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1.
《天文和天体物理学研究(英文版)》2020,(9)
We consider the coplanar planetary four-body problem, where three planets orbit a large star without the cross of their orbits. The system is stable if there is no exchange or cross of orbits. Starting from the Sundman inequality, the equation of the kinematical boundaries is derived. We discuss a reasonable situation, where two planets with known orbits are more massive than the third one. The boundaries of possible motions are controlled by the parameter c~2E. If the actual value of c~2E is less than or equal to a critical value(c~2 E)cr, then the regions of possible motions are bounded and therefore the system is stable.The criteria obtained in special cases are applied to the Solar System and the currently known extrasolar planetary systems. Our results are checked using N-body integrator. 相似文献
2.
In this paper, we study the existence of transversal homoclinic orbits in a planar circular restricted four-body problem, based on the perturbation theory of integrable Hamiltonian systems. We start from a planar circular restricted four-body model and regard it as a perturbation of the two-body model. Then, in order to conveniently study unbounded orbits, we transform the infinite points to finite points by a non-canonical transformation, arriving at a non-Hamiltonian system with degenerate fixed points. According to the extended Melnikov method, we finally prove that there exist transversal homoclinic orbits in this four-body model. 相似文献
3.
M. Michalodimitrakis 《Astrophysics and Space Science》1981,75(2):289-305
By generalizing the restricted three-body problem, we introduce the restricted four-body problem. We present a numerical study of this problem which includes a study of equilibrium points, regions of possible motion and periodic orbits. Our main motivation for introducing this problem is that it can be used as an intermediate step for a systematic exploration of the genral four-body problem. In an analogous way, one may introduce the restrictedN-body problem. 相似文献
4.
Keiko Atobe 《Icarus》2007,188(1):1-17
We have investigated the obliquity evolution of terrestrial planets in habitable zones (at ∼1 AU) in extrasolar planetary systems, due to tidal interactions with their satellite and host star with wide varieties of satellite-to-planet mass ratio (m/Mp) and initial obliquity (γ0), through numerical calculations and analytical arguments. The obliquity, the angle between planetary spin axis and its orbit normal, of a terrestrial planet is one of the key factors in determining the planetary surface environments. A recent scenario of terrestrial planet accretion implies that giant impacts of Mars-sized or larger bodies determine the planetary spin and form satellites. Since the giant impacts would be isotropic, tilted spins (sinγ0∼1) are more likely to be produced than straight ones (sinγ0∼0). The ratio m/Mp is dependent on the impact parameters and impactors' mass. However, most of previous studies on tidal evolution of the planet-satellite systems have focused on a particular case of the Earth-Moon systems in which m/Mp?0.0125 and γ0∼10° or the two-body planar problem in which γ0=0° and stellar torque is neglected. We numerically integrated the evolution of planetary spin and a satellite orbit with various m/Mp (from 0.0025 to 0.05) and γ0 (from 0° to 180°), taking into account the stellar torques and precessional motions of the spin and the orbit. We start with the spin axis that almost coincides with the satellite orbit normal, assuming that the spin and the satellite are formed by one dominant impact. With initially straight spins, the evolution is similar to that of the Earth-Moon system. The satellite monotonically recedes from the planet until synchronous state between the spin period and the satellite orbital period is realized. The obliquity gradually increases initially but it starts decreasing down to zero as approaching the synchronous state. However, we have found that the evolution with initially tiled spins is completely different. The satellite's orbit migrates outward with almost constant obliquity until the orbit reaches the critical radius ∼10-20 planetary radii, but then the migration is reversed to inward one. At the reversal, the obliquity starts oscillation with large amplitude. The oscillation gradually ceases and the obliquity is reduced to ∼0° during the inward migration. The satellite eventually falls onto the planetary surface or it is captured at the synchronous state at several planetary radii. We found that the character change of precession about total angular momentum vector into that about the planetary orbit normal is responsible for the oscillation with large amplitude and the reversal of migration. With the results of numerical integration and analytical arguments, we divided the m/Mp-γ0 space into the regions of the qualitatively different evolution. The peculiar tidal evolution with initially tiled spins give deep insights into dynamics of extrasolar planet-satellite systems and discussions of surface environments of the planets. 相似文献
5.
