  The supersonic motion of gravitating objects through a gaseous medium
constitutes a classical problem in theoretical astrophysics. Its application
covers a broad range of objects and scales from planets up to galaxies.
Especially the dynamical friction, caused by the forming wake behind the
object, plays an important role for the dynamics of the system. To calculate
the dynamical friction, standard formulae, based on linear theory are often
used. It is our goal to check the general validity of these formulae and
provide suitable expressions for the dynamical friction acting on the moving
object, based on the basic physical parameters of the problem. We perform
sequences of high resolution numerical studies of rigid bodies moving
supersonically through a homogeneous medium, and calculate the total drag
acting on the object, which is the sum of gravitational and hydro drag. We
study cases without gravity with purely hydrodynamical drag, as well as
gravitating objects. From the final equilibrium state of the simulations, we
compute for gravitating objects the dynamical friction by direct numerical
integration of the gravitational pull acting on the embedded object. The
numerical experiments confirm the known scaling laws for the dependence of the
dynamical friction on the basic physical parameters as derived in earlier
semi-analytical studies. As a new important result we find that the shock's
stand-off distance is revealed as the minimum spatial interaction scale of
dynamical friction. Below this radius, the gas settles into a hydrostatic
state, which causes no net gravitational pull onto the moving body. Finally, we
derive an analytic estimate for the stand-off distance that can be used to
calculate the dynamical friction force.
