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.     

Some special restricted four-body problems—II. From Caledonia to Copenhagen

Special analytical solutions are determined for restricted, coplanar, four-body equal mass problems, including the Caledonian problem, where the masses M_i = M for i = 1,2,3,4. Most of these solutions are shown to reduce to the Lagrange solutions of the Copenhagen problem of three bodies by reducing two of the masses (m_i = m for i = 1,2) in the four-body equal mass problem to zero while maintaining their equality of mass. In so doing, families of special solutions to the four-body problem are shown to exist for any value of the mass ratio μ = m/M.     

New stacked central configurations for the planar 5-body problem

A stacked central configuration in the n-body problem is one that has a proper subset of the n-bodies forming a central configuration. In this paper we study the case where three bodies with masses m ₁, m ₂, m ₃ (bodies 1, 2, 3) form an equilateral central configuration, and the other two with masses m ₄, m ₅ are symmetric with respect to the mediatrix of the segment joining 1 and 2, and they are above the triangle generated by {1, 2, 3}. We show the existence and non-existence of this kind of stacked central configurations for the planar 5-body problem.     

A class of periodic solutions of the N-body problem

We prove existence and multiplicity of T-periodic solutions (for any given T) for the N-body problem in ^m (any m 2) where one of the bodies has mass equal to 1 and the others have masses ₂,..., _N , small. We find solutions such that the body of mass 1 moves close to x = 0 while the body of mass _i moves close to one of the circular solutions of the two body problem of period T/k _i, where ki is any odd number. No relation has to be satisfied by k ₂,...,k _N.     

Collinear Equilibrium Solutions of Four-body Problem

We discuss the equilibrium solutions of four different types of collinear four-body problems having two pairs of equal masses. Two of these four-body models are symmetric about the center-of-mass while the other two are non-symmetric. We define two mass ratios as μ ₁ = m ₁/M _T and μ ₂ = m ₂/M _T, where m ₁ and m ₂ are the two unequal masses and M _T is the total mass of the system. We discuss the existence of continuous family of equilibrium solutions for all the four types of four-body problems.     

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.     

Restricted N+1-Body Problem: Existence and Stability Of Relative Equilibria

We consider n bodies (with equal mass m) disposed at the vertices of a regular n-gon and rotating rigidly around an additional mass m ₀(at its center) with a constant angular velocity (relative equilibrium). In the present paper, we prove results on the existence and on the linear stability of equilibrium positions for a zero-mass particle submitted to the gravitational field generated by the previous system. This revised version was published online in July 2006 with corrections to the Cover Date.     

Symmetric and Asymmetric Librations in Planetary and Satellite Systems at the 2/1 Resonance

Families of asymmetric periodic orbits at the 2/1 resonance are computed for different mass ratios. The existence of the asymmetric families depends on the ratio of the planetary (or satellite) masses. As models we used the Io-Europa system of the satellites of Jupiter for the case m₁>m₂, the system HD82943 for the new masses, for the case m₁=m₂ and the same system HD82943 for the values of the masses m₁<m₂ given in previous work. In the case m₁≥ m₂ there is a family of asymmetric orbits that bifurcates from a family of symmetric periodic orbits, but there exist also an asymmetric family that is independent of the symmetric families. In the case m₁<m₂ all the asymmetric families are independent from the symmetric families. In many cases the asymmetry, as measured by and by the mean anomaly M of the outer planet when the inner planet is at perihelion, is very large. The stability of these asymmetric families has been studied and it is found that there exist large regions in phase space where we have stable asymmetric librations. It is also shown that the asymmetry is a stabilizing factor. A shift from asymmetry to symmetry, other elements being the same, may destabilize the system.     

A highly symmetric restricted nine-body problem and the linear stability of its relative equilibria

We study a highly symmetric nine-body problem in which eight positive masses, called the primaries, move four by four, in two concentric circular motions such that their configuration is always a square for each group of four masses. The ninth body being of negligible mass and not influencing the motion of the eight primaries. We assume all the nine masses are in the same plane and that the masses of the primaries are \(m_{1}=m_{2}=m_{3}=m_{4}=\tilde{m}\) and m ₅=m ₆=m ₇=m ₈=m and the radii associated to the circular motion of the bodies with mass \(\tilde{m}\) is λ∈[λ ₀,1] and for the bodies with mass m is 1. We prove the existence of central configurations which characterize such arrangement of the primaries and we study the influence of the parameter λ, the ratio of the radii of the two circles, on the masses m and \(\tilde{m}\) . We use a synodical system of coordinates to eliminate the time dependence on the equations of motion. We show the existence of equilibria solutions symmetrically distributed on the four quadrants and their dependence on the parameter λ. Finally, we show that there can be 13, 17 or 25 equilibria solutions depending on the size of λ and we investigate their linear stability.     

