# The Macroporosity of Rubble Pile Asteroid Ryugu and Implications for the
Origin of Chondrules

William Herbst Department of Astronomy, Wesleyan University, Middletown, CT,
06457, USA  James P. Greenwood  Department of Earth & Environmental Sciences,
Wesleyan University, Middletown, CT, 06457, USA  Teng Ee Yap  Department of
Physics and Astronomy, Colgate University, Hamilton, NY, 13346 USA

###### Abstract

We use the known surface boulder-size distribution of the C-type rubble pile
asteroid Ryugu (NEA 162173) to determine its macroporosity, assuming it is a
homogeneous granular aggregate. We show that the volume-frequency distribution
of its boulders, cobbles and pebbles, is well represented by a lognormal
function with $\sigma=2.4\pm 0.1$ and $\mu=0.2\pm 0.05$. Application of
linear-mixture packing theory yields a value for the macroporosity of
$\phi=0.14\pm 0.04$. Given its low bulk density of 1.19 gm cm-3, this implies
an average density for Ryugu’s rocks of $1.38\pm 0.07$ gm cm-3 throughout its
volume, consistent with a recent determination for surface boulders based on
their thermal properties. This supports the spectrum-based argument that IDP’s
may be the best analog material available on Earth and suggests that high-
density, well-lithified objects such as chondrules and chondrule-bearing
chondrites may be rare on Ryugu. Implications of this result for the origin of
chondrules, a long-standing problem in cosmochemistry, are discussed. We
propose that chondrules and most chondrites formed together in rare
lithification events, which occurred during the accretion of chondritic
envelopes to large, differentiated planetesimals at a time when they were
still hot from 26Al decay.

asteroids, chondrites, chondrules

††journal: PSJ

##  1 Introduction

Asteroids and meteorites can be associated with one another by their
reflectance spectra (Greenwood, Burbine & Franchi, 2020). However, while
overall composition may be similar, the bulk density ($\rho_{bulk}$) of small
asteroids is often significantly less than the density of their analog
meteorites (Britt et al., 2002). For example, the meteorite analogs of 162173
Ryugu and 101955 Bennu, are carbonaceous chondrites (CC), whose densities,
with the exception of the rare CI class, range from 2.2 to 5.3 gm cm-3 (Macke,
Consolmagno & Britt, 2011), well above the asteroids’ bulk densities of
$1.19\pm 0.02$ and $1.19\pm 0.013$ gm cm-3, respectively (Watanabe et al.,
2019; Lauretta et al., 2019). Similarly, the S-type asteroid 25143 Itokawa has
$\rho_{bulk}=1.95\pm 0.14$ gm cm-3 (Abe et al., 2006), well below its
meteorite analog, LL ordinary chondrites (OC), which have densities in the
range 3.15 - 3.4 gm cm-3 (Consolmagno, et al., 2006). The standard explanation
(Britt et al., 2002) has been that the asteroids are ‘rubble piles’, i.e.
loosely packed piles of gravitationally reassembled pieces of a pre-existing
object that suffered a major impact. An assumption behind this interpretation
is that asteroidal rocks can be assigned the same density as their associated
meteorites, which were, of course, collected on the Earth’s surface.

This assumption is challenged by recent results from the ongoing sample return
missions Hayabusa2 (Watanabe et al., 2019) and OSIRIS-REx (Lauretta et al.,
2017), which visited Ryugu and Bennu, respectively. Based on its thermal
properties one may infer that a Ryugu boulder studied by Grott et al. (2019)
and Hamm et al. (2020) has a porosity of between 0.30 and 0.52. This
translates to a boulder density of between 1.2 and 1.7 gm cm-3 if we assume a
grain density appropriate to the closest spectral match to Ryugu, CI
meteorites (Kitazato et al., 2019). Global images of Ryugu indicate that such
high porosity boulders are very much the rule, not the exception on its
surface (Okada et al., 2020). Grott et al. (2020) found an average density of
around 1.4 gm cm-3 for Ryugu rocks in their study of its packing density.
During the sample retrievals it has become quite clear that the tensile
strength of the surface rocks is very low compared with most, if not all,
meteorites. Grott et al. (2019) have shown that the Ryugu boulder they studied
probably does not have the tensile strength to survive passage through the
Earth’s atmosphere. It may be, therefore, that these asteroids are not
composed of material already familiar to us from meteorite collections, even
though they have similar reflectance spectra. In particular, meteorites appear
to have higher densities and greater tensile strength than common rocks on
these asteroids. If confirmed, this may have implications for the densest and
most abundant component of such meteorites, their chondrules.

Chondrules are dense, mm-scale, igneous spherules that comprise 20-80% of the
volume of nearly all of their eponymous meteorite type, the chondrites (Sears,
2004; Connolly and Jones, 2016). Since $\sim$90% of all meteorites in our
collections are chondrites, there is a common opinion that chondrules are
abundant in primitive asteroids and in the early Solar System (Krot et al.,
2018). They are often referred to in text books as the ‘building blocks’ of
terrestrial planets and could play an important role in that process (Johansen
et al., 2015). It is disappointing, therefore, that despite decades of
concerted effort and an abundance of fine detail, there remains no consensus
among cosmochemists on how they formed (Ciesla, 2005; Desch et al., 2012;
Connolly and Jones, 2016; Russell, Connolly & Krot, 2018). Their textures
indicate that they were born in one or more short blasts of heat, lasting only
minutes to hours, but intense enough to melt pre-chondrule matter (requiring
T$\geqslant$1600 K) without fully vaporizing it (Hewins and Radomsky, 1990;
Jones et al., 2018). But the community is divided on whether these heat blasts
occurred in a ‘nebular’ setting or a ‘planetary’ setting (Connolly and Jones,
2016; Russell, Connolly & Krot, 2018). That is, was chondrule formation a
widespread phenomenon within the protoplanetary disk that preceded asteroid
and planet formation, or a much smaller scale phenomenon associated with the
late stages of planetesimal formation? In the first case, one might expect
primitive asteroids to be as full of chondrules as chondrites are. In the
second, chondrules may be rare or absent on many such objects.

Sears (1998) pointed out decades ago that an extremely rare class of
chondrites on Earth, CI’s, which contain no chondrules at all, may actually be
the most common type of chondrite in space. Their low density and tensile
strength make them much less likely to survive passage through the Earth’s
atmosphere and terrestial weathering, should they reach the Earth’s surface. A
space-based sample of primitive asteroidal material might, therefore, contain
a much lower percentage of chondrules than the Earth-based sample represented
by our museum collections. This point of view is buttressed by the fact that
the flux of meteorites reaching the surface of the Earth is only $\sim 0.15$
metric tonnes per day, or less than 0.1-0.5% of the 30 - 180 t of material
impacting the Earth daily at the top of the atmosphere (Bland et al., 1996;
Zolensky et al., 2006; Drolshagen et al., 2017). Our meteorite collections
clearly sample only a tiny fraction of the interplanetary material incident on
the Earth, with most of it arriving in the form of interplanetary dust
particles (IDP’s) that are much smaller than chondrites or chondrules and
could have an asteroidal or cometary origin.

Here we attempt to illuminate these issues by directly calculating the amount
of void space in Ryugu’s interior, based on what we know about granular piles
on Earth, which then allows us to estimate the average density of its rocks.
Space between constituent rocks in a pile is quantified by porosity ($\phi$),
defined as

$\phi=\left(1-{V_{rock}\over V_{bulk}}\right)$ (1)

where Vrock is the volume occupied by the rocks (grains) themselves and Vbulk
is the volume occupied by the pile (asteroid). In planetary science this is
generally referred to as the ‘macroporosity’ of the pile to distinguish it
from the porosity of the rocks themselves, known as ‘microporosity’. Assuming
that the constituent rocks have a single density ($\rho_{rock}$), and that the
pore spaces are empty, equation 1 may be rewritten in terms of bulk and rock
densities, as

$\phi=\left(1-{\rho_{bulk}\over\rho_{rock}}\right).$ (2)

Application of this equation yields a porosity of about 0.4 for Itokawa (Abe
et al., 2006) and 0.48 - 0.62 for Ryugu and Bennu if they are composed of
rocks with the same density as LL OC’s and common, chondrule-bearing CC’s,
respectively. If Ryugu and/or Bennu are composed of non-chondrule bearing CI
material then their inferred macroporosity is much lower ($\sim 0.24$), since
CI density, based on a single measurement, is only 1.57 gm/cm3 (Macke,
Consolmagno & Britt, 2011). Grott et al. (2020) recently reported a value of
$\phi=0.16\pm 0.03$ for Ryugu based on a similar approach to the one adopted
here.

In this paper, we first calculate $\phi$ for Ryugu and then use Equation 2 to
determine its average rock density, which can be compared with chondrules,
meteorites and with the surface boulder measurement. We model the asteroid as
a self-gravitating, homogeneous granular aggregate, whose grain size
distribution can be inferred from surface boulder counts. We proceed by first
deriving the boulder-size distribution that would apply to a representative
volume within the surface layer of the asteroid, based on the surface boulder
counts of Michikami et al. (2019). We show that the distribution is well-
represented by a lognormal function. This allows us to estimate the
macroporosity of Ryugu by application of linear-mixture packing theory for a
continuous frequency distribution (Yu & Standish, 1991). We then address the
broader question of what the relationship between chondrites and their host
asteroids may be. This impacts discussion of the long-standing and vexing
problem of the origin of chondrules, which has implications for the structure
and origin of asteroids and terrestrial planets, as we discuss.

