The Mass Deficit of the Cold Population

The scenario proposed in [60] (see Sect. 4.2) confines the problem of the mass depletion of the Kuiper belt to just the cold population. In fact, in [60] only —0.2% of the bodies initially in the disk swept by Neptune remained in the Kuiper belt on stable high-« orbits at the end of Neptune's migration. This naturally explains the current low mass of the hot population. However, the population originally in the 40-50 AU range - which would constitute the cold population in the scenario of [60] - should have been only moderately excited and not dynamically depleted, so that it should have preserved most of its primordial mass.

Two general mechanisms have been proposed for the mass depletion: the dynamical ejection of most of the bodies from the Kuiper belt to the Neptune-crossing region, and the collisional comminution of most of the mass of the Kuiper belt into dust.

The dynamical depletion mechanism was proposed in [127] and later revisited in [141]. In this scenario, a planetary embryo, with mass comparable to that of Mars or the Earth, was scattered by Neptune onto a high-eccentricity orbit that crossed the Kuiper belt for ~ 108 years. The repeated passage of the embryo through the Kuiper belt excited the eccentricities of the Kuiper belt bodies, the vast majority of which became Neptune crossers and were subsequently dynamically eliminated by the planets' scattering action. The integrations in [141], however, treated the Kuiper belt bodies as test particles, and therefore, their encounters with Neptune did not alter the position of the planet. Thus, similar simulations have been re-run in [62], in the framework of a more self-consistent model accounting for planetary migration in response to planetesimal scattering. As expected, the dynamical depletion of the Kuiper belt greatly enhanced Neptune's migration. The reason for this is that, thanks to the dynamical excitation of the distant disk provided by the embryo, Neptune interacted not only with the portion of the disk in its local neighborhood, but with the entire mass of the disk at the same time. As shown in Fig. 29, even a low mass disk of 30 Me between 10 and 50 AU (just 7.5M® in the Kuiper belt) could drive Neptune well beyond 30 AU. Halting Neptune's migration at ~30AU requires a disk mass of ~15Me or less (depending on Neptune's initial location). Such a mass and density profile would imply only 3.75 Me of material between 40 and 50 AU as the Kuiper belt formed, which is less than the mass required (10-30 Me) by models of the accretion of Kuiper belt bodies [94,154].

A priori, for the migration of Neptune, there is no evident difference between the case where the Kuiper belt is excited to Neptune-crossing orbits by a planetary embryo or by some other mechanism, such as the primordial secular resonance sweeping proposed in [135]. Therefore, we conclude that Neptune never "saw" the missing mass of the Kuiper belt. The remaining possibility for a dynamical depletion of the Kuiper belt is that the Kuiper belt objects were kicked directly to hyperbolic or Jupiter-crossing orbits and consequently were eliminated without interacting with Neptune. Only the passage of a star through the Kuiper belt seems to be capable of such an extreme excitation [99].

The collisional grinding scenario was proposed in [29, 30,155] and then pursued in [93,95,97]. In essence, a massive Kuiper belt with large eccentricities and inclinations would experience a very intense collisional activity. Consequently, most of the mass originally in bodies smaller than 50-100 km in

N0 100 200 300 400

N0 100 200 300 400

Fig. 29. A self-consistent simulation of the scenario proposed in [141] for the excitation and dynamical depletion of the Kuiper belt (from [62]). Neptune is originally placed at ~23 AU and an Earth-mass embryo at ~27 AU. Both planets are embedded in a 30 M® disk, extending from 10 to 50 AU with an r-1 surface density profile (7.5 M® between 40 and 50 AU). The black curve shows the evolution of Neptune's semi-major axis (its eccentricity remains negligible), while the gray curves refer to the perihelion and aphelion distances of the embryo. Notice that the embryo is never scattered by Neptune, unlike in [141]. It migrates through the disk faster than Neptune, up to the disk's outer edge. Neptune interacts with the entire mass of the disk, thanks to the dynamical excitation of the disk because of the presence of the embryo. Therefore, it migrates much further than it would if the embryo were not present, reaching a final position well beyond 30 AU (40 AU after 1 Gy)

size could be comminuted into dust and then evacuated by radiation pressure and Poynting-Robertson drag, causing a substantial mass depletion.

