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1.
Massive planets form within the lifetime of protoplanetary disks, and therefore, they are subject to orbital migration due to planet–disk interactions. When the first planet reaches the inner edge of the disk, its migration stops and consequently the second planet ends up locked in resonance with the first one. We detail how the resonant trapping works comparing semi-analytical formulae and numerical simulations. We restrict to the case of two equal-mass coplanar planets trapped in first-order resonances, but the method can be easily generalized. We first describe the family of resonant stable equilibrium points (zero-amplitude libration orbits) using series expansions up to different orders in eccentricity as well as a non-expanded Hamiltonian. Then we show that during convergent migration the planets evolve along these families of equilibrium points. Eccentricity damping from the disk leads to a final equilibrium configuration that we predict precisely analytically. The fact that observed multi-exoplanetary systems are rarely seen in resonances suggests that in most cases the resonant configurations achieved by migration become unstable after the removal of the protoplanetary disk. Here we probe the stability of the resonances as a function of planetary mass. For this purpose, we fictitiously increase the masses of resonant planets, adiabatically maintaining the low-amplitude libration regime until instability occurs. We discuss two hypotheses for the instability, that of a low-order secondary resonance of the libration frequency with a fast synodic frequency of the system, and that of minimal approach distance between planets. We show that secondary resonances do not seem to impact resonant systems at low amplitude of libration. Resonant systems are more stable than non-resonant ones for a given minimal distance at close encounters, but we show that the latter nevertheless play the decisive role in the destabilization of resonant pairs. We show evidence that as the planetary mass increases and the minimal distance between planets gets smaller in terms of mutual Hill radius, the region of stability around the resonance center shrinks, until the equilibrium point itself becomes unstable.  相似文献   

2.
F. MarzariH. Scholl 《Icarus》2002,159(2):328-338
We have numerically explored the mechanisms that destabilize Jupiter's Trojan orbits outside the stability region defined by Levison et al. (1997, Nature385, 42-44). Different models have been exploited to test various possible sources of instability on timescales on the order of ∼108 years.In the restricted three-body model, only a few Trojan orbits become unstable within 108 years. This intrinsic instability contributes only marginally to the overall instability found by Levison et al.In a model where the orbital parameters of both Jupiter and Saturn are fixed, we have investigated the role of Saturn and its gravitational influence. We find that a large fraction of Trojan orbits become unstable because of the direct nonresonant perturbations by Saturn. By shifting its semimajor axis at constant intervals around its present value we find that the near 5:2 mean motion resonance between the two giant planets (the Great Inequality) is not responsible for the gross instability of Jupiter's Trojans since short-term perturbations by Saturn destabilize Trojans, even when the two planets are far out of the resonance.Secular resonances are an additional source of instability. In the full six-body model with the four major planets included in the numerical integration, we have analyzed the effects of secular resonances with the node of the planets. Trojan asteroids have relevant inclinations, and nodal secular resonances play an important role. When a Trojan orbit becomes unstable, in most cases the libration amplitude of the critical argument of the 1:1 mean motion resonance grows until the asteroid encounters the planet. Libration amplitude, eccentricity, and nodal rate are linked for Trojan orbits by an algebraic relation so that when one of the three parameters is perturbed, the other two are affected as well. There are numerous secular resonances with the nodal rate of Jupiter that fall inside the region of instability and contribute to destabilize Trojans, in particular the ν16. Indeed, in the full model the escape rate over 50 Myr is higher compared to the fixed model.Some secular resonances even cross the stability region delimited by Levison et al. and cause instability. This is the case of the 3:2 and 1:2 nodal resonances with Jupiter. In particular the 1:2 is responsible for the instability of some clones of the L4 Trojan (3540) Protesilaos.  相似文献   

3.
Massimiliano Guzzo 《Icarus》2006,181(2):475-485
The motion of the giant planets from Jupiter to Neptune is chaotic with Lyapunov time of approximately 10 Myr. A recent theory explains the presence of this chaos with three-planet mean-motion resonances, i.e. resonances among the orbital periods of at least three planets. We find that the distribution of these resonances with respect to the semi-major axes of all the planets is compatible with orbital instability. In particular, they overlap in a region of 10−3 AU with respect to the variation of the semi-major axes of Uranus and Neptune. Fictitious planetary systems with initial conditions in this region can undergo systematic variations of semi-major axes. The true Solar System is marginally in this region, and Uranus and Neptune undergo very slow systematic variations of semi-major axes with speed of order 10−4 AU/Gyr.  相似文献   

