Photogravitational Restricted Three-body Problem Research Articles

Perturbing and Oblateness's Impact on the Generalized Photogravitational Restricted Three-Body Problem

In this research work, the effects of perturbing and oblateness on the generalised photogravitational restricted three body problem (RTBP) are examined when the primary are regarded as radiation sources and oblate spheroid. The perturbing forces were included in the equations of motion that we developed for the system. The triangular libration points were derived in terms of the perturbing parameters using linear approximations, and their influence was seen. Applying Murray's criteria, we determined that the triangular libration points remained unstable due to the nature of the characteristics equation matching to the variational equations of motion of the system generated.

Out-of-plane equilibria in the perturbed photogravitational restricted three-body problem with Poynting-Robertson (P-R) drag

We consider the primaries of the circular restricted three-body problem (CR3BP) to be luminous and study the effects of small perturbations in the Coriolis and centrifugal forces together with Poynting-Robertson (P-R) drag from both primaries on the motion of an infinitesimal body near the out-of-plane equilibrium points (OEPs). It is found that these points appear in pairs and, depending on the values of the parameters of the system, their number may be zero, two, L6,7 or four, L6,7,8,9. It is observed that the positions of these points depend on all the system parameters except small perturbation in the Coriolis force. This has been shown for binary systems RW-Monocerotis and Krüger-60. The linear stability of the out-of-plane equilibria is also studied and it is found that stability of some of these points significantly depends on the perturbing forces. Specifically, the motion of the infinitesimal body around the equilibria is conditionally stable only at points L6 and L7 in the absence of P-R drag effect in both binary systems. However, all the equilibria are unstable in the presence of the P-R drag effect. We may conclude therefore, that P–R effect destroys stability of the out-of-plane equilibria.

Dynamics of Infinitesimal Particle in the Framework of Photo-Gravitational Restricted Three-Body Problem

Dynamics of Infinitesimal Particle in the Framework of Photo-Gravitational Restricted Three-Body Problem

Lie-Series Solution of Restricted Three-Body Problem: Application to Binary Stellar Systems

The purpose of this present paper is to find the Lie-series solutions of the photo-gravitational restricted three-body problem and to apply this Lie-series theory in binary stellar systems. In this paper, we have taken four stellar binary systems namely Kepler-34, Kepler-35, Kepler-413 and Kepler-16. Firstly, the zero-velocity curves are studied in the four binary stellar systems. The Lie-integration method is a concept to deal with the system of ordinary differential equations(ODEs) with the help of Lie-series. We have applied this method to solve the equations of motion of restricted three-body problem with radiating primaries and obtained the solutions of the equations of motion. Then the solution obtained by Lie-series method is compared with that of obtained from the well known Runge-Kutta method. In addition, we have shown the absolute errors graphically for Lie-series method and Runge-Kutta method.

On merging of resonant periodic orbits 4:3; 3:2 and 2:1 in Sun-Jupiter photo gravitational restricted three-body problem

We explore the merging of resonant periodic orbits in the frame work of planar circular restricted three body problem with the help of Poincaré surface of section. We have studied the effect of solar radiation pressure on 4:3, 3:2 and 2:1 periodic orbits. It is found that radiation pressure helps in merging these orbits (4:3 and 3.2 into 1:1 resonance and 2:1 into 1:1 resonance). At the time of merging these orbits become near-circular. The period and size of these orbits reduce with the increase in radiation pressure. Â

Motion around the out-of-plane equilibrium points in the photogravitational Copenhagen elliptic restricted three-body problem with oblateness

This paper studies the motion of an infinitesimal mass near the out-of-plane equilibrium points (OEPs) in the elliptic restricted three-body problem (ER3BP) in the case of two equally heavy bodies (Copenhagen problem) where one of the two primaries is a radiation source and the other an oblate spheroid. We found, as in the photogravitational circular restricted three body problem, that the equations of motion of the three dimensional photogravitational ER3BP allow the existence of OEPs if the radiation parameter has negative values. There are two out of plane equilibria that lie in the (ξ, ζ) plane in symmetrical positions with respect to the (ξ, η) plane. The positions of the OEPs are affected by the parameters involved in the systems' dynamics. In particular, the positions change with increase in the radiation pressure, oblateness, eccentricity and semi-major axis of the orbits. As an application, the positions and linear stability of the problem are investigated numerically for the binary system B1534+12. The OEPs are found unstable.

