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Astron. Astrophys. 342, 34-40 (1999)

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4. Dynamical friction in inhomogeneous systems

The introduction of the notion of dynamical friction is due to CN43. In the stochastic formalism developed by CN43 the dynamical friction is discussed in terms of [FORMULA]:

[EQUATION]

where

[EQUATION]

and [FORMULA]. As shown by CN43, the origin of dynamical friction is due to the asymmetry in the distribution of relative velocities. As previously told, if a test star moves with velocity [FORMULA] in a spherical distribution of field stars, namely [FORMULA] then we have that:

[EQUATION]

The asymmetry in the distribution of relative velocities is conserved in the final Eq. (25). In fact from Eq. (25) we have:

[EQUATION]

(CN43). This means that when [FORMULA] then [FORMULA]; while when [FORMULA] then [FORMULA]. As a consequence, when [FORMULA] has a positive component in the direction of [FORMULA], [FORMULA] increases on average; while if [FORMULA] has a negative component in the direction of [FORMULA], [FORMULA] decreases on average. Moreover, the star suffers a greater amount of acceleration in the direction [FORMULA] when [FORMULA] than in the direction [FORMULA] when [FORMULA].

In other words the test star suffers, statistically, an equal number of accelerating and decelerating impulses. Being the modulus of deceleration larger than that of acceleration the star slows down.

At this point we may show how dynamical friction changes due to inhomogeneity. From Eq. (18) we see that [FORMULA] differs from that obtained in homogeneus system only for the presence of a dependence on the inhomogeneity parameter p. If we divide Eq. (18) for Eq. (25) we obtain:

[EQUATION]

If we consider a homogeneous system, [FORMULA], the previous equation reduces to:

[EQUATION]

In the case of an inhomogeneous system, [FORMULA], we see that:

[EQUATION]

where

[EQUATION]

As we show in Fig. 1, this last equation is an increasing function of p. This means that for increasing values of p the star suffers an even greater amount of acceleration in the direction [FORMULA] when [FORMULA] than in the direction [FORMULA] when [FORMULA], with respect to the homogeneous case. This is due to the fact that the difference between the amplitude of the decelerating impulses and the accelerating ones is, as in homogeneous systems, statistically negative, but now larger, being the scale factor greater. This finally means that, for a given value of n, the dynamical friction increases with increasing inhomogeneity in the space distribution of stars (it is interesting to note that this effect is fundamentally due to the inhomogeneity of the distribution of the stars and not to the density n). In other words two systems having the same n will have their stars slowed down differently according to the value of p. This is strictly connected to the asymmetric origin of the dynamical friction.

[FIGURE] Fig. 1. The function [FORMULA] for several values of the inhomogeneity parameter p; solid line [FORMULA], dashed line [FORMULA], dotted line [FORMULA].

In addition, by increasing n the dynamical friction increases, just like in the homogeneous systems, but the increase is larger than the linear increase observed in homegeneous systems.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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