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1.
The critical inclination in artificial satellite theory 总被引：1，自引：0，他引：1
Shannon L. Coffey André Deprit Bruce R. Miller 《Celestial Mechanics and Dynamical Astronomy》1986,39(4):365-406
Certain it is that the critical inclination in the main problem of artificial satellite theory is an intrinsic singularity. Its significance stems from two geometric events in the reduced phase space on the manifolds of constant polar angular momentum and constant Delaunay action. In the neighborhood of the critical inclination, along the family of circular orbits, there appear two Hopf bifurcations, to each of which there converge two families of orbits with stationary perigees. On the stretch between the bifurcations, the circular orbits in the planes at critical inclinmation are unstable. A global analysis of the double forking is made possible by the realization that the reduced phase space consists of bundles of two-dimensional spheres. Extensive numerical integrations illustrate the transitions in the phase flow on the spheres as the system passes through the bifurcations.A delicacy so very susceptible of offence...—Hester Lynch PIOZZI,Observations and Reflections made in the Course of a Journey through France, Italy and Germany (1789)NAS/NRC Postgraduate Research Associate in 1984–1985. 相似文献
2.
Giuseppe Pucacco 《Monthly notices of the Royal Astronomical Society》2009,399(1):340-348
We investigate an analytical treatment of bifurcations of families of resonant 'thin' tubes in axisymmetric galactic potentials. We verify that the most relevant bifurcations are due to the (1:1) resonance producing the 'inclined' orbits through two different mechanisms: from the disc orbit and from the 'thin' tube associated with the vertical oscillation. The closest resonances occurring after these are the (4:3) resonance in the oblate case and the (2:1) resonance in the prolate case. The (1:1) resonances are treated in a straightforward way using a second-order truncated normal form. The higher order resonances are instead cumbersome to investigate, because the normal form has to be truncated to a high degree and the number of terms grows very rapidly. We therefore adopt a further simplification giving analytical formulae for the values of the parameters at which bifurcations ensue and compare them with selected numerical results. Thanks to the asymptotic nature of the series involved, the predictions are reliable well beyond the convergence radius of the original series. 相似文献
3.
Normalization of a perturbed elliptic oscillator, when executed in Lissajous variables, amounts to averaging over the elliptic anomaly. The reduced Lissajous variables constitute a system of cylindrical coordinates over the orbital spheres of constant energy, but the pole-like singularities are removed by reverting to the subjacent Hopf coordinates. The two-parameter coupling that is a polynomial of degree four admitting the symmetries of the square is studied in detail. It is shown that the normalized elliptic oscillator in that case behaves everywhere in the parameter plane like a rigid body in free rotation about a fixed point, and that it passes through butterfly bifurcations wherever its phase flow admits non isolated equilibria. 相似文献
4.
We consider a system of a harmonic and an unharmonic oscillator with a weak cubic coupling. We study the non-degenerate bifurcations of periodic orbits for the resonant tori of the unperturbed system for which the twist condition holds. We demonstrate that this system also exhibits for certain values of the small parameter non-twist bifurcations, where the rotation number of the Poincaré map attains a minimum value. 相似文献
5.
P.G. Kazantzis 《Astrophysics and Space Science》1997,249(1):7-29
In this work we consider four families of plane periodic orbits direct around the Sun which approach Jupiter but they are sufficiently far from it so as to be considered as predominantly two body orbits of the Sun-asteroid system. We study their horizontal and vertical stabilities and we give the exact orbits of bifurcations of these families with three-dimensional families of the same multiplicity or twice the multiplicity of the above families of plane symmetric periodic orbits. Moreover, we give the first segments of the three dimensional families of symmetric periodic orbits which emanate from these plane bifurcations and we study their stability relating it with the stability of the plane bifurcations. 相似文献
6.
Message and Taylor (1978) have given values of the mean eccentricities and commen-surabilities which correspond to bifurcation orbits of families of symmetric periodic orbits with families of asymmetric periodic orbits in the limit as the mass ratio tends to zero. These bifurcations have been given in a way that they seem to be isolated and unrelated from the whole structure of the periodic orbits of the system.In this paper a numerical investigation of the horizontal stability of the family I and its branches reveals the above bifurcations orbits in the Sun-Jupiter case of the restricted three-body problem and associates these orbits with the whole structure of the system, giving extensive information on them. 相似文献
7.
