Zero-velocity Curves Research Articles

Out-of-Plane Equilibrium Points in the Photogravitational Hill Three-Body Problem

This paper investigates the movement of a negligible mass body (third body) in the vicinity of the out-of-plane equilibrium points of the Hill three-body problem under the effect of radiation pressure of the primaries. We study the effect of the radiation parameters through the factors qi,i=1,2 on the existence, position, zero-velocity curves and stability of the out-of-plane equilibrium points. These equilibrium positions are derived analytically under the action of radiation pressure exerted by the radiating primary bodies. We determined that these points emerge in symmetrical pairs, and based on the values of the radiation parameters, there may be two along the Oz axis and either none or two on the Oxz plane (outside the axes). A thorough numerical investigation found that both radiation factors have a strong influence on the position of the out-of-plane equilibrium points. Our results also reveal that the parameters have impact on the geometry of the zero-velocity curves. Furthermore, the stability of these points is examined in the linear sense. To do so, the spatial distribution of the eigenvalues on the complex plane of the linearized system is visualized for a wide range of radiation parameter combinations. By a numerical investigation, it is found that all equilibrium points are unstable in general.

Investigation of motion around out-of-plane points in the restricted three-body problem with variable shape and masses

The paper examines out-of-plane equilibrium points (OEPs) of the restricted three-body problem with variable masses and shape. The bigger primary varies it shape as the lengths of the semi-axes vary with time. For the autonomized system, two pair of OEPs L6,7κ and L8,9κ, are obtained and differ from those of the non-autonomous system due to time t. The stability of OEPs of both systems is found to be unstable. Further, numerical illustrations is provided when variations in shape of the bigger primary is, a triaxial prolate, a sphere and a triaxial oblate shape. The positions, stability and zero velocity curves (ZVC) of the particle around the OEPs are explored. It is seen that when the bigger primary is a triaxial prolate body, the OEPs L8,9κ are closer to the primaries than L6,7κ. However, the converse happens when it is a triaxial oblate body. Also, when the bigger primary is a triaxial oblate body, the OEPs are farther away from the primaries than when it is a triaxial prolate. In the case of the ZVC, it is seen that when the bigger primary is a triaxial prolate body, there is a petal around it, and region of allowed motion of the particle increases, while the region reduces when the bigger primary evolves from a sphere to a triaxial oblate body. This study can be used to describe motion of a dust grain in the vicinity of Betelgeuse, a red giant star whose mass and shape changes with time and its stellar companion.

Model of Close Orbit Dynamics Around Asteroid (338) Budrosa

Abstract In this paper, we revisited the model of gravitational potential generated by irregular shapes. A finite straight segment model was used to investigate the motion of particles around a massive straight segment of length 2l, mass M, and uniformly rotates with angular velocity ω. We extended this model by converting it to the cylindrical coordinate and applied this model to the asteroid (338) Budrosa. We obtained the position of equilibrium points in this system. We also estimate the zero-velocity curve and determine the Poincare map and its periodic orbit around equilibrium points.

Zero velocity curves of a dust grain around equilibrium points under effects of radiation, perturbations and variable Kruger 60

The paper unveils zero velocity curves of a dust grain particle around eleven equilibrium points under effects of radiation, perturbations and variable Kruger 60. The study is carried out using the model formulations of the restricted three-body problem in which the primaries are assumed to be radiation sources and vary their masses with time coupled with small perturbations in the Coriolis and centrifugal forces due to circular motion. The equations of motion of the non-autonomous system and those of the autonomized equations with constant coefficients have been stated and the equilibrium points explored. Numerical illustrations have been provided to support the study for a particular case when the dust grain moves under the influence of the central binary Kruger 60. It is seen that there exist a pair of triangular points, five collinear equilibrium points, and four out-of-plane equilibrium points, which are defined by the radiation pressure of the Kruger 60, centrifugal perturbation and the arbitrary constant K which defines the sum of the masses of Kruger 60. Finally, the zero velocity curves of the dust grain around the equilibrium points are explored and the effects of the parameters divulged. In the case of the collinear equilibrium points, as K increases the region where motion is allowed decreases, while in the case of the triangular points, the dust grain is permitted to move around the primaries and the triangular points and is also allowed access to the exterior realm as K increases. In the case of the out-of-plane equilibrium points, as K increases the region where motion is allowed decreases while the presence of the perturbing forces due to radiation and centrifugal perturbation also reduces the region where motion of the dust grain particle is allowed. It is seen that the Coriolis perturbation do not affect the locations and zero velocity curves around all the equilibrium points.

