An Analysis of the Lifetimes of Large Elliptical Orbits of Space Debris

The lunisolar perturbation is the principal factor that causes variation in the orbit of a GTO (geosychronous transfer orbit) space debris. In the present article the analytical expression of the long-periodic term of the altitude of the GTO orbital perigee caused by the lunisolar perturbation is derived, and the basic characteristics of the variation of the perigee caused by the lunisolar perturbation are analyzed. A method of selecting a launch window that will help to improve the space debris situation is given, and its correctness is tested and verified through simulation calculation.     

Co-accretion of the Earth-Moon system after the giant impact: reduction of the total angular momentum by lunar impact ejecta

Though the Moon is considered to have been formed by the so-called giant impact, the mass of the Earth immediately after the impact is still controversial. If the Moon was formed during the Earth's accretion, a subsequent accretion of residual heliocentric planetesimals onto the protoearth and the protomoon must have occurred. In this co-accretion stage, a significant amount of lunar-impact-ejecta would be ejected to circumterrestrial orbits, since the mean impact velocity of the planetesimals with the protomoon is much larger than the escape velocity of the protomoon. Orbital calculations of test particles ejected from the protomoon, whose semimajor axis is smaller than that of the present Moon, reveal that most of the particles escaping from the protomoon also escape from the Hill sphere of the protoearth and reduce the planetocentric angular momentum of the primordial Earth-Moon system. Using the results of the ejecta simulations, we investigate the evolution of the mass ratio and the total angular momentum (Earth's spin angular momentum + Moon's orbital angular momentum) of the Earth-Moon system during the co-accretion. We find that the mass of the protomoon is almost constant or rather decreases and the total angular momentum decreases significantly, if the random velocity of planetesimals is as large as the escape velocity of the protoearth. On the other hand, if the random velocity is the half of the escape velocity of the protoearth, the mass ratio is kept to be almost as large as the present value and the decrease of the total angular momentum is not so significant. Comparing with the results of giant impact simulations, we find that the mass of the protoearth immediately after the Moon-forming impact was 0.7-0.8 times the present value if the impactor-to-target mass ratio was 3:7, whereas the giant impact occurred almost in the end of the Earth's accretion if the impactor-to-target mass ratio was 1:9.     

Solar and planetary destabilization of the Earth-Moon triangular Lagrangian points

In the restricted circular three-body problem, two massive bodies travel on circular orbits about their mutual center of mass and gravitationally perturb the motion of a massless particle. The triangular Lagrange points, L₄ and L₅, form equilateral triangles with the two massive bodies and lie in their orbital plane. Provided the primary is at least 27 times as massive as the secondary, orbits near L₄ and L₅ can remain close to these locations indefinitely. More than 2200 cataloged asteroids librate about the L₄ and L₅ points of the Sun-Jupiter system, and five bodies have been discovered around the L₄ point of the Sun-Neptune system. Small satellites have also been found librating about the L₄ and L₅ points of two of Saturn's moons. However, no objects have been discovered around the Earth-Moon L₄ and L₅ points. Using numerical integrations, we show that orbits near the Earth-Moon L₄ and L₅ points can survive for over a billion years even when solar perturbations are included, but the further addition of the far smaller perturbations from other planets destabilize these orbits within several million years. Thus, the lack of observed objects in these regions cannot be used as a constraint on Solar System formation, nor on the tidal evolution of the Moon's orbit.     

定点在日-地(月)系L1点附近的探测器的发射及维持

在限制性三体问题中共线平动点附近的运动虽然是不稳定的,但可以是有条件稳定的,该动力学特征使得一些有特殊目的的探测器只需消耗较少的能量即可定点在这些点附近(如ISEE-3、SOHO).以日-地(月)系的L1点为例,根据其附近的运动特征,探讨定点探测器的发射与轨道控制问题,给出了相应的数值模拟结果,为工程上的实现提供理论依据.     

