On the rotation and the figure of celestial bodies

Abstract

AbstractDetermining the form of celestial bodies one usually assumes according to celestial mechanics, that they rotate as solid bodies, i. e. with a constant angular velocity over the whole body. The equations of rotation of a free fluid subjected to the influence of the forces of its own gravitation and of pressure considered as a solid body are not the general solution of EULER's hydrodynamical equations for the case of rotation. As POINCARÉ and JEANS have shown there exists an infinite number of other solutions describing the movement of a rotating fluid with angular velocity depending on the distance from the axis of rotation. We examine in the present paper these more general solutions generalizing the well known POINCARÉ theorem of the maximum possible angular velocity. Within the framework of these generalizations we examine ROCHE's model which treats the whole mass of the fluid concentrated in the center of gravity. Some of our results are shown to be applicable for the determination of the form and the gravitational field of non‐uniformly rotating celestial bodies, for instance the Sun, Jupiter and Saturn.On the basis of LIAPOUNOFF's classical works (1903 and 1904) we prove the existence of unique solutions of the generalized problem of a non‐uniformly rotating fluid at small angular velocities. We arrive at a system of integrodifferential equations which generalize the CLAIRAUT‐LIAPOUNOFF equations for the case of a non‐uniformly rotating fluid. By means of these exact solutions we examine a model which enables us, assuming some restrictive conditions, to refute the objection put forward against KANT‐LAPLACE's theory concerning the distribution of moment of momentum in the Solar system.