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Astron. Astrophys. 350, 685-693 (1999)

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1. Introduction

The current model for comets in the solar system supposes that a vast cloud of cometary objects orbits the Sun. This cloud consists of three components. The inner one, referred to as the Kuiper Belt (hereafter KB) (Edgeworth 1949; Kuiper 1951), is a disc like structure of [FORMULA] comets extending from [FORMULA] from the Sun (Weissman 1995; Luu et al. 1997). The KB has been proposed as the source of the Jupiter-family short-period (hereafter SP) comets. The second component, referred to as the Oort inner cloud, or the Hills cloud (Hills 1981), is supposed to be a disc, thicker than KB, containing [FORMULA] objects lying [FORMULA] from the Sun. It is proposed as a source of long-period (hereafter LP) and Halley-type SP comets (Levison 1996). The last component, the Oort cloud (Oort 1950), is a spherical cloud of [FORMULA] cometary objects with nearly isotropic velocity distribution extending from [FORMULA] to [FORMULA]. Even if the Oort cloud was considered in the past as the fundamental reservoir of LP comets which have been brought into the inner solar system by perturbations due to the galactic tidal field, molecular clouds and passing stars, nowadays it has been shown that it can contribute only for a small part to the LP population of comets (Duncan et al. 1988; Wiegert & Tremaine 1997).

Observational confirmation of the KB was first achieved with the discovery of object 1992QB1 by Jewitt & Luu (1993). To date over 40 KB objects (hereafter KBO) with diameters between 100 and 400 km have been discovered and the detection statistics obtained suggest that a complete ecliptic survey would reveal [FORMULA] such bodies orbiting between 30 and [FORMULA]. Such a belt of distant icy planetesimals could be a more efficient source of SP comets than the Oort cloud (Fernandez 1980; Duncan et al. 1988). Dynamical simulations have shown that a cometary source with low initial inclination distribution was more consistent with the observed orbits of SP comets than the randomly distributed inclinations typical of the comets in the Oort cloud (Quinn et al. 1990; Levison & Duncan 1993). According to other simulations the greatest part of objects in the KB should be stable for the age of the solar system. However if the population of the KB is [FORMULA] objects, weak gravitational perturbations provide a large enough influx to explain the current population of SP comets (Levison & Duncan 1993; Holman & Wisdom 1993; Duncan et al. 1995). In particular Levison & Duncan (1993) and Holman & Wisdom (1993), studied the long term stability of test particles in low-eccentricity and low-inclination orbits beyond Neptune, subject only to the gravitational perturbations of the giant planets. They found orbital instability on timescales [FORMULA] interior to [FORMULA], regions of stability and instability in the range [FORMULA] and stable orbits beyond [FORMULA].

A study by Malhotra (1995a) showed that the KB is characterized by a highly non uniform distribution: most of the small bodies in the region between Neptune and [FORMULA] would have been swept into narrow regions of orbital resonance with Neptune (the 3:2 and 2:1 orbital resonances, respectively located at distances from the Sun of [FORMULA] and [FORMULA]). The orbital inclinations i of many of these objects would remain low ([FORMULA]) but the eccentricities e would have values from 0.1 to 0.3. At the same time many of the trans-neptunian objects discovered lie in low-inclination orbits, as predicted by the dynamical models of Holman & Wisdom (1993) and Levison & Duncan (1993). A more detailed analysis of this distribution reveals that most objects inside [FORMULA] reside in higher-e, i orbits locked in mean motion resonance with Neptune, but most objects beyond this distance reside in non-resonant orbits with significantly lower eccentricities and inclinations.

After the previously quoted discovery of [FORMULA] km sized objects (Jewitt & Luu 1993; Jewitt & Luu 1995; Weissmann & Levison 1997), proving that the KB is populated, Cochran et al. (1995) have reported Hubble Space Telescope results giving the first direct evidence for comets in the KB. Cochran's observations imply that there is a large population ([FORMULA]) of Halley-sized objects (radii [FORMULA] km) within [FORMULA] of the Sun, made up of low inclination objects ([FORMULA]). At the time of the publication, Cochran's et al. (1995) results were criticized on two grounds:

1) the detections were statistical in nature, and the authors were not able to fit orbits to their objects;

2) the number of detections did not agree with extrapolation of the size distribution of large KBOs determined from early ground-based observations (but Weissman & Levison 1997 showed that Cochran's et al. 1995 results were in agreement with the number of KBOs needed to populate the Jupiter-family comets); moreover Brown et al. 1997 contended that detections reported in Cochran et al. (1995) were not possible, based on an analysis of the noise properties of the data. In a recent paper, Cochran et al. 1998, confirmed the early results by means of a new analysis.

The spatial dimensions and mass distribution in the KB are poorly known. Yamamoto et al. (1994) have applied a planetesimal model to the trans-neptunian region, finding that the maximum number density of the planetesimal population should be about at [FORMULA] and the planetesimal disc itself can extend up to distances [FORMULA]. This is in agreement with detection by IRAS of discs around main sequence stars, Vega (Aumann et al. 1984), [FORMULA] Pictoris (Smith & Terrile 1984), extending to several hundred AU. From the available radio and infrared data, Beckwith & Sargent (1993) conclude that disc masses may range between [FORMULA] to [FORMULA] and extend from a few hundred AU to more than [FORMULA]. In short, both theoretical arguments and observations strengthen the view that our solar system is surrounded by a flattened structure of planetesimals, extending perhaps to several hundred AU.

Although originally it had been thought that the population might be collisionless, recent work (Stern 1995) has shown that the collisional effects cannot be neglected over 4.5 Gyr. As shown by Stern (1995, 1996a,b) and Stern & Colwell (1997a,b), the collisional evolution is an important evolutionary process in the disc as a whole, and moreover, it is likely to be the dominant evolutionary process beyond [FORMULA]. In the case of larger planetesimals the evolution is connected to the energy loss due to dynamical friction, which transfers kinetic energy from the larger planetesimals to the smaller ones. This mechanism, in the early solar system, provides an energy source for the small planetesimals that is comparable to that provided by the viscous stirring process (Stewart & Wetherill 1988; Weidenschilling et al. 1997).

The objective of this paper is to examine the role of dynamical friction, in the primordial KB, in the orbital evolution of the largest planetesimals that lie at a heliocentric distance [FORMULA] (at this distance the effects of the planets decline rapidly to zero and only a small fraction of objects is influenced by planetary perturbations - Wiegert & Tremaine 1999; Stern & Colwell 1997b). While the importance of dynamical friction in planetesimal dynamics was demonstrated in several papers, (Stewart & Kaula 1980; Horedt 1985; Stewart & Wetherill 1988) and in particular in the case of the planetary accumulation process, the role of this effect on the orbital evolution of the largest planetesimals and the consequent change of mass distribution in KB was never studied.

The paper is organized as follows. In Sect. 2, we review the role of encounters and collisions in KB. In Sect. 3, we introduce the equations to calculate dynamical friction effects. In Sect. 4 we describe how we use these equations to determine the evolution of the largest bodies population in KB. In Sect. 5 we discuss the results of the calculation and we also show (supposing the scenario proposed by Stern & Colwell 1997b of Pluto formation beyond [FORMULA] to be correct) how dynamical friction is able to transport an object of the size of Pluto from [FORMULA] to the actual position. In Sect. 6 we give our conclusions.

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© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999