The fate of primordial lunar Trojans

Multiple large impact basins on the lunar nearside formed in a relatively-short interval around 3.8-3.9 Gyr ago, in what is known as the Lunar Cataclysm (LC; also known as Late Heavy Bombardment). It is widely thought that this impact bombardment has affected the whole Solar System or at least all the inner planets. But with non-lunar evidence for the cataclysm being relatively weak, a geocentric cause of the Lunar Cataclysm cannot yet be completely ruled out [Ryder, G., 1990. Eos 71, 313, 322-323]. In principle, late destabilization of an additional Earth satellite could result in its tidal disruption during a close lunar encounter (cf. [Asphaug, E., Agnor, C.B., Williams, Q., 2006. Nature 439, 155-160]). If the lost satellite had D>500 km, the resulting debris can form multiple impact basins in a relatively short time, possibly explaining the LC. Canup et al. [Canup, R.M., Levison, H.F., Stewart, G.R., 1999. Astron. J. 117, 603-620] have shown that any additional satellites of Earth formed together with (and external to) the Moon would be unable to survive the rapid initial tidally-driven expansion of lunar orbit. Here we explore the fate of objects trapped in the lunar Trojan points, and find that small lunar Trojans can survive the Moon's orbital evolution until they and the Moon reach 38 Earth radii, at which point they are destabilized by a strong solar resonance. However, the dynamics of Trojans containing enough mass to cause the LC (diameters >150 km) is more complex; we find that such objects do not survive the passage through a weaker solar resonance at 27 Earth radii. This distance was very likely reached by the Moon long before the LC, which seems to rule out the disruption of lunar Trojans as a cause of the LC.     

Current bombardment of the Earth-Moon system: Emphasis on cratering asymmetries

We calculate the current spatial distribution of projectile delivery to the Earth and Moon using numerical orbital dynamics simulations of candidate impactors drawn from a debiased Near-Earth Object (NEO) model. We examine the latitude distribution of impactor sites and find that for both the Earth and Moon there is a small deficiency of time-averaged impact rates at the poles. The ratio between deliveries within 30° of the pole to that of a 30° band centered on the equator is small for Earth (<5%) (0.958±0.001) and somewhat greater for the Moon (∼10%) (0.903±0.005). The terrestrial arrival results are examined to determine the degree of AM/PM asymmetry to compare with the PM excess shown in meteorite fall times. We find that the average lunar impact velocity is 20 km/s, which has ramifications in converting observed crater densities to impactor size distributions. We determine that current crater production on the leading hemisphere of the Moon is 1.28±0.01 that of the trailing when considering the ratio of craters within 30° of the apex to those within 30° of the antapex and that there is virtually no nearside-farside asymmetry, in agreement with observations of rayed craters. As expected, the degree of leading-trailing asymmetry increases when the Moon's orbital distance is decreased.     

平动点线性稳定条件及阻力对限制性三体问题三角平动点线性稳定性的影响

给出了受摄限制性三体问题平动点线性稳定性的一些判断条件，条件只与相应的平动点切映像的特征方程系数有关，使用方便，用这些判断条件，讨论了一些阻力对经典限制性三体问题三角平动点线性稳定性的影响，改进了Murray等的一些结果。     

月球物理天平动对环月轨道器运动的影响

月球物理天平动是月球赤道在空间真实的摆动，会导致月球引力场在空间坐标系中的变化，从而引起环月轨道器(以下称为月球卫星)的轨道变化，这与地球的岁差章动现象对地球卫星轨道的影响类似．采用类似对地球岁差章动的处理方法，讨论月球物理天平动对月球卫星轨道的影响，给出相应的引力位的变化及卫星轨道的摄动解，清楚地表明了月球卫星轨道的变化规律，并和数值解进行了比对，从定性和定量方面作一讨论．     

Impact of the Quadrupole Moment of the Sun on the Dynamics of the Earth-Moon System