4.
We study the interaction of a satellite and a nearby ringlet on eccentric and inclined orbits. Secular torques originate from mean motion resonances and the secular interaction potential which represents the m  = 1 global modes of the ring. The torques act on the relative eccentricity and inclination. The resonances damp the relative eccentricity. The inclination instability owing to the resonances is turned off by a finite differential eccentricity of the order of 0.27 for nearly coplanar systems. The secular potential torque damps the eccentricity and inclination and does not affect the relative semi-major axis; also, it suppresses the inclination instability that persists at small differential eccentricities. The damping of the relative eccentricity and inclination forces an initially circular and planar small mass ringlet to reach the eccentricity and inclination of the satellite. When the planet is oblate, the interaction of the satellite damps the proper precession of a small mass ringlet so that it precesses at the satellite's rate independently of their relative distance. The oblateness of the primary modifies the long-term eccentricity and inclination magnitudes and introduces a constant shift in the apsidal and nodal lines of the ringlet with respect to those of the satellite. These results are applied to Saturn's F-ring, which orbits between the moons Prometheus and Pandora.  相似文献   

5.
A new semianalytical theory of asteroid motion is presented. The theory is developed on the basis of Kaula's expansion of the disturbing function including terms up to the second order with respect to the masses of disturbing bodies. The theory is constructed in explicit form that gives the possibility to study separately the influence of different perturbations in the dynamics of minor planets. The mean-motion resonances with major planets as well as mixed three-body resonances can also be taken into account. For the non-resonant case the formulas obtained can be used for deriving the second transformation to calculate the proper elements of an asteroid orbit in closed form with respect to inclinations and eccentricities. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

6.
We study the position and the stability of the equilibria for a generic Hamiltonian function developed up to the second harmonic and depending on two parameters; we describe the topology of the phase space for fixed values of these parameters. We show that for some values of the parameters asymmetric equilibria (unstable or stable) may appear. We deduce the conditions of capture into first order resonances for slowly drifting systems. We apply this model to the restricted three-body problem.  相似文献   

7.
In papers (Godziewski and Maciejewski, 1998a, b, 1999), we investigate unrestricted, planar problem of a dynamically symmetric rigid body and a sphere. Following the original statement of the problem by Kokoriev and Kirpichnikov (1988), we assume that the potential of the rigid body is approximated by the gravitational field of a dumb-bell. The model is described in terms of a 2D Hamiltonian depending on three parameters.In this paper, we investigate the stability of triangular equilibria permissible by the dynamics of the model, under the assumption of low-order resonances. We analyze all resonances of order smaller than four, and we examine the stability with application of theorems by Markeev and Sokolsky. These are the possible following cases: the non-diagonal resonance of the first order with two null characteristic frequencies (unstable); resonances of the first order with one nonzero frequency (diagonal and non-diagonal, stable and unstable); the second-order resonance, which is non-diagonal and stable, and the third-order resonance which is generically unstable, except for three points in the parameters' space, corresponding to stable equilibria.We discuss a perturbed version of Kokoriev and Kirpichnikov model, and we find that if the perturbation is small and depends on the coordinates only, the triangular equilibria persist, except if for the unperturbed equilibria the first-order resonance occurs. We show that the resonances of the order higher than two are also preserved if the perturbation acts.  相似文献   

8.
The differential equations of planetary theory are solved analytically to first order for the two-dimensional case, using only Jacobian elliptic functions and the elliptic integrals of the first and second kind. This choice of functions leads to several new features potentially of importance for planetary theory. The first of these is that the solutions do not require the expansion of the reciprocal of the distance between two planets, even for those variables which depend on two angular arguments. A second result is that the solution is free from small divisors with the exception of two special resonances. In fact, not only are the solutions for resonant orbits free from small divisors, the perturbations for all variables are expressible in closed form. A subset of the resonant orbits maintains this form and in addition has the remarkable feature that the first order perturbations are purely periodic; they contain no secular terms. A solution for the 13 resonance case is given as an example.  相似文献   

9.
This paper concerns a model problem illustrating the techniques needed to analyze weakly nonlinear oscillator systems wherein two small divisors (resonances) may occur separately or simultaneously. The method of multiple scales in combination with singular perturbation methods is used to construct a uniformly valid asymptotic solution to the proposed model equation. It is shown that matching is necessary in order to establish a composite solution uniformly valid for the entire phase plane. The model equation does not encompass the class of problems where occurrence of two simultaneous small divisors leads to Kolmogorov-Arnol'd-Moser amplitude instability.This work was supported in part by the National Science Foundation under NSF Grant Number GP-32030.  相似文献   

