Zero-velocity Surface Research Articles

A study of the equilibrium dynamics of the test particle in the collinear circular restricted four-body problem with non-spherical central primary

We consider the collinear restricted four-body problem (CR4BP), where the test particle of infinitesimal mass is moving under the gravitational influence of the three primary bodies. It is further assumed that the central primary is a non-spherical body, particularly either an oblate or prolate spheroid, whereas the peripheral primaries are spherical in shape. A numerical analysis is presented to unveil the effect of the oblateness and prolateness parameters on the position of equilibrium points (EPs) and their linear stability in the CR4BP. Moreover, the permissible regions of possible motion as determined by the zero-velocity surface and associated equipotential curves and the basins of convergence linked with the EPs on the orbital plane are presented. The existence and number of collinear EPs and non-collinear EPs in the problem depend on the combination of the mass parameter of the primaries and the oblateness/prolateness parameter. Additionally, the application of the problem in the Saturn-Moon(1)-Moon(2)-System has been presented.

Motion properties of the variable mass smallest body in cyclic kite configuration with kerr-like oblate heterogeneous primaries

The aim of this paper is to motion of the smallest body in the 5-body problem where 4 bodies construct the cyclic kite configuration. Out of five bodies four bodies are placed at the vertices of a cyclic kite and the shapes of these bodies are considered as oblate heterogeneous with the supposition that these bodies are spinning around their own axes. It is also supposed that the smallest and fifth body is varying its mass according to Jeans law. The motion of smallest body is also affected by the coriolis and centrifugal forces. The equations of motion are determined under the above said perturbations. And then we numerically illustrated the locations of fixed points, their stability, the zero-velocity surfaces with projections and Poincare surfaces of section.

Identification of Galaxy Protoclusters Based on the Spherical Top-hat Collapse Theory

We propose a new method for finding galaxy protoclusters that is motivated by structure formation theory and also directly applicable to observations. We adopt the conventional definition that a protocluster is a galaxy group whose virial mass M vir < M cl at its epoch, where M cl = 1014 M ⊙, but would exceed that limit when it evolves to z = 0. We use the critical overdensity for complete collapse at z = 0 predicted by the spherical top-hat collapse model to find the radius and total mass of the regions that would collapse at z = 0. If the mass of a region centered at a massive galaxy exceeds M cl, the galaxy is at the center of a protocluster. We define the outer boundary of a protocluster as the zero-velocity surface at the turnaround radius so that the member galaxies are those sharing the same protocluster environment and showing some conformity in physical properties. We use the cosmological hydrodynamical simulation Horizon Run 5 (HR5) to calibrate this prescription and demonstrate its performance. We find that the protocluster identification method suggested in this study is quite successful. Its application to the high-redshift HR5 galaxies shows a tight correlation between the mass within the protocluster regions identified according to the spherical collapse model and the final mass to be found within the clusters at z = 0, meaning that the regions can be regarded as the bona fide protoclusters with high reliability. We also confirm that the redshift-space distortion does not significantly affect the performance of the protocluster identification scheme.

Equilibrium points and Lyapunov families in the circular restricted three-body problem with an oblate primary and a synchronous rotating dipole secondary: Application to Luhman-16 binary system

We numerically study a version of the synchronous circular restricted three-body problem, where an infinitesimal mass body is moving under the Newtonian gravitational forces of two massive bodies. The primary body is an oblate spheroid while the secondary is an elongated asteroid of a combination of two equal masses forming a rotating dipole which is synchronous to the rotation of the primaries of the classic circular restricted three-body problem. In this paper, we systematically examine the existence, positions, and linear stability of the equilibrium points for various combinations of the model's parameters. We observe that the perturbing forces have significant effects on the positions and stability of the equilibrium points as well as the regions where the motion of the particle is allowed. The allowed regions of motion as determined by the zero-velocity surface and the corresponding isoenergetic curves as well as the positions of the equilibrium points are given. Finally, we numerically study the binary system Luhman-16 by computing the positions of the equilibria and their stability as well as the allowed regions of motion of the particle. The corresponding families of periodic orbits emanating from the collinear equilibrium points are computed along with their stability properties.

