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1.
C. N. Frangakis 《Astrophysics and Space Science》1973,22(2):421-440
All possible long period motions of the planar circular restricted three-body problem were studied by phase-plane analysis of the disturbing function when a commensurability relation exists. The short period argument was eliminated by an averaging process using an electronic computer. Initial conditions of periodic solutions are determined and the existence of asymmetric ones is proved for certain commensurabilities.The numerical computations performed cover the cases of large eccentricities and the curves of the mean value of the Hamiltonian were studied for the cases which correspond to collision periodic solutions. 相似文献
2.
A. P. Markeev 《Astronomy Letters》2001,27(7):475-479
The centers of the gaps observed in the asteroid belt are displaced toward Jupiter from their positions that correspond to the exact commensurability between the mean motions of an asteroid and Jupiter. Using the current theory of stability and nonlinear oscillations of Hamiltonian systems, we point out the dynamical causes of this asymmetry. Our analysis is performed in terms of the plane circular restricted three-body problem. The orbits that correspond to Poincaré periodic solutions of the first kind are taken as unperturbed asteroid orbits. 相似文献
3.
A. T. Sinclair 《Celestial Mechanics and Dynamical Astronomy》1970,2(3):350-350
Various families of periodic solutions are shown to exist in the three body problem, in which two of the bodies are close to a commensurability in mean motions about the third body, the primary, which is considerably more massive than the other two. The cases considered are
- The non-planar circular restricted problem (in which one of the secondary bodies has zero mass, and the other moves in a fixed circular orbit about the primary).
- The planar non-restricted problem (in which the three bodies move in a plane, and both secondaries have finite mass).
- The planar elliptical restricted problem (in which the three bodies move in a plane, one of the secondary bodies has zero mass, and the other moves in a fixed elliptical orbit about the primary).
4.
The surfaces of section in a harmonic oscillator potential, perturbed by quartic terms, are obtained analytically. A succession
of action‐angle, Lissajous and Lie transformations near the 1:1 commensurability, reduces the three‐dimensional motion to
a one‐dimensional one. The latter is solved in terms of Jacobi's elliptic functions. Existence conditions for periodic orbits
are found and two general families of such solutions are introduced. Two examples of regular motions in oblate and prolate
spheroids are discussed.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
5.
S. G. Zhuravlev 《Celestial Mechanics and Dynamical Astronomy》1981,25(3):297-315
In our article (Zhuravlev, 1979) a formal method of constructing conditionally periodic solutions of canonical systems of differential equations with a quick-rotating phase in the case of sharp commensurability was presented. The existence of stationary (or periodic) solutions of an averaged system of differential equations corresponding to the initial system of differential equations is necessary for an effective application of the method for different problems.Evidently, the stationary solutions do not always exist but in numerous papers on stationary solutions (oscillations or motions), the conditions of existence of such solutions are very often not considered at all. Usually a simple assumption is used that the stationary solutions do exist.Otherwise it is well known that Poincaré's theory of periodic solutions (Poincaré, 1892) let one set up conditions of existence of periodic solutions in different systems of differential equations. Particularly, in papers,Mah (1949, 1956), see alsoexmah (1971), the necessary and sufficient conditions of the existence of periodic solutions of (non-canonical) systems of differential equations which are close to arbitrary non-linear systems are given. For canonical autonomous systems of differential equations the conditions of existence of periodic solutions and a method of calculation are presented in the paperMepmah (1952).In our paper another approach is given and the conditions of existence of stationary solutions of canonical systems of differential equations with a quick-rotating phase are proved. For this purpose Delaunay-Zeipel's transformation and Poincaré's small parameter method are used. 相似文献
6.
I. Stellmacher 《Celestial Mechanics and Dynamical Astronomy》1982,28(4):381-400
In a previous paper (Stellmacher, 1981, hereafter mentioned as Paper I), we have given an algorithm for the construction of periodic orbits in a rotating frame, for satellites around an oblate planet. In the present paper, we apply this theory to the Mimas-Tethys case; we obtain the following results:
- Without resonance, it is possible to find a rotating system in which the solution is a periodic one. The angular velocity of this rotating frame is calculated as function of the masses of the two satellites.
