0.13 for α=2.8 and 3.5, respectively. Forα=0.0, which means that the number of any size ofplanetesimals is the same, Fratio (Tth ) is mainly determinedby larger planetesimals: that is Fratio (Tth  600 C)=0.28
at tint=2.0 Myr (Figure 9(a)). This is comparable with
Fratio (Tth  600 C)=0.27 at tint=2.0 Myr for α=2.8 and
Rpl=100–500 km (Figure 8(d)). When we take α=4.5,
smaller planetesimals regulate the result and Fratio (Tth 
600 C) is lower: Fratio (Tth  600 C) at tint=2.0 Myr is
0.05 for α=4.5 at tint=2.0 Myr (Figure 9(d)).
The value of Fratio (Tth ) depends largely on the size
distribution in the extreme case of α=0.0 or 4.5. However,
the effect of the size distribution is moderate, as long as α ∼3.0.

formation rate decreases on the timescale of τf (see Section 2.2
and Table 1).
The results for τf=1.0 and 4.0 Myr are shown in Figure 10,
where the radius of planetesimals ranges from 1 to 500 km
with the power-law index of α=2.8. When τf=1 Myr,
Fratio (Tth  600 C) gets larger than that for the constant rate
(see Figures 8(a) and 10(a)). This means that, when theformation rate of planetesimals rapidly decreases with time, thecontribution from earlier formed planetesimals, which experience higher temperatures, becomes stronger. For τf=4 Myr,the results are close to those with the constant formation rate
(see Figures 8(a) and 10(b)).
From these results, we conclude that there is the distinguishable difference between the results for the constant formation
rate and the decreasing formation rates with short τf.
4. Discussions
In this section, we compare our results with the abundance ofeach petrologic type of ordinary chondrites and discuss theformation and evolution histories of planetesimals.
The fall statistics of ordinary chondrites indicates that the
number abundance of each petrologic type is 5.2% for type 3,
16.5% for type 4, 38.5% for type 5, 37.9% for type 6, and3.4.4. Dependence on Formation Rate of Planetesimals

In the previous sections, we have assumed that the formation
rate of planetesimals is constant. Here, we consider that the8The Astrophysical Journal, 863:100 (13pp), 2018 August 10

Wakita, Hasegawa, & Nozawa

Figure 10. Same as in Figure 8(a) but with a different formation timescale τf(see also Section 3.4.4). (a) τf=1.0 Myr and (b) τf=4.0 Myr

peak metamorphic temperatures are overlapped between types
4 and 5 (Scott & Krot 2005). Nevertheless, in comparison to
our results, we assign the following temperatures to each type:100–600 °C to type 3, 600–800 °C to types 4 and 5, and
800–1000 °C to type 6 (Scott & Krot 2005; Huss et al. 2006).
In our simulations, we have computed the mass fraction ofplanetesimals (Fratio (Tth )) for a variety of initial radii andformation times of planetesimals (see Equation (11)). Although
the fall statistics is given by the number of meteorites (bypercentages), the statistics may represent the original (mass)abundance in their parent bodies, as a results of (thanks to)the huge number of meteorite samples. Hence, we compare
Fratio (Tth ) with the fall statistics.
First, we ﬁnd that, for all models considered in this study,
Fratio (100C  Tth < 600 C) is higher than Fratio (600 C 
Tth < 800 C) and Fratio (800 C  Tth  1000 C), meaningthat type 3 ordinary chondrites are the most abundant in ourmodels. This result is consistent with previous work (McSween
et al. 2002), but it is opposite to the fall statistics of type 3chondrites, whose mass fraction is much lower than those oftypes 4 and 5 and type 6 (Grady et al. 2014). We will discuss
this point later.

Figure 9. Same as in Figure 8(a) but with different power-law indexes (seeSection 3.4.3). (a) α=0.0, (b) α=3.5, and (c) α=4.5.

