近地小行星轨道演化的数值研究与辛算法有效性的探讨

本文采用改进的显式辛算法（symplecticalgorithm）和嵌套的RKF7（8）积分器对43颗已命名（或编号）的近地小行星的轨道演化进行数值研究．在力学模型上,除考虑各大行星的引力振动外,还增加了后牛顿效应,而在算法上则着重探索辛算法在近地小行星轨道演化研究中的应用前景,特别是当小行星与某一大行星靠近时辛算法的有效性．本文的结果可为了解近地小行星的轨道演化状况和对它们进行监测提供可靠的信息．     

小天体离散元仿真软件的搭建与测试

小天体探测是当今太阳系探测的一大热点, 对小行星演化的研究有助于人们了解太阳系的起源. 演化研究的一项重要内容是小天体结构的演化, 即小天体在多种力学机制作用下自身形状与结构的演化. 在小天体碎石堆结构的假设之下, 一种比较常见的模拟小天体结构演化的方法是离散元仿真算法(Discrete Element Method, DEM), 目前国内外有数个团队开发了相关算法软件. 本文介绍了团队开发的《基于DEM仿真算法的多粒子系统模拟软件》的基础理论、实现方法以及加速算法, 并使用二体接触模型、声速传播仿真、小行星内部压强、小行星旋转稳定性仿真算例验证了算法的可靠性.     

主带小行星的动力学模拟

采用接近真实太阳系的动力学模型,对主带小行星的动力学演化进行了数值模拟。计算的起始时间是儒略日JD=2.4540005×10~6,计算的时间长度为100万年。力学模型采用n+m体模型,计算程序基于小行星轨道演化的软件Orbit9。对演化结果进行分析可以发现测试粒子与木星的平运动共振对测试粒子稳定性的不同作用,以及在2:3、3:4共振处不同初始ω值对测试粒子演化结果的影响。     

关于近地小行星轨道演化的初步探索

本文采用改进的显式辛算法和嵌套的PKF7（8）积分器同时对86颗已命名（或编号）的近地小行星的轨道演化进行了数值研究，在103－104年的时间尺度上，给出了这些小行星轨道演化的状况以及它们与几颗大行星靠近的最小距离，特别是与地球接近的最小距离可小于0．01天文单位，甚至可能比月球还更靠近地球．     

随机森林算法实现小行星分类

随着小行星光谱和测光数据不断增加,以此为基础已有多种分类方法对小行星进行分类.使用随机森林算法对Sloan Digital Sky Survey(SDSS)Moving Object Catalogue(MOC)的观测结果进行小行星分类.根据小行星g、r、i、z 4个波段的星等,结合Tholen、Bus、Lazzaro、De Meo等人已有的分类工作和主成分分析,对多达48642颗的小行星进行了分类,实现了大数据集的小行星类别判定,把这些小行星分为8个类别(C、X、S、B、D、K、L和V).     

小行星(360)Carlova自转参数的测定

小行星自转参数的资料不仅可以为小行星碰撞演化提供数据,还可以为太阳系的演化研究提供依据．利用云南天文台1米望远镜对主带碳类小行星(360)Carlova进行了新的CCD测光观测,结合前人的测光资料,利用Epoch-方法对(360)的自转参数进行了反演计算,得到该小行星自转的恒星周期为0．25780417±0．00000003天,自转轴的黄道坐标为(95°±3,°40°±1°)．新的结果与前人的结果相比较为一致,其精度稍高于以前的估算结果．     

关于太阳系小天体轨道演化的算法研究

太阳系小天体的运动对应—哈密顿(Hamilton)系统,对其轨道演化的数值研究宜采用哈密顿算法(即辛算法)。本文将仔细讨论这一问题,并以主带小行星的运动为例,较系统地介绍几种辛算法对应的显式辛差分格式。     

行星大样本演化的研究进展

随着系外行星观测样本的大量积累,大样本演化方法作为天体物理的重要方法,继在恒星物理中广泛应用后,在行星统计特征的研究中开始变得更加重要。行星大样本演化方法就是通过简化的行星形成演化模型,对大量初始条件(原行星盘参数)和边界条件(星周环境)同时进行模拟,给出行星的各种统计性质,并直接与观测上的行星样本统计特征做比较。该方法对于限制行星形成和演化的理论模型具有重要意义。对行星大样本演化模型进行概述,简要介绍了行星大样本演化的主要统计结果和最新进展,并对其未来发展作出了展望。     

