New Formula for the Angular Velocity of Rotation of Liquid Equilibrium Figures

The aim of the work is to derive a new dynamic formula for the angular velocity of rotation of equilibrium figures of a gravitating fluid with a polytropic equation of state. In this formula, the angular velocity of rotation depends not only on the polytropic index $$0 \leqslant n \leqslant 5,$$ but, most importantly, on the components of the internal and external gravitational energy of the figure. When solving the problem, the integration constant in the full potential was expressed through three global characteristics: mass, full gravitational energy and rotation energy of the equilibrium figure. The validity of the new formula was confirmed by the limiting transition at $$n = 0$$ to classical homogeneous Maclaurin spheroids and Jacobi ellipsoids. The results of the work expand the scope of application of the theory of equilibrium figures.