##  2 The Porosity of a Granular Aggregate

There is considerable evidence now that small, rubble pile asteroids such as
Itokawa, Ryugu and Bennu are ‘granular aggregates’, composed entirely of
mobile, unconsolidated material in the form of boulders, cobbles, pebbles and
perhaps finer grained material, held together by self-gravity and containing
no separate, monolithic core (Abe et al., 2006; Sharma, 2013, 2014; Jawin et
al., 2020; Polishook and Aharonson, 2020). Granular aggregates on Earth are
common in river beds and beaches and have been studied extensively in a wide
variety of contexts. They do not fit within the usual solid/liquid/gas
classification of states of matter but exhibit a range of phenomena unique to
themselves, including granular flow or convection (Jaeger & Nagel, 1992). For
example, large boulders, even when denser than their surroundings, rise to the
top of shaken piles instead of sinking to their bottoms, as they would in a
liquid or gas, a phenomenon known as the ‘Brazil Nut Effect’ (Rosato et al.,
1987; Matsumura et al., 2014).

The principal factors affecting the porosity of a granular aggregate on Earth
are well known. Of greatest importance are grain size distribution, grain
shape and how the grains are packed (Yu & Standish, 1991). In terrestrial
piles, liquid often fills the pores and that can affect porosity. In the case
of very small grains (a few hundred microns or less), cohesive effects can
play a role due to the relative importance of electrostatic forces between
grains (Yang, Zou & Yu, 2000; Zou et al., 2011). Here we assume that the pores
in a granular aggregate asteroid are empty and that cohesive forces between
them are negligible. We return to a discussion of the latter point after
establishing the size of the grains controlling porosity on Ryugu. As a
geometric entity, porosity is independent of absolute scale unless forces
other than gravity are involved (Yu & Standish, 1991; Frings, Schüttrumpf &
Vollmer, 2011). While the surface gravity of a small asteroid is only about
$10^{-5}$ g, its porosity-controlling grains (rocks) are meter-scale, or 1000
times larger than the mm-scale grains composing typical terrestrial
aggregates. There is no reason of which we are aware that terrestrial results
on porosity would not be applicable to granular piles in space, despite these
scale differences.

The porosity of a pile with a single grain size depends on the grain shape and
how they are packed. Spheres are best studied and experimental, analytic and
computational techniques give consistent results. The theoretical lower limit
for spheres of a single size is

$\phi={1-{\pi\over 3\sqrt{2}}}=0.26,$

a result that was conjectured by Kepler and proven by Carl Friedrich Gauss in
1831 (Hales, 1998). When spheres are randomly poured into a container the
porosity of the pile will initially vary between 0.37 and 0.445 and if the
container is shaken sufficiently it will eventually achieve a condition known
as random close packing (RCP) characterized by $\phi=0.3634\pm 0.0005$ (Scott
& Kilgour, 1969). Non-spherical particles in RCP typically have a porosity of
around 0.40 unless the particle shapes are extreme (e.g. needles or flat
disks) (Yu & Zou, 1998).

In binary mixtures or piles with a range of sizes $\phi$ can be lowered
substantially by two factors that come into play. First, small grains tend to
filter down through the spaces between larger grains, filling them and,
thereby, reducing porosity. Second, large grains embedded in a sea of smaller
ones displace not only the grains themselves but also the pore space between
them, again lowering the overall porosity of the pile. The first effect is
commonly known as infiltration or sifting and the second as occupation (Yu &
Standish, 1991). The magnitude of these effects depends on the relative sizes
and abundances of the grain mixtures, but the direction is always the same –
mixtures only lower the porosity of a pile, sometimes dramatically. A well-
chosen size distribution, such as the ‘Fuller distribution’ can reduce
porosity to nearly zero, a fact that has been known for more than a century
and is exploited commercially in making cement (Fuller & Thompson, 1907).

Westman & Hugill (1930) showed that, for binary mixtures of spherical
particles with r1 $>$ r2, $\phi$ decreases rapidly to a minimum value of 0.16
as the ratio r1/r2 increases. The minimum value is achieved when the larger
granules make up about 2/3 of the total volume. In that case, they create a
framework with $\sim$36% void space where the small particles congregate.
Adding additional components has the effect of lowering the minimum porosity
limit even further. For example, Westman & Hugill (1930) find that a three
component mixture with a ratio of granule radii in the proportion 50.5:8:1 has
a minimum porosity of only 0.053 when mixed in a volume proportion of 66%
large, 25% medium and 9% small particles. Mixtures with nearly equal volumes
yield a porosity of 0.22. It is important to note that not only the grain size
distribution, but also the degree to which the particles are mixed, determines
the overall porosity of a pile. In a completely layered structure, for
example, neither sifting nor occupation occur so the porosity of the pile
assumes the value for a uniform size distribution. Interestingly, shaking a
pile with a distribution of grain sizes can actually raise its porosity, not
lower it, because of the Brazil Nut Effect, which tends to separate large
particles (at the top) from smaller ones (at the bottom), at least in the
laboratory within walled containers.

Ideally, one would like to know the grain size and shape distribution as a
function of depth within a pile to most accurately predict its macroporosity.
For asteroids, such information will not be known for a long time, if ever.
However, under the assumptions that the object is well mixed and homogeneous
we can use surface boulder counts and shape determinations to estimate
reasonable limits on $\phi$. Then, equation 2 can be used to infer the grain
(rock) density, which can be compared with direct measurements of surface
rocks from Ryugu brought back by the Hayabysa2 mission, when they become
available. If there is consistency between surface measurements and the ‘whole
body’ average determined by this approach it would support the assumption of
homogeneity. That is the approach adopted here, and it begins with a
determination of the boulder size distribution applicable to the surface layer
of the asteroid.

##  3 The Boulder-Size Distribution within Ryugu’s Surface Layer

Michikami et al. (2019) report rock counts by size on the surface of Ryugu in
the range 0.02 to 140 m. From the counts they derive the relative size-
frequency distribution, R, which applies to a unit area on the surface of the
asteroid (see their figures 2b and 5b). Since the asteroid does not have an
independent solid surface on which the boulders lie, but is a granular
aggregate, the areal counts do not directly reflect what the size distribution
in a unit volume of the surface layer would be. In particular, larger boulders
are over-represented in areal counts because they can be seen to greater
depths within the surface layer than smaller boulders, cobbles or pebbles.

Figure 1: The normalized volume frequency distribution for Ryugu boulders,
derived from the areal counts of Michikami et al. (2019), compared to a
lognormal distribution with $\sigma=2.4\pm 0.1$ and $\mu=0.2\pm 0.05$. The fit
is clearly quite good and characterized by a reduced $\chi^{2}=1.13$. Error
bars based on counting statistics alone are shown. Where they do not appear,
the error bars are smaller than the data points.

In Appendix A we derive the relationship between areal counts, quantified by
R(d), and volume counts, quantified by $f(d)$, where $f(d)$ is the size-
frequency distribution that applies to a unit volume of the asteroid’s surface
(see Equation A6). Since the boulder counts are done on images of vastly
different scales, taken from a range of heights above the asteroid, they must
be normalized before being compared with one another. In carrying out this
process, we noticed the strong resemblance of $f(d)$, based on the boulder
counts, to a lognormal distribution (see Fig. 5 of Appendix A).

Since a lognormal distribution is the result of a natural process that
involves many small, uncorrelated multiplicative/divisive events and since it
appears as commonly in nature as its better-known analog the normal (Gaussian)
distribution that results from a large number of uncorrelated
addition/subtraction events (Limpert, Stahel & Abbt, 2001), this is not
surprising. The surface of Ryugu is presumably frequently hit by meteoroids,
which may fragment its rocks, leading to a representative volume sample of its
surface having a lognormal boulder-size distribution no matter what their
initial size distribution may have been. Or, the asteroid may have reassembled
from a catastrophic distruction that endowed its remnants with a lognormal
distribution. While the lognormal function is one of the most commonly
employed representations in all fields of science, it has been particularly
associated with fragmentation processes on a variety of scales, from rock
crushing to the formation of stars in clusters and associations (Miller and
Scalo, 1979; Brown, 1989). It is worthwhile, therefore, to investigate exactly
how closely $f(d)$ follows a lognormal distribution.

Figure 2: $d\times f(d)$ versus log (d) showing the Gaussian form explicitly.
The peak of the distribution is at boulders with d = exp(0.2) = 100.087 = 1.2
m. Error bars on the points reflect counting statistics alone and are, in some
cases, smaller than the points themselves.