Strategies for plane change of Earth orbits using lunar gravity and derived trajectories of family G
C. F. de Melo E. E. N. Macau O. C. Winter 《Celestial Mechanics and Dynamical Astronomy》2009,103(4):281-299
The dynamics of the circular restricted three-body Earth-Moon-particle problem predicts the existence of the retrograde periodic
orbits around the Lagrangian equilibrium point L1. Such orbits belong to the so-called family G (Broucke, Periodic orbits
in the restricted three-body problem with Earth-Moon masses, JPL Technical Report 32–1168, 1968) and starting from them it
is possible to define a set of trajectories that form round trip links between the Earth and the Moon. These links occur even
with more complex dynamical systems as the complete Sun-Earth-Moon-particle problem. One of the most remarkable properties
of these trajectories, observed for the four-body problem, is a meaningful inclination gain when they penetrate into the lunar
sphere of influence and accomplish a swing-by with the Moon. This way, when one of these trajectories returns to the proximities
of the Earth, it will be in a different orbital plane from its initial Earth orbit. In this work, we present studies that
show the possibility of using this property mainly to accomplish transfer maneuvers between two Earth orbits with different
altitudes and inclinations, with low cost, taking into account the dynamics of the four-body problem and of the swing-by as
well. The results show that it is possible to design a set of nominal transfer trajectories that require ΔV
Total less than conventional methods like Hohmann, bi-elliptic and bi-parabolic transfer with plane change. 相似文献
6.
7.
The study of the evolution of planetary systems, primarily of the Solar System, is one of the basic problems of celestial mechanics. The stability of motion of giant planets on cosmogonic time scales was established by numerical and analytical methods, but the question about the evolution of orbits of terrestrial planets and arbitrary solar-type planetary systems remained open. This work initiates a series of papers allowing one to advance in solving the problem of the evolution of the solar-type planetary systems on cosmogonic time scales by using powerful analytical tools. In the first paper of this series, we choose the optimum reference system and obtain the Poisson series expansion of the Hamiltonian of the problem in all Keplerian elements. We propose to use the integral representation of the corresponding coefficients or the Poisson processor means instead of conventionally addressing any possible special functions. This approach extremely simplifies the algorithm. The next paper of this series deals with the calculation of the expansion coefficients. 相似文献
8.
Planar central configurations of four different masses are analyzed theoretically and computed numerically. We follow Dziobek’s
approach to four-body central configurations with a straightforward implicit (in the masses and distances) method of our own
in which the fundamental quantities are each the quotient of a directed area divided by the corresponding mass. We apply a
new simple numerical algorithm to construct general four-body central configurations. We use this tool to obtain new properties
of the symmetric and non-symmetric central configurations. The explicit continuous connection between three-body and four-body
central configurations where one of the four masses approaches zero is clarified. Some cases of coorbital 1+3 problems are
also considered. 相似文献
9.
This paper contains a numerical study of the stability of resonant orbits in a planetary system consisting of two planets,
moving under the gravitational attraction of a binary star. Its results are expected to provide us with useful information
about real planetary systems and, at the same time, about periodic motions in the general four-body problem (G4) because the
above system is a special case of G4 where two bodies have much larger masses than the masses of the other two (planets).
The numerical results show that the main mechanism which generates instability is the destruction of the Jacobi integrals
of the massless planets when their masses become nonzero and that resonances in the motion of planets do not imply, in general,
instability. Considerable intervals of stable resonant orbits have been found. The above quantitative results are in agreement
with the existing qualitative predictions 相似文献
10.
The restricted (equilateral) four-body problem consists of three bodies of masses m 1, m 2 and m 3 (called primaries) lying in a Lagrangian configuration of the three-body problem i.e., they remain fixed at the apices of an equilateral triangle in a rotating coordinate system. A massless fourth body moves under the Newtonian gravitation law due to the three primaries; as in the restricted three-body problem (R3BP), the fourth mass does not affect the motion of the three primaries. In this paper we explore symmetric periodic orbits of the restricted four-body problem (R4BP) for the case of two equal masses where they satisfy approximately the Routh’s critical value. We will classify them in nine families of periodic orbits. We offer an exhaustive study of each family and the stability of each of them. 相似文献
11.
