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1.
The resonant structure of the restricted three body problem for the Sun- Jupiter asteroid system in the plane is studied, both for a circular and an elliptic orbit of Jupiter. Three typical resonances are studied, the 2 : 1, 3 : 1 and 4 : 1 mean motion resonance of the asteroid with Jupiter. The structure of the phase space is topologically different in these cases. These are typical for all other resonances in the asteroid problem. In each case we start with the unperturbed two-body system Sun-asteroid and we study the continuation of the periodic orbits when the perturbation due to a circular orbit of Jupiter is introduced. Families of periodic orbits of the first and of the second kind are presented. The structure of the phase space on a surface of section is also given. Next, we study the families of periodic orbits of the asteroid in the elliptic restricted problem with the eccentricity of Jupiter as a parameter. These orbits bifurcate from the families of the circular problem. Finally, we compare the above families of periodic orbits with the corresponding families of fixed points of the averaged problem. Different averaged Hamiltonians are considered in each resonance and the range of validity of each model is discussed.  相似文献   

2.
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  相似文献   

3.
木星探测轨道分析与设计   总被引：3，自引：0，他引：3  
研究了与木星探测相关的轨道设计问题.重点关注木星探测轨道与火星、金星等类地行星探测轨道的不同及由此带来的轨道设计难点.首先分析了绕木星探测任务轨道的选择.建立近似模型讨论了向木星飞行需要借助多颗行星的多次引力辅助,对地木转移的多种行星引力辅助序列,使用粒子群算法搜索了2020年至2025年之间的燃料最省飞行方案并对比得到了向木星飞行较好的引力辅助方式为金星-地球-地球引力辅助.结合多任务探测,研究了航天器在飞向木星途中穿越主小行星带飞越探测小行星的轨道设计.最后,给出2023年发射完整的结合引力辅助与小行星多次飞越的木星探测轨道设计算例.  相似文献   

4.
Trajectory Analysis and Design for A Jupiter Exploration Mission   总被引：1，自引：0，他引：1  
The trajectory design for a Jupiter exploration mission is investigated in this paper. The differences between the Jupiter exploration trajectory and the Mars or Venus exploration trajectory are mainly concerned about. Firstly, the selection of the Jupiter-centered orbit is analyzed based on the Galileo Jupiter mission. As for the Earth-Jupiter transfer orbit, the fuel consumption of the direct transfer is too large. So the energy-saving technologies such as the planetary gravity assist should be used for the trajectory to the Jupiter. The different sequences of planetary gravity assists are examined by applying the Particle Swarm Optimization (PSO). According to the searched result, the Venus-Earth-Earth sequence (VEEGA) is the most effective one for the Jupiter mission. During the Jupiter mission, the spacecraft will pass though the main asteroid belt between the orbits of Mars and Jupiter, and may encounter multiple asteroids. Therefore the Jupiter mission is able to combine with the main-belt asteroid flyby mission. The design method of the intermediate asteroid flyby trajectory is also considered. At last, an entire trajectory for the Jupiter mission launched in 2023 is presented.  相似文献   

5.
The orbital evolutions of the asteroid 3040 Kozai and model asteroids with similar orbits have been investigated. Their osculating orbits for an epoch 1991 December 10 were numerically integrated forward within the interval of 20,000 years, using a dynamical model of the solar system consisting of all inner planets, Jupiter, and Saturn.The orbit of the asteroid Kozai is stable. Its motion is affected only by long-period perturbations of planets. With change of the argument of perihelion of the asteroid Kozai, the evolution of the model asteroid orbits changes essentially, too. The model orbits with the argument of perihelion changed by the order of 10% show that asteroids with such orbital parameters may approach the Earth orbit, while asteroids with larger changes may even cross it, at least after 10,000 years. Long-term orbital evolution of asteroids with these orbital parameters is very sensitive on their angular elements.  相似文献   

