# The hot main Kuiper belt size distribution from OSSOS

Jean-Marc Petit Institut UTINAM UMR6213, CNRS, OSU Theta F25000 Besançon,
France  Brett Gladman Department of Physics and Astronomy, University of
British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada  J. J.
Kavelaars Herzberg Astronomy and Astrophysics Research Centre, National
Research Council of Canada, 5071 West Saanich Rd, Victoria, British Columbia
V9E 2E7, Canada  Department of Physics and Astronomy, University of Victoria,
Elliott Building, 3800 Finnerty Rd, Victoria, BC V8P 5C2, Canada  Department
of Physics and Astronomy, University of British Columbia, 6224 Agricultural
Road, Vancouver, BC V6T 1Z1, Canada  Michele T. Bannister School of Physical
and Chemical Sciences — Te Kura Matū, University of Canterbury, Private Bag
4800, Christchurch 8140, New Zealand  Mike Alexandersen Center for
Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138,
USA  Kathryn Volk Lunar and Planetary Laboratory, The University of Arizona,
1629 E University Blvd, Tucson, AZ 85721, USA  Planetary Science Institute,
11700 East Fort Lowell, Suite 106, Tucson, AZ 85719, USA  Ying-Tung Chen
Institute of Astronomy and Astrophysics, Academia Sinica, No. 1, Sec. 4,
Roosevelt Rd., Taipei 10617, Taiwan

(Received 22-Dec-2022; Revised 16-March-2023; Accepted 16-March-2023)

###### Abstract

Using the absolute detection calibration and abundant detections of the OSSOS
(Outer Solar System Origin Survey) project, we provide population measurements
for the main Kuiper Belt. For absolute magnitude $H_{r}<8.3$, there are 30,000
non-resonant main-belt objects, with twice as many hot-component objects than
cold, and with total mass of 0.014 $M_{\earth}$, only 1/7 of which is in the
cold belt (assuming a cold-object albedo about half that of hot component
objects). We show that transneptunian objects with $5.5<H_{r}<8.3$ (rough
diameters 400–100 km) have indistinguishable absolute magnitude (size)
distributions, regardless of being in the cold classical Kuiper belt (thought
to be primordial) or the ‘hot’ population (believed to be implanted after
having been formed elsewhere). We discuss how this result was not apparent in
previous examinations of the size distribution due to the complications of
fitting assumed power-law functional forms to the detections at differing
depths. This shared size distribution is surprising in light of the common
paradigm that the hot population planetesimals formed in a higher density
environment much closer to the Sun, in an environment that also (probably
later) formed larger (dwarf planet and bigger) objects. If this paradigm is
correct, our result implies that planetesimal formation was relatively
insensitive to the local disk conditions and that the subsequent planet-
building process in the hot population did not modify the shape of the
planetesimal size distribution in this 50–300 km range.

††journal: ApJL

=9

CC-BY 4.0 (https://creativecommons.org/licenses/by/4.0/)

##  1 Introduction

The size distribution of objects produced at various stages of the planet
formation process is a topic of intense interest (eg. Kenyon, 2002; Kobayashi
et al., 2016; Ormel, 2017; Schaller & Brown, 2007; Schlichting & Sari, 2011;
Shannon et al., 2016) . One must conceptually separate the size distribution
of objects directly built by some planetesimal formation process from those
that are then created by either collisional grinding or accumulation (Kenyon
et al., 2008; Morbidelli & Nesvorný, 2020) . In the main asteroid belt there
has been heavy collisional modification which has greatly obscured the initial
size distribution, although arguments that many asteroids were ‘born big’ have
been made (Johansen et al., 2007; Morbidelli et al., 2009). The non-saturated
cratering record on Pluto/Charon and Arrokoth (Greenstreet et al., 2019;
Singer et al., 2019; Spencer et al., 2020) argues that in the transneptunian
region the size distribution has not been modified by collisional and
accretional effects since the surfaces of these bodies formed. If true this
would mean that the currently essentially collisionless environment of the
Kuiper Belt (Abedin et al., 2021; Greenstreet et al., 2019; McKinnon et al.,
2020; Petit & Mousis, 2004) has persisted for the Solar System’s age and that
the size distributions are thus primordial and preserve the outcome of the
planetesimal formation and planet building process.

The recent study of Kavelaars et al. (2021) used an ensemble of survey samples
(Kavelaars et al., 2009; Petit et al., 2011; Alexandersen et al., 2016; Petit
et al., 2017; Bannister et al., 2018), referred to collectively as OSSOS++, to
show that the dynamically cold classical Kuiper belt’s size distribution
follows a power-law with an exponential cutoff at the large size end. This
‘exponential taper’ shape is compatible with the initial mass function
obtained in simulations of planetesimal formation in a streaming instability
scenario (Schäfer et al., 2017; Li et al., 2019) and while other low-density
formation scenarios exist (eg. Shannon et al., 2016), which mechanism created
planetesimals is irrelevant for this present manuscript. Several independent
facts (Petit & Mousis, 2004; Parker & Kavelaars, 2010; Tegler et al., 2003;
Pike et al., 2017; Schwamb et al., 2019; Greenstreet et al., 2019; Abedin et
al., 2021; McKinnon et al., 2020) also hint at an in-situ formation of the
cold belt in a low density environment.

The region where the cold belt resides, between the 3:2 and 2:1 mean motion
resonance with Neptune, also contains resonant Trans-Neptunian Objects (TNOs)
and other non-resonant, yet excited TNOs (objects with either large
eccentricity $e$ or inclination $i$, or both) forming the hot belt. These
objects are commonly stated to have been formed in a region closer to the Sun
and then implanted in the Kuiper belt during late planetary migration, based
on dynamical (reviewed by Nesvorný, 2018) and compositional (Schwamb et al.,
2019) arguments. The goal of this manuscript is to use the OSSOS++ sample and
compare the absolute magnitude distribution of the hot belt to that of the
cold belt; we find that the two distributions are extremely similar over the
magnitude range where we have high accuracy.