The Integral Variation (IV) method is a technique to generate an approximate solution to initial value problems involving systems of first-order ordinary differential equations. The technique makes use of generalized Fourier expansions in terms of shifted orthogonal polynomials. The IV method is briefly described and then applied to the problem of near Earth satellite orbit prediction. In particular, we will solve the Lagrange planetary equations including the first three zonal harmonics and drag. This is a highly nonlinear system of six coupled first-order differential equations. Comparison with direct numerical integration shows that the IV method indeed provides accurate analytical approximations to the orbit prediction problem.Advanced Systems Studies; Bldg. 254EElectro-Optical Systems Laboratory; Bldg. 201. 相似文献
12.
H. Waalkens A. Burbanks S. Wiggins 《Monthly notices of the Royal Astronomical Society》2005,361(3):763-775
In this paper we use recently developed phase-space transport theory coupled with a so-called classical spectral theorem to develop a dynamically exact and computationally efficient procedure for studying escape from a planetary neighbourhood. The 'planetary neighbourhood' is a bounded region of phase space where entrance and escape are only possible by entering or exiting narrow 'bottlenecks' created by the influence of a saddle point. The method therefore immediately applies to, for example, the circular restricted three-body problem and Hill's lunar problem (which we use to illustrate the results), but it also applies to more complex, and higher-dimensional, systems possessing the relevant phase-space structure. It is shown how one can efficiently compute the mean passage time through the planetary neighbourhood, the phase-space flux in, and out, of the planetary neighbourhood, the phase-space volume of initial conditions corresponding to trajectories that escape from the planetary neighbourhood, and the fraction of initial conditions in the planetary neighbourhood corresponding to bound trajectories. These quantities are computed for Hill's problem. We study the dependence of the proportions of these quantities on energy and dimensionality (two-dimensional planar and three-dimensional spatial Hill's problem). The methods and quantities presented are of central interest for many celestial and stellar dynamical applications such as, for example, the capture and escape of moons near giant planets, the formation of binaries in the Kuiper belt and the escape of stars from star clusters orbiting about a galaxy. 相似文献
13.
Cosmogonical theories as well as recent observations allow us to expect the existence of planets around many stars other than the Sun. On an other hand, double and multiple star systems are established to be more numerous than single stars (such as the Sun), at least in the solar neighborhood. We are then faced to the following dynamical problem: assuming that planets can form in a binary early environment (I do not deal here with), does long-term stability for planetary orbits exist in double star systems.Although preliminary studies were rather pessimistic about the possibility of existence of stable planetary orbits in double or multiple star systems, modern computation have shown that many such stable orbits do exist (but possible chaotic behavior), either around the binary as a whole (P-type) or around one component of the binary (S-type), this latter being explored here.The dynamical model is the elliptic plane restricted three-body problem; the phase space of initial conditions is systematically explored, and limits for stability have been established. Stable S-type planetary orbits are found up to distance of their "sun" of the order of half the periastron distance of the binary; moreover, among these stable orbits, nearly-circular ones exist up to distance of their "sun" of the order of one quarter the periastron distance of the binary; finally, among the nearly-circular stable orbits, several stay inside the "habitable zone", at least for two nearby binaries which components are nearly of solar type.Nevertheless, we know that chaos may destroy this stability after a long time (sometimes several millions years). It is therefore important to compute indicators of chaos for these stable planetary orbits to investigate their actual very long-term stability. Here we give an example of such a computation for more than a billion years. 相似文献
14.
Luis Benet 《Celestial Mechanics and Dynamical Astronomy》2001,81(1-2):123-128
We consider the system of planetary rings with shepherds as a restricted three or four-body problem, neglecting interactions between ring particles. We show that the generic occurrence of rings for the case of rotating short-range potentials can be extended to the case of gravitational potentials. The consecutive collision periodic orbits created by saddle-center bifurcations are of central importance. 相似文献
15.
