  We have recently proved the impossibility of imposing the condition of local
charge neutrality in a self-gravitating system of degenerate neutrons, protons
and electrons in $\beta$-equilibrium. The coupled system of the general
relativistic Thomas-Fermi equations and the Einstein-Maxwell equations have
been shown to supersede the traditional Tolman-Oppenheimer-Volkoff equations.
Here we present the Newtonian limit of the new equilibrium equations. We also
extend the treatment to the case of finite temperatures and finally we give the
explicit demonstration of the constancy of the Klein potentials in the case of
finite temperatures generalizing the condition of constancy of the general
relativistic Fermi energies in the case of zero temperatures.
