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This study analyzes the motion of two infinitesimal bodies in the restricted problem of bodies, including an oblate spheroid and straight segment as primary bodies, along with the photogravitational effect of the more massive primary. The pertinent equations of motion of the infinitesimal bodies in the dimensionless variables are obtained by incorporating the effect caused by the oblateness, radiation pressure, and straight segment. On close examination of the parameters involved in the problem, a study of the equilibrium points of the dynamical system reveals that six of them are collinear with primary bodies' centers, while the remaining eight are not. An in-depth study is done on how the position of the equilibrium points changes with respect to the radiation parameter. The permissible and forbidden regions of motion are shown, and it is inferred that when the Jacobian constant is decreased, the permissible regions of motion expand. Furthermore, a linear stability analysis of the generated equilibrium points is done, and we observe that all the equilibrium points are unstable for a set of values of the model's parameters. [ABSTRACT FROM AUTHOR]

The present study investigates the combined effects of the oblateness and straight segment on the positions and linear stability of the equilibrium points in the restricted 2 + 2 body problem. The present model holds fourteen equilibrium points, out of which six are collinear with the centers of the primaries and the rest are non-collinear. It has been observed that the positions of all the equilibrium points are subsequently affected by the oblateness and length of the primary bodies. The linear stability of the equilibrium points is also presented by slightly perturbing the position of the equilibrium points. It is observed that for a considered set of parameters, all the fourteen equilibrium points are unstable. An application of the present model is also studied, for which the position and stability of the equilibrium points are investigated for the Earth-22 Kalliope-dual satellite system. It has been observed that for this system, all the equilibrium points are unstable except for two non-collinear equilibrium points that are found to be stable. [ABSTRACT FROM AUTHOR]

The fractal basins of convergence (BoC) linked with the equilibrium points are explored in the (N + 1)− body ring problem under the effect of small perturbations in the Coriolis and centrifugal forces when the bodies are sources of radiation. The evolution of the positions and the stability of the libration points and the possible regions of motion are determined as the function of mass parameter, the Coriolis, and centrifugal and radiation parameters. In addition, the parametric variation of the effect of the small perturbations in the Coriolis and centrifugal forces on the stability of the equilibrium points are also illustrated numerically. The multivariate version of the Newton–Raphson(NR) iterative method is used to analyze the effect of the radiation parameters and mass parameter on the topology of the BoC. [ABSTRACT FROM AUTHOR]

In non-linear stability analysis, the effects of non-linear terms in the equations of motion are considered. The presence of these non-linear terms can cause the results of linear stability to change dramatically. Therefore, the Arnold–Moser theorem (Kolmogorov–Arnold–Moser theory) has been used to study the non-linear stability of the equilibrium point in the planar Robe's restricted three-body problem when the second primary is a finite straight segment of length . The density parameter is considered zero. The equilibrium point has been found to be stable in non-linear sense for all mass ratios in the range of linear stability except possibly for six critical values of the mass ratio . The critical values depend on the length parameter , which shows that the length parameter has significant effect on the non-linear stability of the equilibrium point. The obtained results are applied to predict the stability of equilibrium point for Jupiter–Amalthea system. It is observed that the equilibrium point is unstable for Jupiter‒Amalthea system. [ABSTRACT FROM AUTHOR]

In this problem, one of the primaries of mass m1 is a Roche ellipsoid filled with a homogeneous incompressible fluid of density ρ1. The smaller primary of mass m2 is an oblate body outside the Ellipsoid. The third and the fourth bodies (of mass m3 and m4 respectively) are small solid spheres of density ρ3 and ρ4 respectively inside the Ellipsoid, 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 m1. The masses m3 and m4 mutually attract each other, do not influence the motions of m1 and m2 but are influenced by them. We have extended the Robe’s restricted three-body problem to 2+2 body problem under the assumption that the fluid body assumes the shape of the Roche ellipsoid (Chandrashekhar in Ellipsoidal figures of equilibrium, Chap. 8, Dover, New York, 1987 ). We have taken into consideration all the three components of the pressure field in deriving the expression for the buoyancy force viz (i) due to the own gravitational field of the fluid (ii) that originating in the attraction of m2 (iii) that arising from the centrifugal force. In this paper, equilibrium solutions of m3 and m4 and their linear stability are analyzed. We have proved that there exist only six equilibrium solutions of the system, provided they lie within the Roche ellipsoid. In a system where the primaries are considered as Earth-Moon and m3, m4 as submarines, the equilibrium solutions of m3 and m4 respectively when the displacement is given in the direction of x1-axis or x2-axis are unstable. [ABSTRACT FROM AUTHOR]

The belief that celestial bodies as perfect spheres results in certain idealistic conditions because most often they are in irregular shapes. With this in mind, we conduct an analysis to study the restricted three-body problem by taking the primaries as oblate spheroids. The modified mean motion expression [1] is used by incorporating the secular perturbation effect due to oblateness of the primaries on mean anomaly, argument of perigee and right ascension of ascending node. The model possesses five equilibrium points, which are subsequently affected by the oblateness parameters. Furthermore, we have studied the stability of the equilibrium points and it is observed that the collinear equilibrium points are always unstable. However, the non-collinear equilibrium points are stable for some combinations of the involved parameters. We have also plotted the zero velocity curves of the infinitesimal body for different values of the Jacobian constant and oblateness parameter. It has been observed that the value of the Jacobian constant has been playing a vital role in obtaining the permissible regions of motion of the infinitesimal body. Further, the results obtained of the study are applied to study the motion of a satellite in the Saturn–Titan system. [ABSTRACT FROM AUTHOR]

