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1.
We study analytically the orbits along the asymptotic manifolds from a complex unstable periodic orbit in a symplectic 4-D Froeschlé map. The orbits are given as convergent series. We compare the analytic results by truncating the series at various orders with the corresponding numerical results and we find agreement along a more extended length, as the order of truncation increases. The agreement is improved when the parameters approach those of the stability domain. Along the manifolds no terms with small divisors appear in the series. The same result is found if we use a parametrization method along the asymptotic curves. In the case of orbits starting close to the manifolds small divisors appear, but the orbits remain close to the manifolds for an extended period of time. If the parameters of the map are close to the stable domain the orbits recede and approach the origin several times and remain confined in a certain volume around the origin for a long time before escaping to large distances. For special sets of parameters we see resonance phenomena and the orbits take particular forms near every resonance.  相似文献   

2.
Using the theory of the inverse problem, we construct galactic type potentials based on perturbed harmonic oscillators, producing exact analytic trajectories represented by second-order curves. The various families of orbits are examined in these potentials. It is found, that there are cases where our galactic models produce exact analytic orbits and chaotic orbits as well. The coexistence of analytic and chaotic orbits, happens when the energy of the test particle is close to the energy of escape. Comparison with previous work is also made. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

3.
The second species periodic solutions of the restricted three body problem are investigated in the limiting case of μ=0. These orbits, called consecutive collision orbits by Hénon and generating orbits by Perko, form an infinite number of continuous one-parameter families and are the true limit, for μ→0, of second species periodic solutions for μ>0. By combining a periodicity condition with an analytic relation, for criticality, isolated members of several families are obtained which possess the unique property that the stability indexk jumps from ±∞ to ?∞ at that particular orbit. These orbits are of great interest since, for small μ>0, ‘neighboring’ orbits will then have a finite (but small) region of stability.  相似文献   

4.
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   

5.
Close planetary encounters play an important role in the evolution of the orbits of small Solar system bodies and are usually studied with the help of numerical integrations. Here we study close encounters in the framework of an analytic theory, focusing on the so-called b-plane, which is the plane centred on the planet and perpendicular to the planetocentric velocity at infinity of the small body. As shown in previous papers, it is possible to identify the initial conditions on the b-plane that lead to post-encounter orbits of given semimajor axis. In this paper we exploit analytical relationships between b-plane coordinates and pre-encounter orbital elements and compute the probability of transition to these post-encounter states, and numerically check the validity of the analytic approach.  相似文献   

6.
We present an improved analytic calculation for the tidal radius of satellites and test our results against N -body simulations.
The tidal radius in general depends upon four factors: the potential of the host galaxy, the potential of the satellite, the orbit of the satellite and the orbit of the star within the satellite . We demonstrate that this last point is critical and suggest using three tidal radii to cover the range of orbits of stars within the satellite. In this way we show explicitly that prograde star orbits will be more easily stripped than radial orbits; while radial orbits are more easily stripped than retrograde ones. This result has previously been established by several authors numerically, but can now be understood analytically. For point mass, power-law (which includes the isothermal sphere), and a restricted class of split power-law potentials our solution is fully analytic. For more general potentials, we provide an equation which may be rapidly solved numerically.
Over short times (≲1–2 Gyr ∼1 satellite orbit), we find excellent agreement between our analytic and numerical models. Over longer times, star orbits within the satellite are transformed by the tidal field of the host galaxy. In a Hubble time, this causes a convergence of the three limiting tidal radii towards the prograde stripping radius. Beyond the prograde stripping radius, the velocity dispersion will be tangentially anisotropic.  相似文献   

