Infinitesimal Particle Research Articles

The vicinity of Earth–Moon [formula omitted] and [formula omitted] in the Hill restricted 4-body problem

The Hill restricted 4-body problem (HR4BP) is a periodic time-dependent Hamiltonian dynamical system that is both a generalization of the circular restricted 3-body problem (CR3BP) and Hill’s problem. In this work, the HR4BP is used to model the motion of an infinitesimal particle under the mutual gravitation of the Sun, Earth, and Moon. In the HR4BP, the equilibria L1 and L2 of the CR3BP are replaced by π-periodic orbits, the so-called EM L1,2 dynamical equivalents. We compute center manifold reductions about EM L1,2 to obtain a local overview of the dynamics in their vicinity. In this way, we show the existence of families of 2- and 3-dimensional quasi-periodic invariant tori around EM L1,2. Additionally, we exploit the first integral constructed in the center manifold reduction to compute hyperbolic invariant manifolds. We draw comparisons to the bicircular and quasi-bicircular restricted 4-body problems.

Basins of Escape of the Particle’s Planar Motion in the Rectilinear (3 varvec{+} 1)-Body Ring Problem

The objective of this work is to study the motion of an infinitesimal particle in the gravitational field of three big bodies in a ring configuration consisting of two peripheral and one central bodies, when the energy of the particle does not allow the escape from the potential well of the system. We have numerically determined the basins of escape using a new surface of section. Additionally, we have computed and analyzed the geometry of the set of asymptotic trajectories of the periodic orbit that governs the escape from the neighborhood of one of the two satellites, which also defines the limiting curves of the basins of escape from this region.

On the restricted (N + 1)-body problem on surfaces of constant curvature

In this paper, we consider a restricted (N+1)-body problem on surfaces Mκ2, where the constant κ≠0 is the Gaussian curvature, which by means of a rescaling can be reduced to κ=±1. This problem consists in the study of the dynamics of an infinitesimal mass particle attracted by N primaries of identical masses describing elliptic relative equilibria of the N-body problem on Mκ2, i.e., a solution where the primaries are rotating uniformly with constant angular velocity ω on a fixed parallel of S2 or H2 and placed at the vertices of a regular polygon. In a rotating frame, this problem yields a two degrees of freedom Hamiltonian system. The goal of this paper is to describe analytically some dynamics features for κ=±1. Precisely, we study the relative location of equilibria, obtaining, in particular, that the poles of S2 and vertex of H2 represent equilibrium points for any value of the parameters. Thus, analysis of the nonlinear stability of these equilibria is carried out, as well as two types of bifurcations are detected: Hamiltonian-Hopf and N-bifurcation. Additionally, we prove the existence of a family of Hill's orbits and a family of periodic orbits when the primaries are located near the poles of S2 or the vertex of H2. Finally we prove the existence of KAM 2-tori related to these periodic orbits.

Invariant manifolds near L_1 and L_2 in the quasi-bicircular problem

The quasi-bicircular problem (QBCP) is a periodic time-dependent perturbation of the Earth–Moon restricted three-body problem (RTBP) that accounts for the effect of the Sun. It is based on using a periodic solution of the Earth–Moon–Sun three-body problem to write the equations of motion of the infinitesimal particle. The paper focuses on the dynamics near the L_1 and L_2 points of the Earth–Moon system in the QBCP. By means of a periodic time-dependent reduction to the center manifold, we show the existence of two families of quasi-periodic Lyapunov orbits around L_1 (resp. L_2) with two basic frequencies. The first of these two families is contained in the Earth–Moon plane and undergoes an out-of-plane (quasi-periodic) pitchfork bifurcation giving rise to a family of quasi-periodic Halo orbits. This analysis is complemented with the continuation of families of 2D tori. In particular, the planar and vertical Lyapunov families are continued, and their stability analyzed. Finally, examples of invariant manifolds associated with invariant 2D tori around the L_2 that pass close to the Earth are shown. This phenomenon is not observed in the RTBP and opens the room to direct transfers from the Earth to the Earth–Moon L_2 region.

Numerical reproduction of the spiral wave visualized experimentally in a wide-gap spherical Couette flow

Spherical Couette flow experiments were conducted according to the work of Egbers and Rath [Acta Mech. 111, 125–140 (1995)]. While the value of the critical Reynolds number obtained by the previous experiments was in good agreement with the numerical prediction, it has remained a question why a spiral wave bifurcating over the critical Reynolds number can be visualized even by a classical flow visualization technique like the mixing of a small amount of aluminum flakes to the working fluid. In the present study, through visualization using aluminum flakes drifting on a horizontal plane illuminated by a laser sheet, the flow was identified as a spiral wave with azimuthal wavenumber m = 3, using the experimentally obtained and numerically deduced comparison between phase velocities. By solving the equation of motion for the infinitesimal planar particles advecting in the flow field of the spiral wave, a visual distribution of reflected light was virtually reproduced, which is in good agreement with the experimentally obtained picture.

