  Amplifying on a proposal by O'Dell et al. for the realization of
Bose-Einstein condensates of neutral atoms with attractive $1/r$ interaction,
we point out that the instance of self-trapping of the condensate, without
external trap potential, is physically best understood by introducing
appropriate "atomic" units. This reveals a remarkable scaling property: the
physics of the condensate depends only on the two parameters $N^2 a/a_u$ and
$\gamma/N^2$, where $N$ is the particle number, $a$ the scattering length,
$a_u$ the "Bohr" radius and $\gamma$ the trap frequency in atomic units. We
calculate accurate numerical results for self-trapping wave functions and
potentials, for energies, sizes and peak densities, and compare with previous
variational results. As a novel feature we point out the existence of a second
solution of the extended Gross-Pitaevskii equation for negative scattering
lengths, with and without trapping potential, which is born together with the
ground state in a tangent bifurcation. This indicates the existence of an
unstable collectively excited state of the condensate for negative scattering
lengths.
