Normal Modes in Rotation of the Earth-like Planets

Introduction: In many theoretical investigations [e.g. 1–8] the normal modes of the linearized equations of rotation are computed, yielding both the periods and the eigenspaces of three librations. Libration in longitude decouples from the other two, vertical modes. There is a fast vertical mode (polar motion) with a period very close to the average rotational period. It corresponds to tilting the body around a horizontal axis while retaining nearly principal-axis rotation. In the inertial frame, this mode appears as nutation and free precession. The other vertical mode, a slow one, is the free wobble. The modern view of internal structure of the planet takes into account a complex twoor three-layer model. For a planet with a solid inner core and a liquid outer core, there are four rotational normal modes. This numbers is reduced to two for a planet without inner core, and to one for a planet without liquid core [4, 5, 6]. All types of modes are result of non-coincidence of rotation axes and of the main inertia moments of mantle, outer and inner core [2, 9, 10]. It was theoretically shown, that the periods of the rotational modes are depend on the characteristics of planet’s interior [11]. Because of this, the searching for the fine effects in polar rotation of the celestial bodies is of great importance. Obtained parameters from the observation might be used as a key to the internal nature of the planet. Analytical description of the free rotation: Let the Ω is a mean angular velocity of the planet, the A, B, C and those with indexes “m” , “c” are the main inertia moments of the whole planet, mantle and core, the α and β are dynamical flattening of the mantle. For the case of a two-layer planet the periods of the polar modes – Chandler Wobble and Free Core Nutation – may be written in the form: