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

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4. Parameters used in the simulation

To calculate the effects of dynamical friction, introduced in the previous section, on a disc-like structure as KB, we cannot use the classical Chandrasekhar (1943) theory but we need equations specified for distributions like KB. These equations were written in the previous section (Eqs. 8 - 11). To calculate the effect of dynamical friction on the orbital evolution of the largest bodies we suppose that [FORMULA] and [FORMULA] are constants and that [FORMULA]=[FORMULA]. We need also the mass density distribution in the disc. We assume a heliocentric (R) distribution in surface mass density [FORMULA] and a total primordial mass, [FORMULA] in the region [FORMULA] (Stern & Colwell 1997b). To evaluate the dynamical friction force we need the spatial distribution of the field objects, [FORMULA]. To reproduce the quoted surface density we need a number density decreasing as a function of distance from the Sun, R, to the third power (Levison & Duncan 1990):

[EQUATION]

We remember also that the KB is a disc and then we use the mass distribution given by a Myiamoto-Nagai disc (Binney & Tremaine 1987; Wiegert & Tremaine 1997):

[EQUATION]

which in the disc plane reproduces the quoted surface density and Eq. (15) [FORMULA]. Here M is the disc mass, a and b are parameters describing the disc characteristic radius and thickness. Because there is presently no information on the way in which ensemble-averaged inclinations ([FORMULA]) and eccentricities ([FORMULA]) vary in the KB, we adopt a disc with a width [FORMULA] (Stern 1996b). The equations of motion were integrated in heliocentric coordinates using the Bulirsch-Stoer method.

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© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999
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