7.
We investigate the dynamics of putative Earth-mass planets in the habitable zone (HZ) of the extrasolar planetary system OGLE-2006-BLG-109L, a close analogue of the Solar system. Our work is inspired by the work of Malhotra & Minton. Using the linear Laplace–Lagrange theory, they identified a strong secular resonance that may excite large eccentricity of orbits in the HZ. However, due to uncertain or unconstrained orbital parameters, the subsystem of Jupiters may be found in a dynamically active region of the phase space spanned by low-order mean-motion resonances. To generalize this secular model, we construct a semi-analytical averaging method in terms of the restricted problem. The secular orbits of large planets are approximated by numerically averaged osculating elements. They are used to calculate the mean orbits of terrestrial planets by means of a high-order analytic secular theory developed in our previous works. We found regions in the parameter space of the problem in which stable, quasi-circular orbits in the HZ are permitted. The excitation of eccentricity in the HZ strongly depends on the apsidal angle of jovian orbits. For some combinations of that angle, eccentricities and semimajor axes consistent with the observations, a terrestrial planet may survive in low eccentric orbits. We also study the effect of post-Newtonian gravity correction on the innermost secular resonance.  相似文献   

8.
The purpose of this work is to show that chaos control techniques (OGY, in special) can be used to efficiently keep a spacecraft around another body performing elaborate orbits. We consider a satellite and a spacecraft moving initially in coplanar and circular orbits, with slightly different radii, around a heavy central planet. The spacecraft, which is the inner body, has a slightly larger angular velocity than the satellite so that, after some time, they eventually go to a situation in which the distance between them becomes sufficiently small, so that they start to interact with one another. This situation is called as an encounter. In previous work we have shown that this scenario is a typical situation of a chaotic scattering for some well-defined range of parameters. Considering this scenario, we first show how it is possible to find the unstable periodic orbits that are located in the chaotic invariant set. From the set of unstable periodic orbits, we select the ones that can be combined to provide the desired elaborate orbit. Then, chaos control technique based on the OGY method is used to keep the spacecraft in the desired orbit. Finally, we analyze the results and make considerations regarding a realistic scenario of space exploration.  相似文献   

9.
In the current study, the existence of periodic orbits around a fixed homogeneous cube is investigated, and the results have powerful implications for examining periodic orbits around non-spherical celestial bodies. In the two different types of symmetry planes of the fixed cube, periodic orbits are obtained using the method of the Poincaré surface of section. While in general positions, periodic orbits are found by the homotopy method. The results show that periodic orbits exist extensively in symmetry planes of the fixed cube, and also exist near asymmetry planes that contain the regular Hex cross section. The stability of these periodic orbits is determined on the basis of the eigenvalues of the monodromy matrix. This paper proves that the homotopy method is effective to find periodic orbits in the gravity field of the cube, which provides a new thought of searching for periodic orbits around non-spherical celestial bodies. The investigation of orbits around the cube could be considered as the first step of the complicated cases, and helps to understand the dynamics of orbits around bodies with complicated shapes. The work is an extension of the previous research work about the dynamics of orbits around some simple shaped bodies, including a straight segment, a circular ring, an annulus disk, and simple planar plates.  相似文献   

10.
The conjecture of Strömgren according to which, in the restricted problem, a class of doubly asymptotic orbits are limit members of families of periodic orbits is examined in the more general framework of analytic Hamiltonian system with two degrees of freedom. Sufficient conditions for the conjecture to become a theorem are established. Theses conditions amount to a transversality condition for the doubly asymptotic orbits and are likely to be verified in the cases considered in the literature of numerical explorations of the restricted problem.  相似文献   

11.
The dynamics of the two Jupiter triangular libration points perturbed by Saturn is studied in this paper. Unlike some previous works that studied the same problem via the pure numerical approach, this study is done in a semianalytic way. Using a literal solution, we are able to explain the asymmetry of two orbits around the two libration points with symmetric initial conditions. The literal solution consists of many frequencies. The amplitudes of each frequency are the same for both libration points, but the initial phase angles are different. This difference causes a temporary spatial asymmetry in the motions around the two points, but this asymmetry gradually disappears when the time goes to infinity. The results show that the two Jupiter triangular libration points should have symmetric spatial stable regions in the present status of Jupiter and Saturn. As a test of the literal solution, we study the resonances that have been extensively studied in Robutel and Gabern (Mon Not R Astron Soc 372:1463–1482, 2006). The resonance structures predicted by our analytic theory agree well with those found in Robutel and Gabern (Mon Not R Astron Soc 372:1463–1482, 2006) via a numerical approach. Two kinds of chaotic orbits are discussed. They have different behaviors in the frequency map. The first kind of chaotic orbits (inner chaotic orbits) is of small to moderate amplitudes, while the second kind of chaotic orbits (outer chaotic orbits) is of relatively larger amplitudes. Using analytical theory, we qualitatively explain the transition process from the inner chaotic orbits to the outer chaotic orbits with increasing amplitudes. A critical value of the diffusion rate is given to separate them in the frequency map. In a forthcoming paper, we will study the same problem but keep the planets in migration. The time asymmetry, which is unimportant in this paper, may cause an observable difference in the two Jupiter Trojan groups during a very fast planet migration process.  相似文献   

