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

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2. Encounters and collisions in KB

As shown in several N-body simulations (Stern 1995), the structure of KB in the region [FORMULA] is fundamentally due to two processes:

  1. dynamical erosion due to resonant interactions with Neptune (Holman & Wisdom 1993; Levison & Duncan 1993; Duncan et al. 1995; Malhotra 1995a). In this region the KB has a complex structure. Objects with perihelion distances [FORMULA] are unstable. For orbits with [FORMULA] and with semi-major axis [FORMULA] the only stable orbits are those in Neptune resonances. Between [FORMULA] and [FORMULA] at low inclinations and between [FORMULA] and [FORMULA] with [FORMULA] the orbits are unstable.

  2. collisional erosion (Stern 1995, 1996a,b; Stern & Colwell 1997a,b). Starting from a primordial disc having a mass of [FORMULA] collisions are able to reduce its mass to [FORMULA] in [FORMULA] if [FORMULA].

Then the [FORMULA] zone is both collisionally and dynamically evolved, since dynamics acted to destabilize most orbits with [FORMULA] and were able to induce eccentricities that caused collisions out to almost [FORMULA]. The dynamically and collisionally evolved zone might extend as far as [FORMULA], if Malhotra's (1995a) mean motion resonance sweeping mechanism is important. Beyond this region one expects there to be a collisionally evolved zone where accretion has occurred but eccentricity perturbations by the giant planets have been too low to initiate erosion. Beyond that region a primordial zone is expected in which the accretion rates have hardly modified the initial population of objects. Supposing that the last assumption is correct and that the radial distribution of heliocentric surface mass density in the disc, [FORMULA], can be described by [FORMULA] (Tremaine 1990; Stern 1996a,b) and supposing that in the zone [FORMULA] of the primordial disc [FORMULA] of matter was present, we expect [FORMULA] in the zone [FORMULA] of the present KB. As the effects of planets are negligible for planetesimals beyond [FORMULA], the orbital motion there can be considered not far from Keplerian and circular (Brunini & Fernandez 1996). This last assumption is more strictly satisfied by the largest objects, which should, through energy equipartition, evolve to the lowest eccentricity in the swarm (Stewart & Wetherill 1988; Stern & Colwell 1997b)

Although the main motion of KB objects (KBOs) is Keplerian rotation around the Sun, the motion is perturbed by encounters with other objects and by collisions. Encounters influence the structure of the system in several ways:

a) Relaxation;

b) Equipartition;

c) Escape;

d) Inelastic encounters.

Each of the quoted effects has greater or smaller importance in a system evolution according to the system characteristics. In the case of a system like KB, inelastic encounters have a fundamental role because KBOs have on average a much smaller escape speed than the rms velocity dispersion, [FORMULA]. If [FORMULA] is the radius of a KBO, [FORMULA] is the escape speed from the KBO surface, [FORMULA] is the Safronov number, n the number density and [FORMULA] the velocity dispersion, then the collision time [FORMULA] for a population of planetesimals with a Gaussian distribution in dispersion velocity is given by:


(Binney & Tremaine 1987; Palmer et al. 1993).

Within [FORMULA] from the Sun, a population of a few hundred km-sized planetesimals, with several Earth masses in total, would have [FORMULA], while at distances [FORMULA] [FORMULA] becomes comparable with the age of the solar system. Indeed by means of N-body simulations, Stern (1995) showed how collisional evolution plays an important role through KB and that it has a dominant role at [FORMULA]. The effect of collisions is that of inducing energy dissipation in the system but at the same time collisions are important in the growth of QB1 objects, Pluto-scale and larger objects starting from 1 to 10 km building blocks in a time that in some plausible circumstances is as little as [FORMULA].

Mutual gravitational scattering induces random velocity in two different ways: one is viscous stirring which converts solar gravitational energy into random kinetic energy of planetesimals. Energy is transferred from circular, co-planar orbits with zero random velocities to eccentric, mutually inclined orbits with nonzero random velocities. The other is dynamical friction which transfers random kinetic energy from the larger planetesimals to the smaller ones (Stewart & Wetherill 1988; Ida 1990; Ida & Makino 1992; Palmer et al. 1993). Unlike viscous stirring, the exchange of energy does not depend on the differential rotation of the mean flow for its existence. Dynamical friction would drive the system to a state of equipartition of kinetic energy but viscous stirring opposes this tendency. In the dispersive regime, the time scales of stirring and dynamical friction are almost equal to the two-body relaxation time


where [FORMULA] is the gravitational radius, n the number density,


[FORMULA] being the largest impact parameter and [FORMULA] the mean square velocity of the objects, which for the typical values of the parameters in the KB is of the order of [FORMULA] and [FORMULA].

In conclusion the random velocity of the smaller planetesimals is increased by viscous stirring while the larger planetesimals suffer dynamical friction due to smaller planetesimals.

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© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999