The kelvin-poincare problem of stellar evolution

The evolution of a rotating liquid mass with an assigned angular momentum about a central axis has been sketched by Poincaré in his celebrated memoir on a rotating liquid mass ("Acta Math.,” 1885). The liquid continually contracts so as not to violate the principle of degradation of mechanical energy which we take for the criterion of the trend of evolution. For the sake of mathematical simplicity we make the assumption that the contraction permits the liquid to lie always homogeneous. Initially a sphere with a “small” angular velocity about a diameter, the liquid mass acquires greater and greater velocity as it contracts with constant momentum and passes through the stable Maclaurin and Jacobian forms. This is the order up to the critical Jacobian ellipsoid, if for conserved momentum the total mechanical energy becomes continually smaller so that the principle of degradation of energy is not contradicted. The next type in the succession, a pear-shaped figure, is unstable, and so it is presumed that a cataclysm involving tumultuous change sets in at this stage. In view of the slowness of the evolution, it is perhaps to be expected that the cataclysm would be smooth in its earlier stages. We examine the general features of these earlier stages and show that contraction subject to degradation of energy permit the Darwin-Poincaré pyriform solution, unlike that of Jeans. The former seems to have been found to be stable while the latter, the more developed pear, has been shown by Jeans to be unstable. Without discussing the cataclysm further, we pass next to a series of double-figure configurations (both components spherical), which is characterized by two parameter, λ the mass-ratio and r the separating distance. In the diagram of the course of evolution which summarizes the investigation all the configurations are brought in. If the cataclysm ends in such a smooth state of fission, we must have λ < 3·4 which is in keeping with the observation of double star. As regards the curlier stages, the Maclaurin spheroid shrinks in every direction, but not the Jacobian ellipsoid.