6.
李培俊  周济林 《天文学报》2006,47(4):394-401
介绍了N体模拟的Hermite算法,并利用该算法研究了不同质量行星在小行星主带上轨道的演化情况．采用的演化模型是太阳系N体模型(N=7),即把水星、金星、地球的质量加到太阳上,忽略冥王星,同时在小行星主带附近增加一个假想行星,系统演化时间为1亿年．数值模拟显示能够稳定存在于小行星主带上的单个天体的质量上限其量级为10~(25)kg．模拟同时还显示在某些情况下,假想行星与木星之间的低阶共振可以增强系统的稳定性．  相似文献   

7.
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.  相似文献   

8.
The existing explanations for the asteroid distribution in the main belt (between the orbits of Mars and Jupiter) are based on numerical integration of resonance orbits in models with more than two degrees of freedom. We suggest an approach based on the investigation of the families of periodic solutions of the planar circular restricted three-body problem, i.e., a model with two degrees of freedom. This work shows that (a) the distribution of asteroids near the (p + 1)/p resonances and position of the outer boundary of the main asteroid belt can be explained within the planar circular restricted three-body problem and (b) this problem does not explain the asteroid distribution near other resonances.  相似文献   

9.
An essential role in the asteroidal dynamics is played by the mean motion resonances. Two-body planet–asteroid resonances are widely known, due to the Kirkwood gaps. Besides, so-called three-body mean motion resonances exist, in which an asteroid and two planets participate. Identification of asteroids in three-body (namely, Jupiter–Saturn–asteroid) resonances was initially accomplished by Nesvorný and Morbidelli (Nesvorný D., Morbidelli, A. [1998]. Astron. J. 116, 3029–3037), who, by means of visual analysis of the time behaviour of resonant arguments, found 255 asteroids to reside in such resonances. We develop specialized algorithms and software for massive automatic identification of asteroids in the three-body, as well as two-body, resonances of arbitrary order, by means of automatic analysis of the time behaviour of resonant arguments. In the computation of orbits, all essential perturbations are taken into account. We integrate the asteroidal orbits on the time interval of 100,000 yr and identify main-belt asteroids in the three-body Jupiter–Saturn–asteroid resonances up to the 6th order inclusive, and in the two-body Jupiter–asteroid resonances up to the 9th order inclusive, in the set of ～250,000 objects from the “Asteroids – Dynamic Site” (AstDyS) database. The percentages of resonant objects, including extrapolations for higher-order resonances, are determined. In particular, the observed fraction of pure-resonant asteroids (those exhibiting resonant libration on the whole interval of integration) in the three-body resonances up to the 6th order inclusive is ≈0.9% of the whole set; and, using a higher-order extrapolation, the actual total fraction of pure-resonant asteroids in the three-body resonances of all orders is estimated as ≈1.1% of the whole set.  相似文献   

10.
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.  相似文献   

11.
Earlier work indicates a comparatively rapid chaotic evolution of the orbits of some Hilda asteroids that move at the border of the domain occupied by the characteristic parameters of the objects at the 3/2 mean motion resonance. A simple Jupiter–Saturn model of the forces leads to numerical results on some of these cases and allows a search for additional resonances that can contribute to the chaotic evolution. In this context the importance of the secondary resonances that depend on the period of revolution of the argument of perihelion is pointed out. Among the studied additional resonances there are three-body resonances with arguments that depend on the mean longitudes of Jupiter, Saturn, and asteroid, but on slowly circulating angular elements of the asteroid as well, and the frequency of these arguments is close to a rational ratio with respect to the frequency of the libration due to the basic resonance.  相似文献   

12.
We consider dynamics of a Sun–Jupiter–Asteroid system, and, under some simplifying assumptions, show the existence of instabilities in the motions of an asteroid. In particular, we show that an asteroid whose initial orbit is far from the orbit of Mars can be gradually perturbed into one that crosses Mars’ orbit. Properly formulated, the motion of the asteroid can be described as a Hamiltonian system with two degrees of freedom, with the dynamics restricted to a “large” open region of the phase space reduced to an exact area preserving map. Instabilities arise in regions where the map has no invariant curves. The method of MacKay and Percival is used to explicitly rule out the existence of these curves, and results of Mather abstractly guarantee the existence of diffusing orbits. We emphasize that finding such diffusing orbits numerically is quite difficult, and is outside the scope of this paper.  相似文献   

