On the origin of the unusual orbit of Comet 2P/Encke

The orbit of Comet 2P/Encke is difficult to understand because it is decoupled from Jupiter—its aphelion distance is only 4.1 AU. We present a series of orbital integrations designed to determine whether the orbit of Comet 2P/Encke can simply be the result of gravitational interactions between Jupiter-family comets and the terrestrial planets. To accomplish this, we integrated the orbits of a large number of objects from the trans-neptunian region, through the realm of the giant planets, and into the inner Solar System. We find that at any one time, our model predicts that there should be roughly 12 objects in Encke-like orbits. However, it takes roughly 200 times longer to evolve onto an orbit like this than the typical cometary physical lifetime. Thus, we suggest that (i) 2P/Encke became dormant soon after it was kicked inward by Jupiter, (ii) it spent a significant amount of time inactive while rattling around the inner Solar System, and (iii) it only became active again as the ν₆ secular resonance drove down its perihelion distance.     

The Role of Hyperbolic Invariant Sets in Stickiness Effects

With the standard map model, we study the stickiness effect of invariant tori, particularly the role of hyperbolic sets in this effect. The diffusion of orbits originated from the neighborhoods of hyperbolic points, periodic islands and torus is studied. We find that they possess similar diffusion rules, but the diffusion of orbits originated from the neighborhood of a torus is faster than that originated near a hyperbolic set. The numerical results show that an orbit in the neighborhood of a torus spends most of time around hyperbolic invariant sets. We also calculate the areas of islands with different periods. The decay of areas with the periods obeys a power law, and the absolute values of the exponents increase monotonously with the perturbation parameter. According to the results obtained, we conclude that the stickiness effect of tori is caused mainly by the hyperbolic invariant sets near the tori, and the diffusion speed becomes larger when orbits diffuse away from the torus.     

Orbital evolution of giant comet-like objects

We investigated by numerical integrations the long-term orbital evolution of four giant comets or comet-like objects. They are Chiron, P/Schwassmann-Wachmann 1 (SW1), Hidalgo, and 1992AD (5145), and their orbits were traced for 100–200 thousand years (kyr) toward both the past and the future. For each object, 13 orbits were calculated, one for the nominal orbital elements and other 12 with slightly modified elements based on the rms residual of the orbit determination and on the number of observations. As past studies indicate, their orbital evolution is found to be very chaotic, and thus can be described only in terms of probability. Plots of the semi-major axis (a) and perihelion distance (q) of the objects treated here seem to cross each other frequently, suggesting a possibility of their common evolutionary paths. About a half of all the calculated orbits showedq- ora-decreasing evolution. This indicates that, at least on the time scale in question, the giant comet-like objects are possibly on a dynamical track that can lead to capture from the outer solar system. We could hardly find the orbits with perihelia far outside the orbit of Saturn (q>15 AU). This is perhaps because the evolution of the orbits beyond Saturn is so slow that substantial orbital changes do not take place within 100–200 kyr.     

A novel algorithm for generating libration point orbits about the collinear points

This paper presents a numerical algorithm that can generate long-term libration points orbits (LPOs) and the transfer orbits from the parking orbits to the LPOs in the circular-restricted three-body problem (CR3BP) and the full solar system model without initial guesses. The families of the quasi-periodic LPOs in the CR3BP can also be constructed with this algorithm. By using the dynamical behavior of LPO, the transfer orbit from the parking orbit to the LPO is generated using a bisection method. At the same time, a short segment of the target LPO connected with the transfer orbit is obtained, then the short segment of LPO is extended by correcting the state towards its adjacent point on the stable manifold of the target LPO with differential evolution algorithm. By implementing the correction strategy repeatedly, the LPOs can be extended to any length as needed. Moreover, combining with the continuation procedure, this algorithm can be used to generate the families of the quasi-periodic LPOs in the CR3BP.     

Motion near frozen orbits as a means for mitigating satellite relative drift

Generally, any initially-close satellites—chief and deputy—moving on orbits with slightly different orbital elements, will depart each other on locally unbounded relative trajectories. Thus, constraints on the initial conditions must be imposed to mitigate the chief-deputy mutual departure. In this paper, it is analytically proven that choosing the chief’s orbit to be a frozen orbit can mitigate the natural relative drift of the satellites. Using mean orbital element variations, it is proven that if the chief’s orbit is frozen, then the mean differential eccentricity is periodic, leading to a periodic variation of the differential mean argument of latitude. On the other hand, if the chief’s orbit is non-frozen, a secular growth in the differential mean argument of latitude leads to a concomitant along-track separation of the deputy from the chief, thereby considerably increasing the relative distance evolution over time. Long-term orbital simulation results indicate that the effect of choosing a frozen orbit vis-à-vis a non-frozen orbit can reduce the relative distance drift by hundreds of meters per day.     