Range of values of the Sun's mass quadrupole moment of coefficient J₂ arising both from experimental and theoretical determinations enlarge across literature on two orders of magnitude, from around 10^-7 until to 10^-5. The accurate knowledge of the Moon's physical librations, for which the Lunar Laser Ranging data reach an outstanding precision level, prove to be appropriate to reduce the interval of J₂ values by giving an upper bound of J₂. A solar quadrupole moment as high as 1.1 10^-5 given either from the upper bounds of the error bars of the observations, or from the Roche's theory, is not compatible with the knowledge of the lunar librations accurately modeled and observed with the LLR experiment. The suitable values of J₂ have to be smaller than 3.0 10^-6. As a consequence, this upper bound of 3.0 10^-6 is accepted to study the impact of the Sun's quadrupole moment of mass on the dynamics of the Earth-Moon system. Such as effect (with J₂ = 5.5±1.3 × 10^-6) has been already tested in 1983 by Campbell & Moffat using analytical approximate equations, and thus for the orbits of Mercury, Venus, the Earth and Icarus. The approximate equations are no longer sufficient compared with present observational data and exact equations are required. As if to compute the effect on the lunar librations, we have used our BJV relativistic model of solar system integration including the spin-orbit coupled motion of the Moon. The model is solved by numerical integration. The BJV model stems from general relativity by using the DSX formalism for purposes of celestial mechanics when it is about to deal with a system of n extended, weakly self-gravitating, rotating and deformable bodies in mutual interactions. The resulting effects on the orbital elements of the Earth have been computed and plotted over 160 and 1600 years. The impact of the quadrupole moment of the Sun on the Earth's orbital motion is mainly characterized by variations of , , and . As a consequence, the Sun's quadrupole moment of mass could play a sensible role over long time periods of integration of solar system models. This revised version was published online in July 2006 with corrections to the Cover Date.     

LAGEOS卫星精密定轨及残差分析

介绍LAGEOS卫星用卫星激光测距(SLR)资料精密定轨和在精密定轨基础上的残差分析处理．SLR资料的分析处理方法、采用的力学模型和解参数的多少根据目标的不同而不同．对不同的方案进行了详细分析、比较．其关键是对LAGEOS卫星进行精密定轨．得出了目前上海天文台在SLR资料的分析处理中已采用的解算模式．作为例证，该解算模式分析处理了1998年12月31日至1999年6月29日LAGEOS卫星的SLR资料，得到每3天一个弧段的精密轨道．结果显示每3天一个弧段的定轨残差rms都小于2厘米．上海天文台从1999年10月1日开始了全球LAGEOS—l和LAGEOS—2资料的快速分析服务，结果可从APSG的网址：http：／／center．shao．ac．cn／APSG／result获得．     

潮汐能量耗散与地月系统演化

对地月系统而言, 在很大程度上角动量守恒是正确的. 地月距离的变化主要是受到月球引起的潮汐能量耗散的影响. 根据月球的平均运动和它的长期加速度, 就可以计算出月潮能量耗散的数值. 海洋是潮汐能量耗散的主要区域. 由于潮汐的高度正比于月球对潮汐隆起的万有引力, 由此可导出总的月球潮汐摩擦力正比于月球平均运动的平方. 如果采用月球平均加速度数值-20.72$''\cdot$cy^-2, 就可以推算出35亿年来地月之间的距离以及回归年日数和朔望月日数的演化. 此理论结果与古生物钟的数据进行比对, 两者符合较好.     

New analysis of Lunar Prospector radio tracking data brings the nearside gravity field of the Moon with an unprecedented resolution

A new analysis of the Doppler tracking data from the Lunar Prospector mission in 1999 revealed a number of previously-unseen gravity anomalies at spatial scales as small as 27 km over the nearside. The tracking data at low altitudes (50 km or below) were better analyzed to resolve the nearside features without dampening from a power law constraint, by partitioning the gravity parameters concentrated on either the nearside or farside. The resulting model presents gravity anomalies correlated with topography with a correlation coefficient of 0.7 or higher from degree 50 to 150, the widest bandwidth yet. The gravity-topography admittance of ∼70 mGal/km is found from numerous craters of which diameters are 60 km or less. In addition, the new model produces orbits that fit to independent radio tracking data from the Lunar Reconnaissance Orbiter and Kaguya (SELENE) better than previous gravity models. This high-resolution model can be of immediate use to geophysical analysis of small craters. Our technique could be applied to an upcoming mission, the Gravity Recovery And Interior Laboratory and useful to extract short wavelength signals from the MESSENGER Doppler data.     