The The Stability of Triangular Lagragian Points in the Photogravitational Elliptic Restricted Three – Body Problem (ER3BP) with Triaxial Primaries Surrounded by a Belt with A P-R Drag Force

This paper studies the motion of an infinitesimal particle near the triangular equilibrium points (TEPs) in the elliptic restricted three body problem (ER3BP) when the primaries are radiating- triaxial rigid bodies and are affected by the Poynting-Robertson (P-R) drag force enclosed in a circumbinary disc (disc,belt). We present the equations of motion, obtained the positions of (TEPs) and found that there exist two Lagragian equilibria points L4,5 which lies  in the ξη- plane in symmetrical positions with respect to the orbital plane.The parameters involved in the system affect their positions.The position changes with an increase in triaxiality, radiation, P-R drag force and the disc. The positions and linear stability of the  TEPs are investigated numerically using  the  binary systems Archid  and it was observed that the effect  of P-R drag force of the smaller primary is not sufficient in  causing instability at the EPs  but  the radiation pressure force.We observed that when the triaxiality coefficient, values of the belt and P-R drag force are varied increasingly in the absence of radiation equilibrium points (EPs) are stable in the linear sense but becomes unstable on introducing  radiation.Thus radiation is the cause of instability when both primaries are radiating and triaxial with the smaller primary having an effective P-R drag force enclosed in a circumbinary disc and not the P-R drag.

Stability and bifurcation in the circular restricted $ (N+2) $-body problem in the sphere $ \mathbb{S}^2 $ with logarithmic potential

In this paper we study part of the dynamics of a circular restricted $ (N+2) $-body problem on the sphere $ \mathbb{S}^2 $ and considering the logarithmic potential, where $ N $ primaries remain in a ring type configuration (identical masses placed at the vertices of a regular polygon in a fixed parallel and rotating uniformly with respect to the $ Z $-axis) and a $ (N+1) $-th primary of mass $ M\in \mathbb{R} $ fixed at the south pole of $ \mathbb{S}^2 $. Such a particular configuration will be called ring-pole configuration (RP). An infinitesimal mass particle has an equilibrium position at the north pole for any value of $ M $, any parallel where the ring has been fixed (we use as parameter $ z = \cos\theta $, where $ \theta $ is the polar angle of the ring) and any number $ N\geq 2 $ of masses forming the ring. We study the non-linear stability of the north pole in terms of the parameters $ (z, M, N) $ and some bifurcations near the north pole.

Mathematical Model to Determine Motion of Infinitesimal Mass When Primaries Are both Radiating and Triaxial

This model has examined the motion of an infinitesimal particle when both primaries are radiating and triaxial in the framework of the Circular Restricted Three Body Problem (CR3BP). It was observed that the equations of motion were affected by perturbing forces (triaxiality and radiation pressures) of the primary bodies. Analytically, the locations of equilibrium points were obtained and applied to the binary systems Kruger 60 (AB) and Achird to obtain the numerical results with the help of MathLab software. Numerical investigations reveal that locations of equilibrium points of this problem have been determined and the effect of increasing triaxiality on the position of equilibrium point were obtained with graphical results. It reveal that the infinitesimal mass moves in the direction of the bigger primary towards the line joining the primary bodies. The linear stability of the equilibrium points has also been examined and a numerical solution of the analytical result was applied to the binary stars Kruger 60. It was found that the triangular points are unstable which means that the instability exist as a result of the presence of perturbation forces (Radiation and Triaxiality).
 Keywords: Model, Motion, Infinitesimal, Radiating, Triaxial.

Lie series solution of the bicircular problem

The present work performs a semi–analytical solution for the orbit of an infinitesimal particle in the framework of the bicircular Sun–Earth–Moon system. In particular, Lie series technique is applied to find the solution of the equations of motion of bicircular Sun–Earth–Moon system with radiating bigger primary. To apply Lie-series technique, the second order system of ordinary differential equations has been reduced to the corresponding first order system. Then, a set of recurrence relations is obtained in the Lie series solutions of the bicircular model (BCM) and graphical representations of the orbit for short, intermediate and long time are shown. Moreover, we study the effect of the radiation parameter on the orbit of the massless body and demonstrate that this parameter as well as the initial conditions affect its size. Specifically, it is observed that the trajectory enlarges further when the values of the radiation parameter increase while additionally its size enlarges or compacts according to the selected set of initial conditions.

