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.     

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.     

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.     

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.     

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.     

Satellite de-orbiting via controlled solar radiation pressure

We analyze the families of central configurations of the spatial 5-body problem with four masses equal to 1 when the fifth mass m varies from 0 to \(+\infty \). In particular we continue numerically, taking m as a parameter, the central configurations (which all are symmetric) of the restricted spatial (\(4+1\))-body problem with four equal masses and \(m=0\) to the spatial 5-body problem with equal masses (i.e. \(m=1\)), and viceversa we continue the symmetric central configurations of the spatial 5-body problem with five equal masses to the restricted (\(4+1\))-body problem with four equal masses. Additionally we continue numerically the symmetric central configurations of the spatial 5-body problem with four equal masses starting with \(m=1\) and ending in \(m=+\infty \), improving the results of Alvarez-Ramírez et al. (Discrete Contin Dyn Syst Ser S 1: 505–518, 2008). We find four bifurcation values of m where the number of central configuration changes. We note that the central configurations of all continued families varying m from 0 to \(+\infty \) are symmetric.     

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.     

Global Bifurcations of Periodic Solutions of the Restricted Three Body Problem

We describe global bifurcations from the libration points of non-stationary periodic solutions of the restricted three body problem. We show that the only admissible continua of non-stationary periodic solutions of the planar restricted three body problem, bifurcating from the libration points, can be the short-period families bifurcating from the Lagrange equilibria L ₄, L ₅. We classify admissible continua and show that there are possible exactly six admissible continua of non-stationary periodic solutions of the planar restricted three body problem. We also characterize admissible continua of non-stationary periodic solutions of the spatial restricted three body problem. Moreover, we combine our results with the Déprit and Henrard conjectures (see [8]), concerning families of periodic solutions of the planar restricted three body problem, and show that they can be formulated in a stronger way. As the main tool we use degree theory for SO(2)-equivariant gradient maps defined by the second author in [25].This revised version was published online in October 2005 with corrections to the Cover Date.     

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.     

Effect of perturbations in Coriolis and centrifugal forces on the stability of libration points in the restricted problem

The location and the stability of the libration points in the restricted problem have been studied when small perturbation and are given to the Coriolis and the centrifugal forces respectively. It is seen that the pointsL ₄ andL ₅ form nearly equilateral triangles with the primaries and the pointsL ₁,L ₂,L ₃ remain collinear. It is further observed that for the pointsL ₄ andL ₅, the range of stability increases or decreases depending upon whether the point (, ) lies in one or the other of the two parts in which the (, ) plane is divided by the line 36-19=0 and the stability of the collinear points is not influenced by the perturbations and they remain unstable.     

Periodic solutions of circular-elliptic type in the planarN-body problem

Letn2 mass points with arbitrary masses move circularly on a rotating straight-line central-configuration; i.e. on a particular solution of relative equilibrium of then-body problem. Replacing one of the mass points by a close pair of mass points (with mass conservation) we show that the resultingN-body problem (N=n+1) has solutions, which are periodic in a rotating coordinate system and describe precessing nearlyelliptic motion of the binary and nearlycircular collinear motion of its center of mass and the other bodies; assuming that also the mass ratio of the binary is small.     

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.     

Robe’s restricted problem of 2+2 bodies with one of the primaries an oblate body

In this problem, one of the primaries of mass \(m^{*}_{1}\) is a rigid spherical shell filled with a homogeneous incompressible fluid of density ρ ₁. The smaller primary of mass m ₂ is an oblate body outside the shell. The third and the fourth bodies (of mass m ₃ and m ₄ respectively) are small solid spheres of density ρ ₃ and ρ ₄ respectively inside the shell, with the assumption that the mass and the radius of the third and the fourth body are infinitesimal. We assume that m ₂ is describing a circle around \(m^{*}_{1}\) . The masses m ₃ and m ₄ mutually attract each other, do not influence the motions of \(m^{*}_{1}\) and m ₂ but are influenced by them. We also assume that masses m ₃ and m ₄ are moving in the plane of motion of mass m ₂. In the paper, equilibrium solutions of m ₃ and m ₄ and their linear stability are analyzed. There are two collinear equilibrium solutions for the system. The non collinear equilibrium solutions exist only when ρ ₃=ρ ₄. There exist an infinite number of non collinear equilibrium solutions of the system, provided they lie inside the spherical shell. In a system where the primaries are considered as earth-moon and m ₃,m ₄ as submarines, the collinear equilibrium solutions thus obtained are unstable for the mass parameters μ,μ ₃,μ ₄ and oblateness factor A. In this particular case there are no non-collinear equilibrium solutions of the system.     

