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Y. V. Barkin (Sternberg Astronomical Institute, Moscow; Alicante University, Spain)
The theory of the unperturbed rotational motion of the deformable celestial bodies is developed. This motion describes the rotation of an isolated celestial body weakly deformed by its own rotation. On the base of equations in Andoyer variables describing rotational motion of the celestial bodies with a changeable in the time tensor of inertia (Barkin, 1979, 1984) the problem is reduced to the classical Euler-Poinsot problem for a rigid body, but with another set of constant moments of inertia. The unperturbed motion describes Chandler’s pole motion and we have called it as the Chandler or Euler-Chandler motion (Barkin, 1992, 1998). The statement of the unperturbed theory is given in a exhaustive and detailed form. The solution of the Chandler problem (Andoyer’s variables, components of angular velocity w.r.t to the body and space reference systems, direction cosines and their different functions) is presented in elliptical and theta-functions, and in the form of Fourier series in the angle-action variables of unperturbed motion. The construction of Fourier series for the products and squares of the direction cosines of the body has a central role. On the base of these results the Fourier series of the second harmonic of the Earth-Moon force function has been obtained in angle-action variables. The coefficients of these series are expressed through the complete and incomplete elliptical integrals of the first, second and third kinds with modulus which is the function of the action variables and moments of inertia of the body. As an illustration we give an application of unperturbed theory to the study of the Earth’s rotation (the principal properties of the Earth’s rotation and perturbations).
The well known phenomenon of a distinction of Euler and Chandler periods is confirmed by unperturbed theory (433.2 and 304.4 days). A new phenomenon of a distinction of eccentricities of Euler and Chandler pole trajectories has been established (corresponding dynamical eccentricities are 0.0807 and 0.0958; corresponding geometrical eccentricities are 0.00328 and 0.00462). Also the new phenomenon of the non-uniformity of pole motion on the ellipse was described. The periods 437.1 and 433.1 days(of the Chandler motion) and periods 305.4 and 303.4 days(of the Euler motion) correspond to the extreme values of velocity of pole motion along the ellipse.
Theory of the perturbed rotational motion of the Earth is constructed on the basis of the special forms of equations of the rotation of the deformable body (in angle-action variables and their modifications for the Chandler-Euler problem). The analytical formulae for perturbations of the first and second orders in the Earth’s rotation have been obtained. Secular perturbations in the Earth’s rotation due to second harmonic of the force function have been studied (the determination of the constant of precession; constant additives to the angular velocities of the Chandler and axial motions of the Earth). All results of the paper are presented in analytical form and are applicable for studies of the unperturbed and perturbed rotational motions of other celestial bodies (Venus, asteroids, satellites etc.), for studies of the tidal evolution of the rotation of celestial bodies.
Barkin's work was accepted by grant SAB2000-0235 of Ministry of Education of Spain and RFBR grant 02-05-64176.
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Bulletin of the American Astronomical Society, 36 #2
© 2004. The American Astronomical Soceity.