国际小行星搜寻活动

2008年2月1日至3月14日和3月17日至4月30日,历时三个月共两期国际小行星搜寻活动（International Asteroid Search Campaign,简称IASC）已经圆满结束。中国、德国、意大利、日本、摩洛哥、波兰、葡萄牙、美国、加拿大等9个国家的几十所学校参加了活动。到目前为止,两期活动共发现主带小行星6颗,证认了近地小行星34颗,获得虚拟撞击体观测结果6次。中国学生在此次活动中发现主带小行星1颗,证认了近地小行星4颗,获得虚拟撞击体观测结果1次。     

行星科学与深空探测（三）——太阳系小天体研究和行星深空探测的国际现状与发展趋势

小行星与慧星研究 小行星和彗星是45亿年前太阳系形成时遗留的原始残骸,包含着太阳系早期的物理和化学信息,对研究太阳系起源和行星系统形成具有重要科学意义,同时精密确定其轨道动力学演化又是目前空间环境和地球安全方面的重要现实需求。     

An Upper Limit of Mass for a Stable Minor Planet in the Main Asteroid Belt

The stability of an imaginary planet located in the present main asteroid belt is studied with a 7-body model (Sun, Mars, Jupiter, Saturn, Uranus, Neptune and the imaginary planet). The fourth-order Hermite algorithm P(EC)³ is used, which has a very small secular energy error for the integration of periodic orbits with a constant time-step. The evolution of orbits is followed up to 10⁸ years. Our numerical results show that the low-order resonances with Jupiter can enhance the stability of the imaginary planet in some cases. The survival probability of the imaginary planet decreases with the planet mass. The upper limit of the imaginary planet's mass that can survive in the main belt is around 10²⁵ kg, i.e., about the Earth's mass.     

HD 12661行星系统构形的形成与稳定性

最近的多普勒观测表明恒星HD 12661周围存在两颗中等偏心率轨道上运行的行星,内行星的最小质量为2．3木星质量,轨道周期为263．6天;外行星的最小质量为1．57木星质量,轨道周期为1444．5天．该系统的稳定性要求两颗行星处在平运动轨道共振．用数值方法研究了该系统形成初期在恒星气体盘作用下的轨道迁移与稳定性,计算了行星在迁移中被平运动共振俘获的概率．发现这两颗行星目前很可能正处在11:2平运动共振边缘,且运动是混沌的,从而澄清了关于系统目前构形的不同说法,并且很可能在系统形成后行星迁移到目前构形时,气体盘几乎消失了．     

Building the terrestrial planets: Constrained accretion in the inner Solar System

To date, no accretion model has succeeded in reproducing all observed constraints in the inner Solar System. These constraints include: (1) the orbits, in particular the small eccentricities, and (2) the masses of the terrestrial planets - Mars’ relatively small mass in particular has not been adequately reproduced in previous simulations; (3) the formation timescales of Earth and Mars, as interpreted from Hf/W isotopes; (4) the bulk structure of the asteroid belt, in particular the lack of an imprint of planetary embryo-sized objects; and (5) Earth’s relatively large water content, assuming that it was delivered in the form of water-rich primitive asteroidal material. Here we present results of 40 high-resolution (N = 1000-2000) dynamical simulations of late-stage planetary accretion with the goal of reproducing these constraints, although neglecting the planet Mercury. We assume that Jupiter and Saturn are fully-formed at the start of each simulation, and test orbital configurations that are both consistent with and contrary to the “Nice model”. We find that a configuration with Jupiter and Saturn on circular orbits forms low-eccentricity terrestrial planets and a water-rich Earth on the correct timescale, but Mars’ mass is too large by a factor of 5-10 and embryos are often stranded in the asteroid belt. A configuration with Jupiter and Saturn in their current locations but with slightly higher initial eccentricities (e = 0.07-0.1) produces a small Mars, an embryo-free asteroid belt, and a reasonable Earth analog but rarely allows water delivery to Earth. None of the configurations we tested reproduced all the observed constraints. Our simulations leave us with a problem: we can reasonably satisfy the observed constraints (except for Earth’s water) with a configuration of Jupiter and Saturn that is at best marginally consistent with models of the outer Solar System, as it does not allow for any outer planet migration after a few Myr. Alternately, giant planet configurations which are consistent with the Nice model fail to reproduce Mars’ small size.     

