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The question of how much surplus of electric charge (“surcharge”) fits on an object is generally very difficult to answer. Here, it is shown that it is easy to answer when the object is a failed white dwarf star (a brown dwarf in its ground state) made of protons and electrons: Given the number of protons, how many electrons can there be? Surprisingly, the answer (in the form: as few as A and as many as B) is independent of the speed of light c and the Planck quantum h, even when the star is stabilized against collapse by relativistic quantum mechanics.

We present an energy functional for a Thomas-Fermi type two-fluid model of a self-gravitating nonrotating charged body, with a non-relativistic kinetic energy. We prove that, under certain conditions on the total number of positively charged and negatively charged particles, a minimizer exists and both fluids have compact support. We prove the same result for special relativistic kinetic energy, assuming further conditions on the total number of particles. In the non-relativistic kinetic energy case, we further prove the uniqueness of the minimizer, as well as present results relating the general shape of the minimizer to the total number of particles.

We generalize the recent Newtonian two-component charged fluid models for white dwarf stars of Krivoruchenko, Nadyozhin and Yudin and of Hund and Kiessling to the context of general relativity. We compare the equations and numerical solutions of these models. We extend to the general relativistic setting the non-neutrality results and bounds on the stellar charge obtained by Hund and Kiessling.