Classifying self-gravitating radiations

We study a static system of self-gravitating radiations confined in a sphere by using numerical and analytical calculations. Because of the scaling symmetry of radiations, most of the main properties of a solution can be represented as a segment of a solution curve on a plane of two-dimensional scale invariant variables. We define an "approximate horizon" (AH) from the analogy with an apparent horizon. Any solution curve contains a unique point that corresponds to the AH. A given solution is uniquely labeled by three parameters representing the solution curve, the size of the AH, and the sphere size, which are an alternative to the data at the outer boundary. Various geometrical properties including the existence of an AH and the behaviors around the center can be identified from the parameters. We additionally present an analytic solution of the radiations on the verge of forming a black hole. Analytic formulas for the central mass of the naked singularity are given.