Drag on a Satellite Moving across a Spherical Galaxy: Tidal and Frictional Forces in Short-lived Encounters

The drag force on a satellite of mass M moving with speed V in the gravitational field of a spherically symmetric background of stars is computed. During the encounter, the stars are subject to a time-dependent force that alters their equilibrium. The resulting distortion in the stellar density field acts back to produce a force FΔ that decelerates the satellite. This force is computed using a perturbative technique known as linear response theory. In this paper, we extend the formalism of linear response to derive the correct expression for the back-reaction force FΔ that applies when the stellar system is described by an equilibrium one-particle distribution function. FΔ is expressed in terms of a suitable correlation function that couples the satellite dynamics to the unperturbed dynamics of the stars. At time t, the force depends upon the whole history of the composite system. In the formalism, we account for the shift of the stellar center of mass resulting from linear momentum conservation. The self-gravity of the response is neglected since it contributes to a higher order in the perturbation. Linear response theory applies also to the case of a satellite orbiting outside the spherical galaxy. The case of a satellite moving on a straight line, at high speed relative to the stellar dispersion velocity, is explored. We find that the satellite during its passage raises (1) global tides in the stellar distribution and (2) a wake, i.e., an overdense region behind its trail. If the satellite motion is external to the galaxy, it suffers a dissipative force that is not exclusively acting along V but acquires a component along R, the position vector relative to the center of the spherical galaxy. We derive the analytical expression of the force in the impulse approximation. In penetrating short-lived encounters, the satellite moves across the stellar distribution and the transient wake excited in the density field is responsible for most of the deceleration. We find that dynamical friction arises from a memory effect involving only those stars perturbed along the path. The force can be written in terms of an effective Coulomb logarithm that now depends upon time. The value of ln Λ is computed for two simple equilibrium density distributions; it is shown that the drag increases as the satellite approaches the denser regions of the stellar distribution and attains a maximum after pericentric passage. When the satellite crosses the edge of the galaxy, the force does not vanish since the galaxy keeps memory of the perturbation induced and declines on a time comparable to the dynamical time of the stellar system. In the case of a homogeneous cloud, we compute the total energy loss. In evaluating the contribution resulting from friction, we derive self-consistently the maximum impact parameter, which is found to be equal to the length traveled by the satellite within the system. Tides excited by the satellite in the galaxy reduce the value of the energy loss by friction; in close encounters, this value is decreased by a factor of ~1.5.