Triangular Points Research Articles

Effects of Variable Mass, Disk-Like Structure, and Radiation Pressure on the Dynamics of Circular Restricted Three-Body Problem

In this paper, we intend to investigate the dynamics of the Circular Restricted Three-Body Problem. Here we assumed the primaries as the source of radiation and have variable mass. The gravitational perturbation from disk-like structure are also considered in this study. There exist five equilibrium points in this system. By considering the combined effect of disk–like structure and the mass transfer, we found that the classical collinear equilibrium points depart from the x–axis. We called these equilibrium points as quasi–collinear equilibrium points. Meanwhile, this combined effect also breaks the symmetry of triangular equilibrium point positions. We noted that the quasi–equilibrium points are unstable whereas the triangular equilibrium points are stable if the mass ratio μ is smaller than critical mass μc. Besides the mass ratio, the stability of triangular equilibrium points depend on time.

Transfers to Earth-Moon triangular libration points by Sun-perturbed dynamics

This paper is devoted to the construction of transfers to Earth-Moon triangular libration points based on their Sun-perturbed dynamics in the Sun-Earth-Moon system. In this bi-circular problem (BCP), the solar gravity perturbation breaks the stability of the Earth-Moon triangular libration points with three remaining periodic orbits (two stable and one unstable) around them as their dynamical substitutions. As the stable region, the invariant tori full of quasi-periodic orbits around the two stable periodic orbits can be used as the destination of triangular libration point transfers for a low station-keeping cost. However, the instability of the unstable periodic orbit is so mild that its stable and unstable manifolds are folded around the invariant tori surrounding the other two stable orbits. Thus, the authors approximate the stable and unstable manifolds of the unstable orbit through the parameterization method, which emerge from the Sun-perturbed Earth-Moon triangular libration points vicinity. They act as a bridge between outer space and the stable region around triangular libration points. First, the transfers connecting the Earth’s vicinity are constructed using the manifolds. Subsequently, a new connection mechanism near L4 and L5 based on the N:M multi-loop transfers is developed by extending the conventional patching skill to time-dependent situations. The quickest one is achieved by a 1:1 L4-L5 heteroclinic-based connection with a TOF of less than 26.9 days at a fuel cost of 117.88 m/s. Note that such Sun-perturbed transfers cannot be produced by continuing any trajectory in the Earth-Moon circular restricted three-body problem but they can generate two-impulsive transfers in the ephemeris model by continuation.

Perturbing and Oblateness's Impact on the Generalized Photogravitational Restricted Three-Body Problem

In this research work, the effects of perturbing and oblateness on the generalised photogravitational restricted three body problem (RTBP) are examined when the primary are regarded as radiation sources and oblate spheroid. The perturbing forces were included in the equations of motion that we developed for the system. The triangular libration points were derived in terms of the perturbing parameters using linear approximations, and their influence was seen. Applying Murray's criteria, we determined that the triangular libration points remained unstable due to the nature of the characteristics equation matching to the variational equations of motion of the system generated.

Spherical Magnetic Vector Forwarding of Isoparametric DGGS Cells with Natural Superconvergent Points

With the rapid advancement of satellite remote sensing technology, many scientists and organizations, including NASA, ESA, NAOC, and Roscosmos, observe and study significant changes in the geomagnetic field, which has greatly promoted research on the geomagnetic field and made it an important research direction in Earth system science. In traditional geomagnetic field research, tesseroid cells face degradation issues in high-latitude regions and accuracy limitations. To overcome these limitations, this paper introduces the Discrete Global Grid System (DGGS) to construct a geophysical model, achieving seamless global coverage through multi-level grid subdivision, significantly enhancing the processing capability of multi-source and multi-temporal spatial data. Addressing the challenges of the lack of analytical solutions and clear integration limits for DGGS cells, a method for constructing shape functions of arbitrary isoparametric elements is proposed based on the principle of isoparametric transformation, and the shape functions of isoparametric DGGS cells are successfully derived. In magnetic vector forwarding, considering the potential error amplification caused by Poisson’s formula, the DGGS grid is divided into six regular triangular sub-units. The triangular superconvergent point technique is adopted, and the positions of integration points and their weight coefficients are accurately determined according to symmetry rules, thereby significantly improving the calculation accuracy without increasing the computational complexity. Finally, through the forward modeling algorithm based on tiny tesseroid cells, this study comprehensively compares and analyzes the computational accuracy of the DGGS-based magnetic vector forwarding algorithm, verifying the effectiveness and superiority of the proposed method and providing new theoretical support and technical means for geophysical research.

