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1.
A strategy is formulated to design optimal time-fixed impulsive transfers between three-dimensional libration-point orbits in the vicinity of the interiorL
1 libration point of the Sun-Earth/Moon barycenter system. The adjoint equation in terms of rotating coordinates in the elliptic restricted three-body problem is shown to be of a distinctly different form from that obtained in the analysis of trajectories in the two-body problem. Also, the necessary conditions for a time-fixed two-impulse transfer to be optimal are stated in terms of the primer vector. Primer vector theory is then extended to non-optimal impulsive trajectories in order to establish a criterion whereby the addition of an interior impulse reduces total fuel expenditure. The necessary conditions for the local optimality of a transfer containing additional impulses are satisfied by requiring continuity of the Hamiltonian and the derivative of the primer vector at all interior impulses. Determination of the location, orientation, and magnitude of each additional impulse is accomplished by the unconstrained minimization of the cost function using a multivariable search method. Results indicate that substantial savings in fuel can be achieved by the addition of interior impulsive maneuvers on transfers between libration-point orbits.An earlier version was presented as Paper AAS 92–126 at the AAS/AIAA Spaceflight Mechanics Meeting, Colorado Springs, Colorado, February 24–26, 1992. 相似文献
2.
Donald J. Jezewski 《Celestial Mechanics and Dynamical Astronomy》1976,14(1):105-111
A method for developing the missing general K/S (Kustaanheimo/Stiefel) boundary conditions is presented, with use of the formalism of optimal control theory. As an illustrative example, the method is applied to the K/S Lambert problem to derive the missing terminal condition. The necessary equations are developed for a solution to this problem with the generalized eccentric anomaly,E, as the independent variable. This formulation, requiring the solution of only one nonlinear, well-behaved equation in one unknown,E, results in considerable simplification of the problem.Presented also at the AAS/AIAA Astrodynamics Specialist Conference, Nassau, Bahamas, July 1975 (AAS Paper No. 75-032). 相似文献
3.
George H. Born 《Celestial Mechanics and Dynamical Astronomy》1970,2(1):103-113
In this paper, a combination analytical-numerical integration method for solving the differential equations of a modified set of Lagrange's planetary equations is described. The integration method is an Encke-type method because it involves integrating the deviations between the actual trajectory and a reference trajectory. The reference trajectory is obtained from an analytical solution containing the dominant secular and periodic effects of the gravitational field of the primary body. A set of nonsingular elements is used so that the method will be valid for all circular and elliptical motions. It is shown that the method is an accurate and efficient means of satellite ephemeris generation.This paper was presented at the AIAA/AAS Meeting, Princeton University, August 1969. 相似文献
4.
George Contopoulos Christos Efthymiopoulos 《Celestial Mechanics and Dynamical Astronomy》2008,102(1-3):219-239
We discuss the issue of ordered and chaotic trajectories in the Bohmian approach of Quantum Mechanics from points of view relevant to the methods of Celestial Mechanics. The Bohmian approach gives the same results as the orthodox (Copenhagen) approach, but it considers also underlying trajectories guided by the wave. The Bohmian trajectories are rather different from the corresponding classical trajectories. We give examples of a classical chaotic system that is ordered quantum-mechanically and of a classically ordered system that is mostly chaotic quantum mechanically. Then we consider quantum periodic orbits and ordered orbits, that can be represented by formal series of the “third integral” type, and we study their asymptotic properties leading to estimates of exponential stability. Such orbits do not approach the “nodal points” where the wavefunction ψ vanishes. On the other hand, when an orbit comes close to a nodal point, chaos is generated in the neighborhood of a hyperbolic point (called X-point). The generation of chaos is maximum when the X-point is close to the nodal point. Finally we remark that high order periodic orbits may behave as “effectively ordered” or “effectively chaotic” for long times before reaching the period. 相似文献
5.
Victor A. Brumberg Eugene V. Brumberg: John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》1999,75(4):301-301
The aim of this book is to present techniques for the study of motion of solar system objects in highly eccentric orbits. Instead of using the usual anomalies (mean, true, eccentric), the authors define and use a new kind of anomaly, the elliptic anomaly.In this way, it is possible, in a theory using perturbation series expansions, to make the ratio: (accuracy)/(number of needed terms), higher than in the classical techniques. The book consists of six chapters. The first chapter deals with the elliptic anomaly in the two-body problem and the second chapter presents the general technique to construct first-order perturbation theory in elliptic function expansions. The next three chapters deal with applications of the new technique to artificial satellites and asteroids, in highly eccentric orbits. The last chapter describes the basic algorithms of the theory.The tools developed in the book demand the use of computer algebra, which is implemented by means of Mathematica 3.0.The book is well written and the new technique is clearly presented and related to the existing techniques, making it useful to all those who use analytical or semi-analytical methods for the study of highly eccentric motion.
