Oblateness Factor Research Articles

Out-of-plane equilibrium points in the elliptic restricted three-body problem under albedo effect

The aim of this research is to show the significant effects of albedo on the existence of out-of-plane equilibria in the elliptic restricted three-body problem under an oblate primary model. We computed out-of-plane equilibria numerically and graphically for different values of the parameters μ, α, e, k and σ where μ, α, e, k and σ are mass parameter, albedo factor, eccentricity, ratio of the luminosity of smaller primary to luminosity of bigger primary considered as constant and oblateness factor due to smaller primary, respectively. Further, we examined the stability of out-of-plane equilibria and found that these equilibria are unstable in linear sense for all parameters μ, α, e, k and σ. Finally, the three-dimensional periodic orbits are analyzed for different values of albedo factor α and oblateness factor σ.

The analysis of periodic orbits generated by Lagrangian solutions of the restricted three-body problem with non-spherical primaries

The present paper deals with the periodic orbits generated by Lagrangian solutions of the restricted three-body problem when both the primaries are oblate bodies. We have illustrated the periodic orbits for different values of μ, h, σ1 and σ2 (h is energy constant, μ is mass ratio of the two primaries, σ1 and σ2 are oblateness factors). These orbits have been determined by giving displacements along the tangent and normal to the mobile coordinates as defined by (Karimov and Sokolsky, 1989). We have applied the predictor-corrector algorithm to construct the periodic orbits in an attempt to unveil the effect of oblateness of the primaries by taking the fixed values of parameters μ, h, σ1 and σ2.

Trajectory of asteroid 2017 SB20 within the CRTBP

Regular monitoring the trajectory of asteroids to a future time is a necessity, because the variety of known probably unsafe near-Earth asteroids are increasing. The analysis is perform to avoid any incident or whether they would have a further future threat to the Earth or not. Recently a new Near Earth Asteroid (2017 SB20) has been observed to cross the Earth orbit. In view of this we obtain the trajectory of Asteroid in the circular restricted three body problem with radiation pressure and oblateness. We examine nature of Asteroid’s orbit with Lyapunov Characteristic Exponents (LCEs) over a finite intervals of time. LCE of the system confirms that the motion of asteroid is chaotic in nature. With the effect of radiation pressure and oblateness the length of curve varies in both the planes. Oblateness factor is found to be more perturbative than radiation pressure. To see the precision of result obtain from numerical integration we show the error propagation and the numerical stability is assured around the singularity by applying regularized equations of motion for precise long-term study.

Non-collinear libration points in ER3BP with albedo effect and oblateness

In this paper we establish a relation between direct radiations (generally called radiation factor) and reflected radiations (albedo) to show their effects on the existence and stability of non-collinear libration points in the elliptic restricted three-body problem taking into account the oblateness of smaller primary. It is discussed briefly when \(\alpha =0\) and \(\sigma =0\), the non-collinear libration points form an isosceles triangle with the primaries and as e increases the libration points \(L_{4,5}\) move vertically downward (\(\alpha \), \(\sigma \) and e represents the radiation factor, oblateness factor and eccentricity of the primaries respectively). If \(\alpha = 0\) but \(\sigma \ne 0\), the libration points slightly displaced to the right-side from its previous location and form scalene triangle with the primaries and go vertically downward as e increases. If \(\alpha \ne 0\) and \(\sigma \ne 0\), the libration points \(L_{4,5}\) form scalene triangle with the primaries and as e increases \(L_{4,5}\) move downward and displaced to the left-side. Also, the libration points \(L_{4,5}\) are stable for the critical mass parameter \(\mu \le \mu _{c}\).

Pulsating Zero Velocity Surfaces and Fractal Basin of Oblate Infinitesimal in the Elliptic Restricted Three Body Problem

The pulsating surfaces of zero velocity of the elliptic restricted three body problem when both the primaries are luminous oblate spheroids is investigated considering the effect of the oblateness of the infinitesimal. It is observed that if the third participating body in the restricted problem has sufficiently high value of the oblateness factor, the shape of the pulsating zero velocity surfaces changes for certain values of true anomaly. The projection of the zero velocity surfaces on xy- and xz-plane is also studied along with the low-velocity sub-regions in the respective planes. Employing the multivariate Newton–Raphson iterative scheme, the basins of attraction of the equilibrium points on the xy- and xz-plane is determined. Simulation technique is applied to demonstrate the influence of the oblateness of the infinitesimal on the basins of attraction.