12.
The purpose of this paper is to make a numerical search for natural orbits that can be used for a spacecraft to study a possible small moon of Pallas. There are many speculations about the existence of a small companion around this large asteroid, so finding and classifying orbits around this possible celestial body is an interesting problem in astrodynamics and that can be used for a spacecraft to observe this body. It is assumed that this moon has a radius that can vary from 0.125 to 1 km and that is located 750 or 500 km away from the center of Pallas. The idea is to show the effects of this parameter in the orbits around this moon. It means that the moon is much smaller than Pallas, so Keplerian orbits are not possible around it. To solve this problem, it is possible to find some special orbits that are called "Quasi Satellite Orbits" (QSO). They are orbits dominated by the gravity of Pallas, but that use the smaller perturbation from the moon to keep the spacecraft close to it. The present work searches for orbits that make the spacecraft to remain at given limits in its distance from the moon, like in the range from 3 to 50 km, the values used as an example in the present paper. This value is used because it is a good range to observe the body without getting to close to it, so reducing the risks of collisions. Each trajectory can be identified by the initial conditions of the spacecraft with respect to the moon, which means its initial position and velocity. The dynamics considers the restricted three-body problem and the influence of the solar radiation pressure, because some spacecraft may have higher values for the area-to-mass ratio, which gives a non-negligible effect in the trajectory of the spacecraft.  相似文献   

13.
Man Hoi Lee  S.J. Peale 《Icarus》2006,184(2):573-583
Two small satellites of Pluto, S/2005 P1 (hereafter P1) and S/2005 P2 (hereafter P2), have recently been discovered outside the orbit of Charon, and their orbits are nearly circular and nearly coplanar with that of Charon. Because the mass ratio of Charon-Pluto is ∼0.1, the orbits of P2 and P1 are significantly non-Keplerian even if P2 and P1 have negligible masses. We present an analytic theory, with P2 and P1 treated as test particles, which shows that the motion can be represented by the superposition of the circular motion of a guiding center, the forced oscillations due to the non-axisymmetric components of the potential rotating at the mean motion of Pluto-Charon, the epicyclic motion, and the vertical motion. The analytic theory shows that the azimuthal periods of P2 and P1 are shorter than the Keplerian orbital periods, and this deviation from Kepler's third law is already detected in the unperturbed Keplerian fit of Buie and coworkers. In this analytic theory, the periapse and ascending node of each of the small satellites precess at nearly equal rates in opposite directions. From direct numerical orbit integrations, we show the increasing influence of the proximity of P2 and P1 to the 3:2 mean-motion commensurability on their orbital motion as their masses increase within the ranges allowed by the albedo uncertainties. If the geometric albedos of P2 and P1 are high and of order of that of Charon, the masses of P2 and P1 are sufficiently low that their orbits are well described by the analytic theory. The variation in the orbital radius of P2 due to the forced oscillations is comparable in magnitude to that due to the best-fit Keplerian eccentricity, and there is at present no evidence that P2 has any significant epicyclic eccentricity. However, the orbit of P1 has a significant epicyclic eccentricity, and the prograde precession of its longitude of periapse with a period of 5300 days should be easily detectable. If the albedos of P2 and P1 are as low as that of comets, the large inferred masses induce significant short-term variations in the epicyclic eccentricities and/or periapse longitudes on the 400-500-day timescales due to the proximity to the 3:2 commensurability. In fact, for the maximum inferred masses, P2 and P1 may be in the 3:2 mean-motion resonance, with the resonance variable involving the periapse longitude of P1 librating. Observations that sample the orbits of P2 and P1 well on the 400-500-day timescales should provide strong constraints on the masses of P2 and P1 in the near future.  相似文献   

