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1.
We use a three dimensional generalization of Szebehely’s invariant relation obtained by us (Makó and Szenkovits, Celest. Mech.
Dyn. Astron. 90, 51, 2004) in the elliptic restricted three-body problem, to establish more accurate criterion of the Hill stability. By
using this criterion, the Hill stability of four extrasolar planets (γ Cephei Ab, Gliese 86 Ab, HD 41004 Ab and HD 41004 Bb) is investigated. 相似文献
2.
We study the capture and crossing probabilities in the 3:1 mean motion resonance with Jupiter for a small asteroid that migrates from the inner to the middle Main Belt under the action of the Yarkovsky effect. We use an algebraic mapping of the averaged planar restricted three-body problem based on the symplectic mapping of Hadjidemetriou (Celest Mech Dyn Astron 56:563–599, 1993), adding the secular variations of the orbit of Jupiter and non-symplectic terms to simulate the migration. We found that, for fast migration rates, the captures occur at discrete windows of initial eccentricities whose specific locations depend on the initial resonant angles, indicating that the capture phenomenon is not probabilistic. For slow migration rates, these windows become narrower and start to accumulate at low eccentricities, generating a region of mutual overlap where the capture probability tends to 100 %, in agreement with the theoretical predictions for the adiabatic regime. Our simulations allow us to predict the capture probabilities in both the adiabatic and non-adiabatic cases, in good agreement with results of Gomes (Celest Mech Dyn Astron 61:97–113, 1995) and Quillen (Mon Not RAS 365:1367–1382, 2006). We apply our model to the case of the Vesta asteroid family in the same context as Roig et al. (Icarus 194:125–136, 2008), and found results indicating that the high capture probability of Vesta family members into the 3:1 mean motion resonance is basically governed by the eccentricity of Jupiter and its secular variations. 相似文献
3.
Alessandra Celletti Daniel J. Scheeres 《Celestial Mechanics and Dynamical Astronomy》2017,127(1):1-18
We revisit the relegation algorithm by Deprit et al. (Celest. Mech. Dyn. Astron. 79:157–182, 2001) in the light of the rigorous Nekhoroshev’s like theory. This relatively recent algorithm is nowadays widely used for implementing closed form analytic perturbation theories, as it generalises the classical Birkhoff normalisation algorithm. The algorithm, here briefly explained by means of Lie transformations, has been so far introduced and used in a formal way, i.e. without providing any rigorous convergence or asymptotic estimates. The overall aim of this paper is to find such quantitative estimates and to show how the results about stability over exponentially long times can be recovered in a simple and effective way, at least in the non-resonant case. 相似文献
4.
Families of Periodic Orbits Emanating From Homoclinic Orbits in the Restricted Problem of Three Bodies 总被引：2，自引：1，他引：1
We describe and comment the results of a numerical exploration on the evolution of the families of periodic orbits associated
with homoclinic orbits emanating from the equilateral equilibria of the restricted three body problem for values of the mass
ratio larger than μ
1. This exploration is, in some sense, a continuation of the work reported in Henrard [Celes. Mech. Dyn. Astr. 2002, 83, 291]. Indeed it shows how, for values of μ. larger than μ
1, the Trojan web described there is transformed into families of periodic orbits associated with homoclinic orbits. Also we describe how families
of periodic orbits associated with homoclinic orbits can attach (or detach) themselves to (or from) the best known families
of symmetric periodic orbits.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
5.
Anne-Sophie Libert Jacques Henrard 《Celestial Mechanics and Dynamical Astronomy》2005,93(1-4):187-200
We analyse the secular interactions of two coplanar planets which are not in mean motion resonances. The analysis is based
on a high order (order 12) expansion of the perturbative potential in powers of the eccentricities. The model depends on only
two parameters (the ratio of semi-major axis and the mass ratio of the planets) and can be reduced to a one degree of freedom
system, allowing for an exhaustive parametric analysis. Following Pauwels [Pauwels T.: 1983, Celet. Mech. & Dyn. Astro. 30, 229–247] we map the phase space on a sphere, avoiding in this way the artificial singularities introduced by other mappings.
We show that the 12 order expansion is able to describe correctly most of the exosolar planetary systems discovered so far,
even if the eccentricities of these planets are considerably larger than the eccentricities of our own solar system. The expansion
is even able to reproduce, at moderate eccentricities, the secular resonances discovered numerically by Michtchenko and Malhotra
[Michtchenko, T. A. and Malhotra, R.: 2004, Icarus 168, 237–248] at moderate to large eccentricities.
FNRS Research Fellow. 相似文献
6.