Motion and zero velocity curves of a dust grain around collinear libration points for the binary IRAS 11472-0800 and G29-38 with a triaxial star and variable masses

This paper investigates motion and zero velocity curves of a dust grain around collinear libration points (CLPs) of the circular restricted three-body problem (R3BP) when both stars IRAS 11472-0800 and G29-38 vary their masses according to the unified Mestschersky law (UML) and their motion is described by the Gylden-Mestschersky problem (GMP) with further assumption that the bigger body is a variable triaxial star. The non-autonomous equations of motion of the model are derived and transformed to the autonomized forms using the Mestschersky transformation (MT), the UML, the particular integral and solutions of the GMP, and a transformation we introduced which enable us to convert the time dependent triaxiality of the bigger star to constant triaxiality. The CLPs are examined and it is seen that when motion takes place on the plane joining the stars, there are three CLPs. To further validate the existence of the CLPs, we obtain three polynomial equations of degree seven for the three CLPs, respectively and we numerically explored the roots. It is seen that only three CLPs exists for all values of the mass parameters coupled with triaxiality of the bigger star. The stability of these points is studied and it is seen that CLPsL1 andL2 are unstable for all mass parameters throughout the interval 0<κ<∞, whereκ is a constant of the particular integral of the GMP and depicts the mass variation parameter of the stars. However, CLPL3can be stable and unstable due to the triaxiality of the bigger star, mass parameters and the mass variation parameter. Further, the zero velocity curves (ZVCs) of the dust grain around the CLPs are investigated and the effects of the system parameters divulged. It is seen that, the mass parameters and mass variation parameter can increase or decrease the region where motion of the dust grain is dynamically forbidden, while triaxiality of the bigger star increases the region where motion is forbidden around the CLPs and it allows the appearance of a petal-shaped ZVC around the bigger star. Finally, the orbits of the dust grain around the CLPs is explored, and it is seen that the paths around each collinear point differs. The deviations in the paths are due to the perturbing effects of the mass parameters, triaxiality of the bigger star and the mass variation parameter of the stars. Our problem could have important physical applications when studying celestial bodies of triaxial shape with changing masses of the bodies. In particular, we can assume that the binary system may describe a host star along with an exoplanet, or a massive exoplanet with an exomoon whose masses vary with time.

Unveiling the intricacies of attracting zones in magnetic binary systems: Investigating the impact of Yukawa correction

This study delves into the restricted three-body problem with a Yukawa correction to Newtonian gravitational forces, focusing on magnetic binary systems. We scrutinize the influence of Yukawa correction parameters (α, β) and the ratio of magnitude of magnetic moments (λ) on the system’s equilibrium points and their stability, zero-velocity curves. In our case, there exist of five and seven equilibrium points and all are found to be unstable for given range of parameters. Our examination extends to the basins of convergence and the existence of fractal under the influence of α and λ. Graphs drawn with the help of Wolfram Mathematica software vividly portray the parameter-driven evolution of equilibrium points, zero-velocity curves and basins of convergence. Furthermore, we explore the fractal characteristics within the basins of convergence, offering valuable insights into the complex dynamics of magnetic binary systems with Yukawa correction.

