So far we have treated galaxies as isolated, non-interacting systems. However, in the hierarchical scenario of structure formation, galaxies and their associated dark matter halos undergo frequent interactions with each other. As we have seen in §7.3.6, a large fraction of dark matter halos are expected to be dynamically young (i.e. to have experienced a merger event in their recent history). In fact, as shown by Li et al. (2007), each halo, independent of its mass, experiences about three major mergers (defined as mergers with a progenitor mass ratio larger than 1/3) after its main progenitor has acquired 1% of its present-day mass. Hence, galaxies and their associated dark matter halos cannot be considered isolated ‘island universes', but are constantly influenced by gravitational interactions with other systems. These interactions may have dramatic impact on the morphologies and star-formation histories of galaxies, making the study of their nature and frequency an important part of galaxy formation and evolution. Consider a body S which has an encounter with a perturber P with impact parameter b and initial velocity v ∞ (in the limit of infinite, initial separation between S and P). Let q be a particle (e.g. a star) in S, at a distance r(t) from the center of S, and let R(t) be the position vector of P from S (see Fig. 12.1 for an illustration).