Dynamics around the Earth–Moon triangular points in the Hill restricted 4-body problem

This paper investigates the motion of a small particle moving near the triangular points of the Earth–Moon system. The dynamics are modeled in the Hill restricted 4-body problem (HR4BP), which includes the effect of the Earth and Moon as in the circular restricted 3-body problem (CR3BP), as well as the direct and indirect effect of the Sun as a periodic time-dependent perturbation of the CR3BP. Due to the periodic perturbation, the triangular points of the CR3BP are no longer equilibrium solutions; rather, the triangular points are replaced by periodic orbits with the same period as the perturbation. Additionally, there is a 2:1 resonant periodic orbit that persists from the CR3BP into the HR4BP. In this work, we investigate the dynamics around these invariant objects by performing a center manifold reduction and computing families of 2-dimensional invariant tori and their linear normal behavior. We identify bifurcations and relationships between families. Mechanisms for transport between the Earth, L4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_4$$\\end{document}, and the Moon are discussed. Comparisons are made between the results presented here and in the bicircular problem (BCP).

Study of periodic orbits around the triangular points in the circular restricted three-body problem when both the primaries are oblate spheroids and sources of radiation

Study of periodic orbits around the triangular points in the circular restricted three-body problem when both the primaries are oblate spheroids and sources of radiation

On the effect of the sun on Kordylewski clouds

In this paper, we focus on the existence of dust clouds moving near the triangular points of the Earth–Moon system, the so-called Kordylewski clouds. The study is based on using some simplified planar models to find possible locations for these clouds. The validity of these predictions is tested by means of numerical simulations on a realistic model. The simplified models are based on the Earth–Moon restricted three-body problem plus the direct gravitational effect of the Sun on the particles (this is the so-called bicircular model), the solar radiation pressure and the Poynting–Robertson effect. The analysis of these models shows that there are some stability regions in the Earth–Moon plane, at some distance of the triangular points. The stability of these regions has been tested numerically in realistic (nonplanar) models. The results show that particles in these regions persist for some time (about a century), but it is very remarkable that many of these particles also escape the Earth–Moon system. If we perform backwards in time numerical simulations we obtain a similar result: particles also escape the Earth–Moon system after a similar time. From this point of view, the clouds are not a stable region in the classical sense of the term, but a region with “slow diffusion” where interplanetary particles stay for some years.

Structure tensor-based Gaussian kernel edge-adaptive depth map refinement with triangular point view in images

<span lang="EN-US">Image reconstruction is the process of restoring the image resolution. In 3D image reconstruction, the objects in different viewpoints are processed with the triangular point view (TPV) method to estimate object geometry structure for 3D model. This work proposes a depth refinement methodology in preserving the geometric structure of objects using the structure tensor method with a Gaussian filter by transforming a series of 2D input images into a 3D model. The computation of depth map errors can be found by comparing the masked area/patch with the distribution of the original image's greyscale levels using the error pixel-based patch extraction algorithm. The presence of errors in the depth estimation could seriously deteriorate the quality of the 3D effect. The depth maps were iteratively refined based on histogram bins number to improve the accuracy of initial depth maps reconstructed from rigid objects. The existing datasets such as the dataset tanks and unit (DTU) and Middlebury datasets, were used to build the model out of the object scene structure. The results of this work have demonstrated that the proposed patch analysis outperformed the existing state of the art models depth refinement methods in terms of accuracy.</span>

Dynamics around small irregularly shaped objects modeled as a mass dipole

In this work, we investigate the dynamics of a spacecraft near two primary bodies. The massive body is considered to have a spherical shape, while the less massive one is elongated and modeled as a dipole. The dipole consists of two connected masses, one is spherical and the other is an oblate spheroid. The gravitational potential of the elongated body is determined by four independent parameters. To study the dynamics, we construct the equations of motion of a spacecraft with negligible mass under the effect of the current force model. The existence and locations of the equilibrium points are analyzed for various values of the system parameters. We found that the existence and locations of the points are affected by the system parameters. Also, we studied the linear stability of the equilibrium points. We found some stable collinear points when the oblateness parameter is negative, otherwise the points are not stable. We used the curves of zero velocity to identify the regions of allowed motion. Furthermore, we discussed the 2001 SN263 asteroid system and found some stable collinear points when the oblateness parameter is negative. In addition, the triangular points of the system are stable in a linear sense.

