Photogravitational Restricted Three-body Problem Research Articles

Bifurcations of plane to 3D periodic orbits in the photogravitational restricted three-body problem

We consider bifurcation of 3D periodic orbits from the plane ofmotion of the primaries in the photogravitational restricted three-bodyproblem. The simplest periodic 3D orbits branch from the plane periodicorbits of indifferent vertical stability. We compute the first few suchorbits of the basic families l, m, i, h, a, b, c forvarying mass parameter and for varying radiation coefficient of thelarger primary. The horizontal stability of the orbits is also computedleading to predictions about possible stability of the 3D orbits.

The Restricted Three -Body Problem. Comments on the `Spatial' Equilibrium Points

In the photogravitational restricted three-body problem, the role of theradiation-pressure is reviewed. By the analytical considerations, the existence of the `spatial' equilibrium points is analysed. In order to avoid the unknown parameters of the infinitesimal body, for the two component stars an analytical function β2 =f(β1 ) is established.In contrast to results given by other authors, here only one pair of suchpoints is found. A numerical simulation for RW Monocerotis is undertakenand it is found L6(−0.055; 0; +1.07) and L7(−0.055; 0; −1.07).

The stability of triangular libration points in the photogravitational three-body problem

The photogravitational restricted three-body problem in which a passively gravitating point, in addition to the Newtonian force of gravitation the main bodies, also experiences the force of light pressure from each of them is considered. This problem provides a fairly adequate model, for example, of the motion of a particle of a gas-dust cloud which is in the field of two gravitating and radiating stars. In the case of the elliptic restricted problem when the main bodies are rotating about one another in elliptic orbits, the existence of a family of positions of relative equilibrium is established which are analogous to the Lagrangian libration points of the classical restricted three-body problem. The necessary conditions for the orbital stability of the triangular libration points which have been found are derived using the linearized equations of motion. It is shown that, in the configuration space of the system, the stability domain has a fairly simple geometrical meaning in the circular version of the problem. Conditions for the existence of parametric resonance, which lead to instability in the elliptic version of the problem, are established for small values of the eccentricity.

A NOTE ON PARAMETER VALUES FOR STABLE LOW-INCLINATION PERIODIC MOTION IN THE RESTRICTED THREE-BODY PROBLEM WITH RADIATION

The intervals of the mass parameter (µ) values for possible stability of the basic families of 3D periodic orbits in the restricted three-body problem determined elsewhere are now extended into regions of theµ - q1 parameter space of the photogravitational restricted three-body problem, where q1 is the radiation factor of m1 and it is assumed that m2 does not radiate. Several 3D periodic orbits corresponding to these regions are computed and tested for stability and seven regions, corresponding to the vertical-critical orbits l1v, l'1v, l6v, m1v, m2v and i1v, survive this stability test, emerging as the regions allowing the simplest types of stable low inclination 3D motion of the infinitesimal particle.

Families of periodic orbits in the photogravitational three-body problem

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 (q1=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 pointL1 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.

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 (ξ, η, ζ) 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 theξ 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.

The libration points in photogravitational restricted three-body problem

The photogravitational restricted three-body problem in which the mass reduction factors of two primaries q1, q2 e −∞,1] are studied and an analytic method to estimate the number of libration points and to calculate their location is given in this paper. The results show that in photogravitational restricted three-body problem, the number of libration points is from one to seven for different q1 and q2. As application, the motion of dust grain like comet tail in the solar system is also discussed.

The photogravitational restricted three-body problem

The photogravitational restricted three-body problem is reviewed and the case of the “out-of-plane” equilibrium points is analysed. It is found that, when the motion of an infinitesimal body is determined only by the gravitational forces and effects of the radiation pressure, there are no “out-of-plane” stable equilibrium points.

On the existence of the ?out of plane? equilibrium points in the photogravitational restricted three-body problem

The existence of equilibrium points laying out of the plane of the two primaries in the photogravitational restricted three-body problem is discussed. It is verified that such points do exist.

Stability of the Triangular Lagrangian Solutions of the Photo Gravitational Restricted Three-Body Problem in the Three-Dimensional Case

AbstractThe stability of the triangular Lagrangian solutions for the photo-gravitational restricted three-body problem in the three-dimensional case is investigated for the case when the resonances are absent and also when the resonances are present. Stability is proved for most (in the sense of Lebesgue) initial conditions for all μ < μ0 except for the resonance cases.

Remarks on the photogravitational restricted three-body problem

The effects of the radiation pressure in the restricted three-body problem are considered and the existence of the “out-of-plane” equilibrium points is analyzed. It is found that within the framework of the stellar stability, the five Lagrangian points are the only equilibrium points, at least as far as the force of the radiation pressure is taken into account.

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 solutions around the collinear langrangian points in the photo gravitational restricted three-body problem: Sun-Jupiter case

The evolution of the periodic orbits around the collinear equilibrium positions, belonging to the Stromgren families a, b and c, with the radiation pressure parameter of the more massive body is studied in the Sun-Jupiter system. These families are determined for a single value of the radiation pressure parameter and particularly when the radiation force of the more massive body is equal to one half of the gravitational attraction. Then the critical stability orbits of each family are transferred with the radiation parameter. The stability of each periodic solution is also studied.

The zero velocity surfaces in the photogravitational restricted three-body problem

The surfaces of zero velocity of the restricted three-body problem when the more massive body is luminous, are studied. The properties of the function Ω which determines these surfaces are given. It is found that the topological properties of the zero velocity surfaces while not affected by the variation of the mass parameter, are essentially varied when the radiation pressure parameter changes values. Closed regions where the motion can be trapped are described while periodic motions about the “out of plane” equilibrium points seem to be probable.

Periodic solutions about the ?out of plane? equilibrium points in the photogravitational restricted three-body problem

The regions of stability for the ‘out of plane’ equilibrium points of the photogravitational restricted three-body problem are given. Second order expansions of periodic solutions around these points are constructed and the corresponding families are computed. It is found that two such families exist. One of them originates and terminates on the same equilibrium point while the other terminates by flattening on the orbital plane.

The linear stability of libration points of the photogravitational restricted three-body problem when the smaller primary is an oblate spheroid

This paper deals with the stationary solutions of the planar restricted three-body problem when the more massive primary is a source of radiation and the smaller primary is an oblate spheroid with its equatorial plane coincident with the plane of motion. The collinear equilibria have conditional retrograde elliptical periodic orbits around them in the linear sense, while the triangular points have long- or short-periodic retrograde elliptical orbits for the mass parameter 0 ≤ μ < μcrit, the critical mass parameter, which decreases with the increase in oblateness and radiation force. Through special choice of initial conditions, retrograde elliptical periodic orbits exist for the case μ = μcrit, whose eccentricity increases with oblateness and decreases with radiation force for non-zero oblateness.

On the stability of triangular libration points of the photogravitational restricted circular three-body problem

This paper considers the restricted circular three-body problem with respect to the radiation repulsion force acting upon a particle on the part of one of the main bodies (the Sun). The characteristic of the family of stationary particular solutions of the problem (libration points) representing the relative equilibrium positions in a rotating Cartesian system is given. On the basis of the KAM theory with the help of a computer a nonlinear analysis of the triangular libration points stability for the planar case is carried out. These libration points are proved to be strictly stable by Liapunov practically in the whole area of fulfilling the necessary stability conditions. Instability is discovered at the resonant curve of the third order and at the greater part of the resonant curve of the fourth order. The plotted results of the investigation allowed us to draw a conclusion about the Liapunov stability of the triangular libration points in a problem with respect to the radiation pressure for all the planets of the Solar system.