The hot main Kuiper belt size distribution from OSSOS

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, %<<BG added clause 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.


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 OS-SOS++ 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.

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 au < a < 47.7 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 found 1 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 f ree < 4.5 • 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 , 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++.

Sample characterization
We use the method in Kavelaars et al. (2021) to debias the orbit and H r distributions of the OSSOS++ detections 2 . The most difficult TNO orbits to detect, at a given H r , are those with near-circular orbits at the a ≃ 47 au outer edge of the main belt. Objects on such orbits always remain at distance d ≃ 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 fraction 3 of their orbital longitude. Because OSSOS++ reached apparent magnitude m r ≳ 25 for some blocks, H r ≃ 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.
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 M P C − 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).
2 Divide 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. 3 Using 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 ≃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. 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 ≃ 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 ≃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. Figure 2 shows a clearer representation of the shape similarity of the two populations. Note that OS-SOS++ 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.

Comparison of the hot and cold distributions
Given the uncertainties shown by the 95% confidence ranges and the expectation to have a small part of this range at ≃ 2σ 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 χ 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 ≃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 χ 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.  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 α = 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 ∝ 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. 4 An asymptotic power of 0.4H r is supported for H r ≃ 12 − 17 by the crater record on Charon , 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 ≳ 5.5. But if the cold classical belt is formed in situ in a lowdensity environment (exhibiting an exponential cutoff at a size scale set by local conditions at ≃ 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.

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   Figure 1) and the hot (Figure 1) populations down to H r ≃ 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 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 ≃ 5.3. The abrupt change at H r ≃ 3.3 in both distributions has been evident ) since large shallow surveys first covered most of the sky and continues to signal an 4 Despite 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. 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' regime 5 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 5 Although 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. study's depth at H ≃ 12). The hot-cold comparison indicates that consistency of the H r -magnitude shape for H r ≳ 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.

CONSIDERATION ON PAST SURVEYS
Because of the curved shape of the exponential cutoff, representing the H-distribution by a single exponential N (< H r ) ∝ 10 αHr yields different logarithmic slopes α 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 ∼0.6 in the range H ∈ [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 ≲ 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, likely more so for the cold population than for the hot one. 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 ∼300 km range.

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. Table 1 gives estimated TNO numbers at the faintest absolute magnitude we confidently debias (H r ≤ 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 ν 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 ± 2) × 10 3 , in agreement with our current N (H r < 7.3) = 6.7 +1.3 −1.6 × 10 3 if one uses an averaged g − r ≃ 0.7 color.

Population estimates
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  (2016) and  state implantation efficiencies of (7 ± 3) × 10 −4 and (7.0 ± 0.7) × 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 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 α = 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.) 6 Starting from H ∼ 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 ∼ 17 as found by Greenstreet et al. (2015Greenstreet et al. ( , 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 (ν r ∼ 0.15) than for hot objects (ν r ∼ 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. 6 Petit et al. (2011) used an albedo ν g = 0.05 which means ν r ∼ 0.08 for cold objects (assuming ⟨g − r⟩ ∼ 0.9) and

Mass estimates
ν r ∼ 0.06 for hot objects (for ⟨g − r⟩ ∼ 0.6). If one instead uses ν r = 0.15 for cold and ν 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.
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 ρ = 1000 kg/m 3 and albedos ν r,c = 0.15 and ν r,h = 0.08 for cold and hot TNOs, respectively (Fraser et al. 2014;Lacerda et al. 2014)  Because the tapered exponential lacks an abrupt break between a steep slope power-law (steeper than α = 0.6 for H r distribution or than q = 4 for D distribution 8 ) 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 7 Note that Vilenius et al. (2014) reported very similar values of ν v,c ∼ 0.14 and ν v,h ∼ 0.085 in band v from Herschel and Spitzer observations. 8 The relations N (< H) ∝ 10 αH and dN/dD(D) = n(D) ∝ D −q are related by q = 5α + 1 for a fixed albedo ν.
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 α = 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 ∼ 10 −3 retention efficiency of the scattering out process , 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.

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 ≃ 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 ≃ 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 ≃ 100 km bodies. Benavidez et al. (2022) supports this picture, showing that the D ≳ 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 ≲ 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 ≃ 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 ∼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 ≃80-85 km with width ∼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 ∼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 ∼300 km down to a few 10s km or even less.

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(Hr−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.

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 ′ here) is related to its radius r, in km, its geometric albedo ν r , its distance to the Sun R and to the observer on Earth ∆ (both in au), its phase angle γ, the phase function Φ(γ), 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 ν r r 1 km 2 Φ(γ)f (t) + 2.5 log 10 2.25 × 10 16 R 2 ∆ 2 au 4 (B1) Here we assume f (t) = 1 as the average of a rapid periodic function. From H r definition, we have H r = m r + 2.5 log 10 (Φ(γ)) − 2.5 log 10 R 2 ∆ 2 au 4 ) = m r,Sun + 2.5 log 10 2.25 × 10 16 − 2.5 log 10 ν r r 1 km 2 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 ν r r 1 km with B = 4π 3 10 13.96 3 5 kg = 9.685 × 10 8 kg, and ρ given in kg.km −3 (1 g.cm −3 = 10 12 kg.km −3 ).
The computation of TNO mass from its absolute magnitude H r thus strongly depends on the two unknowns ρ and ν r . Conservatively assuming that ρ can vary from 0.5 g.cm −3 to 2 g.cm −3 , and ν r from 0.06 (as seen from the comets), to 0.24 (Arrokoth), we can formally have a variation by a factor of ∼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 ν r = 0.08 (Lacerda et al. 2014), ρ = 1g/cm 3 (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 ⊕ when its actual mass is 5.2 10 −4 M ⊕ (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 ν ∝ r β . 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 ρν −3/2 which we then model as ρν −3/2 = A(M/1 kg) γ . We select γ = 0.77 with resulting A ∼ 6.7×10 28 for ρ expressed in kg.km −3 to roughly match the known masses of Makemake,  (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 ρν −3/2 factor reaches its nominal value for ρ = 1g/cm 3 and ν 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 ρ and ν.
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 ⊕ .

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 ρ = 1000kg/m 3 with more reflective ν 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 ⊕ .
For cold TNOs with H r > 8.3, we use the exponential cutoff parameterization from Kavelaars et al.
where N is the total population for TNOs with absolute magnitude less than H r ; H o = −2.6 is a normalization factor, α = 0.4 is the asymptotic logarithmic slope at large H r , β = 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 magnitudes 9 . 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 ⊕ for cold H r > 8.3 TNOs, assuming α = 0.4 continues.
Thus the total mass of the cold belt is 0.0021 M ⊕ , for the assumed albedo and bulk density, or about 1/6 of our hot-belt mass estimate.