13.
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.  相似文献   

14.
The depletion of an initially uniform distribution of asteroids extending form Mars to Saturn, caused by the gravitational perturbations of Jupiter and Saturn, is calculated by numerical integration of the asteroid orbits. Almost all (about 85%) the asteroids between Jupiter and Saturn are ejected in the first 6000 years Most of the asteroids between the 23 Jupiter resonance (4.0 A.U.) and Jupiter are ejected in the first 2400 years with the exception of the stable librators (e.g., the Hilda group). Interior to the 23 resonance the depletion was small, and interior to the 12 resonance (3.3 A.U.) no asteroids were ejected in the first 2400 years.  相似文献   

15.
When the precessional rate of the orbital plane of an asteroid is nearly equal to that of Jupiter, the orbital inclination of the asteroid changes quite largely due to this near equality of their precessional rates, which is called a secular resonance. In the vicinity of the exact resonance the difference of their longitudes of nodes librates with quite a long period of order of 1×106 yr. In this paper we treat this secular resonance by a method of semianalytical secular perturbations with use of numerical averaging for both non-resonant and resonant asteroids and show that the results by the semi-analytical treatment agrees qualitatively with those obtained by direct numerical integrations of asteroid's orbits.  相似文献   

16.
The 2/1 mean motion resonance with Jupiter, intersecting the main asteroid belt at ≈3.27  au, contains a small population of objects. Numerical investigations have classified three groups within this population: asteroids residing on stable orbits (i.e. Zhongguos), those on marginally stable orbits with dynamical lifetimes of the order of 100 Myr (i.e. Griquas), and those on unstable orbits. In this paper, we reexamine the origin, evolution and survivability of objects in the 2/1 population. Using recent asteroid survey data, we have identified 100 new members since the last search, which increases the resonant population to 153. The most interesting new asteroids are those located in the theoretically predicted stable island A, which until now had been thought to be empty. We also investigate whether the population of objects residing on the unstable orbits could be resupplied by material from the edges of the 2/1 resonance by the thermal drag force known as the Yarkovsky effect (and by the YORP effect, which is related to the rotational dynamics). Using N -body simulations, we show that test particles pushed into the 2/1 resonance by the Yarkovsky effect visit the regions occupied by the unstable asteroids. We also find that our test bodies have dynamical lifetimes consistent with the integrated orbits of the unstable population. Using a semi-analytical Monte Carlo model, we compute the steady-state size distribution of magnitude   H < 14  asteroids on unstable orbits within the resonance. Our results provide a good match with the available observational data. Finally, we discuss whether some 2/1 objects may be temporarily captured Jupiter-family comets or near-Earth asteroids.  相似文献   

17.
We investigate the orbital evolution of 10(13)- to 10(25) -g planetesimals near 1 AU and in the asteroid belt (near 2.6 AU) prior to the stage of evolution when the mutual perturbations between the planetesimals become important. We include nebular gas drag and the effects of Jupiter and Saturn at their present masses and in their present orbits. Gas drag introduces a size-dependent phasing of the secular perturbations, which leads to a pronounced dip in encounter velocities (Venc) between bodies of similar mass. Plantesimals of identical mass have Venc approximately 1 and approximately 10 m s-1 (near 1 and 2.6 AU, respectively) while bodies differing by approximately 10 in mass have Venc approximately 10 and approximately 100 m s-1 (near 1 and 2.6 AU, respectively). Under these conditions, growth, rather than erosion, will occur only by collisions of bodies of nearly the same mass. There will be essentially no gravitational focusing between bodies less than 10(22) to 10(25) g, allowing growth of planetary embryos in the terrestrial planet region to proceed in a slower nonrunaway fashion. The environment in the asteroid belt will be even more forbidding and it is uncertain whether even the severely depleted present asteroid belt could form under these conditions. The perturbations of Jupiter and Saturn are quite sensitive to their semi-major axes and decrease when the planets' heliocentric distances are increased to allow for protoplanet migration. It is possible, though not clearly demonstrated, that this could produce a depleted asteroid belt but permit formation of a system of terrestrial planet embryos on a approximately 10(6)-year timescale, initially by nonrunaway growth and transitioning to runaway growth after approximately 10(5) years. The calculations reported here are valid under the condition that the relative velocities of the bodies are determined only by Jupiter and Saturn perturbations and by gas drag, with no mutual perturbations between planetesimals. If, while subject to these conditions, the bodies become large enough for their mutual perturbations to influence their velocity and size evolution significantly, the problem becomes much more complex. This problem is under investigation.  相似文献   