10.
Using our non-local time-dependent theory of convection, the linear non-adiabatic oscillations of 10 evolutionary model series with masses of  1–3 M⊙  are calculated. The results show that there is a red giant instability strip in the lower temperature side of the Hertzsprung–Russell diagram which goes along the sequences of the red giant branch and the asymptotic giant branch. For red giants of lower luminosities, pulsation instability is found at high order overtones; the lower order modes from the fundamental to the second overtone are stable. Towards higher luminosity and lower effective temperature, instability moves to lower order modes, and the amplitude growth rate of oscillations also grows. At the high luminosity end of the strip, the fundamental and the first overtone become unstable, while all the modes above the fourth order become stable. The excitation mechanisms have been studied in detail. It is found that turbulent pressure plays a key role for excitation of red variables. The frozen convection approximation is unavailable for the low temperature stars with extended convective envelopes. In any case, this approximation can explain neither the red edge of the Cepheid instability strip, nor the blue edge of the pulsating red giant instability strip. An analytic expression of a pulsation constant as a function of stellar mass, luminosity and effective temperature is presented from this work.  相似文献   

11.
The twenty most chaotic objects found among first hundred of numbered asteroids are studied. Lyapunov time calculated with and without inner planets indicates that for eleven of those asteroids the strongest chaotic effect results from the resonances with Mars. The filtered semimajor axis displays an abrupt variation only when a close approach to Mars takes place. The study of the behaviour of the critical argument for candidate resonances can reveal which is responsible for the semimajor axis variation. We have determined these resonances for the asteroids in question. For the asteroids chaotic even without the inner planets we have determined the most important resonances with Jupiter, or three-body resonances. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

12.
From a comparison of the 2:1 and 3:2 resonances (in the asteroidal belt) two possible explanations to the absence of 3:2 apocentric librators are suggested. The first one is that such 3:2 resonant motion is dynamically unstable. The second interpretation requires the absence of nearcircular orbits originally at 4 AU. The latter view, if correct, is inconsistent with cosmogonic models which predict the original orbits of the asteroids to be nearly circular.  相似文献   

13.
Mean orbital elements are obtained from osculating ones by removing the short periodic perturbations. Large catalogues of asteroid mean elements need to be computed, as a first step in the computation of proper elements, used to study asteroid families. The algorithms for this purpose available so far are only accurate to first order in the masses of the perturbing planet; the mean elements have satisfactory accuracy for most of the asteroid belt, but degraded accuracy in the neighbourhoods of the main mean motion resonances, especially the 2:1. We investigate a number of algorithms capable of improving this approximation; they belong to the two classes of Breiter-type methods and iterative methods. The former are obtained by applying some higher order numerical integration scheme, such as Runge–Kutta, to the differential equation whose solution is a transformation removing the fast angular variables from the equations; they can be used to compute a full second order theory, however, only if the full second order determining function is explicitly computed, and this is computationally too cumbersome for a complicated problem such as the N-body. The latter are fixed point iterative schemes, with the first order theory as an iteration step, used to compute the inverse map from mean to osculating elements; formally the method is first order, but because they implement a fixed frequency perturbation theory, they are more accurate than conventional single iteration methods; a similar method is already in use in our computation of proper from mean elements. Many of these methods are tested on a sample of asteroid orbits taken from the Themis family, up to the edge of the 2:1 resonance, and the dispersion of the values of the computed mean semimajor axis over 100 000 years is used as quality control. The results of these tests indicate that the iterative methods are superior, in this specific application, to the Breiter methods, in accuracy and reliability. This is understood as the result of the cancellations occurring between second order perturbation terms: the incomplete second order theory, resulting from the use of a Breiter method with the first order determining function only, can be less accurate than complete, fixed frequency theories of the first order. We have therefore computed new catalogues of asteroid mean and proper elements, incorporating an iterative algorithm in both steps (osculating to mean and mean to proper elements). This new data set, significantly more reliable even in the previously degraded regions of Themis and Cybele, is in the public domain. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

14.
Phenomena of bifurcation in hydrodynamic stellar models of radial pulsation are reviewed. By changing control parameters of models, we can see qualitatively different pulsation behaviors in hydrodynamic models with transitions due to various types of bifurcation.In weakly dissipative models (classical Cepheids), the bifurcation is induced by modal resonances. Two types of the modal resonances found in models are discussed: The higherharmonic resonances of the second overtone mode in the fundamental mode pulsator and of the fourth overtone mode in the first overtone pulsator are relevant to observations. The subharmonic resonance between the fundamental and first overtone modes is confirmed in classical Cepheid models.In strongly dissipative models (less-massive supergiant stars), the bifurcation of nonlinear pulsation is induced by the hydrodynamics of ionization zones as well as modal resonances. The sequence of the bifurcation sometimes leads to chaotic behaviors in nonlinear pulsation. The transition routes from regular to the chaotic pulsations found in models are discussed with respect to the theory of chaos in simple dynamical systems: The cascade of period-doubling bifurcation is confirmed to cause chaotic pulsation in W Virginis models. For models of higher luminosity, the tangent bifurcation is found to lead intermittent chaos.Finally, hydrodynamic models for chaotic pulsation with small amplitudes observed in the post-AGB stars are briefly discussed.  相似文献   

