中心体自转对天体轨道要素变化的后牛顿效应

本文给出了在三种引力理论为中心自转对天体轨道要素变化产生的后牛顿摄动效应的研究结果。研究结果表明：六个轨道要素除长钾不受摄动影响外其它五个要素均有周期摄动，特别升交点经度和近星点经度还有长期摄动效应。最后将文中的理论结论同前人的工作做了比较还应用于行星自转对卫星轨道要素变化的摄动效应计算上。作者在文[1]中研究了天体轨道要素变化的后牛顿效应，但在该文中并没有考虑中心体自转的影响。本文研究了三种引力理论（Einstein,Brans-Dick和Nordtvedt)中的这方面效应，并给出理论和数值的研究结果。     

限制性三体问题中摄动运动方程的坐标系选择

针对限制性三体问题,分别选取以中心天体和摄动体质心为坐标原点的惯性系,及以中心天体为坐标原点的非惯性系,讨论了不同坐标系下天体运动轨道描述的异同。利用运动天体轨道能量E的大小,可以确定受摄运动方程采用椭圆轨道根数还是采用双曲线轨道根数进行描述。为此,推导出一个关于轨道半长径和偏心率满足的临界关系判别式。结果表明,在摄动天体质量较大的情况下,非惯性系中存在大量轨道,这些轨道在原惯性坐标系中是稳定的椭圆轨道,转换到非惯性系中后却无法用椭圆轨道根数进行描述。只能引入双曲线轨道根数来描述轨道,由此将产生非惯性系下摄动运动方程轨道根数类型选择问题。最后,指出选择雅可比坐标系可以避免上述问题,并推导出适用于任意运动区域的具有统一形式的摄动函数展开式。     

制约卫星轨道寿命的另一种机制(续)

对于制约低轨人造地球卫星轨道寿命的耗散机制，人们已有足够的重视，但在深空探测中，另一种制约低轨卫星轨道寿命的引力机制同样应予重视，前文讨论了高轨卫星的情况，在第三体引力作用下，有可能导致卫星轨道偏心率产生变幅较大的长周期变化。特别是极轨卫星，其轨道偏心率在一定的时间内可增大到使其近星距rp=a(1-e)≈Re（Re是中心天体的赤道半径），从而落到中心天体上，结束其轨道寿命，目前对低轨卫星作了详尽的理论分析，研究表明，与高轨卫星有类似结果，但其力学机制却不相同，低轨卫星的轨道寿命与第三体引力无关。它取决于中心天体非球形引力位中的扁率项（即J2项）与其他带谐项之间的相对大小，这不仅是一个纯理论结果，也有实际背景，在太阳系中慢自传天体（月球和金星等）的低轨卫星就存在这一问题，还给出了有关判据，并以计算实例作了验证。     

金星凌日与天文科学

我们的太阳系内,恒星太阳位于中心,是最主要的天体。在它的周围有八颗大行星、许多矮行星和小行星,还有许多彗星和流星体。其中六颗大行星拥有自己的卫星。这些天体在太阳系内有规律地运动着,构成了丰富多姿的太阳系家族。也正是这些天体的有规律运动,形成了太阳系内天体的各种天文现象,如日食（日全食、日环食和日偏食）,月食（月全食和月偏食）,     

关于数值求解天体运动方程的几个问题

本文讨论三个问题：1.在采用各种非辛（Symplectic）的数值积分器积分天体运动方程时，截断误差将引起人为的能量耗散，这一问题是不能用简单地在相应的力模型中加进一个人为的阻力因子而得以解决的，被歪曲的能量（或数值轨道）必须在积分过程的每一步用能量关系来进行校正，此即能量控制方法．2.当摄动加速度涉及到坐标轴的旋转时，如何在各种积分器中采用能量控制方法．3.对于大偏心率轨道，用数值方法求解相应运动方程时，积分步长必须随运动天体与中心天体之间的距离变化而改变，显然，这对所有积分器都是不方便的，特别是多步积分器．本义给出了一种步长均匀化的处理，可以使上述大偏心率轨道积分问题按定步长计算．     

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.     