Jacques Laskar 《Celestial Mechanics and Dynamical Astronomy》2005,91(3-4):351-356
Recently, Breiter et al. [Celest. Mech. Dyn. Astron., 2004, 88, 153–161] reported the computation of Hansen coefficients X
k
γ ,m
for non-integer values of γ. In fact, the Hansen coefficients are closely related to the Laplace b
s
(m), and generalized Laplace coefficients b
s,r
(m) [Laskar and Robutel, 1995, Celest. Mech. Dyn. Astron., 62, 193–217] that do not require s,r to be integers. In particular, the coefficients X
0
γ ,m
have very simple expressions in terms of the usual Laplace coefficients b
γ +2
(m), and all their properties derive easily from the known properties of the Laplace coefficients. 相似文献
7.
In this paper we have examined the stability of triangular libration points in the restricted problem of three bodies when the bigger primary is an oblate spheroid. Here we followed the time limit and computational process of Tuckness (Celest. Mech. Dyn. Mech. 61, 1–19, 1995) on the stability criteria given by McKenzie and Szebehely (Celest. Mech. 23, 223–229, 1981). In this study it was found that in comparison to other studies the value of the critical mass μ c has been reduced due to oblateness of the bigger primary, i.e. the range of stability of the equilateral triangular libration points reduced with the increase of the oblateness parameter I and hence the order of commensurability was increased. 相似文献
8.
Alberto Escapa 《Celestial Mechanics and Dynamical Astronomy》2011,110(2):99-142
We explore the evolution of the angular velocity of an elastic Earth model, within the Hamiltonian formalism. The evolution
of the rotation state of the Earth is caused by the tidal deformation exerted by the Moon and the Sun. It can be demonstrated
that the tidal perturbation to spin depends not only upon the instantaneous orientation of the Earth, but also upon its instantaneous
angular velocity. Parameterizing the orientation of the Earth figure axis with the three Euler angles, and introducing the
canonical momenta conjugated to these, one can then show that the tidal perturbation depends both upon the angles and the
momenta. This circumstance complicates the integration of the rotational motion. Specifically, when the integration is carried
out in terms of the canonical Andoyer variables (which are the rotational analogues to the orbital Delaunay variables), one
should keep in mind the following subtlety: under the said kind of perturbations, the functional dependence of the angular
velocity upon the Andoyer elements differs from the unperturbed dependence (Efroimsky in Proceedings of Journées 2004: Systèmes
de référence spatio-temporels. l’Observatoire de Paris, pp 74–81, 2005; Efroimsky and Escapa in Celest. Mech. Dyn. Astron. 98:251–283, 2007). This happens because, under angular velocity dependent perturbations, the requirement for the Andoyer elements to be canonical
comes into a contradiction with the requirement for these elements to be osculating, a situation that parallels a similar
antinomy in orbital dynamics. Under the said perturbations, the expression for the angular velocity acquires an additional
contribution, the so called convective term. Hence, the time variation induced on the angular velocity by the tidal deformation
contains two parts. The first one comes from the direct terms, caused by the action of the elastic perturbation on the torque-free
expressions of the angular velocity. The second one arises from the convective terms. We compute the variations of the angular
velocity through the approach developed in Getino and Ferrándiz (Celest. Mech. Dyn. Astron. 61:117–180, 1995), but considering the contribution of the convective terms. Specifically, we derive analytical formulas that determine the
elastic perturbations of the directional angles of the angular velocity with respect to a non-rotating reference system, and
also of its Cartesian components relative to the Tisserand reference system of the Earth. The perturbation of the directional
angles of the angular velocity turns out to be different from the evolution law found in Kubo (Celest. Mech. Dyn. Astron.
105:261–274, 2009), where it was stated that the evolution of the angular velocity vector mimics that of the figure axis. We investigate comprehensively
the source of this discrepancy, concluding that the difference between our results and those obtained in Ibid. stems from an oversimplification made by Kubo when computing the direct terms. Namely, in his computations Kubo disregarded
the motion of the tide raising bodies with respect to a non-rotating reference system when compared with the Earth rotational
motion. We demonstrate that, from a numerical perspective, the convective part provides the principal contribution to the
variation of the directional angles and of length of day. In the case of the x and y components in the Tisserand system, the convective contribution is of the same order of magnitude as the direct one. Finally,
we show that the approximation employed in Kubo (Ibid.) leads to significant numerical differences at the level of a hundred micro-arcsecond. 相似文献
9.
A symplectic mapping for Trojan-type motion has been developed in the secularly changing elliptic restricted three-body problem. The mapping describes well the characteristics of Trojan-type dynamics at small eccentricities. By using this mapping the boundary of the stability region has been studied for different values of the initial eccentricities of hypothetical Jupiter's Trojans. It has been found that in the secularly changing elliptic case the chaotic diffusion at the border of the stability region is stronger than simply in the elliptic case. An explanation of this observation might be the destruction of the chain of islands of the 13:1 secondary resonance between the short and long period component of the Trojan-like motion, caused possibly by the indirect perturbations of Saturn. 相似文献
10.
