The time-transformed leapfrog scheme of Mikkola ＆ Aarseth was specifi- cally designed for a second-order differential equation with two individually separable forms of positions and velocities. It can have good numer...The time-transformed leapfrog scheme of Mikkola ＆ Aarseth was specifi- cally designed for a second-order differential equation with two individually separable forms of positions and velocities. It can have good numerical accuracy for extremely close two-body encounters in gravitating few-body systems with large mass ratios, but the non-time-transformed one does not work well. Following this idea, we develop a new explicit symplectic integrator with an adaptive time step that can be applied to a time-dependent Hamiltonian. Our method relies on a time step function having two distinct but equivalent forms and on the inclusion of two pairs of new canonical con- jugate variables in the extended phase space. In addition, the Hamiltonian must be modified to be a new time-transformed Hamiltonian with three integrable parts. When this method is applied to the elliptic restricted three-body problem, its numerical pre- cision is explicitly higher by several orders of magnitude than the nonadaptive one＇s, and its numerical stability is also better. In particular, it can eliminate the overestima- tion of Lyapunov exponents and suppress the spurious rapid growth of fast Lyapunov indicators for high-eccentricity orbits of a massless third body. The present technique will be useful for conservative systems including N-body problems in the Jacobian coordinates in the the field of solar system dynamics, and nonconservative systems such as a time-dependent barred galaxy model in a rotating coordinate system.展开更多
As for a double pyramidal central configuration in 6-body problems, the case when its base is a concave polygon is studied. By advancing several assumptions according to the definition of double pyramidal central conf...As for a double pyramidal central configuration in 6-body problems, the case when its base is a concave polygon is studied. By advancing several assumptions according to the definition of double pyramidal central configuration and deducing two theorems and two corollaries on this subject, the essential and sufficient conditions to form a double pyramidal central configuration with a concave quadrilateral base are demonstrated.展开更多
We intend to study a modified version of the planar Circular Restricted Three-Body Problem(CRTBP) by incorporating several perturbing parameters. We consider the bigger primary as an oblate spheroid and emitting radia...We intend to study a modified version of the planar Circular Restricted Three-Body Problem(CRTBP) by incorporating several perturbing parameters. We consider the bigger primary as an oblate spheroid and emitting radiation while the small primary has an elongated body. We also consider the perturbation from a disk-like structure encompassing this three-body system. First, we develop a mathematical model of this modified CRTBP.We have found there exist five equilibrium points in this modified CRTBP model, where three of them are collinear and the other two are non-collinear. Second, we apply our modified CRTBP model to the Sun–Haumea system by considering several values of each perturbing parameter. Through our numerical investigation, we have discovered that the incorporation of perturbing parameters has resulted in a shift in the equilibrium point positions of the Sun–Haumea system compared to their positions in the classical CRTBP. The stability of equilibrium points is investigated. We have shown that the collinear equilibrium points are unstable and the stability of non-collinear equilibrium points depends on the mass parameter μ of the system. Unlike the classical case, non-collinear equilibrium points have both a maximum and minimum limit of μ for achieving stability. We remark that the stability range of μ in non-collinear equilibrium points depends on the perturbing parameters. In the context of the Sun–Haumea system, we have found that the non-collinear equilibrium points are stable.展开更多
The Purple Mountain Observatory of the Chinese Academy of Sciences and the Department of Astronomy, Nanjing University, cooperating with the Albert Einstein Institute in Germany, put forward a model of the planar co-o...The Purple Mountain Observatory of the Chinese Academy of Sciences and the Department of Astronomy, Nanjing University, cooperating with the Albert Einstein Institute in Germany, put forward a model of the planar co-orbital circular restricted three-body problem.展开更多
