Effects of dynamical evolution on the distribution of substructures

We develop a semi-analytical model that determines the evolution of the mass, position and internal structure of dark matter substructures orbiting in dark matter haloes. We apply this model to the case of the Milky Way. We focus in particular on the effects of mass loss, dynamical friction and substructure--substructure interactions, the last of which has previously been ignored in analytic models of substructure evolution. Our semi-analytical treatment reproduces both the spatial distribution of substructures and their mass function as obtained from the most recent N-body cosmological calculations of Gao et al. (2004). We find that, if mass loss is taken into account, the present distribution of substructures is practically insensitive to dynamical friction and scatterings from other substructures. Implementing these phenomena leads to a slight increase (~5%) in the number of substructures at r<0.35 r_vir, whereas their effects on the mass function are negligible. We find that mass loss processes lead to the disruption of substructures before dynamical friction and gravitational scattering can significantly alter their orbits. Our results suggest that the present substructure distribution at r>0.35 r_vir reflects the orbital properties at infall and is, therefore, purely determined by the dark matter environment around the host halo and has not been altered by dynamical evolution.