1.9 % for others (Grady et al. 2014). The petrologic types ofchondrites are divided by chemical compositions and texture,
as well as their peak metamorphic temperature. Thus, the peakmetamorphic temperature would not necessarily be adequate toseparate each type rigorously; especially because the estimated9The Astrophysical Journal, 863:100 (13pp), 2018 August 10

Wakita, Hasegawa, & Nozawa

formation condition of planetesimals; (1) planetesimals shouldhave started to form at t2.0 Myr after CAIs when theirprimordial sizes were larger than 60 km and/or (2) planetesimals whose size distribution was weighted toward small radiiof 1–50 km should have started to form at t 1.5 Myr after
CAI formation. These conclusions are schematically depicted
in Figure 12.
Figure 11 also shows that the results for size distributions
with α=0 and 3.5 lead to a slightly worse agreement with thefall statistics than that for the ﬁducial case with α=2.8 (seethe legends Fig. 8(a), 9(a), and 9(b) in Figure 11 and Table 2).
However, the difference is not so large that we cannot impose
any constraint on the primordial size distribution of planetesimals. Meanwhile, our results can provide some implications
to the formation rate of planetesimals (see the legends Fig. 8(a),10(a), and 10(b) in Figure 11 and Table 2). When the formation
rate of planetesimals decreases slowly (τf = 4.0 Myr), the mass
ratios of type 6 to types 4 and 5 can reproduce the fall statistics
relatively well (see the legend Fig. 10(b) in Figure 11 andTable 2). Even for the rapid (τf=1.0 Myr) case whose result isslightly higher than the fall statistics, it is in a reasonable range(see the legend Fig. 10(a) in Figure 11 and Table 2). Although
the preferable formation rate of planetesimals can be a constant
one or a relatively long formation timescale (τf=4.0 Myr), wecannot rule out the rapid formation case (τf=1.0 Myr).
Our results show that the mass ratios of type 6 to types 4 and5 are produced by the thermal evolution model of planetesimals. However, the mass fraction of type 3 ordinary chondrites
cannot be explained. As shown in Figures 8–10, even if
planetesimals start to form as early as 0.1 Myr, about 70%of the total mass of the planetesimals cannot experiencetemperatures higher than 600 ◦C. Therefore, to lower theabundance of type 3 chondrites than those of types 4–6 ones,
planetesimals have to be heated more than the present
calculations. However, we do not know other internal sourcesand effective mechanisms of heating the whole planetesimals(except for the decay of short-lived radionuclides).
Type 3 chondrites are divided into petrologic subtypes from
3.0 to 3.9 based on their characteristics of thermoluminescence sensitivity and chemical compositions of minerals, suchas iron content of olivines (Sears et al. 1980; Scott & Jones
1990; Scott et al. 1994; Grossman & Brearley 2005; Bonalet al. 2006). Although our model cannot reproduce theabundance of type 3 chondrites relative to types 4 and 5 andtype 6, it is still interesting to see what implications our results
have for the fall statistics of type 3 subtypes. It is suggested
that types 3.0–3.2 chondrites experienced much lower peaktemperatures than the other type 3s and that type 3.9 would
have experienced almost the same temperature as type 4 (e.g.,
Huss et al. 2006; Cody et al. 2008; Vernazza et al. 2014). It isvery hard to distinguish the subtypes only by their peakmetamorphic temperatures, but we here roughly divide type 3
subtypes into three subgroups, assigning peak metamorphic
temperatures as follows: 100–400 °C for types 3.0–3.2,
400–500 °C for types 3.3–3.6, and 500–600 °C for types
3.7–3.9. The relative fall statistics of type 3 subtypes arederived based on the Meteoritical Bulletin Database (http://
www.lpi.usra.edu/meteor/metbull.php): 0.132, 0.398, and,
0.470 for types 3.0–3.2, 3.3–3.6, and 3.7–3.9, respectively.
Note that we use only the samples of type 3 ordinarychondrites that are clearly classiﬁed as subtypes. We
summarize the normalized fall statistics of type 3 subtypes

Figure 11. Relative abundance ratios of type 6 to types 4 and 5 ordinarychondrites. Each line comes from our results (see legends and corresponding
ﬁgures). The fall statistics of ordinary chondrites (dashed line) with theuncertainty±30% (shaded region) are also shown.