The lognormal function may be written as:

${f(x)}={1\over{\sigma x\sqrt{2\pi}}}\ {\rm
exp}{\left(-{{(ln(x)-\mu)^{2}}\over{2\sigma^{2}}}\right)}.$ (3)

It has two free parameters, $\sigma$ and $\mu$, and is normalized such that
$\int_{0}^{\infty}f(x)dx=1.$ In Figure 1 we compare our measurements of
$f(d)$, renormalized to a common scale (see Appendix B), based on the surface
boulder counts, to a best-fit lognormal distribution, determined by a non-
linear Levenberg-Marquardt fitting technique. The data points are weighted by
the number of boulders counted in each bin and error bars reflect only
counting statistics. The fitting process returns the following values and
error limits on the parameters: $\mu=0.20\pm 0.05$ and $\sigma=2.4\pm 0.1$. By
eye, the lognormal distribution is clearly a good fit to the data over its
full range, approaching four orders of magnitude in size. This impression is
verified quantitatively by the reduced $\chi^{2}$ value that applies to the
fit, $\chi^{2}=1.13$. The reduced chi-square is the chi-square per degree of
freedom and a value of 1 is expected of a perfect fit to data whose error bars
are accurately known. Since we have only included Poisson (counting) effects
in the estimation of errors on the individual data bins here, and since there
are only two free parameters ($\mu$ and $\sigma$) involved in the fit, this is
about as good as one could expect. It would seemingly be hard to improve on
the lognormal function as an appropriate representation of these data,
especially given its physical underpinning.

In Figure 2 we display the same data and fit in a different manner that
highlights its Gaussian nature and makes the quality of the fit more apparent.
In this case, the ordinate ($d\times f(d)$) is unitless and is the fraction of
the sample that would be contained within a bin centered on the specified
boulder size. The distribution peaks at a value of exp(0.2) = 1.2 m,
indicating that in a representative sample of the surface layer of Ryugu,
boulders a little larger than 1 m are the most common contributors to the
total volume or mass. The fact that its surface layer is so well represented
as lognormal greatly simplifies the task of estimating the porosity of Ryugu
under the assumption that it is an homogeneous object. The size scale of the
boulders controlling the porosity ($\sim$1 m) also informs the discussion of
the extent to which mobility issues and/or cohesion among grains may or may
not be important to consider in estimating the macroporosity of Ryugu, a task
to which we now turn.

##  4 The Macroporosity of Ryugu and Average Density of its Boulders

With $f(d)$ established for the surface layer we can proceed to estimate the
porosity of Ryugu under the assumption that it is an homogeneous object.
Frings, Schüttrumpf & Vollmer (2011) have evaluated a number of methods
developed for terrestrial piles and found one, the ‘linear-mixture’ packing
theory of Yu & Standish (1991), to be the most reliable and we employ it here.
It is a semi-empirical theory that can be applied to either discrete or
continuous frequency distributions. The authors have provided a convenient
representation of the predictions of their model for the lognormal
distribution because of its widespread appearance as a descriptor applicable
to terrestrial piles. As they show in their Figure 12, the macroporosity
($\phi$) of a pile with a lognormal grain-size distribution depends on only
two parameters, the geometric standard deviation, $\sigma_{G}$, and the
‘initial porosity’, $\phi_{0}$, of the grains. Note, in particular, that the
macroporosity is independent of the median particle size, $\mu$, as long as
the initial porosity is independent of grain size.

Figure 3: Macroporosity ($\phi$) versus geometric standard deviation
($\sigma_{G}$) of a lognormal distribution for samples of varying initial
porosity ($\phi_{0}$). The lines indicate the predictions of linear-mixture
theory; this figure is an adaptation of Fig. 12 of Yu & Standish (1991). Data
for analog samples were obtained by us (see Appendix C) and the data for Ryugu
boulders, are based on the surface measurements of Michikami et al. (2019) as
analyzed in this paper.

Yu & Standish (1991) present the porosity predictions of linear-mixture theory
for a lognormal distribution (Fig. 12 of their paper) in terms of packing
density (1 - $\phi$) and inverse geometric standard deviation
($\sigma_{G}^{-1}$). We display their results in Figure 3 in terms of $\phi$
and $\sigma_{G}$. Third order polynomial fits were used to transform the
solution lines on their Fig. 12 to those on our Fig. 3. These lines represent
the results for different values of the ‘initial porosity’, $\phi_{0}$, which
is the macroporosity of a pile of single-sized grains of the appropriate
shape. As may be seen on Fig. 3, linear-mixture theory predicts that the
macroporosity ($\phi$) of a pile decreases as the dispersion in grain sizes
within it, measured by $\sigma_{G}$, increases. Again, this is due to the
combined effects of sifting and occupation, which become more effective at
eliminating pore space as the range of particle sizes increases. It is clear
from the theory and its empirical confirmation that a granular aggregate with
a large dispersion in grain sizes will have a smaller macroporosity than an
aggregate of similarly shaped grains with a smaller range in sizes. It is also
clear that to achieve a porosity in excess of about 0.25 in any object with a
dispersion in grain size as large as Ryugu’s, would require rather extremely
shaped objects ($\phi_{0}>0.6$).

The value of $\sigma_{G}$ for Ryugu is obtained from the lognormal fit to its
boulder-size distribution discussed in Section 3. According to the formulation
used by Yu & Standish (1991) it is $\sigma_{G}=exp(\sigma)=11.0\pm 1.0$.
Therefore, we only need to determine $\phi_{0}$ to predict Ryugu’s
macroporosity from linear-mixture theory. Initial porosity can be determined
empirically from laboratory simulations using analog particles with a similar
shape distribution to what is measured for the asteroid. Again, unless the
grain shapes are extreme or there are non-gravitational forces at play,
initial porosity is generally around 0.4, only a little larger than what is
found for spheres in RCP (Yu & Zou, 1998).

As described in Appendix C, we assembled four sets of analog grains with small
grain size dispersions and shapes similar to those measured for Ryugu’s rocks
by Michikami et al. (2019). We measured their values of porosity in order to
constrain the value of $\phi_{0}$ for the asteroid. The results are given in
Table 2 of Appendix C and plotted in Fig. 3 as the x’s. As expected they yield
values for initial porosity between 0.36 and 0.48. On the basis of those
measurements we estimate $\phi_{0}=0.43\pm 0.07$ for the asteroid, where the
error bar is a reasonably conservative outer bound, as evident on the figure.
The solid circle on Figure 3 represents Ryugu based on these numbers. As can
be seen on the figure, linear-mixture theory predicts a macroporosity of
$\phi=0.14\pm 0.04$ for the asteroid.

There are several factors that could undermine the reliability of this result
and deserve discussion before moving on. One, of course, is the assumption of
homogeneity. Due to granular convection and the Brazil Nut Effect, it is
possible that the surface layer of Ryugu is overpopulated with large boulders
compared to its interior. Stratification of that sort would tend to increase
$\phi$, by reducing the impacts of sifting and occupation. If this affects
only the largest boulders, e.g. Otohime, which perches precariously on the
very edge of the surface layer, it would not have much effect on $\phi$ since,
as Fig. 2 shows, the largest boulders account for only a very small fraction
of the total mass. In linear-mixture theory, it is the intermediate-size
grains that primarily control the macroporosity of the pile and it is unclear
whether granular convection and shock-induced shaking and mixing could
stratify an asteroid sufficiently to have a significant effect on $\phi$.
Exactly how a granular aggregate asteroid evolves, given repeated impacts that
can create smaller fragments on the surface, which will sift downward,
displacing larger boulders that are forced upward, is an interesting question.
Computer simulations of this process have been reported by Matsumura et al.
(2014). Depending on how long Ryugu has existed as a granular aggregate it may
have ‘turned over’ once or thousands of times. Rocks on its current surface
have not necessarily been there since its formation and the object likely
transforms itself constantly in response to impacts.

Unfortunately, there is no simple way at present to test whether Ryugu is
actually homogeneous or, more precisely, to determine exactly how
inhomogeneous it is. In a granular aggregate there is reason to expect that it
will evolve toward inhomogeneity since repeated shaking, driven by impacts,
may lead to stratification, with the smaller particles sifting to the bottom.
This would increase porosity both there and at the surface due to a smaller
range in particle sizes. Scheeres (2020) has presented some weak ($\sim
3\sigma$) evidence of such an effect on Bennu, which is about half the size of
Ryugu. On Itokawa, one finds a ‘beach” of smaller grains at the gravitational
low point between its two lobes, indicating that a sifting-driven
stratification effect is present on its surface. If Ryugu’s interior has
evolved to a more stratified state, that would increase its macroporosity over
the value determined here on the basis of an assumption of homogeneity. On the
other hand, repeated surface impacts might reduce the macroporosity at the
surface by increasing the size dispersion through fracturing. While we see no
way at present to correct our estimate for potential effects of inhomogeneity,
if the density of surface rocks in the sample returned by the Hayabusa2
mission matches our prediction of $1.38\pm 0.07$ gm cm-3 for the average
density of rocks through its entire volume, it will support the assumption of
homogeneity. If the density of surface rocks in the sample return is much
different from the prediction then a likely cause would be inhomogeneity of
the asteroid, although it would probably require another spacecraft mission to
prove that.