This review article aims to give a comprehensive review of periodic orbits in the circular restricted three-body problem(CRTBP),which is a standard ideal model for the Earth-Moon system and is closest to the practical...This review article aims to give a comprehensive review of periodic orbits in the circular restricted three-body problem(CRTBP),which is a standard ideal model for the Earth-Moon system and is closest to the practical mechanical model.It focuses the attention on periodic orbits in the Earth-Moon system.This work is primarily motivated by a series of missions and plans that take advantages of the three-body periodic orbits near the libration points or around two gravitational celestial bodies.Firstly,simple periodic orbits and their classification that is usually considered to be early work before 1970 are summarized,and periodic orbits around Lagrange points,either planar or three-dimensional,are intensively studied during past decades.Subsequently,stability index of a periodic orbit and bifurcation analysis are presented,which demonstrate a guideline to find more periodic orbits inspired by bifurcation signals.Then,the practical techniques for computing a wide range of periodic orbits and associated quasi-periodic orbits,as well as constructing database of periodic orbits by numerical searching techniques are also presented.For those unstable periodic orbits,the station keeping maneuvers are reviewed.Finally,the applications of periodic orbits are presented,including those in practical missions,under consideration,and still in conceptual design stage.This review article has the function of bridging between engineers and researchers,so as to make it more convenient and faster for engineers to understand the complex restricted three-body problem(RTBP).At the same time,it can also provide some technical thinking for general researchers.展开更多
Based on large quantities of co-orbital phenomena in the motion of natural bodies and spacecraft, a model of the co-orbital restricted three-body problem is put forward. The fundamental results for the planar co-orbit...Based on large quantities of co-orbital phenomena in the motion of natural bodies and spacecraft, a model of the co-orbital restricted three-body problem is put forward. The fundamental results for the planar co-orbital circular restricted three-body problem are given, which include the selection of variables and equations of motion, a set of approximation formulas, and an approximate semi-analytical solution. They are applied to the motion of the barycenter of the planned gravitational observatory LISA constellation, which agrees very well with the solution of precise numerical integration.展开更多
Newton＇s gravitational law and its derived Kapler＇s orbit of planetary motions are the very first triumph of science, which confirms that a purely human＇s invention of mathematics is able to accurately describe obs...Newton＇s gravitational law and its derived Kapler＇s orbit of planetary motions are the very first triumph of science, which confirms that a purely human＇s invention of mathematics is able to accurately describe observations in nature. For the past three centuries, the frontier of science has moved to almost all areas of human experience, yet some basic problem in celestial mechanics has remained unsolved, despite the effort of many great minds like Newton （1687）, Euler （1760） and Lagrange （1776）, among others.展开更多
基金Supported by the National Natural Science Foundation of China
文摘The time-transformed leapfrog scheme of Mikkola ＆ Aarseth was specifi- cally designed for a second-order differential equation with two individually separable forms of positions and velocities. It can have good numerical accuracy for extremely close two-body encounters in gravitating few-body systems with large mass ratios, but the non-time-transformed one does not work well. Following this idea, we develop a new explicit symplectic integrator with an adaptive time step that can be applied to a time-dependent Hamiltonian. Our method relies on a time step function having two distinct but equivalent forms and on the inclusion of two pairs of new canonical con- jugate variables in the extended phase space. In addition, the Hamiltonian must be modified to be a new time-transformed Hamiltonian with three integrable parts. When this method is applied to the elliptic restricted three-body problem, its numerical pre- cision is explicitly higher by several orders of magnitude than the nonadaptive one＇s, and its numerical stability is also better. In particular, it can eliminate the overestima- tion of Lyapunov exponents and suppress the spurious rapid growth of fast Lyapunov indicators for high-eccentricity orbits of a massless third body. The present technique will be useful for conservative systems including N-body problems in the Jacobian coordinates in the the field of solar system dynamics, and nonconservative systems such as a time-dependent barred galaxy model in a rotating coordinate system.
基金Funded by the Natural Science Foundation of China (No. 19871096).
文摘As for a double pyramidal central configuration in 6-body problems, the case when its base is a concave polygon is studied. By advancing several assumptions according to the definition of double pyramidal central configuration and deducing two theorems and two corollaries on this subject, the essential and sufficient conditions to form a double pyramidal central configuration with a concave quadrilateral base are demonstrated.
