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1.
We study some simple periodic orbits and their bifurcations in the Hamiltonian
. We give the forms of the orbits, the characteristics of the main families, and some existence diagrams and stability diagrams. The existence diagram of the family 1a contains regions that are stable (S), simply unstable (U), doubly unstable (DU) and complex unstable (). In the regionsS andU there are lines of equal rotation numberm/n. Along these lines we have bifurcations of families of periodic orbits of multiplicityn. When these lines reach the boundary of the complex unstable region, they are tangent to it. Inside the region there are linesm/n, along which the orbits 1a, describedn-times, are doubly unstable; however, along these lines there are no bifurcations ofn-ple periodic orbits. The families bifurcating from 1a exist only in certain regions of the parameter space (, ). The limiting lines of these regions join at particular points representing collisions of bifurcations. These collisions of bifurcations produce a nonuniqueness of the various families of periodic orbits. The complicated structure of the various bifurcations can be understood by constructing appropriate stability diagrams. 相似文献
2.
The analysis of the fine structure of the Asteroidal Belt evidenciates a group of asteroids next to the resonance 4/9 with Jupiter. In this group and in other groups associated to the Hirayama families there are indications that their orbital parameters can be represented by quantum numbers as defined here and in two of our previous works. Together with this the distribution of the eccentricities and inclinations of the orbital planes of short period comets and diverse type of asteroids indicates that they can be classified as objects with e > sin i and objects with e > sin i with a limit e = sin i which determinates geometrical properties of the orbits related with discrete states in the solar system. This study lets open the possibility of following studies in order to confirm the quantum characteristics of the Asteroidal Belt being these characteristics common to all the solar system and depending of the same fundamental constant of action per mass unit H
0 = 1/2
0 × T
0 (potential × time) because only a small part of all the available data in the Asteroid Belt is used here. 相似文献
3.
The theory specifying the change i in a satellite's orbital inclination due to atmospheric rotation, in terms of the decrease in orbital period T, has been extended to an atmosphere with sinusoidal variation of density between day and night. It is found that with certain special sets of values for the orbital parameters, the day-to-night variation in the Earth's atmosphere can alter the equation for i/T by as much as 25% though only for a few days. Appreciable changes in i/T persisting for several months can only occur for certain resonant orbits: the maximum change is then about 8%. Near-resonance is very unlikely, but the resonance conditions are derived so that orbits can be recognised and avoided. 相似文献
4.
P. Delibaltas 《Astrophysics and Space Science》1976,45(1):207-233
Several families of planar planetary-type periodic orbits in the general three-body problem, in a rotating frame of reference, for the Sun-Jupiter-Saturn mass-ratio are found and their stability is studied. It is found that the configuration in which the orbit of the smaller planet is inside the orbit of the larger planet is, in general, more stable.We also develop a method to study the stability of a planar periodic motion with respect to vertical perturbations. Planetary periodic orbits with the orbits of the two planets not close to each other are found to be vertically stable. There are several periodic orbits that are stable in the plane but vertically unstable and vice versa. It is also shown that a vertical critical orbit in the plane can generate a monoparametric family of three-dimensional periodic orbits. 相似文献
5.
This work considers periodic solutions, arc-solutions (solutions with consecutive collisions) and double collision orbits of the plane elliptic restricted problem of three bodies for =0 when the eccentricity of the primaries,e
p
, varies from 0 to 1. Characteristic curves of these three kinds of solutions are given. 相似文献
6.
V. Kudielka 《Celestial Mechanics and Dynamical Astronomy》1994,60(4):455-470
An analytic model for third-body perturbations and for the second zonal harmonic of the central body's gravitational field is presented. A simplified version of this model applied to the Earth-Moon-Sun system indicates the existence of high-altitude and highly-inclined orbits with their apsides in the equator plane, for which the apsidal as well as the nodal motion ceases. For special positions of the node, secular changes of eccentricity and inclination disappear too (balanced orbits). For an ascending node at vernal equinox, the inclination of balanced orbits is 94.56°, for a node at autumnal equinox 85.44°, independent of the eccentricity of the orbit. For a node perpendicular to the equinox, there exist circular balanced orbits at 90° inclination. By slightly adjusting the initial inclination as suggested by the simplified model, orbits can be found — calculated by the full model or by different methods — that show only minor variations in eccentricity, inclination, argument of perigee, and longitude of the ascending node for 105 revolutions and more. Orbits near the unstable equilibria at 94.56° and 85.44° inclination show very long periodic librations and oscillations between retrogade and prograde motion.Retired from IBM Vienna Software Development Laboratory. 相似文献
7.
