Dynamical friction and the evolution of satellites in virialized halos: The theory of linear response

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, the tidal deformation, and the shift of the barycenter of the primary, thereby unifying the local and global interpretations of dynamical friction. The N-body simulations follow the evolution of both rigid and live satellites within larger systems. Sizes, masses, orbital energies, and eccentricities are chosen as expected in hierarchical clustering models for the formation of structures. Results from this coupled approach are applicable to a vast range of astrophysical problems, from galaxies in galaxy clusters to small satellites of individual galaxies. The main contribution to the drag results from the gravitational pull of the overdensity region trailing the satellite's path, since the stellar response to the external perturbation remains correlated over a time shorter than the typical orbital period. The analytical approach and the N-body experiments demonstrate that there is no significant circularization of the orbits and that the dynamical friction timescale is weakly dependent of the circularity, epsilon. While the theory and the N-body simulations give a complete description of the orbital decay of satellites, a good fitting formula for the orbital decay time is tau(DF) = 1.2 J(cir)r(cir)/(GM(sat)/e)ln(M-halo/M-sat) (epsilon)0.4, where J(cir) and r(cir) are, respectively, the initial orbital angular momentum and the radius of the circular orbit with the same energy as the actual orbit. Tidal stripping can reduce the satellite's mass by 60% after the first pericentric passage, increasing the orbital decay time. The e factor takes that effect into account and should be removed in the simplified case of rigid satellites. In cosmologically relevant situations, our model gives orbital decay times larger by a factor of 2 than most previous estimates. For peripheral orbits in which the apocenter is larger than the virial radius of the primary decay, the tidal held and the shift of the barycenter become important. In this case, tau(DF) needs to be further increased by at least similar or equal to 50%. The final fate of a satellite is determined by its robustness against the effect of tides. While low-density satellites are disrupted over a time comparable to the decay time of their rigid counterparts, satellites with small cores can survive up to a Hubble time within the primary, regardless of the initial choice of orbital parameters. Dwarf spheroidal satellites of the Milky Way, such as Sgr 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.