The locations of triangular equilibrium points in elliptic restricted three-body problem under the oblateness and radiation Effects

The Restricted Three-Body Problem (R3BP) considers motion of a third infinitesimal object under the gravitational influences of the primaries (bigger and smaller massive bodies) whose orbits are around the center of mass. If the orbits are elliptical, this belongs to Elliptic R3BP (ER3BP). In planar case it possesses five equilibrium points consisting of three collinear (L1, L2, and L3) and two triangular (L4 and L5). To mimic a better astrophysical R3BP, such as motion of a satellite in star-planet system, the classical problem can be generalized by considering the effects of radiation pressure and oblate spheroid shape on the primaries. We study analytically the locations and the stability of L4 and L5 equilibrium points in the frame of ER3BP with incorporating the effects of radiation for bigger primary and oblateness for smaller primary. Our study suggests that the oblateness factor (A2) and the radiation factor (q1) shift the positions of L4 and L5 points compared with the classical ones. We also find that there is a stability limit for motion of the third body around these points.

Stability of triangular points in the relativistic R3BP when the bigger primary is an oblate spheroid

In this paper, we study the effect of oblateness of the more massive primary in the relativistic R3BP. We observe that the locations of the triangular points and their stability are affected by the relativistic and oblateness factors. It is also noticed that the oblateness factor possesses destabilizing behavior. Therefore, the size of the region of stability decreases with increase in the value of the oblateness factor.Further, a numerical study on the locations of the triangular points and the critical mass for the Earth-Moon , Jupiter and its Moons, Saturn and its Moons systems is given.

Combined effects of Finite Straight Segment and Oblateness on the Libration Points in the Restricted-Three Body Problem

This paper deals with the existence and stability of the libration points in the restricted-three body problem when the smaller primary is a finite straight segment and bigger primary is an oblate spheroid. We have observed that five libration points do exist in this problem, out of which three are collinear and two are non-collinear with the primaries. The collinear libration points are unstable for all values of mass parameter /u and the non-collinear libration points are stable if jU < /Jc, where jU is a critical value of mass parameter /u. The range of stability depends length of the finite straight segment and oblateness factor.

On the photogravitational R4BP when the third primary is an oblate/prolate spheroid

The present paper deals with the photogravitational restricted four-body problem, when the third primary placed at the triangular libration point of the restricted three-body problem is an oblate/prolate body. The third primary $m_{3}$ is not influencing the motion of the dominating primaries $m_{1}$ and $m_{2}$ . We have studied the motion of $m_{4}$ , moving under the influence of the three primaries $m_{i}, i=1,2,3$ , but the motion of the primaries is not being influenced by infinitesimal mass $m_{4}$ . The aim of this study is to find the locations of the libration points and their stability. We obtain three collinear and five non-collinear libration points when the third body is an oblate spheroid and source of radiation. The collinear libration points are unstable for all the mass parameter. The non-collinear libration points are stable for different mass parameters and oblateness factors. Further there exist 12 libration points, depending on the mass ratio of dominating primaries, the prolateness, and the radiation pressure of the third primary. We have drawn the zero velocity surface to determine the possible allowed boundary region. We observed that for increasing values of the oblateness coefficient $A$ , the corresponding possible boundary region increases where the particle can freely move from one side to another side. Further, for different values of the Jacobi constant $C$ , we can find the boundary region where the particle can move in the possible allowed partitions. The stability region of the libration points expanded due to the presence of the oblateness coefficient and various values of $C$ . We further apply these findings to the Sun–Jupiter–asteroid–spacecraft system.

