  In this work, we discuss the relation between the strength of the
self-interaction of dark matter particles and the predicted properties of the
inner density distributions of dark matter haloes. We present the results of
$N$-body simulations for 28 haloes performed with the same initial conditions
for cold dark matter and for self-interacting dark matter, for a range of
cross-sections. We provide a simple phenomenological description of these
results and compare with the semi-analytical model typically used in the
literature. Using these results, we then predict how the inner dark matter
surface density and core radius should depend on the self-interaction
cross-section for observed haloes.