Although our model cannot reproduce the fall statistics oftype 3 ordinary chondrites, it is interesting to compare ourresults with the observed relative abundance of type 6 to types
4 and 5 chondrites. This is because these types of chondrites
are dominant (90% of all chondrites). Additionally, whenthermal evolution of planetesimals is the main cause to
generate types 4, 5, and 6 chondrites in their parent bodies,
these chondrites should have been present deep inside of theplanetesimals (see Figure. 1). Consequently, the relative
abundance of type 6 to types 4 and 5 chondrites could be a
key to specifying formation times of planetesimals and theirprimordial sizes. Figure 11 represents the mass ratios of type 6
to types 4 and 5 chondrites obtained from our results as afunction of the onset time of planetesimal formation (seelegends). In this plot, we consider some representative cases.
Note that Fratio (Tth  600 C) becomes zero at tint>2.5 Myr,thus we cannot plot the result of tint 3.0 Myr. Table 2
summarizes our results for the case of tint=1.5 Myr, together
with a set of parameters.
One of our important ﬁndings is that the parent bodies ofordinary chondrites must begin to form before t∼2.0 Myr, toproduce type 6 chondrites that undergo thermal metamorphismwith the temperature of 800 °C or higher. In other words,
if formation of planetesimals would be initiated aftert∼2.0 Myr, no planetesimal could be heated high enough tocontain type 6 chondrites, which contradicts the observational fact.
It should be emphasized that the calculated mass ratios oftype 6 to types 4 and 5 are in agreement with the fall statistics
within the uncertainty of 30% for most of our models whentint2 Myr. The planetesimal formation time of 2.0 Myr is
consistent with the estimation from our previous work for theparent bodies of the asteroid Itokawa (Wakita et al. 2014).
However, adopting tint=2.0 Myr, the mass ratios of type 6 totypes 4 and 5 from the small size range (Rpl=1–50 km) of
planetesimals become lower than the fall statistics (see thelegend Fig. 8(b) in Figure 11 and Table 2). The result withα=4.5, wherein small planetesimals dominate the total
mass, also underproduces the fall statistics (see the legend
Fig. 9(c) in Figure 11 and Table 2). On the other hand, whenplanetesimal formation begins at t1.5 Myr, these cases can
successfully explain the fall statistics of type 6 ordinarychondrites. Therefore, our results suggest two scenarios for the10

The Astrophysical Journal, 863:100 (13pp), 2018 August 10

Wakita, Hasegawa, & Nozawa

Table 2
Relative Abundance Ratios of Type 6 to Types 4 and 5 Ordinary Chondrites Calculated in This Study
Type 4

Type 5

Type 6

Type 6/Types 4and5

0.18

0.41

0.41

0.69

Fall Statistics

0.41
0.37
0.42
0.42
0.42
0.39
0.36
0.45
0.42

0.69
0.58
0.72
0.72
0.72
0.64
0.55
0.82
0.72

Figure 8(a): Rpl=1–500 km, α=2.8
Figure 8(b): Rpl=1–50 km, α=2.8
Figure 8(c): Rpl=60–90 km, α=2.8
Figure 8(d): Rpl=100–500 km, α=2.8
Figure 9(a): Rpl=1–500 km, α=0.0
Figure 9(b): Rpl=1–500 km, α=3.5
Figure 9(c): Rpl=1–500 km, α=4.5
Figure 10(a): Rpl=1–500 km, α=2.8, τf=1.0 Myr
Figure 10(b): Rpl=1–500 km, α=2.8,τf=4.0 Myr

0.59
0.63
0.58
0.58
0.58
0.61
0.64
0.55
0.58

Fall or Figure and Parameters

Note.Abundance ratio of each type is given for the case of tint=1.5 Myr and is normalized by the sum of the values for Types 4 to 6.

