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1.
Eugene Brumberg Victor A. Brumberg Thomas Konrad Michael Soffel 《Celestial Mechanics and Dynamical Astronomy》1995,61(4):369-387
Kaula's satellite linear perturbation theory has been extended for the case of highly eccentric orbits by using elliptic function expansions.On leave from Institute of Applied Astronomy, St.-Petersburg 相似文献
2.
Ashraf Hamdy Mostafa K. Ahmed Mohamad Radwan Fawzy Ahmed Abd El-Salam 《Earth, Moon, and Planets》2003,93(4):261-273
The objective of the present work is to develope explicit analytical expressions for the small amplitude orbits of the infinitesimal
mass about the equilibrium points in the elliptic restricted three body problem. To handle this dynamical problem, the Hamiltonian
for the elliptic problem is formed with the true anomaly and then with the eccentric anomaly as independent variables. The
origin is then transformed to a fixed point and the Hamiltonian is developed up to O(4) in the eccentricity, e, (which plays the role of the small parameter of the problem) of the primaries. The integration of the model problem under
consideration is undertaken by means of a perturbation technique based on Lie series developments, which leads to the solution
of the canonical equations of motion. 相似文献
3.
The techniques of Brumberg and Brumberg (1999) based on the use of elliptic anomaly are specified in this paper in two aspects. The iteration technique (Broucke, 1969) to construct short-term semi-analytical theories of motion in rectangular coordinates in lines of Encke and Hill is reelaborated in terms of elliptic anomaly resulting in extending this technique for high-eccentricity orbits. In constructing long-term semi-analytical theories the key point is to integrate trigonometric functions of several angular arguments related to time by different differential expressions. This problem is reduced in the paper to linear algebraic recurrence relations admitting efficient solution by iterations. 相似文献
4.
K. Uldall Kristiansen P. L. Palmer M. Roberts 《Celestial Mechanics and Dynamical Astronomy》2010,106(4):371-390
This paper builds upon the work of Palmer and Imre exploring the relative motion of satellites on neighbouring Keplerian orbits.
We make use of a general geometrical setting from Hamiltonian systems theory to obtain analytical solutions of the variational
Kepler equations in an Earth centred inertial coordinate frame in terms of the relevant conserved quantities: relative energy,
relative angular momentum and the relative eccentricity vector. The paper extends the work on relative satellite motion by
providing solutions about any elliptic, parabolic or hyperbolic reference trajectory, including the zero angular momentum
case. The geometrical framework assists the design of complex formation flying trajectories. This is demonstrated by the construction
of a tetrahedral formation, described through the relevant conserved quantities, for which the satellites are on highly eccentric
orbits around the Sun to visit the Kuiper belt. 相似文献
5.
Janis Britals 《Celestial Mechanics and Dynamical Astronomy》1992,54(4):305-316
It is shown in this paper how to build a canonical transformation of variables, so that the eccentric anomaly becomes the new independent variable. In the case of eccentric elliptical orbits it changes the equations of motion so, that they can be integrated analytically to any order of approximation comparatively easy. 相似文献
6.
Sandro Da Silva Fernandes 《Celestial Mechanics and Dynamical Astronomy》1995,62(4):305-321
Some classic expansions of the elliptic motion — cosmE and sinmE — in powers of the eccentricity are extended to highly eccentric orbits, 0.6627...<e<1. The new expansions are developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients are given in terms of the derivatives of Bessel functions with respect to the eccentricity. The expansions have the same radius of convergence (e*) of the extended solution of Kepler's equation, previously derived by the author. Some other simple expansions — (a/r), (r/a), (r/a) sinv, ..., — derived straightforward from the expansions ofE, cosE and sinE are also presented. 相似文献
7.
Sandro da Silva Fernandes 《Celestial Mechanics and Dynamical Astronomy》1995,63(3-4):375-408
Expansions of the functions (r/a)cos
jv and (r/a)m
sin
jv of the elliptic motion are extended to highly eccentric orbits, 0.6627 ... <e<1. The new expansions are developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients of these expansions are expressed in terms of the derivatives of Hansen's coefficients with respect to the eccentricity. The new expansions are convergent for values of the eccentricity such that |e–e*|<(e*), where the radius of convergence (e*) is the same of the extended solution of Kepler's equation. The new expansions are intrinsically related to Lagrange's series. 相似文献
8.
