Polytropes

While it seems possible to present a fairly complete uni. ed theory of undistorted polytropes, as attempted in the previous chapter, the theory of distorted polytropes is much more extended and - phisticated, so that I present merely a brief overview of the theories that seem to me most interesting and important. Basically, the methods proposed to study the hydrostatic equilibrium of a distorted self-gravitating mass can be divided into two major groups (Blinnikov 1975): (i) Analytic or semia- lytic methods using a small parameter connected with the distortion of the polytrope. (ii) More or less accurate numerical methods. Lyapunov and later Carleman (see Jardetzky 1958, p. 13) have demonstrated that a sphere is a unique solution to the problem of hydrostatic equilibrium for a. uid mass at rest in tridimensional space. The problem complicates enormously if the sphere is rotating rigidly or di. erentially in space round an axis, and/or if it is distorted magnetically or tidally. Even for the simplest case of a uniformly rotating. uid body with constant density not all possible solutions have been found (Zharkov and Trubitsyn 1978, p. 222). The sphere becomes an oblate. gure, and we have no a priori knowledge of its strati. cation, boundary shape, planes of symmetry, transfer of angular momentum in di. erentially rotating bodies, etc.