they formed. However, planetesimals that were not incorporated into (proto)planets would have suffered from destructive
collisions with their surrounding planetesimals over the age ofthe solar system. In fact, destructive collisions are one of
the important ingredients for reproducing the size distributionof the current main asteroid belt (Morbidelli et al.
2009). When planetesimals experience destructive collisions,materials located in their surface regions are ejected. Suchmaterials can potentially serve as the parent bodies of type 3chondrites due to a low level of metamorphism. It nonethelesscan be expected that they would not contribute to the (current)
fall statistics very much. This is because they would probably
be accreted onto nearby (proto)planets and/or be gone beyond
the solar system on a relatively short timescale. Furthermore,
the removal of surface materials via destructive collisions
would increase the chance for the core region of planetesimals
to be exposed to subsequent collisions. Given that materials in
the core region can become the parent bodies of types 4–6
chondrites, destructive collisions can eventually enhance thepossibility that types 4–6 chondrites fall onto the Earth asmeteorites. It is interesting that the asteroid Itokawa would be
mainly composed of types 5–6 chondrites, and its parent bodyis considered to have been originally larger than 20 km in
radius before being catastrophically destroyed by a big impact(e.g., Nakamura et al. 2011; Wakita et al. 2014). Thus, thedestructive collisions that planetesimals have underwent during
their long life may effectively decrease the amount of type 3chondrites. In other words, the fall statistics of type 3chondrites may hold the key information on the collisionhistories of planetesimals.

Figure 12. Formation time of planetesimals and their sizes that are consistent
with the ratio of type 6 to types 4 and 5 ordinary chondrites obtained from thefall statistics. We adopt α=2.8 and the constant formation rate. The parameter
space that can reproduce the fall statistics is shown in the white region and theone that cannot is in the gray region (see Figures 8 in Table 2).

and our results for the case of tint = 1.5 Myr in Table 3. Our
results show that the lower numbers of type 3 subtypes
are more abundant than the higher numbers of them (i.e.,
types 3.0–3.2 > 3.3–3.6 > 3.7–3.9), which is opposite to
the trend that appeared for the fall statistics (types
3.0–3.2 < 3.3–3.6 < 3.7–3.9). In particular, our models predict much higher abundances of types 3.0–3.2 (>0.8) than its
fall statistics (0.13, see Table 3). Even if we take the ratio oftypes 3.3–3.6 to types 3.7–3.9, almost all of our results
overestimate the observed values out of the uncertainty of
30% (Figure 13). This disagreement might partially stem fromthe facts that the fall statistics of type 3 subtypes used here arefar from complete (only 1/3 of type 3 ordinary chondrites issorted out into subtypes) and/or that we do not consider
the heat transport through volatile materials, which would
affect the abundances of the least metamorphosed ones
(types 3.0–3.2).
One strong possibility to account for the small fall statistics
of type 3 ordinary chondrites (as well as the tendency that lessmetamorphosed type 3 subtypes are less abundant) is impact
events for planetesimals. In this study, we assume thatplanetesimals do not undergo any destructive processes once

5. Conclusions
In this paper, we systematically examine the thermal
evolution of planetesimals with a wide range of initial sizes
and formation times. Our numerical results show thatplanetesimals can reach higher temperatures when they format earlier times and/or have larger initial radii. There is aconsiderable difference in maximum temperatures between asmall size range (from 1 to 50 km) and a larger size range (from60 to 90 km and/or from 100 to 500 km) (see Figure 8).
We also examine the effect of the formation rate ofplanetesimals. We found that, in the case of the continuous
formation of planetesimals, their mass fraction is signiﬁcantly
affected by the onset time of planetesimal formation (Figure 8).
This trend is prominent when we focus on the planetesimal
11

The Astrophysical Journal, 863:100 (13pp), 2018 August 10

Wakita, Hasegawa, & Nozawa

Table 3
Relative Abundance Ratios in Type 3 Ordinary Chondrites Calculated in This Study
Types 3.0–3.2