The capture of small near-Earth asteroids in a bound binary pair in Earth's orbit

Hill's approximation models the motion of two small masses gravitationally interacting with each other and perturbed by a large central body. An application of the model is employed in this paper to manoeuvre the relative motion of two asteroids by a small impulse to capture them in bound binary motion in Earth's orbit. The initial conditions prior to the capture manoeuvre are restricted in a parameter space termed the gateway region. The gateway region is produced by applying the constraint that the capture impulse is a real-valued function, and the zero-velocity curve closes, enveloping both asteroids. A full mission scenario is designed with three impulses. The first impulse transfers one asteroid from a far field region to the gateway region, where there is mutual interaction with the second asteroid, which is assumed to be the origin of the relative motion. The second impulse changes the trajectory of the asteroid in the gateway region to a linear drift approaching the second asteroid. The third impulse is calculated by means of increasing the Jacobi integral to reach the critical value required to close the zero-velocity curve. It is demonstrated in principle that the triple impulses enable the candidate asteroids to be captured in a bound pair in Earth's orbit. It is also shown that the thruster on-time for the capture impulse as applied for the different small asteroids considered is short relative to the natural time-scale of the orbital dynamics of the problem, which shows the feasibility of the impulse approximation with the typical thruster forces applied. This strategy could provide a basis for parking small captured near-Earth asteroids in Earth's orbit.

The Role of Oblateness in Shaping Stability and Zero Velocity Curves in the Perturbed Restricted Three-Body Problem

The Role of Oblateness in Shaping Stability and Zero Velocity Curves in the Perturbed Restricted Three-Body Problem

Effect of Stokes drag in the restricted four-body problem with variable mass

The aim of this paper is to study the existence, locations and stability of the equilibrium points as well as zero velocity curves (ZVCs) under the effect of dissipative force (Stokes drag) in the restricted four-body problem (R4BP) with a variable mass. We have studied the dynamics of the infinitesimal body whose mass is variable and changing according to Jeans’ law. We have determined the equations of motion of the infinitesimal body by using the space time transformation of Meshcherskii when the change in mass is non-isotropic. We have determined the locations of equilibrium points in the special case of non-isotropic variation of mass under the effect of the dissipative force and found that all the equilibrium points are non-collinear. The effect of parameter k(0<k<1), which controls the dissipative force, on the locations of the libration points and their stability are investigated. It has been observed that the energy integral is time dependent due to the presence of the Stokes drag force. Further, it has been observed that all the equilibrium points are unstable for all values of the parameters used.

On the rhomboidal restricted five-body problem: Analysis of the basins of convergence

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∈(13,3) 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.

Floquet stability analysis of equilibrium points and numerical exploration of pulsating zero-velocity curves and Newton–Raphson basins of attraction

An important and pivot aspect of a dynamical system is the stability property and its range, which play a crucial role towards the stabilisation of a mission design. Present paper deals with Floquet stability analysis of equilibrium points and estimation of pulsating zero velocity curves and Newton–Raphson basins of attraction associated to these equilibrium points under the radiation pressure effect in the photo-gravitational planar elliptic restricted four body problem. Floquet stability test for the equilibrium points is performed with the help of the transition curves, which are generated under the influence of radiation pressure. It is noticed that stability range of the respective equilibrium points have deviated due to radiation pressure. To observe the possible regions of motion for restricted body, the pulsating zero velocity curves are numerically explored after establishing an invariant relation. It is found that not only the radiation parameter but also eccentricity and true anomaly reflect a considerable impact on the shape and size of forbidden regions. To see the tendency of randomly selected points in the phase space, Newton–Raphson basins of attraction associated to each attractor are estimated and effects of perturbing parameters are analysed. It is observed that effects of radiation and mass parameters are considerable however, that of eccentricity and true anomaly are negligible. Moreover, from probability distribution bar diagram, most probable number of iterations is 10 for the case when two primaries are of equal masses and third one is different, whereas it is 20 for the case when all primaries are of identical mass. These results will be helpful to study the generalised problem along with other kind of perturbations.

