(2011) predict that $q\approx 4$ over a large range of planetesimal sizes,
which is largely inconsistent with the most recent observational constraints.
An updated model that includes collisional evolution (_Schlichting et al._ ,
2013) (shown in Fig. 9) also does not agree with observations. As with the SI,
there are too many planetesimals compared with that from _Kavelaars et al._
(2021). In particular, this model produced an increasingly large number of
$D<2$ km bodies (not shown on the graph). This excess of small bodies is
inconsistent with recent measurements of cratering records on Pluto and Charon
showing a turn over in the size distribution (_Singer et al._ , 2019) as well
as even more recent work that, while suggesting a turn over does not exist,
limits the slope to being shallower than predicted (_Morbidelli et al._ ,
2021). Beyond this, the general shape of the _Schlichting et al._ (2013) size
distribution is inconsistent with observations, as shown in the figure.

The numerical calculations of _Kenyon and Bromley_ (2012) more broadly
demonstrated that a wide range of size distribution slopes $q$ (their results
had different $q$ values for different planetesimal size ranges) were possible
depending on their initial conditions and chosen parameters. However, they
generally found that $q\approx 3$ for $D>200$km; again, this is inconsistent
with observations. However, it is worth noting that at smaller planetesimal
diameters, $q\approx 3$–3.5, which is  approximately consistent.

Finally, the turbulent clustering mechanism makes specific predictions for the
initial size distribution of planetesimals. In particular, as outlined in
_Hartlep and Cuzzi_ (2020), the initial size distribution has a well defined
peak of between 10 and 100 km in diameter, depending on the relevant model
parameters171717The total number of planetesimals formed via this mechanism is
also largely dependent on these parameters (_Hartlep and Cuzzi_ , 2020)., and
does not have a power law shape at smaller scales. While the characteristic
size of planetesimals formed via this mechanism is of approximately the same
order of magnitude as observed objects, there is still disagreement in the
shape of the distribution compared with observations.

In summary, the various gravitational collapse mechanisms that have quantified
the size distribution as well as the coagulation model have yet to completely
reproduce the typical sizes of planetesimals, their size distributions, and
the total number of bodies formed, strongly motivating future work in this
area. However, as we will now see, another diagnostic has proven to be a
powerful discriminant between two of these models.

###  4.2 Constraints from KBO Binaries

Figure 10: The inclination distribution of binary orbits in a snapshot of the
$\mathrm{St}=2,Z=0.1$, higher resolution run of (_Nesvorný et al._ , 2019)
(blue solid line) and observations of binary KBOs (red dashed line). The
shaded area is a 68% envelope of the model, which results when a sample of
model orbits is randomly drawn from a sample size equal to that of real
binaries with known inclinations. The binary orbits are predominantly prograde
with approximately 80% having inclination < 90°. The competing gravitational
coagulation model predicts either a 50% split between prograde and retrograde
(black dash-dot line, which was calculated assuming a uniform distribution of
inclinations) or 3% prograde and 97% retrograde (see orange horizontal
marker). The SI model is in excellent agreement with observations, in contrast
to the coagulation model.

It is now known that many observed KBOs are actually in a binary configuration
(e.g., _Noll et al._ 2008). More recently, _Grundy et al._ (2019) analyzed 35
such binaries and found that $\approx 80\%$ of them orbit each other in a
prograde sense.

_Nesvorný et al._ (2019) showed that the SI mechanism produces a nearly
identical distribution of binary orbital orientations: 80% of the simulated
planetesimals orbit prograde. This astonishingly strong agreement is shown in
the cumulative distribution (see Fig. 10) of orbital inclination for both SI
models and observations. Even more encouragingly, _Nesvorný et al._
(2021)181818Also, see _Nesvorný et al._ (2010) and _Robinson et al._ (2020)
for similar work on pebble cloud collapse extended this and found that the
angular momentum magnitude of SI-produced binaries also agrees with observed
KBO binaries.

Future work to test the SI hypothesis for planetesimal formation includes a
further exploration of parameter space as well as extending to a larger number
of simulated particles in Pkdgrav. The latter is important to numerically
resolving the tight and contact binaries, such as Arrokoth (_McKinnon et al._
, 2020).

In contrast, the gravitational coagulation model described in Section 4.1.3
makes predictions that are inconsistent with these observations. More
specifically, if the KBO velocities during this coagulation stage are on the
order of or greater than the so-called “Hill velocity" (see _Schlichting and
Sari_ 2008 for a definition), this model predicts roughly equal number of
prograde and retrograde orbits (this line is also included in Fig. 10). For
KBO velocities less than 10% of the Hill velocity, retrograde orbits dominate
by 97% (_Schlichting and Sari_ , 2008). Clearly, these considerations taken
together strongly support the SI-induced gravitational collapse paradigm.

###  4.3  Comparison with Comet 67P

Moving from constraints from KBOs to those from comets, we now consider recent
observations of comet 67P Churyumov-Gerasmineko (hereafter comet 67P) and
compare these with model predictions of planetesimal-formation and -evolution
models. This comparison will include internal properties, like porosity,
granularity, cohesive and tensile strength, and activity patterns.