LOCATIONS OF OUT-OF-PLANE EQUILIBRIUM POINTS IN THE ELLIPTIC RESTRICTED THREE-BODY PROBLEM UNDER RADIATION AND OBLATENESS EFFECTS

This study deals with the generalization of the Elliptic Restricted Three-Body Problem (ER3BP) by considering the effects of radiation and oblate spheroid primaries. This may illustrate a gas giant exoplanet orbiting its host star with eccentric orbit. In the three dimensional case, this generalization may possess two additional equilibrium points (<TEX>$L_{6,7}$</TEX>, out-of-plane). We determine the existence of <TEX>$L_{6,7}$</TEX> in ER3BP under the effects of radiation (bigger primary) and oblateness (small primary). We analytically derive the locations of <TEX>$L_{6,7}$</TEX> and assume initial approximations of (<TEX>${\mu}-1$</TEX>, <TEX>${\pm}\sqrt{3A_2}$</TEX>), where <TEX>${\mu}$</TEX> and <TEX>$A_2$</TEX> are the mass parameter and oblateness factor, respectively. The fixed locations are then determined. Our results show that the locations of <TEX>$L_{6,7}$</TEX> are periodic and affected by <TEX>$A_2$</TEX> and the radiation factor (<TEX>$q_1$</TEX>).

A study of non-collinear libration points in restricted three body problem with stokes drag effect when smaller primary is an oblate spheroid

The existence of non-collinear libration points and their stability (in linear sense) are examined in the circular restricted three body problem, in which we have considered the smaller primary as an oblate spheroid and bigger one a point mass including the effect of dissipative force specially Stokes drag. Two non-collinear libration points are found but they are unstable for the given range of dissipative constant $k$ and oblateness factor $A$ (i.e. $0 < k < 1$ and $0 < A < 1$ respectively). Stability of non-collinear libration points are discussed using a different analytical approach. We have also shown analytically the non-existence of collinear libration points due to effect of Stokes drag.

On the R4BP when third primary is an oblate spheroid

The present paper deals with the restricted four-body problem, when the third primary placed at the triangular libration point of the restricted three-body problem is an oblate body. The third primary m 3 is not influencing the motion of the dominating primaries m 1 and m 2. We have studied the motion of m 4, moving under the influence of the three primaries m i , i=1,2,3, but the motion of the primaries is not being influenced by infinitesimal mass m 4. The aim of this study is to find the locations of equilibrium points and their stability. We obtain three collinear and five non-collinear equilibrium points. The collinear equilibrium points are unstable for all the mass parameter. The non-collinear points are stable for different mass parameter and oblateness factor. Also, we have considered the autonomous coplanar circular restricted four-body problem with the infinitesimal mass as a low-thrust spacecraft. Artificial equilibrium points are created with the use of continuous low-thrust propulsion. The obtained results show that, in absence of thrust there are unstable equilibrium points close to the third primary. Also, using the constant low-thrust, the artificial equilibrium points can be generated, which move from the natural equilibrium points. Further, it is proved that in certain region closed to the third primary, these points are stable. We have drawn the zero velocity surfaces to determine the possible allowed boundary regions. We observed that for increasing values of oblateness coefficient A, the corresponding possible boundary regions increase where the particle can freely move from one side to another side. Further, for different values of Jacobi constant C, we can find the boundary region where the particle can move in possible allowed partitions. The stability regions of the equilibrium points expanded due to presence of oblateness coefficient and various values of C.

Robe’s restricted problem of 2+2 bodies with one of the primaries an oblate body

In this problem, one of the primaries of mass $m^{*}_{1}$ is a rigid spherical shell filled with a homogeneous incompressible fluid of density ρ1. The smaller primary of mass m2 is an oblate body outside the shell. The third and the fourth bodies (of mass m3 and m4 respectively) are small solid spheres of density ρ3 and ρ4 respectively inside the shell, with the assumption that the mass and the radius of the third and the fourth body are infinitesimal. We assume that m2 is describing a circle around $m^{*}_{1}$ . The masses m3 and m4 mutually attract each other, do not influence the motions of $m^{*}_{1}$ and m2 but are influenced by them. We also assume that masses m3 and m4 are moving in the plane of motion of mass m2. In the paper, equilibrium solutions of m3 and m4 and their linear stability are analyzed. There are two collinear equilibrium solutions for the system. The non collinear equilibrium solutions exist only when ρ3=ρ4. There exist an infinite number of non collinear equilibrium solutions of the system, provided they lie inside the spherical shell. In a system where the primaries are considered as earth-moon and m3,m4 as submarines, the collinear equilibrium solutions thus obtained are unstable for the mass parameters μ,μ3,μ4 and oblateness factor A. In this particular case there are no non-collinear equilibrium solutions of the system.