Celestial Mechanics at High Eccentricities, Gordon and Breach Publishers, US$95, GBP 59, EUR 79, ISBN 90-5699-212-0 相似文献
6.
Harold S. Morton Jr. John L. Junkins Jeffrey N. Blanton 《Celestial Mechanics and Dynamical Astronomy》1974,10(3):287-301
Two new analytical solutions for Poinsot motion in terms of Euler parameters are derived. The first solution is a straightforward ‘universal’ (no branches) time series practical for short time motion calculations or as a basis for analytical continuation. The second, more involved solution is also universal but is not restricted to short times; it is in terms of circular, hyperbolic, and elliptic functions and elliptic integrals. 相似文献
7.
Victor R. Bond 《Celestial Mechanics and Dynamical Astronomy》1974,10(3):303-318
The Newtonian differential equations of motion for the two-body problem can be transformed into four, linear, harmonic oscillator equations by simultaneously applying the regularizing time transformation dt/ds=r and the Kustaanheimo-Stiefel (KS) coordinate transformation. The time transformation changes the independent variable from time to a new variables, and the KS transformation transforms the position and velocity vectors from Cartesian space into a four-dimensional space. This paper presents the derivation of uniform, regular equations for the perturbed twobody problem in the four-dimensional space. The variation of parameters technique is used to develop expressions for the derivatives of ten elements (which are constants in the unperturbed motion) for the general case that includes both perturbations which can arise from a potential and perturbations which cannot be derived from a potential. These element differential equations are slightly modified by introducing two additional elements for the time to further improve long term stability of numerical integration.Originally presented at the AAS/AIAA Astrodynamics Specialists Conference, Vail, Colorado, July 1973 相似文献
8.
A new simple method for the closed-form solution of nonlinear algebraic and transcendental equations through integral formulae is proposed. This method is applied to the solution of the famous Kepler equation in the two-body problem for elliptic orbits. The resulting formulae are quite elementary and, beyond their analytical interest, they can also provide quite accurate numerical results by using Gausstype quadrature rules. 相似文献
9.
Colette Edelman 《Celestial Mechanics and Dynamical Astronomy》1985,37(4):351-358
Dans un système d'axes fixes le problème gravitationnel des n. corps possède quatre groupes d'invariance (rectifications). Aucun de ces groupes ne peut échanger une solution non bornée et une solution bornée.Dans le cas du problème non circulaire et non rectilinéaire des deux corps, une transformation paramétrique peut-être définie, changeant seulement l'exentricité et l'horaire. Cette transformation est de type homographique et son expression anlytique dépend des valeurs de l'exentricité par rapport à l'unité. Par conséquent, une solution hyperbolique ou parabolique peut-être changée en une solution elliptique. Les applications et l'utilité d'une telle transformation concerne les captures des comètes. Finalement, une hypothétique extension est indiquée pour le problème des n. corps.
Invariant transformation of the two-body problem associated with eccentricity
In an absolute reference frame the gravitational n-body problem possesses four groups of invariant transformations (rectifications). But no one can change an unbounded solution into a bounded solution.For the non-circular two-body problem, having non-zero angular momentum a parametric transformation may be defined changing only the eccentricity and the time. This transformation is of homographic type, and it is an analytical expression depends on the value of the eccentricity with respect to unity. Therefore an hyperbolic or parabolic solution may be changed into an elliptic solution. The application and usefulness of this transformation is concerned with the capture of comets [5].Finally, an hypothetic extension is indicated to the n-body problem.相似文献
10.
Transition from elliptic to hyperbolic orbits in the two-body problem with slowly decreasing mass is investigated by means of asymptotic approximations.Analytical results by Verhulst and Eckhaus are extended to construct approximate solutions for the true anomaly and the eccentricity of the osculating orbit if the initial conditions are nearly-parabolic. It becomes clear that the eccentricity will monotonously increase with time for all mass functions satisfying a Jeans-Eddington relation and even for a larger set of functions. To illustrate these results quantitatively we calculate the eccentricity as a function of time for Jeans-Eddington functionsn=0(1) 5 and 18 nearly-parabolic initial conditions to find that 93 out of 108 elliptic orbits become hyperbolic. 相似文献
11.