Another concern relevant to the determination of $\phi$ is the mobility of
grains. If cohesive forces prevented sifting one could, perhaps, end up with
an asteroid having a hard outer shell and substantial voids within. On Earth,
cohesive forces do not become important until d $\leq$ 0.15 mm (Zou et al.,
2011), but in the much weaker gravity field of an asteroid, it is possible
that cohesion is important for larger objects. Grott et al. (2020) discuss
this issue in depth and estimate 0.52 m as the maximum size of boulders that
could be affected by cohesion on Ryugu. They describe its effects on their
calculation to be of secondary importance at most. One way of testing this is
to search for evidence of mobility on the surfaces of rubble pile asteroids
and examples of this are growing. Bennu, for example, has been observed to
eject small particles from its surface and shows other clear examples of the
mobility of its surface material (Jawin et al., 2020). Richardson et al.
(2019) have noted the relaxed nature of surface features on many rubble pile
asteroids, which suggests there are no mobility issues, at least among the
meter-scale boulders that control macroporosity on Ryugu. Cohesion effects can
formally be taken into account in linear-mixture theory by adjusting the
initial porosity upwards to account for them. The conservative error bars that
we place on $\phi_{0}$ in Fig. 3 reflect an allowance for that. Absent any
definitive demonstration that mobility or cohesion effects are important on
asteroids at any scale, we cannot go further than that in adjusting our
results to accommodate them.

In the late stages of this study we learned of a similar effort to determine
the macroporosity of Ryugu by Grott et al. (2020), and their results are now
published. They report a value of $\phi=0.16\pm 0.03$, in perfect agreement
with what we have found. As in this paper, they employed the linear-mixture
theory of Yu & Standish (1991), but in its form appropriate to discrete,
rather than continuous, frequency distributions. They also employed the
surface boulder counts of Michikami et al. (2019), but use a different
approach to the transformation from areal to volume counts. As noted above,
they also consider in detail the possible influence of cohesive forces on
their results, and find them to be negligible.

To summarize, we find the macroporosity of Ryugu, based on its surface layer
grain-size distribution and an assumption of homogeneity to be $\phi=0.14\pm
0.04$. This corresponds to an average density of $1.38\pm 0.07$ gm cm-3 for
its rocks (grains), throughout its volume, which is consistent with the
results obtained for its surface rocks, based on their thermal properties and
an independent study of the macroporosity of Ryugu (Grott et al., 2019, 2020;
Okada et al., 2020). These results indicate that the low bulk density of Ryugu
is due to a very low density for its constituent rocks, not to an inefficient
packing of them. Since the required rock density is much lower than any
chondrule-bearing meteorite and only marginally consistent with any meteorite
class at all (CI’s), this raises the issue of how representative Earth-based
meteorites may be as samples of primitive asteroidal material. In particular,
it appears that our collections on Earth may overrepresent dense, better-
lithified objects such as most chondrite classes (and the chondrules they
contain) by a substantial amount. One consequence of this may be that we have
overestimated the abundance of chondrule-bearing chondrites and the chondrules
themselves within the Solar System, thereby exaggerating their importance to
asteroid and terrestrial planet formation. In the next section we explore this
line of reasoning in a bit more detail, especially as it impacts an
outstanding problem in cosmochemisty – the origin of chondrules.

##  5 The Abundance of Chondrules in Primitive Asteroidal Material

Chondrules are well-lithified, hard spherules with measured densities,
$\rho_{ch}\sim$3.0-3.4 gm cm-3 (Gooding, 1983; Friedrich et al., 2015), so it
is clear that they cannot be the dominant component of rocks whose overall
density is only 1.38 gm cm-3. The rare CI’s are the only chondrite class known
with a member approaching such a low value, 1.57 gm cm-3 (Macke, Consolmagno &
Britt, 2011), and they have no chondrules. While other factors, such as
chemical composition, certainly affect the bulk density of chondrites, the
importance of chondrule abundance is obvious. Most classes of chondrites are
quite rich in chondrules, with volume fractions, x, in the range 50 - 80%,
corresponding to mass fractions, fch, of 67 - 90%. The exceptions are the CM
class, with x $\sim$ 10% (fch $\sim$30%) and the CI class with x = fch = 0%
(Sears, 2004).

Fig. 4 shows the rock densities measured for several classes of CC and OC
meteorites by Macke, Consolmagno & Britt (2011) along with an indication of
their chondrule content. The expected correlation between them is clear. We
also display on the figure the range of possible average rock densities
applicable to the three rubble pile asteroids that have been visited by
spacecraft. These depend on the macroporosity of the asteroids, through
Equation 2. Open bars represent the range of possibilities corresponding to
$0.0<\phi<0.4$, while the solid portion of Ryugu’s bar shows the allowed
portion of its range found above, $0.10<\phi<0.18$. These data suggest that
chondrules may be much less common on Ryugu than they are within any
chondrule-bearing meteorites, including the CM class.

It is important to note that the implied low density for Ryugu surface rocks
does not guarantee that they are deficient in chondrules, since their matrix
may be unlike any reference material on Earth. For a two-component
chondrule/matrix system, we may write that

$\rho_{rock}=f_{ch}\rho_{ch}+(1-f_{ch})\rho_{matrix}.$ (4)

Since the inferred Ryugu rock density is even less than that of CI meteorites
(the lowest density chondritic material on Earth) $f_{ch}=0$ would be required
if Earth-based samples were an accurate guide. However, Ryugu rocks could have
even lower matrix density than CI meteorites, if their microporosity were
especially high or their grain density especially low, or both (Grott et al.,
2020). Hence, we must await the analysis of Hayabusa2 samples to provide any
firmer constraint on the fractional mass abundance of chondrules in primitive
asteroidal material. While we have not analyzed Bennu in detail, its overall
similar appearance to Ryugu and identical bulk density suggest it is also a
granular aggregate with a large value of $\sigma_{G}$. If so, its
macroporosity and average rock density should be similar to that of Ryugu, and
one might expect a similar value of $f_{ch}$.

Figure 4: The density of some classes of carbonaceous chondrites (CC) and
ordinary chodrites (OC) from Macke, Consolmagno & Britt (2011) are compared to
the average density of rocks on three rubble pile asteroids as a function of
their macroporosity ($\phi$; open bars). The result from this study that
$\phi=0.14\pm 0.04$ is indicated by the solid portion of Ryugu’s bar and
corresponds to a density for its rocks of $1.38\pm 0.07$.

The situation with respect to the S-type asteroid Itokawa and the OC’s is less
clear. The power law fit to its areal boulder counts is much different than
for Ryugu (Grott et al., 2020), but its macroporosity may still be in the
range 0.1-0.2, since this depends only on $\sigma_{G}$ and $\phi_{0}$ if
$f(d)$ is lognormal. That would imply an average rock density of $\sim$2.3 gm
cm-3, which is significantly higher than Ryugu, but still much lower than any
OC meteorite class. The higher density of OC meteorites in general (see Fig.
4) reflects not only their greater abundance of chondrules, but also a
difference in their overall composition, which in turn reflects their
formation in a warmer zone of the solar nebula, closer to the Sun (Vollstaedt
et al., 2020). The relative lack of volatiles in OC material and S-type
asteroids may be responsible for the absence of classes of aqueously altered
meteorites among the OC’s that would be analogous to the CM and CI classes.

Is the small fractional abundance of chondrules suggested by our results for
Ryugu at variance with the meteorite record, which clearly shows that
chondrules dominate Earth-based samples of primitive asteroidal material? Not
necessarily, since as pointed out in the introduction, the flux of chondrule-
bearing meteorites, measured by what makes it to the Earth’s surface (Bland et
al., 1996; Zolensky et al., 2006), is much less than the flux of material at
the top of the atmosphere (Drolshagen et al., 2017). Most of the flux of
solids at 100 km above the Earth’s surface is in the form of small
interplanetary dust particles (IDPs). If, as has been argued, for example, by
Dermott et al. (2002), these are mostly asteroidal in origin, then there is no
conflict. It would simply mean that our Earth-based sample is heavily biased
towards more lithified, chondrule-bearing material, as Sears (1998) has
argued. In support of this view, we note that the spectrum of surface material
on Ryugu, while not exactly matching any meteorite class, does most closely
resemble heated CI’s and, possibly, IDPs (Kitazato et al., 2019), and that
Grott et al. (2019) have found, based on an analysis of their thermal data,
that most of Ryugu’s boulders would be too fragile to survive passage through
the Earth’s atmosphere.

To summarize, it appears that Ryugu, and likely Bennu, are composed primarily
of lower density, less well-lithified and possibly less chondrule-rich
material than commonly found in our meteorite collections. If most IDPs are
asteroidal in origin, an issue which is unresolved because some could also be
of cometary origin, then the ratio of meteorite to IDP impacts at the top of
the Earth’s atmosphere ($\sim 10^{-3}$) could indicate the relative abundance
of well-lithified to not-so-well-lithified primitive asteroidal material
available in the Solar System at 1 AU today. We note that chondrules were
probably even fractionally less abundant in the asteroid belt 4.6 Gyr ago
since their enhanced density and hardness probably increases their odds of
survival, as Rivkin and Stickle (2019), for example, have discussed. It may
be, therefore, that only a very minor fraction of the preserved chondritic
material in the Solar System, perhaps $\sim 10^{-3}$ or less, was processed
into chondrules, despite their commonness in Earth-based meteorites.

