  We develop the suggestion that dark matter could be a Bose-Einstein
condensate. We determine the mass-radius relation of a Newtonian
self-gravitating Bose-Einstein condensate with short-range interactions
described by the Gross-Pitaevskii-Poisson system. We numerically solve the
equation of hydrostatic equilibrium describing the balance between the
gravitational attraction and the pressure due to quantum effects (Heisenberg's
uncertainty principle) and short-range interactions (scattering). We connect
the non-interacting limit to the Thomas-Fermi limit. We also consider the case
of attractive self-interaction. We compare the exact mass-radius relation
obtained numerically with the approximate analytical relation obtained with a
Gaussian ansatz. An overall good agreement is found.