- Including the resonant terms and assuming an exact commensurability of the implied frequencies, we demonstrate that the condition for periodic solutions in the rotating system as defined in (a) is: the initial position of the satellites at conjunction lies on an axis defined by (Ω1+Ω2)/2 or (Ω1+Ω2)/2 + π/2;Ω1 and Ω2 are the longitudes of the ascending nodes of the satellite's orbits. The solution still is a periodic one, thus all the conjunction occur in either axis.
- In the Mimas Tethys case there is only approximately commensurability between these frequencies. The two satellites are considered as oscillators whose amplitudes and phases are functions of time. The equation of the libration can be established; we find the usual form, but for each satellite the generating solution is a periodic solution (as defined in Paper I), but not a Keplerian one. It follows a determination of the masses which slightly differs from that given by Kozai (1957), when the same values of the observed quantities are used for calculations.
- The equation of the libration is: $$\ddot z + n_1^2 h^2 \sin z + n_1 q\dot z\sin z = 0$$
7.
Mohamad Radwan 《Astrophysics and Space Science》2002,282(3):551-562
The present work deals with constructing a conditionally periodic solution for the motion of an Earth satellite taking into
consideration the oblateness of the Earth and the Luni-Solar attractions. The oblatenessof the Earth is truncated beyond the
second zonal harmonic J
2. The resonance resulting from the commensurability between the mean motions of the satellite, the Moon, and the Sun is analyzed. 相似文献
8.
P. J. Message 《Celestial Mechanics and Dynamical Astronomy》1987,43(1-4):119-125
In the presence of a single small-integer near commensurability of orbital period, the construction of a complete formal solution of the equations for the mutual perturbations in a planetary or satellite system, entirely in periodic terms, can be carried out after the use of a transformation of the variables which brings the quadratic terms of the Hamiltonian to a suitable normal form. A method for finding such a transformation is described. 相似文献
9.
In the present paper we analyse the motion of a massless particle during the capture process in an exterior mean-motion resonance
under the effects of an external dissipative force. In particular, we study the orbital evolution from its initial approach
to the commensurability up to the final nesting place in the periodic orbit around the equilibrium solution.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
10.
分析了上两个世纪发生在淮河的洪水事件的可公度性,根据其可公度值及其黄金分割点指出1991年与2003年淮河洪水的不可避免,最后讨论了可公度性的局限及淮河洪水可公度值的可能机制。 相似文献
11.
A first-order, semi-analytical method for the long-term motion of resonant satellites is introduced. The method provides long-term solutions, valid for nearly all eccentricities and inclinations, and for all commensurability ratios. The method allows the inclusion of all zonal and tesseral harmonics of a nonspherical planet.We present here an application of the method to a synchronous satellite includingonly theJ
2 andJ
22 harmonics. Global, long-term solutions for this problem are given for arbitrary values of eccentricity, argument of perigee and inclination. 相似文献
12.
Abdul Ahmad 《Celestial Mechanics and Dynamical Astronomy》1995,61(2):181-196
The periodic solutions of the restricted three-body problem representing analytic continuations of Keplerian rectilinear periodic motions are well known (Kurcheeva, 1973). Here the stability of these solutions are examined by applying Poncaré's characteristic equation for periodic solutions. It is found that the isoperiodic solutions are stable and all other solutions are unstable. 相似文献
13.
The character of orbital evolution for bodies moving near the if 1 : 3 commensurability with Jupiter was studied by model calculations for the time interval of ~500 years. A comparison of oscillations of the orbital elements a, e, q and q′ is made for ensembles of bodies along three starting orbits in the vicinity of the sharp commensurability with Jupiter. These orbits are eccentric ones of low inclinations having perihelia near the Earth's orbit. Examples of a deceleration of the rate of orbital evolution near the sharp commensurability are revealed. The existence of a group of asteroids connected with the Kirkwood gap, i.e., being in a resonant motion with Jupiter, is suggested. A connection of asteroids 887 Alinda and 1915 Quetzalcoatl with this gap is confirmed. 相似文献
14.
R. R. Allan 《Planetary and Space Science》1967,15(12):1829-1845
For a satellite in a nominally circular orbit at arbitrary inclination whose mean motion is commensurable with the Earth's rotation, the dependence of gravity on longitude leads to a resonant variation in eccentricity as well as the long-period oscillation in longitude. Provided forces capable of processing perigee are present, it is shown that the change in eccentricity for a satellite captured in librational resonance is not secular but periodic.
