Stability Of Libration Points Research Articles

On Libration Points in the Asteroid–Research Probe System

This paper is dedicated to a study of relative motion in the field of homogeneous spheroid bodies. It is assumed that an asteroid’s satellite does not affect its translational and rotational motion by inertia. The latter motion has the form of rotation in the mode of regular precession. The case in which the asteroid is an ellipsoid of revolution is considered. A system of conditions that the libration points must satisfy is constructed; it is shown that its solution is two types of libration points, which differ in their location with respect to the constant vector of the angular momentum and the axis of symmetry of the ellipsoid. Thereafter, the stability of libration points is investigated numerically and analytically in the first approximation.

Capricious basins of attraction in photogravitational magnetic binary problem

The present paper deals with the magnetic binary problem in the framework of restricted problem of three bodies. The bigger primary is taken as the source of radiation pressure and smaller primary is taken as point mass. The parametric evolution of the position and existence of libration points are explored. The study of zero velocity curves and the stability of libration points are one of the critical aspects of this investigation. A systematic probe to find out the influence of solar radiation pressure q, ratio of magnetic moments λ and mass ratio μ on the geometry of basins of attraction is carried out. The shape of the basins of attraction is significantly affected by the variation of these parameters. It is also found that the presence of solar radiation pressure and ratio of magnetic moments increase the unpredictability of basins of attraction. Variation of the mass ratio μ also affects the evolution of libration points as well as the geometry of basins of attraction.

Libration points and periodic orbit families near a binary asteroid system with different shapes of the secondary

Due to the limited capability of ground-based observations, the shapes of asteroids cannot be determined precisely, especially for the secondary of a binary system. In this paper, libration points and periodic orbit families near a binary asteroid system with different shapes of the secondary are investigated, with the binary 66391 Moshup, previously known as 1999 KW4, as an example. The gravity fields of the primary and secondary of the binary system are modelled as those of a polyhedron and a triaxial ellipsoid, respectively. With a nominal shape model for the secondary, the positions and stability of libration points are determined, and periodic orbit families emanating from each libration point are generated by numerical methods. The connections and bifurcations of periodic orbit families are analyzed through comparisons with the classic results of the circular restricted three body problem (CR3BP). Then, the effect of the shape uncertainty is investigated by changing the shape of the secondary. Libration points and associated periodic orbit families are calculated and analyzed with different shapes of the secondary. It has been found that the libration points have slight offsets on position, but their topological structures remain the same. In some cases, the halo orbit family bifurcates from the vertical Lyapunov orbit family rather than the planar Lyapunov orbit family, indicating that bifurcations of periodic orbit families are different and new connections between periodic orbit families can exist. The results will provide general insights into variations of the dynamical environments near a binary asteroid system caused by different shapes of the secondary, and will be helpful for the planetary science research and exploration mission design.

POSITIONS AND STABILITY OF LIBRATION POINTS IN THE CIRCULAR RESTRICTED THREE-BODY PROBLEM WHEN THE BIGGER PRIMARY IS RADIATING AND OBLATING

This paper investigates the positions and stability of libration points in the framework of the circular restricted three-body problem for the systems: Luyten726-8 and HD98800. The position of the third body lie in the plane almost directly above and below the center of the oblate primary. It is found that radiations and oblateness of the primary have destabilizing effects; the presence of any one or more of the latter makes weak the stabilizing ability of the former, consequently the overall effect is that the range of stability of the libration points decreases. Considering the range of stability and instability, that is and , the libration points are respectively stable and unstable for HD98800 and Luyten 762-8 systems. Our results show that, all the roots are real, and for each set of values, there exist at least a positive real part and hence in the Lyapunov sense, the stability of the libration points are unstable for the systems HD98800 and Luyten 762-8.

Central-body square configuration of restricted six-body problem

This paper deals with the existence and stability of libration points in linear sense in the central-body square configuration of restricted six-body problem. It is found that there exist twelve libration points, four collinear and eight non-collinear. All libration points lie on the concentric circles C1, C2 and C3 centered at origin. The libration points L1,3,5,7 lie on circle C1, L9,10,11,12 on C2 and L2,4,6,8 on C3. This is also observed that the eight libration points are on the axes and four are off the axes, i.e., L1,2,3,4 are on x-axis, L5,6,7,8 on y-axis and rest are off the axes. The libration points on circles C1 and C3 are unstable for all values of mass parameter µ while the libration points on circle C2 are stable for the critical mass parameter µc = 0.00910065.