The common trend of saltation particles on the surface of fast-rotating asteroids

Context. An asteroid spun up to its critical limit has unique surface mechanical properties that its gravity and the centrifugal force largely balance, creating a relaxation environment where low-energy events such as mass shedding may trigger subsequent long complex motion of an asteroid’s regolith grains. Exploring such an evolution process may provide key clues for understanding the early formation of multi-asteroid systems. Aims. This paper investigates the complex evolution process of loose particles becoming triggered by shedding events and the dependency of their dynamical propagation on the contact mechanical properties of the asteroid surface. Methods. We present a numerical model for tracking the trajectory of a shed particle that considers the collision between the particle and the surface of an asteroid. Monte Carlo simulations are performed to reflect the statistical behavior of shed particles. We also introduce zero-velocity surfaces to our data-based analysis in order to reveal the intrinsic invariance of the evolutionary processes. We used the average mechanical energy of the particle cloud to check the connection between contact property and the temporal-spatial distribution of the shed particles. Results. We sketch a common evolutionary path of the particle in the vicinity of a fast-rotating asteroid, that is, particles dislodged from the unstable region will eventually enter, through several collisions with the surface, non-return orbits that launch from the minimum geopotential area of the unstable region. The common trend is independent of any particular asteroid morphology, and all shed particles (no matter where they originate from) enter the same evolutionary path. We also find that the orbital energy of the particle cloud is statistically independent of the surface contact property, meaning that the collision coefficient of restitution is a nonsensitive parameter in the outward spreading process of the shed particles.

Zero velocity surface in the general three-body-problem

The zero-velocity surfaces of the general planar three-body problem are constructed in the form space, the factor-space of the configuration space by transfer and rotation. Such a space is the space of congruent triangles, and the sphere in this space is similar triangles. The integral of energy in the form space gives the equation of the zero velocity surface. These surfaces can also be obtained based on the Sundman inequality. Such surfaces separate areas of possible movement from areas where movement is impossible. Without loss of generality, we can assume that the constant energy is −1/2 and the sought for surfaces depend only on the magnitude of the angular momentum of the problem, J. Depending on this value, five topologically different types of surfaces can be distinguished. For small J, the surface consists of two separate surfaces, internal and external ones, the motion is possible only between them. With J increasing the inner surface increases, the outer surface decreases, the surfaces first have a common point at some value of J, with a further increasing of J, their topological type changes and finally the zero-velocity surface splits into three disjoint surfaces, the motion is possible only inside them. Examples of the corresponding surfaces are given for each of these types, their sections in the plane xy and in the plane xz and the surfaces themselves are constructed, their properties are studied.

Analysis of the equilibrium points and orbits stability for the asteroid 93 Minerva

Abstract In this article, we study the orbital dynamics with the gravitational potential of the asteroid 93 Minerva using an irregular shape model from observations. We calculate its physical size, physical mass, surface height, and zero-velocity surface. Meanwhile, we recognize that there are five equilibrium points around Minerva, four of which are external, and one is internal. Two of the external equilibrium points are stable and near the y y -axis, while two external equilibrium points are unstable and near the x x -axis. In addition, we study the changes in the number, position, and topological case of the equilibrium points when changing the spin speed and the density. We calculate the gravitational force acceleration of the polyhedron model, and we back up our calculations by simulating the orbit of one moonlet under the gravitational force acceleration of Minerva. With the simulation result, we demonstrate the existence of stable orbits around Minerva.

Motion of Stars in Layered Inhomogeneous Elliptical Galaxies

The problem of the spatial motion of a passively gravitating body (PGB) in the gravitational field of a layered inhomogeneous elliptical galaxy (LIEG) is considered on the basis of the previously developed model. It is assumed that a LIEG consists of baryonic mass (BM) and dark matter (DM), which have different laws of density distribution. A star or the center of mass of a globular cluster is taken as the PGB, the motion of which considers the BM and DM attraction. To obtain accurate results, the BM and DM attraction potentials are not expanded in a series, but their exact expressions are taken. An analogue of the Jacobi integral is found, the region of the possible motion of the PGB is determined, and the zero-velocity surfaces are constructed. The stationary solutions (libration points) are found to be stable in the sense of Lyapunov. The results are applied to the elliptical galaxies NGC 4472 (M 49), NGC 4697, and NGC 4374 (M 84).

Determination of the Local Hubble Constant from Virgo Infall Using TRGB Distances

Abstract An independent determination of H 0 is crucial given the growing tension between the Hubble constant, H 0, derived locally and that determined from the modeling of the cosmic microwave background (CMB) originating in the early universe. In this work, we present a new determination of H 0 using velocities and tip of the red giant branch (TRGB) distances to 33 galaxies located between the Local Group and the Virgo cluster. We use a model of the infall pattern of the local Hubble flow modified by the Virgo mass, which is given as a function of the cosmological constants (H 0, ΩΛ), the radius of the zero-velocity surface R 0, and the intrinsic velocity dispersion, σ v . Fitting velocities and TRGB distances of 33 galaxies to the model, we obtain H 0 = 65.8 ± 3.5 (stat) ± 2.4 (sys) km s−1 Mpc−1 and R 0 = 6.76 ± 0.35 Mpc. Our local H 0 is consistent with the global H 0 determined from CMB radiation, showing no tension. In addition, we present new TRGB distances to NGC 4437 and NGC 4592, which are located near the zero-velocity surface: D = 9.28 ± 0.39 Mpc and D = 9.07 ± 0.27 Mpc, respectively. Their spatial separation is 0.29 Mpc, suggesting that they form a physical pair.

