Triaxial Rigid Body Research Articles

The The Stability of Triangular Lagragian Points in the Photogravitational Elliptic Restricted Three – Body Problem (ER3BP) with Triaxial Primaries Surrounded by a Belt with A P-R Drag Force

This paper studies the motion of an infinitesimal particle near the triangular equilibrium points (TEPs) in the elliptic restricted three body problem (ER3BP) when the primaries are radiating- triaxial rigid bodies and are affected by the Poynting-Robertson (P-R) drag force enclosed in a circumbinary disc (disc,belt). We present the equations of motion, obtained the positions of (TEPs) and found that there exist two Lagragian equilibria points L4,5 which lies  in the ξη- plane in symmetrical positions with respect to the orbital plane.The parameters involved in the system affect their positions.The position changes with an increase in triaxiality, radiation, P-R drag force and the disc. The positions and linear stability of the  TEPs are investigated numerically using  the  binary systems Archid  and it was observed that the effect  of P-R drag force of the smaller primary is not sufficient in  causing instability at the EPs  but  the radiation pressure force.We observed that when the triaxiality coefficient, values of the belt and P-R drag force are varied increasingly in the absence of radiation equilibrium points (EPs) are stable in the linear sense but becomes unstable on introducing  radiation.Thus radiation is the cause of instability when both primaries are radiating and triaxial with the smaller primary having an effective P-R drag force enclosed in a circumbinary disc and not the P-R drag.

On the equilibria of the restricted three-body problem with a triaxial rigid body, II: prolate primary

The main goal of this work is to elucidate the effect of the inclusion of a rigid triaxial prolate primary body on the dynamics of the circular restricted three-body problem. The position, type, and linear stability of the coplanar points of equilibria are determined by numerical methods. Specifically, we conduct a systematic numerical study for elucidating the influence on the dynamics of the system of the triaxial prolate shape. Our findings are conclusive and indicate that the triaxiality parameters are very influential on the equilibria of the system, modifying not only the total number of equilibrium points but also their stability and nature. We also compare our findings with those of the previous paper of the series, where we have studied the case of triaxial oblate primary bodies.

On the Equilibria of the Restricted Three-Body Problem with a Triaxial Rigid Body - Ii. Prolate Primary

On the Equilibria of the Restricted Three-Body Problem with a Triaxial Rigid Body - Ii. Prolate Primary

On the equilibria of the restricted three-body problem with a triaxial rigid body - I. Oblate primary

The restricted three-body problem, with a rigid triaxial oblate primary body, is being under investigation. Using numerical methods we reveal how the triaxility parameters affect the coordinates and the linear stability of the libration points. We perform a systematic and rigorous analysis for understanding the effect of the parameters σ1 and σ2 on the dynamical properties of the system. Our study suggests that these parameters have a strong impact on the dynamics of the equilibrium points of the system.

Periodic orbits in the restricted problem of three bodies in a three-dimensional coordinate system when the smaller primary is a triaxial rigid body

Abstract In this paper, we have studied the equations of motion for the problem, which are regularised in the neighbourhood of one of the finite masses and the existence of periodic orbits in a three-dimensional coordinate system when μ = 0. Finally, it establishes the canonical set (l, L, g, G, h, H) and forms the basic general perturbation theory for the problem.

On the equilibria of the restricted four-body problem with triaxial rigid primaries - I. Oblate bodies

• We numerically explore the equilibria of the restricted four-body problem with triaxial rigid primaries. • We compute the linear stability of the libration points. • We determine how the triaxility parameters affect the dynamics of the equilibrium points. The equilateral restricted four-body problem, with triaxial rigid oblate bodies, is investigated. Using numerical methods we examine how the linear stability and the positions of the coplanar libration points are affected by the triaxility parameters of the primaries. In each case, we perform a rigorous and systematic analysis for determining the influence of the triaxility parameters σ 1 and σ 2 on the dynamics of the system. Our computations suggest the strong influence of these parameters by revealing additional cases, regarding the total number of equilibria, thus improving the findings of previous related works.

A study of non-collinear libration points under the combined effects of radiation pressure and Stokes drag

We study the non-collinear libration points in the frame work of photo-gravitational circular restricted three-body problem with Stokes drag acting as a dissipative force and considering the more massive primary as a radiating body and the less massive primary as a triaxial rigid body. The combined effects of radiation pressure and Stokes drag on the existence and stability of non-collinear libration points is analyzed. It is found that there exist two non-collinear libration points and are asymptotically stable in the interval 0.6149 ≤ q ≤ 1 for μ = 0.01, where q and μ are the radiation factor and mass ratio, respectively.

