Stability Of Triangular Equilibrium Points Research Articles

Effects of Variable Mass, Disk-Like Structure, and Radiation Pressure on the Dynamics of Circular Restricted Three-Body Problem

In this paper, we intend to investigate the dynamics of the Circular Restricted Three-Body Problem. Here we assumed the primaries as the source of radiation and have variable mass. The gravitational perturbation from disk-like structure are also considered in this study. There exist five equilibrium points in this system. By considering the combined effect of disk–like structure and the mass transfer, we found that the classical collinear equilibrium points depart from the x–axis. We called these equilibrium points as quasi–collinear equilibrium points. Meanwhile, this combined effect also breaks the symmetry of triangular equilibrium point positions. We noted that the quasi–equilibrium points are unstable whereas the triangular equilibrium points are stable if the mass ratio μ is smaller than critical mass μc. Besides the mass ratio, the stability of triangular equilibrium points depend on time.

Nonlinear stability of triangular equilibrium points in non-resonance case with perturbations

Nonlinear stability of triangular equilibrium points in non-resonance case with perturbations

Triangular Equilibria in R3BP under the Consideration of Yukawa Correction to Newtonian Potential

We study the triangular equilibrium points in the framework of Yukawa correction to Newtonian potential in the circular restricted three-body problem. The effects of α and λ on the mean-motion of the primaries and on the existence and stability of triangular equilibrium points are analyzed, where α ∈ − 1 , 1 is the coupling constant of Yukawa force to gravitational force, and λ ∈ 0 , ∞ is the range of Yukawa force. It is observed that as λ ⟶ ∞ , the mean-motion of the primaries n ⟶ 1 + α 1 / 2 and as λ ⟶ 0 , n ⟶ 1 . Further, it is observed that the mean-motion is unity, i.e., n = 1 for α = 0 , n > 1 if α > 0 and n < 1 when α < 0 . The triangular equilibria are not affected by α and λ and remain the same as in the classical case of restricted three-body problem. But, α and λ affect the stability of these triangular equilibria in linear sense. It is found that the triangular equilibria are stable for a critical mass parameter μ c = μ 0 + f α , λ , where μ 0 = 0.0385209 ⋯ is the value of critical mass parameter in the classical case of restricted three-body problem. It is also observed that μ c = μ 0 either for α = 0 or λ = 0.618034 , and the critical mass parameter μ c possesses maximum ( μ c max ) and minimum ( μ c min ) values in the intervals − 1 < α < 0 and 0 < α < 1 , respectively, for λ = 1 / 3 .

Stability of triangular equilibrium points of oblate infinitesimal influenced by oblate and radiating primaries in Elliptic Restricted Three Body Problem based on Floquet’s theory

This paper deals with the stability analysis of the triangular equilibrium points for the generalized problem of the photogravitational restricted three body where both the primaries are radiating. The problem is generalized in the sense that the eccentricity of the orbits and the oblateness due to both the primaries and infinitesimal are considered. The stability in the case of linear resonance are analyzed based on the Floquet’s theory for finding the characteristic exponent for a system containing periodic coefficients. It was found that the critical value of μ for the stability boundary for parametric excitation is dependent on the oblateness of the primaries as well as infinitesimal.

Effects of Triaxiality and Radiation Pressure on Existence of Resonance Stability of Triangular Equilibrium Points in Elliptical Restricted Three-Body Problem

This paper deals with the effects of radiation and triaxiality on linear stability of the triangular librations points for small value of eccentricity ‘\(e\)’ in presence of resonance. The study is carried out for various values of radiation pressure and Triaxiality parameter of both Primaries. It is found that the parametric resonance is only possible at the resonance frequency \(\omega_2 =1/2\) in elliptical case.

