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1.
The stability of co-orbital motions is investigated in such exoplanetary systems, where the only known giant planet either moves fully in the habitable zone, or leaves it for some part of its orbit. If the regions around the triangular Lagrangian points are stable, they are possible places for smaller Trojan-like planets. We have determined the nonlinear stability regions around the Lagrangian point L4 of nine exoplanetary systems in the model of the elliptic restricted three-body problem by using the method of the relative Lyapunov indicators. According to our results, all systems could possess small Trojan-like planets. Several features of the stability regions are also discussed. Finally, the size of the stability region around L4 in the elliptic restricted three-body problem is determined as a function of the mass parameter and eccentricity. 相似文献
2.
The orbits of fictitious bodies around Jupiter’s stable equilibrium points L 4 and L 5 were integrated for a fine grid of initial conditions up to 100 million years. We checked the validity of three different dynamical models, namely the spatial, restricted three body problem, a model with Sun, Jupiter and Saturn and also the dynamical model with the Outer Solar System (Jupiter to Neptune). We determined the chaoticity of an orbit with the aid of the Lyapunov Characteristic Exponents (=LCE) and used also a method where the maximum eccentricity of an orbit achieved during the dynamical evolution was examined. The goal of this investigation was to determine the size of the regions of motion around the equilibrium points of Jupiter and to find out the dependance on the inclination of the Trojan’s orbit. Whereas for small inclinations (up to i=20°) the stable regions are almost equally large, for moderate inclinations the size shrinks quite rapidly and disappears completely for i>60°. Additionally, we found a difference in the dynamics of orbits around L 4 which – according to the LCE – seem to be more stable than the ones around L 5. 相似文献
3.
B. Érdi I. Nagy Zs. Sándor Á. Süli G. Fröhlich 《Monthly notices of the Royal Astronomical Society》2007,381(1):33-40
The size distribution of the stability region around the Lagrangian point L 4 is investigated in the elliptic restricted three-body problem as the function of the mass parameter and the orbital eccentricity of the primaries. It is shown that there are minimum zones in the size distribution of the stability regions, and these zones are connected with the secondary resonances between the frequencies of librational motions around L 4 . The results can be applied to hypothetical Trojan planets for predicting values of the mass parameter and the eccentricity for which such objects can be expected or their existence is less probable. 相似文献
4.
O. Ragos K. E. Papadakis C. G. Zagouras 《Celestial Mechanics and Dynamical Astronomy》1997,67(4):251-274
The regions of quasi-periodic motion around non-symmetric periodic orbits in the vicinity of the triangular equilibrium points
are studied numerically. First, for a value of the mass parameter less than Routh's critical value, the stability regions
determined by quasi-periodic motion are examined around the existing families of short (Ls
4) and long (Ll
4) period solutions. Then, for two values of μ greater than the Routh value, the unified family Lsl
4, to which, in these cases, Ls
4 and Ll
4 merge, is considered. It is found that such regions surround in general the linearly stable segments of the corresponding
families and become smaller as the mass ratio increases.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
5.
R. Schwarz Á. Süli R. Dvorak E. Pilat-Lohinger 《Celestial Mechanics and Dynamical Astronomy》2009,104(1-2):69-84
Today there are more than 340 extra-solar planets in about 270 extra-solar systems confirmed. Besides the observed planets there exists also the possibility of a Trojan planet moving in the same orbit as the Jupiter-like planet. In our investigation we take also into account the habitability of a Trojan planet and whether such a terrestrial planet stays in the habitable zone. Its stability was investigated for multi-planetary systems, where one of the detected giant planets moves partly or completely in the habitable zone. By using numerical computations, we studied the orbital behaviour up to 107 years and determined the size of the stable regions around the Lagrangian equilibrium points for different dynamical models for fictitious Trojans. We also examined the interaction of the Trojan planets with a second or third giant planet, by varying its semimajor axis and eccentricity. We have found two systems (HD 155358 and HD 69830) that can host habitable Trojan planets. Another aim of this work was to determine the size of the stable region around the Lagrangian equilibrium points in the restricted three body problem for small mass ratios μ of the primaries μ ≤ 0.001 (e.g. Neptune mass of the secondary and smaller masses). We established a simple relation for the size depending on μ and the eccentricity. 相似文献
6.