Relative equilibria of an intermediary model for the roto-orbital dynamics. The low rotation regime

We address the attitude dynamics of a triaxial rigid body in a circular orbit. This task is done by means of an intermediary model, which is obtained by splitting the Hamiltonian in the form H=H0+H1, where H0 is required to be a non-degenerate integrable Hamiltonian system. A numerical study is presented comparing the dynamics of the new intermediary model with the full system (MacCullagh’s truncation) and showing a competitive performance for the cases Sun-asteroid and Earth-spacecraft. This model defines a Poisson flow endowed with invariants defining a SM2×SM2 reduced space. We analyze the coupling between the orbital mean motion and rotational variables. The key role played by the moments of inertia and the value of the angular momentum is shown in detail. The analysis of the intermediary shows that, under slow rotation regime, the classic dynamics of the free rigid body is no longer maintained: bifurcations with changes of stability are displayed for several critical inclinations of the rotational angular momentum plane and for critical orientations of the body frame. Moreover, the evolution of the angular momentum plane is given by a time dependent harmonic oscillator.

Libration Points in the R3BP with a Triaxial Rigid Body as the Smaller Primary and a Variable Mass Infinitesimal Bod

The paper deals with the existence of the coplanar libration points in the restricted three-body problem when the smaller primary is a triaxial rigid body and the infinitesimal body is of variable mass. Following small parameter method, the coordinates of collinear libration points are established whereas the coordinates of triangular libration points are established by classical method. It is found that the mass reduction factor has small effect but triaxiality parameters of the smaller primary have great effects on the coordinates of the libration points.

Complex variables approach to the short-axis-mode rotation of a rigid body

Abstract Decomposition of the free (triaxial) rigid body Hamiltonian into a “main problem” and a perturbation term provides an efficient integration scheme that avoids the use of elliptic functions and integrals. In the case of short-axis-mode rotation, it is shown that the use of complex variables converts the integration of the torque-free motion by perturbations into a simple exercise of polynomial algebra that can also accommodate the gravity-gradient perturbation when the rigid body rotation is close enough to the axis of maximum inertia.

Libration points in the restricted three-body problem: Euler angles, existence and stability

The objective of the present paper is to study in an analytical way the existence and the stability of the libration points, in the restricted three-body problem, when the primaries are triaxial rigid bodies in the case of the Euler angles of the rotational motion are equal to \begin{document}$ θ_i = π/2, \, ψ_i = 0, \,\varphi_i = π/2 $\end{document} , \begin{document}$ i = 1, 2 $\end{document} . We prove that the locations and the stability of the triangular points change according to the effect of the triaxiality of the primaries. Moreover, the solution of long and short periodic orbits for stable motion is presented.

Semianalytical Technique for Six-Degree-of-Freedom Space Object Propagation

The cannonball assumption of three-degree-of-freedom (3DOF) space object state prediction can lead to large inaccuracies if the sphericity assumption is violated, while numerical propagation of both the translational and rotational (6DOF) equations of motion is computationally expensive. In this paper, a middle ground is proposed, in which the translational equations of motion are propagated numerically using approximate attitude predictions obtained via a closed-form perturbation solution. The capabilities of this semianalytical “hybrid” of special and general perturbation techniques are illustrated using a specific attitude solution, which assumes a fast-rotating, triaxial rigid body in an elliptical orbit subject to gravity-gradient torque. Even when these assumptions are mildly violated—such as when modeling higher-fidelity forces and torques—the approximate attitude predictions allow for more accurate modeling of body forces than a 3DOF cannonball propagation. In numerical examples in which attitude-dependent dynamics are considered, the hybrid method produces position predictions one or more orders of magnitude more accurate than a 3DOF cannonball propagation while requiring approximately one-third of the CPU time of a full 6DOF propagation for certain accuracy tolerance levels. Relative speedups achievable by the hybrid method are shown to increase as the rotation rate of the body increases.

Roto-orbital dynamics of a triaxial rigid body around a sphere. Relative equilibria and stability

We study the roto-orbital motion of a triaxial rigid body around a sphere, which is assumed to be much more massive than the triaxial body. The associated dynamics of this system, which consists of a normalized Hamiltonian with respect to the fast angles (partial averaging), is investigated making use of variables referred to the total angular momentum. The first order approximation of this model is integrable. We carry out the analysis of the relative equilibria, which hinges principally in the dihedral angle between the orbital and rotational planes and the ratio among the moments of inertia ρ=(B-A)/(2C-B-A). In particular, the dynamics of the body frame, though formally given by the classical Euler equations, experiences changes of stability in the principal directions related to the roto-orbital coupling. When ρ=1/3, we find a family of relative equilibria connected to the unstable equilibria of the free rigid body.

