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1.
We consider particles with low free or proper eccentricity that are orbiting near planets on eccentric orbits. Through collisionless particle integration, we numerically find the location of the boundary of the chaotic zone in the planet's corotation region. We find that the distance in semimajor axis between the planet and boundary depends on the planet mass to the 2/7 power and is independent of the planet eccentricity, at least for planet eccentricities below 0.3. Our integrations reveal a similarity between the dynamics of particles at zero eccentricity near a planet in a circular orbit and with zero free eccentricity particles near an eccentric planet. The 2/7th law has been previously explained by estimating the semimajor at which the first-order mean motion resonances are large enough to overlap. Orbital dynamics near an eccentric planet could differ due to first-order corotation resonances that have strength proportional to the planet's eccentricity. However, we find that the corotation resonance width at low free eccentricity is small; also the first-order resonance width at zero free eccentricity is the same as that for a zero-eccentricity particle near a planet in a circular orbit. This accounts for insensitivity of the chaotic zone width to planet eccentricity. Particles at zero free eccentricity near an eccentric planet have similar dynamics to those at zero eccentricity near a planet in a circular orbit.  相似文献   

2.
Jiang & Yeh proposed gas-drag-induced resonant capture as a mechanism able to explain the dominant 3:2 resonance observed in the trans-Neptunian belt. Using a model of a disc–star–planet system they concluded that gaseous drag in a protoplanetary disc can trap trans-Neptunian object (TNO) embryos into the 3:2 resonance rather easily although it could not trap objects into the 2:1 resonance. Here we further investigate this scenario using numerical simulations within the context of the planar restricted four-body problem by including both present-day Uranus and Neptune. Our results show that mean motion and corotation resonances are possible and trapping into both the 3:2 and 2:1 resonances as well as other resonances is observed. The associated corotation centres may easily form larger planetesimals from smaller ones. Corotation resonances evolve into pure Lindblad resonances in a time-scale of 0.5 Myr. The non-linear corotation and mean motion resonances produced are very size selective. The 3:2 resonance is dominant for submetric particles but for larger particles the 2:1 resonance is stronger. In summary, our calculations show that confined chaotic motion around the resonances not only increases trapping efficiency but also the orbital eccentricities of the trapped material, modifying the relative abundance of trapped particles in different resonances. If we assume a more compact planetary system, instead of using the present-day values of the orbital elements of Uranus and Neptune, our results remain largely unchanged.  相似文献   

3.
In the framework of the planar and circular restricted three-body problem, we consider an asteroid that orbits the Sun in quasi-satellite motion with a planet. A quasi-satellite trajectory is a heliocentric orbit in co-orbital resonance with the planet, characterized by a nonzero eccentricity and a resonant angle that librates around zero. Likewise, in the rotating frame with the planet, it describes the same trajectory as the one of a retrograde satellite even though the planet acts as a perturbator. In the last few years, the discoveries of asteroids in this type of motion made the term “quasi-satellite” more and more present in the literature. However, some authors rather use the term “retrograde satellite” when referring to this kind of motion in the studies of the restricted problem in the rotating frame. In this paper, we intend to clarify the terminology to use, in order to bridge the gap between the perturbative co-orbital point of view and the more general approach in the rotating frame. Through a numerical exploration of the co-orbital phase space, we describe the quasi-satellite domain and highlight that it is not reachable by low eccentricities by averaging process. We will show that the quasi-satellite domain is effectively included in the domain of the retrograde satellites and neatly defined in terms of frequencies. Eventually, we highlight a remarkable high eccentric quasi-satellite orbit corresponding to a frozen ellipse in the heliocentric frame. We extend this result to the eccentric case (planet on an eccentric motion) and show that two families of frozen ellipses originate from this remarkable orbit.  相似文献   

