Compact stars with a small electric charge: the limiting radius to mass relation and the maximum mass for incompressible matter

One of the stiffest equations of state for matter in a compact star is constant energy density and this generates the interior Schwarzschild radius to mass relation and the Misner maximum mass for relativistic compact stars. If dark matter populates the interior of stars, and this matter is supersymmetric or of some other type, some of it possessing a tiny electric charge, there is the possibility that highly compact stars can trap a small but non-negligible electric charge. In this case the radius to mass relation for such compact stars should get modifications. We use an analytical scheme to investigate the limiting radius to mass relation and the maximum mass of relativistic stars made of an incompressible fluid with a small electric charge. The investigation is carried out by using the hydrostatic equilibrium equation, i.e., the Tolman-Oppenheimer-Volkoff (TOV) equation, together with the other equations of structure, with the further hypothesis that the charge distribution is proportional to the energy density. The approach relies on Volkoff and Misner's method to solve the TOV equation. For zero charge one gets the interior Schwarzschild limit, and supposing incompressible boson or fermion matter with constituents with masses of the order of the neutron mass one finds that the maximum mass is the Misner mass. For a small electric charge, our analytical approximating scheme, valid in first order in the star's electric charge, shows that the maximum mass increases relatively to the uncharged case, whereas the minimum possible radius decreases, an expected effect since the new field is repulsive, aiding the pressure to sustain the star against gravitational collapse.