Deprit and Miller have conjectured that normalization of integrable Hamiltonians may produce normal forms exhibiting degenerate equilibria to very high order. Several examples in the class of coupled elliptic oscillators are known. In order to test the utility of normalization as a detector of integrability we normalize, to high order, a perturbed Keplerian system known to have several integrable limits; the generalized van der Waals Hamiltonian for a hydrogen atom. While the separable limits give rise to high order degeneracy we find a non-separable, integrable limit for which the normal form does not exhibit degeneracy. We conclude that normalization may, in certain cases, indicate integrability but is not guaranteed to uncover all integrable limits. 相似文献
8.
Despina Voyatzi Simos Ichtiaroglou 《Celestial Mechanics and Dynamical Astronomy》2005,93(1-4):331-342
We study the problem of the motion of a unit mass on the unit sphere
and examine the relation between integrability and certain monoparametric families of orbits. In particular we show that
if the potential is compatible with a family of meridians, it is integrable with an integral linear in the velocities, while
a family of parallels guarantees integrability with an integral quadratic in the velocities. 相似文献
9.
Three-dimensional motions in the Chermnykh restricted three-body problem are studied. Specifically, families of three-dimensional periodic orbits are determined through bifurcations of the family of straight-line periodic oscillations of the problem which exists for equal masses of the primaries. These rectilinear oscillations are perpendicular to the plane of the primaries and give rise to an infinite number of families consisting entirely of periodic orbits which belong to the three-dimensional space except their respective one-dimensional bifurcations as well as their planar terminations. Many of the computed branch families are continued in all mass range that they exist. 相似文献
10.
We discuss existence and bifurcations of non-collinear (Lagrangian) relative equilibria in a generalized three body problem. Specifically, one of the bodies is a spheroid (oblate or prolate) with its equatorial plane coincident with the plane of motion where only the “J 2” term from its potential expansion is retained. We describe the bifurcations of relative equilibria as function of two parameters: J 2 and the angular velocity of the system formed by the mass centers. We offer the values of the parameters where bifurcations in shape occur and discuss their physical meaning. We conclude with a general theorem on the number and the shape of relative equilibria. 相似文献
11.
Dane Quinn Brett Gladman Phil Nicholson Richard Rand 《Celestial Mechanics and Dynamical Astronomy》1997,67(2):111-130
We study the rotational evolution under tidal torques of axisymmetric natural satellites in inclined, precessing orbits. In
the spin- and orbit-averaged equations of motion, we find that a global limit cycle exists for parameter values near the stability
limit of Cassini state
. The limit cycle involves an alternation between states of near-synchronous spin at low obliquity, and strongly subsynchronous
spin at an obliquity near 90°. This dynamical feature is characterized as a relaxation oscillation, arising as the system
slowly traverses two saddle-node bifurcations in a reduced system. This slow timescale is controlled by ε, the nondimensional
tidal dissipation rate. Unfortunately, a straightforward expansion of the governing equations for small ε is shown to be insufficient
for understanding the underlying structure of the system. Rather, the dynamical equations of motion possess a singular term,
multiplied by ε, which vanishes in the unperturbed system. We thus provide a demonstration that a dissipatively perturbed
conservative system can behave qualitatively differently from the unperturbed system.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
12.
G. Contopoulos 《Celestial Mechanics and Dynamical Astronomy》1987,42(1-4):239-250
We study the evolution of families of periodic orbits of simple 3-dimensional models representing the central parts of deformed galaxies. In some cases the evolution is non-unique, i.e. if we follow a closed path in the parameter space we do not return with the same periodic orbit. This happens when the path surrounds a critical point. We found that critical points are generated at particular collisions of bifurcations in limiting cases when the 3-D system is separated into a 2-D system and an independent oscillation along the third axis. The regions of stability and instability of some families of periodic orbits change in remarkable ways near the various collisions of bifurcations and around the critical points. 相似文献
13.