Close encounters of a rotating star with planets in parabolic orbits of varying inclination and the formation of hot Jupiters

In this paper we extend the theory of close encounters of a giant planet on a parabolic orbit with a central star developed in our previous work (Ivanov and Papaloizou in MNRAS 347:437, 2004; MNRAS 376:682, 2007) to include the effects of tides induced on the central star. Stellar rotation and orbits with arbitrary inclination to the stellar rotation axis are considered. We obtain results both from an analytic treatment that incorporates first order corrections to normal mode frequencies arising from stellar rotation and numerical treatments that are in satisfactory agreement over the parameter space of interest. These results are applied to the initial phase of the tidal circularisation problem. We find that both tides induced in the star and planet can lead to a significant decrease of the orbital semi-major axis for orbits having periastron distances smaller than 5?C6 stellar radii with tides in the star being much stronger for retrograde orbits compared to prograde orbits. Assuming that combined action of dynamic and quasi-static tides could lead to the total circularisation of orbits this corresponds to observed periods up to 4?C5 days. We use the simple Skumanich law to characterise the rotational history of the star supposing that the star has its rotational period equal to one month at the age of 5 Gyr. The strength of tidal interactions is characterised by circularisation time scale, t _ev , which is defined as a typical time scale of evolution of the planet??s semi-major axis due to tides. This is considered as a function of orbital period P _obs , which the planet obtains after the process of tidal circularisation has been completed. We find that the ratio of the initial circularisation time scales corresponding to prograde and retrograde orbits, respectively, is of order 1.5?C2 for a planet of one Jupiter mass having P _obs ~ 4 days. The ratio grows with the mass of the planet, being of order five for a five Jupiter mass planet with the same P _orb . Note, however, this result might change for more realistic stellar rotation histories. Thus, the effect of stellar rotation may provide a bias in the formation of planetary systems having planets on close orbits around their host stars, as a consequence of planet?Cplanet scattering, which favours systems with retrograde orbits. The results reported in the paper may also be applied to the problem of tidal capture of stars in young stellar clusters.     

Populating the asteroid belt from two parent source regions due to the migration of giant planets—“The Grand Tack”

Abstract– The asteroid belt is found today in a dramatically different state than that immediately following its formation. It is estimated that it has been depleted in total mass by a factor of at least 1000 since its formation, and that the asteroids’ orbits evolved from having near‐zero eccentricity and inclination to the complex distributions we find today. The asteroid belt also hosts a wide range of compositions, with the inner regions dominated by S‐type and other water‐poor asteroids and the outer regions dominated by C‐type and other primitive asteroids. We discuss a model of early inner solar system evolution whereby the gas‐driven migration of Jupiter and Saturn brings them inwards to 1.5 AU, truncating the disk of planetesimals in the terrestrial planet region, before migrating outwards toward their current locations. This model, informally titled “The Grand Tack,” examines the planetary dynamics of the solar system bodies during the final million years of the gaseous solar nebula lifetime—a few million years (Myr) after the formation of the first solids, but 20–80 Myr before the final accretion of Earth, and approximately 400–600 Myr before the Late Heavy Bombardment of the inner solar system. The Grand Tack attempts to solve some outstanding problems for terrestrial planet formation, by reproducing the size of Mars, but also has important implications for the asteroid population. The migration of Jupiter causes a very early depletion of the asteroid belt region, and this region is then repopulated from two distinct source regions, one inside the formation region of Jupiter and one between and beyond the giant planets. The scattered material reforms the asteroid belt, producing a population the appropriate mass, orbits, and with overlapping distributions of material from each parent source region.     

Jupiter and Super-Earth embedded in a gaseous disc

In this paper we investigate the evolution of a pair of interacting planets – a Jupiter-mass planet and a Super-Earth with a mass of  5.5 M _⊕  – orbiting a Solar-type star and embedded in a gaseous protoplanetary disc. We focus on the effects of type I and II orbital migrations, caused by the planet–disc interaction, leading to the capture of the Super-Earth in first-order mean-motion resonances by the Jupiter. The stability of the resulting resonant system in which the Super-Earth is on the internal orbit relative to the Jupiter is studied numerically by means of full 2D hydrodynamical simulations. Our main aim is to determine the Super-Earth behaviour in the presence of the gas giant in the system. It is found that the Jupiter captures the Super-Earth into the interior 3:2 or 4:3 mean-motion resonance, and that the stability of such configurations depends on the initial positions of the planets and on the evolution of the eccentricity. If the initial separation of the orbits of the planets is larger than or close to that required for the exact resonance, the final outcome is the migration of the pair of planets at a rate similar to that of the gas giant, at least for the time of our simulations. Otherwise, we observe a scattering of the Super-Earth from the disc. The evolution of planets immersed in a gaseous disc is compared with their behaviour in the case of the classical three-body problem when the disc is absent.     

