  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 gets 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.

Solar and Stellar Astrophysics;General Relativity and Quantum Cosmology;High Energy Physics - Theory