Anoop Sivasankaran Bonnie A. Steves Winston L. Sweatman 《Celestial Mechanics and Dynamical Astronomy》2010,107(1-2):157-168
Several papers in the last decade have studied the Caledonian symmetric four-body problem (CSFBP), a restricted four-body system with a symmetrically reduced phase space. During these studies, difficulties have arisen when the system approaches a close encounter. These are due to collision singularities causing numerical integration algorithms to fail. In this paper, we give the full details of a regularisation approach that now enables us to study these close encounters and collision events. The resulting equations of motion can be efficiently integrated by a high-order integrator. The results from numerical testing of the algorithm verify that the regularisation is advantageous in preserving numerical stability. The effectiveness of the approach is illustrated for a range of CSFBP orbits. Numerical experiments show that the newly developed regularisation algorithm has excellent energy conservation properties. 相似文献
16.
Keiji Ohtsuki 《Icarus》2004,172(2):432-445
We examine the rotation of a small moonlet embedded in planetary rings caused by impacts of ring particles, using analytic calculation and numerical orbital integration for the three-body problem. Taking into account the Rayleigh distribution of particles' orbital eccentricities and inclinations, we evaluate both systematic and random components of rotation, where the former arises from an average of a large number of small impacts and the latter is contribution from large impacts. Calculations for parameter values corresponding to inner parts of Saturn's rings show that a moonlet would spin slowly in the prograde direction if most impactors are small particles whose velocity dispersion is comparable to or smaller than the moonlet's escape velocity. However, we also find that the effect of the random component can be significant, if the velocity dispersion of particles is larger and/or impacts of large particles comparable to the moonlet's size are common: in this case, both prograde and retrograde rotations can be expected. In the case of a small moonlet embedded in planetary rings of equal-sized particles, we find that the systematic component dominates the moonlet rotation when m/M?0.1 (m and M are the mass of a particle and a moonlet, respectively), while the random component is dominant when m/M?0.3. We derive the condition for the random component to dominate moonlet rotation on the basis of our results of three-body orbital integration, and confirm agreement with N-body simulation. 相似文献
17.
John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》1987,43(1-4):371-390
A review is presented of periodic orbits of the planetary type in the general three-body problem and fourbody problem and the restricted circular and elliptic tnreebody problem. These correspond to planetary systems with one Sun and two or three planets (or a planet and its satellites), the motion of asteoids and also planetary systems with two Suns. The factors which affect the stability of the above configurations are studied in connection with resonance or additional perturbations. Finally, the correspondence of the periodic orbits in the restricted three-body problem with the fixed points obtained by the method of averaging or the method of surface of section is indicated. 相似文献
18.
19.
J?rg Waldvogel 《Celestial Mechanics and Dynamical Astronomy》2012,113(1):113-123
We consider the planar symmetric four-body problem with two equal masses m 1?=?m 3?>?0 at positions (±x 1(t),?0) and two equal masses m 2?=?m 4?>?0 at positions (0, ±x 2(t)) at all times t, referred to as the rhomboidal symmetric four-body problem. Owing to the simplicity of the equations of motion this problem is well suited to study regularization of the binary collisions, periodic solutions, chaotic motion, as well as the four-body collision and escape manifolds. Furthermore, resonance phenomena between the two interacting rectilinear binaries play an important role. 相似文献
20.
Lennard F. Bakker Tiancheng Ouyang Duokui Yan Skyler Simmons Gareth E. Roberts 《Celestial Mechanics and Dynamical Astronomy》2010,108(2):147-164
We apply the analytic-numerical method of Roberts to determine the linear stability of time-reversible periodic simultaneous
binary collision orbits in the symmetric collinear four-body problem with masses 1, m, m, 1, and also in a symmetric planar
four-body problem with equal masses. In both problems, the assumed symmetries reduce the determination of linear stability
to the numerical computation of a single real number. For the collinear problem, this verifies the earlier numerical results
of Sweatman for linear stability with respect to collinear and symmetric perturbations. 相似文献