Stability of triangular points in the photogravitational CR3BP with Poynting-Robertson drag and a smaller triaxial primary

The stability of triangular equilibrium points in the framework of the circular restricted three-body problem (CR3BP) is investigated for a test particle of infinitesimal mass in the vicinity of two massive bodies (primaries), when the bigger primary is a source of radiation and the smaller one is a triaxial rigid body with one of the axes as the axis of symmetry and its equatorial plane coinciding with the plane of motion, under the Poynting-Robertson (P-R) drag effect as a result of the radiating primary. It is found that the involved parameters influence the position of triangular points and their linear stability. It is noted that these points are unstable in the presence of Poynting-Robertson drag effect and conditionally stable in the absence of it.     

Nonlinear stability in the generalised photogravitational restricted three body problem with Poynting-Robertson drag

The nonlinear stability of triangular equilibrium points has been discussed in the generalised photogravitational restricted three body problem with Poynting-Robertson drag. The problem is generalised in the sense that smaller primary is supposed to be an oblate spheroid. The bigger primary is considered as radiating. We have performed first and second order normalization of the Hamiltonian of the problem. We have applied KAM theorem to examine the condition of non-linear stability. We have found three critical mass ratios. Finally we conclude that triangular points are stable in the nonlinear sense except three critical mass ratios at which KAM theorem fails.     

Analysis on the stability of triangular points in the perturbed photogravitational restricted three-body problem with variable masses

This paper examines the effect of a constant κ of a particular integral of the Gylden-Meshcherskii problem on the stability of the triangular points in the restricted three-body problem under the influence of small perturbations in the Coriolis and centrifugal forces, together with the effects of radiation pressure of the bigger primary, when the masses of the primaries vary in accordance with the unified Meshcherskii law. The triangular points of the autonomized system are found to be conditionally stable due to κ. We observed further that the stabilizing or destabilizing tendency of the Coriolis and centrifugal forces is controlled by κ, while the destabilizing effects of the radiation pressure remain unchanged but can be made strong or weak due to κ. The condition that the region of stability is increasing, decreasing or does not exist depend on this constant. The motion around the triangular points L _4,5 varying with time is studied using the Lyapunov Characteristic Numbers, and are found to be generally unstable.     

The equilibrium points in the perturbed R3BP with triaxial and luminous primaries

This study explores the effects of small perturbations in the Coriolis and centrifugal forces, radiation pressures and triaxiality of the two stars (primaries) on the position and stability of an infinitesimal mass (third body) in the framework of the planar circular restricted three-body problem (R3BP). it is observed that the positions of the usual five (three collinear and two triangular) equilibrium points are affected by the radiation, triaxiality and a small perturbation in the centrifugal force, but are unaffected by that of the Coriolis force. The collinear points are found to remain unstable, while the triangular points are seen 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 ratio influenced by the small perturbations in the Coriolis and centrifugal forces, radiation and triaxiality. It is also noticed that the former one and all the latter three posses stabilizing and destabilizing behavior respectively. Therefore, the overall effect is that the size of the region of stability decreases with increase in the values of the parameters involved.     

About stability of libration points in the restricted photogravitational three body problem

Nonlinear stability of the triangular libration point in the photogravitational restricted three body problem was investigated in the whole range of the parameters. Some results obtained earlier are corrected. The method for proper determination of cases when stability cannot be determined by four order terms of the hamiltonian was proposed.     

Nonlinear stability of the libration points in the photogravitational restricted three body problem

The stability of the triangular libration points in the case when the first and the second order resonances appear was investigated. It was proved that the first order resonances do not cause instability. The second order resonances may lead to instability. Domains of the instability in the two-dimensional parameter space were determined.     

On bridges B(pL, qS) around triangular libration points

When μ is smaller than Routh’s critical value μ ₁ = 0.03852 . . . , two planar Lyapunov families around triangular libration points exist, with the names of long and short period families. There are periodic families which we call bridges connecting these two Lyapunov families. With μ increasing from 0 to 1, how these bridges evolve was studied. The interval (0,1) was divided into six subintervals (0, μ ₅), (μ ₅, μ ₄), (μ ₄, μ ₃), (μ ₃, μ ₂), (μ ₂, μ ₁), (μ ₁, 1), and in each subinterval the families B(pL, qS) were studied, along with the families B(qS, qS′). Especially in the interval (μ ₂, μ ₁), the conclusion that the bridges B(qS, qS′) do not exist was obtained. Connections between the short period family and the bridges B(kS, (k + 1)S) were also studied. With these studies, the structure of the web of periodic families around triangular libration points was enriched.     