Owing to the ground-breaking Rosetta mission of the European Space Agency, our
knowledge about the inner makeup of cometary nuclei has increased enormously.
We are now in the situation to use these findings as diagnostic tools for the
formation of these bodies (see _Blum et al._ , 2017; _Blum_ , 2018; _Weissman
et al._ , 2020). The most striking physical property of comets for which we
now possess empirical evidence is their extremely low mechanical strength,
claimed to be essential for the sustaining activity of comets (_Kuehrt and
Keller_ , 1994; _Skorov and Blum_ , 2012; _Blum et al._ , 2014). Measurements
of the tensile strength of the cometary material have shown that the tensile
strength indeed is $\lesssim 1$ Pa (_Attree et al._ , 2018). Numerical studies
suggest that such a requirement is mandatory for a sustained cometary activity
(_Skorov and Blum_ , 2012; _Gundlach et al._ , 2020). Extremely low
compressive-strength values were also inferred from the bouncing of the Philae
lander (_O’Rourke et al._ , 2020).

Rosetta also allowed precise mass and volume determinations of the nucleus of
comet 67P (_Pätzold et al._ , 2019). In combination with assumptions on the
material composition, this allows the derivation of global porosity values.
Different approaches result in values for the porosity of 73%–76% (_Herique et
al._ , 2019), 65%–80% (_Pätzold et al._ , 2019), and 63%–79% (_Fulle et al._ ,
2016), which correspond to global volume filling factors of the nucleus of
comet 67P of $\Phi_{\mathrm{67P}}=0.24-0.27$, $\Phi_{\mathrm{67P}}=0.20-0.35$,
and $\Phi_{\mathrm{67P}}=0.21-0.37$, respectively. As pointed out in Sect.
3.1.4, the expected volume filling factor for small ($\lesssim 100$ km)
planetesimals is $\Phi_{\mathrm{pl}}\approx 0.18-0.37$, which is well matched
by the values estimated for comet 67P. _Güttler et al._ (2019) showed that
such volume filling factors indicate a makeup consisting of two hierarchical
levels (e.g. solid dust grains and fluffy pebbles). Fewer levels lead to too
high densities, whereas more levels render the planetesimal too porous.

A "smoking gun" for the primordial nature of comet 67P could be the discovery
of extremely fluffy dust particles by the Giada and Midas instruments of
Rosetta (see the review by _Güttler et al._ , 2019, and original references
therein). These particles possess porosities $>95\%$ and have pre-detection
sizes on the order of 1 cm, but electrostatically fragment in the vicinity of
the Rosetta spacecraft. Even under the most optimistic conditions, such
particles could not have formed in the coma of comet 67P. To see this,
consider the maximum measured dust mass flux of comet 67P:
$F=10^{-8}\,\mathrm{kg\,m^{-2}\,s^{-1}}$ (_Merouane et al._ , 2016). This
result, along with the assumption that the monomer grains have sizes of $\sim
100$ nm (see Sect. 2.3) and masses of $m=10^{-18}\,\mathrm{kg}$, a particle
flux (per unit mass) of $F^{\prime}=F/m=10^{10}\,\mathrm{m^{-2}\,s^{-1}}$
results. The dust monomers are so small that they couple to the gas outflow on
short timescales and, thus, obtain typical outflow velocities of
$v_{\mathrm{out}}=100\,\mathrm{m\,s^{-1}}$. Hence, a maximum number density of
the monomer grains in the coma of
$n=F^{\prime}/v_{\mathrm{out}}=10^{8}\,\mathrm{m^{-3}}$ results. To determine
the collision time scale, we assume that all collisions occur at the sticking-
bouncing threshold, ignoring that fractal growth requires much lower collision
speeds. From the models described in Sect. 2.3, the sticking-bouncing
threshold velocity of 100 nm grains should be on the order of
$v_{\mathrm{r}}=10\,\mathrm{m\,s^{-1}}$. With a collision cross section for
100 nm grains of $\sigma\approx 10^{-13}\,\mathrm{m^{2}}$, a collision
timescale of $\tau_{\mathrm{coll}}=\frac{1}{n\sigma
v_{\mathrm{r}}}=10^{4}\,\mathrm{s}$ follows. At a distance of Rosetta from the
surface of comet 67P of $d=10-30\,\mathrm{km}$ (_Della Corte et al._ , 2015),
the time between emission of the dust and detection by the Rosetta/Giada
instrument was $\tau_{\mathrm{trans}}=d/v_{\mathrm{out}}=100-300\,\mathrm{s}$,
much too short to allow any collisional growth of the grains. Thus, the
observed fluffy grains must have been present inside the nucleus of comet 67P
before they were ejected into the coma.

_Fulle and Blum_ (2017) showed that these fluffy particles could be the
remnants of the pre-pebble growth stage of protoplanetary dust, captured and
preserved between the pebbles during the gravitational collapse. If that
scenario was correct, the cometary nuclei in the Solar System could not be
rubble piles, re-assembled after a catastrophic collision. If post-formation
collisions were unavoidable, then they had to be sub-catastrophic so that
major parts of the colliding planetesimals would remain intact (see _Schwartz
et al._ , 2018, for an example).

