Equations of motion of elliptically-restricted problem of a body with variable mass and two tri-axial bodies

The differential equations of motion of the elliptic restricted problem of three bodies, an infinitesimal spherical body with decreasing mass and two tri-axial bodies are derived. We have applied Jeans's law and the space-time of Meshcherskii in the special case whenn=1,k=0,q=1/2. Also Nechvíle's transformation for the elliptic problem be applied for this case.     

Rigid body dynamics in the restricted ring problem of N + 1 bodies

The dynamic behavior of a small tri-axial body acted upon by the Newtonian forces of N major bodies of spherical symmetry which forma planar ring configuration is studied in this paper. The equations of the translational-rotational motion of the minor body are derived and its equilibrium states as well as their stability are investigated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Existence of particular solutions in the generalised restricted problem of three bodies with variable masses

The restricted problem of three bodies with variable masses is considered. It is assumed that the infinitesimal body is axisymmetric with constant mass and the finite bodies are spherical with variable masses such that the ratio of their masses remains constant. The motion of the finite bodies are determined by the Gyldén-Meshcherskii problem. It is seen that the collinear, triangular, and coplanar solutions not exist, but these solutions exist when the infinitesimal body be a spherical.     

On the attitude dynamics of perturbed triaxial rigid bodies

Attitude dynamics of perturbed triaxial rigid bodies is a rather involved problem, due to the presence of elliptic functions even in the Euler equations for the free rotation of a triaxial rigid body. With the solution of the Euler–Poinsot problem, that will be taken as the unperturbed part, we expand the perturbation in Fourier series, which coefficients are rational functions of the Jacobian nome. These series converge very fast, and thus, with only few terms a good approximation is obtained. Once the expansion is performed, it is possible to apply to it a Lie-transformation. An application to a tri-axial rigid body moving in a Keplerian orbit is made.     

Hamiltonian Dynamics of a Gyrostat in the N-Body Problem: Relative Equilibria

We consider the non-canonical Hamiltonian dynamics of a gyrostat in Newtonian interaction with n spherical rigid bodies. Using the symmetries of the system we carry out two reductions. Then, working in the reduced problem, we obtain the equations of motion, a Casimir function of the system and the equations that determine the relative equilibria. Global conditions for existence of relative equilibria are given. Besides, we give the variational characterization of these equilibria and three invariant manifolds of the problem; being calculated the equations of motion in these manifolds, which are described by means of a canonical Hamiltonian system. We give some Eulerian and Lagrangian equilibria for the four body problem with a gyrostat. Finally, certain classical problems of Celestial Mechanics are generalized.     

The general motion of three mutually-interacting,axisymmetric spherical rigid bodies

In this paper, the translational-rotational motions of an axisymmetric rigid body and two spherical rigid bodies under the influence of their mutual gravitational attraction are considered. The equations of motion in the canonical elements of Delaunay-Andoyer are obtained. The elements of motion in the zero and first approximations can be determined.     

The libration points of a spherical satellite in the photogravitational restricted problem of three bodies

In this paper the photogravitational circular restricted problem of three bodies is considered. We have assumed that one of the finite bodies be a spherical luminous and the other be a triaxial nonluminous body. The possibility of existence of the libration points be studied.     

Resonant and non-resonant gravity-gradient perturbations of a tumbling tri-axial satellite

Gravity-gradient perturbations of the attitude motion of a tumbling tri-axial satellite are investigated. The satellite center of mass is considered to be in an elliptical orbit about a spherical planet and to be tumbling at a frequency much greater than orbital rate. In determining the unperturbed (free) motion of the satellite, a canonical form for the solution of the torque-free motion of a rigid body is obtained. By casting the gravity-gradient perturbing torque in terms of a perturbing Hamiltonian, the long-term changes in the rotational motion are derived. In particular, far from resonance, there are no long-period changes in the magnitude of the rotational angular momentum and rotational energy, and the rotational angular momentum vector precesses abound the orbital angular momentum vector.At resonance, a low-order commensurability exists between the polhode frequency and tumbling frequency. Near resonance, there may be small long-period fluctuations in the rotational energy and angular momentum magnitude. Moreover, the precession of the rotational angular momentum vector about the orbital angular momentum vector now contains substantial long-period contributions superimposed on the non-resonant precession rate. By averaging certain long-period elliptic functions, the mean value near resonance for the precession of the rotational angular momentum vector is obtained in terms of initial conditions.     

