Revisiting relativistic electrically charged polytropic spheres

We revisit the problem of the structure and physical properties of electrically charged static spherically symmetric solutions of the Einstein-Maxwell system of equations where the matter model is a polytropic gas. We consider a relativistic polytrope equation of state and take the electric charge density to be proportional to the rest mass density. We construct the families of solutions corresponding to various sets of parameters and analyze their stability and compliance with the causality requirement, with special emphasis on the possibility of constructing black hole mimickers. Concretely, we want to test how much electric charge a given object can hold and how compact it can be. We conclude that there is a microscopic bound on the charge density to rest mass density ratio coincident with the macroscopic bound regarding the extremal Reissner-Nordst\"om black hole. The macroscopic charge to mass ratio for the object can exceed the corresponding microscopic ratio if the object is non-extremal. Crucially, the only way to obtain a black hole mimicker is by taking a subtle limit in which an electrically counterpoised dust solution is obtained.