Exact solutions for charged spheres and their stability. I. Perfect Fluids

We study exact solutions of the Einstein-Maxwell equations for the interior gravitational field of static spherically symmetric charged compact spheres. The spheres are composed of a perfect fluid with a charge distribution that creates a static radial electric field. The inertial mass density of the fluid has the form $\rho (r) = \rho_o + \alpha r^2$ ($\rho_o$ and $\alpha$ are constants) and the total charge $q(r)$ within a sphere of radius $r$ has the form $q = \beta r^3$ ($\beta$ is a constant). We evaluate the critical values of $M/R$ for these spheres as a function of $Q/R$ and compare these values with those given by the Andr\'{e}asson formula.