基金funded partially by BRIN’s research grant Rumah Program AIBDTK 2023。
文摘We intend to study a modified version of the planar Circular Restricted Three-Body Problem(CRTBP) by incorporating several perturbing parameters. We consider the bigger primary as an oblate spheroid and emitting radiation while the small primary has an elongated body. We also consider the perturbation from a disk-like structure encompassing this three-body system. First, we develop a mathematical model of this modified CRTBP.We have found there exist five equilibrium points in this modified CRTBP model, where three of them are collinear and the other two are non-collinear. Second, we apply our modified CRTBP model to the Sun–Haumea system by considering several values of each perturbing parameter. Through our numerical investigation, we have discovered that the incorporation of perturbing parameters has resulted in a shift in the equilibrium point positions of the Sun–Haumea system compared to their positions in the classical CRTBP. The stability of equilibrium points is investigated. We have shown that the collinear equilibrium points are unstable and the stability of non-collinear equilibrium points depends on the mass parameter μ of the system. Unlike the classical case, non-collinear equilibrium points have both a maximum and minimum limit of μ for achieving stability. We remark that the stability range of μ in non-collinear equilibrium points depends on the perturbing parameters. In the context of the Sun–Haumea system, we have found that the non-collinear equilibrium points are stable.
基金Funding from the National Natural Science Foundation of China (Grant Nos. 10503013, 10933004)the Foundation of Minor Planets of Purple Mountain Observatory supported this research
文摘The Purple Mountain Observatory of the Chinese Academy of Sciences and the Department of Astronomy, Nanjing University, cooperating with the Albert Einstein Institute in Germany, put forward a model of the planar co-orbital circular restricted three-body problem.
文摘This review article aims to give a comprehensive review of periodic orbits in the circular restricted three-body problem(CRTBP),which is a standard ideal model for the Earth-Moon system and is closest to the practical mechanical model.It focuses the attention on periodic orbits in the Earth-Moon system.This work is primarily motivated by a series of missions and plans that take advantages of the three-body periodic orbits near the libration points or around two gravitational celestial bodies.Firstly,simple periodic orbits and their classification that is usually considered to be early work before 1970 are summarized,and periodic orbits around Lagrange points,either planar or three-dimensional,are intensively studied during past decades.Subsequently,stability index of a periodic orbit and bifurcation analysis are presented,which demonstrate a guideline to find more periodic orbits inspired by bifurcation signals.Then,the practical techniques for computing a wide range of periodic orbits and associated quasi-periodic orbits,as well as constructing database of periodic orbits by numerical searching techniques are also presented.For those unstable periodic orbits,the station keeping maneuvers are reviewed.Finally,the applications of periodic orbits are presented,including those in practical missions,under consideration,and still in conceptual design stage.This review article has the function of bridging between engineers and researchers,so as to make it more convenient and faster for engineers to understand the complex restricted three-body problem(RTBP).At the same time,it can also provide some technical thinking for general researchers.
基金support by the National Natural Science Foundation of China (Grant No. 10503013)the Foundation of Minor Planets of Purple Mountain Observatory
文摘Based on large quantities of co-orbital phenomena in the motion of natural bodies and spacecraft, a model of the co-orbital restricted three-body problem is put forward. The fundamental results for the planar co-orbital circular restricted three-body problem are given, which include the selection of variables and equations of motion, a set of approximation formulas, and an approximate semi-analytical solution. They are applied to the motion of the barycenter of the planned gravitational observatory LISA constellation, which agrees very well with the solution of precise numerical integration.
文摘Newton＇s gravitational law and its derived Kapler＇s orbit of planetary motions are the very first triumph of science, which confirms that a purely human＇s invention of mathematics is able to accurately describe observations in nature. For the past three centuries, the frontier of science has moved to almost all areas of human experience, yet some basic problem in celestial mechanics has remained unsolved, despite the effort of many great minds like Newton （1687）, Euler （1760） and Lagrange （1776）, among others.
