  When a dark matter halo moves through a background of dark matter particles,
self-interactions can lead to both deceleration and evaporation of the halo and
thus shift its centroid relative to the collisionless stars and galaxies. We
study the magnitude and time evolution of this shift for two classes of dark
matter self-interactions, viz. frequent self-interactions with small momentum
transfer (e.g. due to long-range interactions) and rare self-interactions with
large momentum transfer (e.g. contact interactions), and find important
differences between the two cases. We find that neither effect can be strong
enough to completely separate the dark matter halo from the galaxies, if we
impose conservative bounds on the self-interaction cross-section. The majority
of both populations remain bound to the same gravitational potential and the
peaks of their distributions are therefore always coincident. Consequently any
apparent separation is mainly due to particles which are leaving the
gravitational potential, so will be largest shortly after the collision but not
observable in evolved systems. Nevertheless the fraction of collisions with
large momentum transfer is an important characteristic of self-interactions,
which can potentially be extracted from observational data and provide an
important clue as to the nature of dark matter.