In this paper we explore the dynamical stability of the Mars Trojan region applying mainly Laskar's Frequency Map Analysis. This method yields the chaotic diffusion rate of orbits and allows to determine the most stable regions. It also gives the frequencies which are responsible for the instability of orbits. The most stable regions are found for inclinations between about 15° and 30°. For inclinations smaller than 15°, we confirm, by applying a synthetic secular theory, that the secular resonances ν3, ν4, ν13, ν14 rapidly excite asteroid orbits within a few Myrs, or even faster. The asteroids are removed from the Trojan region after a close encounter with Mars. For large inclinations, the secular resonance ν5 clears a small region around 30° while the Kozai resonance rapidly removes bodies for inclinations larger than 35°. The dynamical lifetimes of the three L5 Trojans, (5261) Eureka, 1998 VF31, 2001 DH47, and the only L4 Trojan 1999 UJ7 are determined by numerically integrating clouds of corresponding clones over the age of the Solar System. All four Trojans reside in the most stable region with smallest diffusion coefficients. Their dynamical half-lifetime is of the order of the age of the Solar System. The Yarkovsky force has little effect on the known Trojans but for bodies smaller than about 1-5 m the drag is strong enough to destabilize Trojans on a timescale shorter than 4.5 Gyr. 相似文献
7.
We consider the motion of a test particle around a triaxial primary and an oblate companion orbiting each other in elliptic orbits about their common barycenter in the neighborhood of triangular libration points. The positions and stability of these points are influenced by the triaxiality and oblateness of the primary and secondary, and by the semi-major axis and eccentricity of the orbits. The triangular points are stable for 0<μ<μ c ; where μ is the mass ratio (μ≤1/2) and μ c is the critical mass value influenced by the eccentricity, oblateness, semi major axis and triaxiality factors. The size of the region of stability increases with decreasing values of triaxiality and oblateness. An application of the results obtain to double neutron star binaries results show that the positions and stability of the triangular points of PSR J1518+4904, PSR B1534+12, PSR B1914+16 and PSR B2127+11c are affected by the parameters in the systems’ dynamics. 相似文献
8.
Judit Györgyey 《Celestial Mechanics and Dynamical Astronomy》1985,36(3):281-285
The non-linear stability of motions around L5 in the elliptic restricted problem of the three bodies is investigated numerically with emphasis on the effect of the orbital eccentricity of the primaries on the shape of the established stability regions. It is shown that with increasing eccentricity, the width of these regions is decreasing. 相似文献
9.
Navin Chandra 《Planetary and Space Science》2004,52(8):747-760
The non-linear stability of the triangular libration point L4 of the restricted three-body problem is studied under the presence of third- and fourth-order resonances, when the more massive primary is a triaxial rigid body and source of radiation. In this study, Markeev's theorems are applied with the help of Moser's theorem. It is found that the stability of the triangular libration point is unstable in the third-order resonance case and in the fourth-order resonance case, this is stable or unstable depending on A1 and A2, and a source of radiation parameter α, where A1, A2 depend upon the lengths of the semi-axes of the triaxial rigid body. 相似文献
10.
We investigated the stable area for fictive Trojan asteroids around Neptune’s Lagrangean equilibrium points with respect to their semimajor axis and inclination. To get a first impression of the stability region we derived a symplectic mapping for the circular and the elliptic planar restricted three body problem. The dynamical model for the numerical integrations was the outer Solar system with the Sun and the planets Jupiter, Saturn, Uranus and Neptune. To understand the dynamics of the region around L 4 and L 5 for the Neptune Trojans we also used eight different dynamical models (from the elliptic problem to the full outer Solar system model with all giant planets) and compared the results with respect to the largeness and shape of the stable region. Their dependence on the initial inclinations (0° < i < 70°) of the Trojans’ orbits could be established for all the eight models and showed the primary influence of Uranus. In addition we could show that an asymmetry of the regions around L 4 and L 5 is just an artifact of the different initial conditions. 相似文献
11.