Jack Wisdom 《Celestial Mechanics and Dynamical Astronomy》1986,38(2):175-180
The solution by Sessin and Ferraz-Mello (Celes. Mech.
32, 307–332) of the Hori auxiliary system for the motion of two planets with periods nearly commensurate in the ratio 21 is considerably simplified by the introduction of canonical variables. An analogous canonical transformation simplifies the elliptic restricted problem. 相似文献
11.
We revisit a set of symplectic variables introduced by Andre Deprit (Celest Mech 30, 181–195, 1983), which allows for a complete symplectic reduction in rotation invariant Hamiltonian systems, generalizing to arbitrary dimension
Jacobi’s reduction of the nodes. In particular, we introduce an action-angle version of Deprit’s variables, connected to the
Delaunay variables, and give a new hierarchical proof of the symplectic character of Deprit’s variables. 相似文献
12.
K. G. Hadjifotinou John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》2002,84(2):135-157
A symplectic mapping is constructed for the study of the dynamical evolution of Edgeworth-Kuiper belt objects near the 2:3 mean motion resonance with Neptune. The mapping is six-dimensional and is a good model for the Poincaré map of the real system, that is, the spatial elliptic restricted three-body problem at the 2:3 resonance, with the Sun and Neptune as primaries. The mapping model is based on the averaged Hamiltonian, corrected by a semianalytic method so that it has the basic topological properties of the phase space of the real system both qualitatively and quantitatively. We start with two dimensional motion and then we extend it to three dimensions. Both chaotic and regular motion is observed, depending on the objects' initial inclination and phase. For zero inclination, objects that are phase-protected from close encounters with Neptune show ordered motion even at eccentricities as large as 0.4 and despite being Neptune-crossers. On the other hand, not-phase-protected objects with eccentricities greater than 0.15 follow chaotic motion that leads to sudden jumps in their eccentricity and are removed from the 2:3 resonance, thus becoming short period comets. As inclination increases, chaotic motion becomes more widespread, but phase-protection still exists and, as a result, stable motion appears for eccentricities up to e = 0.3 and inclinations as high as i = 15°, a region where plutinos exist. 相似文献
13.
Rocío Isabel Páez Christos Efthymiopoulos 《Celestial Mechanics and Dynamical Astronomy》2018,130(2):20
One of the most interesting features in the libration domain of co-orbital motions is the existence of secondary resonances. For some combinations of physical parameters, these resonances occupy a large fraction of the domain of stability and rule the dynamics within the stable tadpole region. In this work, we present an application of a recently introduced ‘basic Hamiltonian model’ \(H_\mathrm{b}\) for Trojan dynamics (Páez and Efthymiopoulos in Celest Mech Dyn Astron 121(2):139, 2015; Páez et al. in Celest Mech Dyn Astron 126:519, 2016): we show that the inner border of the secondary resonance of lowermost order, as defined by \(H_\mathrm{b}\), provides a good estimation of the region in phase space for which the orbits remain regular regardless of the orbital parameters of the system. The computation of this boundary is straightforward by combining a resonant normal form calculation in conjunction with an ‘asymmetric expansion’ of the Hamiltonian around the libration points, which speeds up convergence. Applications to the determination of the effective stability domain for exoplanetary Trojans (planet-sized objects or asteroids) which may accompany giant exoplanets are discussed. 相似文献
14.
Jacques Henrard 《Celestial Mechanics and Dynamical Astronomy》2005,93(1-4):101-112
In a previous paper (The Rotation of Europa, Henrard, Celest. Mech. Dyn. Astr., 91, 131–149, 2005) we have developed a semi-analytical theory of Europa, one of the Galilean satellites of Jupiter. It is based
on a synthetic theory of the orbit of Europa and is developed in the framework of Hamiltonian formalism. It was assumed that
Europa is a rigid body and Jupiter a point mass. Several additional effects should be investigated in order to complete the
theory. The present contribution considers the effect of the shape of Jupiter and of the gravitational pull of Io. The sensitivity
of the main theory to a change in the values of the moments of inertia of Europa is also considered. 相似文献
15.
D. Viswanath 《Celestial Mechanics and Dynamical Astronomy》2006,94(2):213-235
The restricted three-body problem describes the motion of a massless particle under the influence of two primaries of masses
1− μ and μ that circle each other with period equal to 2π. For small μ, a resonant periodic motion of the massless particle
in the rotating frame can be described by relatively prime integers p and q, if its period around the heavier primary is approximately 2π p/q, and by its approximate eccentricity e. We give a method for the formal development of the stable and unstable manifolds associated with these resonant motions.