We develop a new and fast method to estimate perturbations by a planet on cometary orbits. This method allows us to identify accurately the cases of large perturbations in a set of fictitious orbits. Hence, it can be used in constructing perturbation samples for Monte Carlo simulations in order to maximize the amount of information. Furthermore, the estimated perturbations are found to yield a good approximation to the real perturbation sample. This is shown by a comparison of the perturbations obtained by the new estimator with the results of numerical integration of regularized equations of motion for the same orbits in the same dynamical model: the three-dimensional elliptic restricted three-body problem (Sun-Jupiter-comet). 相似文献
9.
The rectilinear elliptic restricted three-body problem (TBP) is the limiting case of the elliptic restricted TBP when the motion of the primaries is described by a Keplerian ellipse with eccentricity \(e'=1\), but the collision of the primaries is assumed to be a non-singular point. The rectilinear model has been proposed as a starting model for studying the dynamics of motion around highly eccentric binary systems. Broucke (AIAA J 7:1003–1009, 1969) explored the rectilinear problem and obtained isolated periodic orbits for mass parameter \(\mu =0.5\) (equal masses of the primaries). We found that all orbits obtained by Broucke are linearly unstable. We extend Broucke’s computations by using a finer search for symmetric periodic orbits and computing their linear stability. We found a large number of periodic orbits, but only eight of them were found to be linearly stable and are associated with particular mean motion resonances. These stable orbits are used as generating orbits for continuation with respect to \(\mu \) and \(e'<1\). Also, continuation of periodic solutions with respect to the mass of the small body can be applied by using the general TBP. FLI maps of dynamical stability show that stable periodic orbits are surrounded in phase space with regions of regular orbits indicating that systems of very highly eccentric orbits can be found in stable resonant configurations. As an application we present a stability study for the planetary system HD7449. 相似文献
10.
J. Baumgarte 《Celestial Mechanics and Dynamical Astronomy》1972,5(4):490-501
A stabilization of the classical equations of two-body motion is offered. It is characterized by the use of the regularizing independent variable (eccentric anomaly) and by the addition of a control-term to the differential equations. This method is related to the KS-theory (Stiefel, 1970) which performed for the first time a stabilization of the Kepler motion. But in contrast to the KS-theory our method does not transform the coordinates of the particle. As far as the theory of stability and the numerical experiments are concerned we restrict ourselves to thepure Kepler motion. But, of course, the stabilizing devices will also improve the accuracy of the computation of perturbed orbits. We list, therefore, also the equations of the perturbed motion. 相似文献
11.
The discovery of extra-solar planetary systems with multiple planets in highly eccentric orbits (∼0.1-0.6), in contrast with our own Solar System, makes classical secular perturbation analysis very limited. In this paper, we use a semi-numerical approach to study the secular behavior of a system composed of a central star and two massive planets in co-planar orbits. We show that the secular dynamics of this system can be described using only two parameters, the ratios of the semi-major axes and the planetary masses. The main dynamical features of the system are presented in geometrical pictures that allows us to investigate a large domain of the phase space of this three-body problem without time-expensive numerical integrations of the equations of motion, and without any restriction on the magnitude of the planetary eccentricities. The topology of the phase space is also investigated in detail by means of spectral map techniques, which allow us to detect the separatrix of a non-linear secular apsidal resonance. Finally, the qualitative study is supplemented by direct numerical integrations. Three different regimes of secular motion with respect to the secular angle Δ? are possible: they are circulation, oscillation (around 0° and 180°), and high eccentricity libration in a non-linear secular resonance. The first two regimes are a continuous extension of the classical linear secular perturbation theory; the last is a new feature, hitherto unknown, in the secular dynamics of the three-body problem. We apply the analysis to the case of the two outer planets in the υ Andromedae system, and obtain its periodic and ordinary orbits, the general structure of its secular phase space, and the boundaries of its secular stability; we find that this system is secularly stable over a large domain of eccentricities. Applying this analysis to a wide range of planetary mass and semi-major axis ratios (centered about the υ Andromedae parameters), we find that apsidal oscillation dominates the secular phase space of the three-body coplanar system, and that the non-linear secular resonance is also a common feature. 相似文献
12.