Minimal Velocity Surface in a Restricted Circular Three-Body Problem

In a restricted circular three-body problem, the concept of the minimum velocity surface $$\mathscr{S}$$ is introduced, which is a modification of the zero-velocity surface (Hill surface). The existence of the Hill surface requires the occurrence of the Jacobi integral. The minimum velocity surface, apart from the Jacobi integral, requires conservation of the sector velocity of a zero-mass body in the projection on the plane of motion of the main bodies. In other words, there must exist one of the three angular momentum integrals. It is shown that this integral exists for a dynamic system obtained after a single averaging of the original system over the longitude of the main bodies. The properties of $$\mathscr{S}$$ are investigated. We highlight the most significant issues. The set of possible motions of the zero-mass body bounded by surface $$\mathscr{S}$$ is compact. As an example, surfaces $$\mathscr{S}$$ for four small moons of Pluto are considered within the averaged Pluto–Charon–small satellite problem. In all four cases, $$\mathscr{S}$$ is a topological torus with a small cross section, having a circumference in the plane of motion of the main bodies as the center line.

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.

SEARCH FOR STABLE ORBITS AROUND THE BINARY ASTEROID SYSTEMS 1999 KW4 AND DIDYMOS

This work includes analytical and numerical studies of spacecrafts orbiting two binary asteroid systems: 1999 KW4 and Didymos. The binary systems are modeled as full irregular bodies, such that the whole evolution of the results will show the impact of the irregular gravity field in the lifetime and dynamics of the spacecraft’s orbit. The equations of motion of the binary system and the spacecraft are derived from Lagrange Equations. The solar radiation pressure is consired in the dynamics of the spacecraft.Two distinct methods are used to search for stable orbits around the binary systems. One is called the grid search method, which defines the main body as a point mass to estimate the initial state of the spacecraft based on a circular Keplerian orbit. The second method is the search for periodic orbits based on zero-velocity surfaces.

Bifurcation Analysis and Periodic Solutions of the HD 191408 System with Triaxial and Radiative Perturbations

The nonlinear orbital dynamics of a class of the perturbed restricted three-body problem is studied. The two primaries considered here refer to the binary system HD 191408. The third particle moves under the gravity of the binary system, whose triaxial rate and radiation factor are also considered. Based on the dynamic governing equation of the third particle in the binary HD 191408 system, the motion state manifold is given. By plotting bifurcation diagrams of the system, the effects of various perturbation factors on the dynamic behavior of the third particle are discussed in detail. In addition, the relationship between the geometric configuration and the Jacobian constant is discussed by analyzing the zero-velocity surface and zero-velocity curve of the system. Then, using the Poincaré–Lindsted method and numerical simulation, the second- and third-order periodic orbits of the third particle around the collinear libration point in two- and three-dimensional spaces are analytically and numerically presented. This paper complements the results by Singh et al. [Singh et al., AMC, 2018]. It contains not only higher-order analytical periodic solutions in the vicinity of the collinear equilibrium points but also conducts extensive numerical research on the bifurcation of the binary system.

Stability of generalized elliptic restricted four body problem with radiation and oblateness effects

This paper studies the existence and stability of non-collinear equilibrium points in the elliptic restricted four body problem with bigger primary as a source of radiation and other two primaries having equal masses as oblate spheroid. In the elliptic restricted four body problem, three of the bodies are moving in elliptical orbit around their common centre of mass fixed at the origin of the coordinate system, while the fourth one is infinitesimal. Three pairs of non-collinear points are obtained symmetric with respect to x-axis. We found the equilibrium points are stable in linear sense. We also investigate the pulsating zero velocity surfaces and basin of attraction for varying value of oblateness coefficient and radiation pressure parameter.

Minimal velocity surface in the restricted circular Three-Body-Problem

In the framework of the restricted circular Three-Body-Problem, the concept of the minimum velocity surface S is introduced, which is a modification of the zero-velocity surface (Hill surface). The existence of Hill surface requires occurrence of the Jacobi integral. The minimum velocity surface, other than the Jacobi integral, requires conservation of the sector velocity of a zero-mass body in the projection on the plane of the main bodies motion. In other words, there must exist one of the three angular momentum integrals. It is shown that this integral exists for a dynamic system obtained after a single averaging of the original system by longitude of the main bodies. Properties of S are investigated. Here is the most significant. The set of possible motions of the zero-mass body bounded by the surface S is compact. As an example the surfaces S for four small moons of Pluto are considered in the framework of the averaged problem Pluto — Charon — small satellite. In all four cases, S represents a topological torus with small cross section, having a circumference in the plane of motion of the main bodies as the center line.

