Gravitating configurations with magnetic field. II. Riemann's theorem and rigid-body rotation

Gravitating configurations that rotate with angular velocity rvec ω and have a magnetic field and internal motion with constant vorticity rvec ξ are investigated. It is shown that Riemann's theorem, according to which rvec ω and rvec ξ lie in one of the principal planes of the ellipsoid, remains valid. Rigidly rotating configurations are regarded as limiting cases of objects with internal motion. A theorem is proved which establishes that the vectors rvec ω and the electric current density rvec J are collinear. Conditions of equilibrium of ellipsoidal forms are obtained. It is established that there exists a limiting oblateness f = 1 - (a3/a1) approx-lt 0.4. The magnetic field and angular velocity can be uniquely expressed in terms of the ratios of the semiaxes of the ellipse; in the case a1 ≠ a2 ≠ a3 the magnetic field does not have axial symmetry. It is shown that a toroidal field does not admit the existence of triaxial ellipsoids. The magnetic moments produced by the surface current are found