任意倾角的共轨运动研究

共轨运动天体与摄动天体的半长径相同,处于1:1平运动共振中.太阳系内多个行星的特洛伊天体即为处于蝌蚪形轨道的共轨运动天体,其中一些高轨道倾角特洛伊天体的轨道运动与来源仍未被完全理解.利用一个新发展的适用于处理1:1平运动共振的摄动函数展开方式,对三维空间中的共轨运动进行考察,计算不同初始轨道根数情况下共轨轨道的共振中心、共振宽度,分析轨道类型与初始轨道根数的关系.并将分析方法所得结果与数值方法的结果相互比较验证,得到了广阔初始轨道根数空间内共轨运动的全局图景.     

太阳质量损失对流星群轨道改变的长期影响

将作者在变质量天体力学所得理论结果应用于太阳质量损失对流星群轨道根数变化的长期效应上。太阳质量损失包括光子辐射和太阳风造成的质量损失。利用G—M型变质量天体轨道根数变化方程的一阶和二阶解对15个流星群轨道半长轴、近日点距离、轨道周期和近日点经度因太阳质量损失造成的每世纪的长期改变效应做了数值计算,并得出计算结果。其计算结果表明,太阳质量损失使流星群轨道半长轴每世纪的改变效应较明显,它们同太阳距离的扩大影响值得关注,但对轨道周期的拉长每世纪的影响甚小,对近日点经度只有量级变化小到可以略而不计。     

Detailed survey of the phase space around Nix and Hydra

We present a detailed survey of the dynamical structure of the phase space around the new moons of the Pluto–Charon system. The spatial elliptic restricted three-body problem was used as model and stability maps were created by chaos indicators. The orbital elements of the moons are in the stable domain on the semimajor axis, eccentricity and inclination spaces. The structures related to the 4:1 and 6:1 mean motion resonances are clearly visible on the maps. They do not contain the positions of the moons, confirming previous studies. We showed the possibility that Nix might be in the 4:1 resonance if its argument of pericentre or longitude of node falls in a certain range. The results strongly suggest that Hydra is not in the 6:1 resonance for arbitrary values of the argument of pericentre or longitude of node.     

Electromagnetic Radiation and Motion of a Particle

We consider the motion of uncharged dust grains of arbitrary shape including the effects of electromagnetic radiation and thermal emission. The resulting relativistically covariant equation of motion is expressed in terms of standard optical parameters. Explicit expressions for secular changes of osculating orbital elements are derived in detail for the special case of the Poynting-Robertson effect. Two subcases are considered: (i) central acceleration due to gravity and the radial component of radiation pressure independent of the particle velocity, (ii) central acceleration given by gravity and the radiation force as the disturbing force. The latter case yields results which may be compared with secular orbital evolution in terms of orbital elements for an arbitrarily shaped dust particle. The effects of solar wind are also presented. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the stability of the satellites of asteroid 87 Sylvia

The triple asteroidal system (87) Sylvia is composed of a 280-km primary and two small moonlets named Romulus and Remus ( Marchis et al. 2005b ). Sylvia is located in the main asteroid belt, with semi-major axis of about 3.49 au, eccentricity of 0.08 and 11° of orbital inclination. The satellites are in nearly equatorial circular orbits around the primary, with orbital radius of about 1360 km (Romulus) and 710 km (Remus). In this work, we study the stability of the satellites Romulus and Remus. In order to identify the effects and the contribution of each perturber, we performed numerical simulations considering a set of different systems. The results from the three-body problem, Sylvia–Romulus–Remus, show no significant variation of their orbital elements. However, the inclinations of the satellites present a long-period evolution with amplitude of about 20° when the Sun is included in the system. Such amplitude is amplified to more than 50° when Jupiter is included. These evolutions are very similar for both satellites. An analysis of these results shows that Romulus and Remus are librating in a secular resonance and their longitude of the nodes are locked to each other. Further simulations show that the amplitude of oscillation of the satellites' inclination can reach higher values depending on the initial values of their longitude of pericentre. In those cases, the satellites get caught in an evection resonance with Jupiter, their eccentricities grow and they eventually collide with Sylvia. However, the orbital evolutions of the satellites became completely stable when the oblateness of Sylvia is included in the simulations. The value of Sylvia's J ₂ is about 0.17, which is very high. However, even just 0.1 per cent of this value is enough to keep the satellite's orbital elements with no significant variation.     

