The parameter planes of the spherically symmetric and static relativistic solutions for polytropes

We explore the parameter space of the family of static and spherically symmetric solutions of the Einstein field equations for polytropes, that were presented in a previous paper. This is a four-parameter family of exact solutions, of which one parameter can be factored out, so that there are only three essential free parameters. The solutions are exact in the sense that no approximations are involved, other than those implied by the numerical precision limitations. The primary objectives of this exploration are to establish directly the existence of large collections of specific solutions, and to determine some of their most important properties. For each value of one of the three essential free parameters of the family of solutions, the polytropic index n, taken here, for the sake of simplicity, to be either an integer or a half-integer, we define and explore the parameter planes spanned by the other two essential free parameters. In this way, besides establishing their existence, we are also able to classify the solutions according to their overall matter energy density, as well as in terms of their proximity to solutions that display an event horizon. For four values of n we successfully establish the allowed regions of the parameter planes, where the solutions not only exist but also correspond to physically acceptable matter. We find that there are solutions within these regions with overall matter energy densities varying all the way from very low to very high, including some that are as close as one may wish to solutions that display an event horizon, and that therefore represent black holes, or extremely dense objects very similar to them.