The perturbed photogravitational restricted three-body problem: Analysis of resonant periodic orbits

In the framework of the perturbed photo-gravitational restricted three-body problem, the first order exterior resonant orbits and the first, third and fifth order interior resonant periodic orbits are analyzed. The location, eccentricity and period of the first order exterior and interior resonant orbits are investigated in the unperturbed and perturbed cases for a specified value of Jacobi constant C.It is observed that as the number of loops increases successively from one loop to five loops, the period of infinitesimal body increases in such a way that the successive difference of periods is either 6 or 7 units. It is further observed that for the exterior resonance, as the number of loops increases, the location of the periodic orbit moves towards the Sun whereas for the interior resonance as the number of loops increases, location of the periodic orbit moves away from the Sun. Thereby we demonstrate that the location of resonant orbits of the given order moves away from the Sun when perturbation is included.The evolution of interior first order resonant orbit with three loops is studied for different values of Jacobi constant C. It is observed that when the value of C increases, the size of the loop decreases and degenerates finally into a circle, the eccentricity of periodic orbit decreases and location of the periodic orbit moves towards the second primary body.

Dynamics of the spatial restricted three-body problem stabilized by Hamiltonian structure-preserving control

The local invariant (stable, unstable, and center) manifolds of a saddle $$\times $$ center $$\times $$ center equilibrium point can be used to construct a three-dimensional Hamiltonian structure-preserving (HSP) controller. The linear stability of the controlled Hamiltonian system is verified when one of the proposed criteria is satisfied. The nonlinear dynamics of a linearly stabilized Hamiltonian system is analyzed by normalizing the Hamiltonian high-order perturbation terms using the Lie series method. The analytical solutions of the invariant tori are then obtained in trigonometric series form. A theorem for the Nekhoroshev stability of the controlled Hamiltonian system is provided. When the constructed HSP controller is applied to a photogravitational restricted three-body problem with oblateness, a hyperbolic artificial equilibrium point at which a criterion is satisfied can be stabilized, and bounded Lissajous trajectories near the equilibrium point are obtained. The allocation law demonstrates that the attitude angles and lightness number of the solar sail can serve as control parameters to provide the required acceleration of the HSP controller.

Diagonalization of Hamiltonian in the photogravitation-al restricted three body problem with P-R drag

In this paper, restricted, three-body problem (RTBP) is generalised to study the non-linear stability of equilibrium points in the photogravitational RTBP with P-R drag. In the present study, both primaries are considered as a source of radiation and effect of P-R drag. Hence the problem will contain four parameters q1, q2, W1 and W2. At first, the Lagrangian and the Hamiltonian of the problem were computed, then the Lagrangian function is expanded in power series of the coordinates of the triangular equilibrium points x and y. Lastly, diagonalized the quadratic term of the Hamiltonian of the problem, which is obtained by expanding original Lagrangian or Hamiltonian by Taylor's series about triangular equilibrium point. Finally, the study concluded that the diagonalizable Hamiltonian is H2=ω1I1-ω2I2.

Progression of bifurcated family f type periodic orbits in the circular restricted three-body problem

Progression of f-type family of periodic orbits, their nature, stability and location nearer the smaller primary for different mass ratios in the framework of circular restricted three-body problem is studied using Poincaré surfaces of section. The orbits around the smaller primary are found to decrease in size with increase in Jacobian Constant C, and move very close towards the smaller primary. The orbit bifurcates into two orbits with the increase in C to 4.2. The two orbits that appear for this value of C belong to two adjacent separate families: one as direct orbit belonging to family g of periodic orbits and other one as retrograde orbit belonging to family f of periodic orbits. This bifurcation is interesting. These orbits increase in size with increase in mass ratio. The elliptic orbits found within the mass ratio 0 < µ ≤ 0.1 have eccentricity less than 0.2 and the orbits found above the mass ratio µ > 0.1 are elliptical orbits with eccentricity above 0.2. Deviations in the parameters: eccentricity, semi-major axis and time period of these orbits with solar radiation pressure q are computed in the frame work of photogravitational restricted Three-body problem in addition to the restricted three-body problem. These parameters are found to decrease with increase in the solar radiation pressure.

