Co-Orbital Motion with Slowly Varying Parameters

We consider the dynamics of a test particle co-orbital with a satellite of mass m _s which revolves around a planet of mass M ₀ m _s with a mean motion n _s and semi-major axis a _s. We study the long term evolution of the particle motion under slow variations of (1) the mass of the primary, M ₀, (2) the mass of the satellite, m _s and (3) the specific angular momentum of the satellite J _s. The particle is not restricted to small harmonic oscillations near L ₄ or L ₅, and may have any libration amplitude on tadpole or horseshoe orbits. In a first step, no torque is applied to the particle, so that its motion is described by a Hamiltonian with slowly varying parameters. We show that the torque applied to the satellite, as measured by _s = j_s/(n _s J _s) induces an distortion of the phase space which is entirely described by an asymmetry coefficient = _s/, where = m _s/M. The adiabatic invariance of action implies furthermore that the long term evolution of the particle co-orbital motion depends only on the variation of m _s a _s with time. Applying a constant torque to the particle, as measured by _s = j_s/(n _s J _p) is then merely equivalent to replacing = _s/ by = (_s – _p)/. However, if the torque acting on the particle exhibits a radial gradient, then the action is no more conserved and the evolution of the particle orbit is no more controlled by m _s a _s only. We show that even mild torque gradients can dominate the orbital evolution of the particle, and eventually decide whether the latter will be pulled towards the stable equilibrium points L ₄ or L ₅, or driven away from them. Finally, we show that when the co-orbital bodies are two satellites with comparable masses m ₁ and m ₂, we can reduce the problem to that of a test particle co-orbital with a satellite of mass m ₁ + m ₂. This new problem has then parameters varying at rates which are combinations, with appropriate coefficients, of the changes suffered by each satellite.     

A Perturbative Treatment of The Co-Orbital Motion

We develop a formalism of the non-singular evaluation of the disturbing function and its derivatives with respect to the canonical variables. We apply this formalism to the case of the perturbed motion of a massless body orbiting the central body (Sun) with a period equal to that of the perturbing (planetary) body. This situation is known as the co-orbital motion, or equivalently, as the 1/1 mean motion commensurability. Jupiter's Trojan asteroids, Earth's co-orbital asteroids (e.g., (3753) Cruithne, (3362) Khufu), Mars' co-orbital asteroids (e.g., (5261) Eureka), and some Jupiter-family comets are examples of the co-orbital bodies in our solar system. Other examples are known in the satellite systems of the giant planets. Unlike the classical expansions of the disturbing function, our formalism is valid for any values of eccentricities and inclinations of the perturbed and perturbing body. The perturbation theory is used to compute the main features of the co-orbital dynamics in three approximations of the general three-body model: the planar-circular, planar-elliptic, and spatial-circular models. We develop a new perturbation scheme, which allows us to treat cases where the classical perturbation treatment fails. We show how the families of the tadpole, horseshoe, retrograde satellite and compound orbits vary with the eccentricity and inclination of the small body, and compute them also for the eccentricity of the perturbing body corresponding to a largely eccentric exoplanet's orbit.This revised version was published online in October 2005 with corrections to the Cover Date.     

Analytical Approach to the Secular Behaviour of Exoplanetary Systems

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.     

Symplectic Mappings of Co-orbital Motion in the Restricted Problem of Three Bodies

The dynamics of co-orbital motion in the restricted three-body problem are investigated by symplectic mappings. Analytical and semi-numerical mappings have been developed and studied in detail. The mappings have been tested by numerical integration of the equations of motion. These mappings have been proved to be useful for a quick determination of the phase space structure reflecting the main characteristics of the dynamics of the co-orbital problem.     

Stability of Surface Motion on a Rotating Ellipsoid

The dynamical environment on the surface of a rotating, massive ellipsoid is studied, with applications to surface motion on an asteroid. The analysis is performed using a combination of classical dynamics and geometrical analysis. Due to the small sizes of most asteroids, their shapes tend to differ from the classical spheroids found for the planets. The tri-axial ellipsoid model provides a non-trivial approximation of the gravitational potential of an asteroid and is amenable to analytical computation. Using this model, we study some properties of motion on the surface of an asteroid. We find all the equilibrium points on the surface of a rotating ellipsoid and we show that the stability of these points is intimately tied to the conditions for a Jacobi or MacLaurin ellipsoid of equilibria. Using geometrical analysis we can define global constraints on motion as a function of shape, rotation rate, and density, we find that some asteroids should have accumulation of material at their ends, while others should have accumulation of surface material at their poles. This study has implications for motion of a rover on an asteroid, and for the distribution of natural material on asteroids, and for a spacecraft hovering over an asteroid.     