Combined effect of Poynting-Robertson (P-R) drag, oblateness and radiation on the triangular points in the elliptic restricted three-body problem

This study investigates the motion of a test particle around triangular equilibrium points in the elliptic restricted three-body problem (ER3BP) under the influence of the two oblate and radiating primaries having Poynting-Robertson (P-R) drag. It is observed that the position of triangular points of the problem is affected by oblateness, radiation pressure, eccentricity, semi-major axis and Poynting-Robertson (P-R) drag. The stability of these points is demonstrated analytically by the Routh-Hurwitz criterion. It is seen that they are unstable under the combined effect of involved parameters. The effect of these parameters on the position of triangular points is examined numerically using the binary systems, 61 Cygni and Archird. The results obtained by these binary systems can be used to broaden the scope of interest in astronomy, astrophysics, space science and celestial mechanics in general.

Hrot typu Emireh z lokality bohunicienu Ořechov IV–Kabáty (jižní Morava, Česká republika)

The Emireh point is a triangular point, which distinctive bulb of percussion was eliminated with bifacial thinning applied to proximal part of the artefact. This point is a characteristic tool for the Emiran industries and up to yet reported from Levantine sites only. Recently, an isolated Emireh point was excavated at Bohunician site Ořechov IV in South Moravia, ca 3,000 km as the crow flies from Levantine sites.

Analysis of motion in RTBP with variable mass based on Loglogistic distribution

Recent literature has stated that the physical characteristics of natural satellites in Jupiter’s Ananke and Carme groups, such as mass, exhibit the same statistical law, with the best-fitting distribution being a Loglogistic distribution. This paper establishes dynamical equations for the variable-mass restricted three-body problem based on this new statistical law. In addition to discussing the influence of the Jacobian constant related to the system energy on the accessible region of the third body, it also uses the Lindstedt–Poincaré perturbation method to give an analytical periodic orbit of second- and third-order around the triangular Lagrangian point. Numerical simulations show that as the value of the variable mass parameter of the third body becomes smaller, the corresponding periodic orbit is affected less and less.

Two-dimensional silicon-based dielectric column photonic crystal point defect microcavity Neural network modelling

In order to solve the problem that it is difficult to predict the microcavity photonic energy bands of point defects in two-dimensional silicon-based dielectric column photonic crystals, this paper proposes a method to predict the microcavity photonic energy bands of point defects in two-dimensional silicon-based photonic crystals by using an artificial neural network model. In this paper, the energy band structure of triangular lattice point defects at different radii is calculated using MPB, and a neural network model is established based on this data and the accuracy of the established model is verified.

LOCATIONS OF TRIANGULAR EQUILIBRIUM POINTS OF THE RESTRICTED THREE-BODY PROBLEM WITH POYNTING-ROBERTSON DRAG AND VARIABLE MASSES

The restricted three-body problem (R3BP) is a fascinating problem that has been receiving attentions of astronomers and scientists because of its vast implications in diverse scientific fields, including among others; celestial mechanics, galactic dynamics, chaos theory and molecular physics. In this paper, we examine the locations of the triangular equilibrium points of the R3BP with Poynting-Robertson (P-R) drag forces and variable masses. The primaries are assumed to vary under the unified Mestschersky law and their dynamics defined by the Gylden-Mestschersky equation, while the smaller primary is assumed to be a radiation emitter with P-R drag. The dynamical equations are obtained for both the non-autonomous with variable coefficients and autonomized system with constant coefficients. Further, the locations of the triangular points of the autonomized systems are obtained using perturbation method. It is seen that the positions are defined by the mass parameter, radiation pressure and P-R drag of the smaller primary. The triangular points of the non-autonomous equations are obtained with help of the Mestschersky transformation, and differ from those of the autonomized system due to a function of time. The equilibrium points have several applications in space missions, satellites constellations and station-keeping.