On the stability of a planet between Mars and the asteroid belt: Implications for the Planet V hypothesis

The stability of an additional planet between the orbit of Mars and the asteroid belt is examined in the context of the Planet V hypothesis. In this model, the Solar System initially contained a fifth terrestrial planet, “Planet V,” which was removed after ∼700 Myr, a possible trigger for the late heavy bombardment on the inner planets. The model is investigated using 96 N-body integrations of the 8 major planets with an additional body between Mars and the asteroid belt. In more than 1/4 of simulations, Planet V survives for 1000 Myr. In most other cases, Planet V collides with the Sun or hits another planet after several hundred Myr, leaving 4 surviving terrestrial planets. In 24/96 simulations, Planet V is lost by ejection or collision with the Sun while the other four terrestrial planets survive without undergoing a collision. In 18 cases, Planet V is removed at least 200 Myr after the beginning of the simulation. The endstate depends sensitively on the mass of Planet V. Collision with the Sun is likely when Planet V's mass is 0.25 Mars masses or less. When Planet V is more massive than this, collisions involving it and/or other terrestrial planets become commonplace. In unstable systems, the times of first encounter and first collision/ejection depend on the initial aphelion distance of Mars. Reducing Mars's aphelion distance increases these times and also increases the fraction of systems surviving for 1000 Myr. When Mars's current orbit is used, the stability of Planet V increases when these two planets are widely separated initially. Planet V's aphelion distance Q typically begins to cross the asteroid belt within a few tens to a few hundred Myr, and its orbit last leaves the belt several hundred Myr later in most cases. The total time spent with Q>2.1 AU is typically less than 200 Myr.     

单位矢量法的数学模型(MMUVM)及其简化形式(PUVM1)的迭代法的收敛性

利用人造地球卫星观测资料测定有摄初轨的单位矢量法(PUVM1),已得到了非常广泛地实际应用.为了对单位矢量法作进一步地完善和改进,首先在考虑测量误差模型的基础上,建立单位矢量法所对应的数学模型MMUVM.它本质上就是一个非线性最优化问题.针对MMUVM,先分别使用多圈仿真数据和实测数据,形成了与之相对应的目标函数,再利用求解最优化问题的一种三对角二次插值模型的直接搜索方法,分别对其进行了数值处理.计算结果表明,所建立的优化模型MMUVM正确合理,所采用的直接搜索方法实用有效.其次,进一步指明了PUVM1和MMUVM之间关系,即:从本质上讲,PUVM1就是MMUVM的一种简化形式.从数学原理上,清楚地解释了利用PUVM1的准法化方程,只能使用单圈短弧段数据进行初始轨道确定,而不能使用长弧段多圈资料进行轨道确定或轨道改进的根本原因.最后,对PUVM1的迭代算法的收敛性问题进行了初步的理论分析,并给出了相应的数值验证实例,指出了PUVM1的迭代格式是条件收敛的,即:只有在满足一定条件后,才能收敛.这也就意味着:有的时候,尽管准法化方程是合情合理的,但是,此时该迭代法却是发散的,无法迭代求出所要的解.     

地月系潮汐演化长期趋势的估计

根据角动量守恒原理,计算了地月系经潮汐演化到达平衡状态时的旋转周期和地月距离.并根据当前与平衡状态时地月系的总能量差,计算了到达平衡状态的时间.进而估计了地月距离变化和地球自转速率变化的长期趋势.     

