Body Problem Research Articles

Periodic orbit in the photo-gravitational restricted three body problem around the collinear Lagrangian points when more massive primary is an oblate spheroid and source of radiation

The circular Restricted Three Body Problem is considered with the more massive primary as an oblate spheroid and source of radiation. A new mean motion expression given by n2=1+6A is used in the present study, when the secular effect of the oblateness on the mean motion, argument of perigee and right ascension of the ascending node is considered. The locations of the collinear Lagrangian points are found. It is found to have some variation from the previous study conducted on the same because of the new mean motion that is considered in this study. The variations of the location of the Lagrangian points due to the unperturbed as well as the perturbed problem due to oblateness and radiation pressure are studied. A study on the eccentricity e and angular frequency s at L1, L2 and L3 is carried out. It is observed how the change in effect of oblateness and radiation pressure has affected the location, angular frequency and eccentricity at L3, though there are only small changes noticed in the case of L1 and L2. Â

Remarks on the inverse problem of the collinear central configurations in the $ N $-body problem

<abstract><p>For a fixed configuration in the collinear $ N $-body problem, the existence of the central configurations is determined by a system of linear equations, which in turn is determined by certain Pfaffians for even or odd $ N $ in literatures. In this short note, we prove that the Pfaffians of the associate matrices for all even number collinear configurations are nonzero if and only if the extended Pfaffians of the associate matrices for all odd number collinear configurations are nonzero. Therefore, the inverse problem of the collinear central configurations can be answered and each collinear configuration determines a one-parameter family of masses with a fixed total mass if the Pfaffians of the associate matrix for all collinear even number bodies are nonzero. We also make some remarks on the super central configurations and the number of collinear central configurations under different equivalences, especially a lower bound for the number of collinear central configurations under the geometric equivalence.</p></abstract>

On the Dynamics of the Solar System I: Orbital Inclination and Nodal Precession

The dynamic equations of the $n$-body problem are solved in relative coordinates and applied to the solar system, whence the mean variation rates of the longitudes of the ascending nodes and of the inclinations of the planetary orbits at J2000 have been calculated with respect to the ecliptic and to the Laplace invariable plane under the approximation of circular orbits. The theory so obtained supersedes the Lagrange-Laplace secular evolution theory. Formulas for the change from the equatorial and ecliptic coordinates to those of the Laplace invariable plane are also provided.

Symmetric Solutions to the N Body Problem

Symmetric Solutions to the N Body Problem

Notes on spatial twisted central configurations for $2N$-body problem

We are concerned with the spatial twisted central configuration formed by two paralleled regular \(N\)-polygons (\(N\geqslant 3\)) with twist angle \(\theta \), and we prove that the values of the masses in each separate regular \(N\)-polygon are equal if and only if \(\theta =0\) or \(\theta =\pi /N\). Moreover, suppose that all of the sides in one regular \(N\)-polygon have length \(a\) and all of the sides in the other regular \(N\)-polygon have length 1, then for the case of the twist angle is \(\theta =\pi /N\), if \(3\leqslant N\leqslant 8\), for any \(a>0\), the spatial twisted central configuration for \(2N\)-body problem exists. Furthermore, whether the twist angle is \(\theta =0\) or \(\theta =\pi /N\): for every \(N\geqslant 3\), if all the masses of the \(2N\) bodies or the lengths of all the sides of the two regular \(N\)-polygons are equal to each other, then there is only one spatial central configuration, and the distance between the two paralleled regular \(N\)-polygons with \(\theta =0\), must be larger than the case of \(\theta =\pi /N\).

NCC research and the problem of consciousness

One of the reasons why the Neural Correlates of Consciousness Program could appear attractive in the 1990s was that it seemed to disentangle theoretical and empirical problems. Theoretical disagreements could thus be sidestepped in order to focus on empirical research regarding the neural substrate of consciousness. One of the further consequences of this dissociation of empirical and theoretical questions was that fundamental questions regarding the Mind Body Problem or the “Hard Problem of Consciousness” could remain unresolved even if the search for the neural correlates had been successful.Drawing on historical examples, a widely held consensus in the philosophy of science, and actual NCC research we argue that there is no such independence. Moreover, as the dependence between the theoretical and the empirical level is mutual, empirical progress will go hand in hand with theoretical development. Thus, contrary to what the original NCC program suggested, we conclude that NCC research may significantly take advantage from and contribute to theoretical progress in our explanation and understanding of consciousness. Eventually, this might even contribute to a solution of the Hard Problem of Consciousness.