This issue will be informed soon by the contents of the samples being returned
from Ryugu and Bennu. Since there is no reason, of which we are aware, that
these samples should be unrepresentative of the full asteroid in terms of
chondrule content, we predict on the basis of Fig. 4 that chondrules will make
up significantly less than 10% of the volume of those samples, and that
chondrules could, indeed, be entirely missing from them. The OSIRIS-REx
mission (Lauretta et al., 2017) has apparently obtained a large sample, of
order a few hundred grams of material, from Bennu. If its rocks are
representative of meteorites in our collections this would imply that they
will have collected at least $\sim$100 grams of chondrules, i.e. millions of
them. By contrast, based on the low rock densities found here, we predict the
sample will contain thousands of chondrules, at most, and possibly none at
all. Both asteroids, of course, should have had some higher-density,
chondrule-bearing meteoroids implanted by collisions over their lifetimes, and
there is evidence of such exogenous objects in the thermal maps of Ryugu
(Okada et al., 2020).

##  6 Discussion

If chondrules were as common in the early asteroid belt as they are in
chondrites, then one requires an efficient, global mechanism for their
formation, and that has been the objective of most work to date. If they are
actually rare, as argued above, one cannot dismiss inefficient or local
mechanisms, just on that account. The current lack of consensus on chondrule
formation may result from a decades-long community preference for something
that does not exist – an efficient, global mechanism. The central question is
whether the meteorites found on Earth are representative samples of the
primitive material from which the asteroids formed, or are an unrepresentative
population that experienced higher than normal lithification, which was
critical to their survival? Here we discuss how a simple, but inefficient,
local mechanism for chondrule and simultaneous chondrite formation can account
for a broad range of observed phenomena that have challenged global models,
including complementarity (see Section 6.2) in the composition of chondrules
and matrix within chondrites, uniformity of chondrule sizes within meteorite
classes, the existence of compound chondrules and cluster chondrites, and
chondrule formation in the presence of a weak magnetic field. Admittedly, only
a small fraction of primitive matter could have been processed in the manner
we describe, but that may be what is required of a successful theory.

###  6.1 Inefficient and Local Chondrule Formation

Decades of work has yielded many constraints, but so far no widely accepted
mechanism for chondrule formation (Connolly and Jones, 2016; Russell, Connolly
& Krot, 2018). A critical first step is to identify the source of the energy
within the asteroid belt that was capable of elevating the temperature of pre-
chondrule material to around 1600 - 1800 K for times on the order of an hour,
since that is what is demanded by chondrule textures (Hewins and Radomsky,
1990; Jones et al., 2018). Other fundamental constraints are the ages of
chondrules, which sets their formation epoch, the composition of the gaseous
environment in which they formed, and the complementarity that exists in the
composition of chondrules and matrix within chondrites.

The oldest solids known to have formed within the Solar System are the CAI’s.
Pb-Pb and Al-Mg dating techniques yield absolute and relative ages,
respectively, that are consistent with an age spread of $\leq$ 0.3 Myr, and
that is generally used to define t = 0 for the Solar System (Connelly et al.,
2012; Kita et al., 2013; Connelly & Bizzarro, 2018; Nagashima, Kita & Lu,
2018). CAI’s are likely to have condensed from the solar nebula at a time when
it was still hot from accretion, a time which is only expected to have lasted
for $\sim$0.1 Myr (Chiang and Youdin, 2010). CAI’s are highly refractory and
fortunately many have survived, embedded in chondrites. Grossman et al. (2008)
have shown that they formed in a reducing atmosphere at a pressure and
temperature consistent with expectation for a protoplanetary disk dominated by
H.

Chondrules, on the other hand, formed in an O-rich gas that was also enhanced
in Na and perhaps other volatiles (Alexander et al., 2008; Grossman et al.,
2008). Unlike CAI’s they exhibit a range of ages that extends to $\sim$4 Myr,
which marks the end of the chondrule formation epoch. There is some debate
about when it began. The most widely used and precise chronometer for the
early solar system is based on the 26Al $\rightarrow$ 26Mg decay, which has a
half-life of 7.17 $\times$ 105 yr. It yields ages relative to the CAI’s, under
the assumption that 26Al was uniformly distributed within the CAI and
chondrule formation zone of the solar nebula. Over 150 chondrules from a
variety of meteorites have now been dated by this technique and none yield a
formation age earlier than t = 1.8 Myr, indicating that there was a
significant gap in time between the formation of the CAI’s and the formation
of the first chondrules (Villeneuve, Chaussidon, & Libourel, 2009; Kita et
al., 2013; Nagashima, Kita & Lu, 2018; Pape et al., 2019). This result is
supported by work employing the 182Hf $\rightarrow$ 182Ta $\rightarrow$ 182W
decay chain, which has a half-life of 8.9 Myr (Budde et al., 2016; Kleine et
al., 2018).

The Pb-Pb system of chronometry (Bollard et al., 2017) yields absolute ages
under reasonable assumptions, but with somewhat less precision and, perhaps,
more susceptibility to systematic error due to the possibility of
environmental contamination (Pape et al., 2019). While there is good agreement
with the short-lived isotope results on the main points that CAI’s are the
oldest objects and have a much narrower age spread than chondrules, there is
controversy over whether chondrule formation began at t = 0. At present, there
are only 22 chondrules with Pb-Pb ages reported and they yield a mean age of
1.4 $\pm$ 1.3 Myr, consistent with results from the short-lived isotopes
(Connelly et al., 2012; Bollard et al., 2017; Connelly & Bizzarro, 2018). The
fact that some chondrules yield dates consistent with t = 0, leads to the
claim that chondrule formation began at that time, but given the small number
of objects and the large error bars involved this is not yet convincing to
all. In addition, reconciliation with the 26Al data requires that the isotope
was not distributed evenly in the early solar system (Bollard et al., 2017;
Connelly & Bizzarro, 2018), which has its own set of issues (Krot et al.,
2018; Pape et al., 2019). While the matter is unresolved, a gap in time
between CAI and chondrule formation of $\sim$1 Myr is not ruled out by
currently available Pb-Pb ages, in our opinion.

Accepting that chondrules formed in the asteroid belt between about 1.8 and 4
Myr in an oxidizing, not reducing, environment by a relatively inefficient
process narrows down the possible energy sources. ‘Nebular’ models, in which
some event, such as shocks or lightning, heats the gas in which pre-chondrule
material is suspended, are challenged by the required composition of the gas,
the fact that the gaseous disk may have already dissipated by then, and the ad
hoc nature of the posited events. ‘Planetary’ models have mostly invoked
collisional heating between colliding planetesimals (Johnson et al., 2018).
They are supported by the O-rich gas requirement, but challenged by many
details of chondrule textures. In particular, it has not yet been demonstrated
either experimentally or by computation that igneous spherules with the
characteristics of most chondrules do actually form in a collisional process.
And, since collisions between objects have continued over the entire 4.6 Gyr
history of the Solar System, the mechanism begs the question of what put an
end to chondrule formation.

One potential source of energy for chondrule formation that we know was
present in the asteroid belt between t = 1 and 4 Myr is the decay of 26Al.
Given its canonical abundance at t = 0, there is more than enough energy in
the decay of this isotope to fully melt (several times over) any object larger
than 5-10 km in diameter (Lee et al., 1977; Sanders and Scott, 2018), assuming
it forms early enough to trap that energy. Sanders and Scott (2012, 2018) have
developed a chondrule formation theory that makes use of this accumulated
energy. They showed, based on the models of Hevey and Sanders (2006), that
50-100 km radii planetesimals formed near t = 0 would be expected to develop
fully liquid interiors encased by very thin crusts. They are the likely parent
bodies of the iron meteorites, whose ages cluster near t = 0 (Scherstén et
al., 2006; Kruijer et al., 2017). In the Sanders and Scott scenario, impacts
onto these large, differentiated planetesimals (LDPs) fracture their thin
crusts releasing molten material as ‘splash’, which cools and solidifies into
chondrules. The model neatly accounts for the gap between CAI formation and
the start of chondrule formation as the time needed to form an LDP and fully
melt its interior. It further identifies the end of the chondrule formation
epoch near t = 4 Myr as the time by which the 26Al abundance has dropped
sufficiently that LDP crusts will thicken and can no longer release molten
material during an impact. The fact that one probably cannot form a large
number of chondrules by this mechanism is not necessarily an argument against
it, given the results of this study that chondrules may not be all that common
in the Solar System, but, again, it remains to be proven that objects with
chondrule textures and compositions can form from a splash into space of a
molten asteroid’s interior. And, the mechanism provides no clear pathway to
explaining complementarity (see Section 6.2).