Time evolution of orbital uncertainties for the impactor candidate 2004 AS1

We evaluate the asteroid impact risk from the discovery night onwards using six-dimensional statistical orbit computation techniques to examine the a posteriori probability density of the orbital elements. Close to the discovery moment the observational data of an object are typically exiguous: the number of observations is very small and/or the covered orbital arc is very short. For such data, the covariance matrices computed in the linear approximation (e.g., with the least-squares technique) are known to fail to describe the uncertainties in the orbital parameters. The technique of statistical ranging gives us rigorous means to assess the orbital uncertainties already on the discovery night. To examine the time evolution of orbital uncertainties, we make use of a new nonlinear Monte Carlo technique of phase-space sampling using volumes of variation, which complements the ranging technique for exiguous data and the least-squares technique for extensive observational data. We apply the statistical techniques to the near-Earth Asteroid 2004 AS₁, which grabbed the attention of asteroid scientists because, for one day, it posed the highest and most immediate impact risk so far recorded. We take this extreme case to illustrate the ambiguities in the impact risk assessment for short arcs. We confirm that the weighted fraction of the collision orbits at discovery was large but conclude that this was mostly due to the discordance of the discovery-night observations. This case study highlights the need to introduce a regularization in terms of an a priori probability density to secure the invariance of the probabilistic analysis especially in the nonlinear orbital inversion for short arcs. We remark that a predominant role of the a priori can give indications of the feasibility of the probabilistic interpretation, that is, how reliable the results derived from the a posteriori probability density are. Nevertheless, the strict mathematical definition of, e.g., the collision probability remains valid, and our nonlinear statistical techniques give us the means to always deduce, at the very least, order-of-magnitude-estimates for the collision probability.     

The occurrence of high-order resonances and Kozai mechanism in the scattered disk

By means of numerical methods we explore the relevance of the high-order exterior mean motion resonances (MMR) with Neptune that a scattered disk object (SDO) can experience in its diffusion to the Oort cloud. Using a numerical method for estimate the strength of these resonances we show that high-eccentricity or high-inclination resonant orbits should have evident dynamical effects. We investigate the properties of the Kozai mechanism (KM) for non-resonant SDO's and the conditions that generate the KM inside a MMR associated with substantial changes in eccentricity and inclination. We found that the KM inside a MMR is typical for SDO's with Pluto-like or greater inclinations and is generated by the oscillation of ω inside the mixed (e,i) resonant terms of the disturbing function. A SDO diffusing to the Oort cloud should experience temporary captures in MMR, preferably of the type 1:N, and when evolving inside a MMR and experiencing the KM it can reach regions where the strength of the resonance drops and consequently there is a possibility of being decoupled from the resonance generating by this way a long-lived high-perihelion scattered disk object (HPSDO).     

GPS星历对LEO星载GPS精密定轨精度的影响

目前,越来越多的低轨卫星上都搭载了用于精密定轨的星载GPs接收机,星载GPS已成为低轨卫星精密定轨的主要手段之一.星载GPS精密定轨精度依赖于GPS星历及钟差精度.基于SHORDE-Ⅲ非差动力学定轨功能,以2005年8月1日至8月7日一周的GRACE卫星实测数据为例,采用事后精密轨道(igs)、快速轨道(igr)和超快速轨道(igu)三种GPS星历在同等条件下定轨,估计GPS星历精度对低轨卫星定轨精度的影响,实际计算结果表明igs和igr两类GPS星历定轨精度相当,约为9.5 cm,igu星历定轨精度略低于igs和igr星历,约为10.5cm:高频GPS卫星钟差数据对定轨精度会产生1-6cm影响.     

The influence of shapes of the bodies on the dynamics of planetary and star systems

The dynamics of planetary and star systems including perturbing forces due to the flattering and quadrupole distortion of the bodies is studied. The analytical model describing the perturbations which influence on the orbital motion of extrasolar planetary systems is presented. The calculations of the secular evolution of the mean orbital elements have shown that the effects related to the shape of the body are more important than the ones due to the quadrupole distortion.     

An Estimate of the Long-term Tendency of Tidal Evolution of Earth-lunar System

According to the conservation principle of angular momentum, we calculate in this paper the revolution period and the distance between the Earth and the Moon in the equilibrium state of the tidal evolution in the Earth-Moon system. The difference of energy between the current state and the equilibrium state is used to compute the time needed to fulfil the equilibrium state. Then the long-term variations of the Earth-Moon distance and of the Earth rotation rate are further estimated.