Abstract:
Within the framework of restricted four-body problem, we study the motion of an infinitesimal mass by assuming that the primaries of the system are radiating-oblate spheroids surrounded by a circular cluster of material points. In our model, we assume that the two masses of the primaries $m_2$ and $m_3$ are equal to $\mu$ and the mass $m_1$ is $1-2\mu$. By using numerical approach, we have obtained the equilibrium points and examined their linear stability. The effect of potential created by the circular cluster and oblateness coefficients for the more massive primary and the less massive primary, on the existence and linear stability of the libration point have been critically examine via numerical computation. The stability of these points examined shows that the collinear and the non-collinear equilibrium points are unstable. The result presented in this paper have practical application in astrophysics.

Abstract:
In this paper, we extend the basic model of the restricted four-body problem introducing two bigger dominant primaries $m_1$ and $m_2$ as oblate spheroids when masses of the two primary bodies ($m_2$ and $m_3$) are equal. The aim of this study is to investigate the use of zero velocity surfaces and the Poincar\'{e} surfaces of section to determine the possible allowed boundary regions and the stability orbit of the equilibrium points. According to different values of Jacobi constant $C$, we can determine boundary region where the particle can move in possible permitted zones. The stability regions of the equilibrium points expanded due to presence of oblateness coefficient and various values of $C$, whereas for certain range of $t$ ($100 \le t \le 200$), orbits form a shape of cote's spiral. For different values of oblateness parameters $A_1~ (0

Abstract:
The oblateness and the photogravitational effects of both the primaries on the location and the stability of the triangular equilibrium points in the elliptical restricted three-body problem have been discussed. The stability of the triangular points under the photogravitational and oblateness effects of both the primaries around the binary systems Achird, Lyeten, Alpha Cen-AB, Kruger 60, and Xi-Bootis, has been studied using simulation techniques by drawing different curves of zero velocity. 1. Introduction The present paper is devoted to the analysis of the photogravitational and the oblateness effects of both primaries on the stability of triangular equilibrium points of the planar elliptical restricted three-body problem. The elliptical restricted three-body problem describes the dynamical system more accurately on account that the realistic assumptions of the motion of the primaries are subjected to move along the elliptical orbit. We have attempted to investigate the stability of triangular equilibrium points under the photogravitational and oblateness effects of both the primaries. The bodies of the elliptical restricted three-body problem are generally considered to be spherical in shape, but in actual situations, we have observed that several heavenly bodies are either oblate spheroid or triaxial rigid bodies. The Earth, Jupiter, and Saturn are examples of the oblate spheroid. The lack of sphericity in heavenly bodies causes large perturbation. In addition to the oblateness of heavenly bodies, the triaxiality, the radiation forces of the bodies, the atmospheric drag, and the solar wind are also causes of perturbation. This motivates studies of stability of triangular equilibrium points under the influence of oblateness and radiation of the primaries in the elliptical restricted three-body problem. The stability of the infinitesimal around the triangular equilibrium points in the elliptical restricted three-body problem described in considerable details is due to [1] and the problem was also studied [2–9]. The stability of motion of infinitesimal around one of the triangular equilibrium points ( ) also depends on and . Nonlinear stability of the triangular equilibrium points of the elliptical restricted three-body problem with or without radiation pressure was studied [10–12]. Furthermore, the nonlinear stability of the infinitesimal in the orbits or the size of the stable region around was studied numerically by [11] and the parametric resonance stability around in the elliptical restricted three-body problem has been studied [10]. The

Abstract:
In this paper we have examined the linear stability of triangular equilibrium points in the generalised photogravitational restricted three body problem with Poynting-Robertson drag. We have found the position of triangular equilibrium points of our problem. The problem is generalised in the sense that smaller primary is supposed to be an oblate spheroid. The bigger primary is considered as radiating. The equations of motion are affected by radiation pressure force, oblateness and P-R drag. All classical results involving photogravitational and oblateness in restricted three body problem may be verified from this result. With the help of characteristic equation, we discussed the stability. Finally we conclude that triangular equilibrium points are unstable.

Abstract:
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.

Abstract:
In the present work, the collinear equilibrium points of the restricted three-body problem are studied under the effect of oblateness of the bigger primary using an analytical and numerical approach. The periodic orbits around these points are investigated for the Earth-Moon system. The Lissajous orbits and the phase spaces are obtained under the effect of oblateness.