Types 3.3–3.6

Types 3.7–3.9

Types 3.3–3.6/Types 3.7–3.9

Fall or Figure and Parameters

0.13

0.40

0.47

0.84

Fall Statistics

0.83
0.88
0.82
0.81
0.81
0.88
0.93
0.76
0.81

0.09
0.07
0.10
0.10
0.10
0.07
0.04
0.12
0.10

0.08
0.05
0.08
0.09
0.09
0.05
0.03
0.12
0.09

1.16
1.36
1.21
1.10
1.06
1.29
1.52
1.07
1.14

Figure 8(a): Rpl=1–500 km, α=2.8
Figure 8(b): Rpl=1–50 km, α=2.8
Figure 8(c): Rpl=60–90 km, α=2.8
Figure 8(d): Rpl=100–500 km, α=2.8
Figure 9(a): Rpl=1–500 km, α=0.0
Figure 9(b): Rpl=1–500 km, α=3.5
Figure 9(c): Rpl=1–500 km, α=4.5
Figure 10(a): Rpl=1–500 km, α=2.8, τf=1.0 Myr
Figure 10(b): Rpl=1–500 km, α=2.8, τf=4.0 Myr

Note.Abundance ratio of each subtype is given for the case of tint=1.5 Myr and is normalized by the sum of the values for types 3.0 to 3.9.

We thank the referee, Andrew M. Davis, for helpfulcomments and suggestions. Numerical computations werecarried out on the PC cluster at the Center for Computational
Astrophysics, National Astronomical Observatory of Japan.
T.N. has been supported in part by a JSPS Grant-in-Aid for
Scientiﬁc Research (26400223). Part of this research was
carried out at the Jet Propulsion Laboratory, California Instituteof Technology, under a contract with the National Aeronautics
and Space Administration. Y.H. is supported by JPL/Caltech.
ORCID iDs
Shigeru Wakita
Takaya Nozawa
Figure 13. Relative abundance ratios of types 3.3–3.6 to types 3.7–3.9 ofordinary chondrites. Each line comes from our results (see legends andcorresponding ﬁgures). The fall statistics of ordinary chondrites (dashed line)with the uncertainty±30% (shaded region) are also shown.

https://orcid.org/0000-0002-3161-3454
https://orcid.org/0000-0002-6153-7915
References

Amelin, Y., Kaltenbach, A., Iizuka, T., et al. 2010, E&PSL, 300, 343
Blackburn, T., Alexander, C. M. O., Carlson, R., & Elkins-Tanton, L. T. 2017,
GeCoA, 200, 201
Bland, P. A., & Travis, B. J. 2017, SciA, 3, e1602514
Bollard, J., Connelly, J. N., & Bizzarro, M. 2015, M&PS, 50, 1197
Bonal, L., Quirico, E., Bourot-Denise, M., & Montagnac, G. 2006, GeCoA,
70, 1849
Bottke, W. F., Durda, D. D., Nesvorný, D., et al. 2005a, Icar, 175, 111
Bottke, W. F., Durda, D. D., Nesvorný, D., et al. 2005b, Icar, 179, 63
Cody, G. D., Alexander, C. M. O., Yabuta, H., et al. 2008, E&PSL, 272, 446
Connelly, J. N., Bizzarro, M., Krot, A. N., et al. 2012, Sci, 338, 651
Cuzzi, J. N., Hogan, R. C., Paque, J. M., & Dobrovolskis, A. R. 2001, ApJ,
546, 496
Davis, A. M., Alexander, C. M. O., Ciesla, F. J., et al. 2014, in Protostars andPlanets VI, ed. H. Beuther et al. (Tucson, AZ: Univ. Arizona Press), 809
Davis, A. M., & McKeegan, K. D. 2014, in Meteorites and Cosmochemical
Processes, Vol. 1, Treatise of Geochemistry, ed. A. M. Davis (2nd ed.; New
York: Elsevier), 361
Dodd, R. T. 1981, Meteorites–A Petrologic-chemical Synthesis (Cambridge:
Cambridge Univ. Press)
Gail, H.-P., Trieloff, M., Breuer, D., & Spohn, T. 2014, in Protostars andPlanets VI, ed. H. Beuther et al. (Tucson, AZ: Univ. Arizona Press), 571
Ghosh, A., & McSween, H. Y., Jr. 2000, M&PSA, 35, A59
Grady, M., Pratesi, G., & Moggi Cecchi, V. 2014, Atlas of Meteorites(Cambridge: Cambridge Univ. Press)
Grimm, R. E., & McSween, H. Y., Jr. 1989, Icar, 82, 244
Grossman, J. N., & Brearley, A. J. 2005, M&PS, 40, 87
Hasegawa, Y., Turner, N. J., Masiero, J., et al. 2016a, ApJL, 820, L12
Hasegawa, Y., Wakita, S., Matsumoto, Y., & Oshino, S. 2016b, ApJ, 816, 8
Henke, S., Gail, H.-P., Trieloff, M., & Schwarz, W. H. 2013, Icar, 226, 212
Huss, G. R., Rubin, A. E., & Grossman, J. N. 2006, in Meteorites and the Early
Solar System II, ed. D. S. Lauretta & H. Y. McSween, Jr. (Tucson, AZ:
Univ. Arizona Press), 567
Ida, S., & Lin, D. N. C. 2004, ApJ, 604, 388
Johansen, A., Blum, J., Tanaka, H., et al. 2014, in Protostars and Planets VI,
ed. H. Beuther et al. (Tucson, AZ: Univ. Arizona Press), 547

