Analysis of regular precession conditions for asymmetrical liquid-filled rigid bodies

Abstract

To describe the rotation of a “rigid mantle + liquid core” system, the Poincare–Hough–Zhukovsky equations are used. An analysis is made of the previously obtained (Ol’shanskii in Celest Mech Dyn Astron 131(12):Article number:57, 2019) conditions for regular precession of a system that does not have an axial symmetry. Upon receipt of the conditions, it is considered that the external torque can be neglected as, for example, for free-floating planetary bodies. In the case when the axis of proper rotation is one of the principal axes of inertia, the formulas for the rates of precession and proper rotation have been simplified. For a particular case, when the shape of the core differs little from spherical, it is shown that the precession and proper rotation rates differ from the rates of empty axisymmetric rigid mantle by values of the first order of smallness. The ratio of these rates differs from the ratio for the rigid mantle by the second-order value. For a system with the axis of proper rotation deviated from a principal axis of inertia, the expression of the principal moments of inertia of the mantle through the moments of inertia of the core and one arbitrary parameter is found. The formulas for finding the angular velocities and for determining the position of the axis of proper rotation relative to the mantle are written in simple parametric form. The possibility of regular precession with the axis of proper rotation, which does not coincide with any principal axes, is studied in the case when the shape of the core differs little from the spherical one. Examples of elongated non-axisymmetric systems that allow precession with the axis of proper rotation deviated from principal axis of inertia are given.