Hyperbolic regularization of the restricted three–body problem on curved spaces

Hyperbolic regularization of the restricted three–body problem on curved spaces

Effects of oblateness of the primaries on natural periodic orbit-attitude behaviour of satellites in three body problem

• Equations of Motion for the Perturbed Circular Restricted Three Body Problem (P-CR3BP) with considering A 2 - A 6 primaries oblate coefficients have derived for the first time. • It was shown that considering primaries zonal harmonic coefficients j 2 - j 6 has a significant effect on satellite orbit-attitude behavior. It is observed that considering A 2 - A 6 primaries oblate coefficients amplify the effect of oblate perturbations and the zonal harmonic coefficients after j 6 have negligible effect. • Proper initial guess of periodic solution of perturbed model identified by Poincaré maps. • As an innovation, finding the appropriate value of inertia ratio K to find periodic response of orbit-attitude was performed using Poincaré mapping for the first time. • It was shown for the first time, by more accurately simulating the study environment by considering oblate perturbations, the density of points on the islands created on Poincaré map appeared to increase, and in fact, it became easier to identify the initial guess of attitude states. • Increasing the inertia ratio and increasing the orbital period reduces the chances of finding periodic responses. This study was done to investigate the effect of perturbations of both oblate massive primaries on the periodic orbit-attitude behavior of satellites at the three-body problem. Governing equations of the perturbed coupled orbit-attitude were derived using principles of the Lagrangian mechanics. As initial conditions of the coupled model play a key role in finding periodic responses of both orbit and attitude, so the perturbed circular restricted three-body problem (P-CR3BP) coupled orbit-attitude correction algorithm was proposed to correct the initial guess of the coupled model. The correction algorithm requires an appropriate initial guess vector as the number of periodic solutions is limited. This initial guess vector consists of two parts: orbital and attitude dynamic state parameters. A suitable initial guess was suggested for orbital parameters considering a small error at the initial conditions of the unperturbed periodic orbits. These initial conditions were extracted by another orbital correction algorithm to correct orbital states of periodic orbits in the unperturbed circular restricted three-body problem (U-CR3BP). Furthermore, the initial guess of attitude dynamic parameters was identified by the Poincaré mapping method. Considering oblate perturbations would change the initial conditions of periodic orbits at P-CR3BP relative to U-CR3BP. Also, because the proposed model of the coupled orbit-attitude governing equations is such that only orbital state parameters influence attitude dynamic parameters, so the behavior of attitude dynamics will also change by altering orbital motion. Comparison of patterns in Poincaré maps, satellites angular velocity, and initial conditions of periodic coupled orbit-attitude response in U-CR3BP and P-CR3BP indicated the effect of both oblate primaries perturbations. Consideration of these perturbations resulted in a more accurate simulation of the study environment that helps to understand the natural motion of the satellite.

The circular restricted 4-body problem with three equal primaries in the collinear central configuration of the 3-body problem

We study the dynamics of the circular restricted 4-body problem with three primaries with equal masses at the collinear configuration of the 3-body problem with an infinitesimal mass. We calculate the equilibrium points and study their linear stability. By applying the Lyapunov theorem, we prove the existence of periodic orbits bifurcating from the equilibrium points and, further, prove that they continue in the full 4-body problem. Moreover, we prove analytically the existence of Hill and of comet-like periodic orbits.

Transfers from the Earth to L_2\xa0Halo orbits in the Earth\u2013Moon bicircular problem

This paper deals with direct transfers from the Earth to Halo orbits related to the translunar point. The gravitational influence of the Sun as a fourth body is taken under consideration by means of the Bicircular Problem (BCP), which is a periodic time dependent perturbation of the Restricted Three Body Problem (RTBP) that includes the direct effect of the Sun on the spacecraft. In this model, the Halo family is quasi-periodic. Here we show how the effect of the Sun bends the stable manifolds of the quasi-periodic Halo orbits in a way that allows for direct transfers.

Some properties of lemaitre regularization. II isosceles trajectories and figure‐eight

AbstractSome properties of the Lemaitre regularization in the planar 3 body problem are considered. Application of these properties to the degenerated case of an isosceles problem in the space of forms is studied. A number of the isosceles orbits is obtained using a suitable parametrization. Also, properties of such orbits are analyzed. The figure‐eight orbit is considered.

Applying the symmetry groups to study the n body problem

We introduce an algebraic method to study local stability in the Newtonian n-body problem when certain symmetries are present. We use representation theory of groups to simplify the calculations of certain eigenvalue problems. The method should be applicable in many cases, we give two main examples here: the square central configurations with four equal masses, and the equilateral triangular configurations with three equal masses plus an additional mass of arbitrary size at the center. Using representation theory of finite groups, we explicitly found the eigenvalues of certain 8×8 Hessians in these examples, with only some simple calculations of traces. We also studied the local stability properties of corresponding relative equilibria in the four-body problems.

