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Shatina A. V., Djioeva M. I., Osipova L. S.
Mathematical Model of Satellite Rotation near Spin-Orbit Resonance 3:2
2022, Vol. 18, no. 4, pp. 651-660
This paper considers the rotational motion of a satellite equipped with flexible viscoelastic rods in an elliptic orbit. The satellite is modeled as a symmetric rigid body with a pair of flexible viscoelastic rods rigidly attached to it along the axis of symmetry. A planar case is studied, i. e., it is assumed that the satellite’s center of mass moves in a Keplerian elliptic orbit lying in a stationary plane and the satellite’s axis of rotation is orthogonal to this plane. When the rods are not deformed, the satellite’s principal central moments of inertia are equal to each other. The linear bending theory for thin inextensible rods is used to describe the deformations. The functionals of elastic and dissipative forces are introduced according to this model. The asymptotic method of motions separation is used to derive the equations of rotational motion reflecting the influence of the fluctuations, caused by the deformations of the rods. The method of motion separation is based on the assumption that the period of the autonomous oscillations of a point belonging to the rod is much smaller than the characteristic time of these oscillations’ decay, which, in its turn, is much smaller than the characteristic time of the system’s motion as a whole. That is why only the oscillations induced by the external and inertial forces are taken into account when deriving the equations of the rotational motion. The perturbed equations are described by a third-order system of ordinary differential equations in the dimensionless variable equal to the ratio of the satellite’s absolute value of angular velocity to the mean motion of the satellite’s center of mass, the angle between the satellite’s axis of symmetry and a fixed axis and the mean anomaly. The right-hand sides of the equation depend on the mean anomaly implicitly through the true anomaly. A new slow angular variable is introduced in order to perform the averaging for the perturbed system near the 3:2 resonance, and the averaging is performed over the mean anomaly of the satellite’s center of mass orbit. In doing so the wellknown expansions of the true anomaly and its sine and cosine in powers of the mean anomaly are used. The steady-state solutions of the resulting system of equations are found and their stability is studied. It is shown that, if certain conditions are fulfilled, then asymptotically stable solutions exist. Therefore, the 3:2 spin-orbital resonance capture is explained.
Vilke V. G., Shatina A. V., Osipova L. S.
The Effect of the Mutual Gravitational Interactions on the Perihelia Displacement of the Orbits of the Solar System’s Planets
2018, Vol. 14, no. 3, pp. 291-300
The classical $N$-body problem in the case when one of the bodies (the Sun) has a much larger mass than the rest of the mutually gravitating bodies is considered. The system of equations in canonical Delaunay variables describing the motion of the system relative to the barycentric coordinate system is derived via the methods of analitical dynamics. The procedure of averaging over the fast angular variables (mean anomalies) leads to the equation describing the evolution of a single Solar system planet’s perihelion as the sum of two terms. The first term corresponds to the gravitational disturbances caused by the rest of the planets, as in the case of a motionless Sun. The second appears because the problem is considered in the barycentric coordinate system and the orbits’ inclinations are taken into account. This term vanishes if all planets are assumed to be moving in one static plane. This term contributes substantially to the Mercury’s and Venus’s perihelion evolutions. For the rest of the planet this term is small compared to the first one. For example, for Mercury the values of the two terms in question were calculated to be 528.67 and 39.64 angular seconds per century, respectively.