Potential energy of gravitationally interacting disk galaxies

Methods are developed for analysing the gravitational properties of disks having circularly symmetric distribution of matter. It is shown how this can be conveniently done by assuming that the surface density distribution may be approximated by a polynomial in ascending powers of the distance from the centre of the configuration. A theory has been developed to determine the gravitational potential of a single disk at any point in space in terms of the coefficients of the polynomial defining the surface distribution of matter, and the potential energy of two disks of arbitrary separation and orientation due to their mutual gravitational attraction. The basic functions, required for obtaining the potential in the plane of the disk and the mutual potential energy of two coplanar disks, have been tabulated. Two overlapping coplanar disks attract just like mass-points at a certain separation,r _c , of their centres. The force of attraction of disks is less than the force of attraction of mass-points having masses equal to the masses of the disks, if the separation of the centres is less thanr _c , and greater if the separation is greater thanr _c . For typical galaxies of equal radiiR,r _c ≈R.     

Potential energy of gravitationally interacting spherical galaxies

A theory has been developed for obtaining the potential energy of two interpenetrating spherically symmetric galaxies of unequal dimensions due to their mutual gravitational interaction. The mass distribution in both the galaxies is assumed to be that of a polytrope of integral index. A basic function that occurs in the theory has been tabulated for the cases of polytropes of indicesn=0 and 4 for four ratios of the radii.     

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 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.     

Spin axis evolution of two interacting bodies

We consider the solid-solid interactions in the two body problem. The relative equilibria have been previously studied analytically and general motions were numerically analyzed using some expansion of the gravitational potential up to the second order, but only when there are no direct interactions between the orientation of the bodies. Here we expand the potential up to the fourth order and we show that the secular problem obtained after averaging over fast angles, as for the precession model of Boué and Laskar [Boué, G., Laskar, J., 2006. Icarus 185, 312-330], is integrable, but not trivially. We describe the general features of the motions and we provide explicit analytical approximations for the solutions. We demonstrate that the general solution of the secular system can be decomposed as a uniform precession around the total angular momentum and a periodic symmetric orbit in the precessing frame. More generally, we show that for a general n-body system of rigid bodies in gravitational interaction, the regular quasiperiodic solutions can be decomposed into a uniform precession around the total angular momentum, and a quasiperiodic motion with one frequency less in the precessing frame.     

The Sitnikov problem for several primary bodies configurations

In this paper we address an \(n+1\)-body gravitational problem governed by the Newton’s laws, where n primary bodies orbit on a plane \(\varPi \) and an additional massless particle moves on the perpendicular line to \(\varPi \) passing through the center of mass of the primary bodies. We find a condition for the described configuration to be possible. In the case when the primaries are in a rigid motion, we classify all the motions of the massless particle. We study the situation when the massless particle has a periodic motion with the same minimal period as the primary bodies. We show that this fact is related to the existence of a certain pyramidal central configuration.     

Central configurations of four bodies with one inferior mass

The number of equivalence classes of central configurations (abbr. c.c.) in the planar 4-body problem with three arbitrary and a fourth small mass is investigated. These c.c. are derived according to their generic origin in the 3-body problem. It is shown that each 3-body collinear c.c. generates exactly 2 non-collinear c.c. (besides 4 collinear ones) of 4 bodies with smallm ₄0; and that any 3-body equilateral triangle c.c. generates exactly 8 or 9 or 10 (depending onm ₁,m ₂,m ₃) planar 4-body c.c. withm ₄=0. Further, every one of these c.c. can be continued uniquely to sufficiently smallm ₄>0 except when there are just 9; then exactly one of them is degenerate, and we conjecture that it is not continuable tom ₄>0.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.     

Point-connected rigid bodies in a topological tree

Identical equations of motion are shown to emerge for a system ofn+1 rigid bodies all interconnected byn points, each of which is common to two bodies, by means of each of the following derivation procedures, all of which employ a kinematical identity developed by Hooker and Margulies: The Hooker-Margulies/Hooker equations; Kane's quasicoordinate formulation of D'Alembert's principle; the combination of Lagrange's generalized coordinate equations and Lagrange's quasicoordinate equations; and the combination of Lagrange's generalized coordinate equations and the vector rotational equationM=H applied to the total system and resolved into a vector basis fixed in a reference body of the system. Thus the previously published Hooker-Margulies/Hooker equations are shown to be the natural result of several derivation procedures other than the Newton-Euler method originally used, provided that the central kinematical identity of the original derivation of Hooker and Margulies is employed.     

Extended two-body problem for rotating rigid bodies

Given the interest in future space missions devoted to the exploration of key moons in the solar system and that may involve libration point orbits, an efficient design strategy for transfers between moons is introduced that leverages the dynamics in these multi-body systems. The moon-to-moon analytical transfer (MMAT) method is introduced, comprised of a general methodology for transfer design between the vicinities of the moons in any given system within the context of the circular restricted three-body problem, useful regardless of the orbital planes in which the moons reside. A simplified model enables analytical constraints to efficiently determine the feasibility of a transfer between two different moons moving in the vicinity of a common planet. In particular, connections between the periodic orbits of such two different moons are achieved. The strategy is applicable for any type of direct transfers that satisfy the analytical constraints. Case studies are presented for the Jovian and Uranian systems. The transition of the transfers into higher-fidelity ephemeris models confirms the validity of the MMAT method as a fast tool to provide possible transfer options between two consecutive moons.