Periodic perturbations in close binary systems due to tidal distortion (non-lagging tides)

In a previous paper (Zafiropoulos and Kopal, 1982; hereafter referred to as Paper I) we have studies the effects of rotational distortion on the orbital elements. The aim of the present paper is to investigate the secular and periodic perturbations of the orbital elements due to tidal distortion. For tidal distortion when tides do not lag, the Gaussian form of Lagrange's planetary equations has been employed to yield the first- and second-order approximations. The results obtained include the effects produced by the second, third and fourth harmonic distortions. The first order approximation for non-lagging tides has been expressed by means of Hansen coefficients.     

Orbital YORP and asteroid orbit evolution, with application to Apophis

Photon thrust from shape alone can produce quasi-secular changes in an asteroid's orbital elements. An asteroid in an elliptical orbit with a north–south shape asymmetry can steadily alter its elements over timescales longer than one orbital trip about the Sun. This thrust, called here orbital YORP (YORP = Yarkovsky–O'Keefe–Radzievskii–Paddack), operates even in the absence of thermal inertia, which the Yarkovsky effects require. However, unlike the Yarkovsky effects, which produce secular orbital changes over millions or billions of years, the change in an asteroid's orbital elements from orbital YORP operates only over the precession timescale of the orbit or of the asteroid's spin axis; this is generally only thousands or tens of thousands of years. Thus while the orbital YORP timescale is too short for an asteroid to secularly journey very far, it is long enough to warrant investigation with respect to 99942 Apophis, which might conceivably impact the Earth in 2036. A near-maximal orbital YORP effect is found by assuming Apophis is without thermal inertia and is shaped like a hemisphere, with its spin axis lying in the orbital plane. With these assumptions orbital YORP can change its along-track position by up to ±245 km, which is comparable to Yarkovsky effects. Though Apophis' shape, thermal properties, and spin axis orientation are currently unknown, the practical upper and lower limits are liable to be much less than the ±245 km extremes. Even so, the uncertainty in position is still likely to be much larger than the 0.5 km “keyhole” Apophis must pass through during its close approach in 2029 in order to strike the Earth in 2036.     

An analytical solution for Kepler's problem

In this paper, we present a framework which provides an analytical (i.e. infinitely differentiable) transformation between spatial coordinates and orbital elements for the solution of the gravitational two-body problem. The formalism omits all singular variables which otherwise would yield discontinuities. This method is based on two simple real functions for which the derivative rules are only required to be known, all other applications – e.g. calculating the orbital velocities, obtaining the partial derivatives of radial velocity curves with respect to the orbital elements – are thereafter straightforward. As it is shown, the presented formalism can be applied to find optimal instants for radial velocity measurements in transiting explanatory systems to constrain the orbital eccentricity as well as to detect secular variations in the eccentricity or in the longitude of periastron.     

太阳多方模型对行星轨道要素变化的长期摄动影响

本文利用天体力学中的摄动理论和天体物理学中的气体星多方模型理论研究了太阳多方模型对行星轨道要素变化的长期摄动影响。文中给出了太阳日多方指数ｎ＝３的模型由于自转、扁度和内部密度分布等因素对行星轨道要素变化的长期摄动效应的理论结果。研究结果表明：行星轨道要素除长轴、偏心率和轨道倾角不受长期摄动外升交点经度、近日点经度以及平近点角均受长期摄动的影响。最后利用理论结果对行星轨道要素的长期投动效应做了数值计算，数值结果在表１中给出。     

Parameterized post-Newtonian orbital effects in extrasolar planets

Perturbative post-Newtonian variations of the standard osculating orbital elements are obtained by using the two-body equations of motion in the parameterized post-Newtonian theoretical framework. The results obtained are applied to the Einstein and Brans–Dicke theories. As a results, the semi-major axis and eccentricity exhibit periodic variation, but no secular changes. The longitude of periastron and mean longitude at epoch experience both secular and periodic shifts. The post-Newtonian effects are calculated and discussed for six extrasolar planets.     

Trojans in Stable Chaotic Motion

The orbits of 13 Trojan asteroids have been calculated numerically in the model of the outer solar system for a time interval of 100 million years. For these asteroids Milani et al. (1997) determined Lyapunov times less than 100 000 years and introduced the notion "asteroids in stable chaotic motion". We studied the dynamical behavior of these Trojan asteroids (except the asteroid Thersites which escaped after 26 million years) within 11 time intervals - i.e. subintervals of the whole time - by means of: (1) a numerical frequency analysis (2) the root mean square (r.m.s.) of the orbital elements and (3) the proper elements. For each time interval we compared the root mean squares of the orbital elements (a, e and i) with the corresponding proper element. It turned out that the variations of the proper elements e_p in the different time intervals are correlated with the corresponding r.m.s.(e); this is not the case for sin I_p with r.m.s.(i). This revised version was published online in July 2006 with corrections to the Cover Date.     

