  The evolution of a small satellite inside a more massive truncated isothermal
spherical halo is studied using both the Theory of Linear Response for
dynamical friction and N-Body simulations. The analytical approach includes the
effects of the gravitational wake, of the tidal deformation and the shift of
the barycenter of the primary, so unifying the local versus global
interpretation of dynamical friction. Sizes, masses, orbital energies and
eccentricities are chosen as expected in hierarchical clustering models. We
find that in general the drag force in self-gravitating backgrounds is weaker
than in uniform media and that the orbital decay is not accompanied by a
significant circularization. We also show that the dynamical friction time
scale is weakly dependent on the initial circularity. We provide a fitting
formula for the decay time that includes the effect of mass and angular
momentum loss. Live satellites with dense cores can survive disruption up to an
Hubble time within the primary, notwithstanding the initial choice of orbital
parameters. Dwarf spheroidal satellites of the Milky Way, like Sagittarius A
and Fornax, have already suffered mass stripping and, with their present
masses, the sinking times exceed 10 Gyr even if they are on very eccentric
orbits.