4.
We examine the orbital evolution of planetesimals under the influence of Jupiter's perturbations and nebular gas drag, under the assumption that gas persisted in the asteroid region for some time after Jupiter attained its final mass. Two distinct mechanisms, associated with the 2 : 1 and 3 : 2 mean motion resonances, can excite eccentricities to high values, despite the damping effect of drag. If Jupiter's eccentricity was comparable to its present value, planetesimals can be temporarily trapped in the 2 : 1 resonance. Bodies crossing the 3 : 2 resonance can enter a region of phase space with overlapping high-order resonances. Both mechanisms can produce eccentricities greater than 0.5 for asteroid-sized planetesimals. The combination of resonant perturbations and drag causes secular decay of semimajor axes, resulting in migration of bodies from the outer to inner belt. Inclinations remain low, implying significant collisional evolution during this migration. Velocities of resonant bodies relative to the gas are highly supersonic; these would have been a source of shock waves in the solar nebula.This revised version was published online in October 2005 with corrections to the Cover Date.  相似文献   

5.
We analyse the global structure of the phase space of the planar planetary 2/1 mean-motion resonance in cases where the outer planet is more massive than its inner companion. Inside the resonant domain, we show the existence of two families of periodic orbits, one associated to the librational motion of resonant angle (σ-family) and the other related to the circulatory motion of the difference in longitudes of pericentre (  Δϖ  -family). The well-known apsidal corotation resonances (ACR) appear as intersections between both families. A complex web of secondary resonances is also detected for low eccentricities, whose strengths and positions are dependent on the individual masses and spatial scale of the system.
The construction of dynamical maps for various values of the total angular momentum shows the evolution of the families of stable motion with the eccentricities, identifying possible configurations suitable for exoplanetary systems. For low–moderate eccentricities, several different stable modes exist outside the ACR. For larger eccentricities, however, all stable solutions are associated to oscillations around the stationary solutions.
Finally, we present a possible link between these stable families and the process of resonance capture, identifying the most probable routes from the secular region to the resonant domain, and discussing how the final resonant configuration may be affected by the extension of the chaotic layer around the resonance region.  相似文献   

6.
A Hamiltonian model is constructed for the spin axis of a planet perturbed by a nearby planet with both planets in orbit about a star. We expand the planet–planet gravitational potential perturbation to first order in orbital inclinations and eccentricities, finding terms describing spin resonances involving the spin precession rate and the two planetary mean motions. Convergent planetary migration allows the spinning planet to be captured into spin resonance. With initial obliquity near zero, the spin resonance can lift the planet’s obliquity to near 90\(^\circ \) or 180\(^\circ \) depending upon whether the spin resonance is first or zeroth order in inclination. Past capture of Uranus into such a spin resonance could give an alternative non-collisional scenario accounting for Uranus’s high obliquity. However, we find that the time spent in spin resonance must be so long that this scenario cannot be responsible for Uranus’s high obliquity. Our model can be used to study spin resonance in satellite systems. Our Hamiltonian model explains how Styx and Nix can be tilted to high obliquity via outward migration of Charon, a phenomenon previously seen in numerical simulations.  相似文献   

7.
Third and fourth order mean motion resonances are studied in the model of the restricted three-body problem by numerical methods for mass parameters corresponding approximately to the Sun?CJupiter and Sun?CNeptune systems. In the case of inner resonances, it is shown that there are two regions of libration in the 8:5 and 7:4 resonances, one at low, the other at high eccentricities. In the 9:5 and 7:3 resonances libration can exist only in one region at high eccentricities. The 5:2 and 4:1 resonances are very regular, with one librational zone existing for all eccentricities. There is no visible region of libration at any eccentricities in the 5:1 resonance, the transition between the regions of direct and retrograde circulation is very sharp. In the case of outer resonances, the 8:5 and 7:4 resonances have also two regions of libration, but the 9:5 resonance has three, the 7:3 resonance two librational zones. The 5:2 resonance is again very regular, but it is parted for two regions of libration at high eccentricities. Libration is possible in the 4:1 resonance only at high eccentricities. The 5:1 resonance is very symmetric. In the case of outer resonances, a comparison is made with trans-Neptunian objects (TNO) in higher order mean motion resonances. Several new librating TNOs are identified.  相似文献   

