Impacts of Poynting–Robertson Drag and Dynamical Flattening Parameters on Motion around the Triangular Equilibrium Points of the Photogravitational ER3BP

Using an analytical and numerical study, this paper investigates the equilibrium state of the triangular equilibrium points L 4 , 5 of the Sun-Earth system in the frame of the elliptic restricted problem of three bodies subject to the radial component of Poynting–Robertson (P–R) drag and radiation pressure factor of the bigger primary as well as dynamical flattening parameters of both primary bodies (i.e., Sun and Earth). The equations of motion are presented in a dimensionless-pulsating coordinate system ξ − η , and the positions of the triangular equilibrium points are found to depend on the mass ratio μ and the perturbing forces involved in the equations of motion. A numerical analysis of the positions and stability of the triangular equilibrium points of the Sun-Earth system shows that the perturbing forces have no significant effect on the positions of the triangular equilibrium points and their stability. Hence, this research work concludes that the motion of an infinitesimal mass near the triangular equilibrium points of the Sun-Earth system remains linearly stable in the presence of the perturbing forces.

Beyond-Newtonian dynamics of a planar circular restricted three-body problem with Kerr-like primaries

ABSTRACT The dynamics of the planar circular restricted three-body problem with Kerr-like primaries in the context of a beyond-Newtonian approximation is studied. The beyond-Newtonian potential is developed by using the Fodor–Hoenselaers–Perjés procedure. An expansion in the Kerr potential is performed and terms up to the first non-Newtonian contribution of both the mass and spin effects are included. With this potential, a model for a test particle of infinitesimal mass orbiting in the equatorial plane of the two primaries is examined. The introduction of a parameter, ϵ, allows examination of the system as it transitions from the Newtonian to the beyond-Newtonian regime. The evolution and stability of the fixed points of the system as a function of the parameter ϵ is also studied. The dynamics of the particle is studied using the Poincaré map of section and the Maximal Lyapunov Exponent as indicators of chaos. Intermediate values of ϵ seem to be the most chaotic for the two cases of primary mass ratios (=0.001, 0.5) examined. The amount of chaos in the system remains higher than the Newtonian system as well as for the planar circular restricted three-body problem with Schwarzschild-like primaries for all non-zero values of ϵ.

Three-Axis Theorem in Moment of Inertia Computation

A very important property in the study of rigid body dynamics, moment of inertia describes the resistance of an object to any change in its angular velocity, given a certain amount of torque. Although many novel methods have been developed to simplify its calculation, this paper presents a remarkable theorem in moment of inertia that has never been widely used, the three-axis theorem. The theorem provides an alternative way for moment of inertia computation and better visualization in integrating each infinitesimal constituent mass element of a rigid body. The key idea is to focus on the distance from this infinitesimal mass to the intersection of the three axes, instead of its distance to a certain rotational axis.

On Motion around the Collinear Equilibrium Points in the Relativistic R3BP with a Smaller Triaxial Primary

This paper studies the motion of an infinitesimal mass near the collinear equilibrium points in the framework of relativistic restricted three-body problem (R3BP) when the smaller primary is a triaxial body. It is observed that the positions of the collinear points are affected by the relativistic and triaxiality factors. The collinear points are found to remain unstable. Numerical studies in this connection, with the Sun-Earth, Sun-Pluto and Earth-Moon systems have been carried out to show the relativistic and triaxiality effects.

Pulsating curves of zero velocity for infinitesimal mass around oblate and triaxial rigid body of triangular equilibrium points in elliptical restricted three body problem

This paper explore pulsating Curves of zero velocityof the infinitesimal mass around the triangular equilibrium points with oblate and triaxial rigid body in the elliptical restricted three body problem(ER3BP).

A general solution to non-collinear equilibria in terms of largest root (κ) of confocal oblate spheroid

<p>This paper deals with the existence of non-collinear equilibria in restricted three-body problem when less massive primary is an oblate spheroid and the potential of oblate spheroid is in terms of largest root of confocal oblate spheroid. This is found that the non-collinear equilibria are the solution of the equations <em>r</em>₁ = <em>n</em>^-2/3 and κ = 1 – <em>a</em>², where <em>r</em>₁ is the distance of the infinitesimal mass from more massive primary, <em>n</em> is mean-motion of primaries, <em>a</em> is semi axis of oblate spheroid and κ is the largest root of the equation of confocal oblate spheroid passes through the infinitesimal mass.</p>

The Stability of Solutions of Sitnikov Restricted Problem of three Bodies When the Primaries are Triaxial Rigid Bodies

The paper deals with the stability of the solutions of Sitnikov's restricted problem of three bodies if the primaries are triaxial rigid bodies. The infinitesimal mass is moving in space and is being influenced by motion of two primaries (m1>m2). They move in circular orbits without rotation around their centre of mass. Both primaries are considered as axis symmetric bodies with one of the axes as axis of symmetry whose equatorial plane coincides with motion of the plane. The synodic system of co-ordinates initially coincides with inertial system of co-ordinates. It is also supposed that initially the principal axis of the body m1 is parallel to synodic axis and are of the axes of symmetry is perpendicular to plane of motion.Journal of Institute of Science and Technology, 2014, 19(2): 76-78

Trajectories of the infinitesimal mass around the triangular equilibrium points in elliptical restricted three bodies problem under oblate and radiating primaries for the binary systems

<p>This paper describes the trajectory of the infinitesimal mass around L₄ of the triangular equilibrium points for the binary systems in the elliptical restricted three body’s problem (ERTBP), where both oblate primaries are radiating. The solutions for the perturbed motion in the vicinity of L₄ is given by u(f) and v(f) function .The stability of the infinitesimal mass around the triangular points is also studied by plotting u(f) and v(f) curve. It is found that radiation pressure, oblateness and eccentricity show a significant effect on the trajectory and stability of the infinitesimal mass around the triangular equilibrium points. Simulation technique has been used to design the trajectory of the binary systems (Achird, Luyten, α Cen AB, Kruger-60 and Xi-Bootis).</p>

Robe's Restricted Three-Body Problem with Variable Masses and Perturbing Forces

The linear stability of equilibrium points of a test particle of infinitesimal mass in the framework of Robe's circular restricted three-body problem, as in Hallan and Rana, together with effect of variation in masses of the primaries with time according to the combined Meshcherskii law, is investigated. It is seen that, due to a small perturbation in the centrifugal force and an arbitrary constant of a particular integral of the Gylden-Meshcherskii problem, every point on the line joining the centers of the primaries is an equilibrium point provided they lie within the shell. Further, a number of pairs of equilibrium points lying on the -plane and forming triangles with the centers of the shell and the second primary exist, for some values of . The points collinear with the center of the shell are found to be stable under some conditions and the range of stability depends on the small perturbations and , while the triangular points are unstable. Illustrative numerical exploration is given to indicate significant improvement of the problem in Hallan and Rana.