14.
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15.
A method for determining the main families of isolated periodic orbits and their characteristic exponents in planar potentials which are separated by a point transformation is proposed. Since these orbits are continued analytically with the same stability, these results are persistent under small perturbations. The method is applied to the two fixed centers problem, the Paul trap and the dipole expansion of an electrostatic potential. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

16.
Periodic Orbits of a Collinear Restricted Three-Body Problem   总被引：6，自引：0，他引：6  
In this paper we study symmetric periodic orbits of a collinear restricted three-body problem, when the middle mass is the largest one. These symmetric periodic orbits are obtained from analytic continuation of symmetric periodic orbits of two collinear two-body problems.  相似文献   

17.
A proof is offered of the existence of periodic solutions of the general problem of three bodies, of the third sort envisaged by Poincaré, that is, arising by analytic continuation from unperturbed keplerian motion of each of two bodies about a primary, in which the two orbits are of commensurable periods, of zero eccentricity, but lying in different planes, provided that the inclination of the two planes is sufficiently small (but not zero).Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.  相似文献   

18.
Comets must form a major part of the interstellar medium. The solar system provides a flux of comets into the interstellar space and there is no reason to suspect that many other stars and their surrounding cometary systems would not make a similar contribution. Occasionally interstellar comets must pass through the inner solar system, but Whipple (1975) considers it unlikely that such a comet is among the known cases of apparently hyperbolic comets. Even so the upper limit for the density of unobserved interstellar comets is relatively high.In addition, we must consider the possibility that comets are a genuine component of interstellar medium, and that the Oort Cloud is merely a captured part of it (McCrea, 1975). Here we review various dynamical possibilities of two-way exchange of comet populations between the Solar System and the interstellar medium. We describe ways in which a traditional Oort Cloud (Oort, 1950) could be captured from the interstellar medium. However, we note that the so called Kuiper belt (Kuiper, 1951) of comets cannot arise through this process. Therefore we have to ask how necessary the concept of the yet unobserved Kuiper belt is for the theory of short period comets.There has been considerable debate about the question whether short period comets can be understood as a captured population of the Oort Cloud of comets or whether an additional source has to be postulated. The problem is made difficult by the long integration times of comet orbits through the age of the Solar System. It would be better to have an accurate treatment of comet-planet encounters in a statistical sense, in the form of cross sections, and to carry out Monte Carlo studies. Here we describe the plan of action and initial results of the work to derive cross sections by carrying out large numbers of comet — planet encounters and by deriving approximate analytic expressions for them. Initially comets follow parabolic orbits of arbitrary inclination and perihelion distance; cross sections are derived for obtaining orbits of given energy and inclination after the encounter. The results are used in subsequent work to make evolutionary models of the comet population.  相似文献   

19.
Intermediate perturbed orbits, which were proposed earlier by the first author and are calculated based on three position vectors and three measurements of angular coordinates of a small celestial body, are examined. Provided that the reference time interval encompassing the measurements is short, these orbits are close in the accuracy of approximation of actual motion to an orbit with fourth-order tangency. The shorter the reference time interval is, the better is the approximation. The laws of variation of the errors of methods for constructing such intermediate orbits with the length of the reference time interval are formulated. According to these laws, the rate of convergence of the methods to an exact solution in the process of shortening of the reference time interval is, in general, three orders of magnitude higher than that of conventional methods relying on an unperturbed Keplerian orbit. The considered orbits are among the most accurate of their class that is defined by the order of tangency. The obtained theoretical results are verified by numerical experiments on determining the orbit of 99942 Apophis.  相似文献   

20.
There is a very large number of small bodies in the Solar System and their orbits are varied and complicated. Some types of orbits and events are so rare that they occur in numerical simulations only when millions or billions of orbits have been calculated. In order to study these orbits or events an efficient Monte Carlo simulation is useful. Here we describe a new Monte Carlo simulation method and test it against some existing simulations of orbits of small bodies which have been obtained by different methods. We find good agreement with many earlier calculations, and study briefly the possibility of the Oort Cloud capture origin of short period comets. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

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