A New Application of the Sumudu Transform for the Falling Body Problem

In this study, we investigate the falling body problem with three different fractional derivatives. We acquire the solutions of the model by the Sumudu transform. We show the accuracy of the Sumudu transform by some theoretic results and implementations.

Continuation of relative equilibria in the n–body problem to spaces of constant curvature

We prove that all non-degenerate relative equilibria of the planar Newtonian n–body problem can be continued to spaces of constant curvature κ, positive or negative, for small enough values of this parameter. We also compute the extension of some classical relative equilibria to curved spaces using numerical continuation. In particular, we extend Lagrange's triangle configuration with different masses to both positive and negative curvature spaces.

Approximation Solution of the Nonlinear Circular Sitnikov Restricted Four–Body Problem

In this paper, the approximated periodic solutions of the circular Sitnikov restricted four–body problem (RFBP) were constructed using the Lindstedt–Poincaré method, by removing the secular terms, and compared with numerical solution. It can be observed that, in the numerical as well as approximated solutions patterns, the initial conditions are important. In the sense of a numerical solution, the motion is periodic in a certain interval, but beyond this interval, the motion is not periodic. But, the Lindstedt–Poincaré method constantly gives regular and periodic motion all time. Finally, we observed that the solution obtained by the Lindstedt–Poincaré method gives the true motion of the circular Sitnikov RFBP and the fourth approximate solution has more accuracy than the first, second, and third approximate solutions.

Apartment purchase under Shirkah-ul Milk and shariah compliance in Islamic banks: the perception of bankers and clients in Bangladesh

PurposeApartment purchase is an increasing investment trend in Islamic banks as clients trusted to eradicate the interest from financial dealings to capture the Islamic Principles. The paper aims to investigate the Shariah compliance in the investment of apartment purchases under Shirkah-ul milk in Bangladesh Islamic banks.Design/Methodology/ApproachThis study examined the perception of 125 clients and 25 managers from five key Islamic banks on the investment practices of apartment purchases. This study applied two data analysis methods: reflective measurement model was operated for examining the consistency, reliability, multicollinearity problems and validity; and Smart PLS-SEM (structural equation model) was used to investigate the direct impact of each explanatory variable on Shari’ah compliance in Islamic banks.FindingsThe results indicated that Islamic banks could not strictly maintain the Shariah compliance in apartment purchase. Such Shari’ah non-compliance is due to lack of knowledge and understanding with regard to bank’s objectives and philosophy, weak Shariah board, audit, weak regulatory body and supervision problem. These findings also firmly match clients’ and managers’ perceptions.Research Limitations/ImplicationsA limited size of clients and bankers were involved in this study. The study focuses only on perceptions of Islamic banks’ clients and managers and avoids Muslim clients who involved conventional banks.Originality/ValueThis study provides Shariah-compliant alternatives in investment avenue for apartment purchases. These Shariah modes include Apartment Purchase Musharakah Mutanaqasa; Bai-Muajjal; and Apartment Purchases under Shirkah-ul Milk, which captures various limitations against current apartment purchase practices as well as to develop clients’ trust in Islamic banks.

Symmetric Constellations of Satellites Moving Around a Central Body of Large Mass

We consider a \((1+N)\)-body problem in which one particle has mass \(m_0 \gg 1\) and the remaining N have unitary mass. We can assume that the body with larger mass (central body) is at rest at the origin, coinciding with the center of mass of the N bodies with smaller masses (satellites). The interaction force between two particles is defined through a potential of the form $$\begin{aligned} U \sim \frac{1}{r^\alpha }, \end{aligned}$$where \(\alpha \in [1,2)\) and r is the distance between the particles. Imposing symmetry and topological constraints, we search for periodic orbits of this system by variational methods. Moreover, we use \(\Gamma \)-convergence theory to study the asymptotic behaviour of these orbits, as the mass of the central body increases. It turns out that the Lagrangian action functional \(\Gamma \)-converges to the action functional of a Kepler problem, defined on a suitable set of loops. In some cases, minimizers of the \(\Gamma \)-limit problem can be easily found, and they are useful to understand the motion of the satellites for large values of \(m_0\). We discuss some examples, where the symmetry is defined by an action of the groups \({\mathbb {Z}}_4\) , \({\mathbb {Z}}_2 \times {\mathbb {Z}}_2\) and the rotation groups of Platonic polyhedra on the set of loops.