In the next section, we present the OSSOS++ sample of hot objects and compare
its size distribution to that of the cold population. We extend the OSSOS++
sample with the MPC database to the large size side of the distribution. Next
we present some cosmogonic implications of our findings.

##  2 The OSSOS++ hot population absolute magnitude distribution

We use the OSSOS++ sample (see Bannister et al., 2018, for full details) to
determine the absolute magnitude distribution of the populations of the main
classical Kuiper belt. The main-belt classicals are made up of the non-
resonant, non-scattering objects with semimajor axes
$39.4~{}\textrm{au}<a<47.7~{}\textrm{au}$ (that is, between the 3:2 and 2:1
resonances, rejecting all resonant objects in this range).

As explained in Van Laerhoven et al. (2019) and Kavelaars et al. (2021), the
best single parameter to discriminate between the cold and the hot populations
of the main classical Kuiper belt is the free inclination with respect to the
$a$-dependent Laplace plane. Huang et al. (2022a) provide an improved
determination of the local Laplace plane by double-averaging over the two fast
angles rather than from the classical first order secular theory; the
resulting free inclinations are very stable over time and can be found111The
main-belt free inclinations can also be found at http://yukunhuang.com. in
Huang et al. (2022b). Given the distribution of the free inclinations shown in
Fig. 3 of Huang et al. (2022a), in our manuscript we elect to use
$i_{free}<4.5^{\circ}$ as an acceptable split between the main-belt cold and
hot populations (understanding there will be interlopers at some level). The
cold classical belt only exists at $a>42.4$ au (Kavelaars et al., 2008), and
is heavily concentrated to perihelia $q>39$ au (Petit et al., 2011) and (see
Fig. 5 of Gladman & Volk, 2021). We here adopt these simple cuts to define the
cold classical belt region, arguing that the higher eccentricities for TNOs
with $q<39$ au and $a>42.4$ au indicate they have suffered dynamical
excitation. The hot main-belt population is then all objects not in the cold
sample. This yields 327 cold and 219 hot objects in OSSOS++.

###  2.1 Sample characterization

We use the method in Kavelaars et al. (2021) to debias the orbit and $H_{r}$
distributions of the OSSOS++ detections222Divide the phase space in small
$(a,q,sin(i),H)$ cells; for any cell with a detection in it, determine the
detection biais of this cell using the Survey Simulator.. The most difficult
TNO orbits to detect, at a given $H_{r}$, are those with near-circular orbits
at the $a\simeq 47$ au outer edge of the main belt. Objects on such orbits
always remain at distance $d\simeq 47$ au, while those with either lower $a$
or larger eccentricity $e$ will spend some time closer to the Sun and be
visible for some fraction333Using $H_{r}=9.0$ as an example, $a=47$ au
circular orbits can never be observed by OSSOS, while $H_{r}=9$ objects
distributed along an $e\simeq$0.1 orbit rise above the flux threshold close to
perihelion and give one sensitivity to that set of $a,q$ orbital elements for
the purposes of debiasing (where debiasing is essentially taking into account
Kepler’s Second Law). The method has no ability, however, to debias the orbit
for which there is never any possibility of detection. of their orbital
longitude. Because OSSOS++ reached apparent magnitude $m_{r}\gtrsim 25$ for
some blocks, $H_{r}\simeq 8.3$ TNOs are visible at 47 au; Kavelaars et al.
(2021) used this $H_{r}$ for the limit down to which we trust our debiasing
for the cold population; even if the higher $e$’s of hot TNOs give a mild
increase in sensitivity to $H_{r}>8.3$ hot-object orbits, we maintain the
limit at 8.3 since our goal is to compare the two populations as a function of
absolute magnitude. Because this debiasing method uses only the detections,
this model will certainly increasingly underestimate TNO numbers beyond these
sensitivity limits.

Figure 1: Calibrated differential $H_{r}$ distribution of the main-belt hot
(blue bold curve) and cold (bold red curve) components from debiasing the
OSSOS++ sample for $H_{r}\in[3;8.3]$. Less secure estimates (see text) for
$8.3<H_{r}<9$, which are only lower limits, are shown by dash-dotted curves.
These curves are Epanechnikov Kernel Density Estimates (appendix A) of the
debiased $H_{r}$ absolute magnitude distributions of the main classical belt
objects in OSSOS++. The dotted lines represent 95% confidence intervals from
Poisson statistics. Up to $H_{r}\simeq 5.3$ the OSSOS absolute calibration
reproduces the MPC numbers, indicating the latter is now essentially complete
in the main-belt.

Simple binned histograms to represent a differential distribution are subject
to large fluctuations when the sample is not very numerous. To avoid this
shortcoming, we use a Kernel Density Estimator (KDE) method which spreads out
each detection over a kernel of some size; each detection thus contributes to
the differential distribution not only at its exact position but on some
interval with a varying weight (see appendix A). Figure 1 presents these
debiased differential TNO numbers (per 0.25 magnitudes, in order to compare
with the known sample in the Minor Planet Center, MPC) as the hot and cold
populations implied by the OSSOS++ detections. MPC $H$ magnitudes are
converted to $H_{r}=H_{MPC}-0.2$; this is the known shift between OSSOS++
$H_{r}$ magnitudes and the $H$ stated in the MPC. The rollover in the cold
distribution for $H_{r}>8.3$ where we expected to insensitivity to begin is
clear; the rollover in the hot distribution begins a few tenths of a magnitude
fainter due to the ability to detect the lower perihelion objects present in
that population (as mentioned above).

