Effect of peturbations on the location of equilibrium points in the restricted problem of three bodies with variable mass

The effect of small perturbations in the coriolis and the centrifugal forces on the location of equilibrium points in the restricted problems of three bodies with variable mass has been studied. It is found that the points L₄ and L₅ form nearly equilateral triangles with the primaries and the points L₁, L₂, L₃ remain collinear and lie on the line joining the primaries.     

Bifurcation points and intersections of families of periodic orbits in the three-dimensional restricted three-body problem

Intersections of families of three-dimensional periodic orbits which define bifurcation points are studied. The existence conditions for bifurcation points are discussed and an algorithm for the numerical continuation of such points is developed. Two sequences of bifurcation points are given concerning the family of periodic orbits which starts and terminates at the triangular equilibrium pointsL ₄,L ₅. On these sequences two trifurcation points are identified forµ = 0.124214 andµ = 0.399335. The caseµ = 0.5 is studied in particular and it is found that the space families originating at the equilibrium pointsL ₂,L ₃,L ₄,L ₅ terminate on the same planar orbitm _1v of the familym.     

Effect of Stellar Wind and Poynting–Robertson Drag on Photogravitational Elliptic Restricted Three Body Problem

The existence and linear stability of the planar equilibrium points for photogravitational elliptical restricted three body problem is investigated in this paper. Assuming that the primaries, one of which is radiating are rotating in an elliptical orbit around their common center of mass. The effect of the radiation pressure, forces due to stellar wind and Poynting–Robertson drag on the dust particles are considered. The location of the five equilibrium points are found using analytical methods. It is observed that the collinear equilibrium points L₁, L₂ and L₃ do not lie on the line joining the primaries but are shifted along the y-coordinate. The instability of the libration points due to the presence of the drag forces is demonstrated by Lyapunov’s first method of stability.     

Effect of oblateness, perturbations, radiation and varying masses on the stability of equilibrium points in the restricted three-body problem

This paper investigates the combined effect of small perturbations ε,ε′ in the Coriolis and centrifugal forces, radiation pressure q _i , and changing oblateness of the primaries A _i (t) (i=1,2) on the stability of equilibrium points in the restricted three body problem in which the primaries is a supergiant eclipsing binary system which consists of a pair of bright oblate stars having the appearance of a giant peanut in space and their masses assumed to vary with time in the absence of reactive forces. The equations of motion are derived and the equilibrium points are obtained. For the autonomized system, it is seen that there are more than a pair of the triangular points as κ→∞; κ being the arbitrary sum of the masses of the primaries. In the case of the collinear points, two additional equilibrium points exist on the line joining the primaries when simultaneously κ+ε′<0 and both primaries are oblate, i.e., 0<α _i ?1. So there are five collinear equilibrium points in this case. Two non-planar equilibrium points exist for κ>1. Hence, there are at least nine equilibrium points of the system. The stability of these points is explored analytically and numerically. It is seen that the collinear and triangular points are stable with respect to certain conditions controlled by κ while the non-planar equilibrium points are unstable.     

Integration theory for the restricted problem of three axisymmetric bodies

In this paper the circular planar restricted problem of three axisymmetric ellipsoids S _i (i = 1, 2, 3), such that their equatorial planes coincide with the orbital plane of the three centres of masses, be considered. The equations of motion of infinitesimal body S ₃ be obtained in the polar coordinates. Using iteration approach we have given an approximation for another integral, which independent of the Jacobian integral, in the case of P-type orbits (near circular orbits surrounding both primaries).     

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.     

Motion in the generalized restricted three-body problem

This paper investigates the motion of an infinitesimal body in the generalized restricted three-body problem. It is generalized in the sense that both primaries are radiating, oblate bodies, together with the effect of gravitational potential from a belt. It derives equations of the motion, locates positions of the equilibrium points and examines their linear stability. It has been found that, in addition to the usual five equilibrium points, there appear two new collinear points L _n1, L _n2 due to the potential from the belt, and in the presence of all these perturbations, the equilibrium points L ₁, L ₃ come nearer to the primaries; while L ₂, L ₄, L ₅, L _n1 move towards the less massive primary and L _n2 moves away from it. The collinear equilibrium points remain unstable, while the triangular points are stable for 0<μ<μ _c and unstable for $\mu_{c} \le\mu\le\frac{1}{2}$ , where μ _c is the critical mass ratio influenced by the oblateness and radiation of the primaries and potential from the belt, all of which have destabilizing tendency. A practical application of this model could be the study of the motion of a dust particle near the oblate, radiating binary stars systems surrounded by a belt.     