ρ-Libration point in the three body problem

Herein, the restricted circular three-body problem in homogeneous and inhomogeneous media is considered. Particular attention is paid to libration points. The conditions of their existence or non-existence in the Newtonian and post-Newtonian approximations of the general theory of relativity are derived. Several regularities, new Newtonian and relativistic effects arising due to the impact of the additional relativistic forces on bodies of gravitational fields of mediums in the differential equations of the motion of bodies are indicated. Using the previously derived equations of the motion of two bodies A1, A2 in the medium, the authors substantiated the following statements. In a homogeneous medium (density of the medium ρ = const) in the Newtonian approximation of the general theory of relativity there are ρ-libration points , 1,...,5, moving along the same circles as the Euler and Lagrangian libration points Li but with an angular velocity 0 , greater than the angular velocity ω0 of libration points Li in a vacuum. Bodies A1, A2 also move along their circles with an angular velocity 0 > w When passing from the Newtonian approximation of the general theory of relativity to the post-Newtonian approximation of the general theory of relativity, the centre of mass of two bodies, resting in a homogeneous medium in the Newtonian approximation of the general theory of relativity, must move along a cycloid. The trajectories of the bodies can not be circles, the libration points Li disappear. In the case of an inhomogeneous medium distributed, for example, spherically symmetrically, the centre of mass of two bodies, already in the Newtonian approximation of the general theory of relativity, must move along the cycloid, despite it was at rest in the void. Therefore, bodies A1, A2 must describe loops that form, figuratively speaking, a «lace», as in the case of a homogeneous medium in the post-Newtonian approximation of the general theory of relativity. The figure illustrating the situation is provided. Due to the existence of the «lace» effect, the libration point Li movements are destroyed. In the special case, when the masses of bodies A1, A2 are equal (m1 = m2), the cycloids disappear and all the ρ-libration points exist in homogeneous and inhomogeneous media in the Newtonian and post-Newtonian approximations of the general theory of relativity. Numerical estimates of the predicted patterns and effects in the Solar and other planetary systems, interstellar and intergalactic mediums are carried out. For example, displacements associated with these effects, such as the displacement of the centre of mass, can reach many billions of kilometres per revolution of the two-body system. The possible role of these regularities and effects in the theories of the evolution of planetary systems, galaxies, and their ensembles is discussed. A brief review of the studies carried out by the Belarusian scientific school on the problem of the motion of bodies in media in the general theory of relativity is given.

Mass-spectroscopy of [$$bb][{\bar{b}}{\bar{b}}$$] and [$$bq][{\bar{b}}{\bar{q}}$$] tetraquark states in a diquark–antidiquark formalism

In this article, we utilise the non-relativistic potential model to calculate the mass-spectra of all bottom [ $$bb][{\bar{b}}{\bar{b}}$$ ] and heavy-light bottom [ $$bq][{\bar{b}}{\bar{q}}$$ ] (q = u,d) tetraquark states in diquark–antidiquark approximation. The four-body problem is reduced into two body problem by numerically solving the $$Schr\ddot{o}dinger$$ equation using a Cornell-inspired potential along with relativistic correction term. The splitting structure of the tetraquark spectrum is described using spin-dependent terms (spin-spin, spin-orbit, and tensor). We have successfully calculated and predicted the masses of bottom mesons, diquarks and tetraquarks. The masses of S and P-wave tetraquark states [ $$bb][{\bar{b}}{\bar{b}}$$ ] and [ $$bq][{\bar{b}}{\bar{q}}$$ ], respectively, are found to be between 18.7–19.4 GeV and 10.4–11.3 GeV, in which the masses of S-wave [ $$bb][{\bar{b}}{\bar{b}}$$ ] states are less than the 2 $$\eta _{b}$$ , $$\eta _{b}\Upsilon $$ , and 2 $$\Upsilon $$ threshold. Additionally, we have investigated the $$Z_b(10610)$$ and $$Z_ b(10650)$$ states in the current model and found that they are 150 MeV below the $$BB^{*}$$ and $$B^{*}B^{*}$$ thresholds.

LOCATION AND STABILITY OF THE TRIANGULAR POINTS IN THE TRIAXIAL ELLIPTIC RESTRICTED THREE-BODY PROBLEM

The main goal of the present paper is to evaluate the perturbed locations and investigate the linear stability of the triangular points. We studied the problem in the elliptic restricted three body problem frame of work. The problem is generalized in the sense that the two primaries are considered as triaxial bodies. It was found that the locations of these points are affected by the triaxiality coefficients of the primaries and the eccentricity of orbits. Also, the stability regions depend on the involved perturbations. We also studied the periodic orbits in the vicinity of the triangular points.