Several important results follow immediately from Fig. 1. First, the MPC
inventory (shown as histograms) of the main belt is very close to complete for
$H_{r}<5.3$; this was already clear for cold TNOs (Kavelaars et al., 2021) but
our estimates reinforce the idea that the hot main-belt population is also now
essentially complete. Sheppard et al. (2011) already suggested that the TNO
inventory was nearly complete to apparent magnitude $M_{R}\simeq 21$,
corresponding to $H_{r}\simeq 6.4$ at 30 au and $H_{r}\simeq 4.2$ at 50 au.
Despite a burst of $H_{r}<5.3$ discoveries during 2013-2015 by Pan-STARRS
(Weryk et al., 2016) no bright main-belt TNOs have been discovered despite
continued operations, signifying completeness to this magnitude. The agreement
of our estimates with the complete population at the bright end shows that our
debiasing method yields correct number estimates; these estimates then
establish that $H_{r}>5.8$ is largely incomplete (the known TNOs are below our
95% confidence intervals). The up-coming LSST will survey the main classical
Kuiper belt down to $H_{r}\simeq 7.5$ at 47 au; Figure 1 indicates that LSST
will mostly discover TNOs at $H$ magnitudes for which completeness in 2022 is
still only $\simeq$5%. Secondly, one is struck that the hot and cold
populations have rather similar $H_{r}$ distribution shape in the range [5.8,
8.3]. This is surprising because many papers (Bernstein et al., 2004; Elliot
et al., 2005; Petit et al., 2011; Adams et al., 2014; Fraser et al., 2014)
conclude that the hot and cold components have different absolute magnitude
distributions. Here we suggest that these differences might be dominantly
confined only to the largest ($H_{r}<6$) TNOs of the populations; if true,
this has major implications for the planetesimal formation and planet-building
process.

###  2.2 Comparison of the hot and cold distributions

Figure 2 shows a clearer representation of the shape similarity of the two
populations. Note that OSSOS++ has a sufficiently large number of detections
that construction of a differential $H$-magnitude and good resolution is
possible; many past analyses have shown cumulative distributions. Here, we
have multiplied the number of cold TNOs by a factor of 2.2, which matches the
curves at $H_{r}=6$. Given the uncertainties shown by the 95% confidence
ranges and the expectation to have a small part of this range at $\simeq
2\sigma$ discrepancy, the shapes are nearly identical in the range from
5.5$<H_{r}<$8.3, with the most compelling indication of a difference at
$H_{r}<6$ where the exponential taper cuts off the cold population. For
$H_{r}<5$, Fig. 1 makes it clear that the hot population contains large
objects while the cold belt has none.

A $\chi^{2}$ test can quantitatively evaluate if one can reject the null
hypothesis that the two observed distributions are drawn from the same
underlying distribution in the range from $\simeq$5.5–8.3. In order to
mitigate binning effects, we varied differential bin sizes (0.2, 0.3, or 0.4
mag) and the histogram starting magnitude (by steps of 0.1 mag), yielding
probabilities that the two distributions are from the same underlying
distribution ranging from 55% to 90%; the idea that the distributions are
identical is clearly plausible. Fig. 2 also shows the exponential taper
function derived in Kavelaars et al. (2021); another $\chi^{2}$ test to
compare the hot $H_{r}$ distribution to that function (only shifting the
function to match the debiased hot number at the single value of $H_{r}=8.0$)
yields probabilities between 50% and 80% of drawing the hot population from
the same functional form as the cold in this same $H_{r}$ range.

Figure 2: Similar to Fig. 1, where here the debiased cold component estimates
and uncertainties have been scaled upward by a factor of 2.2. The green dashed
line shows the Kavelaars et al. (2021) cold population exponential cutoff fit
(with large-$H$ asymptotic slope of 0.4), also scaled upward by a factor of
2.2. The inset shows the continuation of the scaled exponential cutoff fit at
larger/fainter $H_{r}$; the red error-bar gives the 95% confidence range from
the detection of three cold TNOs using HST by Bernstein et al. (2004), and is
compatible with the above extrapolation. The inset’s blue arrow shows the 95%
confidence upper limit from the non-detection of hot population TNOs in that
same HST study. For $H_{r}>5.5$ the shape of the hot and cold population’s are
surprisingly similar, which hints at similar formation processes. The black
dotted line is a reference exponential with logarithmic slope $\alpha=0.4$, to
help see how the exponential taper deviates from a single exponential.

Kavelaars et al. (2021) showed that if one extends the $H_{r}$-distribution
beyond 8.3 with a functional form asymptotic to an exponential law
$dN/dH\propto 10^{0.4H}$, the resulting estimated differential numbers are
consistent with the detection of three $H_{r}<12$ cold TNOs in the HST search
of Bernstein et al. (2004). A similar extrapolation of our hot main-belt
population estimate is consistent with no hot detections in the HST
search.444Despite the hot population having a factor of two more TNOs than the
cold population at each $H_{r}>5.8$ magnitude, non-detection in the HST survey
is understood when one considers that these hot objects are spread out over an
order of magnitude more sky area due to their larger orbital inclinations. An
asymptotic power of $0.4H_{r}$ is supported for $H_{r}\simeq 12-17$ by the
crater record on Charon (Singer et al., 2019), which has been estimated to be
dominated by hot population projectiles (Greenstreet et al., 2015).

Thus, there is as yet no firm evidence that the shape of the $H$ magnitude
distributions of the hot and cold are different for $H_{r}\gtrsim 5.5$. But if
the cold classical belt is formed in situ in a low-density environment
(exhibiting an exponential cutoff at a size scale set by local conditions at
$\simeq 44$ au) and the hot object implanted after being formed much closer to
the Sun (having a different formation, collision, and dynamical history), one
might reasonably expect them to have very different $H$ magnitude
distributions. We will return to this issue in the discussion section after
discussing the small-$H$ regime, where the two populations differ markedly.

###  2.3 The large size tail

The sample of large bodies from the main classical Kuiper belt in the MPC
database is very close to complete for the cold (Kavelaars et al., 2021, and
Figure 1) and the hot (Figure 1) populations down to $H_{r}\simeq 5.3$. Hence
the $H_{r}$ distribution at large sizes can be directly obtained from the MPC
database. The debiased OSSOS++ sample fails to explore the $H_{r}<4$ range due
to limited sky area covered and the resulting tiny number of detections (only
one hot main-belt object brighter than $H_{r}=5$ and three with $H_{r}<5.5$).