Abstract:
This paper studies the existence and stability of the artificial equilibrium points (AEPs) in the low-thrust restricted three-body problem when both the primaries are oblate spheroids. The artificial equilibrium points (AEPs) are generated by canceling the gravitational and centrifugal forces with continuous low-thrust at a non-equilibrium point. Some graphical investigations are shown for the effects of the relative parameters which characterized the locations of the AEPs. Also, the numerical values of AEPs have been calculated. The positions of these AEPs will depend not only also on magnitude and directions of low-thrust acceleration. The linear stability of the AEPs has been investigated. We have determined the stability regions in the xy, xz and yz-planes and studied the effect of oblateness parameters A_{1}(0<A_{1}<1)？and ？A_{2}(0<A_{2}<1) on the motion of the spacecraft. We have found that the stability regions reduce around both the primaries for the increasing values of oblateness of the primaries. Finally, we have plotted the zero velocity curves to determine the possible regions of motion of the spacecraft.

The positions and linear stability of the equilibrium
points of the Robe’s circular restricted three-body problem, are generalized
to include the effect of mass variations of the primaries in accordance with
the unified Meshcherskii law, when the motion of the primaries is determined by
the Gylden-Meshcherskii problem. The autonomized dynamical system with constant
coefficients here is possible, only when the shell is empty or when the
densities of the medium and the infinitesimal body are equal. We found that the
center of the shell is an equilibrium point. Further, when k﹥1; k beingthe constant
of a particular integral of the Gylden-Meshcherskii problem; a pair of
equilibrium point, lying in the -plane with each forming
triangles with the center of the shell and the second primary exist. Several of
the points exist depending on k; hence every point inside the shell
is an equilibrium point. The linear stability of the equilibrium points is
examined and it is seen that the point at the center of the shell of the
autonomized system is conditionally stable; while that of
the non-autonomized system is unstable. The triangular equilibrium points on
the -planeof both systems are
unstable.

Abstract:
We consider the modified restricted three body problem with power-law density profile of disk, which rotates around the center of mass of the system with perturbed mean motion. Using analytical and numerical methods we have found equilibrium points and examined their linear stability. We have also found the zero velocity surfaces for the present model. In addition to five equilibrium points there is a new equilibrium point on the line joining the two primaries. It is found that $L_2$ and $L_3$ are stable for some values of inner and outer radius of the disk while collinear points are unstable, but $L_4$ is conditionally stable for mass ratio less than that of Routh's critical value. Lastly we have obtained the effects of radiation pressure, oblateness and mass of the disk.

Abstract:
We study the effects of oblateness up to $J_4$ of the primaries and power-law density profile (PDP) on the linear stability of libration location of an infinitesimal mass within the framework of restricted three body problem (R3BP), by using a more realistic model in which a disc with PDP is rotating around the common center of the system mass with perturbed mean motion. The existence and stability of triangular equilibrium points have been explored. It has been shown that triangular equilibrium points are stable for $0<\mu<\mu_c$ and unstable for $\mu_c\leq\mu\leq1/2$, where $\mu_c$ denotes the critical mass parameter. We find that, the oblateness up to $J_2$ of the primaries and the radiation reduces the stability range while the oblateness up to $J_4$ of the primaries increases the size of stability both in the context where PDP is considered and ignored. The PDP has an effect of about $\approx0.01$ reduction on the application of $\mu_c$ to Earth-Moon and Jupiter-Moons systems. We find that the comprehensive effects of the perturbations have a stabilizing proclivity. However, the oblateness up to $J_2$ of the primaries and the radiation of the primaries have tendency for instability, while coefficients up to $J_4$ of the primaries have stability predisposition. In the limiting case $c=0$, and also by setting appropriate parameter(s) to zero, our results are in excellent agreement with the ones obtained previously. Libration points play very important role in space mission and as a consequence, our results have a practical application in space dynamics and related areas. The model may be applied to study the navigation and stationkeeping operations of spacecraft (infinitesimal mass) around the Jupiter (more massive) -Callisto (less massive) system, where PDP account for the circumsolar ring of asteroidal dust, which has a cloud of dust permanently in its wake.