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.

我们探索了多变体爱因斯坦场方程组静态和球对称解的参数空间，这些解在之前的论文中已经介绍过。这是一个四参数族的精确解，其中一个参数可以被分解出来，因此只有三个必要的自由参数。这些解是精确的，因为除了数值精度限制所暗示的之外，不涉及任何近似值。这种探索的主要目标是直接确定大量特定解决方案的存在，并确定它们的一些最重要的属性。对于解族的三个基本自由参数之一的每个值，为简单起见，此处采用的多变指数 n无论是整数还是半整数，我们定义并探索由其他两个基本自由参数跨越的参数平面。通过这种方式，除了确定它们的存在之外，我们还可以根据它们的整体物质能量密度以及它们与显示事件视界的解的接近程度对解进行分类。对于 n 的四个值，我们成功建立了参数平面的允许区域，其中解不仅存在，而且对应于物理上可接受的物质。我们发现这些区域内存在总物质能量密度从非常低到非常高变化的解，其中一些解非常接近显示事件视界的解，因此代表黑洞，或与它们非常相似的极其致密的物体。