Coupled motion of rigid bodies about their center of mass

A study is made of the motion of a system consisting of two rigid bodies coupled by a massless rigid boom. Relative translational and rotational motions are examined with the assumption that no external forces are acting on the system. For specific sets of initial conditions and assumptions on the symmetries of the two bodies, nontrivial analytic solutions are observed. The stability and the internal torques are also examined for a few selected cases.This research was conducted while the author was a senior research associate of the National Research Council at the National Aeronautics and Space Administration (NASA) Lyndon B. Johnson Space Center.     

The relative motion of two spheroidal rigid bodies

We consider two spheroidal rigid bodies of comparable size constituting the components of an isolated binary system. We assume that (1) the bodies are homogeneous oblate ellipsoids of revolution, and (2) the meridional eccentricities of both components are small parameters.We obtain seven nonlinear differential equations governing simultaneously the relative motion of the two centroids and the rotational motion of each set of body axes. We seek solutions to these equations in the form of infinite series in the two meridional eccentricities.In the zero-order approximation (i. e., when the meridional eccentricities are neglected), the equations of motion separate into two independent subsystems. In this instance, the relative motion of the centroids is taken as a Kepler elliptic orbit of small eccentricity, whereas for each set of body axes we choose a composite motion consisting of a regular precession about an inertial axis and a uniform rotation about a body axis.The first part of the paper deals with the representation of the total potential energy of the binary system as an infinite series of the meridional eccentricities. For this purpose, we had to (1) derive a formula for representing the directional derivative of a solid harmonic as a combination of lower order harmonics, and (2) obtain the general term of a biaxial harmonic as a polynomial in the angular variables.In the second part, we expound a recurrent procedure whereby the approximations of various orders can be determined in terms of lower-order approximations. The rotational motion gives rise to linear differential equations with constant coefficients. In dealing with the translational motion, differential equations of the Hill type are encountered and are solved by means of power series in the orbital eccentricity. We give explicit solutions for the first-order approximation alone and identify critical values of the parameters which cause the motion to become unstable.The generality of the approach is tantamount to studying the evolution and asymptotic stability of the motion.Research performed under NASA Contract NAS5-11123.     

Equilibrium solutions of the restricted problem of 2+2 bodies

Fourteen equilibrium solutions of the restricted problem of 2+2 bodies are shown to exist. Six of these solutions are located about the collinear Lagrangian points of the classical restricted problem of three bodies. Eight solutions are found in the neighborhood of the triangular Lagrangian points. Linear stability analysis reveals that all of the equilibrium solutions are unstable with the exception of four solutions; two in the vicinity of each of the triangular Lagrangian points. These four solutions are found to be stable provided the mass parameter of the primary masses is less than a critical value which depends also on the mass of the minor bodies.     

Stability of binary asteroids

The restricted problem of 2+2 bodies is applied to the study of the stability and dynamics of binary asteroids in the solar system. Numerical investigation of the behavior of the orbital elements and the maximal Lyapunov characteristic number of binary asteroids reveal extensive regions where bounded quasiperiodic motion is possible. These regions are compared to the bounded regions which are predicted by the classical restricted problem of three bodies. Regions of bounded chaotic solutions are also found.     

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.).     

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.     

The formal integrals of the restricted planar problem of the three bodies

The first integrals of motion of the restricted planar circular problem of three bodies are constructed as the formal power series in r^1/2, r being the distance of a moving particle from the primary. It is shown that the coefficients of these series are trigonometric polynomials of an angular variable. Some particular solutions have been found in a closed form. The proposed method for constructing the formal integrals can be generalized to a spatial problem of three bodies.     

Analytical solutions for J 2-perturbed unbounded equatorial orbits

While solutions for bounded orbits about oblate spheroidal planets have been presented before, similar solutions for unbounded motion are scarce. This paper develops solutions for unbounded motion in the equatorial plane of an oblate spheroidal planet, while taking into account only the J ₂ harmonic in the gravitational potential. Two cases are distinguished: A pseudo-parabolic motion, obtained for zero total specific energy, and a pseudo-hyperbolic motion, characterized by positive total specific energy. The solutions to the equations of motion are expressed using elliptic integrals. The pseudo-parabolic motion unveils a new orbit, termed herein the fish orbit, which has not been observed thus far in the perturbed two-body problem. The pseudo-hyperbolic solutions show that significant differences exist between the Keplerian flyby and the flyby performed under the the J ₂ zonal harmonic. Numerical simulations are used to quantify these differences.     

Equilibria and stability of the restricted photogravitational problem of 2 + 2 bodies

The restricted problem of 2 + 2 bodies when one of the infinitesimal masses is further acted upon by the light pressure of the two primaries, is considered. The stationary solutions of this problem are found out. A short discussion is devoted to the stability of these solutions.     

The oblate spheroids version of the restricted photogravitational 2+2 body problem

The paper deals with the restricted photogravitational 2+2 body problem when the primaries are oblate spheroids. A study of the effect of the oblateness on the equilibrium positions and on the areas of the permissible motion of the minor bodies, is also made.     

Gravitational attraction until relativistic equipartition of internal and translational kinetic energies

Translational ordering of the internal kinematic chaos provides the Special Relativity referents for the geodesic motion of warm thermodynamical bodies. Taking identical mathematics, relativistic physics of the low speed transport of time-varying heat-energies differs from Newton’s physics of steady masses without internal degrees of freedom. General Relativity predicts geodesic changes of the internal heat-energy variable under the free gravitational fall and the geodesic turn in the radial field center. Internal heat variations enable cyclic dynamics of decelerated falls and accelerated takeoffs of inertial matter and its structural self-organization. The coordinate speed of the ordered spatial motion takes maximum under the equipartition of relativistic internal and translational kinetic energies. Observable predictions are discussed for verification/falsification of the principle of equipartition as a new basic for the ordered motion and self-organization in external fields, including gravitational, electromagnetic, and thermal ones.     