Application of binary pulsars to axisymmetric bodies in the Elliptic R3BP

Binary systems hosting astrophysical compact objects such as white dwarfs and/or neutron stars provide excellent test beds for studying the impact of the oblateness of the main bodies in the restricted three-body problem (R3BP). The case is investigated when the primary bodies are non-luminous, non-spherical (oblate) bodies and the third body of infinitesimal mass is also an oblate spheroid. The existence of extra solar planets orbiting these systems constitutes a three-body problem which makes them excellent models for this axisymmetric ER3BP. The positions of the equilibrium points are affected by the oblateness parameters of the three-bodies; this is shown for double neutron star binaries. The triangular points are stable for 0<μ<μc; where μ is the mass ratio (μ≤1/2) and μc is the critical mass value influenced by the eccentricity, semi major axis and oblateness factors. The size of the region of stability increases with decreasing values of the oblateness. The oblateness of the system’s bodies does not affect the nature of the stability of the collinear points since they remain unstable. Due to the almost equal masses of the primaries, our study shows that even the triangular points of these systems are unstable.

On the stability of triangular points in the elliptic R3BP under radiating and oblate primaries

This paper investigates the stability of triangular equilibrium points (L4,5) in the elliptic restricted three-body problem (ER3BP), when both oblate primaries emit light energy simultaneously. The positions of the triangular points are seen to shift away from the line joining the primaries than in the classical case on account of the introduction of the eccentricity, semi-major axis, radiation and oblateness factors of both primaries. This is shown for the binary systems Achird, Luyten 726-8, Kruger 60, Alpha Centauri AB and Xi Bootis. We found that motion around these points is conditionally stable with respect to the parameters involved in the system dynamics. The region of stability increases and decreases with variability in eccentricity, oblateness and radiation pressures.

Stability of Collinear Points in the Generalized Photogravitational Robes Restricted Three-Body Problem

In studying the effects of radiation and oblateness of the primaries on the stability of collinear equilibrium points in the Robes restricted three-body problem we observed the variations of the density parameter k with the mass parameter μ for constant radiation and oblateness factors on the location and stability of the collin-ear points L1, L2and L3. It is also discovered that the collinear points are unstable for k > 0 and stable for k < 0.

Periodic orbits generated by Lagrangian solutions of the restricted three body problem when one of the primaries is an oblate body

We have studied periodic orbits generated by Lagrangian solutions of the restricted three body problem when one of the primaries is an oblate body. We have determined the periodic orbits for different values of μ, h and A (h is energy constant, μ is mass ratio of the two primaries and A is an oblateness factor). These orbits have been determined by giving displacements along the tangent and normal to the mobile coordinates as defined by Karimov and Sokolsky (Celest. Mech. 46:335, 1989). These orbits have been drawn by using the predictor-corrector method. We have also studied the effect of oblateness by taking some fixed values of μ, A and h. As starters for our method, we use some known periodic orbits in the classical restricted three body problem.

Effect of oblateness on the non-linear stability of the triangular liberation points of the restricted three-body problem in the presence of resonances

The non-linear stability of the triangular libration points of the restricted three-body problem is studied under the presence of third and fourth order resonance's, when the more massive primary is an oblate spheroid. In this study Markeev's theorem are utilised with the help of KAM theorem. It is found that the stability of the triangular libration points are unstable in the third order resonance case and stable in the fourth order resonance case, for all the values of oblateness factor A1.

Trajectories of satellites under the combined influences of earth oblateness and air drag

Trajectories of satellites under the influences of earth oblateness and air drag are derived by the asymptotic method in nonlinear mechanics. Based on the assumptions: (1) the dominant oblateness factor of the earth is the second harmonic (J 2), (2) a non-rotating, spherically symmetric atmosphere and an exponential distribution of atmospheric density, (3) original elliptical orbits being of small eccentricity, closed-form solutions for the improved first order approximation are obtained. After finding the osculating orbital elements of the resulting trajectories, we expose the behavior of osculating orbits at various inclinations.