Figure 3: Cumulative $H_{r}$ distribution (black curve) of all hot TNOs (here,
$a>29.5$ au and not in the cold population) in the MPC database as of 13
January 2022. The blue curve shows the main-belt subset. The vertical line at
$H_{r}=5.3$ represents where our analysis indicates that the the hot main-belt
population is nearly complete. The vertical line at $H_{r}=5.8$ shows where
the MPC sample is certainly incomplete (95% confidence level). The two samples
share very similar $H_{r}$ distributions brighter than $H_{r}=5.3$ (in
particular both distributions show an obviously shallower slope for
$H_{r}<3$), supporting the claim that all hot objects have the same origin.

Figure 3 shows the cumulative number of objects present in the MPC database
for hot main classical belt (blue line) and all hot (black line) objects. It
is noteworthy that the full hot sample, although obviously incomplete (because
even the largest objects cannot be seen at the large distances reached by the
scattering and resonant populations), is parallel to the hot main-belt sample,
which is complete bright-ward of $H_{r}\simeq 5.3$. The abrupt change at
$H_{r}\simeq 3.3$ in both distributions has been evident (Brown, 2008) since
large shallow surveys first covered most of the sky and continues to signal an
abrupt change of exponential index between 0.14 and 0.6 at this magnitude
(Ashton et al., 2021). We will refer to the $H_{r}<3.3$ range as the ‘dwarf
planet’ regime555Although the current IAU dwarf planet definition involves
difficult to know properties related to ‘roundness’ (e.g. Tancredi & Favre,
2008; Grundy et al., 2019), we point out that using a simple $H_{V}<3.5$
definition would make the terminology adapt to some obvious transition in the
object distribution. below.

Fortuitously, the bright end of OSSOS++ happens to lie at the hot main-belt
completeness limit. The argument above indicates we can graft our debiased hot
distribution onto the MPC sample and, with the parallel distribution the full
hot sample allows one to have access to the $H_{r}$-magnitude distribution
from Pluto/Eris scale down to $H_{r}=8.3$ (and further by reasonable extension
to the HST study’s depth at $H\simeq 12$). The hot-cold comparison indicates
that consistency of the $H_{r}$-magnitude shape for $H_{r}\gtrsim 5.5$ is not
also true at bright absolute magnitudes. The cold population exhibits the
exponential cutoff while the hot population (both in the main belt and in
general) not only lacks the exponential cutoff but has very shallow
distribution as one goes up in size into the dwarf planet regime. We will
interpret this in the Discussion section below.

##  3 Consideration on past surveys

Because of the curved shape of the exponential cutoff, representing the
$H$-distribution by a single exponential $N(<H_{r})\propto 10^{\alpha H_{r}}$
yields different logarithmic slopes $\alpha$ depending on the area and depth
of the survey. A shallow, wide survey will probe the steep part of the curve,
while a fainter, narrower survey will probe a shallower part of the curve. The
case of the hot component is even worse, due to the change between the
logarithmic slopes of $\sim$0.6 in the range $H\in[3.5;6]$ and 0.14 for
brighter objects (Ashton et al., 2021). A single exponential representation
will thus require a very shallow slope for wide, shallow survey, to medium
steep for intermediate surveys, and then again again shallow slopes for
fainter surveys. Many of the early works did not separate between the hot and
cold populations. Surveys close to the ecliptic were dominated by the cold
component, while surveys mostly out of the ecliptic missed much or all of the
cold population. To blur things even more, these studies used the apparent
magnitude, thus convolving the $H$-distribution with the distance distribution
(see Petit et al., 2008, for a review).

More recent studies directly determined the $H$ distribution, using either a
single exponential when the range of $H$ magnitudes is small (e.g. Petit et
al., 2011; Adams et al., 2014) or a double exponential when the range of $H$
magnitudes was large (Fraser et al., 2014). Petit et al. (2011) separately
modeled the cold and hot components to their limit of $H_{r}\lesssim 7.5$ and
obtained different size distributions for the two components, with population
measurements to that $H$-magnitude limit. The number of $D>100$ km TNOs in the
hot and cold main belt given in Petit et al. (2011) were extrapolations with
these single slopes (0.8 for hot and 1.2 for cold), beyond the range where
they were measured. It is now clear that these slopes do not represent the
$H$-magnitude distribution beyond 7.5 (which flattens steadily) and thus the
extrapolation overestimated the $D>100$ km populations. When including small
(faint) TNO surveys, Fraser et al. (2014) used a double exponential with
breaks around $H_{r}$=7–8 where the bright part was essentially the same
single exponentials for the two components as Petit et al. (2011).

The sparseness of data from homogeneous characterized surveys and the use of
particular functional forms prevented the recognition of the similarity of the
hot and cold size distributions in the few 10s km to $\sim$300 km range.

##  4 Population and mass estimates

Using our derived $H_{r}$ distribution, we can provide an accurate debiased
main-belt population (which does not include resonant or scattering TNOs) as a
function of $H_{r}$ and also number and mass estimates (which are more
uncertain) for TNOs larger than a certain diameter.

###  4.1 Population estimates

Table 1 gives estimated TNO numbers at the faintest absolute magnitude we
confidently debias ($H_{r}\leq 8.3$) and for $H_{r}$ values corresponding to a
diameter $D$=100 km for various commonly-used values of the albedo (eq. B2).
We added a column with $\nu_{r}=0.24$, the albedo of Arrokoth determined by
Hofgartner et al. (2021) in $r$ band filter. Note that here all estimates are
based on direct debiasing, which is very uncertain for $H_{r}=8.9$.
Alternately, to estimate numbers of $H_{r}<8.9$ TNOs, one could use the
exponential cutoff formula (eq. B4) for the cold population and multiply by
2.2 for the hot population; Table 1’s last column would then read 21, 47, and
68 thousand TNOs, which is identical given the uncertainties. This is
consistent with our previous work at the older measurement limit: Petit et al.
(2011, Table 5) estimated $N(H_{g}<8.0)=(8\pm 2)\times 10^{3}$, in agreement
with our current $N(H_{r}<7.3)=6.7^{+1.3}_{-1.6}\times 10^{3}$ if one uses an
averaged $g-r\simeq 0.7$ color.