Numerical Instruments for the Analysis of Secular Dynamics of Exoplanetary Systems

A review is given of modern numerical methods for the analysis of resonant and chaotic dynamics: calculation of the Lyapunov characteristic exponents, the MEGNO method, and the maximum eccentricity method. These methods are used to construct stability diagrams for the planetary systems γ Cep, HD 196885, and HD 41004. The diagrams are analyzed to determine the most probable values taken by the orbital parameters of the exoplanets and obtain estimates for the Lyapunov time of their orbital dynamics. The stability diagrams constructed using the different methods are compared to analyze their effectiveness in the study of secular dynamics of exoplanetary systems.     

Resonant Periodic Motion and the Stability of Extrasolar Planetary Systems

Families of nearly circular periodic orbits of the planetary type are studied, close to the 3/1 mean motion resonance of the two planets, considered both with finite masses. Large regions of instability appear, depending on the total mass of the planets and on the ratio of their masses.Also, families of resonant periodic orbits at the 2/1 resonance have been studied, for a planetary system where the total mass of the planets is the 4% of the mass of the sun. In particular, the effect of the ratio of the masses on the stability is studied. It is found that a planetary system at this resonance is unstable if the mass of the outer planet is smaller than the mass of the inner planet.Finally, an application has been made for the stability of the observed extrasolar planetary systems HD82943 and Gliese 876, trapped at the 2/1 resonance.     

Rotational Dynamics and Evolution of Planetary Satellites in the Solar and Exoplanetary Systems

Solar System Research - In this review we consider the main rotation regimes that are inherent for planetary satellites of the Solar System, satellites of trans-Neptunian objects, and potential...     

A Symplectic Mapping Model for the Study of 2:3 Resonant Trans-Neptunian Motion

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.     

热海王星系统HD 106315的近$2:1$ 平运动共振捕获与轨道演化

通过结合理论分析和数值模拟方法,可以对热海王星系统HD 106315轨道迁移中的近2:1平运动共振捕获机制以及潮汐作用下的演化过程进行研究.在轨道迁移阶段,初始轨道半长径、初始偏心率以及行星c的偏心率衰减系数K会对系统轨道构型产生影响.数值模拟结果显示当初始轨道半长径分别为ab～0.4 au、ac～0.8 au,偏心率eb和ec均小于0.03时, HD 106315b和HD 106315c在中央恒星的引力作用以及原行星盘粘滞作用下向内迁移, 65000 yr左右两颗行星均可迁移至当前观测位置附近并形成近2:1平运动共振捕获.此外,中央恒星的潮汐效应也可能会对行星系统共振构型产生影响,理论分析表明当行星潮汐耗散系数Q=100时,潮汐效应造成的轨道半长径衰减使系统轨道周期比发生的变化可能是系统脱离共振构型的原因.数值模拟结果显示, HD 106315系统内两颗行星Q103时,来自中央恒星的潮汐效应并不会使行星系统产生明显的偏心率和轨道半长径衰减,不足以使HD 106315行星系统在剩余寿命内脱离2:1平运动共振轨道构型.     

Stability Regions and Quasi-Periodic Motion in the Vicinity of Triangular Equilibrium Points

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 (L^s ₄) and long (L^l ₄) period solutions. Then, for two values of μ greater than the Routh value, the unified family L^sl ₄, to which, in these cases, L^s ₄ and L^l 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.     

Invariant Tori in the Secular Motions of the Three-body Planetary Systems

We consider the problem of the applicability of KAM theorem to a realistic problem of three bodies. In the framework of the averaged dynamics over the fast angles for the Sun–Jupiter–Saturn system we can prove the perpetual stability of the orbit. The proof is based on semi-numerical algorithms requiring both explicit algebraic manipulations of series and analytical estimates. The proof is made rigorous by using interval arithmetics in order to control the numerical errors.     

