Linear stability of the triangular equilibrium points in the photogravitational elliptic restricted problem,II

The linear stability of the triangular equilibrium points in the photogravitational elliptic restricted three-body problem is examined and the stability regions are determined in the space of the parameters of mass, eccentricity, and radiation pressure, in the case of equal radiation factors of the two primaries. The full range of values of the common radiation factor is explored, from the gravitational caseq ₁ =q ₂ =q = 1 down to the critical value ofq = 1/8 at which the triangular equilibria disappear by coalescing on the rotating axis of the primaries. It is found that radiation pressure exerts a significant influence on the stability regions. For certain intervals of radiation values these regions become qualitatively different from the gravitational case as well as the solar system case considered in Paper I. There exist values of the common radiation factor, in the range considered, for which the triangular equilibrium points are stable for the entire range of mass distribution among the primaries and for large eccentricities of their orbits.     

The stability of inner collinear equilibrium points in the photogravitational elliptic restricted problem

The linear stability of the inner collinear equilibrium point of the photogravitational elliptic restricted three-body problem is examined and the stability regions are determined in the space of the parameters of mass, eccentricity and radiation pressure. The case of equal radiation factors of the two primaries is considered and the full range of values of the common radiation factor is explored, from the caseq ₁ =q ₂ =q = 1/8 at which the triangular equilibria disappear by coalescing on the rotating axis of the primaries transferring their stability to the collinear point, down toq = 0 at which value the stability regions in theµ - e plane disappear by shrinking down to zero size. It is found that radiation pressure exerts a significant influence on the stability regions. For certain intervals of radiation values these regions become qualitatively different from the gravitational case as well as the solar system case. They evolve as in the case of the triangular equilibrium point considered in a previous paper. There exist values of the common radiation factor, in the range considered, for which the collinear equilibrium point is stable for the entire range of mass distribution among the primaries and for large eccentricities of their orbits.     

Homoclinic and heteroclinic orbits in the photogravitational restricted three-body problem

We study numerically the asymptotic homoclinic and heteroclinic orbits around the hyperbolic Lyapunov periodic orbits which emanate from Euler's critical points L ₁ and L ₂, in the photogravitational restricted plane circular three-body problem. The invariant stable-unstable manifolds associated to these Lyapunov orbits, are also presented. Poincaré surface of sections of these manifolds on appropriate planes and several homoclinic and heteroclinic orbits for the gravitational case as well as for varying radiation factor q ₁, are displayed. Homoclinic-homoclinic and homoclinic-heteroclinic-homoclinic chains which link the interior with the exterior Hill's regions, are illustrated. We adopt the Sun-Jupiter system and assume that only the larger primary radiates. It is found that for small deviations of its value from the gravitational case (q ₁ = 1), the radiation pressure exerts a significant impact on the Hill's regions and on these asymptotic orbits.     

The geometry of the roche coordinates and zero-velocity curves in the photogravitational three-body problem

The aim of the paper is to study the geometry of the Roche curvilinear coordinates (, , ) in the photogravitational circular restricted three-body problem, with varying radiation pressure, and special attention is given to the geometry of zero-velocity curves specified by the coordinate. The radiation pressure exerted by the primary bodies on the infinitesimal third body is considered the same (q ₁ =q ₂), and the primaries are taken to have equal masses (m ₁ =m ₂). The full range of values of the common radiation factor is explored, from the valueq ₁ =q ₂ = 1 (the gravitational three-body problem) down toq ₁ =q ₂ 0. It is found that radiation has a strong influence on the geometry of the Roche coordinates and the zero-velocity curves.     

Asymptotic Orbits at the Triangular Equilibria in the Photogravitational Restricted Three-Body Problem

We study numerically the asymptotic homoclinic and heteroclinic orbits associated with the triangular equilibrium points L ₄ and L ₅, in the gravitational and the photogravitational restricted plane circular three-body problem. The invariant stable-unstable manifolds associated to these critical points, are also presented. Hundreds of asymptotic orbits for equal mass of the primaries and for various values of the radiation pressure are computed and the most interesting of them are illustrated. In the Copenhagen case, which the problem is symmetric with respect to the x- and y-axis, we found and present non-symmetric heteroclinic asymptotic orbits. So pairs of heteroclinic connections (from L ₄ to L ₅ and vice versa) form non-symmetric heteroclinic cycles. The termination orbits (a combination of two asymptotic orbits) of all the simple families of symmetric periodic orbits, in the Copenhagen case, are illustrated.     