18.
We have performed new simulations of two different scenarios for the excitation and depletion of the primordial asteroid belt, assuming Jupiter and Saturn on initially circular orbits as predicted by the Nice Model of the evolution of the outer Solar System [Gomes, R., Levison, H.F., Tsiganis, K., Morbidelli, A., 2005. Nature 435, 466-469; Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F., 2005. Nature 435, 459-461; Morbidelli, A., Levison, H.F., Tsiganis, K., Gomes, R., 2005. Nature 435, 462-465]. First, we study the effects of sweeping secular resonances driven by the depletion of the solar nebula. We find that these sweeping secular resonances are incapable of giving sufficient dynamical excitation to the asteroids for nebula depletion timescales consistent with estimates for solar-type stars, and in addition cannot cause significant mass depletion in the asteroid belt or produce the observed radial mixing of different asteroid taxonomic types. Second, we study the effects of planetary embryos embedded in the primordial asteroid belt. These embedded planetary embryos, combined with the action of jovian and saturnian resonances, can lead to dynamical excitation and radial mixing comparable to the current asteroid belt. The mass depletion driven by embedded planetary embryos alone, even in the case of an eccentric Jupiter and Saturn, is roughly 10-20× less than necessary to explain the current mass of the main belt, and thus a secondary depletion event, such as that which occurs naturally in the Nice Model, is required. We discuss the implications of our new simulations for the dynamical and collisional evolution of the main belt.  相似文献   

19.
《Icarus》1987,72(2):276-303
Proper elements have been calculated for 1227 higher accuracy orbits from the Palomar-Leiden Survey of faint minor planets. Tabulations are given for the special orbits: Earth and deep Mars crossers, Trojans, Hildas, and one 2:1 librator. The frequency distributions of the proper semimajor axis, eccentricity, sine inclination, longitudes of perihelion and node plus their rates, and the closest distances of approach to Mars and Jupiter are displayed as histograms and discussed. The distribution of the closest approach distance to Mars drops off sharply near zero while that for Jupiter vanishes near 1.1 AU. Mars and Jupiter have apparently caused these boundaries and the asteroid belt must have been larger early in the history of the solar system. 3.5% of the sample can impact Mars. Most of these potential impactors are shallow crossers which require occasional fortuitous alignments of the secular terms to intersect Mars' orbit so that long lifetimes result and moderate populations remain. As these fortuitous alignments occur with near simultaneity for many, but not all, asteroids the shallow crossers in the observed size range will episodically bombard Mars. Proper elements have been used to recognize families and 49% of the sample of minor planets fall into these families. The proper elements and family assignments are tabulated.  相似文献   

20.
We have accumulated thousands of orbits of test particles in the Solar System from the asteroid belt to beyond the orbit of Neptune. We find that the time for an orbit to make a close encounter with a perturbing planet, T c ,is a function of the Lyapunov time, T ty .The relation is log (T c /T o )= a + b log (T ly T o )where T o is a fiducial period which we have taken as the period of the principal perturber or the period of the asteroid. There are exceptions to this rule interior to the 2/3 resonance with Jupiter. There, at least in the restricted problem, for sufficiently small Jupiter mass, orbits may have a positive Lyapunov exponent and still be blocked from having a close approach to Jupiter by a zero velocity curve. Of more serious concern is whether the relation holds for purely secular resonances, and if it does, how to choose T o .This is the case of interest for the planets in the solar system.  相似文献   

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