Interior Resonance Periodic Orbits in Photogravitational Restricted Three-body Problem

Interior Resonance Periodic Orbits in Photogravitational Restricted Three-body Problem

Halo Orbits at Sun-Mars L1, L2 in the Photogravitational Restricted Three-Body Problem with Oblateness

Halo Orbits at Sun-Mars L1, L2 in the Photogravitational Restricted Three-Body Problem with Oblateness

Effects of Poynting-Robertson Drag and Oblateness on the Stability of Photogravitational Restricted Three-Body Problem

Effects of Poynting-Robertson Drag and Oblateness on the Stability of Photogravitational Restricted Three-Body Problem

Halo Orbits around Sun-Earth L1 in Photogravitational Restricted Three-Body Problem with Oblateness of Smaller Primary

This paper deals with generation of halo orbits in the three-dimensional photogravitational restricted three-body problem, where the more massive primary is considered as the source of radiation and the smaller primary is an oblate spheroid with its equatorial plane coincident with the plane of motion. Both the terms due to oblateness of the smaller primary are considered. Numerical as well as analytical solutions are obtained around the Lagrangian point L1, which lies between the primaries, of the Sun-Earth system. A comparison with the real time flight data of SOHO mission is made. Inclusion of oblateness of the smaller primary can improve the accuracy. Due to the effect of radiation pressure and oblateness, the size and the orbital period of the halo orbit around L1 are found to increase.

On the collinear point L3 in the generalized relativistic R3BP

The photogravitational oblate restricted three-body problem is carried out within the post-Newtonian framework. The primaries are assumed to be oblate as well as radiant sources. The perturbed location of the collinear point \( L_{3}\) is computed. A series form of \( L_{3}\) location is obtained as a new analytical results. In order to introduce a semi-analytical view, a Mathematica program is constructed so as to draw the locations of collinear point \( L_{3}\)versus the mass ratio \( \mu\in (0, 0.5)\) taking into account one or more of the considered perturbations. All the obtained figures are analyzed.

Unveiling the influence of the radiation pressure in nature of orbits in the photogravitational restricted three-body problem

The case of the planar circular photogravitational restricted three-body problem where the more massive primary is an emitter of radiation is numerically investigated. A thorough numerical analysis takes place in the configuration $(x,y)$ and the $(x,C)$ space in which we classify initial conditions of orbits into three main categories: (i) bounded, (ii) escaping and (iii) collisional. Our results reveal that the radiation pressure factor has a huge impact on the character of orbits. Interpreting the collisional motion as leaking in the phase space we related our results to both chaotic scattering and the theory of leaking Hamiltonian systems. We successfully located the escape as well as the collisional basins and we managed to correlate them with the corresponding escape and collision times. We hope our contribution to be useful for a further understanding of the escape and collision properties of motion in this interesting version of the restricted three-body problem.

Out-of-Plane Equilibrium Points in the Photogravitational CR3BP with Oblateness and P-R Drag

This paper investigates the motion of a test particle around the out-of-plane equilibrium points in the circular photogravitational restricted three-body problem when the effect of radiation pressure from the smaller primary and its Poynting-Robertson (P-R) drag are taken into account, and the bigger primary is modeled as an oblate spheroid. These points lie in the xz-plane almost directly above and below the center of the oblate primary. The equilibrium points are sought, and we observe that, there are two coordinate points L 6,7 which depend on the oblateness of the bigger primary, and the radiation pressure force and P-R drag of the smaller primary. The positions and linear stability of the problem are investigated both analytically and numerically for the binary system Cen X-4. The out-of-plane equilibrium points are found to be unstable in the sense of Lyapunov due to the presence of a positive real root.

Perturbed Location of L1 point in the photogravitational relativistic restricted three-body problem (R3BP) with oblate primaries

The restricted three-body problem is studied in the post-Newtonian framework. The primaries are assumed oblate radiant sources. The perturbed location of the L1 point is computed, and a series form of the location of this point is obtained as a new analytical result. To introduce a semianalytical view, a Mathematica 9 program is constructed so as to draw the location of L1 versus the mass ratio μ ∈ (0, 0.5) taking into account one or more of the considered perturbations. All the obtained illustrations are analyzed.

Quasi-periodic Solutions of a Spiral Type for Photogravitational Restricted Three-Body Problem

A new type of exact solutions for photogravitational restricted three-body problem (a case of spiral motion) is presented here. A key point is that we obtain the appropriate specific case of spiral motions from the Jacobian-type integral of motion for photogravitational restricted three-body problem (when orbit of small 3-rd body is assumed to be like a spiral). Besides, we should especially note that there is a proper restriction to the type of spiral orbital motion of small 3-rd body, which could be possible for choosing as the exact solution of equations for photogravitational restricted three-body problem. The main result, which should be outlined, is that in a case of quasi-planar orbital motion (of the small 3-rd body) the asymptotic expression for component z of motion is proved to be given by the proper elliptical integral.

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.