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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.
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.
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