摄动函数高阶展开与长期共振

研究三体系统的长期共振效应,有助于了解系统的稳定性。在雅可比坐标系下建立了第三体摄动一般运动模型,将摄动函数按照半长径之比展开到十六极矩。通过对摄动函数进行轨道双重平均,消除了内外轨道的短周期项。基于限制性三体模型,分别在内摄和外摄两类情形下进行讨论。外摄情形下十六极矩项对系统结构的演化只有微弱的影响,而内摄情形下系统会出现新的共振和混沌现象。在圆型内摄情形下,系统出现了类似于Lidov-Kozai效应的近点共振。区别于Lidov-Kozai效应只在近心点幅角ω2=±90?时可能存在平衡点,十六极矩近似下,在ω2为0?和180?时也可能存在平衡点。角动量Z轴分量的取值会影响共振平衡点的数量、位置和稳定性。在椭圆型内摄情形下,系统在十六极矩近似下激发出新的轨道翻转,且翻转没有周期性,呈现混沌现象。十六极矩近似下的轨道翻转明显区别于八极矩近似下的轨道翻转,特别是,当半长径之比相当大时,十六极矩近似下偏心率的振幅明显大于八极矩近似下偏心率的振幅。     

Kerr场中缓慢粒子的运动

本文用后－后牛顿近似讨论Ｋｅｒｒ场中缓慢粒子的运动，我们用Ｂｏｙｅｒ－Ｌｉｎｄｑｕｉｓｔ坐标，导出试验粒子的运动方程，把它与有心力场中粒子作二体运动之球坐标形式下的运动方程对比，得出由于Ｋｅｒｒ场的作用而引起的试验粒子的等效摄动加速度，利用球面三角公式把它换算到行星运动摄动方程的形状，对摄动方程进行积分，我们得出了试验粒子绕中心天体运动一周后粒子轨道根数的变化以及单位时间中轨道根数的平均变化，运用     

分点和赤道改正的分离摄动项求解法

本文所提出的分离摄动项求解法,是利用小行星在整个轨道上分布的不少于9次的位置测定值,与用该小行星的轨道根数初值计算出的列表位置进行比较,将由太阳系其它天体摄动力对小行星位置、速度的影响进行分离,求解出分点及赤道改正,小行星轨道根数改正,地球轨道根数改正和由摄动力引起的小行星位置和速度改正。这种方法的优点在于:(1)列表位置仅需根据小行星的轨道根数初值计算出,不考虑摄动力的作用,这样可避免小行星运动理论不完善对确定分点和赤道改正的影响;(2)在解算中,可以单独地求出摄动力对小行星运动速度和位置的影响,通过对摄动函数的数值积分,可求得任一时刻的小行星的真位置。     

天体的视运动

一、天体的周日视运动 在晴朗的夜晚,仰望天空,眼前像有一个半球形的夜幕天穹,上面点缀着无数星星,感觉自己仿佛是处在这个天穹的中心,这就是人们对“天球”的印象。天文学家为了研究天体的位置和天体的运动引入天球的概念,定义天球是以地心为中心,以无穷远为半径的假想球,所有天体都分布在天球上,地球自转轴的延伸叫天轴,天轴与天球的交点叫天极。由于地球自西向东转,人们不觉地球运动,却看到所有天体都围绕着天轴自     

Explicit Semianalytical Theory of Asteroid Motion

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.     

The Parabolic Three-Body Problem

In this communication we present an analytical model for the restricted three-body problem, in the case where the perturber is in a parabolic orbit with respect to the central mass. The equations of motion are derived explicitly using the so-called Global Expansion of the disturbing function, and are valid for any eccentricity of the massless body, as well as in the case where both secondary masses have crossing orbits. Integrating the equations of motion over the complete passage of the perturber through the system, we are then able to construct a first-order algebraic mapping for the change in semimajor axis, eccentricity and inclination of the perturbed body.Comparisons with numerical solutions of the exact equations show that the map yields precise results, as long as the minimum distance between both bodies is not too small. Finally, we discuss several possible applications of this model, including the evolution of asteroidal satellites due to background bodies, and simulations of passing stars on extra-solar planets.     

A note on the tidally induced potential of a satellite in eccentric orbit

A planetary body moving on an eccentric orbit around the primary is subject to a periodic perturbing potential, affecting its internal mass distribution. In a previous paper (Rappaport et al., 1997, Icarus 126, 313), we have calculated the periodic modulation of the gravity coefficients of degree 2, for a body on a synchronous orbit. Here, the previous analysis is extended by considering also non-synchronous orbits, and by properly accounting for the apparent motion of the primary due to the non uniform motion along the elliptical orbit. The cases of Titan and Mercury are briefly discussed.     