Stability analysis of Lagrangian points of geo-synchronous satellite including the resistive force and earth’s equatorial ellipticity

In this paper, we have investigated the existence of the Lagrangian points and their stability in the problem of geo-synchronous satellite including the effect of resistive force and earth’s equatorial ellipticity parameter γ. Equations of motion of satellite are expressed in spherical coordinates (r,θ,ϕ) including the terms of earth’s equatorial ellipticity parameter γ and the resistive force. It is observed that there exists two collinear and two non-collinear Lagrangian points for different values of γ. It is found that the effect of γ on the position of the Lagrangian points is very small. Zero velocity curves are also drawn at different values of Jacobi constant for different values of γ. Finally, stability of Lagrangian point is discussed. We have found that all the Lagrangian points are unstable for different values of γ.

Study of Lagrange Points in the Earth–Moon System with Continuation Fractional Potential

In this work, the restricted three-body system is studied in the framework of the continuation fractional potential with its application on the Earth–Moon system. With the help of a numerical technique, we obtained thirteen equilibrium points, such that nine of them are collinear while the remaining four are non-collinear points. We found that the collinear points near the smaller primary were shifted outward from the Moon, whereas the points near the bigger primary were shifted towards the Earth as the value of the continuation fractional parameter increased. We analyzed the zero-velocity curves and discussed the perturbation of the continuation fractional potential effect on the possible regions of the motion. We also discussed the linear stability of all the equilibrium points and found that out of thirteen only two were stable. Due to such a prevalence, the continuation fractional potential is a source of significant perturbation, which embodies the lack of sphericity of the body in the restricted three-body problem

Dynamics of the restricted three-body problem having elongated smaller primary with disc-like structure

In this paper, we have analysed the dynamical behavior of the restricted three-body problem having elongated smaller primary with disc-like structure. We discuss the zero-velocity curve, equilibrium points, and their stability by considering the different segment-length of the elongated primary and mass ratio of the primary. The impacts of disc-like structure and elongated primary body are observed on the equilibrium points and their stability analysis. We demonstrate that all collinear equilibrium points are unstable by varying the mass parameter μ and the segment length l. Further, the critical point for mass parameter μc at non-collinear equilibrium points is calculated. Furthermore, we emphasize our discussion through an example by taking mass parameter μ=0.01<μc and segment-length l=0.01. Finally, we conclude that the proposed problem is a credible model for describing the infinitesimal body’s motion in a disc-like structure with an elongated smaller primary.

APOPHIS – effects of the 2029 Earth’s encounter on the surface and nearby dynamics

ABSTRACT The 99942 Apophis close encounter with Earth in 2029 may provide information about asteroid’s physical characteristics and measurements of Earth’s effects on the asteroid surface. In this work, we analysed the surface and the nearby dynamics of Apophis. The possible effects of its 2029 encounter on the surface and environment vicinity are also analysed. We consider a 340 m polyhedron with a uniform density (1.29, 2.2, and 3.5 g cm−3). The slope angles are computed, as well their variation that arises during the close approach. Such variation reaches 4° when low densities are used in our simulations and reaches 2° when the density is high. The zero-velocity curves, the equilibrium points, and their topological classification are obtained. We found four external equilibrium points and two of them are linearly stable. We also perform numerical simulations of bodies orbiting the asteroid, taking into account the irregular gravitational field of Apophis and two extra scenarios of perturbations: the solar radiation pressure and the Earth’s perturbation during the close approach. The radiation pressure plays an important role in the vicinity of the asteroid, only cm-sized particles survived for the time of integration. For densities of 2.2 and 3.5 g cm−3, a region of 5 cm radius particles survived for 30 yr of the simulation, and for 1.29 g cm−3, only particles with 15 cm of radius survived. The ejections and collisions are about 30–50 times larger when the close encounter effect is added but around 56–59 per cent of particles still survive the encounter.

Hovering of an Electrically Actuated Spacecraft in a Small-Body Plasma Field

This paper presents simulation models and an analysis of the hovering capability of an electrostatic spacecraft around a small celestial body. The hovering capabilities of an electrically actuated spacecraft are evaluated by combining orbital dynamics analysis with 3-D fully kinetic particle-in-cell simulations of asteroid/spacecraft interactions with the solar-wind plasma. The zero-velocity curves obtained from the analysis allow the identification of the equilibrium points for different levels of charge. The analysis of the system equilibria indicates the presence of equilibrium points in the combined gravitational, electrostatic, and solar illumination fields, most of which can be obtained by charging the spacecraft negatively. The charge-to-mass ratio needed to hover is obtained for different orbital positions, and an analysis of the sensitivity of the equilibria with respect to the spacecraft equivalent radius and with respect to the sun-to-main-body distance provides additional insight into the system dynamics.