The equilibrium configurations of the restricted problem of 2+2 triaxial rigid bodies

The restricted 2+2 body problem is considered. The infinitesimal masses are replaced by triaxial rigid bodies and the equations of motion are derived in Lagrange form. Subsequently, the equilibrium solutions for the rotational and translational motion of the bodies are detected. These solutions are conveniently classified in groups according to the several combinations which are possible between the translational equilibria and the constant orientations of the bodies.     

Application of a Fictitious Attracting Center to the Study of the Motion of Minor Bodies Approaching Major Planets

The motion of minor Solar System bodies having close encounters with major planets is described using the model of motion within the framework of the perturbed restricted three-body problem. The actual motion of a minor body is represented as a combination of two motions, namely, the motion of a fictitious attracting center with a variable mass and the motion with respect to the fictitious center. The position and mass of the fictitious center are chosen so that, when the minor body collides with any of the primaries, the fictitious center carries into the center of inertia of the colliding body and the mass of the fictitious center becomes identical to the mass of this body. The regularizing KS-transformation and Sundman’s time transformation were applied to coordinates and velocities. As a result, a system of differential equations of motion that are quasilinear within the nearest vicinity of each of the primary attracting bodies was obtained. These equations are characterized by a numerical behavior during the encounters of the minor body with the primaries that is essentially better than that of the initial equations of motion. The motion of comets Brooks 2 and Gehrels 3, which have fairly close encounters with Jupiter, is simulated.__________Translated from Astronomicheskii Vestnik, Vol. 39, No. 3, 2005, pp. 272–280.Original Russian Text Copyright © 2005 by Shefer.     

On lagrange solutions in the problem of three rigid bodies

The present paper is a direct continuation of the paper (Duboshin, 1973) in which was proved the existence of one kind of Lagrange (triangle) and Euler (rectilinear) solutions of the general problem of the motion of three finite rigid bodies assuming different laws of interaction between the elementary particles of the rigid bodies. In particular, Duboshin found that the general problem of three rigid bodies permits such solutions in which the centres of mass of the bodies always form an equilateral triangle or always remain on one straight line, and each body possesses an axial symmetry and a symmetry with respect to the plane of the centres of mass and rotates uniformly around its axis orthogonal to this plane. The conditions for the existence of such solutions have also been found. The results in Duboshin's paper have greatly interested the author of the present paper. In another paper (Kondurar and Shinkarik, 1972) considering a more special problem, when two of the three bodies are spheres, either homogeneous or possessing a spherically symmetric distribution of the densities or of the material points, and the third is an axially symmetrical body possessing equatorial symmetry, the present author obtained analogous solutions of the ‘float’ type describing the motion of the indicated dynamico-symmetrical body in assuming its passive gravitation. In the present paper new Lagrange solutions of the considered general problems of three rigid bodies of ‘level’ type are found when the axes of geometrical and mechanical symmetry of all three bodies always lie in the triangle plane, and the bodies themselves rotate inertially around the symmetry axis, independently of the parameters of the orbital motion of the centres of mass as in the ‘float’ case. The study of particular solutions of the general problem of the translatory-rotary motion of three rigid bodies, which are a generalization of Lagrange solutions, is in the author's opinion, a novelty of some interest for both theoretical and practical divisions of celestial mechanics. For example, in recent times the problem of the libration points of the Earth-Moon system has acquired new interest and value. A possible application which should be mentioned is that to the orbits of artificial satellites near the triangular libration points to serve as observation stations with the aim of specifying the physical parameters in the Earth-Moon system (e.g., the relation of the Earth's mass to the Moon's mass for investigating the orientation of the satellite, solar radiation, etc.).     

Equations of motion of the restricted problem of three bodies with variable mass

In this paper we have deduced the differential equations of motion of the restricted problem of three bodies with decreasing mass, under the assumption that the mass of the satellite varies with respect to time. We have applied Jeans law and the space time transformation contrast to the transformation of Meshcherskii. The space time transformation is applicable only in the special casen=1,k=0,q=1/2. The equations of motion of our problem differ from the equations of motion of the restricted three body problem with constant mass only by small perturbing forces.     