To work, the collisional erosion scenario requires that two essential conditions are fulfilled. First, it requires a peculiar primordial size distribution, such that all of the missing mass was contained in small, easy-to-break objects, while the number of large objects was essentially identical to that in the current population. Some models support the existence of such a size distribution at the end of the accretion phase [92,94]. However, the collisional formation of the Pluto-Charon binary [20], the capture of Triton onto a satellite orbit around Neptune [2], and the discovery of 2003 UB313 in the Extended Scattered disk [17] suggest that the number of big bodies was much larger in the past, with as many as 1,000 Pluto-sized objects [151]. In principle, it is possible that all of these large bodies were in the planetesimal disk inside 30 AU, swept by Neptune's migration, while the primordial Kuiper belt contained only the number of large bodies inferred from the current discovery statistics, but this would require that the size distribution in the planetesimal disk had a very sensitive dependence on heliocentric distance.

The second essential condition for substantial collisional grinding is that the massive primordial Kuiper belt had a large eccentricity and inclination excitation, comparable to the current one (e ~ 0.25 and/or i ~ 7°). However, as reported at the beginning of this section, in light of [60], the mass depletion problem concerns only the cold Kuiper belt, and the dynamical excitation of the cold population is significantly smaller than that required by the collisional grinding models.

Moreover, it must be said that even assuming that the two conditions above are fulfilled, the collisional grinding models still have problems in reducing the total mass of the belt down to the current values of a few percent of an Earth mass. As the mass decreases, the collisional grinding process progressively slows down and eventually effectively stops when the total mass is still about 1 Me. The most advanced of the collisional models [97] can reduce the total mass to few 0.01 Me only if a very low specific disruption energy Q* is assumed; if more reasonable values of Q* (similar to those obtained in hydro-code experiments [9]) are adopted, the final mass achieved by collisional grinding is at least one-tenth of the initial mass, namely about 1 Me or more.

It is very difficult to reach a firm conclusion on the possibility of collisional grinding of the Kuiper belt from the collisional models alone, because of the sensitivity of these models on the assumed parameters. Perhaps the best strategy is to assume that the collisional grinding was effective, explore its general consequences and compare them with the available constraints. This work is mostly in progress, but I can briefly outline its preliminary results.

First, most of the binaries in the cold population would not have survived the collisional grinding phase [143]. In fact, given that the observed Kuiper belt binaries have large separations, it can be easily computed that the impact of a projectile just 1% the mass of the satellite at 1 kms-1 would give the satellite an impulse velocity sufficient to escape to an unbound orbit. If the collisional activity was strong enough to cause an effective reduction of the overall mass of the Kuiper belt, these kind of collisions had to be extremely common, so that we would not expect a significant fraction of widely separated binary objects in the current population.

Second, if the conditions favorable for collisional grinding in the Kuiper belt are assumed for the entire planetesimal disk (5-50 AU), the Oort cloud would not have formed: the planetesimals would have been destroyed before being ejected as in [156] (Charnoz private communication).

Third, as the Kuiper belt mass decreased during the grinding process, the precession frequencies of Neptune and the planetesimals had to change. Consequently, secular resonances had to move, potentially sweeping the belt. Assuming that, when Neptune reached 30 AU, the disk was already depleted inside 35 AU but was still massive in the 35-50 AU region, [62] showed that the v8 secular resonance would have started sweeping through the disk as soon as the mass decreased below 10 Me. The v8 resonance sweeping would have excited the eccentricity of the bodies to Neptune-crossing values and - given the large mass that the Kuiper belt would have still had when this phenomenon started - Neptune would have continued its radial migration well beyond its current location.

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