  The excess number of atoms around an ion immersed in a Bose-Einstein
condensate is determined as a function of the condensate density far from the
ion. We use thermodynamic arguments to demonstrate that in the limit of low
densities the excess number of atoms is proportional to the ratio of the
atom-ion and atom-atom scattering lengths. For denser systems we calculate the
excess number from solutions of the Gross-Pitaevskii equation using a model
potential that has a $1/r^{4}$ attraction coming from the polarization of the
neutral atoms and a hard core repulsion at short distances. We show that there
exist in general many solutions to the Gross-Pitaevskii equation for a given
condensate density, the maximum number of solutions being related to the number
of bound states of the Schr\"odinger equation for the same potential. With
increasing density, pairs of these solutions merge and disappear, implying a
discontinuous change of the state of the system.
