Charged boson stars revisited

We consider again stationary solutions to the spherically symmetric Einstein--Maxwell--Klein--Gordon system, commonly known as ``charged boson stars'', originally studied by Jetzer and Van Der Bij. We construct families of charged boson stars in the ground state, for different values of the charge parameter $q$, and different values of the central scalar field. Following Jetzer and Van Der Bij, one can define a critical value for the charge $q=q_c$ that corresponds to the value for which the Coulomb repulsion of the bosonic particles exactly cancels their newtonian gravitational attraction. We confirm the claim made by Pugliese et al. that super-critical solutions exist for a limited range of charges above the critical value $q>q_c$ (though we find an even smaller range of $q$ for which this is possible). Our analysis indicates, however, that all such super-critical solutions are gravitationally unbound, and are therefore expected to be unstable. One of the main results of our analysis is the fact that, even though we do find a family of slightly super-critical solutions in the sense that $q>q_c$, there are no super-critical solutions in the sense that the total charge $Q$ is larger than the total mass $M$ of the system.