tokens: scales linearly with the gas mass density and is proportional to the logarithm of the ratio of the spatial extents of the forming wake and the shock’s stand-off distance. 7.2. Minimum spatial interaction scale The numerical experiments allow us to compute the total dynamical friction acting on the body and also to investigate the individual contributions to the dynamical friction induced as a function of the distance to the perturber. The outcome of this analysis reveals that the region within the stand-off distance of the shock does not contribute to the dynamical friction of a gaseous medium (see Fig. 5 and associated main text for details). At radii smaller than the stand-off distance, a spherically symmetric stratified atmosphere forms around the moving body, which (because of its symmetry) induces no net gravitational pull or pressure force. Only at radii above the stand-off distance does a wake of higher density form behind the moving body and induce a gravitational drag force. Hence, in the well-known formulae by Ruderman Spiegel (1971) or Ostriker (1999) , it is the stand-off distance of the shock that determines the minimum length scale required in the so-called Coulomb logarithm : The formation of a spherically symmetric atmosphere (with radius ) around a gravitating object implies that the hydrodynamical drag force vanishes in contrast to the non-gravitating case, but for very high Mach numbers the shock moves very close to the object so that . In this case, no spherically symmetric envelope around the object forms and the hydrodynamical drag becomes important again. 7.3. Relating the stand-off distance to the accretion radius In a second step, to allow an easy computation of the stand-off distance for general dynamical friction problems, we derived the relation of the shock’s stand-off distance to the common definition of the accretion radius, depending only on the mass and velocity of the perturber. The stand-off distance is directly proportional to the accretion radius and the proportionality constant only depends on the Mach number and the adiabatic index of the gas: Additionally, the geometrical radius of the moving body yields a lower limit for the stand-off distance . Via an analytical estimate of the stand-off distance and further fitting of the remaining free parameters, we gave an approximate relation of the dependence of the shock’s stand-off distance on the adiabatic index of the gaseous medium with and values of and . 7.4. Updated formula of dynamical friction Finally, we combined the derived scaling relations and the finding on the stand-off distance to give an update of the well-known formula of dynamical friction acting on a body of mass moving at supersonic speed through a gaseous homogeneous medium of mass density , where denotes the extent of the wake and can be obtained using the derived formulae above. Appendix A: Hydrostatic equilibrium stratification As mentioned in the text, behind the shock front and
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