A case of commensurability induced by oblateness

In the three-dimensional restricted three-body problem, it is known that there exists a near one-to-one commensurability ratio between the planar angular frequencies (s _{1, 2, 3}) and the corresponding angular frequency (S ₂) in thez-direction at the three collinear equilibria (L _{1, 2, 3}), which is significant for small and practically important values of the mass parameter (). When the more massive primary is treated as an oblate spheroid with its equatorial plane coincident with the plane of motion of the primaries, it is established that oblateness induces a one-to-one commensurability at the exterior pointL ₃ (to the right of the more massive primary) and at the interior pointL ₂ for 01/2 and that atL ₁ no such commensurability exists. However, the values of the oblateness coefficient (A ₁) involved atL ₂ are too high to have any practical significance, while those atL ₃ being small for small values of may be useful for generating periodic orbits of the third kind.     

Theorie litterale du neuvieme satellite de Saturne,Phoebe

We study a theory for the ninth satellite of Saturn, Phoebe, based on the literal solution we have obtained in the main problem of the lunar theory.These series were computed by solving, by successive approximations, the Lagrange's equations expressed in variables, functions of the elliptic elements.We may consider the case of Phoebe simpler than a lunar case because we seek less precision (1/10 geocentric) than in the Lunar case, although the eccentricity of Phoebe is stronger.Main problem: our series are computed to the complete seventh order and a great part of the perturbations of the eighth and ninth order, where we have attributed to the small lunar parameters the order 1 tom ₀=n/n ₀,e ₀,e, sin (i ₀/2), the order 2 to ₀=(a ₀/a)((M ₁–)/(M ₁+M)) and the order 4 toµ ₀(a ₀/a)M ₁ M/M ₁ ² –M ².In the case of Phoebe,µ ₀ equal zero and ±₀ is the ratioa ₀/a.We study the further development of these series by using, instead of parameterm ₀, the quantity m ₀=n/n ₀–m ₁ wherem ₁ is an approached value ofm ₀, in order to accelerate the convergence of the series with respect tom ₀.Comparison with a numerical integration we are adjusting a numerical integration to the observations. We have already more than 100 observations, for the period 1900–1957.At first, we compare the series of the main problem to a numerical integration of the Keplerian problem.

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Proceedings of the Conference on Analytical Methods and Ephemerides: Theory and Observations of the Moon and Planets. Facultés universitaires Notre Dame de la Paix. Namur, Belgium, 28–31 July, 1980.     

A family of stacked central configurations in the planar five-body problem

We study planar central configurations of the five-body problem where three bodies, \(m_1, m_2\) and \(m_3\), are collinear and ordered from left to right, while the other two, \(m_4\) and \(m_5\), are placed symmetrically with respect to the line containing the three collinear bodies. We prove that when the collinear bodies form an Euler central configuration of the three-body problem with \(m_1=m_3\), there exists a new family, missed by Gidea and Llibre (Celest Mech Dyn Astron 106:89–107, 2010), of stacked five-body central configuration where the segments \(m_4m_5\) and \(m_1m_3\) do not intersect.     