Bruno Thüring 《Celestial Mechanics and Dynamical Astronomy》1970,2(3):351-351
For 14 values of the mass parameter μ (from 0.0010 until 0.0150) the non-periodic Transtrojan orbits (aroundL 4 and L5) are investigated, which on the plane restricted problem of three bodies pass the point situated opposite to the body μ (looking from the main mass) with zero velocity in the rotating coordinate system. Results: The Transtrojan state contains a finite number of ‘double librations’ (aroundL 4 and L5); this number decreases with growing value of the mass parameter. Above a value of mass parameter between 0.010 and 0.015 no further double-libration takes place. Certain topologic properties of the Transtrojan state are found; for example this state has a phase of narrowing and a phase of widening of the single librations; thereby the amplitudes of the librations fluctuate in a characteristic manner. The investigations will be published inAstronomische Nachrichten in German language. 相似文献
12.
In the restricted circular three-body problem, two massive bodies travel on circular orbits about their mutual center of mass and gravitationally perturb the motion of a massless particle. The triangular Lagrange points, L4 and L5, form equilateral triangles with the two massive bodies and lie in their orbital plane. Provided the primary is at least 27 times as massive as the secondary, orbits near L4 and L5 can remain close to these locations indefinitely. More than 2200 cataloged asteroids librate about the L4 and L5 points of the Sun-Jupiter system, and five bodies have been discovered around the L4 point of the Sun-Neptune system. Small satellites have also been found librating about the L4 and L5 points of two of Saturn's moons. However, no objects have been discovered around the Earth-Moon L4 and L5 points. Using numerical integrations, we show that orbits near the Earth-Moon L4 and L5 points can survive for over a billion years even when solar perturbations are included, but the further addition of the far smaller perturbations from other planets destabilize these orbits within several million years. Thus, the lack of observed objects in these regions cannot be used as a constraint on Solar System formation, nor on the tidal evolution of the Moon's orbit. 相似文献
13.
Michèle Moons Alessandro Morbidelli 《Celestial Mechanics and Dynamical Astronomy》1993,57(1-2):99-108
We study the motion of asteroids in the main mean motion commensurabilities in the frame of the planar restricted three-body problem. No assumption is made about the size of the eccentricity of the asteroid. At small to moderate eccentricity, we recover existing results (shape of the phase space and location of secondary resonances). We also provide global pictures of the dynamics in the region of secondary resonances. At high eccentricity, the phase space portraits of the integrable part of the Hamiltonian show new families of stable orbits for the 3:2 and 2:1 cases and the secular resonances 5 and 6 are located. 相似文献
14.
Richard Schwarz Markus Gyergyovits Rudolf Dvorak 《Celestial Mechanics and Dynamical Astronomy》2004,90(1-2):139-148
The orbits of real asteroids around the Lagrangian points L4 and L 5of Jupiter with large inclinations (i > 20°) were integrated for 50 Myrs. We investigated the stability with the aid of the
Lyapunov characteristic exponents (LCE) but tested also two other methods: on one hand we integrated four neighbouring orbits
for each asteroid and computed the maximum distance in every group, on the other hand we checked the variation of the Delaunay
element H of the asteroid. In a second simulation – for a grid of initial eccentricity versus initial inclination – we examined the
stability of the orbits around both Lagrangian points for 20° < i < 55° and 0.0 < e < 0.20. For the initial semimajor axes
we have chosen the one ofJupiter(a = 5.202 AU). We determined the stability with the aid of the LCEs and also the maximum
eccentricity of the orbits during the whole integration time. The region around L4 turned out to be unstable for large inclinations and eccentricities (i > 55° and e > 0.12). The stable region shrinks for
orbits around L5: we found that they become unstable already for i > 45° and e > 0.10. We interpret it as a first hint why we observe more
Trojans around the leading Lagrangian point. The results confirm the stability behaviour of the real Trojans which we computed
in the first part of the paper. 相似文献
15.