The eccentric frame decomposition of central force fields

The rosette-shaped motion of a particle in a central force field is known to be classically solvable by quadratures. We present a new approach of describing and characterizing such motion based on the eccentricity vector of the two body problem. In general, this vector is not an integral of motion. However, the orbital motion, when viewed from the nonuniformly rotating frame defined by the orientation of the eccentricity vector, can be solved analytically and will either be a closed periodic circulation or libration. The motion with respect to inertial space is then given by integrating the argument of periapsis with respect to time. Finally we will apply the decomposition to a modern central potential, the spherical Hernquist–Newton potential, which models dark matter halos of galaxies with central black holes.     

The celestial bodies motion theories: on the required accuracy of the motion theories of the perturbing bodies

The equation for calculation of the required accuracy of the perturbing bodies motion theories is obtained. The equation relates the accuracy required to take into account perturbing acceleration, acting on the perturbed body, with the accuracy of the motion theory of the perturbing body. The solutions for estimation of the required accuracy both for the inner and the external cases in the spherical coordinates are coincided. The solution for the calculation of the required accuracy for the general case (combining the inner and the external cases) in Cartesian coordinates is obtained. The special cases for the solution in Cartesian coordinates are studied. As an example, the estimations of the required accuracy of the motion theories of the solar system planets for some perturbed bodies (the near-Earth asteroid 4179 Toutatis, the main belt asteroid 208 Larcimosa, the trojan asteroid 588 Achilles, the centaur asteroid 5145 Pholus, the Kuiper belt asteroid 1995 QZ9, the comet Halley) are obtained. The conditions of the use of the obtained results are discussed.     

Restricted four-body problem. The case of a central configuration: Libration points and their stability

We consider the problem of the motion of a zero-mass body in the vicinity of a system of three gravitating bodies forming a central configuration.We study the case where two gravitating bodies of equal mass lie on the same straight line and rotate around the central body with the same angular velocity. Equations for calculating the equilibrium positions in this system have been derived. The stability of the equilibrium points for a system of three gravitating bodies is investigated. We show that, as in the case of libration points for two bodies, the collinear points are unstable; for the triangular points, there exists a ratio of the mass of the central body to the masses of the extreme bodies, 11.720349, at which stability is observed.     

Hill stability in the many-body problem

We established a criterion for the Hill stability of motions in the problem of many spherical bodies with a spherical density distribution. The region of Hill stability was determined. The sizes of this region are comparable to the total volume of all of the bodies in the system, which sharply increases the probability of mutual collisions. This result may be considered as a confirmation that a supermassive core can be formed at the center of a globular star cluster. The motions in the n-body problem are shown to be unstable according to Hill.     

Sur les mouvements réguliers dans le probléme des deux corps solides

In the present paper the problem of translatory-rotatory motion of two rigid bodies is discussed. Author has shown that this problem admits particular solutions, when each body possesses axial symmetry. In these solutions the centre of mass of one body described the circular orbit around the other body and each body keeps the invariable orientation about this orbit.     

Equations of motion of elliptic restricted problem of three bodies with variable mass

The differential equations of motion of the elliptic restricted problem of three bodies with decreasing mass are derived. The mass of the infinitesimal body varies with time. We have applied Jeans' law and the space-time transformation of Meshcherskii. In this problem the space-time transformation is applicable only in the special case whenn=1,k=0,q=1/2. We have applied Nechvile's transformation for the elliptic problem. We find that the equations of motion of our problem differ from that of constant mass only by a small perturbing force.     

Collisional Spin Up of Nonspherical Asteroids

We investigate the spin up of nonspherical asteroids by successive random collisions with other smaller asteroids. The spin-up rates are calculated for ellipsoidal and spherical bodies of the same mass. Then, we derive the relative spin-up efficiency, that is, the ratio of the spin-up rate of the ellipsoidal body to that of the spherical body. We suppose a collisional process in our model different from that adopted in the previous works. The results show that ellipsoidal bodies can spin up more rapidly than spherical bodies.     

The close triple approach

A new method is described for representing the motion in the planar problem of three bodies when all three point masses simultaneously come close to each other. The main results are (1) that the motion during the critical phase of closest approach is intimately connected with triple parabolic escape and (2) that a sufficiently close triple approach generally leads to the escape of one body witharbitrarily high asymptotic velocity.