Abstract
The work is devoted to the study of the evolution of the rotational motion of a planet in the central Newtonian field of forces. The planet is modeled by a body consisting of a solid core and a viscoelastic shell rigidly attached to it. A limited formulation of the problem is considered, when the center of mass of the planet moves along a given Keplerian elliptical orbit. The equations of motion are derived in the form of a system of Routh equations using the canonical Andoyer variables, which are “action-angle” variables in the unperturbed problem and have the form of integro-differential equations with partial derivatives. The technique developed by V.G. Vilke is used for mechanical systems with an infinite number of degrees of freedom. A system of ordinary differential equations is obtained by the method of separation of motions. The system describes the rotational motion of the planet taking into account the perturbations caused by elasticity and dissipation. An evolutionary system of equations for the “action” variables and slow angular variables is obtained by the averaging method. A phase portrait is constructed that describes the mutual change in the modulus of the angular momentum vector G of the rotational motion and the cosine of the angle between this vector and the normal to the orbital plane of the planet’s center of mass. A stationary solution of the evolutionary system of equations is found, which is asymptotically stable. It is shown that in stationary motion, the angular momentum vector G is orthogonal to the orbital plane, and the limiting value of the modulus of this vector depends on the eccentricity of the elliptical orbit. The constructed mathematical model can be used to study the tidal evolution of the rotational motion of planets and satellites. The results obtained in this work are consistent with the results of previous studies in this area.
Highlights
Financial disclosure: The authors have no a financial or property interest in any material or method mentioned
The work is devoted to the study of the evolution of the rotational motion of a planet in the central Newtonian field of forces
The equations of motion are derived in the form of a system of Routh equations using the canonical Andoyer variables, which are “action-angle” variables in the unperturbed problem and have the form of integro-differential equations with partial derivatives
Summary
Что жесткость деформируемой оболочки планеты велика, т.е. Мал безразмерный параметр ε = ω2 (0)ρ1r12E−1 (ω(0) – модуль начальной угловой скорости планеты). В качестве невозмущенной задачи рассмотрим задачу о движении абсолютно твердой сферически симметричной планеты на эллиптической орбите. Уравнения (2.1) описывают равномерное вращение планеты вокруг одного из диаметров с угловой скоростью φ 2 = I2 A. При ε ≠ 0 согласно методу разделения движений [4], после затухания собственных колебаний вязкоупругого шара решение u(r,t) ищется в виде ряда по степеням малого параметра ε : u(r,t) = ε u1(r,t) + ε2u2 (r,t) + ⋅⋅⋅ (2.2). С учетом (1.20), (1.9) уравнение (1.22) для функции u1(r,t) первого приближения примет вид:. Краевая задача для нахождения функции u1(r,t) первого приближения примет вид: ε∇E[u1 + χu 1] =. В равенстве (2.9) дифференцирование по времени производится в силу невозмущенной системы уравнений движения (2.1), а величины G, ξ определяются формулами (1.15), (1.16)
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