Ram Krishan Sharma 《Celestial Mechanics and Dynamical Astronomy》1989,46(4):321-333
Analytical solutions using KS elements are derived. The perturbation considered is the Earth's zonal harmonic J
2. The series expansions include terms of fourth power in the eccentricity. Only two of the nine KS element equations are integrated analytically due to the reasons of symmetry. The analytical solution is suitable for short-term orbit computations. Numerical studies show that reasonably good estimates of the orbital elements can be obtained in one step of 10 to 30 degrees of eccentric anomaly for near-Earth orbits of moderate eccentricity. For application purposes, the analytical solution can be effectively used for onboard computation in the navigation and guidance packages, where the modelling of J
2 effect becomes necessary. 相似文献
13.
Sergei A. Klioner Akmal A. Vakhidov Nickolay N. Vasiliev 《Celestial Mechanics and Dynamical Astronomy》1998,68(3):257-272
Fourier expansions of elliptic motion functions in multiples of the true, eccentric, elliptic and mean anomalies are computed
numerically by means of the fast Fourier transform. Both Hansen-like coefficients and their derivatives with respect to eccentricity
of the orbit are considered. General behavior of the coefficients and the efficiency (compactness) of the expansions are investigated
for various values of eccentricity of the orbit.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
14.
M. J. H. Walker B. Ireland Joyce Owens 《Celestial Mechanics and Dynamical Astronomy》1985,36(4):409-419
Modified equinoctial elements are introduced which are suitable for perturbation analysis of all kinds of orbit. Equations of motion in Lagrangian and Gaussian forms are derived. Identities connecting the partial derivatives of the disturbing function with respect to equinoctial elements are established. Numerical comparisons of the evolution of a perturbed, highly eccentric, elliptic orbit analysed in equinoctial elements and by Cowell's method show satisfactory agreement. 相似文献
15.
Guy Janin 《Celestial Mechanics and Dynamical Astronomy》1974,10(4):451-467
For computing highly eccentric (e0.9) Earth satellite orbits with special perturbation methods, a comparison is made between different schemes, namely the direct integration of the equations of motion in Cartesian coordinates, changes of the independent variable, use of a time element, stabilization and use of regular elements. A one-step and a multi-step integration are also compared.It is shown that stabilization and regularization procedures are very helpful for non or smoothly perturbed orbits. In practical cases for space research where all perturbations are considered, these procedures are no longer so efficient. The recommended method in these cases is a multi-step integration of the Cartesian coordinates with a change of the independent variable defining an analytical step size regulation. However, the use of a time element and a stabilization procedure for the equations of motion improves the accuracy, except when a small step size is chosen. 相似文献
16.
A massless particle may perform a ballistic capture about a primary when two or more gravitational attractions are considered. The dynamics governing the ballistic capture depend on the mutual position of the primaries, if these are let to revolve in eccentric orbits. This paper studies the effect of the primaries true anomaly on the ballistic capture about the smaller primary in the planar elliptic restricted three-body problem. The dynamics of the Hill curves are studied, and the conditions for a favorable capture are derived. It is shown that these lead to regular, quasi-stable post-capture orbits. This is confirmed by numerical simulations implementing the concept of capture set: a set of initial conditions that generates ballistic capture orbits with a prescribed stability number. 相似文献
17.