Asymptotic orbits in the (<Emphasis Type="Italic">N</Emphasis>+1)-body ring problem

In this paper we study the asymptotic solutions of the (N+1)-body ring planar problem, N of which are finite and ν=N−1 are moving in circular orbits around their center of masses, while the Nth+1 body is infinitesimal. ν of the primaries have equal masses m and the Nth most-massive primary, with m ₀=β m, is located at the origin of the system. We found the invariant unstable and stable manifolds around hyperbolic Lyapunov periodic orbits, which emanate from the collinear equilibrium points L ₁ and L ₂. We construct numerically, from the intersection points of the appropriate Poincaré cuts, homoclinic symmetric asymptotic orbits around these Lyapunov periodic orbits. There are families of symmetric simple-periodic orbits which contain as terminal points asymptotic orbits which intersect the x-axis perpendicularly and tend asymptotically to equilibrium points of the problem spiraling into (and out of) these points. All these families, for a fixed value of the mass parameter β=2, are found and presented. The eighteen (more geometrically simple) families and the corresponding eighteen terminating homo- and heteroclinic symmetric asymptotic orbits are illustrated. The stability of these families is computed and also presented.     

The collinear central configuration of n equal masses

Moulton's Theorem says that given an ordering of masses m ₁, ..., m _nthere exists a unique collinear central configuration. The theorem allows us to ask the questions: What is the distribution of n equal masses in the collinear central configuration? What is the behavior of the distribution as n → ∞? These questions are due to R. Moekel (personal conversation). Central configurations are found to be attracting fixed points of a flow — a flow we might call an auxiliary flow (in the text it is denoted F(X)), since it has little to do with the equations of motion. This flow is studied in an effort to characterize the mass distribution. Specifically, for a collinear central configuration of n equal masses, a bound is found for the position of the masses furthest from the center of mass. Also some facts concerning the distribution of the inner masses are discovered.     

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.     

Three-dimensional periodic motion of three finite masses around collinear equilibrium configurations

Three-dimensional periodic motions of three bodies are shown to exist in the infinitesimal neighbourhood of their collinear equilibrium configurations. These configurations and some characteristic quantities of the emanating three-dimensional periodic orbits are given for many values of the two mass parameters, =m ₂/(m ₁+m ₂) andm ₃, of the general three-body problem, under the assumption that the straight line containing the bodies at equilibrium rotates with unit angular velocity. The analysis of the small periodic orbits near the equilibrium configurations is carried out to second-order terms in the small quantities describing the deviation from plane motion but the analytical solution obtained for the horizontal components of the state vector is valid to third-order terms in those quantities. The families of three-dimensional periodic orbits emanating from two of the collinear equilibrium configurations are continued numerically to large orbits. These families are found to terminate at large vertical-critical orbits of the familym of retrograde periodic orbits ofm ₃ around the primariesm ₁ andm ₂. The series of these termination orbits, formed when the value ofm ₃ varies, are also given. The three-dimensional orbits are computed form ₃=0.1.     

Constraining the relative inclinations of the planets B and C of the millisecond pulsar PSR B1257+12

We investigate on the relative inclination of the planets B and C orbiting the pulsar PSR B1257+12. First, we show that the third Kepler’s law does represent an adequate model for the orbital periods P of the planets, because other Newtonian and Einsteinian corrections are orders of magnitude smaller than the accuracy in measuring P _B/C. Then, on the basis of available timing data, we determine the ratio sin i _C/ sin i _B = 0.92±0.05 of the orbital inclinations i _B and i _C independently of the pulsar’s mass M. It turns out that coplanarity of the orbits of B and C would imply a violation of the equivalence principle. Adopting a pulsar mass range 1 ≲ M ≲ 3, in solar masses (supported by present-day theoretical and observational bounds for pulsar’s masses), both face-on and edge-on orbital configurations for the orbits of the two planets are ruled out; the acceptable inclinations for B span the range 36 deg ≲ i _B ≲ 66 deg, with a corresponding relative inclination range 6 deg ≲ (i _C − i _B) ≲ 13 deg.     

Symmetry, bifurcation and stacking of the central configurations of the planar 1+4 body problem

In this work we are interested in the central configurations of the planar $1+4$ body problem where the satellites have different infinitesimal masses and two of them are diametrically opposite in a circle. We can think of this problem as a stacked central configuration too. We show that the configurations are necessarily symmetric and the other satellites have the same mass. Moreover we prove that the number of central configurations in this case is in general one, two or three and, in the special case where the satellites diametrically opposite have the same mass, we prove that the number of central configurations is one or two and give the exact value of the ratio of the masses that provides this bifurcation.     

