Periodic elliptic motions in a planar restricted (N+1)-body problem

LetN2 mass points (primaries) move on a collinear solution of relative equilibrium of theN-body problem; i.e. suitably fixed on a uniformly rotating straight line. Consider the motion of a massless particle in the gravitational field of these primaries with arbitrarily given masses. An existence proof for periodic solutions (i.e. closed trajectories in a rotating coordinate system) will be given, in which the particle performs nearly keplerian elliptic motions about (and close to) any one of the primaries.     

Steady state obliquity of a rigid body in the spin–orbit resonant problem: application to Mercury

In this paper, we consider the elliptic collinear solutions of the classical n-body problem, where the n bodies always stay on a straight line, and each of them moves on its own elliptic orbit with the same eccentricity. Such a motion is called an elliptic Euler–Moulton collinear solution. Here we prove that the corresponding linearized Hamiltonian system at such an elliptic Euler–Moulton collinear solution of n-bodies splits into \((n-1)\) independent linear Hamiltonian systems, the first one is the linearized Hamiltonian system of the Kepler 2-body problem at Kepler elliptic orbit, and each of the other \((n-2)\) systems is the essential part of the linearized Hamiltonian system at an elliptic Euler collinear solution of a 3-body problem whose mass parameter is modified. Then the linear stability of such a solution in the n-body problem is reduced to those of the corresponding elliptic Euler collinear solutions of the 3-body problems, which for example then can be further understood using numerical results of Martínez et al. on 3-body Euler solutions in 2004–2006. As an example, we carry out the detailed derivation of the linear stability for an elliptic Euler–Moulton solution of the 4-body problem with two small masses in the middle.     

TheN-body problem of celestial mechanics

A selective survey of then-body problem of celestial mechanics is given where the emphasis is on the asymptotic behavior of all solutions ast, the possible configurations the particles can assume in phase space and in physical space, and collision and non-collision singularities.Supported in part by NSF Grant MPS 71-03407 A03.     

The pulse-period distribution of binary X-ray pulsars

The pulse-period distribution of binary X-ray pulsars has been considered. A gap in this distribution, in the period rangeP10 s toP100s has been explained in terms of the character of mass transfer in the X-ray binary systems. It is shown that this gap arises because the rotating magnetised neutron stars in these systems are slowed down by accretion torques, either toP10 s when the mass transfer is by means of Roche-lobe overflow in low mass binaries, or toP100 s by stellar winds in massive binaries. The gap is maintained as the slow pulsars (P>100 s) in their spin-up phase cross the gap in a time short compared to their life-time, because of the increase in mass transfer with the evolution of the normal star.     

On the number of central configurations in the N-body problem

Central configurations are critical points of the potential function of the n-body problem restricted to the topological sphere where the moment of inertia is equal to constant. For a given set of positive masses m ₁,..., m _n we denote by N(m ₁, ..., m _n, k) the number of central configurations' of the n-body problem in ^k modulus dilatations and rotations. If m _{n 1},..., m _n, k) is finite, then we give a bound of N(m ₁,..., m _n, k) which only depends of n and k.     

Sur l'existence de certaines configurations d'equilibre relatif dans le probleme desN corps

Résumé On sait que les positions d'équilibre relatif dans le problème des trois corps, où les corps se trouvent aux sommets d'un triangle équilatéral, existent lorsque les masses sont quelconques; tandis que pourn=4 (voir [3]) et pourn=5 (voir [4]), les positions d'équilibre relatifs, où les corps se trouvent aux sommets d'un polygone régulier de n cotés, existent seulement si les masses sont égales. L'objet de cet article est de montrer que ce dernier résultat est vrai pour toutn4.

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It is known that in the three body problem, the equilateral configuration of relative equilibrium exists for all values of masses, while in then-body problem, forn=4 (see [3]) andn=5 (see [4]), the position of relative equilibrium where the bodies are at the vertices of a regular polygon withn sides, exists only if the masses are equal.We prove in this paper, that this last result is true for alln4.

     

Effect of Krein signatures on the stability of relative equilibria

We consider the linear stability of the equilibrium points of the generic rotating potentials U(r), U(r, ), U(r, and U(r, , . The stability analysis is performed using the concept of Krein's signature. This signature is calculated for all eigenvalues of the above potentials. Thereby, the Lagrangian points of the restricted three-body problem and the synchronous satellites of oblate and prolate planets are also studied. We find also the new positions of the eigenvalues for perturbations of the original L ₄ and L ₅ points of Mars, Jupiter, Saturn, Uranus and Neptune. Finally, we study the problem with the mass ratio µ close to the critical value and the stability of geostationary satellites perturbed by the Moon.     

