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.     

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.     

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 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.     

Stability of equilibrium points in the generalized perturbed restricted three-body problem

This paper studies the existence and stability of equilibrium points under the influence of small perturbations in the Coriolis and the centrifugal forces, together with the non-sphericity of the primaries. The problem is generalized in the sense that the bigger and smaller primaries are respectively triaxial and oblate spheroidal bodies. It is found that the locations of equilibrium points are affected by the non-sphericity of the bodies and the change in the centrifugal force. It is also seen that the triangular points are stable for 0<μ<μ _c and unstable for \(\mu_{c}\le\mu <\frac{1}{2}\), where μ _c is the critical mass parameter depending on the above perturbations, triaxiality and oblateness. It is further observed that collinear points remain unstable.     

Instability of triangular libration points in the perturbed photogravitational R3BP with Poynting-Robertson (P-R) drag

We consider the primaries of the circular restricted three-body problem (CR3BP) to be luminous and investigate the influences of small perturbations in the Coriolis and centrifugal forces together with Poynting-Robertson (P-R) drag from both primaries on the triangular points. It is seen, both analytically and numerically, that the positions of triangular points are affected by the radiation pressures, P-R drag and a small perturbation in the centrifugal force. This has been shown for the binary systems Luyten 726-8 and Kruger 60.1. These perturbing forces do not influence the nature of the stability of triangular points in the presence of P-R drag. They remain unstable in the linear sense.     

On ‘out of plane’ equilibrium points in the elliptic restricted three-body problem with radiating and oblate primaries

We locate and examine the stability of the ‘out of plane’ equilibrium points, L _6,7 of an infinitesimal body in the field of stellar-oblate binary systems moving in elliptic orbits around their common center of mass. Their positions and stability depend on the oblateness as well as radiation coefficients of the primaries and the eccentricity of their orbits. A numerical application of this problem for the systems: Gamma Leporis and Altair are given.     

Perturbed Robe’s circular restricted three-body problem under an Oblate Primary

This paper examines the existence and linear stability of equilibrium points in the perturbed Robe’s circular restricted three-body problem under the assumption that the hydrostatic equilibrium figure of the first primary is an oblate spheroid. The problem is perturbed in the sense that small perturbations are given to the Coriolis and centrifugal forces are being considered. Results of the analysis found two axial equilibrium points on the line joining the centre of both primaries. It is further observed that under certain conditions, points on the circle within the first primary are also equilibrium points. And a special case where the density of the fluid and that of the infinitesimal mass are equal (D = 0) is discussed. The linear stability of this configuration is examined; it is observed that the first axial point is unstable while the second one is conditionally stable and the circular points are unstable.     

Combined effects of perturbations, radiation and oblateness on the periodic orbits in the restricted three-body problem

This paper investigates the periodic orbits around the triangular equilibrium points for 0<μ<μ _c , where μ _c is the critical mass value, under the combined influence of small perturbations in the Coriolis and the centrifugal forces respectively, together with the effects of oblateness and radiation pressures of the primaries. It is found that the perturbing forces affect the period, orientation and the eccentricities of the long and short periodic orbits.     

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.     

Equilibrium points and stability in the restricted three-body problem with oblateness and variable masses

The existence and stability of a test particle around the equilibrium points in the restricted three-body problem is generalized to include the effect of variations in oblateness of the first primary, small perturbations ϵ and ϵ′ given in the Coriolis and centrifugal forces α and β respectively, and radiation pressure of the second primary; in the case when the primaries vary their masses with time in accordance with the combined Meshcherskii law. For the autonomized system, we use a numerical evidence to compute the positions of the collinear points L _2κ , which exist for 0<κ<∞, where κ is a constant of a particular integral of the Gylden-Meshcherskii problem; oblateness of the first primary; radiation pressure of the second primary; the mass parameter ν and small perturbation in the centrifugal force. Real out of plane equilibrium points exist only for κ>1, provided the abscissae x < \fracn(k-1)b\xi<\frac{\nu(\kappa-1)}{\beta}. In the case of the triangular points, it is seen that these points exist for ϵ′<κ<∞ and are affected by the oblateness term, radiation pressure and the mass parameter. The linear stability of these equilibrium points is examined. It is seen that the collinear points L _2κ are stable for very small κ and the involved parameters, while the out of plane equilibrium points are unstable. The conditional stability of the triangular points depends on all the system parameters. Further, it is seen in the case of the triangular points, that the stabilizing or destabilizing behavior of the oblateness coefficient is controlled by κ, while those of the small perturbations depends on κ and whether these perturbations are positive or negative. However, the destabilizing behavior of the radiation pressure remains unaltered but grows weak or strong with increase or decrease in κ. This study reveals that oblateness coefficient can exhibit a stabilizing tendency in a certain range of κ, as against the findings of the RTBP with constant masses. Interestingly, in the region of stable motion, these parameters are void for k = \frac43\kappa=\frac{4}{3}. The decrease, increase or non existence in the region of stability of the triangular points depends on κ, oblateness of the first primary, small perturbations and the radiation pressure of the second body, as it is seen that the increasing region of stability becomes decreasing, while the decreasing region becomes increasing due to the inclusion of oblateness of the first primary.     