Stability of resonant planetary orbits in binary stars

This paper contains a numerical study of the stability of resonant orbits in a planetary system consisting of two planets, moving under the gravitational attraction of a binary star. Its results are expected to provide us with useful information about real planetary systems and, at the same time, about periodic motions in the general four-body problem (G4) because the above system is a special case of G4 where two bodies have much larger masses than the masses of the other two (planets). The numerical results show that the main mechanism which generates instability is the destruction of the Jacobi integrals of the massless planets when their masses become nonzero and that resonances in the motion of planets do not imply, in general, instability. Considerable intervals of stable resonant orbits have been found. The above quantitative results are in agreement with the existing qualitative predictions     

On the true circular orbit of a satellite

The osculating orbit of a planetary satellite moving in the equatorial plane of the central body under the influence of a rotational symmetric perturbation force is elliptical in first order approximation even if the true orbit is always circular. The satellite motion is influenced by a resonance effect due to this perturbing force. An inclined true satellite orbit cannot be circular.     

太阳系P型逆行小天体运动的稳定区域问题

本文讨论了太阳系Ｐ型逆行小天体运动的稳定区域问题，首先，指出了Ｐ型逆行小天体顺行小天体的Ｈｉｌｌ稳定区域及数值稳定区域的差别，然后，讨论了大行星间Ｐ型逆行小天体存在的可能性问题。     

On the stability of circular ‘asteroid’ orbits in anN-planetary system

The restricted problem of the motion of a point of negligible mass (asteroid) in anN-planetary system is considered. It is assumed that all the planets move about the central body (Sun) along circular orbits in the same plane and the mean motions of the asteroid and the planets are incommensurable. The asteroid orbit evolution is described as a first approximation by secular equations with the perturbing function averaged by the mean longitudes of the asteroid and the planets. For small values of the asteroid orbit eccentricity an expression for the secular part of the perturbing function has been obtained. This expression holds for the arbitrary values of the asteroid orbit semiaxis which are different from those of the planet orbit radii. The stability of the asteroid circular orbits in a linear approximation with respect to the eccentricity is studied. The critical inclinations for a Solar system model are calculated.     

Analysis of the potential field and equilibrium points of irregular-shaped minor celestial bodies

The equilibrium points of the gravitational potential field of minor celestial bodies, including asteroids, comets, and irregular satellites of planets, are studied. In order to understand better the orbital dynamics of massless particles moving near celestial minor bodies and their internal structure, both internal and external equilibrium points of the potential field of the body are analyzed. In this paper, the location and stability of the equilibrium points of 23 minor celestial bodies are presented. In addition, the contour plots of the gravitational effective potential of these minor bodies are used to point out the differences between them. Furthermore, stability and topological classifications of equilibrium points are discussed, which clearly illustrate the topological structure near the equilibrium points and help to have an insight into the orbital dynamics around the irregular-shaped minor celestial bodies. The results obtained here show that there is at least one equilibrium point in the potential field of a minor celestial body, and the number of equilibrium points could be one, five, seven, and nine, which are all odd integers. It is found that for some irregular-shaped celestial bodies, there are more than four equilibrium points outside the bodies while for some others there are no external equilibrium points. If a celestial body has one equilibrium point inside the body, this one is more likely linearly stable.     

A method for averaging the perturbing function in Hill’s problem

A modified method for averaging the perturbing function in Hill’s problem is suggested. The averaging is performed in the revolution period of the satellite over the mean anomaly of its motion with a full allowance for a variation in the position of the perturbing body. At its fixed position, the semimajor axis of the satellite orbit during the revolution of the satellite is constant in view of the evolution equations, while the remaining orbital elements undergo secular and long-period perturbations. Therefore, when the motion of the perturbing body is taken into account, the semimajor axis of the satellite orbit undergoes the strongest perturbations. The suggested approach generalizes the averaging method in which only the linear (in time) term is included in the perturbing function. This method requires no expansion in powers of time. The described method is illustrated by calculating the perturbations of the semimajor axes for two distant satellites of Saturn, S/2000 S 1 and S/2000 S5. An approximate analytic solution is compared with the results of numerical integration of the averaged system of equations of motion for these satellites.     

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.