In the framework of the planar restricted three-body problem we study a considerable number of resonances associated to the basic dynamical features of Kuiper belt and located between 30 and 48 a.u. Our study is based on the computation of resonant periodic orbits and their stability. Stable periodic orbits are surrounded by regular librations in phase space and in such domains the capture of trans-Neptunian object is possible. All the periodic orbits found are symmetric and there is an indication of the existence of asymmetric ones only in a few cases. In the present work first, second and third order resonances are under consideration. In the planar circular case we found that most of the periodic orbits are stable. The families of periodic orbits are temporarily interrupted by collisions but they continue up to relatively large values of the Jacobi constant and highly eccentric regular motion exists for all cases. In the elliptic problem and for a particular eccentricity value of the primary bodies, the periodic orbits are isolated. The corresponding families, where they belong to, bifurcate from specific periodic orbits of the circular problem and seem to continue up to the rectilinear problem. Both stable and unstable orbits are obtained for each case. In the elliptic problem, the unstable orbits found are associated with narrow chaotic domains in phase space. The evolution of the orbits, which are located in such chaotic domains, seems to be practically regular and bounded for long time intervals. 相似文献
18.
New expansions of elliptic motion based on considering the eccentricitye as the modulusk of elliptic functions and introducing the new anomalyw (a sort of elliptic anomaly) defined byw=u/2K–/2,g=amu–/2 (g being the eccentric anomaly) are compared with the classic (e, M), (e, v) and (e, g) expansions in multiples of mean, true and eccentric anomalies, respectively. These (q,w) expansions turn out to be in general more compact than the classical ones. The coefficients of the (e,v) and (e,g) expansions are expressed as the hypergeometric series, which may be reduced to the hypergeometric polynomials. The coefficients of the (q,w) expansions may be presented in closed (rational function) form with respect toq, k, k=(1–k
2)1/2,K andE, q being the Jacobi nome relatedk whileK andE are the complete elliptic integrals of the first and second kind respectively. Recurrence relations to compute these coefficients have been derived.on leave from Institute of Applied Astronomy, St.-Petersburg 197042, Russia 相似文献
19.
Approximate Analytical Theory of Satellite Orbit Prediction Presented in Spherical Coordinates Frame
Gennadii Alimov Alexhander Greb Eduard Kuznetsov Elena Polyakhova 《Celestial Mechanics and Dynamical Astronomy》2001,81(3):219-234
A comparative review of analytic theories for the motion of Earth satellites in quasi-circular orbits written in the spherical coordinate frame is presented. The theory of motion is developed for satellites in quasi-circular and quasi-equatorial orbits subjected to geopotential, luni-solar and solar radiation pressure force perturbations. The intermediate orbit is Keplerian and the equations of motion are solved by the Lyapunov–Poincaré small parameter method. Both resonant and non-resonant cases are considered. The results can be useful for the development of a complete theory of weakly eccentric orbits. 相似文献
20.
D. Nesvorný F. Thomas S. Ferraz-Mello A. Morbidelli 《Celestial Mechanics and Dynamical Astronomy》2002,82(4):323-361
We develop a formalism of the non-singular evaluation of the disturbing function and its derivatives with respect to the canonical variables. We apply this formalism to the case of the perturbed motion of a massless body orbiting the central body (Sun) with a period equal to that of the perturbing (planetary) body. This situation is known as the co-orbital motion, or equivalently, as the 1/1 mean motion commensurability. Jupiter's Trojan asteroids, Earth's co-orbital asteroids (e.g., (3753) Cruithne, (3362) Khufu), Mars' co-orbital asteroids (e.g., (5261) Eureka), and some Jupiter-family comets are examples of the co-orbital bodies in our solar system. Other examples are known in the satellite systems of the giant planets. Unlike the classical expansions of the disturbing function, our formalism is valid for any values of eccentricities and inclinations of the perturbed and perturbing body. The perturbation theory is used to compute the main features of the co-orbital dynamics in three approximations of the general three-body model: the planar-circular, planar-elliptic, and spatial-circular models. We develop a new perturbation scheme, which allows us to treat cases where the classical perturbation treatment fails. We show how the families of the tadpole, horseshoe, retrograde satellite and compound orbits vary with the eccentricity and inclination of the small body, and compute them also for the eccentricity of the perturbing body corresponding to a largely eccentric exoplanet's orbit.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献