L. M. Perko 《Celestial Mechanics and Dynamical Astronomy》1981,24(2):155-171
This paper illustrates the application of the theory for second species solutions with an 0(
v
), 0<v<1, near-Moon passage to first species-second species bifurcations and to second species-second species bifurcations. It also corrects and improves the asymptotic approximations obtained in the author's previous work on this subject and it establishes a local form of Broucke's Principle for the types of bifurcations studied in this paper.This work was supported by the National Science Foundation under Grant MCS 7703591. 相似文献
14.
Ch. Skokos P. A. Patsis E. Athanassoula 《Monthly notices of the Royal Astronomical Society》2002,333(4):847-860
In this series of papers we investigate the orbital structure of three-dimensional (3D) models representing barred galaxies. In the present introductory paper we use a fiducial case to describe all families of periodic orbits that may play a role in the morphology of three-dimensional bars. We show that, in a 3D bar, the backbone of the orbital structure is not just the x1 family, as in two-dimensional (2D) models, but a tree of 2D and 3D families bifurcating from x1. Besides the main tree we have also found another group of families of lesser importance around the radial 3:1 resonance. The families of this group bifurcate from x1 and influence the dynamics of the system only locally. We also find that 3D orbits elongated along the bar minor axis can be formed by bifurcations of the planar x2 family. They can support 3D bar-like structures along the minor axis of the main bar. Banana-like orbits around the stable Lagrangian points build a forest of 2D and 3D families as well. The importance of the 3D x1-tree families at the outer parts of the bar depends critically on whether they are introduced in the system as bifurcations in z or in z˙ . 相似文献
15.
Thierry Combot 《Celestial Mechanics and Dynamical Astronomy》2012,114(4):319-340
We prove an integrability criterion and a partial integrability criterion for homogeneous potentials of degree ?1 which are invariant by rotation. We then apply it to the proof of the meromorphic non-integrability of the n-body problem with Newtonian interaction in the plane on a surface of equation (H, C) = (H 0, C 0) with (H 0, C 0) ?? (0, 0) where C is the total angular momentum and H the Hamiltonian, in the case where the n masses are equal. Several other cases in the 3-body problem are also proved to be non integrable in the same way, and some examples displaying partial integrability are provided. 相似文献
16.
Els van der Aa 《Celestial Mechanics and Dynamical Astronomy》1983,31(2):163-191
Hamiltonian systems with three degrees of freedom which have a resonance-ratio 1∶2∶ω with ω=1, 2, 3, or 4 are studied here. The periodic orbits are determined together with their stability characteristics. Furthermore we shall study the fundamental question of asymptotic integrability of these systems. 相似文献
17.
We provide a result of non-analytic integrability of the so-called J
2-problem. Precisely by using the Lerman theorem we are able to prove the existence of a region of the phase space, where the dynamical system exhibits chaotic motions. 相似文献
18.
Celestial Mechanics and Dynamical Astronomy - The two asymmetric bifurcations associated with the exterior commensurabilities of the formq+1: 1 are found to exist forq=1, 2, 3, 4 throughout the... 相似文献
19.
SŁawomir Breiter 《Celestial Mechanics and Dynamical Astronomy》2001,80(1):1-20
The resonance C1 occurs when the longitude of the perigee measured from the equinox becomes a slow angle in the doubly averaged equations of motion. This resonance is one of the critical inclination family with I 46°. For prograde Earth satellite orbits, up to five critical points can be identified. Only simple pitchfork bifurcations occur for the single resonance C1. A two degrees of freedom system is studied to check how a coupling of two lunisolar resonances affects the results furnished by the analysis of an isolated resonance case. In the system with two critical angles (g+h and h,+2 , seven types of critical points have been identified. The critical points arise and change their stability through 11 bifurcations. If the initial conditions are selected close to the critical points, the system becomes chaotic as shown in Poincaré maps. 相似文献
20.
Andrzej J. Maciejewski Maria Przybylska 《Celestial Mechanics and Dynamical Astronomy》2003,87(4):317-351
In this paper we analyse the integrability of a dynamical system describing the rotational motion of a rigid satellite under the influence of gravitational and magnetic fields. In our investigations we apply an extension of the Ziglin theory developed by Morales-Ruiz and Ramis. We prove that for a symmetric satellite the system does not admit an additional real meromorphic first integral except for one case when the value of the induced magnetic moment along the symmetry axis is related to the principal moments of inertia in a special way. 相似文献