We prove the validity of this formal development and the existence of homoclinic points in the resonant region. In the study
of the Kirkwood gaps in the asteroid belt, the separatrices of the averaged equations of the restricted three-body problem
are commonly used to derive analytical approximations to the boundaries of the resonances. We use the unaveraged equations
to find values of asteroid eccentricity below which these approximations will not hold for the Kirkwood gaps with q/p equal to 2/1, 7/3, 5/2, 3/1, and 4/1. Another application is to the existence of asymmetric librations in the exterior resonances.
We give values of asteroid eccentricity below which asymmetric librations will not exist for the 1/7, 1/6, 1/5, 1/4, 1/3,
and 1/2 resonances for any μ however small. But if the eccentricity exceeds these thresholds, asymmetric librations will exist
for μ small enough in the unaveraged restricted three-body problem. 相似文献
16.
Michael Nauenberg 《Celestial Mechanics and Dynamical Astronomy》2007,97(1):1-15
Numerical solutions are presented for a family of three dimensional periodic orbits with three equal masses which connects
the classical circular orbit of Lagrange with the figure eight orbit discovered by C. Moore [Moore, C.: Phys. Rev. Lett. 70, 3675–3679 (1993); Chenciner, A., Montgomery, R.: Ann. Math. 152, 881–901 (2000)]. Each member of this family is an orbit with finite angular momentum that is periodic in a frame which rotates
with frequency Ω around the horizontal symmetry axis of the figure eight orbit. Numerical solutions for figure eight shaped
orbits with finite angular momentum were first reported in [Nauenberg, M.: Phys. Lett. 292, 93–99 (2001)], and mathematical proofs for the existence of such orbits were given in [Marchal, C.: Celest. Mech. Dyn. Astron.
78, 279–298 (2001)], and more recently in [Chenciner, A. et al.: Nonlinearity 18, 1407–1424 (2005)] where also some numerical solutions have been presented. Numerical evidence is given here that the family
of such orbits is a continuous function of the rotation frequency Ω which varies between Ω = 0, for the planar figure eight
orbit with intrinsic frequency ω, and Ω = ω for the circular Lagrange orbit. Similar numerical solutions are also found for
n > 3 equal masses, where n is an odd integer, and an illustration is given for n = 21. Finite angular momentum orbits were also obtained numerically for rotations along the two other symmetry axis of the
figure eight orbit [Nauenberg, M.: Phys. Lett. 292, 93–99 (2001)], and some new results are given here. A preliminary non-linear stability analysis of these orbits is given
numerically, and some examples are given of nearby stable orbits which bifurcate from these families. 相似文献
17.
Report of the IAU/IAG Working Group on Cartographic Coordinates and Rotational Elements: 2003 总被引：1，自引：1，他引：0
P. K. Seidelmann B. A. Archinal M. F. A’Hearn D. P. Cruikshank J. L. Hilton H. U. Keller J. Oberst J. L. Simon P. Stooke D. J. Tholen P. C. Thomas 《Celestial Mechanics and Dynamical Astronomy》2005,91(3-4):203-215
Every three years the IAU/IAG Working Group on Cartographic Coordinates and Rotational Elements revises tables giving the
directions of the north poles of rotation and the prime meridians of the planets, satellites, and asteroids. This report introduces
a system of cartographic coordinates for asteroids and comets. A topographic reference surface for Mars is recommended. Tables
for the rotational elements of the planets and satellites and size and shape of the planets and satellites are not included,
since there were no changes to the values. They are available in the previous report (Celest. Mech. Dyn. Astron., 82, 83–110, 2002), a version of which is also available on a web site. 相似文献
18.
Kinoshita and Nakai (Celest. Mech. Dyn. Astr. 75, 125–147, 1999) gave the analytical solution of the Kozai mechanism. In this
solution the eccentricity and the inclination of a disturbed body take any value, but the argument of the pericenter is restricted
to take 0° or 90°. In this paper, we derive the general solution that can be applied for any value of the argument of the
pericenter. 相似文献
19.
Analysis of periodic orbits in the Saturn-Titan system using the method of Poincare section surfaces
We explore the periodic orbits and the regions of quasi-periodic motion around both the primaries in the Saturn-Titan system
in the framework of planar circular restricted three-body problem. The location, nature and size of periodic and quasi-periodic
orbits are studied using the numerical technique of Poincare surface of sections. The maximum amplitude of oscillations about
the periodic orbits is determined and is used as a parameter to measure the degree of stability in the phase space for such
orbits. It is found that the orbits around Saturn remain around it and their stability increases with the increase in the
value of Jacobi constant C. The orbits around Titan move towards it with the increase in C. At C=3.1, the pericenter and apocenter are 358.2 and 358.5 km, respectively. No periodic or quasi-periodic orbits could be found
by the present method around the collinear Lagrangian point L
1 (0.9569373834…). 相似文献
20.
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