On Sitnikov-like motions generating new kinds of 3D periodic orbits in the R3BP with prolate primaries

The existence of new equilibrium points is established in the restricted three-body problem with equal prolate primaries. These are located on the Z-axis above and below the inner Eulerian equilibrium point L ₁ and give rise to a new type of straight-line periodic oscillations, different from the well known Sitnikov motions. Using the stability properties of these oscillations, bifurcation points are found at which new types of families of 3D periodic orbits branch out of the Z-axis consisting of orbits located entirely above or below the orbital plane of the primaries. Several of the bifurcating families are continued numerically and typical member orbits are illustrated.     

Critical symmetric periodic orbits in the photogravitational restricted three-body problem

In this paper, we determine series of horizontally critical symmetric periodic orbits of the six basic families, f,g,h,i,l,m, of the photogravitational restricted three-body problem, and computetheir vertical stability. We restrict our study in the case where only the first primary is radiating, namely q ₁≠1 andq ₂=1. We also compare our results with those of Hénon and Guyot (1970) so as to study the effect of radiation to this kind of orbits. This revised version was published online in July 2006 with corrections to the Cover Date.     

On some new aspects of the photo-gravitational Copenhagen problem

We deal with some new aspects of the photo-gravitational Copenhagen case of the restricted three-body problem; more particularly, the distribution and the attracting domains of the stationary solutions of small particles that move in the neighborhood of two major bodies with equal masses when one or both primaries are radiation sources with constant luminosity. Under these conditions, each particle is subjected not only to gravitational forces but to the radiation emitted from the primaries as well.     

Families of periodic orbits in the restricted four-body problem

In this paper, families of simple symmetric and non-symmetric periodic orbits in the restricted four-body problem are presented. Three bodies of masses m ₁, m ₂ and m ₃ (primaries) lie always at the apices of an equilateral triangle, while each moves in circle about the center of mass of the system fixed at the origin of the coordinate system. A massless fourth body is moving under the Newtonian gravitational attraction of the primaries. The fourth body does not affect the motion of the three bodies. We investigate the evolution of these families and we study their linear stability in three cases, i.e. when the three primary bodies are equal, when two primaries are equal and finally when we have three unequal masses. Series, with respect to the mass m ₃, of critical periodic orbits as well as horizontal and vertical-critical periodic orbits of each family and in any case of the mass parameters are also calculated.     

Periodic orbits in the restricted four-body problem with two equal masses

The restricted (equilateral) four-body problem consists of three bodies of masses m ₁, m ₂ and m ₃ (called primaries) lying in a Lagrangian configuration of the three-body problem i.e., they remain fixed at the apices of an equilateral triangle in a rotating coordinate system. A massless fourth body moves under the Newtonian gravitation law due to the three primaries; as in the restricted three-body problem (R3BP), the fourth mass does not affect the motion of the three primaries. In this paper we explore symmetric periodic orbits of the restricted four-body problem (R4BP) for the case of two equal masses where they satisfy approximately the Routh’s critical value. We will classify them in nine families of periodic orbits. We offer an exhaustive study of each family and the stability of each of them.     

The equilibrium solutions of the restricted problem of three rigid bodies with variable mass

In this paper we consider the circular planar restricted problem of three rigid bodiesS _i(i=1, 2, 3), two of them are axisymmetric ellipsoids and a third bodyS ₃ is a spherical satellite with decreasing mass, under the gravitational forces. The effect of small perturbations in the Coriolis force and the centrifugal forces on the location of equilibrium points has been studied. It is found only in the case when the primaries have equal differences between their respective principal moments of inertial the pointsL ₄ andL ₅ form nearly equilateral tringles with the primaries. The equilibrium pointsL ₁,L ₂,L ₃ remain collinear an ies on the line joining the primaries.     

