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

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3. Dynamical friction in KB

The equation of motion of a KBO can be written as:

[EQUATION]

(Wiegert & Tremaine 1997). The term [FORMULA] represents the force per unit mass from the Sun, [FORMULA] that from planets, [FORMULA] that from the Galactic tide, [FORMULA] that from giant molecular clouds, [FORMULA] that from passing stars, [FORMULA] that from other sources (e.g. non-gravitational forces), while R is the dissipative force (the sum of accretion and dynamical friction terms - see Melita & Woolfson 1996). If we consider KBOs at heliocentric distances [FORMULA] then [FORMULA] may be neglected. We also neglect the effects of non-gravitational forces and the perturbations from Galactic tide, GCMs or stellar perturbations, which are important only for objects at heliocentric distances [FORMULA] (Brunini & Fernandez 1996). We assume that the planetesimals travel around the Sun in circular orbits and we study the orbital evolution of KBOs after they reach a mass [FORMULA]. Moreover we suppose that the role of collisions for our KBOs at distances [FORMULA] can be neglected. We know that the role of collisions would be progressively less important with increasing distance from the Sun because the collision rate, [FORMULA] ([FORMULA] is the collision cross section), decreases due to the decrease in the local space number density, n of KBOs and the local average crossing velocity, v, of the target body. As stated previously, using Eq. (1) at distances larger than [FORMULA] the collision time, [FORMULA], is of the order of the age of the solar system. Besides, the energy damping is not dominated by collisional damping but by dynamical friction damping; also, artificially increasing the collisional dampings hardly change the dynamics of the largest bodies (Kokubo & Ida 1998).

To take account of dynamical friction we need a suitable formula for a disk-like structure such as KB. Following Chandrasekhar & von Neumann's (1942) method, the frictional force which is experienced by a body of mass [FORMULA], moving through a homogeneous and isotropic distribution of lighter particles of mass [FORMULA], having a velocity distribution [FORMULA] is given by:

[EQUATION]

(Chandrasekhar 1943); where [FORMULA] is the Coulomb logarithm, [FORMULA] and [FORMULA] are, respectively, the masses of the test particle and that of the field one, and [FORMULA] and [FORMULA] the respective velocity, [FORMULA] is the number of field particles with velocities between [FORMULA], [FORMULA].

If the velocity distribution is Maxwellian Eq. (5) becomes:

[EQUATION]

(Chandrasekhar 1943, Binney & Tremaine 1987),where [FORMULA] is the density of field particles and [FORMULA], [FORMULA] being the velocity dispersion. Eq. (6) cannot be used for systems not spherically symmetric except for the case of objects moving in the equatorial plane of an axisymmetric distribution of matter. These objects, in fact, have no way of perceiving that the potential in which they move is not spherically symmetric.

We know that KB is a disc and consequently for objects moving away from the disc plane we need a more general formula than Eq. (6). Moreover dynamical friction in discs differs from that in spherical isotropic three dimensional systems. First, in a disc close encounters give a contribution to the friction that is comparable to that of distant encounters (Donner & Sundelius 1993; Palmer et al. 1993). Collective effects in a disc are much stronger than in a three-dimensional system. The velocity dispersion of particles in a disc potential is anisotropic. N-body simulations and observations show that the radial component of the dispersion, [FORMULA], and the vertical one, [FORMULA], are characterized by a ratio [FORMULA] for planetesimals in a Keplerian disc (Ida et al. 1993). The velocity dispersion evolves through gravitational scattering between particles. Gravitational scattering between particles transfers the energy of the systematic rotation to the random motion (Stewart & Wetherill 1988). In other words the velocity distribution of a Keplerian particle disc is ellipsoidal with ratio 2:1 between the radial and orthogonal (z ) directions (Stewart & Wetherill 1988). According with what previously told, we assume that the matter-distribution is disc-shaped, having a velocity distribution:

[EQUATION]

(Hornung & al. 1985, Stewart & Wetherill 1988) where [FORMULA] and [FORMULA] are the velocity and the velocity dispersion in the direction parallel to the plane while [FORMULA] and [FORMULA] are the same in the perpendicular direction. We suppose that [FORMULA] and [FORMULA] are constants and their ratio is simply taken to be 2:1. Then according to Chandrasekhar (1968) and Binney (1977) we may write the force components as:

[EQUATION]

[EQUATION]

where

[EQUATION]

[EQUATION]

and

[EQUATION]

while [FORMULA] is the average spatial density. When [FORMULA] the drag caused by dynamical friction will tend to increase the anisotropy of the velocity distribution of the test particles. The frictional drag on the test particles may be written:

[EQUATION]

where [FORMULA] and [FORMULA] are two unity vectors parallel and perpendicular to the disc plane.

This result differs from the classical Chandrasekhar (1943) formula. Chandrasekhar's result tells that dynamical friction force, [FORMULA], is always directed as [FORMULA]. This means that if a massive body moves, for example, in a disc in a plane different from the symmetry plane, dynamical friction causes it to spiral through the center of the mass distribution always remaining in its own plane. It shall reach the disc plane only when it reaches the centre of the distribution. Eq. (13) shows that a massive object suffers drags [FORMULA] and [FORMULA] in the directions within and perpendicular to the equatorial plane of the distribution. This means that the object shall find itself confined to the plane of the disc before it reaches the centre of the distribution (this means that also inclinations are damped). In other words the dynamical drag experienced by an object of mass [FORMULA] moving through a non-spherical distribution of less massive objects of mass [FORMULA] is not directed in the direction of the relative movement of the massive particle and the centre of mass of the less massive objects (as in the case of spherically symmetric distribution of matter). As a consequence the already flat distribution of more massive objects will be further flattened during the evolution of the system (Binney 1977). The objects lying in the plane, as previously told, have no way to perceive that they are moving in a not spherically symmetric potential. Hence we expect that the dynamical drag is directed in the direction opposite to the motion of the particle:

[EQUATION]

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© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999
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