There are corresponding resonance effects for a satellite in a nominally equatorial but eccentric orbit. Here the commensurability condition is that the longitudes of the apses shall be nearly repetitive relative to the rotating Earth. There will be a long-period oscillation in longitude which can take the form of either a libration (trapped) or a circulation (free), and there will also be an oscillation of the orbital plane having the same period as the precession of perigee relative to inertial space. 相似文献
15.
R. Broucke 《Celestial Mechanics and Dynamical Astronomy》1975,12(4):439-462
We describe two relatively simple reductions to order 6 for the planar general three-body problem. We also show that this reduction leads to the distinction between two types of periodic solutions: absolute or relative periodic solutions. An algorithm for obtaining relative periodic solutions using heliocentric coordinates is then described. It is concluded from the periodicity conditions that relative periodic solutions must form families with a single parameter. Finally, two such families have been obtained numerically and are described in some detail.The present research was carried out partially at the University of California and partially at the Jet Propulsion Laboratory under contract NAS7-100 with NASA. 相似文献
16.
L. G. Luk’yanov 《Astronomy Letters》2012,38(3):201-211
Families of conditionally periodic solutions have been found by a slightly modified Lyapunov method of determining periodic
solutions near the libration points of the restricted three-body problem. When the frequencies of free oscillations are commensurable,
the solutions found are transformed into planar or spatial periodic solutions. The results are confirmed by numerically integrating
the starting nonlinear differential equations of motion. 相似文献
17.
18.
V. Singh 《Celestial Mechanics and Dynamical Astronomy》1976,14(2):167-173
The paper discusses the existence of periodic and quasi-periodic solutions in the space relativistic problem of three bodies with the help of Poincaré's small parameter method starting from non-Keplerian generating solutions, i.e., using Gauss's method. The main peculiarity of these periodic orbits is the fact that they close, in general, after many revolutions. It is worth noticing that these periodic orbits give a new class of periodic solutions of the classical circular problem of three bodies, if relativistic effects are neglected. 相似文献
19.
Carol A. Williams 《Celestial Mechanics and Dynamical Astronomy》1984,34(1-4):395-410
Small divisors caused by certain linear combinations of frequencies appear in all analytical planetary theories. With the exception of the deep resonance between Neptune and Pluto, they can be removed at the expense of introducing secular and mixed secular terms, limiting the domain in which the solution is valid. Because of them classical solutions are known not to converge uniformly; Poincaré referred to them as asymptotic. The KAM theory shows that if one is far enough from exact commensurability and has small enough planetary masses, expansions exist which will converge to quasi-periodic orbits. Solutions showing very small divisors are excluded from this region of convergence. The question of whether they are intrinsic to the problem or are just manifestations of the method of solution is not settled. Problems with a single commensurabily that can be isolated from the rest of the Hamiltonian may have solutions with no small divisors. The problem of two or more commensurabilities remains unsolved. 相似文献
20.
《New Astronomy》2020
The objective of this paper is to find periodic solutions of the circular Sitnikov problem by the multiple scales method which is used to remove the secular terms and find the periodic approximated solutions in closed forms. Comparisons among a numerical solution (NS), the first approximated solution (FA) and the second approximated solution (SA) via multiple scales method are investigated graphically under different initial conditions. We observe that the initial conditions play a vital role in the numerical and approximated solutions behaviour. The obtained motion is periodic, but the difference of its amplitude is directly proportional with the initial conditions. We prove that the obtained motion by the numerical or the second approximated solutions is a regular and periodic, when the infinitesimal body starts its motion from a nearer position to the common center of primaries. Otherwise when the start point distance of motion is far from this center, the numerical solution may not be represent a periodic motion for along time, while the second approximated solution may present a chaotic motion, however it is always periodic all time. But the obtained motion by the first approximated solution is periodic and has regularity in its periodicity all time. Finally we remark that the provided solutions by multiple scales methods reflect the true motion of the Sitnikov restricted three–body problem, and the second approximation has more accuracy than the first approximation. Moreover the solutions of multiple scales technique are more realistic than the numerical solution because there is always a warranty that the motion is periodic all time. 相似文献