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.     

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.     

Determination of the electromagnetic field produced by a magnetic oblique-rotator

The inertial effect on the structure of the magnetosphere of a rotating star is investigated, in the corotation approximation for a surrounding quasi-neutral plasma. The equation of motion reduces to a usual static balance equation between the electromagnetic and the centrifugal forces, in the rotating frame. However the MHD condition, which can be regarded as a special form of the generalized Ohm's law, is modified by the inclusion of inertial effect, with a violation of the frozen-in condition in case of a general (i.e., not restricted to corotation) plasma motion. The inertial effect on the electromagnetic field is summarized in a partial scalar potential named the non-Backus potential, which is proportional to the centrifugal potential in the corotation approximation.An approximate solution of this corotation problem is given, in which another characteristic radiusr _M appears besides the light radiusr _L . This radius defines a distance beyond which the inertial effect becomes dominant over the electromagnetic one, and is useful in estimating the magnitude of the terminal velocity of a centrifugal wind. A few examples of the modification of dipole magnetic field due to the inertial effect are visualized. In an oblique-rotation case, it can be seen that such a warp of the neutral sheet (the surface ofB _r =0) is reproduced as observed in the Jovian magnetosphere.     

A hypothesis to explain the virial discrepancy in clusters of galaxies

On the basis of numerical experiments onn-body binding energies we tentatively consider the following hypothesis: If the distance between two galaxies forming a binary system isa _g, and a cluster of galaxies that is substructured in a hierarchical fashion onall scales froma _g upwards has a total massM, then the total gravitational binding energy of the cluster is _TH = –GM ²/2a _g . As an explanation for missing masses up to order 100 we test this hypothesis in three different ways, finding remarkable agreement with observation, with no need for physical missing mass.     

Mass ratios of three bodies for stable low-inclination three-dimensional periodic motion

The intervals of possible stability, on the -axis, of the basic families of three-dimensional periodie motions of the restricted three-body problem (determined in an earlier paper) are extended into regions of the -m ₃ parameter space of the general three-body problem. Sample three-dimensional periodic motions corresponding to these regions are computed and tested for stability. Six regions, corresponding to the vertical-critical orbitsl1v, m1v,m2v, andilv, survive this preliminary stability test-therefore, emerging as the mass parameters regions allowing the simplest types of stable low inclination three-dimensional motion of three massive bodies.     

The global solution of the N-body problem

The problem of finding a global solution for systems in celestial mechanics was proposed by Weierstrass during the last century. More precisely, the goal is to find a solution of the n-body problem in series expansion which is valid for all time. Sundman solved this problem for the case of n = 3 with non-zero angular momentum a long time ago. Unfortunately, it is impossible to directly generalize this beautiful theory to the case of n > 3 or to n = 3 with zero-angular momentum.A new blowing up transformation, which is a modification of McGehee's transformation, is introduced in this paper. By means of this transformation, a complete answer is given for the global solution problem in the case of n > 3 and n = 3 with zero angular momentum.The main result in this paper has appeared in Chinese in Acta Astro. Sinica. 26 (4), 313–322. In this version some mistakes have been rectified and the problems we solved are now expressed in a much clearer fashion.     

A new kind of 3-body problem

A new kind of restricted 3-body problem is considered. One body,m ₁, is a rigid spherical shell filled with an homogeneous incompressible fluid of density ₁. The second one,m ₂, is a mass point outside the shell andm ₃ a small solid sphere of density ₃ supposed movinginside the shell and subjected to the attraction ofm ₂ and the buoyancy force due to the fluid ₁. There exists a solution withm ₃ at the center of the shell whilem ₂ describes a Keplerian orbit around it. The linear stability of this configuration is studied assuming the mass ofm ₃ to beinfinitesimal. Explicitly two cases are considered. In the first case, the orbit ofm ₂ aroundm ₁ is circular. In the second case, this orbit is elliptic but the shell is empty (i.e. no fluid inside it) or the densities ₁ and ₃ are equal. In each case, the domain of stability is investigated for the whole range of the parameters characterizing the problem.     

Substructure in clusters of galaxies: A clue to the ‘missing’ mass?