Effects of Radiating Oblate Spheroid and Triaxial Rigid Body in the Restricted Three Body Problem

Effects of Radiating Oblate Spheroid and Triaxial Rigid Body in the Restricted Three Body Problem

Periodic solutions around the collinear equilibrium points in the perturbed restricted three-body problem with triaxial and radiating primaries for binary HD 191408, Kruger 60 and HD 155876 systems

The collinear equilibrium points and periodic motion around them are studied in the framework of the restricted three-body problem where the two primaries are triaxial rigid bodies which emit radiation. Firstly, the positions and stability of the collinear equilibria are studied for the HD 191408, Kruger 60 and HD 155876 binary systems. Then, the planar and three-dimensional periodic motion about these points is considered. Our study includes both semi-analytical and numerical determination of these motions. It is found that all families of planar periodic orbits emanating from these points terminate with asymptotic periodic orbits at the triangular equilibrium points while the corresponding families of three-dimensional periodic orbits terminate with planar periodic orbits. Families of Halo orbits bifurcating from the first vertical critical periodic orbit of the three planar Lyapunov families were also considered.

On the triangular points within frame of the restricted three–body problem when both primaries are triaxial rigid bodies

Abstract In the framework of the restricted three–body problem when both primaries are triaxial rigid bodies, for different cases of Euler’s angles, the locations of the triangular points, and the stability conditions of motion in the proximity of these points are constructed. The numerical solution is obtained by using a fourth order Runge–Kutta–Gill integrator with some graphical investigations.

Fractal basins of attraction in the restricted four-body problem when the primaries are triaxial rigid bodies

The planar equilateral restricted four-body problem, formulated on the basis of Lagrange’s triangular solutions is used to determine the existence and locations of libration points and the Newton-Raphson basins of convergence associated with these libration points. We have supposed that all the three primaries situated on the vertices of an equilateral triangle are triaxial rigid bodies. This paper also deals with the effect of these triaxiality parameters on the regions of motion where the test particle is free to move. Further, the regions on the configuration plane filled by the basins of attraction are determined by using the multivariate version of the Newton-Raphson iterative system. The numerical study reveals that the triaxiality of the primaries is one of the most influential parameters in the four-body problem.

Motion around the Triangular Equilibrium Points in the Circular Restricted Three-Body Problem under Triaxial Luminous Primaries with Poynting-Robertson Drag

This paper explores the motion of an infinitesimal body around the triangular equilibrium points in the framework of circular restricted three-body problem (CR3BP) with the postulation that the primaries are triaxial rigid bodies, radiating in nature and are also under the influence of Poynting–Robertson (P-R) drag. We study the linear stability of these triangular points and for the numerical application, the binary stars Kruger 60 (AB) and Archird have been considered. These triangular points are not only perceived to move towards the line joining the primaries in the direction of the bigger primary with increasing triaxiality, they are also unstable owing to the destabilizing influence of P-R drag.

On the libration collinear points in the restricted three – body problem

Abstract In the restricted problem of three bodies when the primaries are triaxial rigid bodies, the necessary and sufficient conditions to find the locations of the three libration collinear points are stated. In addition, the Linear stability of these points is studied for the case of the Euler angles of rotational motion being θ i = 0, ψi + φi = π/2, i = 1, 2 accordingly. We underline that the model studied in this paper has special importance in space dynamics when the third body moves in gravitational fields of planetary systems and particularly in a Jupiter model or a problem including an irregular asteroid.

Collinear Equilibrium Points in the Relativistic R3BP when the Bigger Primary is a Triaxial Rigid Body

This study examines the effect of the relativistic factor as well as the triaxiality effect of the bigger primary on the positions and stability of the collinear points in the frame work of the post-Newtonian approximation. Using semi-analytical and numerical approach the collinear points are found to be unstable. A numerical exploration in this connection, with the Earth-Moon system, reveals that the relativistic factor has an effect on these positions. It is also found that under the combined effect of relativistic factor and triaxiality, the collinear point L1 moves towards the primaries with the increase in triaxiality, while L2 and L3 move away from the bigger primary. It is also seen that in most of the cases in the presence of triaxiality, the effect of relativistic factor on the positions of L1 and L3 is not observable; however it has an observable effect on the position of L2 in the presence of triaxiality except for the case 2.