银河系中球状星团的空间运动

球状星团是银河系中最古老的天体类型之一,其累积光度很大,是银晕中重要的示踪天体。已以发现的银河系球状星团有１４０多个,其中１２０个银心距Ｒ〈４０Ｋｐｃ的星团已被准确地测定了视向速度。根据结数据以及球状星团金属度的统计分析,可以把球状星团次系再进一步分成某些不同的族群。目前已经测定过绝对自行的球状星团只有３８个,尽管这些自行的精度比视向速度和距离的精度差很多,然而,由此可以得出三维的空间速度,在统计     

On the Stability of the Terrestrial Planets as Models for Exosolar Planetary Systems

All results, achieved up to now, show the long term stability of our planetary system, although, especially the inner solar system is chaotic, due to some specific secular resonances. We study, by means of numerical integrations, the dynamical evolution of the planetary system where we concentrate on the stability of motion of the terrestrial planets Venus, Earth and Mars. Our model consists of a simplified planetary system with the inner planets Venus, Earth and Mars as well as Jupiter and Saturn. A mass factor was introduced to uniformly change the masses of the terrestrial planets; Jupiter and Saturn were involved in the system with their actual masses. We integrated the equations of motion with a Lie-integration method for a time interval of 10⁷ years. It turned out that when 220 < < 245 and > 250 the system became unstable due to the strong interactions between the planets. We discuss the model planetary systems for small mass-factors 0.5 10 and large ones 160 270 with the aid of several different numerical tools. These results can be applied to recently discovered exoplanetary systems, which configuration is comparable to our own.     

Stability of Hamiltonian Systems with Three Degrees of Freedom and the Three Body-Problem

Results are obtained about formal stability and instability of Hamiltonian systems with three degrees of freedom, two equal frequencies and the matrix of the linear part is not diagonalizable, in terms of the coefficients of the development in Taylor series of the Hamiltonian of the system. The results are applied to the study of stability of the Lagrangian solutions of the Three Body-Problem in the case in which the center of mass is over the curve ρ_*, on the border of the region of linear stability of Routh. The curve ρ_* is divided symmetrically in three arcs in such a way that if the center of mass of the three particles lies on the central arc, the Lagrangian solution is unstable in the sense of Liapunov (in finite order), while if the center of mass determines one point that lies on one of the other two arcs of ρ_*, then the Lagrangian solution is formally stable.     

中国大陆地壳运动的背景场

利用中国大陆和周围地区的12个IGS站近3年（1995年5月-1998年3月）的GPS观测资料,确定了这些IGS站的水平位移速率,并用其研究中国大陆地壳构造运动的背景,得到了具有重要意义的推论。     

Regular and Chaotic Motion in Globular Clusters

As a first step towards a comprehensive investigation of stellar motions within globular clusters, we present here the results of a study of stellar orbits in a mildly triaxial globular cluster that follows a circular orbit inside a galaxy. The stellar orbits were classified using the frequency analysis code of Carpintero and Aguilar and, as a check, the Liapunov characteristic exponents were also computed in some cases. The orbit families were obtained using different start spaces. Chaotic orbits turn out to be very common and while, as could be expected, they are particularly abundant in the outer parts of the cluster, they are still significant in the innermost regions. Their relevance for the structure of the cluster is discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Long-Term Stability Analysis of Quasi Integrable Degenerate Systems Through the Spectral Formulation of the Nekhoroshev Theorem

We describe a numerical application of the Nekhoroshev theorem to investigate the long-term stability of quasi-integrable systems. We extend the results of a previous paper to a class of degenerate systems, which are typical in celestial mechanics.     

视超光速运动与加速模型

用相对论加速喷流模型对48个具有视超光速的射电源进行了分析，结果不但支持流行的喷流模型而且说明加速模型是合理的。     

A New Apsidal Motion Study of DI Her

B and V photometry of the eccentric eclipsing binary DI Her is reported, with the aim of contributing to a more accurate determination of the timings required for the ongoing apsidal motion studies. The resulting apsidal motion rate is close to the one reported earlier by Khodykin and Volkov (1989), Guinan, Marshal and Maloney (IBVS 4101), and Yildiz et al. (2000). This revised version was published online in July 2006 with corrections to the Cover Date.