The influence of subclustering in rich clusters of galaxies is examined using results from numericaln-body experiments. It is found that, under some conditions, the standard virial theorem is satisfied. No physical missing mass is needed because its role is replaced by the gravitational energy of the subclustering. We find that, in the Coma cluster, this effect masquerades as a missing mass about 7 times that of the physical mass, so that the apparent extant virial discrepancy (M _VT/M8) in this cluster is explained.     

Dynamical effects of dark matter in systems of galaxies

SeveralN-body experiments were performed in order to simulate the dynamical behaviour of systems of galaxies gravitationally dominated by a massive dark background.We discuss mass estimates from the dynamics of the luminous component (M _VT) under the influence of such a background, assuming a constant dark/luminous mass ratio (M _D/M _L) and plausible physical conditions. We extend in this way previous studies (Smith, 1980, 1984) about the dependence ofM _VT on the relative distributions of dark and luminous matter (Limber, 1959). We found that the observed ratio of the virial theorem mass to luminosity (M _VT/L) in systems of galaxies of different sizes could be the result of different stages of their post-virialisation evolution as was previously suggested by White and Rees (1978) and Barnes (1983). This evolution is mainly the result of the dynamical friction that dark matter exerts on the luminous component. Thus our results give support to the idea that compact groups of galaxies are dynamically more evolved than large clusters, which is expected from the hierarchical clustering picture for the formation of such structures.     

How the Method of Minimization of Action Avoids Singularities

The method of minimization of action is a powerful technique of proving the existence of particular and interesting solutions of the n-body problem, but it suffers from the possible interference of singularities. The minimization of action is an optimization and, after a short presentation of a few optimization theories, our analysis of interference of singularities will show that:(A) An n-body solution minimizing the action between given boundary conditions has no discontinuity: all n-bodies have a continuous and bounded motion and thus all eventual singularities are collisions;(B) A beautiful extension of Lambert's theorem shows that, for these minimizing solutions, no double collision can occur at an intermediate time;(C) The proof can be extended to triple and to multiple collisions. Thus, the method of minimization of action leads to pure n-body motions without singularity at any intermediate time, even if one or several collisions are imposed at initial and/or final times.This method is suitable for non-infinitesimal masses only. Fortunately, a similar method, with the same general property with respect to the singularities, can be extended to n-body problems including infinitesimal masses.     

Numerical integration of the <Emphasis Type="Italic">N</Emphasis>-body ring problem by recurrent power series

The aim of this article is to present a method for the integration of the equations of motion of the N-body ring problem by means of recurrent power series. We prove that the solution is convergent for any set of initial conditions, excluding those corresponding to binary collisions.     

The origin of the emission lines shown by RW Tauri and other binary systems

It is proposed that the observed double-component emission lines originate from the triangular Lagrangian points, L4 and L5, of the restricted three-body problem. The light curves of many close binary systems show absorption dips at ±60° of the primary (and sometimes also the secondary) minimum, indicating appreciable accumulation of matter at these points. The orbital velocity of L4 and L5 is derived as a function of period and the masses of the component stars. This equation is an independent relationship for determining the two stellar masses. It also reproduces Struve's empirical finding ofV ³P ^–1. The observed emission line velocity is consistently higher than the calculated orbital velocity of L4 and L5. This is due to the serious erosion of the low velocity sides of the emission components by the stellar and shell absorption lines. There are observational evidences which indicate the intermittent high velocity radial ejection of matter to be a mode of mass loss from the secondary. And the energy of mass motion from this ejection is sufficient, and may be responsible, for heating the gas at L4 and L5. The ionizing radiation emitted by the primary of the Algol systems is many orders of magnitude below that required by the observed strength of the emission lines. Some related discussion is also given to nova and dwarf nova systems.     

Asymptotic solutions in the many-body problem

A small particle moves in the vicinity of two masses, forming a close binary, in orbit about a distant mass. Unique, uniformly valid solutions of this four-body problem are found for motion near both equilateral triangle points of the binary system in terms of a small parameter , where the primaries move in accordance with a uniformly-valid three-body solution. Accuracy is maintained within a constant errorO(⁸), and the solutions are uniformly valid as tends to zero for time intervalsO(^–3). Orbital position errors nearL ₄ andL ₅ of the Earth-Moon system are found to be less than 5% when numerically-generated periodic solutions are used as a standard of comparison.