Table 1: OSSOS++ population estimates from direct debiasing for main-belt non-resonant TNOs (in $10^{3}$ objects). Population estimates marked with a * are likely underestimates (Sec. 2.1) because $D=100$ km corresponds to $H_{r}>8.3$ for $\nu_{r}=0.04$.  | $H_{r}\leq 8.3$ |  $D\geq 100$ km
---|---|---
|  | $\nu_{r}=0.24$ | $\nu_{r}=0.15$ | $\nu_{r}=0.08$ | $\nu_{r}=0.04$
component |  | ($H_{r}\leq 7.1$) | ($H_{r}\leq 7.5$) | ($H_{r}\leq 8.2$) | ($H_{r}\leq 8.9$)
Cold | $11\pm 1$ | $1.2\pm 0.3$ | $2.8\pm 0.5$ | $9\pm 1$ | *$20\pm 2$
Hot | $20\pm 3$ | $2.4\pm 0.7$ | $7\pm 1$ | $18\pm 3$ | *$46\pm 7$
All | $31\pm 4$ | $3.6\pm 1$ | $10\pm 2$ | $27\pm 4$ | *$66\pm 9$

Our debiasing gives an estimate of the total population of main-belt classical
TNOs which we can compare to other well sampled small-body populations. Out of
curiosity, we note that the main classical Kuiper belt contains 340 and 31,000
objects brighter than $H_{r}=6.0$ and $8.3$ respectively, while there are only
10 and 145 Main Belt Asteroids (MBAs) brighter than those magnitudes.
Comparison with the jovian Trojan population has more cosmogonic interest
because the hot classicals and other excited TNOs have been suggested to come
from the same primordial population as the jovian Trojans (Morbidelli et al.,
2005). In the $H_{r}$=7.3–8.3 range, our debiasing shows that the jovian
Trojans have a similar $H_{r}$ distribution to the hot TNO population, with
the MPC Trojan sample being complete in that range (see Hendler & Malhotra,
2020, for completeness limits). There are 8 Jupiter Trojans with $H_{r}\leq
8.3$, and 20,000 hot main classical TNOs, thus implantation models of jovian
Trojans and hot main classical TNOs must account for an efficiency ratio of
order $3\times 10^{3}$. Nesvorný & Vokrouhlický (2016) and Nesvorný et al.
(2016) state implantation efficiencies of $(7\pm 3)\times 10^{-4}$ and
$(7.0\pm 0.7)\times 10^{-7}$ respectively for hot classicals and jovian
Trojans; this model-based factor of 1000 (with an uncertainty of at least a
factor of 2) is thus similar to the observed factor of 3000. With OSSOS++, the
$H_{r}<8.3$ hot main-belt population is now the least fractionally uncertain
number in this chain of reasoning but, as Table 1 shows, an order of magnitude
population uncertainty appears due to the albedo uncertainty.

Using the CFEPS population estimates (Petit et al., 2011; Gladman et al.,
2012) extrapolated to 100 km size, Greenstreet et al. (2015, 2019) and Abedin
et al. (2021) determined the expected number of craters on the surface of
Pluto, Charon, and Arrokoth formed in the last 4 Gyr of bombardment. They
extrapolated the 100 km population down to kilometer-size projectiles using an
exponential of slope $\alpha=0.4$, concluding that the recorded crater numbers
could be produced without a contribution of an early phase (see below) and
that the dominant source of craters on Arrokoth is from cold population
projectiles.

Our current estimates, however, indicate fewer 100 km bodies than the CFEPS
extrapolation down to this size, by a factor of 3–30 depending on which albedo
is used, which determines $H$ for 100-km bodies.)666Petit et al. (2011) used
an albedo $\nu_{g}=0.05$ which means $\nu_{r}\sim 0.08$ for cold objects
(assuming $\langle g-r\rangle\sim 0.9$) and $\nu_{r}\sim 0.06$ for hot objects
(for $\langle g-r\rangle\sim 0.6$). If one instead uses $\nu_{r}=0.15$ for
cold and $\nu_{r}=0.08$ for hot (see Appendix B), this means a brighter
$H_{r}$ magnitude for 100 km, again decreasing again the number of objects at
that size. Starting from $H_{\sim}6$ down to the limit of our survey, we find
that that the hot population is 2.2 times larger than the cold. For smaller
sizes (larger $H$), we assume that the exponential cutoff continues, thus the
hot population remains twice the cold at any given $H_{r}$. Eq. (B4) then
yields similar numbers of projectiles at $H_{r}\sim 17$ as found by
Greenstreet et al. (2015, 2019) and Abedin et al. (2021), with now twice as
many main-belt hot TNOs as cold, while before the extrapolation produced 2 to
3 times as many cold objects as hot due to the incorrect continuation of a
very steep slope for cold objects to $H_{g}$=8. The ratio hot-to-cold could be
even larger at a specific size (km) when one accounts for the possibly larger
albedo for cold objects ($\nu_{r}\sim 0.15$) than for hot objects
($\nu_{r}\sim 0.08$). Remember that all this is an assumption, not directly
based on observational evidence, and should be taken with a pinch of salt.

Thus all workers should now re-evaluate the Arrokoth cratering rate. Pluto and
Charon will continue to be dominated by hot-population projectiles, while
Arrokoth may cease to be dominated by cold-population impactors. Morbidelli et
al. (2021) concluded dominance of the hot population for Arrokoth crater
formation, but their numbers of 100 km and 2 km projectiles are not in line
with our current estimates and should be re-visited.

###  4.2 Mass estimates

In appendix B, we estimate the main-belt mass between the 3:2 and 2:1 mean-
motion resonances, integrated over all sizes. We use bulk density $\rho=1000$
kg/m3 and albedos $\nu_{r,c}=0.15$ and $\nu_{r,h}=0.08$ for cold and hot TNOs,
respectively (Fraser et al., 2014; Lacerda et al., 2014)777Note that Vilenius
et al. (2014) reported very similar values of $\nu_{v,c}\sim 0.14$ and
$\nu_{v,h}\sim 0.085$ in band $v$ from Herschel and Spitzer observations.,
except for objects larger than $\sim$500 km, where both density and albedo are
known to increase (see Appendices B.1 and B.2.