Inverse problem of central configurations and singular curve in the collinear 4-body problem

In this paper, we consider the inverse problem of central configurations of n-body problem. For a given \({q=(q_1, q_2, \ldots, q_n)\in ({\bf R}^d)^n}\), let S(q) be the admissible set of masses denoted \({ S(q)=\{ m=(m_1,m_2, \ldots, m_n)| m_i \in {\bf R}^+, q}\) is a central configuration for m}. For a given \({m\in S(q)}\), let S _m (q) be the permutational admissible set about m = (m ₁, m ₂, . . . , m _n ) denoted

$S_m(q)=\{m^\prime | m^\prime\in S(q),m^\prime \not=m \, {\rm and} \, m^\prime\,{\rm is\, a\, permutation\, of }\, m \}.$

The main discovery in this paper is the existence of a singular curve \({\bar{\Gamma}_{31}}\) on which S _m (q) is a nonempty set for some m in the collinear four-body problem. \({\bar{\Gamma}_{31}}\) is explicitly constructed by a polynomial in two variables. We proved:

1. (1)
   If \({m\in S(q)}\), then either ^# S _m (q) = 0 or ^# S _m (q) = 1.
    
2. (2)
   ^#S _m (q) = 1 only in the following cases:

   1. (i)
      If s = t, then S _m (q) = {(m ₄, m ₃, m ₂, m ₁)}.
       
   2. (ii)
      If \({(s,t)\in \bar{\Gamma}_{31}\setminus \{(\bar{s},\bar{s})\}}\), then either S _m (q) = {(m ₂, m ₄, m ₁, m ₃)} or S _m (q) = {(m ₃, m ₁, m ₄, m ₂)}.
       
    

     

Parameter values for stable low-inclination periodic motion in the restricted three-body problem with oblateness

The known intervals of possible stability, on the mgr-axis, of basicfamilies of 3D periodic orbits in the restricted three-body problem areextended into -A₁ regions for oblate larger primary, A ₁ beingthe oblateness coefficient. Eight regions, corresponding to the basicstable bifurcation orbits l1v, l1v, l2v, l3v, m1v, m1v,m2v, i1v are determined and related branching 3D periodic orbits arecomputed systematically and tested for stability. The regions for l1v,m1v and m2v survive the test emerging as the regions allowing thesimplest types of stable low inclination 3D motion. For l1v, l2v,l3v, m1v and m2v oblateness seems to have a stabilising effect,while stability of i1v survives only for a very small range of A ₁values.     

The critical and the saturation content of magnetic monopoles in rotating relativistic objects

Both the critical content _c ( N _m /N _B , whereN _m ,N _B are the total numbers of monopoles and nucleons, respectively, contained in the object), and the saturation content _s of monopoles in a rotating relativistic object are found in this paper. The results are:

Co-Orbital Motion with Slowly Varying Parameters

We consider the dynamics of a test particle co-orbital with a satellite of mass m _s which revolves around a planet of mass M ₀ m _s with a mean motion n _s and semi-major axis a _s. We study the long term evolution of the particle motion under slow variations of (1) the mass of the primary, M ₀, (2) the mass of the satellite, m _s and (3) the specific angular momentum of the satellite J _s. The particle is not restricted to small harmonic oscillations near L ₄ or L ₅, and may have any libration amplitude on tadpole or horseshoe orbits. In a first step, no torque is applied to the particle, so that its motion is described by a Hamiltonian with slowly varying parameters. We show that the torque applied to the satellite, as measured by _s = j_s/(n _s J _s) induces an distortion of the phase space which is entirely described by an asymmetry coefficient = _s/, where = m _s/M. The adiabatic invariance of action implies furthermore that the long term evolution of the particle co-orbital motion depends only on the variation of m _s a _s with time. Applying a constant torque to the particle, as measured by _s = j_s/(n _s J _p) is then merely equivalent to replacing = _s/ by = (_s – _p)/. However, if the torque acting on the particle exhibits a radial gradient, then the action is no more conserved and the evolution of the particle orbit is no more controlled by m _s a _s only. We show that even mild torque gradients can dominate the orbital evolution of the particle, and eventually decide whether the latter will be pulled towards the stable equilibrium points L ₄ or L ₅, or driven away from them. Finally, we show that when the co-orbital bodies are two satellites with comparable masses m ₁ and m ₂, we can reduce the problem to that of a test particle co-orbital with a satellite of mass m ₁ + m ₂. This new problem has then parameters varying at rates which are combinations, with appropriate coefficients, of the changes suffered by each satellite.