Ming Xu Jiamin Zhu Tian Tan Shijie Xu 《Celestial Mechanics and Dynamical Astronomy》2012,113(4):403-433
The bounded quasi-periodic relative trajectories are investigated in this paper for on-orbit surveillance, inspection or repair, which requires rapid changes in formation configuration for full three-dimensional imaging and unpredictable evolutions of relative trajectories for non-allied spacecraft. A linearized differential equation for modeling J 2 perturbed relative dynamics is derived without any simplified treatment of full short-period effects. The equation serves as a nominal reference model for stationkeeping controller to generate the quasi-periodic trajectories near the equilibrium, i.e., the location of the chief. The developed model exhibits good numerical accuracy and is applicable to an elliptic orbit with small eccentricity inheriting from the osculating conversion of orbital elements. A Hamiltonian structure-preserving controller is derived for the three-dimensional time-periodic system that models the J 2-perturbed relative dynamics on a mean circular orbit. The equilibrium of the system has time-varying topological types and no fixed-dimensional unstable/stable/center manifolds, which are quite different from the two-dimensional time-independent system with a permanent pair of hyperbolic eigenvalues and fixed-dimensions of unstable/stable/ center manifolds. The unstable and stable manifolds are employed to change the hyperbolic equilibrium to elliptic one with the poles assigned on the imaginary axis. The detailed investigations are conducted on the critical controller gain for Floquet stability and the optimal gain for the fuel cost, respectively. Any initial relative position and velocity leads to a bounded trajectory around the controlled elliptic equilibrium. The numerical simulation indicates that the controller effectively stabilizes motions relative to the perturbed elliptic orbit with small eccentricity and unperturbed elliptic orbit with arbitrary eccentricity. The developed controller stabilizes the quasi-periodic relative trajectories involved in six foundational motions with different frequencies generated by the eigenvectors of the Floquet multipliers, rather than to track a reference relative configuration. Only the relative positions are employed for the feedback without the information from the direct measurement or the filter estimation of relative velocity. So the current controller has potential applications in formation flying for its less computation overload for on-board computer, less constraint on the measurements, and easily-achievable quasi-periodic relative trajectories. 相似文献
12.
本文给出考虑后牛顿(PN)效应的二体问题解所对应的基本关系式,并仿照开普勒(Kepler)运动,给出星历表计算方法和相应的计算公式以及适用于数值研究中的简单形式。 相似文献
13.
14.
Kyle T. Alfriend 《Celestial Mechanics and Dynamical Astronomy》1972,5(4):502-511
The question of whether or not there is a transfer of energy between the in-plane motion and out-of-plane motion in the neighborhood ofL
4 in the restricted problem of three bodies is investigated in this paper. The in-plane motion is assumed to be finite and the out-of-plane motion to be infinitesimal. The equation governing the out-of-plane motion becomes one with time varying coefficients. The stability of this equation is then investigated using Lie Series.Presented as a paper AAS No. 70-313, at the AAS/AIAA Astrodynamics Specialists Conference 1971 at Fort Lauderdale Fla., U.S.A. 相似文献
15.
V. A. Brumberg 《Celestial Mechanics and Dynamical Astronomy》2010,106(3):209-234
As we are now approaching 2015, both the General Relativity Theory (GRT) and the relativistic Celestial Mechanics based on
it will soon arrive at their 100 year anniversaries. There is no border between Newtonian and relativistic Celestial Mechanics.
The five-decade period of intensive development of Celestial Mechanics in the second half of the 20th century left many interesting
techniques and problems uncompleted. This lecture reviews some problems of Newtonian and relativistic Celestial Mechanics
worthy of further investigation. Concerning Newtonian mechanics, these problems include general solution of the three-body
problem by means of the series of polynomials, construction of the short-term and long-term theories of motion using the fast
converging elliptic function expansions, and representation of the rotation of the planets in the form compatible with the
General Planetary Theory reducing the problem to the combined secular system for translatory motion and rotation. Relativistic
problems considered here include the determination of the main relativistic effects in the motion of a satellite, e.g. the
Moon, and in the rotation of the primary planet using the Newtonian theories of motion and rotation combined with the relativistic
transformation of the reference systems, the use of the linearized weak-field GRT metric as a basis of relativistic Celestial
Mechanics in the post-Newtonian approximation, and the motion of the Solar System bodies at the cosmological background in
the framework of the basic cosmological models. The exposition of the chosen relativistic problems is preceded by reminding
the basic features of relativistic Celestial Mechanics with discussing some present tendencies concerning the Parametrized
Post-Newtonian formalism, International Astronomical Union resolutions, and standardization of the GRT routines. 相似文献
16.