P. G. Kazantzis 《Astrophysics and Space Science》1979,61(2):357-367
Two new families of three-dimensional simple-symmetric periodic orbits are determined numerically in the Sun-Jupiter case of the restricted three-body problem. These families emanate from the vertical-critical orbits (v = 1,c
v
= 0)of the familiesi andl of plane symmetric simpleperiodic orbits direct around the Sun and the Sun-Jupiter respectively. Further, the numerical technique employed in the determination of these families has been described and interesting results have been pointed out. Also, computer plots of the orbits of these families have been shown in conical projections. 相似文献
8.
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).
9.
10.
Three-dimensional periodic motion of three finite masses around collinear equilibrium configurations
V. V. Markellos 《Celestial Mechanics and Dynamical Astronomy》1981,25(4):319-344
Three-dimensional periodic motions of three bodies are shown to exist in the infinitesimal neighbourhood of their collinear equilibrium configurations. These configurations and some characteristic quantities of the emanating three-dimensional periodic orbits are given for many values of the two mass parameters, =m
2/(m
1+m
2) andm
3, of the general three-body problem, under the assumption that the straight line containing the bodies at equilibrium rotates with unit angular velocity. The analysis of the small periodic orbits near the equilibrium configurations is carried out to second-order terms in the small quantities describing the deviation from plane motion but the analytical solution obtained for the horizontal components of the state vector is valid to third-order terms in those quantities. The families of three-dimensional periodic orbits emanating from two of the collinear equilibrium configurations are continued numerically to large orbits. These families are found to terminate at large vertical-critical orbits of the familym of retrograde periodic orbits ofm
3 around the primariesm
1 andm
2. The series of these termination orbits, formed when the value ofm
3 varies, are also given. The three-dimensional orbits are computed form
3=0.1. 相似文献
11.
Eugene Rabe 《Celestial Mechanics and Dynamical Astronomy》1970,2(3):358-358
In the ordinary restricted problem of three bodies, the first-order stability of planar periodic orbits may be determined by means of their characteristic exponents, as derived from the condition of a vanishing determinant for the coefficients of an infinite system of homogenous linear equations associated with the exponential series solutionu, v representing any initially small oscillations about the periodic solutionx, y. In the elliptic restricted problem, periodic solutions are possible only for periods which are equal to, or integral multiples of, the periodP of the elliptic motion of the two primary masses. It is shown that the infinite determinant approach to the determination of the characteristic exponents can be extended to the treatment of superposed free oscillations in the elliptic problem, and that in generaltwo exponents appear in any complete solutionu, v for eachone existing in the corresponding ordinary restricted problem. The value of each exponent depends on a series proceeding in even powers of the eccentricitye of the relative orbit of the two primaries, in addition to its basic dependence on the mass ratio . For stable periodic orbits, the oscillation frequenciesn
1 (,e
2),n
2 (,e
2) associated with these two exponents tend, withe0, to certain limiting valuesn
1 (),n
2(), which differ from each other by the amount of the frequencyN=2/P of the orbital motion of the primaries. One of the two frequencies, sayn
1(), is identical with the frequency of the corresponding oscillations in the ordinary restricted problem, while the second one gives rise to oscillations only in the elliptic restricted problem, withe0.The method will be described in more detail, together with its application to two families of small periodie librations about the equilateral points of the elliptic restricted problem (E. Rabe: Two new Classes of Periodic Trojan Librations in the Elliptic Restricted Problem and their Stabilities) in theProceedings of the Symposium on Periodic Orbits, Stability and Resonances, held at the University of São Paulo, Brasil, 4–12 September, 1969.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969. 相似文献
12.