mass that experiences higher temperatures ( 600 °C). Furthermore, our results show that planetesimals should form within
2.0 Myr after formation of Ca-Al-rich inclusions (CAIs), in
order to produce type 6 ordinary chondrites, which would
have experienced the peak metamorphic temperature of800∼1000 °C. We also found that the ratio of type 6 totypes 4 and 5 chondrites in fall statistics can be well explainedby thermal evolution models of planetesimals. This strongly
suggests that these types of chondrites are produced mainly bythermal metamorphism inside the parent bodies. Preferable
scenarios to account for the fall statistics are that planetesimals
with radii of 60 km start to form around 2.0 Myr and/or thatplanetesimals of 50 km start to form at 1.5 Myr after the
CAIs formation.
On the other hand, our thermal evolution models ofplanetesimals cannot explain the fall statistics of type 3
ordinary chondrites; we predict that type 3 chondrites areunavoidably more abundant than any of the types 4 to 6
chondrites. This is against the fact that type 3 chondrites are theleast abundant in the fall statistics. The overabundance of type
3 chondrites could be resolved by taking into accountsubsequent destructive processes such as collisions between
planetesimals. In our future work, we will explore the effect ofimpact and subsequent destruction for a better understanding of
the formation and evolution histories of planetesimals.
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The Astrophysical Journal, 863:100 (13pp), 2018 August 10

Wakita, Hasegawa, & Nozawa
Nakamura, T., Noguchi, T., Tanaka, M., et al. 2011, Sci, 333, 1113
Opeil, C. P., Consolmagno, G. J., & Britt, D. T. 2010, Icar, 208, 449
Pollack, J. B., Hubickyj, O., Bodenheimer, P., et al. 1996, Icar, 124, 62
Raymond, S. N., Kokubo, E., Morbidelli, A., Morishima, R., & Walsh, K. J.
2014, in Protostars and Planets VI, ed. H. Beuther et al. (Tucson, AZ: Univ.
Arizona Press), 595
Ricard, Y., Bercovici, D., & Albarède, F. 2017, Icar, 285, 103
Saumon, D., & Guillot, T. 2004, ApJ, 609, 1170
Scott, E. R. D., & Jones, R. H. 1990, GeCoA, 54, 2485
Scott, E. R. D., Jones, R. H., & Rubin, A. E. 1994, GeCoA, 58, 1203
Scott, E. R. D., & Krot, A. N. 2005, in Meteorites, Comets and Planets: Treatiseon Geochemistry, Vol. 1 ed. A. M. Davis (Amsterdam: Elsevier), 143
Sears, D. W., Grossman, J. N., Melcher, C. L., Ross, L. M., & Mills, A. A.
1980, Natur, 287, 791
Simon, J. B., Armitage, P. J., Li, R., & Youdin, A. N. 2016, ApJ, 822, 55
Travis, B. J., & Schubert, G. 2005, E&PSL, 240, 234
Tsirvoulis, G., Morbidelli, A., Delbo, M., & Tsiganis, K. 2018, Icar, 304, 14
Van Schmus, W. R., & Wood, J. A. 1967, GeCoA, 31, 747
Vernazza, P., Zanda, B., Binzel, R. P., et al. 2014, ApJ, 791, 120
Vernazza, P., Zanda, B., Nakamura, T., Scott, E. R. D., & Russell, S. 2015, inAsteroids IV, ed. P. Michel et al. (Tucson, AZ: Univ. Arizona Press), 617
Wakita, S., Matsumoto, Y., Oshino, S., & Hasegawa, Y. 2017a, ApJ,
834, 125
Wakita, S., Nakamura, T., Ikeda, T., & Yurimoto, H. 2014, M&PS, 49, 228
Wakita, S., Nozawa, T., & Hasegawa, Y. 2017b, ApJ, 836, 106
Weisberg, M. K., McCoy, T. J., & Krot, A. N. 2006, in Meteorites and the
Early Solar System II, ed. D. S. Lauretta & H. Y. McSween, Jr. (Tucson,
AZ: Univ. Arizona Press), 19
Wetherill, G. W., & Stewart, G. R. 1989, Icar, 77, 330
Wlotzka, F. 2005, M&PS, 40, 1673
Yomogida, K., & Matsui, T. 1983, Metic, 18, 430
Yoshikawa, M., Kawaguchi, J., Fujiwara, A., & Tsuchiyama, A. 2015, inAsteroids IV, ed. P. Michel et al. (Tucson, AZ: Univ. Arizona Press), 397
Zheng, X., Lin, D. N. C., & Kouwenhoven, M. B. N. 2017, ApJ, 836, 207