8.
We study orbits of planetary systems with two planets, for planar motion, at the 1/1 resonance. This means that the semimajor axes of the two planets are almost equal, but the eccentricities and the position of each planet on its orbit, at a certain epoch, take different values. We consider the general case of different planetary masses and, as a special case, we consider equal planetary masses. We start with the exact resonance, which we define as the 1/1 resonant periodic motion, in a rotating frame, and study the topology of the phase space and the long term evolution of the system in the vicinity of the exact resonance, by rotating the orbit of the outer planet, which implies that the resonance and the eccentricities are not affected, but the symmetry is destroyed. There exist, for each mass ratio of the planets, two families of symmetric periodic orbits, which differ in phase only. One is stable and the other is unstable. In the stable family the planetary orbits are in antialignment and in the unstable family the planetary orbits are in alignment. Along the stable resonant family there is a smooth transition from planetary orbits of the two planets, revolving around the Sun in eccentric orbits, to a close binary of the two planets, whose center of mass revolves around the Sun. Along the unstable family we start with a collinear Euler–Moulton central configuration solution and end to a planetary system where one planet has a circular orbit and the other a Keplerian rectilinear orbit, with unit eccentricity. It is conjectured that due to a migration process it could be possible to start with a 1/1 resonant periodic orbit of the planetary type and end up to a satellite-type orbit, or vice versa, moving along the stable family of periodic orbits.  相似文献   

9.
The influence of gas drag and gravitational perturbations by a planetary embryo on the orbit of a planetesimal in the solar nebula was examined. Non-Keplerian rotation of the gas causes secular decay of the orbit. If the planetesimal's orbit is exterior to the perturber's, resonant perturbations oppose this drag and can cause it to be trapped in a stable orbit at a commensurability of order j/(j + 1), where j is an integer. Numerical and analytical demonstrations show that resonant trapping occurs for wide ranges of perturbing mass, planetesimal size, and j. Induced eccentricities are large, causing overlap of orbits for bodies in different resonances with j > 2. Collisions between planetesimals in different resonances, or between resonant and nonresonant bodies, result in their disruption. Fragments smaller than a critical size can pass through resonances under the influence of drag and be accreted by the embryo. This effect speeds accretion and tends to prevent dynamical isolation of planetary embryos, making gas-rich scenarios for planetary formation more plausible.  相似文献   

10.
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   

11.
This study is concerned with the stability of motion of the circumbinary exoplanet Kepler-413b. The analysis is performed within the framework of a flat restricted three-body problem. The stability diagram is plotted in the plane of initial conditions “pericentric distance—eccentricity” using mass calculations of Lyapunov exponents. According to the diagram, the Kepler-413b planet is located in a stable resonance cell, confined by the mean-motion resonances 6: 1 and 7: 1 with a central binary star, which agrees with the conclusions of Kostov et al. (2014) based on calculations of the MEGNO parameter. It is shown that the value of the critical semimajor axis acquired from the empirical formula of Holman and Wiegert (1999) almost coincides with the value obtained directly from the stability diagram; at low and moderate eccentricities of the planetary orbit, the position of the calculated boundary of the chaos zone is in close agreement with the boundary predicted by Shevchenko’s theory (2015). If the planet were in the instability zone, its characteristic Lyapunov time would be only ~1 year. In accordance with the conclusions of Kostov et al. (2014), it has been shown that the planet Kepler-413b is outside the habitability zone of the system.  相似文献   

