Heliocentric distance AU

Fig. 6. The observed radial distribution of Kuiper belt objects (solid histogram) compared to radial distributions that would be observed for models where the surface density of Kuiper belt objects decreases by r-3/2 beyond 42 AU (dashed curve), by r-11 beyond 42 AU (solid curve), and where the surface density at 100 AU increases by a factor of 100 to the value expected from an extrapolation of the minimum mass solar nebula (dashed-dotted curve). From [131]

developed in [169]. The analysis reported in that work is reproduced in Fig. 6. The drop off of the discovery heliocentric distance distribution of Kuiper belt objects beyond 42 AU is clearly inconsistent with a smooth decline of the surface density distribution proportional to r-3/2. Instead, it can be fitted with a surface density distribution with a much sharper decay, r-11±4 (error bars are 3a), i.e. by assuming the existence of an effective edge in the radial Kuiper belt distribution. This steep radial decay should presumably hold up to ~ 60 AU, beyond which a much flatter distribution because of the Scattered disk objects should be found.

It has been conjectured [153] that, beyond some range of Neptune's influence, the number density of Kuiper belt objects could increase back up to the level expected for the minimum mass solar nebula [69]. Such an increase can be ruled out at the 3a level within 115 AU from the Sun. Beyond this distance, the biases because of the slow motion of the objects also become important; so no definite conclusion can be drawn from the current data about objects beyond this threshold. If the model is slightly modified to make the maximum object mass proportional to the surface density at a particular distance, a 100 times resumption of the Kuiper belt can still be ruled out inside 94 AU.

Although the drop off in the heliocentric distance distribution starts at 42 AU, a visual inspection of Fig. 3 shows that the edge of the Kuiper belt in semi-major axis space is precisely at the location of the 1:2 mean-motion resonance with Neptune. This is a very important feature, which points to a role for Neptune in the final positioning of the edge. I will come back to this in Sect. 4.

The Missing Mass of the Kuiper Belt

The absolute magnitude4 distribution of the Kuiper belt objects can be determined from the so-called cumulative luminosity function, which is given by the number of detections that surveys report as a function of their limiting magnitude, weighted by the inverse area of sky that they covered. If one assumes that the albedo distribution of the Kuiper belt objects is size independent, the slope of the absolute magnitude distribution can be readily converted into the slope of the cumulative size distribution.

The size distribution turns out to be very steep, with exponent of the cumulative power law falling between —3.5 and —3 for bodies larger in diameter than ^200 km [55]. Actually, the size distribution is slightly shallower for the hot population than for the cold population, as shown in a recent analysis [10] (see Fig. 7). This is not surprising, given that - as we have seen above - the hot and the cold populations contain roughly the same total number of objects, but the former hosts the largest members of the Kuiper belt.

The HST survey in [10] also reported the detection of a change in the size distribution for objects fainter than H = 9-10, corresponding to about 100 km in diameter, assuming a standard albedo of ~ 4%. The slopes of the size distribution below this limit, however, remain very uncertain because of small number statistics. Some researchers still dispute the validity of the detection of any turnover in the size distribution [144]. Given these uncertainties, as well as uncertainties on the mean albedo of the Kuiper belt objects (required to convert a given absolute magnitude into a size) and their bulk density, the total mass of the Kuiper belt is uncertain to at least an order of magnitude, estimates ranging from 0.01 M® [10] to 0.1 M® [55].

Whatever the real value (in this range, or slightly beyond), it nevertheless seems certain that the total mass of the Kuiper belt is now very small, in particular, compared with the mass of the disk of solids from which the Kuiper belt objects had to form. There are two lines of argument to estimate this primordial mass.

A first argument follows the reasoning that led Kuiper to conjecture the existence of a band of small planetesimals beyond Neptune. [104] The minimum mass solar nebula inferred from the total planetary mass (plus lost volatiles; [69]) smoothly declines from the orbit of Jupiter until the orbit of

4The absolute magnitude, H, is a measure of the intrinsic brightness of an object. It corresponds to the visual magnitude that an object would have in the paradoxical situation of being simultaneously at 1 AU from the Sun and the Earth, at opposition!

