General Theory of Spherical Self=Gravitating Star Systems in a Steady State

Abstract

AbstractUsually the mechanics of star systems is investigated by constructing models under some special assumptions. This method, however, is not satisfying, because there are only few observational data on the real star systems available, so that those assumptions are often quite arbitrary. Therfore here the opposite method is applied. The problem is: What are the features of a star system, the assumptions being as general as possible, so that they can be expected to be satisfied by the real star systems? The physical assumptions in this investigation are: The system is in a steady state, possesses spherical symmetry and has a finite total mass (§ 1). An assumption of more mathematical character is that its frequency function depends only on the integrals of energy and angular momentum, no further uniform integral being available (and probably not even existing) (§ 2). It follows that in such a star system the density law of the envelope is nearly that of a polytropic gas sphere. The reasons for this, however, are of purely dynamic nature and have by no means any connection with gas theory (§ 3). Now the problem is reversed: Is the necessary condition for a finite total mass also a sufficient one? Apparently, it is not so. But in every case by a slight alteration of the given frequency function, i. e. by an alteration which is as small as we please, a new frequency function can be constructed which yields a model with a finite total mass (§ 5). In a certain sense, these models may be assumed to be stable (§ 6). In § 4, it is shown that the models satisfying the necessary condition for a finite total mass are uniquely determined by their frequency functions. Methods for computing the models are given, too.