Symmetric Periodic Solutions for the Spatial Maxwell Restricted $$N+1$$-Problem with Manev Potential

In the spatial Maxwell restricted $$N+1$$ -body problem, the motion of an infinitesimal particle attracted by the gravitational field of (N) bodies is studied. These bodies are arranged in a planar ring configuration. This configuration consists of $$N-1$$ primaries of equal mass m located at the vertices of a regular polygon that is rotating on its own plane about its center of mass with a constant angular velocity $$\omega $$ . Another primary of mass $$m_0=\beta m$$ ( $$\beta >0$$ parameter) is placed at the center of the ring. Moreover, we assume that the central body may be an ellipsoid, or radiation source, which introduces a new parameter e. The existence of several families of symmetric periodic solutions for the spatial Maxwell restricted $$(N+1)$$ -problem with Manev potential is proved. More precisely, firstly we get symmetric periodic solutions around the central body (attractor or repulsor) close to the equatorial plane and small parameter of oblateness. Secondly, we obtain symmetric periodic solutions far away of the central body and peripherals, close to the equatorial plane with arbitrary oblateness. Furthermore, all these families of periodic solutions are stable.

Dynamics of Infinitesimal Particle in the Framework of Photo-Gravitational Restricted Three-Body Problem

Dynamics of Infinitesimal Particle in the Framework of Photo-Gravitational Restricted Three-Body Problem

Central Configurations of the 5-Body Problem with Four Infinitesimal Particles

We study central configurations of 5-body problem with one dominant particle and four infinitesimal particles. In 2013, Oliveira (Celest Mech Dyn Astron 116:11–20, 2013) showed that the configurations are symmetric when two infinitesimal particles are diametrically opposite. Moreover, in the case of these two particles have the same mass he proved that the number of central configurations is one or two. In this paper, we provide criteria for the number of the central configurations in the general case where these two particles are unequal mass.

The vicinity of the Earth–Moon $$L_1$$ point in the bicircular problem

The bicircular model is a periodic time-dependent perturbation of the Earth–Moon restricted three-body problem that includes the direct gravitational effect of the Sun on the infinitesimal particle. In this paper, we focus on the dynamics in the neighbourhood of the $$L_1$$ point of the Earth–Moon system. By means of a periodic time-dependent reduction to the centre manifold, we show the existence of two families of quasi-periodic Lyapunov orbits, one planar and one vertical. The planar Lyapunov family undergoes a (quasi-periodic) pitchfork bifurcation giving rise to two families of quasi-periodic halo orbits. Between them, there is a family of Lissajous quasi-periodic orbits, with three basic frequencies.

Equality of the Kelvin and Korteweg–Helmholtz Force Densities Inside Dielectric Materials

The net force is enough to predict the motion of an object, but to predict its deformation, the distribution of the force inside the object should be known. In the continuum electromechanics, various expressions are used to calculate the electric forces acting on a dielectric object. The Kelvin and the Korteweg–Helmholtz forces are the most common electric body forces. The net force resulting from both of these “forces” is equal, but the distribution of these two forces inside a dielectric material is different because the expressions of both these forces lose their physical manifestation in the derivation process. This article is primarily aimed at revealing the common mistakes that are made when deriving volumetric force density. Then, the concept of an infinitesimal particle and external electric field is introduced. By using this concept, the electric force density inside a dielectric material is uniquely determined irrespective of the force expression, which is shown analytically and numerically.

Variational principles and generalized Hill’s bounds in micromechanics of linear peridynamic random structure composites

We consider a static problem for statistically homogeneous matrix linear peridynamic composite materials (CMs). The basic feature of the peridynamic model considered is a continuum description of a material behavior as the integrated non-local force interactions between infinitesimal particles. In contrast to these classical local and non-local theories, the peridynamic equation of motion introduced by Silling (J Mech Phys Solids 2000; 48: 175–209) is free of any spatial derivatives of displacement. Estimation of effective moduli of peridynamic CMs is performed by generalization of some methods used in locally elastic micromechanics. Namely, the admissible displacement and force fields are defined. The theorem of work and energy, Betti’s reciprocal theorem, and the theorem of virtual work are proved. Principles of minimum of both potential energy and complimentary energy are generalized. The strain energy bounds are estimated for both the displacement and force homogeneous volumetric boundary conditions. The classical representations of effective elastic moduli through the mechanical influence functions for elastic CM are generalized to the case of peridynamics, and the energetic definition of effective elastic moduli is proposed. Generalized Hill’s bounds on the effective elastic moduli of peridynamic random structure composites are obtained. In contrast to the classical Hill’s bounds, in the new bounds, comparable scales of the inclusion size and horizon are taken into account that lead to dependance of the bounds on both the size and shape of the inclusions. The numerical examples are considered for the 1D case.