     

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.     

Central configurations of four bodies with an axis of symmetry

A complete solution is given for a symmetric case of the problem of the planar central configurations of four bodies, when two bodies are on an axis of symmetry, and the other two bodies have equal masses and are situated symmetrically with respect to the axis of symmetry. The positions of the bodies on the axis of symmetry are described by angle coordinates with respect to the outside bodies. The solution is such, that giving the angle coordinates, the masses for which the given configuration is a central configuration, can be computed from simple analytical expressions of the angles. The central configurations can be described as one-parameter families, and these are discussed in detail in one convex and two concave cases. The derived formulae represent exact analytical solutions of the four-body problem.     

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.     

Symmetric planar central configurations of five bodies: Euler plus two

We study planar central configurations of the five-body problem where three of the bodies are collinear, forming an Euler central configuration of the three-body problem, and the two other bodies together with the collinear configuration are in the same plane. The problem considered here assumes certain symmetries. From the three bodies in the collinear configuration, the two bodies at the extremities have equal masses and the third one is at the middle point between the two. The fourth and fifth bodies are placed in a symmetric way: either with respect to the line containing the three bodies, or with respect to the middle body in the collinear configuration, or with respect to the perpendicular bisector of the segment containing the three bodies. The possible stacked five-body central configurations satisfying these types of symmetries are: a rhombus with four masses at the vertices and a fifth mass in the center, and a trapezoid with four masses at the vertices and a fifth mass at the midpoint of one of the parallel sides.     

Changes in binding energy of interacting galaxies

It is shown that the fractional increase in binding energy of a galaxy in a fast collision with another galaxy of the same size can be well represented by the formula $$\xi _2 = 3({G \mathord{\left/ {\vphantom {G {M_2 \bar R}}} \right. \kern-\nulldelimiterspace} {M_2 \bar R}}) ({{M_1 } \mathord{\left/ {\vphantom {{M_1 } {V_p }}} \right. \kern-\nulldelimiterspace} {V_p }})^2 e^{ - p/\bar R} = \xi _1 ({{M_1 } \mathord{\left/ {\vphantom {{M_1 } {M_2 }}} \right. \kern-\nulldelimiterspace} {M_2 }})^3 ,$$ whereM ₁,M ₂ are the masses of the perturber and the perturbed galaxy, respectively,V _p is the relative velocity of the perturber at minimum separationp, and \(\bar R\) is the dynamical radius of either galaxy.     

Motion of rigid bodies in a set of redundant variables

In this article we study the conditions for obtaining canonical transformationsy=f(x) of the phase space, wherey(y ₁,y ₂,...,y _2n ) andx(x ₁,x ₂,...,x _2m ) in such a way that the number of variables is increased. In particular, this study is applied to the rotational motion in functions of the Eulerian parameters (q ₀,q ₁,q ₂,q ₃) and their conjugate momenta (Q ₀,Q ₁,Q ₂,Q ₃) or in functions of complex variables (z ₁,z ₂,z ₃,z ₄) and their conjugate momenta (Z ₁,Z ₂,Z ₃,Z ₄) defined by means of the previous variables. Finally, our article include some properties on the rotational motion of a rigid body moving about a fixed point.     

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,II

This study constitutes the second phase of an effort devoted to the relative motion of two spheroidal rigid bodies.

An isolated binary system was considered whose components are bodies: (1) of comparable size; (2) of constant density; and (3) having the shape of an oblate ellipsoid of revolution with small meridional eccentricity.

The equations that determine the relative motion of the centroids and the angular motion for the two sets of body axes constitute a simultaneous system of seven nonlinear, second-order differential equations, for which solutions were obtained as power series in the two meridional eccentricities.

A recurrent procedure was formulated to ascertain the various approximations in terms of lower order terms; it gave rise to linear differential equations with constant coefficients for the angular variables and to differential equations of the Hill type for the other coordinates. The zero-order approximation for the motion of the centroids was assumed to be a Kepler elliptic orbit of small eccentricity.

The following contributions were made:

1. (1)

   The general solution to the zero-order approximation of the rotational motion was obtained in terms of elementary functions;

2. (2)

   Certain functionals, related to the Kepler motion and depending on two parameters, were expressed in terms of the mean anomaly up to the sixth power of the orbital eccentricity in order to evaluate the lower order terms of the various approximations;

3. (3)

   The secular terms were eliminated from the first-order approximation;

4. (4)

   The second-order approximation was also obtained; and

5. (5)

   An alternate procedure was suggested that might be more appropriate for achieving higher order approximations.

     

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.