15.
Two applications of von Zeipel's method to the stellar three-body problem eliminate the short period terms and establish two new integrals of the motion beyond the classical integrals. The remaining time averaged problem with only the second order Hamiltonian has one additional integral and can be solved. The motion with the third order averaged Hamiltonian included is more complex, in that there may be additional resonances, and the additional integral does not exist in all cases.  相似文献   

16.
This is the second of a series of papers aimed to look for an explanation on the generation of high frequency quasi-periodic oscillations (QPOs) in accretion disks around neutron star, black hole, and white dwarf binaries. The model is inspired by the general idea of a resonance mechanism in the accretion disk oscillations as was already pointed out by Abramowicz and Klu’zniak (2001). In a first paper (P'etri, 2005a, paper I), we showed that a rotating misaligned magnetic field of a neutron star gives rise to some resonances close to the inner edge of the accretion disk. In this second paper, we suggest that this process does also exist for an asymmetry in the gravitational potential of the compact object. We prove that the same physics applies, at least in the linear stage of the response to the disturbance in the system. This kind of asymmetry is well suited for neutron stars or white dwarfs possessing an inhomogeneous interior allowing for a deviation from a perfectly spherically symmetric gravitational field. After a discussion on the magnitude of this deformation applied to neutron stars, we show by a linear analysis that the disk initially in a cylindrically symmetric stationary state is subject to {three kinds of resonances: a corotation resonance, a Lindblad resonance due to a driven force and a parametric resonance}. In a second part, we focus on the linear response of a thin accretion disk in the 2D limit. {Waves are launched at the aforementioned resonance positions and propagate in some permitted regions inside the disk, according to the dispersion relation obtained by a WKB analysis}. In a last part, these results are confirmed and extended via non linear hydrodynamical numerical simulations performed with a pseudo-spectral code solving Euler's equations in a 2D cylindrical coordinate frame. {We found that for a weak potential perturbation, the Lindblad resonance is the only effective mechanism producing a significant density fluctuation}. In a last step, we replaced the Newtonian potential by the so called logarithmically modified pseudo-Newtonian potential in order to take into account some general-relativistic effects like the innermost stable circular orbit (ISCO). The latter potential is better suited to describe the close vicinity of a neutron star or a black hole. However, from a qualitative point of view, the resonance conditions remain the same. The highest kHz QPOs are then interpreted as the orbital frequency of the disk at locations where the response to the resonances are maximal. It is also found that strong gravity is not required to excite the resonances.  相似文献   

17.
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   

18.
The Vishniac instability is supposed to explain the fragmentation of the thin shell of shocked matter in the radiative phase of supernova remnants. However its implication and its consequence on the morphological evolution of stellar systems is not fully demonstrated. The present paper tackles this subject by numerical simulations and focus on the role of the adiabatic index in the instability growth. The HYDRO-MUSCL 2D hydrodynamics code has been used to simulate the evolution of a supernova remnant thin shell and the triggering of the Vishniac instability in this thin shell. We have studied the temporal behavior of the perturbation. The first result of the numerical study is the existence of the Vishniac instability in the simulations. This result is proved by the overstability process observed in the simulations as predicted by the theoretical analysis. The second important result is the damping of the perturbation at late evolution and for all the set of parameters. Indeed the accretion of matter onto the shock damps the instability when theoretical analysis predicts its occurrence.  相似文献   

19.
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20.
The present paper reviews the Nekhoroshev theorem from the point of view of physicists and astronomers. We point out that Nekhoroshev result is strictly connected with the existence of a specific structure of the phase space, the existence of which can be checked with several numerical tools. This is true also for a degenerate system such as the one describing the motion of an asteroid in the so called main belt. The main difference is that in some parts of the belt, the Nekhoroshev result cannot apply a priori. Mean motion resonances of order smaller than the logarithm of the mass of Jupiter and first order secular resonances must be excluded. In the remaining parts, conversely, the Nekhoroshev theorem can be proved, provided someparameters, such as the masses, the eccentricities and the inclinations of the planets are small enough. At the light of this result, a massive campaign of numerical integrations of real and fictitious asteroids should allow to understand which is the real dynamical structure of the asteroid belt.  相似文献   

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