Problème restreint des trois corps appliqué à un gyrostat

Résumé On étudie le mouvement d'un corpsS ₃ supposé non ponctuel, attiré par deux corps sphériques homogènes dont les masses sont prépondérantes vis à vis de la masse deS ₃. Le corpsS ₃ est muni de rotors et on recherche les cas d'équilibres apparents de ce corps lorsque son centre d'inertie occupe l'une des positions de LagrangeL ₁, ...,L ₅ du cas ponctuel. Des conditions suffisantes de stabilité de certaines solutions particulières sont obtenues.

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This paper is concerned with an extension of the classical restricted problem of three bodies when the smallest body is not considered to be a point mass. We assume that the smallest body consists of a solid hub and symetric rotors rotating at constant relative angular velocities. The mass center of the gyrostat satellite is presumed to occupy one of five librations pointsL ₁, ...,L ₅ of the classical restricted problem of three bodies. Assuming that the gyrostatic moment can have arbitrary constant values, we find the set of positions of relative equilibrium of the gyrostat satellite. We then proceed to define the domains of stability and instability.

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Un sujet proche du problème traité ici a été étudié par V. V. Rumyantsev (1974b).     

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.     

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.     

Three-dimensional periodic oscillations generating from plane periodic ones around the collinear Lagrangian points

The three families of three-dimensional periodic oscillations which include the infinitesimal periodic oscillations about the Lagrangian equilibrium pointsL ₁,L ₂ andL ₃ are computed for the value =0.00095 (Sun-Jupiter case) of the mass parameter. From the first two vertically critical (|a _v |=1) members of the familiesa, b andc, six families of periodic orbits in three dimensions are found to bifurcate. These families are presented here together with their stability characteristics. The orbits of the nine families computed are of all types of symmetryA, B andC. Finally, examples of bifurcations between families of three-dimensional periodic solutions of different type of symmetry are given.     

Existence of equilibrium points and their linear stability in the generalized photogravitational Chermnykh-like problem with power-law profile

We consider the modified restricted three body problem with power-law density profile of disk, which rotates around the center of mass of the system with perturbed mean motion. Using analytical and numerical methods, we have found equilibrium points and examined their linear stability. We have also found the zero velocity surface for the present model. In addition to five equilibrium points there exists a new equilibrium point on the line joining the two primaries. It is found that L ₁ and L ₃ are stable for some values of inner and outer radius of the disk while other collinear points are unstable, but L ₄ is conditionally stable for mass ratio less than that of Routh’s critical value. Lastly, we have studied the effects of radiation pressure, oblateness and mass of the disk on the motion and stability of equilibrium points.     

Non-symmetric doubly-asymptotic orbits at the outer collinear equilibrium pointL 3

In the framework of the solar system case (with only the larger primary radiating) of the photogravitational restricted three-body problem we compute and present some non-symmetric asymptotic orbits connecting the outer collinear equilibrium pointL ₃ with the neighbourhood of one of the triangular equilibrium pointsL _{4, 5}. Such orbits have not been found previously in the restricted problem.     

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 effect of radiation pressure on the equilibrium points in the generalized photogravitational restricted three body problem

The existence of equilibrium points and the effect of radiation pressure have been discussed numerically. The problem is generalized by considering bigger primary as a source of radiation and small primary as an oblate spheroid. We have also discussed the Poynting-Robertson (P-R) effect which is caused due to radiation pressure. It is found that the collinear points L ₁,L ₂,L ₃ deviate from the axis joining the two primaries, while the triangular points L ₄,L ₅ are not symmetrical due to radiation pressure. We have seen that L ₁,L ₂,L ₃ are linearly unstable while L ₄,L ₅ are conditionally stable in the sense of Lyapunov when P-R effect is not considered. We have found that the effect of radiation pressure reduces the linear stability zones while P-R effect induces an instability in the sense of Lyapunov.     