Normalization of Hamiltonian and Nonlinear Stability of Triangular Equilibrium Points in the Photogravitational Restricted Three Body Problem with P–R Drag in Non-resonance Case

Normal forms of Hamiltonian are very important to analyze the nonlinear stability of a dynamical system in the vicinity of invariant objects. This paper presents the normalization of Hamiltonian and the analysis of nonlinear stability of triangular equilibrium points in non-resonance case, in the photogravitational restricted three body problem under the influence of radiation pressures and P-R drags of the radiating primaries. The Hamiltonian of the system is normalized up to fourth order through Lie transform method and then to apply the Arnold-Moser theorem, Birkhoff normal form of the Hamiltonian is computed followed by nonlinear stability of the equilibrium points is examined. Similar to the case of classical problem, we have found that in the presence of assumed perturbations, there always exists one value of mass parameter within the stability range at which the discriminant $D_4$ vanish, consequently, Arnold-Moser theorem fails, which infer that triangular equilibrium points are unstable in nonlinear sense within the stability range. Present analysis is limited up to linear effect of the perturbations, which will be helpful to study the more generalized problem.

Existence of Resonance Stability of Triangular Equilibrium Points in Circular Case of the Elliptical Restricted Three-Body Problem Under Radiating and Triaxial Primaries

The linear stability of the triangular librations points is studied in the presence of resonance considering both the primaries as radiating and triaxial. The study is carried out for various values of radiation pressure and Triaxiality parameter of both Primaries. It is found that the parametric resonance is only possible at the resonance frequency \(\omega_2=1/2\) in circular case.

Linear stability and resonance of triangular equilibrium points in elliptic restricted three body problem with radiating primary and triaxial secondary

The present paper studies the linear stability of the triangular equilibrium points of the system. The system comprises of a radiating primary and a triaxial secondary in elliptic restricted three body problem. The existence of third order resonances has been shown and the linear stability has been analyzed for these resonance cases. For the resonance case, and , the conditions of the linear stability are satisfied and the system is stable. But, for the resonance cases and the system is unstable.

Stability of triangular equilibrium points in the Photogravitational elliptic restricted three body problem with Poynting-Robertson drag

We have examined the stability of triangular equilibrium points in photogravitational elliptic restricted three-body problem with Poynting-Robertson drag. We suppose that smaller primary is an oblate spheroid. We have taken bigger primary as radiating. We have found the location of triangular equilibrium points and characteristic equation of the problem. We conclude that triangular equilibrium points remain unstable, different from classical case.

Stability of Triangular Equilibrium Points in Photogravitational Restricted three body with Oblate Primary

We have examined the linear stability of triangular equilibrium points in restricted three body problem. We supposed bigger primary as an oblate spheroid and other primary as source of radiation. We have found characteristic equation of the problem. All the four roots of characteristic equation are pure imaginary. Hence we conclude that triangular equilibrium points are stable.

Triangular libration points in the R3BP under combined effects of oblateness, radiation and power-law profile

We study the effects of oblateness up to J4 of the primaries and power-law density profile (PDP) on the linear stability of libration location of an infinitesimal mass within the framework of restricted three body problem (R3BP), by using a more realistic model in which a disc with PDP is rotating around the common center of the system mass with perturbed mean motion. The existence and stability of triangular equilibrium points have been explored. It has been shown that triangular equilibrium points are stable for 0<μ<μc and unstable for μc⩽μ≤1/2, where μc denotes the critical mass parameter. We find that, the oblateness up to J2 of the primaries and the radiation reduces the stability range while the oblateness up to J4 of the primaries increases the size of stability both in the context where PDP is considered and ignored. The PDP has an effect of about ≈0.01 reduction on the application of μc to Earth–Moon and Jupiter–Moons systems. We find that the comprehensive effects of the perturbations have a stabilizing proclivity. However, the oblateness up to J2 of the primaries and the radiation of the primaries have tendency for instability, while coefficients up to J4 of the primaries have stability predisposition. In the limiting case c=0, and also by setting appropriate parameter(s) to zero, our results are in excellent agreement with the ones obtained previously. Libration points play a very important role in space mission and as a consequence, our results have a practical application in space dynamics and related areas. The model may be applied to study the navigation and stationkeeping operations of spacecraft (infinitesimal mass) around the Jupiter (more massive) -Callisto (less massive) system, where PDP accounts for the circumsolar ring of asteroidal dust, which has a cloud of dust permanently in its wake.