Ming Xu Jiamin Zhu Tian Tan Shijie Xu 《Celestial Mechanics and Dynamical Astronomy》2012,113(4):403-433
The bounded quasi-periodic relative trajectories are investigated in this paper for on-orbit surveillance, inspection or repair, which requires rapid changes in formation configuration for full three-dimensional imaging and unpredictable evolutions of relative trajectories for non-allied spacecraft. A linearized differential equation for modeling J 2 perturbed relative dynamics is derived without any simplified treatment of full short-period effects. The equation serves as a nominal reference model for stationkeeping controller to generate the quasi-periodic trajectories near the equilibrium, i.e., the location of the chief. The developed model exhibits good numerical accuracy and is applicable to an elliptic orbit with small eccentricity inheriting from the osculating conversion of orbital elements. A Hamiltonian structure-preserving controller is derived for the three-dimensional time-periodic system that models the J 2-perturbed relative dynamics on a mean circular orbit. The equilibrium of the system has time-varying topological types and no fixed-dimensional unstable/stable/center manifolds, which are quite different from the two-dimensional time-independent system with a permanent pair of hyperbolic eigenvalues and fixed-dimensions of unstable/stable/ center manifolds. The unstable and stable manifolds are employed to change the hyperbolic equilibrium to elliptic one with the poles assigned on the imaginary axis. The detailed investigations are conducted on the critical controller gain for Floquet stability and the optimal gain for the fuel cost, respectively. Any initial relative position and velocity leads to a bounded trajectory around the controlled elliptic equilibrium. The numerical simulation indicates that the controller effectively stabilizes motions relative to the perturbed elliptic orbit with small eccentricity and unperturbed elliptic orbit with arbitrary eccentricity. The developed controller stabilizes the quasi-periodic relative trajectories involved in six foundational motions with different frequencies generated by the eigenvectors of the Floquet multipliers, rather than to track a reference relative configuration. Only the relative positions are employed for the feedback without the information from the direct measurement or the filter estimation of relative velocity. So the current controller has potential applications in formation flying for its less computation overload for on-board computer, less constraint on the measurements, and easily-achievable quasi-periodic relative trajectories. 相似文献
16.
We locate and examine the stability of the ‘out of plane’ equilibrium points, L 6,7 of an infinitesimal body in the field of stellar-oblate binary systems moving in elliptic orbits around their common center of mass. Their positions and stability depend on the oblateness as well as radiation coefficients of the primaries and the eccentricity of their orbits. A numerical application of this problem for the systems: Gamma Leporis and Altair are given. 相似文献
17.
This paper investigates the stability of triangular equilibrium points (L 4,5) in the elliptic restricted three-body problem (ER3BP), when both oblate primaries emit light energy simultaneously. The positions of the triangular points are seen to shift away from the line joining the primaries than in the classical case on account of the introduction of the eccentricity, semi-major axis, radiation and oblateness factors of both primaries. This is shown for the binary systems Achird, Luyten 726-8, Kruger 60, Alpha Centauri AB and Xi Bootis. We found that motion around these points is conditionally stable with respect to the parameters involved in the system dynamics. The region of stability increases and decreases with variability in eccentricity, oblateness and radiation pressures. 相似文献
18.
19.
The temporary capture of the dust grains in the exterior resonances with planets is studied in the frames of the planar circular three-body problem with Poynting-Robertson (PR) drag. For the Earth and particles ~ 10 Μm the resonances 4/5, 5/6, 6/7, 7/8 are shown to be most effective. The capture is only temporary (of order 105 years) and the position of resonance may be calculated from semi-analytical model using averaged disturbing function. These semi-analytical results are confirmed by numerical integration. For various planet this picture changes as with increasing planetary mass the more exterior resonances become more important. We showed that for Jupiter (at least in the space between Jupiter and Saturn) the resonance 1/2 plays the dominant role. The capture time is here several myr but again eccentricity is evolving to eccentricity e 0 ~ 0.48 of libration point for this resonance. 相似文献