Under the assumption that comet 67P was formed by the gentle gravitational
collapse of a pebble cloud, _Blum et al._ (2017) used a multi-Rosetta-
instrument approach to estimate the pebble sizes. They found that pebble radii
in the range of 3-6 mm were the most likely, with the upper limit constrained
by the sub-surface temperatures measured by the Miro instrument and the lower
limit given by the size distribution of surface features identified by the
Civa instrument on the Philae lander. These pebble sizes are in good agreement
with estimates of the bouncing barrier at large heliocentric distances (see
_Lorek et al._ , 2018, and Sect. 2.5).

We should note that there are alternative formation models for the nucleus of
comet 67P (see Sect. 3.2 and _Weissman et al._ , 2020, for a recent discussion
on this issue) and that there are open questions with the formation of
cometary nuclei by the gentle gravitational collapse of a pebble cloud. The
most prominent issues to be solved in future work are: (i) What is the
influence of radiogenic heating on the inner makeup of planetesimals? (see
Sect. 3.1.4) (ii) Can the gravitational collapse form comet-sized
planetesimals directly and what is their rate of binarity? (iii) What is the
role of collisions in the post-formation era and how can present-day cometary
nuclei form as a result of these collisions without losing their volatile and
fluffy-particle inventory? That being said, the evidence for a gravitational
collapse origin of cometesimals is reasonably compelling, giving further
support to the SI as the mechanism to form planetesimals (though of course,
other gravitational collapse mechanisms cannot be ruled out).

##  5 CONCLUDING REMARKS

Less than two decades ago, the formation of planetesimals was one of the
biggest unanswered questions in planetary science and astrophysics. While this
issue does remain open and of utmost importance to understanding the formation
of the Solar System (as well as other planetary systems), it is very
encouraging that so much progress has been made in the time since then.

We now have a detailed understanding of how sub-micron grains grow to pebble
sizes, backed by both theoretical calculations and laboratory experiments, and
we have multiple models to explain how pebbles then reach the next stage and
form planetesimals.

Of course, there remain issues that must be solved in order to build towards a
comprehensive understanding of and to determine which mechanisms are
responsible for planetesimal formation. For instance, the largest issue of
contention is the discrepancy in the size distribution between observations
and theoretical models, those based on both the SI and gravitational
coagulation. Will other gravitational collapse mechanisms (e.g., the SGI) show
better agreement? Or is the discrepancy, at least in the SI simulations, the
result of something simpler, such as a more extensive survey of relevant
parameters or collapse below the numerically limited scales in the SI
calculations?

Even with these outstanding issues, however, there are indications that the
gravitational collapse paradigm, and specifically the SI-induced collapse
model, is the correct one. The thermal and structural properties of
planetesimal made available from the Rosetta mission do indeed support a
gravitational collapse model for planetesimal formation. Furthermore, the
excellent agreement in KBO binary properties between SI simulations and
observations strongly support the SI as the mechanism for planetesimal
formation, while strongly disfavoring the gravitational coagulation model.

Furthermore, as numerical models improve and computational resources continue
to be readily accessible, the theoretical side of this question will likely
see significant progress in the coming years. These enhancements will be
further augmented by improved observational capabilities. For example, in
addition to continued observations from OSSOS, the Vera C. Rubin observatory
will drastically expand our database of detected KBO objects, while JWST will
provide another way to peer into the earliest stages of planet formation in
our Solar System. Furthermore, new missions, such as Lucy and Psyche, are
either being built or are already on their way to exploring Solar System
planetesimals.

In closing, with ever improving numerical models, new observational facilities
coming online, and new missions set to explore Solar System planetesimals, we
live in a golden era of planetesimal and comet exploration. As such, there
should be significant optimism that we as a community will solve the
outstanding issues of planetesimal and comet formation in the coming years;
the future of small body studies looks quite bright, indeed.

Acknowledgments.
J.B.S. acknowledges support from NASA under Emerging Worlds through grant
80NSSC21K0037. J.B. acknowledges the support of this work by the Deutsche
Forschungsgemeinschaft (DFG) through grant BL 298/27-1 and thanks for
continuous support by DFG and the Deutsches Zentrum für Luft- und Raumfahrt –
German Space Agency. T.B. acknowledges funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme under grant agreement No 714769 and funding by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Ref no. FOR
2634/1 and under Germany’s Excellence Strategy - EXC-2094 - 390783311. D.N.
acknowledges support from the NASA Emerging Worlds program. We thank Daniel
Carrera, Wladimir Lyra, Andrew Youdin, and Rixin Li for useful discussions
that greatly improved the quality of this chapter. Furthermore, we thank Jono
Squire for providing the necessary materials and information to create Fig. 5.
We also thank Ingo von Borstel for creating the fractal aggregate in Fig. 2.
Finally, the writing of this document was supported in part by the Munich
Institute for Astro- and Particle Physics (MIAPP) which is funded by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany´s Excellence Strategy – EXC-2094 – 390783311, and J.B.S. thanks MIAPP
for their hospitality while he finished much of this chapter.

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