Equilibrium points in the photogravitational restricted four-body problem

We study numerically the photogravitational version of the problem of four bodies, where an infinitesimal particle is moving under the Newtonian gravitational attraction of three bodies which are finite, moving in circles around their center of mass fixed at the origin of the coordinate system, according to the solution of Lagrange where they are always at the vertices of an equilateral triangle. The fourth body does not affect the motion of the three bodies (primaries). We consider that the primary body m ₁ is dominant and is a source of radiation while the other two small primaries m ₂ and m ₃ are equal. In this case (photogravitational) we examine the linear stability of the Lagrange triangle solution. The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves, as well as the positions of the equilibrium points on the orbital plane are given. The existence and the number of the collinear and the non-collinear equilibrium points of the problem depends on the mass parameters of the primaries and the radiation factor q ₁. Critical masses m ₃ and radiation q ₁ associated with the existence and the number of the equilibrium points are given. The stability of the relative equilibrium solutions in all cases are also studied. In the last section we investigate the existence and location of the out of orbital plane equilibrium points of the problem. We found that such critical points exist. These points lie in the (x,z) plane in symmetrical positions with respect to (x,y) plane. The stability of these points are also examined.     

The effect of solar radiation pressure on the Lagrangian points in the elliptic restricted three-body problem

In this paper the effect of solar radiation pressure on the location and stability of the five Lagrangian points is studied, within the frame of elliptic restricted three-body problem, where the primaries are the Sun and Jupiter acting on a particle of negligible mass. We found that the radiation pressure plays the rule of slightly reducing the effective mass of the Sun and changes the location of the Lagrangian points. New formulas for the location of the collinear libration points were derived. For large values of the force ratio β, we found that at β=0.12, the collinear point L₃ is stable and some families of periodic orbits can be drawn around it.     

Combined effect of oblateness,radiation and a circular cluster of material points on the stability of triangular liberation points in the R3BP

This paper studies the motion of an infinitesimal mass in the framework of the restricted three-body problem (R3BP) under the assumption that the primaries of the system are radiating-oblate spheroids, enclosed by a circular cluster of material points. It examines the effects of radiation and oblateness up to J ₄ of the primaries and the potential created by the circular cluster, on the linear stability of the liberation locations of the infinitesimal mass. The liberation points are found to be stable for 0<μ<μ _c and unstable for $\mu_{c}\le\mu\le\frac{1}{2}$ , where μ _c is the critical mass value depending on terms which involve parameters that characterize the oblateness, radiation forces and the circular cluster of material points. The oblateness up to J ₄ of the primaries and the gravitational potential from the circular cluster of material points have stabilizing propensities, while the radiation of the primaries and the oblateness up to J ₂ of the primaries have destabilizing tendencies. The combined effect of these perturbations on the stability of the triangular liberation points is that, it has stabilizing propensity.     

Periodic motions around a collinear equilibrium solution of the general three-body problem

The collinear equilibrium position of the circular restricted problem with the two primaries at unit distance and the massless body at the pointL ₃ is extended to the planar three-body problem with respect to the massm ₃ of the third body; the mass ratio μ of the two primaries is considered constant and the constant angular velocity of the straight line on which the three masses stay at rest is taken equal to 1. As regards periodic motions ‘around’ the equilibrium pointL ₃, four possible extensions from the restricted to the general problem are presented each of them starting with a simple or a doubly periodic orbit of the family α of the Copenhagen category (μ=0.50). Form ₃=0.10, μ=0.50 (i.e. for fixed masses of all three bodies) the characteristic curve of the extended family α is found. The qualitative differences of the families corresponding tom ₃=0 andm ₃=0.10 are discussed.     

The mass-ratio distribution of spectroscopic binary stars

In order to determine the mass-ratio distribution of spectroscopic binary stars, the selection effects that govern the observations of this class of binary systems are investigated. The selection effects are modelled numerically and analytically. The results of the models are compared to the data inThe Eighth Catalogue of the Orbital Elements of Spectroscopic Binary Stars (DAO8) compiled by Battenet al. (1989). The investigations involve binary systems with Main-Sequence primary components only, in order to avoid confusion of evolutionary and selection effects.For single-lined spectroscopic binaries (SBI) it is found that the mass ratios (q=M _sec/M _prim) in general adhere to a distribution _q q ^-2 forq>q ₀, withq ₀=0.3. The observations are consistent with a distribution that is flat forq<q ₀. The turn-over value varies fromq ₀=0.3 for systems with B-type primaries, toq ₀=0.55 for systems with K-type primaries. The semi-major axesa ₁ are distributed according to _a (a ₁)a ₁ ^-a with an average value of _a =1.3. The power varies from _a =1.7 for systems with B-type primaries to _a =0 for systems with K-type primaries. The eccentricitiese of the orbits of SBI systems are distributed according to _e (e)e ^-1.For double-lined spectroscopic binary stars (SBII) it is found that the shape of theq-distribution, as derived from observations, is almost entirely determined by selection effects. It is shown that the distribution is compatible with theq-distribution found for SBI systems. A sub-sample, consisting of the SBII systems from DAO8 with magnitudesm _V 5 ^m , is less hampered by selection effects, and shows the same shape of theq-distribution as the SBI systems, at theq-interval (0.67, 1).It is estimated that 19–45% of the stars in the solar neighbourhood are spectroscopic binary systems.     

