  We provide an approximate analytical expression of the mass-radius relation
of a Newtonian self-gravitating Bose-Einstein condensate (BEC) with short-range
interactions described by the Gross-Pitaevskii-Poisson system. These equations
model astrophysical objects such as boson stars and, presumably, dark matter
galactic halos. Our study connects the non-interacting case studied by Ruffini
& Bonazzola (1969) to the Thomas-Fermi limit studied by B\"ohmer & Harko
(2007). For repulsive short-range interactions (positive scattering lengths),
there exists configurations of arbitrary mass but their radius is always larger
than a minimum value. For attractive short-range interactions (negative
scattering lengths), equilibrium configurations only exist below a maximum
mass. Above that mass, the system is expected to collapse and form a black
hole. We also study the radius versus scattering length relation for a given
mass. We find that stable configurations only exist above a (negative) minimum
scattering length. Our approximate analytical solution, based on a Gaussian
ansatz, provides a very good agreement with the exact solution obtained by
numerically solving a nonlinear differential equation representing hydrostatic
equilibrium. Our treatment is, however, easier to handle and permits to study
the stability problem, and derive an analytical expression of the pulsation
period, by developing an analogy with a simple mechanical problem.