A variation on this model has been proposed by Herbst and Greenwood (2016,
2019) who suggest that it is near-infrared radiation from the ruptured thin
crusts of the LDPs, not the material itself, that results in chondrule
formation. They show that primitive, porous pre-chondrule material in the
vicinity of an LDP with a ruptured crust could be heated sufficiently to
induce chondrule formation as it flew by exposed incandescent lava. They
calculated a series of heating/cooling curves that would be expected as a
small primitive planetesimal (SPP) orbited a LDP under the influence of its
gravity and, importantly, verified experimentally that objects with chondrule-
like, porphyritic textures could be made in laboratory simulations based on
those curves. This model potentially satisfies the constraint that the gaseous
atmosphere be rich in O and Na and accounts for the epoch of chondrule
formation in the same way as the Sanders and Scott (2012) model, The
efficiency of the flyby mechanism depends on whether SPPs typically orbit LDPs
many times before accreting, but if chondrules are rare, then it is probably
efficient enough (Herbst and Greenwood, 2019).

The ‘flyby model’ has several attributes that, arguably, make it uniquely
attractive. The most important is that chondrite lithification accompanies
chondrule formation, thereby satisfying the severe constraints of isotopic
complementarity (Kleine et al., 2018), compound chondrules, and cluster
chondrites; this is discussed in the next subsection. A second is that
radiative heating may provide a natural explanation for the correlation
between chondrule size and chemical composition, as shown by Eisenhour and
Buseck (1995), obviating the need for any size-sorting mechanism. The optical
properties of pre-chondrule aggregates influence their degree of heating and
these, in turn, depend on the chemical composition of the pre-cursors. Indeed,
a size-class relationship is an expected signature of a radiative heating
process. Finally, the clear evidence that chondrules formed in weak magnetic
fields support the view that an LDP was involved, as Elkins-Tanton et al.
(2011) and Shah et al. (2017) have noted. The flyby model provides a heat
source other than collision for chondrule formation in a ‘planetary’ setting.

###  6.2 The Lithification and Metamorphosis of Chondrites

Lithification is the process of reducing the porosity of material by
compaction and/or cementation. It requires heat and pressure and can be
facilitated by liquids or gasses within the pores. In reducing porosity, a
lithification event increases the density and tensile strength of the material
and may also affect its overall chemical composition, for example, if
volatiles are lost during heating. Heat and pressure were rare commodities in
the asteroid belt 4.6 Gyr ago and it is not obvious how objects as well-
lithified as chondrites could have formed (Consolmagno and Britt, 1999). Even
the central pressure of the largest asteroid, Ceres, which is $\sim$1 GPa, is
insufficient for lithification by cold pressing (Consolmagno et al., 2002).
The results reported here support the view that, in fact, most primitive
material never was lithified to the extent of chondrites. This opens the door
for consideration of a mechanism that could lithify a small fraction of
primitive material in space before it accretes to a larger body, namely hot
isostatic pressing (HIP) (Herbst and Greenwood, 2019). This idea is bolstered
by the increasingly strong evidence that chondrite lithification and chondrule
formation occurred simultaneously or nearly simultaneously. If so, chondrites
may have actually formed in space prior to, or in conjunction with, their
accretion to a larger body.

The strongest evidence linking chondrule and chondrite formation comes from
the phenomenon of ‘complementarity’, which is increasingly well established.
Some elements and even isotopes, that are relatively depleted in some
chondrules are found to be enhanced within the matrix of the same meteorites
containing those chondrules (Budde et al., 2016; Hezel et al., 2018; Kleine et
al., 2018). If a large fraction of primitive material is transformed into
chondrules then one can imagine that the elements, or even isotopes, released
from the chondrules during their formation, might be trapped within a zone of
the solar nebula long enough to condense within matrix grains and ultimately
find their way back to meteorites as they form on parent bodies. Clearly, the
longer the time between chondrule and chondrite formation, the more difficult
it would be to keep the atoms sufficiently confined in space. If, as argued
here, only a tiny fraction of material is formed into chondrules, then there
is the additional problem of isolating the relatively small number of released
atoms and isotopes from the general ambient medium, where they would quickly
mix, erasing any detectable signature of complementarity.

If chondrules are rare, then the phenomenon of complementarity appears to
demand that the whole meteorite form simultaneously with chondrule formation,
so that the escaped elements and isotopes can be immediately trapped and
condensed to grains that populate the matrix. Is this conceivable? Yes, if we
allow that chondrite lithification could occur immediately in space via hot
isostatic pressing (HIP), as the chondrules form, rather than later on parent
bodies (Herbst and Greenwood, 2019). In their scenario, porous SPP’s accreting
to LDP’s that are exposed to a radiation blast from extruding lava will
lithify into chondrites, while forming chondrules within them and trapping
enough of the atoms released in the process to account for the complementarity
signatures of the whole meteorites. Support for this version of chondrite
formation comes from the existence of compound chondrules and ‘cluster
chondrites’, i.e. chondrules whose shapes and packings indicate that they were
still hot and plastic when they were lithified into their chondrite (Metzler,
2012). These arguments quantify the allowable time difference between
chondrule and chondrite formation to be, at most, days and possibly just
minutes.

A consistent story then emerges, in which chondrules and chondrites represent
the small fraction of material accreting to LDPs at just the right time to
avoid being subsumed in lava oceans but still have a chance of suffering an
extreme heat blast during their accretion. They must land, of course, on a
permanently solid surface of the LDP or, again, they will lose their
distinctive primitive characteristics. One might expect, of course, that the
surface will only cool gradually with time and metamorphosis of some of the
meteorites will occur, as observed in the record. Meteorites that did not lose
too many volatiles during their lithification heating event, could also suffer
aqueous alteration on the parent body as its surface cools, although aqueous
alteration during the HIP event is also conceivable. Additional work on this
seemingly promising chondrite lithification mechanism may clarify that issue.

###  6.3 The Origin of Rubble Pile Asteroids

We finish by summarizing how one implication of our results, that chondrules
possibly are and always were, a rare form of matter in the Solar System, might
fit within the general picture of asteroid formation, particularly rubble pile
asteroids such as Ryugu. Dating of iron meteorites (Scherstén et al., 2006;
Kruijer et al., 2017) shows that asteroids large enough to melt their cores
formed within a few hundred thousand years of t = 0. Simulations of asteroid
growth by collisional accretion (Weidenschilling, 2019) indicate that most
matter accretes to large bodies within a million years or less, well before
the start of chondrule formation at t = 1.8 Myr, as inferred from their 26Al
ages (Nagashima, Kita & Lu, 2018; Pape et al., 2019). In the standard picture,
these large planetesimals either collide to form embryos that build
terrestrial planets or are ejected from the Solar System by resonance
interaction with Jupiter. According to the simulations of Weidenschilling
(2019), most of the initial mass in the asteroid belt is either already
incorporated into a body large enough to melt or ejected from the Solar System
before the chondrule formation epoch even begins. On the basis of their age
distribution and the pace of planetesimal growth indicated by accretion
models, we can conclude that chondrules (and chondrites) most likely played no
significant role in building asteroids or planets.

Instead, chondrules appear to be a feature of the final sweep-up phase of
planetesimal formation occurring between t = 1.8 and 4 Myr. and involving
$\sim$10% or less of the total mass remaining in the asteroid belt at that
time (Weidenschilling, 2019). This late-arriving material may accrete directly
to the surface of LDP’s or be stored indefinitely in the form of
gravitationally bound moonlets. As Elkins-Tanton et al. (2011) have proposed,
a chondritic crust may accumulate on top of planetesimals that have already
differentiated or are in the process of doing so. We propose that during this
final phase of accretion, within the narrow time frame permitted by the ages
of chondrules, a small fraction of the accreting material ($\leq 10^{-3}$)
suffers a heat blast from hot lava exposed on the LDP’s crust. This chance
occurrence results in the formation of chondrules and chondrites. The evidence
that many chondrules formed in the presence of a weak magnetic field supports
this view, as LDP’s are a natural source of such fields (Elkins-Tanton et al.,
2011; Shah et al., 2017; Fu et al., 2018).

The scenario described here suggests potential solutions to a number of
problems that have challenged cosmochemists over the decades besides just the
origin of chondrules. We discussed above how the phenomenon of
complementarity, the existence of cluster chondrites and the evidence for
chondrule formation in the presence of weak magnetic fields fit in. There is
also a well-known correlation between chondrule size and chondrite class that
has proven difficult to explain in the standard model of chondrite formation
since it requires a highly efficient size-sorting mechanism if chondrites
actually lithified on parent body asteroids from an accumulation of material
that independently accreted to them. In our model, the uniformity of chondrule
size within a meteorite would simply reflect a degree of homogeneity within
the parent SPP. It may also be that the optical properties of the pre-
chondrule material can account for the variation in mean size between
chondrule classes, as Eisenhour and Buseck (1995) have proposed. As they have
argued, a correlation between chondrule size and chemical composition may
actually be a distinctive sign of radiative heating.

Finally, our model has implications for the origin of ‘rubble pile’ asteroids
such as Ryugu. Since its rocks are apparently not well lithified there is no
need for Ryugu to ever have been part of a larger object, which was broken
apart into ‘rubble’ and then reassembled into a gravitationally bound pile,
although it may have been (Neumann et al., 2021). Another possibility is that
the asteroid may simply have assembled itself out of primordial blocks, which
were never lithified to the degree of most meteorites, in which case ‘rubble
pile’ would be inappropriate terminology. Asteroids as small as Ryugu do need
to be gravitationally attached to larger objects in order to avoid transport
within the Solar System. But they could have spent most of their lives as
moonlets around large asteroids. With sample returns on the way and other
asteroid missions in various stages of development, this is an exciting time
for the field and significant advances in our understanding of asteroid
structure and formation may be expected soon.

