## 3. Dynamical friction in KBThe equation of motion of a KBO can be written as: (Wiegert & Tremaine 1997). The term
represents the force per unit mass
from the Sun, that from planets,
that from the Galactic tide,
that from giant molecular clouds,
that from passing stars,
that from other sources (e.g.
non-gravitational forces), while 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 , moving through a homogeneous and isotropic distribution of lighter particles of mass , having a velocity distribution is given by: (Chandrasekhar 1943); where is the Coulomb logarithm, and are, respectively, the masses of the test particle and that of the field one, and and the respective velocity, is the number of field particles with velocities between , . If the velocity distribution is Maxwellian Eq. (5) becomes: (Chandrasekhar 1943, Binney & Tremaine 1987),where is the density of field particles and , 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,
, and the vertical one,
, are characterized by a ratio
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 ( (Hornung & al. 1985, Stewart & Wetherill 1988) where and are the velocity and the velocity dispersion in the direction parallel to the plane while and are the same in the perpendicular direction. We suppose that and 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: and while is the average spatial density. When 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: where and 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, , is always directed as . 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 and 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 moving through a non-spherical distribution of less massive objects of mass 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: © European Southern Observatory (ESO) 1999 Online publication: October 4, 1999 |