Study on Equilibrium and Stability of Binary Asteroid Systems

The unique motion of binary asteroid systems provide important clues for the evolution of planets. Therefore, binary asteroid systems become hot targets in asteroid exploration. In this paper, based on the double ellipsoid model, the relative motion, equilibrium condition and stability of binary asteroid systems are studied. Firstly, the full two-body dynamic equation of the double ellipsoid model is established to describe the relative motion of a binary asteroid system. Secondly, the Lagrangian equation of motion is used to solve the angular momentum and energy of the system, and the equilibrium conditions are obtained. Finally, a method to determine the stability of equilibrium is given by using the zero-velocity state surface and the influence of the asteroid's physical parameter on the stability of equilibrium is analyzed. This research can provide a theoretical reference for the trajectory design and control in future exploration missions to binary asteroid systems.

Effect of variation of charge in the circular restricted three-body problem with variable masses

In the present paper, we are concerned by some investigation on circular restricted three-body problem (CR3BP), where we assume that the primaries have variable masses and variable charges. Among the principal tools used in the present study, we cite the well known Meshcherskii transformation. We have derived the equations of motion and Jacobi integral which differ by variation constant k and charge q from the classical restricted three-body problem. More exactly, in this paper, we have drawn the equilibrium points, the zero-velocity curves, the periodic orbits, the surfaces and the basins of attraction for the different values of charge. We have found one equilibrium point when the charge is q=0.4 and three equilibrium points when its value is q=0.501. We also have drawn the periodic orbits for these two values of charge and found that they are periodic. We have also plotted the zero-velocity surfaces for these two values of charges and found a tremendous variation in these two surfaces. We notice that the Poincaré surfaces of section are shifting away from the origin, when we increase the value of charge. We also got different surfaces for the motion of infinitesimal body, with respect to the variations of charge. The basins of attraction have been drawn for these two values of charge by using Newton-Raphson iterative method. We also noticed that by increasing the values of charge, the basins of attraction are shrinking. For the stability of the equilibrium points that we have studied, we found that, among them, one is stable and three others are unstable.

Dynamical environment in the triple asteroid system 87 Sylvia

We investigate the dynamical environment in the first announced triple asteroid system 87 Sylvia. The 3D shape of the primary is rebuilt from shape data using the polyhedral model method. The dynamical characteristics of the primary, such as the surface height, the surface effective potential, the surface gravitational acceleration, the structure of the zero-velocity surfaces, and the locations and stability of the equilibrium points, have been investigated. Five equilibrium points are in the potential of 87 Sylvia. Both unstable periodic orbits and stable periodic orbits coexist near the surface of the primary of the triple asteroid system 87 Sylvia. Under the full perturbation force of two moonlets, the semi-major axis, the eccentricity, the inclination, and the mechanical energy vary periodically and have no secular terms. Among them, the semi-major axis, the eccentricity, and the mechanical energy have two period terms, namely, a long period term and a short period term; the inclination has three period terms, i.e., a long period term, a short period term, and an intermediate period term. Unlike these orbital parameters, the argument of the pericenter and the longitude of the ascending node have a secular term, a long period term and a short period term. A simulation of the trajectories of 1000 grains generated in the system considering the gravity of the irregular shape and two moonlets of the primary has been presented. The grains may impact on the surface of the primary, escape the system, or move in the system.

Equilibria and orbits in the dynamical environment of asteroid 22 Kalliope

Abstract This paper studies the orbital dynamics of the potential of asteroid 22 Kalliope using observational data of the irregular shape. The zero-velocity surface are calculated and showed with different Jacobian values. All five equilibrium points are found, four of them are outside and unstable, and the other one is inside and linearly stable. The movement and bifurcations of equilibrium points during the variety of rotation speed and density of the body are investigated. The Hopf bifurcations occurs during the variety of rotational speed from ω=1.0ω 0 to 0.5ω 0, and the Saddle-Node bifurcation occurs during the variety of rotational speed from ω=1.0ω 0 to 2.0ω 0. Both unstable and stable resonant periodic orbits around Kalliope are coexisting. The perturbation of an unstable periodic orbit shows that the gravitational field of Kalliope is strongly perturbed.

Equilibria and orbits around asteroid using the polyhedral model

In the current study, we use the polyhedral model to compute the potential of the asteroid. There are five equilibrium points in the gravitational field of the asteroid 283 Emma. We concluded that the zero-velocity surfaces and the equilibrium points change with the suppositive variation of the rotational speed of the asteroid. It is found that if the rotational speed equals a half as it is in present, the number of equilibrium points is also five. However, if the rotational speed equals twice as it is in present, there are only three equilibrium points left. Four different periodic orbits are calculated using the hierarchical grid searching method. We calculated characteristic multipliers of periodic orbits to investigate the stability of these periodic orbits. The orbit near the primary's equatorial plane is more likely to be stable when the separation/ primary-radius is a large number.