In the present paper, a study on the beyond-Newtonian collinear circular restricted (3 + 1) -body problem with spinning primaries is presented. In the setup of the collinear circular restricted four-body problem (CCR4BP), the three primaries are situated in the collinear central configuration where the masses of two peripheral primaries are equal. We have first derived the equations of motion for a system composed of three Kerr-like primaries. Further, we have unveiled how the existence, evolution, and stability of the libration points in the CCR4BP with beyond-Newtonian first-order correction terms and spin of the primaries depend on the mass parameter β and control parameter ε intending to manage the contribution of the beyond-Newtonian terms. It is observed that equilibria of the system are always even and varying from six to eighteen. [ABSTRACT FROM AUTHOR]

Associating the influences of viscosity and oblateness in the finite straight segment model of the Robe's problem, the linear stability of the collinear and non-collinear equilibrium points for a small solid sphere are analyzed. This small solid sphere is moving inside the first primary which is a homogeneous incompressible viscous fluid whose hydrostatic equilibrium figure is an oblate spheroid. The second primary is a finite straight segment. The existence of the equilibrium points is discussed after deriving the pertinent equations of motion of the small solid sphere. It is found that viscosity does not affect the location and number of equilibrium points but affects the stability as it converts the marginal stability to asymptotic stability. However, oblateness affects the locations of the equilibrium points. Applicability of the results of this study to an astrophysical problem is discussed, and we have calculated a lower bound on the ratio of the orbital radius and the total mass of the primaries of an astrophysical problem to which the results obtained may be applied. This ratio is called the scaling factor. [ABSTRACT FROM AUTHOR]

This paper deals with the photogravitational circular restricted three-body problem, including the effect of circular cluster of material points. The parametric evolution of the position of equilibrium points and the existence of a different number of equilibrium points are illustrated using graphs. A detailed investigation to find out the influence of the radiation pressure q , mass ratio μ and the circular cluster of material points M b on the geometry of basins of attraction is carried out. Further, the number of iterations required to obtain the desired level of accuracy is recorded and presented using probability distribution diagram. The correlations between the domain of convergence of equilibrium points and the corresponding number of the iterations required to obtain the desired level of accuracy are explained. We monitor the parametric evolution of the basin entropy to reveal the unpredictability in the basins of attractions. It is found that unpredictable (fractal) regions exist in the vicinity of boundaries of the basins of attraction. [ABSTRACT FROM AUTHOR]

The Newton-Raphson basins of convergence, related to the equilibrium points, in the collinear restricted four-body problem with repulsive Manev potential are numerically investigated. We monitor the parametric evolution of the position as well as of the stability of the equilibrium points, as a function of the parameter e. The multivariate Newton-Raphson optimal method is used for revealing the basins of convergence, by classifying dense grids of initial conditions in several types of two-dimensional planes. We perform a systematic and thorough analysis in an attempt to understand how the parameter e affects the geometry as well as the basin entropy of the convergence regions. The convergence areas are related with the required number of iterations and also with the corresponding probability distributions. [ABSTRACT FROM AUTHOR]

The Newton–Raphson basins of convergence, related to the equilibrium points, in the Sitnikov four-body problem with nonspherical primaries are numerically investigated. We monitor the parametric evolution of the positions of the roots, as a function of the oblateness coefficient. The classical Newton–Raphson optimal method is used for revealing the basins of convergence, by classifying dense grids of initial conditions in several types of two-dimensional planes. We perform a systematic and thorough analysis in an attempt to understand how the oblateness coefficient affects the geometry as well as the basin entropy of the convergence regions. The convergence areas are related with the required number of iterations and also with the corresponding probability distributions. [ABSTRACT FROM AUTHOR]

This manuscript aims to investigate numerically the effect of parameter λ on the basins of convergence (BoCs) associated with the equilibrium points (EPs) of the restricted rhomboidal five-body problem (RR5BP). Moreover, the parametric variation of EPs and zero velocity curves (ZVCs) are also illustrated. Firstly, we have scanned the entire interval for λ ∈ (1 3 , 3) to evaluate the critical value of λ where the number of EPs changes. It is observed that there exist either eleven, thirteen or fifteen EPs in total, however the stability analysis suggests that none of the EPs are linearly stable. The effect of the parameter λ and Jacobian constant C on the regions of possible motion are also illustrated. A systematic numerical investigation is performed to unveil the fact that how the parameter λ affects the geometry of the BoCs. Moreover, we have recorded the total number of iterations needed for each of the initial condition (IC) to converge a specific attractor and shown how the BoCs are related to these iterations and the associated probability distributions. Our numerical results strongly suggest that the parameter λ is indeed very influential factor in this dynamical system. The evolution of attracting regions in this dynamical system is very complicated yet an issue of paramount importance. • The restricted rhomboidal five-body problem are discussed. • Existence of the libration points and zero velocity curves are discussed. • The Newton–Raphson basins of convergence, linked to the libration points are unveiled. • The parametric evolution of the basin entropy and boundary basin entropy are presented. [ABSTRACT FROM AUTHOR]