Understanding that masses are uncertain to a factor of 3 due to poorly-
constrained albedos and densities, we find a cold-belt mass of 0.002
$M_{\earth}$ for an exponential cutoff shape of $H_{r}$ and 0.012 $M_{\earth}$
for the hot belt, for a total mass of the main classical belt of 0.014
$M_{\earth}$. We find the same hot-belt mass as Fraser et al. (2014), but a
cold-belt mass that is 7 times larger. Gladman et al. (2001) estimated a total
mass for the 30–50 au distance range of 0.04-0.1 $M_{\earth}$, using an albedo
$\nu_{r}=0.04$ for all TNOs; this includes all dynamical components, but
because the main belt represents only about half the mass (determined using
the CFEPS model, Petit et al., 2011) in the 30–50 au distance range, a
0.02-0.05 $M_{\earth}$ estimate results. Using this 4% albedo, our current
approach gives a slightly higher total classical main-belt mass 0.06
$M_{\earth}$.

Based on creation of a planetary ephemeris, Di Ruscio et al. (2020) estimated
the total Kuiper Belt mass to be $(0.061\pm 0.001)M_{\earth}$, with unknown
model-based uncertainty. This is $\simeq$4 times higher than our estimate of
the classical belt, but again only half of this $0.06M_{\earth}$ would be
main-belt TNOs. With the same method, but using a slightly different dataset,
Pitjeva & Pitjev (2018) estimated the mass to be $(0.02\pm 0.004)M_{\earth}$,
which when restricted to the main belt is in agreement with our estimate.

Because the tapered exponential lacks an abrupt break between a steep slope
power-law (steeper than $\alpha=0.6$ for $H_{r}$ distribution or than $q=4$
for $D$ distribution888The relations $N(<H)\propto 10^{\alpha H}$ and
$dN/dD(D)=n(D)\propto D^{-q}$ are related by $q=5\alpha+1$ for a fixed albedo
$\nu$.) to a shallow slope for small sizes, there is less of a concentration
of mass to the typical diameter scale where the break occurs. For our
estimated hot belt, the 25th and 75th percentile for mass are $H_{r}<6.7$ and
$H_{r}<9.8$. For the cold population, the 25th percentile is $H_{r}<7.3$ and
the 75th percentile is at $H_{r}<10.1$. Broadly speaking, our precise
knowledge of the size distribution includes an $H$ magnitude range that
contains about half the mass of the classical Kuiper belt.

Note that because of the very shallow $\alpha=0.14$ (Ashton et al., 2021) in
the $H_{r}<3.5$ hot-object tail, it is possible that the largest TNOs in this
tail were planetary scale and contained most of the mass of the hot-population
formation region. These very massive objects have not been retained the hot
main-belt or the hot population in general due to the $\sim 10^{-3}$ retention
efficiency of the scattering out process Nesvorný & Vokrouhlický (2016), and
thus one should remember that much of mass in the hot-formation region may
have been sequestered into very large (now absent) planetary scale objects.

##  5 Discussion of cosmogonic implications

Based on several lines of evidence (see Kavelaars et al., 2021, and reference
therein), the cold population is primordial, meaning its $H_{r}$ distribution
has not evolved since the formation epoch. Given the shape similarity of the
hot and cold populations in the range $H_{r}\simeq 5.5$ to 8.3, we hypothesize
that the hot population has also preserved its primordial shape in that size
range and the same physical mechanism was responsible for the accretion of
bodies of that size in both the hot and cold forming regions. The physical
conditions (e.g., temperature, dynamical time scales) were likely very
different in these two regions, and in particular the surface density in
solids would likely have been orders of magnitude larger in the hot-forming
region than at $\simeq 43$ au (Gomes et al., 2004; Nesvorný et al., 2020). If
true, it follows that the formed planetesimal size distribution is at most a
weak function of the local conditions, at least in the $H_{r}$=5.5–8.3 range;
over this range it is steeper than collisional equilibrium. Coupled with the
match to the cold (believed to be primordial and unevolved) population, we
thus conclude that at this size scale there was no appreciable collisional
modification of the shape. This implies that the initial phase during which
the hot population was in a dense collisional environment was of shorter
duration than the collisional lifetime of $D\simeq 100$ km bodies. Benavidez
et al. (2022) supports this picture, showing that the $D\gtrsim 100$ km size
distribution’s shape does not change. They also conclude that long (100 Myr)
instability phases do not produce successful matches to observational
constraints.

Nesvorný et al. (2020) and Kavelaars et al. (2021) suggested that a candidate
for this initial planetesimal forming phase is the Gravitational/Streaming
Instability (GI/SI). The numerical simulations typically yield a size
distribution that can be fit with an exponential cutoff functional form (Abod
et al., 2019) that is also a good match to the cold population. Here we have
shown that it is also a good match to the hot population with $D\lesssim 300$
km.

Our paradigm is that after planetesimal formation, the conditions in the hot-
population formation region (in particular much higher surface density)
permitted the creation of bigger objects, which thus erased the taper for
$H_{r}<5.5$. An important implication is that dwarf planets and larger objects
then accumulated mass without altering the relative size distribution in the
$H_{r}\simeq 6$ to 9 range (and possibly much smaller sizes). Two main
mechanisms are usually invoked for this latter stage of accretion: runaway
growth (reviewed by Lissauer, 1993) and pebble accretion (e.g., Ormel, 2017).
Runaway growth does not care about the size of objects it sweeps up, while in
pebble accretion the very-big object mass accretion is from tiny pebbles. In
this later case, our result implies that objects with sizes corresponding to
$H_{r}=$5.5–8.3 accumulate negligible further mass via pebbles.