Alessandra Celletti Andrea Chessa John Hadjidemetriou Giovanni Battista Valsecchi 《Celestial Mechanics and Dynamical Astronomy》2002,83(1-4):239-255
We investigate symmetric periodic orbits in the framework of the planar, circular, restricted, three-body problem. Having fixed the mass of the primary equal to that of Jupiter, we determine the linear stability of a number of periodic orbits for different values of the eccentricity. A systematic study of internal resonances, with frequency p/q with 2p 9, 1 q 5 and 4/3 p/q 5, offers an overall picture of the stability character of inner orbits. For each resonance we compute the stability of the two possible periodic orbits. A similar analysis is performed for some external periodic orbits.Furthermore, we let the mass of the primary vary and we study the linear stability of the main resonances as a function of the eccentricity and of the mass of the primary. These results lead to interesting conclusions about the stability of exosolar planetary systems. In particular, we study the stability of Earth-like planets in the planetary systems HD168746, GI86, 47UMa,b and HD10697. 相似文献
13.
P. G. Kazantzis 《Astrophysics and Space Science》1979,61(2):477-486
Families of three-dimensional axisymmetric periodic orbits are determined numerically in the Sun-Jupiter case of the restricted three-body problem. These families bifurcate from the vertical-critical orbits (v = 1,b
v = 0) of the basic plane familiesi andI. Further the predictor-corrector procedure employed to reveal these families has been described and interesting numerical results have been pointed out. Also, computer plots of the orbits of these families have been shown in conical projections. 相似文献
14.
E.A. Perdios 《Astrophysics and Space Science》2001,278(4):405-407
The known intervals of possible stability, on the mgr-axis, of basicfamilies of 3D periodic orbits in the restricted three-body problem areextended into -A1 regions for oblate larger primary, A
1 beingthe oblateness coefficient. Eight regions, corresponding to the basicstable bifurcation orbits l1v, l1v, l2v, l3v, m1v, m1v,m2v, i1v are determined and related branching 3D periodic orbits arecomputed systematically and tested for stability. The regions for l1v,m1v and m2v survive the test emerging as the regions allowing thesimplest types of stable low inclination 3D motion. For l1v, l2v,l3v, m1v and m2v oblateness seems to have a stabilising effect,while stability of i1v survives only for a very small range of A
1values. 相似文献
15.
C. G. Zagouras 《Celestial Mechanics and Dynamical Astronomy》1985,37(1):27-46
The third-order parametric expansions given by Buck in 1920 for the three-dimensional periodic solutions about the triangular equilibrium points of the restricted Problem are improved by fourthorder terms. The corresponding family of periodic orbits, which are symmetrical w.r.t. the (x, y) plane, is computed numerically for =0.00095. It is found that the family emanating from L4 terminates at the other triangular point L5 while it bifurcates with the family of three-dimensional periodic orbits originating at the collinear equilibrium point L3. This family consists of stable and unstable members. A second family of nonsymmetric three-dimensional periodic orbits is found to bifurcate from the previous one. It is also determined numerically until a collision orbit is encountered with the computations. 相似文献
16.
George Voyatzis John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》2005,93(1-4):263-294
Families of asymmetric periodic orbits at the 2/1 resonance are computed for different mass ratios. The existence of the asymmetric
families depends on the ratio of the planetary (or satellite) masses. As models we used the Io-Europa system of the satellites
of Jupiter for the case m1>m2, the system HD82943 for the new masses, for the case m1=m2 and the same system HD82943 for the values of the masses m1<m2 given in previous work. In the case m1≥ m2 there is a family of asymmetric orbits that bifurcates from a family of symmetric periodic orbits, but there exist also an
asymmetric family that is independent of the symmetric families. In the case m1<m2 all the asymmetric families are independent from the symmetric families. In many cases the asymmetry, as measured by
and by the mean anomaly M of the outer planet when the inner planet is at perihelion, is very large. The stability of these asymmetric families has
been studied and it is found that there exist large regions in phase space where we have stable asymmetric librations. It
is also shown that the asymmetry is a stabilizing factor. A shift from asymmetry to symmetry, other elements being the same,
may destabilize the system. 相似文献
17.