##  7 Summary and Conclusions

We have found that the boulder size distribution in the surface layer of Ryugu
is well fit by a lognormal distribution (Eq. 3) with $\mu=0.2\pm 0.05$ and
$\sigma=2.4\pm 0.1$, based on the counts of Michikami et al. (2019). Assuming
that the asteroid is an homogeneous granular aggregate, we employ linear-
mixture packing theory (Yu & Standish, 1991) to predict that its macroporosity
is $\phi=0.14\pm 0.04$, consistent with the result of Grott et al. (2020).
This indicates that Ryugu’s constituent rocks must have a very low density,
1.38 $\pm$ 0.07 gm cm-3, again consistent with the thermal properties of its
surface boulders reported by Grott et al. (2019) and Okada et al. (2020). Such
low rock density is not found among meteorites on Earth that contain
chondrules, and is below even the CI class, which has no chondrules. Given
that chondrule densities range from 3.0 - 3.4 gm cm-3, they may be absent or
very uncommon in Ryugu material. If confirmed by analysis of the sample
returns, this will support the view that chondrules and chondrites, far from
being the ‘building blocks’ of asteroids and terrestrial planets, are actually
a rare form of primitive material preserved in the Solar System. They could be
highly overrepresented on Earth because most primitive material does not have
the tensile strength to survive passage through the Earth’s atmosphere.

Based on these findings, we favor the view that chondrules and chondrites
formed during the late stages of asteroid formation, when a small subset of
primitive material was subjected to intense infrared radiation from the hot
surface of a planetesimal to which it was accreting. Such lithification events
could transform weak, low density IDP-like material into hard, dense
chondrites full of chondrules, which then accrete, fully formed, to the larger
bodies, where they may experience metamorphosis. The chondrule and chondrite
formation epoch is, therefore, defined by that narrow window of time, t = 1.8
- 4 Myr, when large asteroids were nearly fully molten and lava could easily
extrude onto their surfaces, if ruptured. The suggested lithification
mechanism for the chondrites is hot isostatic pressing (HIP) (Herbst and
Greenwood, 2019). This model is consistent with many known properties of
chondrites, including their formation epoch, chondrule textures,
complementarity, cluster chondrites, remnant magnetism, and others. It would
be inconsistent with a widespread occurrence of chondrules in the Solar
System, if that is the ultimate result of ongoing sample return missions.

##  8 Acknowledgments

We gratefully acknowledge funding for this work from NASA research grant
NNX17AE26G (PI: Greenwood) and from seed grants through the NASA CT Space
Grant Consortium to both Herbst and Greenwood. We further thank the NSF for
support of the SEM at Wesleyan through MRI grant 1725491 (PI: Greenwood). One
of us (TEY) was supported by the Keck Northeast Astronomy Consortium (KNAC),
which runs an REU program funded by NSF grant AST - 1559865 to Wesleyan
University (PI: Seth Redfield). Some materials were provided for this work by
a grant from Colgate University to one of us (TEY). We thank Martha Gilmore,
head of the Planetary Science Group at Wesleyan, and Wesleyan students Alex
Henton, Michael Henderson and Tristan Stetson, as well as Colgate students Ian
Bania and Katie Chapman for their assistance. We are further indebted to Joel
LaBella and Jim Zareski of the Wesleyan E&ES department, Kenichi Abe, formerly
of Wesleyan, and Paula Herbst for their support of the work. Finally, we thank
Matthias Grott and Jens Biele of the German Aerospace Center and Tatsuhiro
Michikami of Kindai University, for making their Ryugu boulder count data
available to us in convenient form and Jens Biele for some very helpful
comments on the work and manuscript. We are grateful to two anonymous referees
and to the PSJ editor, Edgard G. Rivera-Valentín, for their comments and
assistance during the refereeing process.

##  Appendix A Converting boulder counts from areal to volumetric

The primary factor affecting $\phi$ for a granular aggregate is the size
distribution of its constituent boulders (Yu & Standish, 1991; Frings,
Schüttrumpf & Vollmer, 2011). In linear or linear-mixture packing models
porosity is calculated in terms of the normalized volume-frequency size
distribution of the particles, f(d), where d is a characteristic size; here d
is the average of the a and b dimensions of a boulder, as described below. If
boulders from a representative volume sample (V) are sorted into $N_{bins}$
size-bins of widths $\Delta d$ and the volume of a single particle of size $d$
is $v(d)$, then the value of $f(d)$ for each bin is given by

$f_{i}(d)\Delta_{i}d={Cn_{i}(d)v(d)},$ (A1)

where $C$ is a unitless normalization constant and $n_{i}$(d) is the number of
boulders per unit volume having a size that puts them in the bin. Note that
$f(d)$ has units of m-1. $C$ is chosen such that $\sum f_{i}(d)\Delta_{i}d=1$,
where the sum extends over all bins, $i=1$ to $N_{bins}$. If the particles
have a common density, independent of size, then $f(d)$ is also the mass-
frequency size distribution of the particles and $f_{i}(d)$ is the fraction of
the total mass of the sample that is in each bin. The normalization constant
is:

$C=\left({\sum_{i=1}^{N_{bins}}n_{i}(d)v(d)}\right)^{-1}.$ (A2)

The main obstacle to using packing theory to estimate the porosity of an
asteroid is that we do not yet have a direct measurement of $f(d)$ based on a
volume sample. However, for at least one asteroid (Ryugu), we do have boulder
surface counts that extend over more than three orders of magnitude in size
based on spacecraft images obtained at a series of different heights above its
surface (Michikami et al., 2019). In Figs. 2b and 5b of their paper, the
authors report the relative size-frequency distribution ($R$), by volume (or
mass for constant density), of boulders within a given area (A) on its surface
as a function of size ($d$). According to their definition,

$R_{i}(d)={N_{i}(d)\left({d^{3}\over\Delta_{i}d}\right)}$ (A3)

where $N_{i}$(d) is the number of boulders per unit area in the $i^{th}$ bin,
which has width $\Delta_{i}d$. Note that, unlike $f(d)$, $R(d)$ is unitless.

Michikami et al. (2019) also provide information on the shape of the boulders
in their images, in terms of their a, b, c dimensions, where a is the longest
dimension, b is the next longest dimension, measured in an orthogonal
direction, and c is the shortest dimension, measured in a direction orthogonal
to the ab plane. The boulder size ($d$) is defined by them as the arithmetic
average of a and b, i.e. $d$ = (a + b)/2.

Figure 5: The volume frequency distribution for Ryugu boulders, derived from
the areal counts of Michikami et al. (2019), compared to lognormal
distributions with $\mu=0.2$ and four different values of $\sigma$, as
labeled. It is apparent that a value of $\sigma$ between 2 and 3 is required
to fit these data.

To derive a volume-based size distribution such as $f_{i}$(d), from area-based
counts, represented by $R_{i}$(d), it is necessary to consider the thickness
($t$) of the surface layer over which boulders have been counted. Clearly,

$n(d)={N(d)\over t}$ (A4)

but $t$ must vary with the size of the boulder, since larger boulders need not
be as close to the surface as smaller boulders (e.g. pebbles) to be seen and
counted. If areal counts were used directly to define the size distribution of
boulders in a representative volume of the asteroid, one would over-estimate
the number of large boulders. In general, $t$ can be written as $t=k\times d$,
where $k$ is a unitless constant that depends on the shape and orientation of
the surface boulders. As an example, if Ryugu’s boulders were preferentially
oriented such that their smallest dimension, c, were uniformly directed
downward (i.e. $t=c$), then, based on the mean ratios of the a, b, c
dimensions ($<b/a>=0.71,<c/a>=0.44$) $k$ would take the value 0.514. If the
boulders are, instead, randomly aligned then $k=1$. In principle, $k$ could be
a function of $d$, but lacking sufficient knowledge to constrain it, here we
simply assume that it is independent of boulder size, which leads to

$n(d)={(kd)^{-1}N(d)}.$ (A5)

Combining equations 1 through 5 and simplifying leads to the desired
relationship between $f_{i}$(d) and $R_{i}$(d), namely

$f_{i}(d)=\left({\sum_{i=1}^{N_{bins}}R_{i}(d)\Delta
ln(d)}\right)^{-1}\left({R_{i}(d)\over d}\right)$ (A6)

Note that this equation applies independently to each of the six areas (A
through F) surveyed by Michikami et al. (2019); i.e. the sum over all bins
refers to all bins within a particular area. Also note that the volume of an
individual boulder depends on its shape, i.e. $v(d)=k_{shape}d^{3}$, so an
implicit assumption in the derivation of Equation 6 is that shape is
independent of size.