A Study of Albedo Effects on Libration Points in the Elliptic Restricted Three-Body Problem

This paper deals with the libration points in the elliptic restricted three-body problem. The combined effects of the primary bodies’ orbital eccentricity and albedo on the existence and stability of libration points is analyzed. It is found that there exist five libration points, three collinear and two non-collinear. The non-collinear libration points are stable for a critical value of mass parameter μc, the collinear libration points are unstable for all values of μ.

Divulging the effect of small perturbations in the Coriolis and centrifugal forces in the photogravitational version of autonomous restricted four‐body problem with oblate primary

AbstractIn the present problem, we have studied the effect of small perturbations in the Coriolis and centrifugal forces on the locations and stability of libration points by taking the smaller primary as radiating oblate spheroid in the restricted four‐body problem. We have supposed that all the primaries are set in an equilateral triangle configuration, moving in circular orbit around their common center of mass. We have observed that the oblateness and radiation of the third primary, and the effect of the small perturbation in centrifugal force have substantial influences on the locations of libration points but the effect of the small perturbation in the Coriolis force has no impact on their locations. However, the stability of the libration points is highly influenced by the effect of the small perturbation in the Coriolis force. Furthermore, it is obtained that the effect of the small perturbation in the centrifugal force has a substantial influence on the regions of possible motion. In addition, we have also discussed how the oblateness and radiation pressure of the third primary, and the small perturbations in the Coriolis and centrifugal forces influence geometry of the basins of convergence by using multivariate version of the Newton–Raphson iterative scheme.

Bifurcation near Boundaries of Regions of Stability of Libration Points in the Three-Body Problem

The construction of regions of stability in a linear approximation for the triangular libration points in the planar, elliptical, restricted three-body problem is considered, together with the main scenarios for bifurcation when the parameters of the system pass through a boundary of such a region. A new scheme for constructing the boundaries of regions of stability is proposed, which leads to approximate formulas describing these boundaries. The resonance properties of the boundary points (from the point of view of the theory of local bifurcations) are studied. It is shown that one of the main scenarios for bifurcation is the appearance of non-stationary, 4π-periodic solutions close to a triangular libration point.

Stability of libration points in the restricted four-body problem with variable mass

We have investigated the stability of the Lagrangian solutions for the restricted four-body problem with variable mass. It has been assumed that the three primaries with masses $m_{1}$ , $m_{2}$ and $m_{3}$ form an equilateral triangle, wherein $m_{2}=m_{3}$ . According to Jeans’ law (Astronomy and Cosmogony, Cambridge University Press, Cambridge, 1928), the infinitesimal body varies its mass $m$ with time. The space–time transformations of Meshcherskii (Studies on the Mechanics of Bodies of Variable Mass, GITTL, Moscow, 1949) are used by taking the values of the parameters $q=1/2$ , $k=0$ , $n=1$ . The equations of motion of the infinitesimal body with variable mass have been determined. The equations of motion of the current problem differ from the ones of the restricted four-body problem with constant mass. There exist eight libration points, out of which two are collinear with the primary $m_{1}$ and the rest are non-collinear for a fixed value of parameters $\gamma (\frac{m \ \text{at time} \ t}{m \ \text{at initial time}}, 0<\gamma\leq1 )$ , $\alpha$ (the proportionality constant in Jeans’ law (Astronomy and Cosmogony, Cambridge University Press, Cambridge, 1928), $0\leq\alpha\leq2.2$ ) and $\mu=0.019$ (the mass parameter). All the libration points are found to be unstable. The zero velocity surfaces (ZVS) are also drawn and regions of motion are discussed.

Libration points of a rotating complexified triangle

Similarities and differences between the force fields of a classical real dipole and a complex dipole are analyzed. The complex dipole is a pair of points equipped with complex conjugate masses and situated in a complex domain. The results of this analysis are used in the problem of motion of a material point in the field of attraction of a triangle uniformly rotating in its plane about its center of mass. It is assumed that a complex dipole is assigned to each vertex of the triangle. The existence and stability of libration points are studied. In particular, it is shown that there exist libration points outside the plane of the triangle.

