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

Jagadish Singh  Oni Leke 《Astrophysics and Space Science》2013,344(1):51-61

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.  相似文献   

Effects of triaxiality of primaries on oblate infinitesimal in elliptical restricted three body problem     

《New Astronomy》2021

This paper examines the effects of triaxiality of both the primaries on the position and stability of the oblate infinitesimal mass in the neighborhood of triangular equilibrium points in the framework of Elliptical restricted three body problem. We have found the solutions for the locations of triangular equilibrium points. We have investigated the stability of infinitesimal mass around the triangular equilibrium points.It is observed that the infinitesimal motion around triangular equilibrium points are stable under certain condition with respect to triaxiality of primaries. We have applied the method of averaging used by Grebenivok, throughout the analysis of the stability of the infinitesimal mass around the triangular equilibrium points. We have exploited simulation technique using MATLAB 15 to analyze the stability of the system. The critical mass ratio depends on the triaxiality, oblateness, semi- major axis and eccentricity of the elliptical orbits.  相似文献   

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

K. B. Bhatnagar  P. P. Hallan 《Celestial Mechanics and Dynamical Astronomy》1979,20(2):95-103

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.

  相似文献   

Libration Points in Schwarzschild's Circular Restricted Three-Body Problem     

Roman Rodica  Mioc Vasile 《Astrophysics and Space Science》2006,304(1-4):101-103

The restricted three-body problem in Schwarzschild's gravitational field is analyzed. The existen- ce of the equilibrium points in the orbital plane is discussed and the corresponding positions are established. There are three collinear libration points, and, if they exist, two triangular libration points (situated in the orbital plane of the primaries). If triangular points exist, they may not form equilateral triangles; the triangles are isosceles for equal masses of the primaries, and scalene else.  相似文献   

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

Jagadish Singh  Oni Leke 《Astrophysics and Space Science》2014,350(1):143-154

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.  相似文献   

Equilibrium points of the restricted three-body problem with equal prolate and radiating primaries,and their stability     

C. N. Douskos 《Astrophysics and Space Science》2011,333(1):79-87

Paper presents a complete discussion of the existence, location and stability of the equilibrium points of the coplanar restricted three-body problem with equal prolate and radiating primaries. Depending on the values of the radiation and negative oblateness parameters, two or four additional collinear equilibrium points exist, in addition to the three Eulerian points of the classical case, making up a total of up to seven collinear points. Four of these additional points, as well as the classical central equilibrium point located at the origin, are stable for certain ranges of the parameters. Also, depending on the values of the parameters, up to six additional non-collinear equilibrium points exist, in addition to the triangular Lagrangian points of the classical case. Two of these additional points are located symmetrically above and below the origin and are stable, while the other four are located symmetrically in the four quadrants and are unstable.  相似文献   

Equilibrium points and Lyapunov families in the circular restricted three-body problem with an oblate primary and a synchronous rotating dipole secondary: Application to Luhman-16 binary system     

《New Astronomy》2024

We numerically study a version of the synchronous circular restricted three-body problem, where an infinitesimal mass body is moving under the Newtonian gravitational forces of two massive bodies. The primary body is an oblate spheroid while the secondary is an elongated asteroid of a combination of two equal masses forming a rotating dipole which is synchronous to the rotation of the primaries of the classic circular restricted three-body problem. In this paper, we systematically examine the existence, positions, and linear stability of the equilibrium points for various combinations of the model's parameters. We observe that the perturbing forces have significant effects on the positions and stability of the equilibrium points as well as the regions where the motion of the particle is allowed. The allowed regions of motion as determined by the zero-velocity surface and the corresponding isoenergetic curves as well as the positions of the equilibrium points are given. Finally, we numerically study the binary system Luhman-16 by computing the positions of the equilibria and their stability as well as the allowed regions of motion of the particle. The corresponding families of periodic orbits emanating from the collinear equilibrium points are computed along with their stability properties.  相似文献   

Application of binary pulsars to axisymmetric bodies in the Elliptic R3BP     

Jagadish Singh  Aishetu Umar 《Astrophysics and Space Science》2013,348(2):393-402

Binary systems hosting astrophysical compact objects such as white dwarfs and/or neutron stars provide excellent test beds for studying the impact of the oblateness of the main bodies in the restricted three-body problem (R3BP). The case is investigated when the primary bodies are non-luminous, non-spherical (oblate) bodies and the third body of infinitesimal mass is also an oblate spheroid. The existence of extra solar planets orbiting these systems constitutes a three-body problem which makes them excellent models for this axisymmetric ER3BP. The positions of the equilibrium points are affected by the oblateness parameters of the three-bodies; this is shown for double neutron star binaries. The triangular points are stable for 0<μ<μ _c ; where μ is the mass ratio (μ≤1/2) and μ _c is the critical mass value influenced by the eccentricity, semi major axis and oblateness factors. The size of the region of stability increases with decreasing values of the oblateness. The oblateness of the system’s bodies does not affect the nature of the stability of the collinear points since they remain unstable. Due to the almost equal masses of the primaries, our study shows that even the triangular points of these systems are unstable.  相似文献   

Non-linear stability in the generalized restricted three-body problem     

B. Ishwar 《Celestial Mechanics and Dynamical Astronomy》1996,65(3):253-289

The non-linear stability of the triangular equilibrium point L ₄ in the generalized restricted three-body problem has been examined. The problem is generalized in the sense that the infinitesimal body and one of the primaries have been taken as oblate spheroids. It is found that the triangular equilibrium point is stable in the range of linear stability except for three mass ratios.  相似文献   

The libration points of axisymmetric satellite in the gravitation field of two triaxial rigid bodies     