Existence of libration points in the generalised photogravitational restricted problem of three bodies

In this paper we have proved the existence of libration points for the generalised photogravitational restricted problem of three bodies. We have assumed the infinitesimal mass of the shape of an oblate spheroid and both of the finite masses to be radiating bodies and the effect of their radiation pressure on the motion of the infinitesimal mass has also been taken into account. It is seen that there is a possibility of nine libration points for small values of oblateness, three collinear, four coplanar and two triangular.     

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.     

The libration points of a spherical satellite in the photogravitational restricted problem of three bodies

In this paper the photogravitational circular restricted problem of three bodies is considered. We have assumed that one of the finite bodies be a spherical luminous and the other be a triaxial nonluminous body. The possibility of existence of the libration points be studied.     

Effects of axis-symmetric body,zonal harmonic and potential from a belt on the stability of triangular libration points in the CR3BP

Within the frame work of the circular restricted three-body problem (CR3BP) we have examined the effect of axis-symmetric of the bigger primary, oblateness up to the zonal harmonic J ₄ of the smaller primary and gravitational potential from a belt (circular cluster of material points) on the linear stability of the triangular libration points. It is found that the positions of triangular libration points and their linear stability are affected by axis-symmetric of the bigger primary, oblateness up to J ₄ of the smaller primary and the potential created by the belt. The axis-symmetric of the bigger primary and the coefficient J ₂ of the smaller primary have destabilizing tendency, while the coefficient J ₄ of the smaller primary and the potential from the belt have stabilizing tendency. The overall effect of these perturbations has destabilizing tendency. This study can be useful in the investigation of motion of a particle near axis-symmetric—oblate bodies surrounded by a belt.     

Effects of zonal harmonics and a circular cluster of material points on the stability of triangular equilibrium points in the R3BP

We have studied a modified version of the classical restricted three-body problem (CR3BP) where both primaries are considered as oblate spheroids and are surrounded 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 effects of oblateness of both primaries up to zonal harmonic J ₄; together with gravitational potential from the circular cluster of material points on the existence and linear stability of the triangular equilibrium points. It is found that, 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 affected by the oblateness up to J ₄ of the primaries and potential from the circular cluster of material points. The coefficient J ₄ has stabilizing tendency, while J ₂ and the potential from the circular cluster of material points have destabilizing tendency. A practical application of this model could be the study of the motion of a dust particle near oblate bodies surrounded by a circular cluster of material points.     

Motion around L 4 in the perturbed relativistic R3BP

This paper studies the motion of a test particle (infinitesimal mass) in the neighborhood of the triangular point L ₄ in the framework of the perturbed relativistic restricted three-body problem (R3BP). The problem is perturbed in the sense that a small perturbation is given to the centrifugal force. It is found that the position and stability of the triangular point are affected by both the relativistic factor and a small perturbation in the centrifugal force.     

On the stability of the triangular libration points for the photogravitational circular restricted problem of three bodies under the resonances of the third and the fourth order

We investigate the stability of the triangular libration points when both the attracting bodies are radiating under the resonance conditions ₁ = 2₂ and 3₂.     

Effect of perturbed potentials on the stability of libration points in the restricted problem

The location and the stability in the linear sense of the libration points in the restricted problem have been studied when there are perturbations in the potentials between the bodies. It is seen that if the perturbing functions satisfy certain conditions, there are five libration points, two triangular and three collinear. It is further observed that the collinear points are unstable and for the triangular points, the range of stability increases or decreases depending upon whetherP> or <0 wherep depends upon the perturbing functions. The theory is verified in the following four cases:

1. There are no perturbations in the potentials (classical problem).
2. Only the bigger primary is an oblate spheroid whose axis of symmetry is perpendicular to the plane of relative motion (circular) of the primaries.
3. Both the primaries are oblate spheroids whose axes of symmetry are perpendicular to the plane of relative motion (circular) of the primaries.
4. The primaries are spherical in shape and the bigger is a source of radiation.

     

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

The linear stability of the triangular equilibrium points in the photogravitational elliptic restricted problem is examined and the stability regions are determined in the space of the parameters of mass, eccentricity, and radiation pressure. It is found that radiation pressure of the larger body for solar system cases exerts only a small quantitative influence on the stability regions.     

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.     

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 motion around the triangular equilibrium points of the photogravitational restricted problem of three bodies

In this article the effect of radiation pressure on the periodic motion of small particles in the vicinity of the triangular equilibrium points of the restricted three body problem is examined. Second order parametric expansions are constructed and the families of periodic orbits are determined numerically for two sets of values of the mass and radiation parameters corresponding to the non-resonant and the resonant case. The stability of each orbit is also studied.