Fig. 7. The H or size distribution in the Kuiper belt (adapted from [10] with the permission of Bernstein). The red and green bands show the uncertainties for the cold and the hot population, respectively (although the definition for hot and cold used in that work do not exactly match those adopted in this paper). Absolute magnitudes have been computed assuming that all detections occurred at 42 AU (the maximum of the radial surface density distribution of the Kuiper belt), and the conversion to diameters uses the assumption that the mean albedo is 4%

Fig. 7. The H or size distribution in the Kuiper belt (adapted from [10] with the permission of Bernstein). The red and green bands show the uncertainties for the cold and the hot population, respectively (although the definition for hot and cold used in that work do not exactly match those adopted in this paper). Absolute magnitudes have been computed assuming that all detections occurred at 42 AU (the maximum of the radial surface density distribution of the Kuiper belt), and the conversion to diameters uses the assumption that the mean albedo is 4%

Neptune; why should it drop abruptly beyond the last planet? The extrapolation and integration of this surface density distribution predicts that the original total mass of solids in the 30-50 AU range should have been —30 .

The second argument for a massive primordial Kuiper belt was first raised in [152], where it was found that the objects currently in the Kuiper belt could not have formed in the present environment: collisions are sufficiently infrequent that 100 km objects cannot be built by pairwise accretion within the current population over the age of the Solar System. Moreover, owing to the large eccentricities and inclinations of Kuiper belt objects - and consequently to their high encounter velocities - the collisions that do occur tend to be erosive rather than accretional, making bodies smaller rather than larger. Stern suggested that the solution of this dilemma is that the primordial Kuiper belt was both more massive and dynamically colder, so that more collisions occurred, and they were gentler and therefore generally accretional.

Following this idea, detailed modeling of accretion in a massive primordial Kuiper belt was performed [92,93,94,153,154,155]. While each model includes different aspects of the relevant physics of accretion, fragmentation, and velocity evolution, the basic results are in approximate agreement. First, with ~10Me or more of solid material in an annulus from about 35 to 50 AU on very low eccentricity orbits (e < 0.001), all models naturally produce a few objects of the size of Pluto and approximately the right number of ~ 100 km objects, on a timescale ranging from several 107 to several 108 years. The models suggest that the majority of mass in the disk was in bodies approximately 10 km and smaller. The accretion stopped when the formation of Neptune (or other dynamical phenomena; see Sect. 4) began to excite eccentricities and inclinations in the population that were high enough to move the collisional evolution from the accretional to the erosive regime.

A massive and dynamically cold primordial Kuiper belt is also required by the models that attempt to explain the formation of the numerous observed binary Kuiper belt objects [6,54,57,175].

Therefore, the general formation picture of an initially massive Kuiper belt appears to be secure, and understanding the ultimate fate of the 99% of the initial mass that appears no longer to be in the Kuiper belt is a crucial step in reconstructing the history of the outer Solar System.

1.3 Dynamics in the Kuiper Belt

In this section, I will give an overview of the dynamical properties of the Kuiper belt. Without any pretension of being exhaustive, the goal is to understand which properties of the Kuiper belt orbital structure can be explained from the evolution of the objects in the framework of the current architecture of the Solar System and which, conversely, require an explanation built on a scenario of primordial sculpting (as in Sect. 4).

Figure 8 shows a map of the dynamical lifetime of trans-Neptunian bodies as a function of their initial semi-major axis and eccentricity, for an inclination of 1° and a random choice of the orbital angles A, w, and i [38]. Similar maps, referring to different choices of the initial inclination or different projections on orbital element space can be found in [103] and [38]. These maps have been computed numerically, by simulating the evolution of massless particles from their initial conditions, under the gravitational perturbations of the giant planets. The latter were assumed to be initially on their current orbits. Each particle was followed until it suffered a close encounter with Neptune. Objects encountering Neptune would then evolve in the Scattered disk for a typical time of order ~108 years (but much longer residence times in the Scattered disk occur for a minority of objects), until they are transported by planetary encounters into the inner Solar System or to the Oort cloud, or are ejected to the interstellar space. This issue is described in more detail in Sect. 2.

In Fig. 8, the colored strips indicate the timespan required for a particle to encounter Neptune, as a function of its initial semi-major axis and eccentricity. Strips that are colored yellow represent objects that survive for the length of the simulation, 4 x 109 years (the approximate age of the Solar System) without encountering the planet. The figure also reports the orbital

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