Formulae containing the elements of the variational matrix are obtained which determine the linear isoenergetic stability parameters of three-dimensional periodic orbits of the general three-boy problem. This requires the numerical integration of the variational equations but produces the stability parameters with the effective accuracy of the numerical integration. The conditions for stability, criticality, and bifurcations are briefly examined and the stability determination procedure is tested in the determination of some three-dimensional periodic orbits of low inclination bifurcating from vertical-critical coplanar orbits. 相似文献
18.
Roger A. Broucke 《Celestial Mechanics and Dynamical Astronomy》2001,81(4):321-341
The present research was motivated by the recent discovery of planets around binary stars. Our initial intention was thus to investigate the 3-dimensional nearly circular periodic orbits of the circular restricted problem of three bodies; more precisely Stromgren's class L, (direct) and class m, (retrograde). We started by extending several of Hénon's vertical critical orbits of these 2 classes to three dimensions, looking especially for orbits which are near circular and have stable characteristic exponents.We discovered early on that the periodic orbits with the above two qualifications are fairly rare and we decided thus to undertake a systematic exploration, limiting ourselves to symmetric periodic orbits. However, we examined all 16 possible symmetry cases, trying 10000 sets of initial values for periodicity in each case, thus 160000 integrations, all with z
o or o equal to 0.1 This gave us a preliminary collection of 171 periodic orbits, all fairly near the xy-plane, thus with rather low inclinations. Next, we integrated a second similar set of 160000 cases with z
o or o equal to 0.5, in order to get a better representation of the large inclinations. This time, we found 167 periodic orbits, but it was later discovered that at least 152 of them belong to the same families as the first set with 0.1Our paper quickly describes the definition of the problem, with special emphasis on the symmetry properties, especially for the case of masses with equal primaries. We also allow a section to describe our approach to stability and characteristic exponents, following our paper on this subject, (Broucke, 1969). Then we describe our numerical results, as much as space permits in the present paper.We found basically only about a dozen families with sizeable segments of simple stable periodic orbits. Some of them are around one of the two stars only but we do not describe them here because of a lack of space. We extended about 170 periodic orbits to families of up to 500 members, (by steps of 0.005 in the parameter), although, in many cases, we do not know the real end of the families. We also give an overview of the different types of periodic orbits that are most often encountered. We describe some of the rather strange orbits, (some of which are actually stable). 相似文献
19.
P. G. Kazantzis 《Astrophysics and Space Science》1979,65(2):493-513
New families of three-dimensional double-symmetric periodic orbits are determined numerically in the Sun-Jupiter case of the restricted three-body problem. These families bifurcate from the vertical-critical orbits (
v
= –1,c
v
),c
v=0) of the basic plane familiesi,g
1,g
2,h,a,m andl. Further the numerical procedure employed in the determination of these families has been described and interesting results have been pointed out. Also, computer plots of the orbits of these families have been shown in conical projections. 相似文献
20.
W. S. Koon M. W. Lo J. E. Marsden S. D. Ross 《Celestial Mechanics and Dynamical Astronomy》2001,81(1-2):27-38
A number of Jupiter family comets such as Otermaand Gehrels 3make a rapid transition from heliocentric orbits outside the orbit of Jupiter to heliocentric orbits inside the orbit of Jupiter and vice versa. During this transition, the comet can be captured temporarily by Jupiter for one to several orbits around Jupiter. The interior heliocentric orbit is typically close to the 3:2 resonance while the exterior heliocentric orbit is near the 2:3 resonance. An important feature of the dynamics of these comets is that during the transition, the orbit passes close to the libration points L
1and L
2, two of the equilibrium points for the restricted three-body problem for the Sun-Jupiter system. Studying the libration point invariant manifold structures for L
1and L
2is a starting point for understanding the capture and resonance transition of these comets. For example, the recently discovered heteroclinic connection between pairs of unstable periodic orbits (one around the L
1and the other around L
2) implies a complicated dynamics for comets in a certain energy range. Furthermore, the stable and unstable invariant manifold tubes associated to libration point periodic orbits, of which the heteroclinic connections are a part, are phase space conduits transporting material to and from Jupiter and between the interior and exterior of Jupiter's orbit. 相似文献