Elliptic Restricted Three Body Problem Under the Effect of Circumstellar Belt

The dynamics of the infinitesimal for the planar elliptic restricted three body model interacting with circumstellar belt is studied. Analytic exploration of the problem reveals the existence of more than three collinear equilibrium points and a pair of triangular equilibrium points. Stability analysis based on Grebnikov’s averaging method gave the value of critical mass ratio for the stable behaviour of triangular equilibrium point. Also the criteria for stability of collinear points are explored analytically. Applying these analytical results to the Sun–Earth system under the influence of Asteroid belt and the star Epsilon–Eridani-its planet Eridani-b and the circumstellar belt show the presence of four or more collinear and a pair of equilibrium points in the pulsating rotating plane. We found that only the triangular equilibrium point is stable for most of the cases. The zero velocity curves (ZVC) and fractal basin of convergence for the two systems are also plotted. It was found that the ZVC are affected by the mass of the belt. On other hand the basin of convergence for particular equilibrium points are found to be reduced on increasing the value of mass of belt.

Impact of radiation pressure and circumstellar dust on motion of a test particle in Manev’s field

In this paper, we present a study on the impact of radiation pressure and circumstellar dust on the motion of a test particle in the framework of the restricted four-body problem under the Manev’s field. We show that the distribution of equilibrium points on the plane of motion is slightly different from that of the classical Newtonian problem. With the aid of the Lyapunov characteristic exponents, we show that the system is sensitive to changes in initial conditions; hence, the orbit of the system is found to be chaotic in the phase space for the given initial conditions. Furthermore, a numerical application of this model to a stellar system (Gliese 667C) is considered, which validates the dependence of the equilibrium points on the mass parameter. We show that the non-collinear equilibrium points of this stellar system are distributed symmetrically about the x-axis, and five of them are linearly stable. The basins of attraction of the system show that the equilibrium points have irregular boundaries, and we use the energy integral and the Manev parameter to illustrate the zero-velocity curves showing the permissible region of motion of the test particle with respect to the Jacobi constant.

Fractal basins of attraction in a binary quasar model

The present paper investigates the binary system of quasars in the framework of the Circular Restricted Three-Body Problem. The parametric evolution of libration points and the geometry of zero-velocity curves are one of the crucial aspects of our study. The multivariate form of NR method is applied to study the basin of attraction connected with libration points. The algorithm for using the Newton-Raphson method is slightly modified in order to avoid the unnecessary delay in the convergence of initial conditions. The impact of parameters on the pattern of the basin of attraction and the number of iterations needed for the convergence of initial conditions are explored. We carry out an exhaustive (numerical) study to show the influence of these parameters on converging regions in basins of attraction. We unveil the existence of fractal structure in the basin of attraction using the method of basin entropy. In almost all cases, the existence of fractal structure is found throughout the basins of attraction.

Approximate Analytical Periodic Solutions to the Restricted Three-Body Problem with Perturbation, Oblateness, Radiation and Varying Mass

Against the background of a restricted three-body problem consisting of a supergiant eclipsing binary system, the two primaries are composed of a pair of bright oblate stars whose mass changes with time. The zero-velocity surface and curve of the problem are numerically studied to describe the third body’s motion area, and the corresponding five libration points are obtained. Moreover, the effect of small perturbations, Coriolis and centrifugal forces, radiative pressure, and the oblateness and mass parameters of the two primaries on the third body’s dynamic behavior is discussed through the bifurcation diagram. Furthermore, the second- and third-order approximate analytical periodic solutions around the collinear solution point L3 in two-dimensional plane and three-dimensional spaces are presented by using the Lindstedt-Poincaré perturbation method.