  Self-interacting dark matter has been proposed to explain the apparent mass
deficit in astrophysical small-scale halos, while observations from galaxy
clusters suggest that the corresponding cross section depends on the velocity.
Accounting for this is often believed to be highly model-dependent with studies
mostly focusing on scenarios with light mediators. Based on the effective-range
formalism, in this work we point out a model-independent approach which
accurately approximates the velocity dependence of the self-interaction cross
section with only two parameters. We illustrate how this parameterization can
be simultaneously interpreted in various well-motivated scenarios, including
self-interactions induced by Yukawa forces, Breit-Wigner resonances and bound
states. We investigate the astrophysical implications and discuss how the
approximation can be improved in certain special regimes where it works poorly.