Stability of equilibrium points in a CR3BP with oblate bodies enclosed by a circular cluster of material points

We have investigated an improved version of the classic restricted three-body problem where both primaries are considered oblate and are enclosed by a homogeneous circular planar cluster of material points centered at the mass center of the system. In this dynamical model we have examined the effect on the number and on the linear stability of the equilibrium locations of the small particle due to both, the primaries’ oblateness and the potential created by the circular cluster. We have drawn the zero-velocity surfaces and we have found that in addition to the usual five Lagrangian equilibrium points of the classic restricted three-body problem, there exist two new collinear points L _n1,L _n2 due to the potential from the circular cluster of material points. Numerical investigations reveal that with the increase in the mass of the circular cluster of material points, L _n2 comes nearer to the more massive primary, while L _n1 moves away from it. Owing to oblateness of the bodies, L _n1 comes nearer to the more massive primary, while L _n2 moves towards the less massive primary. The collinear equilibrium points remain unstable, while the triangular points are stable for 0<μ<μ _c and unstable for $\mu_{c} \le \mu \le \frac{1}{2}$ , where μ _c is the critical mass ratio influenced by oblateness of the primaries and the potential from the circular cluster of material points. The oblateness and the circular cluster of material points have destabilizing tendency.     

2D calculations of the collapse of magnetized gas cloud

All the families of planar symmetric simple-periodic orbits of the photogravitational restricted plane circular three-body problem, are determined numerically in the case when the primaries are of equal mass and radiate with equal radiation factors (q ₁=q₂=q). We obtain a global view of the possible patterns of periodic three-body motion while the full range of values of the common radiation factor is explored, from the gravitational case (q=1) down to near the critical value at which the triangular equilibria disappear by coalescing with the inner equilibrium pointL ₁ on the rotating axis of the primaries. It is found that for large deviations of its value from the gravitational case the radiation factorq can have a strong effect on the structure of the families.     

Motion around the out-of-plane equilibrium points of the perturbed restricted three-body problem

This paper studies the motion of an infinitesimal body near the out-of-plane equilibrium points, L _6,7, in the perturbed restricted three-body problem. The problem is perturbed in the sense that the primaries of the system are oblate spheroids as well as sources of radiation and small perturbations are give to the Coriolis and centrifugal forces. It locates the positions and examines the stability of L _6,7 with a particular application to the binary system Struve 2398. It is observed that their positions are affected by the radiation, oblateness and a small perturbation in the centrifugal force, but is unaffected by that of the Coriolis force. They are also found to be unstable.     

Effect of radiation on the stability of equilibrium points in the binary stellar systems: RW-Monocerotis,Krüger 60

We have studied the stability of location of various equilibrium points of a passive micron size particle in the field of radiating binary stellar system within the framework of circular restricted three body problem. Influence of radial radiation pressure and Poynting-Robertson drag (PR-drag) on the equilibrium points and their stability in the binary stellar systems RW-Monocerotis and Krüger-60 has been studied. It is shown that both collinear and off axis equilibrium points are linearly unstable for increasing value of β ₁ (ratio of radiation to gravitational force of the massive component) in presence of PR-drag for the binary systems. Further we find that out of plane equilibrium points (L _i , i=6,7) may exists for range of values of β ₁>1 for these binary systems in the presence of PR-drag. Our linear stability analysis shows that the motion near the equilibrium points L _6,7 of the binary systems is unstable both in the absence and presence of PR-drag.     

A variable mass of the primaries and librations around the triangular points

The delimitations of the librational motion around the Lagrangian triangular pointsL ₄,L ₅ are investigated within the framework of the restricted circular three body problem according to Brown's and Thüring's theory. The isotropic mass variation of the primaries does not exceed the order of the small primary and the derivatives of the masses with respect to the time are negligible second order quantities. The amplitude of the maximum elongations with respect to the small primary remains unchanged. The expression for the maximum variation of the distance of the particle from the large mass has the same form as in the classical problem with constant masses.