  We investigate the effect of a density gradient on Chandrasekhar's dynamical
friction formula based on the method of 2-body encounters in the local
approximation. We apply these generalizations to the orbit evolution of
satellite galaxies in Dark Matter haloes. We find from the analysis that the
main influence occurs through a position-dependent maximum impact parameter in
the Coulomb logarithm, which is determined by the local scale-length of the
density distribution. We also show that for eccentric orbits the explicit
dependence of the Coulomb logarithm on position yields significant differences
for the standard homogeneous force. Including the velocity dependence of the
Coulomb logarithm yields ambigous results. The orbital fits in the first few
periods are further improved, but the deviations at later times are much
larger. The additional force induced by the density gradient, the inhomogeneous
force, is not antiparallel to the satellite motion and can exceed 10% of the
homogeneous friction force in magnitude. However, due to the symmetry
properties of the inhomogeneous force, there is a deformation and no secular
effect on the orbit at the first order. Therefore the inhomogeneous force can
be safely neglected for the orbital evolution of satellite galaxies. For the
homogeneous force we compare numerical N-body calculations with semi-analytical
orbits to determine quantitatively the accuracy of the generalized formulae of
the Coulomb logarithm in the Chandresekhar approach. With the local
scale-length as the maximum impact parameter we find a significant improvement
of the orbital fits and a better interpretation of the quantitative value of
the Coulomb logarithm.