We calculated $f_{i}$(d) for Ryugu from the original boulder counts of
Michikami et al. (2019), kindly supplied by the authors. We used 8 bins of
equal logarithmic width $\Delta log(d)=0.15$ for each area and chose the lower
size boundary so as to avoid the turnovers that occur in all the cumulative
counts due to incompleteness at the smallest sizes. Our derived values of
$f_{i}$ are plotted in Fig. 5. We caution that these values cannot be directly
compared with each other because the normalization is specific to an area, but
nonetheless the figure reveals the basic character of $f(d)$. It is a broad
distribution rising to the smallest boulders counted. Intriguingly, it also
follows a zero-mean lognormal distribution with a $\sigma$ of around 3 rather
well, albeit with an offset due to the differing normalizations of the
functions. To proceed, we must re-normalize the data, as described in the main
text and Appendix B.

##  Appendix B Renormalization and fit to a lognormal distribution

Table 1: Renormalization Factors Areaaaas defined by Michikami et al. (2019) | d${}_{min}(m)$ | d${}_{max}(m)$ | CLN
---|---|---|---
A | 4.0 | 63.0 | 0.26
B | 0.40 | 6.30 | 0.43
C | 0.20 | 3.20 | 0.43
D | 0.10 | 1.60 | 0.39
E | 0.05 | 0.79 | 0.33
F | 0.035 | 0.56 | 0.30

The similarity of $f_{i}$(d) to a lognormal distribution leads us to consider
more quantitatively how well the data fit this uniquely important function. To
make such a comparison we need to re-normalize the $f_{i}(d)$ values to a
common system, based on the normalization of the lognormal distribution, to
which $\int_{0}^{\infty}f(d^{\prime})dd^{\prime}=1$ applies. We proceed by
multiplying the $f_{i}(d)$ values within a given size range, as appropriate to
the area under consideration, by the fraction of the full size range that they
represent in a lognormal distribution of the appropriate mean and sigma, i.e.
by the constant, CLN, defined as

$C_{LN}=\int_{d_{min}}^{d_{max}}f_{LN}(d;\mu,\sigma)dd={{1\over 2}\left({{\rm
erf}\left({{\rm ln}\ d_{max}-\mu}\over{\sigma\sqrt{2}}\right)}-{\rm
erf}\left({{\rm ln}\ d_{min}-\mu}\over{\sigma\sqrt{2}}\right)\right)}$ (B1)

where $f_{LN}$ is the lognormal probability density function as written in
Equation 3, and erf is the usual error function,

$\rm{erf}(z)\equiv{2\over\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}dt.$ (B2)

The re-normalized values of $f_{i}(d)$ were then fit to a lognormal
distribution using a weighted non-linear least squares procedure based on the
Levenberg-Marquardt algorithm. The data were weighted according to the number
of boulders counted within a bin, i.e. by $\sigma^{-2}$ under the assumption
that counting errors dominate the uncertainties. An iterative process of
fitting, followed by re-calculating the re-normalization factors, was
required, but converged rapidly. Final values of the re-normalization
constants are given in Table 1. The adopted fit is characterized by
$\sigma=2.4\pm 0.1$, $\mu=0.20\pm 0.05$ and a reduced $\chi^{2}=1.13$. It is
shown in Fig. 1. See the main text of the paper for further discussion of this
result.

##  Appendix C Determining Initial Porosity

Table 2: Shape Parameters, Density and Porosity of Ryugu Boulders and Analog Grains Sample |  $<a>$aaRyugu data are from Michikami et al. (2019). Units are mm for analog samples and m for Ryugu. | $<b/a>$ | $<c/a>$ |  $\sigma_{G}$bbThe unites of $\sigma_{G}$, the geometric standard deviation, are mm for the analog samples and m for Ryugu. Errors are estimated for the samples. |  $\rho_{rock}$ (gm cm-3) |  $\phi$ccThe porosity, $\phi$, is measured for the samples and determined for Ryugu from $\rho_{0}$ and $\sigma_{G}$, using Fig. 3, an adaptation of Fig. 12 of Yu & Standish (1991). The estimated errors include an allowance for edge effects and packing efficiency. |  $\phi_{0}$ddThe initial porosity, $\phi_{0}$, is inferred for the samples from $\phi$ and $\sigma_{G}$ using Fig. 3, which is an adaptation of Fig. 12 of Yu & Standish (1991). A mean value from the analog samples is adopted for Ryugu.
---|---|---|---|---|---|---|---
1 | $27.7\pm 3.0$ | $0.72\pm 0.16$ | $0.47\pm 0.17$ | $1.25\pm 0.10$ | $2.67\pm 0.01$ | $0.43\pm 0.03$ | $0.45_{-0.05}^{+0.02}$
2 | $7.9\pm 2.3$ | $0.72\pm 0.17$ | $0.42\pm 0.20$ | $1.34\pm 0.10$ | $2.67\pm 0.01$ | $0.42\pm 0.03$ | $0.43_{-0.05}^{+0.04}$
3 | $10.6\pm 3.0$ | $0.68\pm 0.22$ | $0.36\pm 0.14$ | $1.32\pm 0.10$ | $2.70\pm 0.03$ | $0.46\pm 0.03$ | $0.48_{-0.03}^{+0.02}$
4 | $45.3\pm 6.3$ | $0.76\pm 0.10$ | $0.45\pm 0.10$ | $1.15\pm 0.10$ | $2.66\pm 0.02$ | $0.37\pm 0.03$ | $0.37^{+0.03}_{-0.03}$
Ryugu | $1.40\pm 0.14$ | $0.71\pm 0.14$ | $0.44\pm 0.14$ | $11.0\pm 1.1$ | $1.38\pm 0.07$ | $0.14\pm 0.04$ | $0.43\pm 0.07$
Figure 6: Histograms of the ‘a’ dimensions measured for 100 analog grains in
each of our four samples. Figure 7: Axial ratios for the analog particles in
our four samples, as measured by us, and Ryugu, from Michikami et al. (2019).
Means and standard deviations of each group are shown. Clearly, the two sets
are quite similar in shape by these measures.

The ‘initial porosity’, $\phi_{0}$, is the macroporosity of a pile of single-
sized grains. It depends primarily on their shape and to a lesser degree on
how they are packed, i.e. loosely or densely. One method of estimating it for
Ryugu is to assemble a sample of particles with a small dispersion (measured
by its geometric standard deviation, $\sigma_{G}$) in size that have shape
parameters close to those measured for Ryugu rocks by Michikami et al. (2019).
The measured porosity for such a sample can then be transformed to $\rho_{0}$
using linear-mixture theory (Yu & Standish, 1991). In adopting this approach,
we minimize any possible effects from cohesive forces by considering only
grains that are substantially larger than 150 microns, where such forces are
known to be negligible on Earth (Zou et al., 2011). Practical considerations
of overall sample size and weight, as well as the desire to minimize edge
effects, further limit us to rocks with diameters of, at most, a few cm.

Four samples were acquired, each with a relatively small size dispersion and
shape properties that approximated those reported for Ryugu boulders by
Michikami et al. (2019) (see Table 2 and Figs. 6 and 7). Samples 1 and 2 are
crushed rock from a quarry in Middlefield, CT, where they had been washed and
size-sorted, and are sold mainly for use on road beds and driveways. Sample 3
was purchased over the Web as “crushed gravel” and sample 4 was sold as
‘Mexican beach pebbles’. The beach pebbles are smoother and rounder than the
crushed rock samples and yielded a somewhat smaller initial porosity than the
others, perhaps on that account. No special selection or attention was made to
match the measured shape parameters of Ryugu. All of our samples had axial
ratios that were reasonable matches to Ryugu, regardless of size, as shown in
Table 2 and the figures.

The porosity of each sample was measured in a straightforward way, by pouring
the rocks into a large cylinder, shaking gently, filling to the top and
weighing. From the known density of the grains, the rock volume could be
determined and compared to the measured volume of the cylinder and the
porosity follows from Eq. 1. By using cylinders large compared to the rocks,
with heights comparable to their diameters we were able to minimize edge
effects, i.e. the accumulation of pores at the rock/cylinder interfaces. The
volume of the cylinders employed was 13,300 ml for samples 1 and 2 and 12,072
ml for samples 3 and 4.

We chose a random subset of 100 rocks from each sample and measured their a,
b, c dimensions to compare with Ryugu boulders. While there are small
differences among our bins, the overall match to Ryugu boulders is quite good,
as is evident from Fig. 7. Our analog particles have mean axial ratios of
$<b/a>=0.70\pm 0.01$ and $<c/a>=0.40\pm 0.01$, which are quite close to what
Michikami et al. (2019) find for Ryugu boulders, 0.71 and 0.44, respectively.
As shown in Table 2, we found measured porosity to vary from 0.37 to 0.46 for
our samples. Note that these are likely to be upper limits to the actual
porosity of the samples because no allowance has been made for edge effects
(Zou & Yu, 1996), other than to try to minimize them in the manners described
above. The measured porosities translate to initial porosites of between 0.37
and 0.48 (see Table 2 and Fig. 7). A conservative estimate for the value of
the initial porosity parameter that applies to Ryugu, based on these
measurements, is $\phi_{0}=0.43\pm 0.07$, as shown in Fig. 7. That is what we
have adopted in the calculations reported in the main text.

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