Weak lensing signal in unified dark matter models

This paper is an extension of the work done by Pierens & Nelson in which they investigated the behaviour of a two-planet system embedded in a protoplanetary disc. They put a Jupiter mass gas giant on the internal orbit and a lower mass planet on the external one. We consider here a similar problem taking into account a gas giant with mass in the range 0.5 to  1 M _J  and a Super-Earth (i.e. a planet with mass  ≤10 M _⊕  ) as the outermost planet. By changing disc parameters and planet masses, we have succeeded in getting the convergent migration of the planets which allows for the possibility of their resonant locking. However, in the case in which the gas giant has the mass of Jupiter, before any mean-motion first-order commensurability could be achieved, the Super-Earth is caught in a trap when it is very close to the edge of the gap opened by the giant planet. This confirms the result obtained by Pierens & Nelson in their simulations. Additionally, we have found that, in a very thin disc, an apsidal resonance is observed in the system if the Super-Earth is captured in the trap. Moreover, the eccentricity of the small planet remains low, while the eccentricity of the gas giant increases slightly due to the imbalance between Lindblad and corotational resonances. We have also extended the work of Pierens & Nelson by studying analogous systems in which the gas giant is allowed to take sub-Jupiter masses. In this case, after conducting an extensive survey over all possible parameters, we have succeeded in getting the 1:2 mean-motion resonant configuration only in a disc with low aspect ratio and low surface density. However, the resonance is maintained just for a few thousand orbits. Thus, we conclude that for typical protoplanetary discs the mean-motion commensurabilities are rare if the Super-Earth is located on the external orbit relative to the gas giant.     

On the migration of a system of protoplanets

The evolution of a system consisting of a protoplanetary disc with two embedded Jupiter-sized planets is studied numerically. The disc is assumed to be flat and non-self-gravitating; this is modelled by the planar (two-dimensional) Navier–Stokes equations. The mutual gravitational interaction of the planets and the star, and the gravitational torques of the disc acting on the planets and the central star are included. The planets have an initial mass of one Jupiter mass M _Jup each, and the radial distances from the star are one and two semimajor axes of Jupiter, respectively.
During the evolution a joint wide annular gap is created by the planets. Both planets increase their mass owing to accretion of gas from the disc: after about 2500 orbital periods of the inner planet it has reached a mass of 2.3  M _Jup, while the outer planet has reached a mass of 3.2  M _Jup. The net gravitational torques exerted by the disc on the planets result in an inward migration of the outer planet on time-scales comparable to the viscous evolution time of the disc. The semimajor axis of the inner planet remains constant as there is very little gas left in its vicinity to induce any migration. When the distance of close approach eventually becomes smaller than the mutual Hill radius, the eccentricities increase strongly and the system may become unstable.
If disc depletion occurs rapidly enough before the planets come too close to each other, a stable system similar to our own Solar system may remain. Otherwise the orbits may become unstable and produce systems like υ And.     

Planets in the asteroid belt

Abstract— The main asteroid belt has lost >99.9% of its solid mass since the time at which the planets were forming, according to models for the protoplanetary nebula. Here we show that the primordial asteroid belt could have been cleared efficiently if much of the original mass accreted to form planetsized bodies, which were capable of perturbing one another into unstable orbits. We provide results from 25 N‐body integrations of up to 200 planets in the asteroid belt, with individual masses in the range 0.017–0.33 Earth masses. In the simulations, these bodies undergo repeated close encounters which scatter one another into unstable resonances with the giant planets, leading to collision with the Sun or ejection from the solar system. In response, the giant planets' orbits migrate radially and become more circular. This reduces the size of the main‐belt resonances and the clearing rate, although clearing continues. If ～3 Earth masses of material was removed from the belt this way, Jupiter and Saturn would initially have had orbital eccentricities almost twice their current values. Such orbits would have made Jupiter and Saturn 10–100x more effective at clearing material from the belt than they are on their current orbits. The time required to remove 90% of the initial mass from the belt depends sensitively on the giant planets' orbits, and weakly on the masses of the asteroidal planets. 18 of the 25 simulations end with no planets left in the belt, and the clearing takes up to several hundred million years. Typically, the last one or two asteroidal planets are removed by interactions with planets in the terrestrial region     

On the global stability of single‐planet systems

In this paper, we estimate the global stability properties of single‐planet systems by using a catalogue of stability maps. The data of the catalogue were used to generate probability values on the mass parameter–eccentricity plane for the occurrence of stable orbits. We showed that the probability data can be well approximated by a second order surface. Using the resulted formula the likelihood of finding Earth‐like planets in single‐planet systems can be easily estimated. As an example, we derived estimations for four known exoplanetary systems. Our formula can be useful in selecting target stars for future space missions. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)