Johansen, A., Mac Low, M.-M., Lacerda, P., & Bizzarro, M. 2015, SciA, 1,
1500109
Johansen, A., Oishi, J. S., Mac Low, M.-M., et al. 2007, Natur, 448, 1022
Johnson, B. C., Minton, D. A., Melosh, H. J., & Zuber, M. T. 2015, Natur,
517, 339
Keil, K. 2000, P&SS, 48, 887
Kokubo, E., & Ida, S. 1998, Icar, 131, 171
Kominami, J. D., Daisaka, H., Makino, J., & Fujimoto, M. 2016, ApJ, 819, 30
Kretke, K. A., & Lin, D. N. C. 2007, ApJL, 664, L55
Krot, A. N., Keil, K., Goodrich, C. A., Scott, E. R. D., & Weisberg, M. K.
2005, in Meteorites, Comets and Planets: Treatise on Geochemistry, Vol. 1
ed. A. M. Davis (Amsterdam: Elsevier), 83
Levison, H. F., Thommes, E., & Duncan, M. J. 2010, AJ, 139, 1297
Lichtenberg, T., Golabek, G. J., Gerya, T. V., & Meyer, M. R. 2016, Icar,
274, 350
MacPherson, G. J. 2005, in Meteorites, Comets, and Planets: Treatise on
Geochemistry, Vol. 1 ed. A. M. Davis (Amsterdam: Elsevier), 201
MacPherson, G. J., Davis, A. M., & Zinner, E. K. 1995, Metic, 30, 365
Matsumoto, Y., Oshino, S., Hasegawa, Y., & Wakita, S. 2017, ApJ, 837, 103
McSween, H. Y., Jr. (ed.) 1999, Meteorites and their Parent Planets (Cambridge:
Cambridge Univ. Press), 322
McSween, H. Y., Jr., Ghosh, A., Grimm, R. E., Wilson, L., & Young, E. D.
2002, in Asteroids III, ed. W. F. Bottke, Jr. et al. (Tucson, AZ: Univ.
Arizona Press), 559
Miller, N., & Fortney, J. J. 2011, ApJL, 736, L29
Miyamoto, M., Fujii, N., & Takeda, H. 1982, LPSC, 12, 1145
Monnereau, M., Toplis, M. J., Baratoux, D., & Guignard, J. 2013, GeCoA,
119, 302
Morbidelli, A., Bottke, W. F., Nesvorný, D., & Levison, H. F. 2009, Icar,
204, 558
Mordasini, C., Alibert, Y., & Benz, W. 2009, A&A, 501, 1139
Mordasini, C., van Boekel, R., Mollière, P., Henning, T., & Benneke, B. 2016,
ApJ, 832, 41
Morishima, R., Stadel, J., & Moore, B. 2010, Icar, 207, 517

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