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.     

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.     

Central configurations of the five-body problem with equal masses

In this paper we present a complete classification of the isolated central configurations of the five-body problem with equal masses. This is accomplished by using the polyhedral homotopy method to approximate all the isolated solutions of the Albouy-Chenciner equations. The existence of exact solutions, in a neighborhood of the approximated ones, is then verified using the Krawczyk method. Although the Albouy-Chenciner equations for the five-body problem are huge, it is possible to solve them in a reasonable amount of time.     

Superhyperbolic expansion,noncollision singularities and symmetry configurations

A large class of symmetry solutions of the Newtonian n-body problem cannot end in a noncollision singularity nor expand faster than any constant multiple of time.Following a suggestion from Christian Marchal, we extended the original theme of this essay to include superhyperbolic motion. The work of D. Saari was supported by NSF Grant ISI 9103180; the work of F. Diacu was supported by NSERC Grant 3-48376     

Minimal energy configurations of gravitationally interacting rigid bodies

Consider a collection of n rigid, massive bodies interacting according to their mutual gravitational attraction. A relative equilibrium motion is one where the entire configuration rotates rigidly and uniformly about a fixed axis in \(\mathbb {R}^3\). Such a motion is possible only for special positions and orientations of the bodies. A minimal energy motion is one which has the minimum possible energy in its fixed angular momentum level. While every minimal energy motion is a relative equilibrium motion, the main result here is that a relative equilibrium motion of \(n\ge 3\) disjoint rigid bodies is never an energy minimizer. This generalizes a known result about point masses to the case of rigid bodies.     

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 and a theorem of palmore

The concept of central configuration is important in the study of total collisions or the relative equilibrium state of a rotating system in the N-body problem. However, relatively few such configurations are known. Aided by a new global optimizer, we have been able to construct new families of coplanar central configurations having particles of equal mass, and extend these constructions to some configurations with differing masses and the non-coplanar case. Meyer and Schmidt had shown that a theorem of Palmore concerning coplanar central configurations was incorrect for N equal masses where 6 N 20 but presented a simple analytic argument only for N = 6. Using straightforward analytic arguments and inequalities we also disprove this theorem for 2N equal masses with N 3.     

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.     

Orbits near central configurations for four equal masses

This study relates to equal-mass four-body orbits close to a quadruple central configuration. Locally, these orbits can be approximated by a perturbation from the homothetic quadruple collision/expansion orbit. Appropriate expressions are derived and the equal-mass four-body Siegel exponents and associated eigenmodes are presented.     

Central configurations for the planar Newtonian four-body problem

Planar central configurations of four different masses are analyzed theoretically and computed numerically. We follow Dziobek’s approach to four-body central configurations with a straightforward implicit (in the masses and distances) method of our own in which the fundamental quantities are each the quotient of a directed area divided by the corresponding mass. We apply a new simple numerical algorithm to construct general four-body central configurations. We use this tool to obtain new properties of the symmetric and non-symmetric central configurations. The explicit continuous connection between three-body and four-body central configurations where one of the four masses approaches zero is clarified. Some cases of coorbital 1+3 problems are also considered.     

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.     

Central configurations and hyperbolic-elliptic motion in the three-body problem

A refined classification of motion for the planar three-body problem with zero total energy is presented. In addition, the structure and size of the sets of initial conditions are obtained. Limited results for the spatial problem are also given.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.Supported in part by the Natural Science Fund and the University Research Council of Vanderbilt University.     

Central configurations of the planar 1+N body problem

In this paper, we give a new derivation of the equations for the central configurations of the 1+n body problem. In the case of equal masses, we show that forn large enough there exists only one solution. Our lower bound forn improves by several orders of magnitude the one previously found by Hall.     

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.     

The angle of inclination of the sunspot symmetry axis to the solar surface

The Wilson effect, used before only as a method of determining the physical depression of sunspots, is used here to estimate a quite different parameter - the sunspot symmetry axis inclination angle to the solar surface, this explains the observed negative Wilson effect.On the basis of photoheliograms taken with three telescopes of the High-Altitude Solar observatory Peak Alma-Ata, the Wilson effect for the whole solar disk is investigated, the east and west parts of the disk being studied separately. 111 sunspots of regular shape at different heliocentric angles were measured, eight of them being under observations from one limb to the other. To study the dependence of the Wilson effect on the heliocentric angle, all observations within an angular interval of 10° were averaged. The dependence thus derived is described by two sinusoids having the zero point shifted along both axes. The shift of the zero Wilson effect to the west, i.e., a shift along the heliocentric angle axis, can be caused by the deviation of the sunspot axis to the east from the normal to the solar surface. On the line of sight-normal plane the angle corresponding to this deviation is =34°±14°.     

Massive configurations with constant densities

Two new classes of solutions with constant observed proper and rest mass densities are described. Unlike the well-known solution of constant coordinate mass density, these solutions pertain to realistic physical situations. For these solutions, the various relevant parameters, viz. the redshifts (dP/d)₀ and binding coefficients have been calculated.