Stability of the photogravitational restricted three-body problem with variable masses

This paper investigates the stability of equilibrium points in the restricted three-body problem, in which the masses of the luminous primaries vary isotropically in accordance with the unified Meshcherskii law, and their motion takes place within the framework of the Gylden–Meshcherskii problem. For the autonomized system, it is found that collinear and coplanar points are unstable, while the triangular points are conditionally stable. It is also observed that, in the triangular case, the presence of a constant κ, of a particular integral of the Gylden–Meshcherskii problem, makes the destabilizing tendency of the radiation pressures strong. The stability of equilibrium points varying with time is tested using the Lyapunov Characteristic Numbers (LCN). It is seen that the range of stability or instability depends on the parameter κ. The motion around the equilibrium points L _i (i=1,2,…,7) for the restricted three-body problem with variable masses is in general unstable.     

The manifolds of families of 3D periodic orbits associated to Sitnikov motions in the restricted three-body problem

This paper deals with the Sitnikov family of straight-line motions of the circular restricted three-body problem, viewed as generator of families of three-dimensional periodic orbits. We study the linear stability of the family, determine several new critical orbits at which families of three dimensional periodic orbits of the same or double period bifurcate and present an extensive numerical exploration of the bifurcating families. In the case of the same period bifurcations, 44 families are determined. All these families are computed for equal as well as for nearly equal primaries (μ = 0.5, μ = 0.4995). Some of the bifurcating families are determined for all values of the mass parameter μ for which they exist. Examples of families of three dimensional periodic orbits bifurcating from the Sitnikov family at double period bifurcations are also given. These are the only families of three-dimensional periodic orbits presented in the paper which do not terminate with coplanar orbits and some of them contain stable parts. By contrast, all families bifurcating at single-period bifurcations consist entirely of unstable orbits and terminate with coplanar orbits.     

Stable Manifolds and Homoclinic Points Near Resonances in the Restricted Three-Body Problem

The restricted three-body problem describes the motion of a massless particle under the influence of two primaries of masses 1− μ and μ that circle each other with period equal to 2π. For small μ, a resonant periodic motion of the massless particle in the rotating frame can be described by relatively prime integers p and q, if its period around the heavier primary is approximately 2π p/q, and by its approximate eccentricity e. We give a method for the formal development of the stable and unstable manifolds associated with these resonant motions. We prove the validity of this formal development and the existence of homoclinic points in the resonant region. In the study of the Kirkwood gaps in the asteroid belt, the separatrices of the averaged equations of the restricted three-body problem are commonly used to derive analytical approximations to the boundaries of the resonances. We use the unaveraged equations to find values of asteroid eccentricity below which these approximations will not hold for the Kirkwood gaps with q/p equal to 2/1, 7/3, 5/2, 3/1, and 4/1. Another application is to the existence of asymmetric librations in the exterior resonances. We give values of asteroid eccentricity below which asymmetric librations will not exist for the 1/7, 1/6, 1/5, 1/4, 1/3, and 1/2 resonances for any μ however small. But if the eccentricity exceeds these thresholds, asymmetric librations will exist for μ small enough in the unaveraged restricted three-body problem.     

On the stability of triangular points in the elliptic R3BP under radiating and oblate primaries

This paper investigates the stability of triangular equilibrium points (L _4,5) in the elliptic restricted three-body problem (ER3BP), when both oblate primaries emit light energy simultaneously. The positions of the triangular points are seen to shift away from the line joining the primaries than in the classical case on account of the introduction of the eccentricity, semi-major axis, radiation and oblateness factors of both primaries. This is shown for the binary systems Achird, Luyten 726-8, Kruger 60, Alpha Centauri AB and Xi Bootis. We found that motion around these points is conditionally stable with respect to the parameters involved in the system dynamics. The region of stability increases and decreases with variability in eccentricity, oblateness and radiation pressures.