Multiple resonance in one problem of the stability of the motion of a satellite relative to the center of mass

We investigate the stability of the periodic motion of a satellite, a rigid body, relative to the center of mass in a central Newtonian gravitational field in an elliptical orbit. The orbital eccentricity is assumed to be low. In a circular orbit, this periodic motion transforms into the well-known motion called hyperboloidal precession (the symmetry axis of the satellite occupies a fixed position in the plane perpendicular to the radius vector of the center of mass relative to the attractive center and describes a hyperboloidal surface in absolute space, with the satellite rotating around the symmetry axis at a constant angular velocity). We consider the case where the parameters of the problem are close to their values at which a multiple parametric resonance takes place (the frequencies of the small oscillations of the satellite’s symmetry axis are related by several second-order resonance relations). We have found the instability and stability regions in the first (linear) approximation at low eccentricities.     

Study of resonances for the restricted 3‐body problem

Our aim is to identify and classify mean‐motion resonances (MMRs) for the coplanar circular restricted three‐body problem (CR3BP) for mass ratios between 0.10 and 0.50. Our methods include the maximum Lyapunov exponent, which is used as an indicator for the location of the resonances, the Fast Fourier Transform (FFT) used for determining what kind of resonances are present, and the inspection of the orbital elements to classify the periodicity. We show that the 2:1 resonance occurs the most frequently. Among other resonances, the 3:1 resonance is the second most common, and furthermore both 3:2 and 5:3 resonances occur more often than the 4:1 resonance. Moreover, the resonances in the coplanar CR3BP are classified based on the behaviour of the orbits. We show that orbital stability is ensured for high values of resonance (i.e., high ratios) where only a single resonance is present. The resonances attained are consistent with the previously established resonances for the solar system, i.e., specifically, in regards to the asteroid belt. Previous work employed digital filtering and Lyapunov characteristic exponents to determine stochasticity of the eccentricity, which is found to be consistent with our usage of Lyapunov exponents as an alternate approach based on varying the mass ratio instead of the eccentricity. Our results are expected to be of principal interest to future studies, including augmentations to observed or proposed resonances, of extra‐solar planets in binary stellar systems (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linear stability and shape analysis of spinning three-craft Coulomb formations

This paper describes the discovery of families of multiple invariant shape solutions for collinear three-craft Coulomb formations with set charges, as well as the results of linear stability analysis on such formations. The charged spacecraft are assumed to be spinning about each other in deep space without relevant gravitational forces present. Up to three invariant shape solutions are possible for a single set of craft charges. This behavior, only speculated in previous work, is confirmed through analysis and numerical simulation examples. In fact, distinct regions are analytically described where two or three invariant shape solutions exist for a single charge set. These regions are analyzed to determine what range of trajectories are possible. Linear stability analysis for circular trajectories yields the first examples of marginally stable three-craft invariant shape formations. Linearly stable behavior is only observed when two invariant shape solutions result for one set of charges, where one shape will be unstable and the other marginally stable. Numerical simulation illustrates stability for ten orbital periods when perturbations are confined to the orbital plane. When out of plane motion is considered the shapes are found to be weakly unstable, though the out of plane motion appears to be decoupled from in plane motion to first order.     

On the detection of the Yarkovsky effect on near-Earth asteroids by means of Gaia

The Yarkovsky effect, which causes a slow drifting of the orbital elements (mainly the semimajor axis) of km-sized asteroids and meteors, is the weak non-gravitational force experienced by these bodies due to the emission of thermal photons. This effect is believed to play a role in the delivery of near-Earth asteroids (NEAs) from the main belt, in the spreading of the orbital elements of asteroid families, and in the orbital evolution of potentially hazardous asteroids.Here we present preliminary results of simulationing indicating that the perturbations induced by the Yarkovsky effect on the positions of some tens of NEAs can be observed by means of the high-precision astrometric observations that will be provided by the ESA mission Gaia.     