S. M. Elshaboury  M. O. Shaker  A. E. El-Dessoky  M. A. Eltantawy 《Earth, Moon, and Planets》1991,52(1):69-81

In this paper we consider the restricted problem of three rigid bodies (an axisymmetric satellite in the gravitation field of two triaxial primaries). The collinear and triangular equilibrium solutions are obtained. The effect of the primaries on the location of the libration points of a spherical satellite has been studied numerically.  相似文献   

Reduction the secular solution to periodic solution in the generalized restricted three-body problem     

Elbaz I. Abouelmagd  M. E. Awad  E. M. A. Elzayat  Ibrahim A. Abbas 《Astrophysics and Space Science》2014,350(2):495-505

The aim of the present work is to find the secular solution around the triangular equilibrium points and reduce it to the periodic solution in the frame work of the generalized restricted thee-body problem. This model is generalized in sense that both the primaries are oblate and radiating as well as the gravitational potential from a belt. We show that the linearized equation of motion of the infinitesimal body around the triangular equilibrium points has a secular solution when the value of mass ratio equals the critical mass value. Moreover, we reduce this solution to periodic solution, as well as some numerical and graphical investigations for the effects of the perturbed forces are introduced. This model can be used to examine the existence of a dust particle near the triangular points of an oblate and radiating binary stars system surrounded by a belt.  相似文献   

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

V. V. Markellos  E. Perdios  K. Papadakis 《Astrophysics and Space Science》1993,199(1):139-146

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.  相似文献   

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

Jagadish Singh  Aishetu Umar 《Astrophysics and Space Science》2012,341(2):349-358

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.  相似文献   

Linear stability of equilibrium points in the generalized photogravitational Chermnykh’s problem     

Badam Singh Kushvah 《Astrophysics and Space Science》2008,318(1-2):41-50

The equilibrium points and their linear stability has been discussed in the generalized photogravitational Chermnykh’s problem. The bigger primary is being considered as a source of radiation and small primary as an oblate spheroid. The effect of radiation pressure has been discussed numerically. The collinear points are linearly unstable and triangular points are stable in the sense of Lyapunov stability provided μ<μ _Routh =0.0385201. The effect of gravitational potential from the belt is also examined. The mathematical properties of this system are different from the classical restricted three body problem.  相似文献   

Stability of Triangular Equilibrium Points in the Photogravitational Restricted Three-Body Problem with Oblateness and Potential from a Belt     

Jagadish Singh  Joel John Taura 《Journal of Astrophysics and Astronomy》2014,35(2):107-119

We have examined the effects of oblateness up to J ₄ of the less massive primary and gravitational potential from a circum-binary belt on the linear stability of triangular equilibrium points in the circular restricted three-body problem, when the more massive primary emits electromagnetic radiation impinging on the other bodies of the system. Using analytical and numerical methods, we have found the triangular equilibrium points and examined their linear stability. The triangular equilibrium points move towards the line joining the primaries in the presence of any of these perturbations, except in the presence of oblateness up to J ₄ where the points move away from the line joining the primaries. It is observed that the triangular points are stable for 0 < μ < μ _c and unstable for \(\mu_{\mathrm{c}} \le \mu \le \frac {1}{2},\) where μ _c is the critical mass ratio affected by the oblateness up to J ₄ of the less massive primary, electromagnetic radiation of the more massive primary and potential from the belt, all of which have destabilizing tendencies, except the coefficient J₄ and the potential from the belt. A practical application of this model could be the study of motion of a dust particle near a radiating star and an oblate body surrounded by a belt.  相似文献   

Robe’s Circular Restricted Three-Body Problem Under Oblate and Triaxial Primaries     

Jagadish Singh  Hafsah Laraba Mohammed 《Earth, Moon, and Planets》2012,109(1-4):1-11

This paper analyzes Robe??s circular restricted three-body problem when the hydrostatic equilibrium figure of the first primary is assumed to be an oblate spheroid, the shape of the second primary is considered as a triaxial rigid body, and the full buoyancy force of the fluid is taken into account. It is found that there is an equilibrium point near the center of the first primary, another equilibrium point exists on the line joining the centers of the primaries and there exist infinite number of equilibrium points on an ellipse in the orbital plane of the second primary. It is also observed that under certain conditions, all these equilibrium points can be stable. The most interesting and distinguishable results of this study are the existence of elliptical points and their stability.  相似文献   

Motion in the generalized restricted three-body problem     

Jagadish Singh  Joel John Taura 《Astrophysics and Space Science》2013,343(1):95-106

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.  相似文献   

Periodic orbits in the generalized perturbed restricted three-body problem     

Jagadish Singh  Jessica Mrumun Begha 《Astrophysics and Space Science》2011,332(2):319-324

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.  相似文献   

Equilibrium solutions of the restricted problem of 2+2 bodies     

Arthur L. Whipple 《Celestial Mechanics and Dynamical Astronomy》1984,33(3):271-294

Fourteen equilibrium solutions of the restricted problem of 2+2 bodies are shown to exist. Six of these solutions are located about the collinear Lagrangian points of the classical restricted problem of three bodies. Eight solutions are found in the neighborhood of the triangular Lagrangian points. Linear stability analysis reveals that all of the equilibrium solutions are unstable with the exception of four solutions; two in the vicinity of each of the triangular Lagrangian points. These four solutions are found to be stable provided the mass parameter of the primary masses is less than a critical value which depends also on the mass of the minor bodies.  相似文献   

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

Jagadish Singh  Oni Leke 《Astrophysics and Space Science》2010,326(2):305-314

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.  相似文献