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.     

Periodic orbits in the generalized perturbed restricted three-body problem

This paper studies the existence and stability of equilibrium points under the influence of small perturbations in the Coriolis and the centrifugal forces, together with the non-sphericity of the primaries. The problem is generalized in the sense that the bigger and smaller primaries are respectively triaxial and oblate spheroidal bodies. It is found that the locations of equilibrium points are affected by the non-sphericity of the bodies and the change in the centrifugal force. It is also seen that the triangular points are stable for 0<μ<μ _c and unstable for m_c ￡ m < \frac12\mu_{c}\le\mu <\frac{1}{2}, where μ _c is the critical mass parameter depending on the above perturbations, triaxiality and oblateness. It is further observed that collinear points remain unstable.     

The restricted 3-body problem with radiation pressure

The restricted 3-body problem is generalised to include the effects of an inverse square distance radiation pressure force on the infinitesimal mass due to the large masses, which are both arbitrarily luminous. A complete solution of the problems of existence and linear stability of the equilibrium points is given for all values of radiation pressures of both liminous bodies, and all values of mass ratios. It is shown that the inner Lagrange point, L₁, can be stable, but only when both large masses are luminous. Four equilibrium points, L₆, L₇, L₈, and L₉ can exist out of the orbital plane when the radiation pressure of the smaller mass is very high. Although L₈ and L₉ are always linearly unstable, L₆ and L₇ are stable for a small range of radiation pressures provided that both large masses are luminous.     

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.     

On out of plane equilibrium points in photo-gravitational restricted three-body problem

We have investigated the out of plane equilibrium points of a passive micron size particle and their stability in the field of radiating binary stellar systems Krüger-60, RW-Monocerotis within the framework of photo-gravitational circular restricted three-body problem. We find that the out of plane equilibrium points (L _i , i = 6, 7, 8, 9) may exist for range of β ₁ (ratio of radiation to gravitational force of the massive component) values for these binary systems in the presence of Poynting-Robertson drag (hereafter PR-drag). In the absence of PR-drag, we find that the motion of a particle near the equilibrium points L _6,7 is stable in both the binary systems for a specific range of β ₁ values. The PR-drag is shown to cause instability of the various out of plane equilibrium points in these binary systems.     

CORIOLIS力和离心力摄动对Robe问题平动点位置和线性稳定性的影响

研究了Coriolis力和离心力摄动对Robe限制性三体问题主要平动点位置及线性稳定性的影响，给出了Robe限制性三体问题主要平动点的摄动位置和Coriolis力和离心力摄动对主要平动点位置和线性稳定的影响量级．改进了Shrivastava的结果．     

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.     

Motion in a modified Chermnykh’s restricted three-body problem with oblateness

In this paper, the restricted problem of three bodies is generalized to include a case when the passively gravitating test particle is an oblate spheroid under effect of small perturbations in the Coriolis and centrifugal forces when the first primary is a source of radiation and the second one an oblate spheroid, coupled with the influence of the gravitational potential from the belt. The equilibrium points are found and it is seen that, in addition to the usual three collinear equilibrium points, there appear two new ones due to the potential from the belt and the mass ratio. Two triangular equilibrium points exist. These equilibria are affected by radiation of the first primary, small perturbation in the centrifugal force, oblateness of both the test particle and second primary and the effect arising from the mass of the belt. The linear stability of the equilibrium points is explored and the stability outcome of the collinear equilibrium points remains unstable. In the case of the triangular points, motion is stable with respect to some conditions which depend on the critical mass parameter; influenced by the small perturbations, radiating effect of the first primary, oblateness of the test body and second primary and the gravitational potential from the belt. The effects of each of the imposed free parameters are analyzed. The potential from the belt and small perturbation in the Coriolis force are stabilizing parameters while radiation, small perturbation in the centrifugal force and oblateness reduce the stable regions. The overall effect is that the region of stable motion increases under the combine action of these parameters. We have also found the frequencies of the long and short periodic motion around stable triangular points. Illustrative numerical exploration is rendered in the Sun–Jupiter and Sun–Earth systems where we show that in reality, for some values of the system parameters, the additional equilibrium points do not in general exist even when there is a belt to interact with.     

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.