Perturbed two-body problems play a special role in Celestial Mechanics as they capture the dominant dynamics for a broad range
of natural and artificial satellites. In this paper, we investigate the classic Stark problem, corresponding to motion in
a Newtonian gravitational field subjected to an additional uniform force of constant magnitude and direction. For both the
two-dimensional and three-dimensional cases, the integrals of motion are determined, and the resulting quadratures are analytically
integrated. A complete list of exact, closed-form solutions is deduced in terms of elliptic functions. It is found that all
expressions rely on only seven fundamental solution forms. Particular attention is given to ensure that the expressions are
well-behaved for very small perturbations. A comprehensive study of the phase space is also made using a boundary diagram
to describe the domains of the general types of possible motion. Numerical examples are presented to validate the solutions. 相似文献
17.
Unified analytical solutions to two-body problems with drag 总被引：3，自引：0，他引：3
The two-body problem with a generalized Stokes drag is discussed. The drag force is proportional to the product of the velocity vector and the inverse square of the distance. The generalization consists of allowing two different proportionality constants for the radial and the transverse components of the force. Under the 'generalized Robertson transformation', the equation of the orbit takes the form of the Lommel equation and admits solutions in terms of Bessel and Lommel functions. The exact, analytical solutions for this type of drag reveal a paradoxical effect of increasing eccentricity for all trajectories. The Poynting–Robertson drag and Poynting–Plummer–Danby problems are discussed as particular cases of the general solution. 相似文献
18.
The two-body problem is a twelfth-order time-invariant dynamic system, and therefore has eleven mutually-independent time-independent
integrals, here referred to as motion constants. Some of these motion constants are related to the ten mutually-independent
algebraic integrals of the n-body problem, whereas some are particular to the two-body problem. The problem can be decomposed into mass-center and relative-motion
subsystems, each being sixth-order and each having five mutually-independent motion constants. This paper presents solutions
for the eleventh motion constant, which relates the behavior of the two subsystems. The complete set of mutually-independent
motion constants describes the shape of the state-space trajectories. The use of the eleventh motion constant is demonstrated
in computing a solution to a two-point boundary-value problem. 相似文献
19.
A. V. Borisov L. C. García-Naranjo I. S. Mamaev J. Montaldi 《Celestial Mechanics and Dynamical Astronomy》2018,130(6):43
We consider the two-body problem on surfaces of constant nonzero curvature and classify the relative equilibria and their stability. On the hyperbolic plane, for each \(q>0\) we show there are two relative equilibria where the masses are separated by a distance q. One of these is geometrically of elliptic type and the other of hyperbolic type. The hyperbolic ones are always unstable, while the elliptic ones are stable when sufficiently close, but unstable when far apart. On the sphere of positive curvature, if the masses are different, there is a unique relative equilibrium (RE) for every angular separation except \(\pi /2\). When the angle is acute, the RE is elliptic, and when it is obtuse the RE can be either elliptic or linearly unstable. We show using a KAM argument that the acute ones are almost always nonlinearly stable. If the masses are equal, there are two families of relative equilibria: one where the masses are at equal angles with the axis of rotation (‘isosceles RE’) and the other when the two masses subtend a right angle at the centre of the sphere. The isosceles RE are elliptic if the angle subtended by the particles is acute and is unstable if it is obtuse. At \(\pi /2\), the two families meet and a pitchfork bifurcation takes place. Right-angled RE are elliptic away from the bifurcation point. In each of the two geometric settings, we use a global reduction to eliminate the group of symmetries and analyse the resulting reduced equations which live on a five-dimensional phase space and possess one Casimir function. 相似文献
20.
Hideki Asada 《Celestial Mechanics and Dynamical Astronomy》2007,97(3):151-164
We present an exact solution of the equations for orbit determination of a two body system in a hyperbolic or parabolic motion.
In solving this problem, we extend the method employed by Asada, Akasaka and Kasai (AAK) for a binary system in an elliptic
orbit. The solutions applicable to each of elliptic, hyperbolic and parabolic orbits are obtained by the new approach, and
they are all expressed in an explicit form, remarkably, only in terms of elementary functions. We show also that the solutions
for an open orbit are recovered by making a suitable transformation of the AAK solution for an elliptic case. 相似文献