Central Configurations of the Symmetric Restricted 4-Body Problem

We consider the symmetric planar (3 + 1)-body problem with finite masses m ₁ = m ₂ = 1, m ₃ = µ and one small mass m ₄ = . We count the number of central configurations of the restricted case = 0, where the finite masses remain in an equilateral triangle configuration, by means of the bifurcation diagram with as the parameter. The diagram shows a folding bifurcation at a value consistent with that found numerically by Meyer [9] and it is shown that for small > 0 the bifurcation diagram persists, thus leading to an exact count of central configurations and a folding bifurcation for small m ₄ > 0.     

A two-parameter scheme for the evolution of symmetrical galaxies

In the present paper, a general evolutionary scheme for axisymmetrical rotationally supported equilibrium models for galaxies is considered. Its main phases are: an expansion phase of the initial protogalaxy, assumed to consist into an homogeneous gas sphere structured into clouds, from recombination to maximum expansion, during which it is surmized that angular momentum is acquired by tidal interactions by the expanding configuration; then a violent relaxation collapse phase, following maximum expansion and ending into a virialized deformed polytropic configuration; the reaching of virialization is considered as an adequate initial state for the new phase of virialized contraction of the gaseous component, due to the collisions of the constituent gas clouds, while the stellar component, due to the stars already formed according to a generalized Schmidt-type law during the early expansion and violent relaxation phases, is assumed to have reached a stabilized situation.The initial mean density and radius for both galaxy and component clouds expressed as functions of the density fluctuation spectrum at recombination, act as physical parameters determining the characteristics of the system at maximum expansion, together with the total amount of angular momentum acquired during the expansion phase. The main physical parameters at virialization are then completely specified when the initial distribution of the clouds inside the galaxy is assigned and the constants appearing in it are derived by normalization with the observed data.We find for systems of given mass that the larger the angular momentum per unit mass is: (1) the larger are the equatorial semiaxis at maximum expansion and at virialization and the lower the mean density; (2) the larger is the time elapsed up the maximum expansion and to virialization; while for systems of different mass, we obtain that to the larger mass correspond the larger time elapsed up to maximum expansion and to virialization, and the lower mean density.For the contraction phase following virialization, two limiting cases are considered: (A) either the star component already present at virialization is entirely neglected; (B) or it is thought to contract as the gas component. In such cases, it is found for systems of equal mass that lower angular momenta lead to final configurations characterized by no or small flat gaseous components (which may correspond to lenticulars and early type spirals) while the contrary is true for large angular momenta (corresponding to late type spirals and irregulars). As mass and angular momentum per unit mass decrease, according to an assumed lawj M, the allowed configurations on the late type side of the morphological sequence tend towards earlier and earlier types, until for masses low enough (10¹⁰ m ), only halo type configurations seem to exist. According to this view, the observed lack of spirals with masses below 10¹⁰ m and the wide mass range exibited by the stellar halo type galaxies might be interpreted. In general, it appears that in the limit of the approximations made, a morphological sequence of galaxies can be described by two parameters, mass and angular momentum.     

Rosette Central Configurations, Degenerate Central Configurations and Bifurcations

In this paper we find a class of new degenerate central configurations and bifurcations in the Newtonian n-body problem. In particular we analyze the Rosette central configurations, namely a coplanar configuration where n particles of mass m₁ lie at the vertices of a regular n-gon, n particles of mass m₂ lie at the vertices of another n-gon concentric with the first, but rotated of an angle π /n, and an additional particle of mass m₀ lies at the center of mass of the system. This system admits two mass parameters μ = m₀/m₁ and ε = m₂/m₁. We show that, as μ varies, if n > 3, there is a degenerate central configuration and a bifurcation for every ε > 0, while if n = 3 there is a bifurcation only for some values of ε.     

Long-term motion of resonant satellites with arbitrary eccentricity and inclination

A first-order, semi-analytical method for the long-term motion of resonant satellites is introduced. The method provides long-term solutions, valid for nearly all eccentricities and inclinations, and for all commensurability ratios. The method allows the inclusion of all zonal and tesseral harmonics of a nonspherical planet.We present here an application of the method to a synchronous satellite includingonly theJ ₂ andJ ₂₂ harmonics. Global, long-term solutions for this problem are given for arbitrary values of eccentricity, argument of perigee and inclination.