12.
The most puzzling property of the extrasolar planets discovered by recent radial velocity surveys is their high orbital eccentricities, which are very difficult to explain within our current theoretical paradigm for planet formation. Current data reveal that at least 25% of these planets, including some with particularly high eccentricities, are orbiting a component of a binary star system. The presence of a distant companion can cause significant secular perturbations in the orbit of a planet. At high relative inclinations, large-amplitude, periodic eccentricity perturbations can occur. These are known as “Kozai cycles” and their amplitude is purely dependent on the relative orbital inclination. Assuming that every planet host star also has a (possibly unseen, e.g., substellar) distant companion, with reasonable distributions of orbital parameters and masses, we determine the resulting eccentricity distribution of planets and compare it to observations? We find that perturbations from a binary companion always appear to produce an excess of planets with both very high (?0.6) and very low (e ? 0.1) eccentricities. The paucity of near-circular orbits in the observed sample implies that at least one additional mechanism must be increasing eccentricities. On the other hand, the overproduction of very high eccentricities observed in our models could be combined with plausible circularization mechanisms (e.g., friction from residual gas) to create more planets with intermediate eccentricities (e? 0.1–0.6).  相似文献   

13.
The behaviour of ‘resonances’ in the spin-orbit coupling in celestial mechanics is investigated in a conservative setting. We consider a Hamiltonian nearly-integrable model describing an approximation of the spin-orbit interaction. The continuous system is reduced to a mapping by integrating the equations of motion through a symplectic algorithm. We study numerically the stability of periodic orbits associated to the above mapping by looking at the eigenvalues of the matrix of the linearized map over the full cycle of the periodic orbit. In particular, the value of the trace of the matrix is related to the stability character of the periodic orbit. We denote by ε* (p/q) the value of the perturbing parameter at which a given elliptic periodic orbit with frequency p/q becomes unstable. A plot of the critical function ε* (p/q) versus the frequency at different orbital eccentricities shows significant peaks at the synchronous resonance (for low eccentricities) and at the synchronous and 3:2 resonances (at higher eccentricities) in good agreement with astronomical observations. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

14.
The utilization of chaotic dynamics approaches allowed the identification of many modes of motion in resonant asteroidal dynamics. As these dynamical systems are not integrable, the motion modes are not separated and one orbit may transit from one mode to another. In some cases, as in the \31 resonance, these transitions may lead, in a relatively short time scale, to eccentricities so high that the asteroid may approach the Sun and be destroyed. In the \21 and \32 resonances these transitions are much slower and only indirect estimations of the time which is needed for a generic asteroid to leave the resonance are possible. It may reach hundreds of million years in the more robust regions of the \21 resonance and a time of the order of billions of years in those of the \32 resonance. These values are consistent with the observed depletion of the \21 resonance (only a few asteroids known while almost 60 asteroids are known in the \32 resonance).  相似文献   

15.
张旭东  周济林 《天文学报》2006,47(2):175-185
最近的多普勒观测表明恒星HD 12661周围存在两颗中等偏心率轨道上运行的行星,内行星的最小质量为2．3木星质量,轨道周期为263．6天;外行星的最小质量为1．57木星质量,轨道周期为1444．5天．该系统的稳定性要求两颗行星处在平运动轨道共振．用数值方法研究了该系统形成初期在恒星气体盘作用下的轨道迁移与稳定性,计算了行星在迁移中被平运动共振俘获的概率．发现这两颗行星目前很可能正处在11:2平运动共振边缘,且运动是混沌的,从而澄清了关于系统目前构形的不同说法,并且很可能在系统形成后行星迁移到目前构形时,气体盘几乎消失了．  相似文献   

16.
A migrating planet can capture planetesimals into mean motion resonances. However, resonant trapping can be prevented when the drift or migration rate is sufficiently high. Using a simple Hamiltonian system for first- and second-order resonances, we explore how the capture probability depends on the order of the resonance, drift rate and initial particle eccentricity. We present scaling factors as a function of the planet mass and resonance strength to estimate the planetary migration rate above which the capture probability drops to less than half. Applying our framework to multiple extrasolar planetary systems that have two planets locked in resonance, we estimate lower limits for the outer planet's migration rate, allowing resonance capture of the inner planet.
Mean motion resonances are comprised of multiple resonant subterms. We find that the corotation subterm can reduce the probability of capture when the planet eccentricity is above a critical value. We present factors that can be used to estimate this critical planet eccentricity. Applying our framework to the migration of Neptune, we find that Neptune's eccentricity is near the critical value that would make its 2 : 1 resonance fail to capture twotinos. The capture probability is affected by the separation between resonant subterms and so is also a function of the precession rates of the longitudes of periapse of both planet and particle near resonance.  相似文献   