If GI/SI is the planetesimal-formation mechanism, scalings indicate that the
mass of the largest bodies formed is governed by the local solid surface
density (to the third power) and heliocentric distance (to the 6th power)
(Abod et al., 2019; Li et al., 2019). For a surface density varying with
heliocentric distance to the -1.5 or -2 power between 25 and 45 au, this would
not be a problem as the density drop would roughly compensate for the increase
in distance. However, there are arguments that the surface density drops by a
factor of $\sim$1000 between these locations (i.e. Nesvorný et al., 2020). It
is thus surprising that two populations would have the same exponential cutoff
shape. Some possible solutions to this dilemma are:

(1) the initial planetesimal formation process is actually only mildly
sensitive to the density, but the creation of $H_{r}<5$ bodies in the hot-
population formation region proceeded efficiently, while in the cold belt
these dynamics were not triggered due to the low surface density, or

(2) cold-population formation occurred in a localized over density (perhaps
caused by a pressure bump) in an environment where the formed planetesimals
are not confined to the overdense region, while in contrast the hot population
formed in an extended high-density zone where large-object formation was
possible through another process.

The first possibility is supported by Klahr & Schreiber (2020) who derive a
criterion for the minimum object mass that can be created by GI/SI when
accounting for diffusion due to turbulence. Based on this criterion, they
claim that the size distribution should be a Gaussian centered on this minimum
size ($D\simeq$80–85 km with width $\sim$45 km). Their minimum size is fairly
insensitive to heliocentric distance from 3 to 30 au, but then drops markedly
at larger distances.

The second possibility would have to be the solution if GI/SI is the dominant
mechanism, unless the current theoretical scaling laws are incorrect. In these
studies, there is a trigger density one gets to and then GI/SI forms
planetesimals quickly. This could result in the two regions having the same
shape because they both reached the same critical density, although the way
they achieved that density would have been different between the two
populations.

Summarizing our discussion, we hypothesize that whatever mechanism created the
first planetesimals up to $\sim$400 km in diameter, it is largely insensitive
to the global physical conditions, and produces objects up to that size. In
dense environments, some other process(es) takes over to build bigger objects,
without altering the size distribution from $\sim$300 km down to a few 10s km
or even less.

Acknowledgements: We thank Wes Fraser for useful discussions while preparing
this manuscript, and reviewer David Nesvorny for useful comments. This work
was supported by the Programme National de Planétologie (PNP) of CNRS-INSU co-
funded by CNES. BG ackowledges discovery grant funding support from NSERC.
This research made use of the Canadian Advanced Network for Astronomy Research
(CANFAR) and the facilities of the Canadian Astronomy Data Centre operated by
the National Research Council of Canada with the support of the Canadian Space
Agency. This research has made use of data and/or services provided by the
International Astronomical Union’s Minor Planet Center. Based on observations
obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at
the Canada-France-Hawaii Telescope (CFHT) which is operated by the National
Research Council (NRC) of Canada, the Institut National des Sciences de
l’Univers of the Centre National de la Recherche Scientifique (CNRS) of
France, and the University of Hawaii. The observations at the CFHT were
performed with care and respect from the summit of Maunakea which is a
significant cultural and historic site.

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##  Appendix A Kernel Density Estimator (KDE)

To avoid the shortcomings of classical histograms due to size and location of
the bins, we use a Kernel Density Estimator (KDE) (Rosenblatt, 1956; Parzen,
1962) to estimate the true $H_{r}$ distribution (see
https://en.wikipedia.org/wiki/Kernel_density_estimation for an easy and basic
introduction to KDEs). The kernel is a non-negative function that smoothes the
contribution of each datum over an interval which size is determined by the
bandwidth parameter. There exists a variety of kernels; we use an Epanechnikov
finite extent kernel (Epanechnikov, 1969), as it is optimal in a mean square
error sense.

The bandwidth is an important parameter that we determine using the empirical
approach of cross-validation. This empirical approach to model parameter
selection does not depend on (dubious) assumptions about the underlying data’s
distribution and thus is very flexible. For the cold OSSOS++ sample the
optimal bandwidth is 0.4, which we use to plot Fig. 1. For the hot sample, the
optimal bandwidth depends rather strongly on the inclusion or exclusion of the
few objects at the tails of the distribution, especially at the bright end
where the $H_{r}$ spread is very uneven. This means that the best bandwidth is
very different at the large size and the small size ends of the range, so we
renormalize the $H_{r}$ values to make them more uniform
($Y_{r}=10^{0.1(H_{r}-9)}$) and then use an optimal bandwidth of 0.1 in this
new variable.

In our computations, we use the implementation from the scikit-learn python
package (Pedregosa et al., 2011). We then scale the KDE to the total number of
objects in each component of the main classical Kuiper Belt per magnitude and
show the result in Fig. 1.

##  Appendix B Mass of the belt

We first derive the relation between $H_{r}$ and the TNO mass $M$. The
apparent magnitude $m_{r}$ in a given filter band (CFHTLS-$r^{\prime}$ here)
is related to its radius $r$, in km, its geometric albedo $\nu_{r}$, its
distance to the Sun $R$ and to the observer on Earth $\Delta$ (both in au),
its phase angle $\gamma$, the phase function $\Phi(\gamma)$, the rotational
lightcurve function $f(t)$ and the magnitude of the Sun $m_{r,Sun}$ in the
same band, by

$m_{r}=m_{r,Sun}-2.5\log_{10}\left(\nu_{r}\left(\frac{r}{1{\rm~{}km}}\right)^{2}\Phi(\gamma)f(t)\right)+2.5\log_{10}\left(2.25\times
10^{16}\frac{R^{2}\Delta^{2}}{{\rm au}^{4}}\right)$ (B1)

Here we assume $f(t)=1$ as the average of a rapid periodic function. From
$H_{r}$ definition, we have

$\displaystyle H_{r}$ $\displaystyle=$ $\displaystyle
m_{r}+2.5\log_{10}(\Phi(\gamma))-2.5\log_{10}\left(\frac{R^{2}\Delta^{2}}{{\rm
au}^{4}}\right))$ $\displaystyle=$ $\displaystyle
m_{r,Sun}+2.5\log_{10}\left(2.25\times
10^{16}\right)-2.5\log_{10}\left(\nu_{r}\left(\frac{r}{1{\rm~{}km}}\right)^{2}\right)$