Existence and stability of equilibrium points in the photogravitational restricted four-body problem with Stokes drag effect

The restricted four-body problem consists of an infinitesimal particle which is moving under the Newtonian gravitational attraction of three finite bodies, $m_{1}$, $m_{2}$ and $m_{3}$. The three bodies (called primaries) are moving in circular orbits around their common centre of mass fixed at the origin of the coordinate system. Moreover, according to the solution of Lagrange, these primaries are fixed at the vertices of an equilateral triangle. The fourth body does not affect the motion of the three bodies. In this paper, we deal with the photogravitational version of the problem with Stokes drag acting as a dissipative force. We consider the case where all the primaries are sources of radiation and that two of the bodies, $m_{2}$ and $m_{3}$, have equal masses ($m _{2} = m_{3} = \mu $) and equal radiation factors ($q_{2} = q_{3} = q$) while the dominant primary body $m_{1}$ is of mass $1 - 2\mu $. We investigate the dynamical behaviour of an infinitesimal mass in the gravitational field of radiating primaries coupled with the Stokes drag effect. It is found that under constant dissipative force, collinear equilibrium points do not exist (numerically and of course analytically) whereas the existence and positions of the non-collinear equilibrium points depend on the parameters values. The linear stability of the non-collinear equilibrium points ($L_{i},i = 1,2, \ldots ,8$) is also studied and it is found that they are all unstable except $L_{1}$, $L _{7}$ and $L_{8}$ which may be stable for a range of values of $\mu $ and various values of radiation factors. Finally, we justify the relevance of the model in astronomy by applying it to a stellar system (Ross 104-Ross775a-Ross775b), for which all the equilibrium points have been seen to be unstable.

Singularities and Dynamics Aspects of a Particle in a Gravitational Field of a Central Punctual Body Surrounded by a Solid Circular Ring

The problem of attraction of an infinitesimal particle by the gravitational force induced by a massive body composed of a ring (in the form of a circle) of radius $a$ and a punctual body at the cen...

The restricted three body problem on surfaces of constant curvature

We consider a symmetric restricted three-body problem on surfaces Mκ2 of constant Gaussian curvature κ≠0, which can be reduced to the cases κ=±1. This problem consists in the analysis of the dynamics of an infinitesimal mass particle attracted by two primaries of identical masses describing elliptic relative equilibria of the two body problem on Mκ2, i.e., the primaries move on opposite sides of the same parallel of radius a. The Hamiltonian formulation of this problem is pointed out in intrinsic coordinates. The goal of this paper is to describe analytically, important aspects of the global dynamics in both cases κ=±1 and determine the main differences with the classical Newtonian circular restricted three-body problem. In this sense, we describe the number of equilibria and its linear stability depending on its bifurcation parameter corresponding to the radial parameter a. After that, we prove the existence of families of periodic orbits and KAM 2-tori related to these orbits.

Investigation of the Stability of a Test Particle in the Vicinity of Collinear Points with the Additional Influence of an Oblate Primary and a Triaxial-Stellar Companion in the Frame of ER3BP

We investigate in the elliptic framework of the restricted three-body problem, the motion around the collinear points of an infinitesimal particle in the vicinity of an oblate primary and a triaxial stellar companion. The locations of the collinear points are affected by the eccentricity of the orbits, oblateness of the primary body and the triaxiality and luminosity of the secondary. A numerical analysis of the effects of the parameters on the positions of collinear points of CEN X-4 and PSR J1903+0327 reveals a general shift away from the smaller primary with increase in eccentricity and triaxiality factors and a shift towards the smaller primary with increase in the semi-major axis and oblateness of the primary on L1. The collinear points remain unstable in spite of the introduction of these parameters.

Oscillatory motions in restricted N-body problems

We consider the planar restricted N-body problem where the N−1 primaries are assumed to be in a central configuration whereas the infinitesimal particle escapes to infinity in a parabolic orbit. We prove the existence of transversal intersections between the stable and unstable manifolds of the parabolic orbits at infinity which guarantee the existence of a Smale's horseshoe. This implies the occurrence of chaotic motions, namely the oscillatory motions, that is, orbits for which the massless particle leaves every bounded region but it returns infinitely often to some fixed bounded region. Our achievement is based in an adequate scaling of the variables which allows us to write the Hamiltonian function as the Hamiltonian of the Kepler problem plus higher-order terms that depend on the chosen configuration. We compute the Melnikov function related to the first non-null perturbative term and characterize the cases where it has simple zeroes. Concretely, for some combinations of the configuration parameters, i.e. mass values and positions of the primaries, and for a specific value of a parameter related to the angular momentum vector, the Melnikov function vanishes, otherwise it has simple zeroes and the transversality condition is satisfied. When the Melnikov function corresponding to the principal part of the perturbation is zero we compute the next non-zero Melnikov function proving that it has simple zeroes. The theory is illustrated for various cases of restricted N-body problems, including the circular restricted three-body problem. No restrictions on the mass parameters are assumed.