17.
We consider orbital resonances in multiplanet systems. These are expected to arise during or just after formation in a gaseous disc. Disc–planet interaction naturally produces orbital migration and circularization through the action of tidal torques which in turn may lead to an orbital resonance. The mass and angular momentum content of the disc is likely to be comparable to that in the planets so that it is essential to fully incorporate the disc in the analysis.We study the orbital evolution of two planets locked in 2:1 commensurability through migration tidally induced by the disc using both analytic methods and numerical hydrodynamic simulations. The planets are assumed to orbit in an inner cavity containing at most only a small amount of disc material. Results are found to be sensitive to initial surface density profile, planet masses and disc parameters. The evolution may range between attaining and subsequently maintaining a resonance lock with two angles librating to divergent migration with no commensurability formed. In the former case eccentricities increase monotonically with time while the system undergoes inward migration. If the migration is halted by loss of the disc leaving the planets in a final configuration, there is likely to be a low probability of seeing resonant planets at small radii as well as a sensitive dependence on past history.We have also considered a multiplanet system in secular apsidal resonance. We consider the system as being in just one secular normal mode and include the effects of a gaseous disc. It is suggested that a normal mode may be selected by adding in some weak dissipative process in the disc and that it may remain, involving only the planets, when the disc is slowly removed.  相似文献   

18.
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 m 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   

19.
We consider a planetary system consisting of two primaries, namely a star and a giant planet, and a massless secondary, say a terrestrial planet or an asteroid, which moves under their gravitational attraction. We study the dynamics of this system in the framework of the circular and elliptic restricted three-body problem, when the motion of the giant planet describes circular and elliptic orbits, respectively. Originating from the circular family, families of symmetric periodic orbits in the 3/2, 5/2, 3/1, 4/1 and 5/1 mean-motion resonances are continued in the circular and the elliptic problems. New bifurcation points from the circular to the elliptic problem are found for each of the above resonances, and thus, new families continued from these points are herein presented. Stable segments of periodic orbits were found at high eccentricity values of the already known families considered as whole unstable previously. Moreover, new isolated (not continued from bifurcation points) families are computed in the elliptic restricted problem. The majority of the new families mainly consists of stable periodic orbits at high eccentricities. The families of the 5/1 resonance are investigated for the first time in the restricted three-body problems. We highlight the effect of stable periodic orbits on the formation of stable regions in their vicinity and unveil the boundaries of such domains in phase space by computing maps of dynamical stability. The long-term stable evolution of the terrestrial planets or asteroids is dependent on the existence of regular domains in their dynamical neighbourhood in phase space, which could host them for long-time spans. This study, besides other celestial architectures that can be efficiently modelled by the circular and elliptic restricted problems, is particularly appropriate for the discovery of terrestrial companions among the single-giant planet systems discovered so far.  相似文献   

20.
We perform numerical simulations to study the secular orbital evolution and dynamical structure of the quintuplet planetary system 55 Cancri with the self-consistent orbital solutions by Fischer and coworkers. In the simulations, we show that this sys-tem can be stable for at least 108 yr. In addition, we extensively investigate the planetary configuration of four outer companions with one terrestrial planet in the wide region of 0.790 AU ≤ a ≤ 5.900 AU to examine the existence of potential asteroid structure and Habitable Zones (HZs). We show that there are unstable regions for orbits about 4:1, 3:1 and 5:2 mean motion resonances (MMRs) of the outermost planet in the system, and sev-eral stable orbits can remain at 3:2 and 1:1 MMRs, which resembles the asteroid belt in the solar system. From a dynamical viewpoint, proper HZ candidates for the existence of more potential terrestrial planets reside in the wide area between 1.0 AU and 2.3 AU with relatively low eccentricities.  相似文献   

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