For the cfhtls-r filter, $m_{r,Sun}=-26.94$ in the AB system (Willmer, 2008)
and therefore

$H_{r}=13.94-2.5\log_{10}\left(\nu_{r}\left(\frac{r}{1{\rm~{}km}}\right)^{2}\right)$
(B2)

Denoting bulk density as $\rho$ , their mass $M=\frac{4\pi}{3}\rho r^{3}$ can
be written, using Eq. B2, as

$M=\frac{4\pi}{3}\frac{\rho}{\nu_{r}^{3/2}}10^{(13.94-H_{r})\frac{3}{5}}=B\frac{\rho}{\nu_{r}^{3/2}}10^{-\frac{3H_{r}}{5}}$
(B3)

with $B=\frac{4\pi}{3}10^{13.96\frac{3}{5}}$ kg $=9.685\times 10^{8}$ kg, and
$\rho$ given in kg.km-3 (1 g.cm-3= 1012 kg.km-3).

The computation of TNO mass from its absolute magnitude $H_{r}$ thus strongly
depends on the two unknowns $\rho$ and $\nu_{r}$. Conservatively assuming that
$\rho$ can vary from 0.5 g.cm-3 to 2 g.cm-3, and $\nu_{r}$ from 0.06 (as seen
from the comets), to 0.24 (Arrokoth), we can formally have a variation by a
factor of $\sim$30 in the mass of individual objects.

###  B.1 The hot belt

For a scenario of fixed values of the albedo and the bulk density, the mass of
the hot belt is dominated by the few largest bodies (such as Makemake, Quaoar,
Varda or Varuna) but the predicted mass of these large bodies is badly
overestimated if using nominal values of these quantities. With the nominal
values of $\nu_{r}=0.08$ (Lacerda et al., 2014), $\rho=1$g/cm3 (Gladman et
al., 2001; Fraser et al., 2014) and $H_{r}=-0.31$, one would find a mass for
Makemake of $1.1\;10^{-2}M_{\earth}$ when its actual mass is
$5.2\;10^{-4}M_{\earth}$ (Parker et al., 2018). The discrepancy decreases with
increasing $H_{r}$ but is still of a factor of 2 for Varuna at $H_{r}=3.59$.

The simple solution of this issues is that Stansberry et al. (2008) showed a
correlation between object size and geometric albedo, which Fraser et al.
(2008) modelled as $\nu\propto r^{\beta}$. It is also likely that the density
of a body increases with its size due to self-compression. In the mass-
to-$H_{r}$ relation, the important factor is $\rho\nu^{-3/2}$ which we then
model as $\rho\nu^{-3/2}=A(M/1{\rm~{}kg})^{\gamma}$. We select $\gamma=0.77$
with resulting $A\sim 6.7\times 10^{28}$ for $\rho$ expressed in kg.km-3 to
roughly match the known masses of Makemake, Quaoar ($M\simeq 1.4\;10^{21}$kg,
Vachier et al. (2012); Fraser et al. (2013)), Varda ($M\simeq
2.45\;10^{20}$kg, Souami et al. (2020)), Varuna ($1.5\;10^{20}$kg, Lacerda &
Jewitt (2007); Lellouch et al. (2013)) and (55637) 2002 UX25
($1.2\;10^{20}$kg, Brown (2013)). We use this dependency for the large bodies
until the $\rho\nu^{-3/2}$ factor reaches its nominal value for $\rho=1$g/cm3
and $\nu_{r}=0.08$, which occurs at $M=5.6\;10^{19}$kg and $H_{r}=4.82$. For
smaller masses (larger $H_{r}$), we use the nominal values for $\rho$ and
$\nu$.

Brightward of $H_{r}=5.47$, we use the raw MPC absolute magnitude
distribution. Between $H_{r}=5.47$ and $H_{r}=8.3$ we use the debiased $H_{r}$
distributions shown in Fig. 1. Faintward of this limit, we use the exponential
cutoff from the cold belt (eq. B4) scaled-up by a factor of 2.2 as in Figure
2. As expected the mass is not concentrated at any given size. The total mass
of the hot main classical belt is 0.012 $M_{\earth}$.

###  B.2 The cold belt

For the cold population, all TNOs are small enough to be in the regime where
albedos and densities that are not correlated with their size. We use the
nominal $\rho=1000$kg/m3 with more reflective $\nu_{r}=0.15$ (Fraser et al.,
2014; Lacerda et al., 2014) estimated for cold objects. The debiased mass of
the cold belt, brighter than $H_{r}=8.3$ comes out as 0.0010 $M_{\earth}$.

For cold TNOs with $H_{r}>8.3$, we use the exponential cutoff parameterization
from Kavelaars et al. (2021),

$N(<H_{r})=10^{\alpha(H_{r}-H_{o})}\;\exp\left[-10^{-\beta(H_{r}-H_{B})}\right]$
(B4)

where $N$ is the total population for TNOs with absolute magnitude less than
$H_{r}$; $H_{o}=-2.6$ is a normalization factor, $\alpha=0.4$ is the
asymptotic logarithmic slope at large $H_{r}$, $\beta=0.25$ is the strength of
the exponential tapering and $H_{B}=8.1$ is the $H_{r}$ value at which the
exponential taper begins to dominate as one moves towards brighter
magnitudes999Note that numbers in Table 1 are obtained from debiasing of OSSOS
data, not from this formula.. We caution that there is degeneracy in this
parameterization, which allows individual parameters to have large variations
as long as the others change in a correlated way, which results in the
cumulative population curve being very similar. This yields a mass of 0.0011
$M_{\earth}$ for cold $H_{r}>8.3$ TNOs, assuming $\alpha=0.4$ continues.

Thus the total mass of the cold belt is 0.0021 $M_{\earth}$, for the assumed
albedo and bulk density, or about 1/6 of our hot-belt mass estimate.