Hip-hop solutions of the 2N-body problem

Hip-hop solutions of the 2N-body problem with equal masses are shown to exist using an analytic continuation argument. These solutions are close to planar regular 2N-gon relative equilibria with small vertical oscillations. For fixed N, an infinity of these solutions are three-dimensional choreographies, with all the bodies moving along the same closed curve in the inertial frame.     

The motion of rotating spheroidal bodies in general relativity

The motion of two rotating spheroidal bodies, constituting the components of a binary system in a weak gravitational field, has been considered up to terms of the second order in the small parameterV/c, whereV denotes the velocity of the bodies andc is the velocity of light.The following simplifying assumptions, consistent with a problem of astronomical interest, have been made: (1) the dimensions of the bodies are small compared with their mutual distance; (2) the bodies consist of matter in the fluid state with internal hydrostatic pressure and their oblateness is due to their own rotation; (3) there exist axial symmetry about the axis of rotation and symmetry with respect to the equatorial plane, the same symmetry properties apply to mass densities and stress tensors.The Fock-Papapetrou method was used to ascertain those terms in the equations of motion which are due to the rotation and to the oblateness of each component. Approximate solutions to the Poisson and wave equations were obtained to express the potential and retarded potential at large distances from the bodies generating them. The explicit evaluation of certain integrals has necessitated the use of the Laplace-Clairaut theory for the equibrium configuration of rotating bodies. The final expressions require the knowledge of the mass density as a function of the mean radius of the equipotential surfaces.As an interpretation of the results, the Lagrangian perturbation equations were employed to evaluate the secular motion of the nodal line for the relative orbit of the two components. The results constitute a generalization of Fock's work and furnish the contribution of the mass distribution to the rotation effect of general relativity.     

Instability of the 2+2 body problem

The equations of motion of the 2+2 body problem (two interacting particles in the gravitational field of two much more massive primaries m₁ and m₂ in circular keplerian orbit) have an integral analogous to the Jacobi integral of the circular 2+1 body problem. We show here that with 2+2 bodies this integral does not give rise to Hill stability, i.e. to confinement for all time in a portion of the configuration space not allowing for some close approaches to occur. This is because all the level manifolds are connected and all exchanges of bodies between the regions surroundingm ₁,m ₂ and infinity do not contradict the conservation of the integral. However, it is worth stressing that some of these exchanges are physically meaningless, because they involve either unlimited extraction of potential energy from the binary formed by the small bodies (without taking into account their physical size) or significant mutual perturbations between the small masses without close approach, a process requiring, for the Sun-Jupiter-two asteroids system, timescales longer than the age of the Solar System.     

Stationary motions in a restricted problem of three rigid bodies

The present paper is a continuation of papers by Shinkaric (1972), Vidyakin (1976), Vidyakin (1977), and Duboshin (1978), in which the existence of particular solutions, analogues to the classic solutions of Lagrange and Euler in the circular restricted problem of three points were proved. These solutions are stationary motions in which the centres of mass of the bodies of the definite structures always form either an equilateral triangle (Lagrangian solutions) or always remain on a straight line (Eulerian solutions) The orientation of the bodies depends on the structure of the bodies. In this paper the usage of the small-parameter method proved that in the general case the centre of mass of an axisymmetric body of infinitesimal mass does not belong to the orbital plane of the attracting bodies and is not situated in the libration points, corresponding to the classical case. Its deviation from them is proportional to the small parameter. The body turns uniformly around the axis of symmetry. In this paper a new type of stationary motion is found, in which the axis of symmetry makes an angle, proportional to the small parameter, with the plane created by the radius-vector and by the normal to the orbital plane of the attracting bodies. The earlier solutions-Shinkaric (1971) and Vidyakin (1976)-are also elaborated, and stability of the stationary motions is discussed.     

On the motion of three rigid bodies: Central configurations

In the present paper, the motion of three rigid bodies is considered. With a set of new variables, and the 10 first integrals of the motion, the problem is reduced to a system of order 25 and one quadrature. The plane motions are characterized, and finally, an equation for the existence of central configurations (in particular, Lagrangian and Eulerian solutions) has been found. Besides, the case of three axisymmetric ellipsoids is studied.     

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.     

On the stability parameters of periodic solutions

The stability parametersa, b, c, d of plane symmetric periodic solutions of non-integrable dynamical systems of two degrees of freedom are obtained in terms of their initial states of motion and elements of their variational matrics. Explicit formulae are given in the cases of the Störmer problem and the restricted problem of three bodies.     

The dynamics of bodies with variable masses

The Lie-series provide a convenient and simple method to solve systems of differential equations, especially in problems dealing with variable masses. A system of n-bodies moving isside a cloud and collecting mass is considered. The equations of motion are derived where-by the interchange of momentum is treated as a perturbation. It is shown that the solutions, represented by Lie-series, can be expressed by binomial expansions plus a perturbation which can be solved by interation. In addition, the motion of massless bodies experiencing frictional forces is briefly discussed.     

The existence and stability of the equilibrium points of a triaxial rigid body moving around another triaxial rigid body

The motion of two mutually attracting triaxial rigid bodies has been considered. Thirty six particular solutions corresponding to the libration points and analogous to the points Spoke, Arrow and Float (Duboshin, 1959) have been found. The stability of these libration points has been discussed in two categories of cases. In the first category, different shapes of the bodies have been taken and in the second category, the mass and the linear dimensions of one of the bodies have been taken small in comparison to the other.