Stability of binary asteroids

The restricted problem of 2+2 bodies is applied to the study of the stability and dynamics of binary asteroids in the solar system. Numerical investigation of the behavior of the orbital elements and the maximal Lyapunov characteristic number of binary asteroids reveal extensive regions where bounded quasiperiodic motion is possible. These regions are compared to the bounded regions which are predicted by the classical restricted problem of three bodies. Regions of bounded chaotic solutions are also found.     

Orbital stability of systems of closely-spaced planets

An investigation of the stability of systems of 1 M_⊕ (Earth-mass) bodies orbiting a Sun-like star has been conducted for virtual times reaching 10 billion years. For the majority of the tests, a symplectic integrator with a fixed timestep of between 1 and 10 days was employed; however, smaller timesteps and a Bulirsch-Stoer integrator were also selectively utilized to increase confidence in the results. In most cases, the planets were started on initially coplanar, circular orbits, and the longitudinal initial positions of neighboring planets were widely separated. The ratio of the semimajor axes of consecutive planets in each system was approximately uniform (so the spacing between consecutive planets increased slowly in terms of distance from the star). The stability time for a system was taken to be the time at which the orbits of two or more planets crossed. Our results show that, for a given class of system (e.g., three 1 M_⊕ planets), orbit crossing times vary with planetary spacing approximately as a power law over a wide range of separation in semimajor axis. Chaos tests indicate that deviations from this power law persist for changed initial longitudes and also for small but non-trivial changes in orbital spacing. We find that the stability time increases more rapidly at large initial orbital separations than the power-law dependence predicted from moderate initial orbital separations. Systems of five planets are less stable than systems of three planets for a specified semimajor axis spacing. Furthermore, systems of less massive planets can be packed more closely, being about as stable as 1 M_⊕ planets when the radial separation between planets is scaled using the mutual Hill radius. Finally, systems with retrograde planets can be packed substantially more closely than prograde systems with equal numbers of planets.     

Dynamical evolution of the Oort cloud: I. A Monte Carlo simulation

We have studied the long-time dynamical evolution of a population of comets surrounding the Solar System at a large distance. Orbital changes are caused by random passing stars. We first emphasize the need for a new simulation because of the lack of completeness of previous analytical and numerical studies. Then the solar neighborhood is modeled by a sphere of 1 pc in radius, which stars cross at random in direction and distance. The geometry of the encounters allows us to compute the impulse gained by the star and the Sun, in the context of an impact approximation. Then we determine the change of orbital elements for a population of comets and follow the evolution of the frequency distribution for the five Keplerian elements. Clouds are selected in such a way that we test the two main hypotheses for the origin of the Oort cloud, and also the regions of stability in an aphelion-eccentricity diagram. We show that stellar perturbations randomize the cloud and prevent one from inferring the initial cloud configuration from the current distribution. Clouds are depleted by the diffusion of comets into the planetary regions, where they become planet-influenced comets or are ejected from the Solar System. The diffusion of aphelion toward interstellar regions proves to be the major source of cometary loss. Direct ejection to hyperbolic orbits amounts to 9% of the originally population over the age of the Solar System. Finally the current and original cloud populations are estimated at 1.8 × 10¹² and 2 × 10¹³ comets and we discuss these results.     

Magnetic braking in magnetic binary stars

The role of an external magnetic field in the magnetic braking of a star with a dipolar field is investigated. In a magnetic cataclysmic variable system (i.e. the primary compact star has a strong magnetic field), the field external to the braking star (a late-type main-sequence star with a dynamo-generated field) originates from the compact star. A closed field region — the system dead zone — is formed within the binary system, and it does not take part in magnetic braking. The overall braking rate depends on the extent of this region and of the open flux, and is dependent on centrifugal effects. In the case of two interacting dipoles, the dipole orientations relative to the spin axes and to each other are found to be important, leading to different amounts of open flux and therefore of magnetic braking, owing to different centrifugal effects on closed field regions. However, in circumstances consistent with observations and dynamo theory, the white dwarf's field reduces the magnetic braking of the secondary significantly, a finding qualitatively similar to the results previously obtained for two anti-aligned dipoles perpendicular to the orbital plane. In the cases where the two dipole axes are not perpendicular to the orbital plane, but are inclined in the plane that links them, the 'cut-off' in magnetic braking is less abrupt, and this effect is more obvious as the inclinations increase. Only in the extreme cases when the two dipole axes are aligned in the orbital plane does the braking increase with white dwarf field strength. We conclude that detailed evolutionary modelling of AM Herculis systems needs to take account of the inclination effect.