Non-linear dynamics of a test particle near the Lagrange points L4 and L5 (Earth-Moon and Sun-Earth case)

The two-bodies problem can be fully solved, and was solved by Kepler (1609) and Newton (1687). The general three-body problem is often given as an example of a mathematical problem that ‘can’t be solved’. So, there is no general analytical solution. This problem can be significant and a special case of this problem is the Circular Restricted Three-Body Problem (CRTBP), which can be applied to the Earth-Moon system with a spacecraft, the Sun-Earth system with an asteroid, etc. In this paper, let’s focus on the motion of a test particle near the triangular Lagrange points L4 and L5 in the Earth-Moon and the Sun-Earth systems. Studying the movement of an object around these points is especially important for space mission design. To generate a trajectory around these points, the non-linear equations of motion for the circular restricted three-body problem were numerically integrated into MATLAB® 2023 software and the results are presented in the plane (x, y) and the phase plane (x, vx) and (y, vy). By numerical orbit integration, it is possible to investigate what happens when the displacement is relatively large or short from the Lagrange points. Then the small astronomical body may vibrate around these points. The results in this paper are shown in the rotating and inertia axes. Various initial positions near the Lagrange points and velocities are used to produce various paths the test particle can take. The same examples of numerical studies of trajectories associated with Lagrange points are shown in the inertial and the rotating coordinates system and are discussed. From the results of the numerical tests performed in MATLAB® 2023, it is possible to saw that there are different types of periodic, quasi-periodic, and chaotic orbits

Using CloudCompare software editing tools for processing a three-dimensional point cloud of an urban development site

This article describes the functions that can be performed by CloudCompare, an open-source triangular mesh and 3D point cloud processing software developed through a collaboration between Telecom ParisTech and EDF R&D. CloudCompare software is used to study and analyze archaeological sites and create their 3D models. Helps engineers monitor the condition of bridges, roads, buildings, etc., identify problems and find solutions to eliminate them. The program provides a set of basic tools for editing and rendering. Some basic editing tools for working in the CloudCompare software are studied and demonstrated, such as cloning, cutting an element from the main object of study, scaling. It is concluded that the CloudCompare software is easy to use and effectively performs the functions contained in it.

Poxyaibamberus Andersen & Dantas, gen. nov. (Diptera, Chironomidae, Orthocladiinae) from Brazil.

Poxyaibamberus Andersen & Dantas, gen. nov. is erected based on the males of two species, P.jamanximensis Andersen & Dantas, sp. nov. from Jamanxim National Park, Pará State, Brazil, and P.ubajarensis Andersen & Dantas, sp. nov. from Ubajara National Park, Ceará State, Brazil. Both species have a comparatively short and wide head, with large eyes and short, five-segmented palps; a strong subapical seta on the ultimate flagellomere; scalpellate acrostichals; no setae on the wing veins except for one seta on the brachiolum; a long costal extension; and a large triangular anal point and a very long heel on the gonostylus. The systematic position of the new genus is briefly discussed.

Nonlinear stability of triangular equilibrium points in non-resonance case with perturbations

Nonlinear stability of triangular equilibrium points in non-resonance case with perturbations

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.

Synthesis of ternary geopolymers using prediction for effective solidification of mercury in tailings

This study used steel slag, fly ash, and metakaolin as raw materials (SFM materials) to create silica-alumina-based geopolymers that can solidify Hg2+ when activated with sodium-based water glass. The experiments began with a triangular lattice point mixing design experiment, and the results were fitted, analyzed, and predicted. The optimum SFM material mass ratio was found to be 70% steel slag, 25% fly ash, and 5% metakaolin. The optimum modulus of the activator was identified by comparing the unconfined compressive strength and solidifying impact on Hg2+of geosynthetics with different modulus. The SFM geopolymer was then applied in the form of potting to cure the granulated mercury tailings. The inclusion of 50% SFM material generated a geosynthetic that reduced mercury transport to the surface soil by roughly 90%. The mercury concentration of herbaceous plant samples was also reduced by 78%. It indicates that the SFM material can effectively attenuate the migration transformation of mercury. Finally, characterization methods such as XPS and FTIR were used to investigate the mechanism of Hg2+ solidification by geopolymers generated by SFM materials. The possible solidification mechanisms were proposed as alkaline environment-induced mercury precipitation, chemical bonding s, surface adsorption of Hg2+ and its precipitates by the geopolymer, and physical encapsulation.