Non-linear Stability of Triangular Equilibrium Points in ER3BP Under Radiating Primaries in Presence of Resonance

The non linear stability of the triangular equilibrium points are studied considering the radiation pressure of both the primaries in the presence of third and fourth order resonance. Markeev (Libration points in celestial mechanics and cosmodynamics, 1978) method is applied throughout our analysis to study the non linear stability of the system. In order to do so, the Birkhoff’s normal form in the vicinity of the triangular equilibrium points up to fourth order terms of the Hamiltonian is computed. The normal forms of the Hamiltonian contains both the resonance cases \(\omega _{1}= 2\omega _{2}\) and \(\omega _{1}= 3\omega _{2}\). It is also observed that the triangular equilibrium points satisfying the conditions \(\mu \le \mu _\mathrm{c} =0.0385209\); are unstable in the third order resonance but stable in the fourth order resonance for different values of radiation pressure taken.

Erratum to: Stability of triangular equilibrium points in the elliptic restricted three body problem with oblate and triaxial primaries

In this work, the elliptic restricted three-body problem (in brief ERTBP) with oblate and triaxial primaries using the pulsating coordinates is treated. Mean motion of the system is derived. The locations of the triangular equilibrium points under the considered model are computed. New expressions for these locations as power series in the mass ratio parameter are obtained. These locations are plotted against the whole domain of the mass ratio for different combinations of perturbations. The equations of motion are linearized near the triangular equilibrium points. The linear stability of the triangular points is discussed. It is found that the stability regions depend on the eccentricity of the orbits, oblateness coefficient and the triaxial parameters of the primaries. It is observed that when e=0.25 the stability region is destroyed completely for the Earth-Moon like system.

Stability of triangular equilibrium points in the elliptic restricted three body problem with oblate and triaxial primaries

Stability of triangular equilibrium points in the elliptic restricted three body problem with oblate and triaxial primaries

Resonance stability of triangular equilibrium points in elliptical restricted three body problem under the radiating primaries

Resonance stability of triangular equilibrium points in elliptical restricted three body problem under the radiating primaries

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.

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

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.

Motion and stability of triangular equilibrium points in elliptical restricted three body problem under the radiating primaries

This paper studies the motion and orbital stability of the infinitesimal mass in the vicinity of the equilateral (triangular) Lagrangian points of the elliptic restricted three body problem, considering photo gravitational effects of both the primaries. The stability of the triangular points is studied under the effects of radiating primaries around the binary system (Achird, Luyten, αCen AB, Kruger-60, Xi-Bootis); using simulation technique by drawing different curves of zero velocity.

Stability of triangular equilibrium points in the elliptic restricted problem of three bodies with radiating and triaxial primaries

This paper studies the stability of infinitesimal motions about the triangular equilibrium points in the elliptic restricted three body problem assuming bigger primary as a source of radiation and the smaller one a triaxial rigid body. The perturbation technique developed by Bennet (Icarus 4:177, 1965b) has been used for determination of characteristic exponents. This technique is based on Floquet’s Theory for determination of characteristic exponents in the system with periodic coefficients. The results of the study are analytical and numerical expressions are simulated for the transition curves bounding the region of stability in the μ–e plane, accurate to O(e 2). The unstable region is found to be divided into three parts. The effect of radiation parameter is significant. For small values of e, the results are in favor with the numerical analysis of Danby (Astron. J. 69:166, 1964), Bennet (Icarus 4:177, 1965b), Alfriend and Rand (AIAA J. 6:1024, 1969). The effect of radiation pressure is significant than the oblateness and triaxiality of the primaries.

Existence of Resonance Stability of Triangular Equilibrium Points in Circular Case of the Planar Elliptical Restricted Three-Body Problem under the Oblate and Radiating Primaries around the Binary System

This paper analyzes the existence of resonance stability of the triangular equilibrium points of the planar elliptical restricted three-body problem when both the primaries are oblate spheroid as well as the source of radiation under the particular case, whene=0. We have derived Hamiltonian function describing the motion of infinitesimal mass in the neighborhood of the triangular equilibrium solutions taken as a convergent series. Hamiltonian function for the system has been derived and also expanded in powers of the generalized components of momenta. We have used canonical transformation to make the Hamiltonian function independent of true anomaly. The most interesting and distinguishable results of this study are establishing the relation for determining the range of stability at and near the resonanceω2=1/2around the binary system.