Effect of elliptic angle φ on the existence and stability of libration points in restricted three-body problem in earth-moon system considering earth as an ellipsoid

This paper deals with the existence and the stability of the earth-moon libration points in the restricted three-body problem. In this paper we have considered the bigger primary as an ellipsoid while the smaller one as a point-mass. This is observed that the collinear and non-collinear libration points exist only in the interval 0˚&lt;φ &lt; 45˚. There exist three collinear libration points and the non-collinear libration points are forming a right triangle with the primaries. Further observed that the libration points either collinear or non-collinear all are unstable in 0˚&lt;φ &lt; 45˚.

Stability and periodicity in the Sitnikov three-body problem when primaries are oblate spheroids

This paper deals with the effect of oblateness of the primaries of equal masses on the series solutions of the Sitnikov problem of three bodies. Effects of oblateness have also been shown on the stability of libration points and Poincare surface of section. Here series solutions have been developed with the help of iteration process of Green’s function and by the Lindstedt-Poincare method. Following Murray and Dermott (Solar System Dynamics, Cambridge University Press, Cambridge, 1999) we have checked the stability of the equilibrium points in the Sitnikov problem. Periodicity and quasi-periodicity have been examined by drawing the Poincare surfaces of section using the mathematical software.

English

This paper deals with the existence and stability of libration points in restricted problem under the effect of dissipative force i.e. Poynting Robertson Light drag. We have determined the equations of motion of infinitesimal mass and investigated the stability of noncollinear libration points in the linear sense and found that there exist only two non-collinear libration points which are unstable for all mass parameter

Effects of perturbations in Coriolis and centrifugal forces on the locations and stability of libration points in Robe’s circular restricted three-body problem under oblate-triaxial primaries

This paper investigates the effect of small perturbations in Coriolis and centrifugal forces on the axial libration points and their stability in Robe’s circular restricted three-body problem when the hydrostatic equilibrium figure of the first primary is an oblate spheroid; the shape of the second primary is a triaxial rigid body. It is observed that the locations of the axial libration points are only influenced by a small change in the centrifugal force due to the force function Ω being dependent of it. The magnitude of the centrifugal force shift the first libration point (p1,0,0) towards the positive side of the x-axis, while second axial libration (x11+p2,0,0) is shifted away from the origin towards the negative part of the x-axis as a result of the increase in magnitude of the triaxiality and centrifugal force. The range of stability is established and is found that the first axial libration point is stable for p1<0, conditionally stable for 0<p1<c2 and unstable for 0<c2<p1, while the second axial libration point is stable whenever x11<0 and if x11>0, then it is conditionally stable.

Existence and Stability of Libration Points in the Restricted Three-Body Problem When the Primaries are Triaxial Rigid Bodies

This paper deals with the stationary solutions of the planar restricted three-body problem when the primaries are triaxial rigid bodies with one of the axes as the axis of symmetry and its equatorial plane coinciding with the plane of motion. It is seen that there are five libration points, two triangular and three collinear. It is further observed that the collinear points are unstable, while the triangular points are stable for the mass parameter 0 ≤ μ < μcrit(the critical mass parameter). It is further seen that the triangular points have long or short periodic elliptical orbits in the same range of μ.

Nonlinear Stability of the Lagrangian Libration Points in the Chermnykh Problem

In this paper we consider the problem of motion of an infinitesimal point mass in the gravity field of an uniformly rotating dumb-bell. The aim of our study is to investigate Liapunov stability of Lagrangian libration points of this problem. We analyze the stability of libration points in the whole range of parameters ω, μ of the problem. In particular, we consider all resonance cases when the order of resonance is not greater than five.

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.

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.

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

Effect of Perturbed Potentials on the Stability of Libration Points in the Restricted Problem

The location and the stability 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 involving the parameters α,α1,α2 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 whether the perturbation point (α,α1,α2) lies on one or the other side of the plane Aα + Bα1 + Cα2 = 0, and it remains the same if the point lies on the plane, where A,B,C depend on the perturbations. 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, (3) both the primaries are oblate spheroids, and (4) the primaries are spherical in shape and the bigger is a source of radiation.