Exploiting Unstable Periodic Orbits of a Chaotic Invariant Set for Spacecraft Control

The purpose of this work is to show that chaos control techniques (OGY, in special) can be used to efficiently keep a spacecraft around another body performing elaborate orbits. We consider a satellite and a spacecraft moving initially in coplanar and circular orbits, with slightly different radii, around a heavy central planet. The spacecraft, which is the inner body, has a slightly larger angular velocity than the satellite so that, after some time, they eventually go to a situation in which the distance between them becomes sufficiently small, so that they start to interact with one another. This situation is called as an encounter. In previous work we have shown that this scenario is a typical situation of a chaotic scattering for some well-defined range of parameters. Considering this scenario, we first show how it is possible to find the unstable periodic orbits that are located in the chaotic invariant set. From the set of unstable periodic orbits, we select the ones that can be combined to provide the desired elaborate orbit. Then, chaos control technique based on the OGY method is used to keep the spacecraft in the desired orbit. Finally, we analyze the results and make considerations regarding a realistic scenario of space exploration.     

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.     

Planet crossing asteroids and parallel computing: Project spaceguard

The orbits of the asteroids crossing the orbit of the Earth and other planets are chaotic and cannot be computed in a deterministic way for a time span long enough to study the probability of collisions. It is possible to study the statistical behaviour of a large number of such orbits over a long span of time, provided enough computing resources and intelligent post processing software are available. The former, problem can be handled by exploiting the enormous power of parallel computing systems. The orbit of the asteroids can be studied as a restricted (N+M)-body problem which is suitable for the use of parallel processing, by using one processor to compute the orbits of the planets and the others to compute the orbits the asteroids. This scheme has been implemented on LCAP-2, an array of IBM and FPS processors with shared memory designed by E. Clementi (IBM). The parallelisation efficiency has been over 80%, and the overall speed over 90 MegaFLOPS; the orbits of all the asteroids with perihelia lower than the aphelion of Mars (410 objects) have been computed for 200,000, years (Project SPACEGUARD). The most difficult step of the project is the post processing of the very large amount of output data and to gather qualitative information on the behaviour of so many orbits without resorting to the traditional technique, i.e. human examination of output in graphical form. Within Project SPACEGUARD we have developed a qualitative classification of the orbits of the planet crossers. To develop an entirely automated classification of the qualitative orbital behaviour-including crossing behaviour, resonances (mean motion and secular), and protection mechanisms avoiding collisions-is a challenge to be met.     

Theoretical meteor radiants for macroscopic Taurid Complex objects

The calculation of theoretical meteor radiants is discussed for comets and asteroids whose orbits pass within, but at present do not necessarily intersect, that of the Earth, in particular from the perspective of developing a suitable method for application to Taurid Complex orbits. The main question addressed here is how to allow for dynamical evolution between epochs when an orbit isnot Earth-intersecting (as at present in most cases for macroscopic bodies) and those when itis (i.e., when meteors can actually be observed). This should be understood in terms of evolution in the past, such that meteoroids released some time ago have evolved differentially from the putative parents, allowing meteors to be detected now. Theoretical radiants for macroscopic Taurid objects are then presented and compared with observations of the nighttime and daytime Taurid meteor showers. These are found to be broadly similar in form, given the sparsity of some of the data, adding weight to the hypothesis that this sub-jovian complex contains kilometre-plus asteroids. A similar conclusion results for the group of objects in similar orbits to (2212) Hephaistos.     

Optimal periodic relative orbit and rectilinear relative orbits with eccentric reference orbits

The problem of two-body linearized periodic relative orbits with eccentric reference orbits is studied in this paper. The periodic relative orbit in the target-orbital coordinate system can be used in fly-around and formation-flying orbit design. Based on the closed-form solutions to the Tschauner–Hempel equations, the initial condition for periodic relative orbits is obtained. Then the minimum-fuel periodic-orbit condition with a single impulse is analytically derived for given initial position and velocity vectors. When considering the initial coasting time, the impulse position of the global minimum-fuel periodic orbit is proved to be near to the perigee of the target and can be obtained by numerical optimization algorithms. Moreover, the condition for a special periodic orbit, i.e., the rectilinear relative orbit in the target-orbital frame, is obtained. Numerical simulations are used to demonstrate the efficacy of the method, and show the geometry of the periodic relative orbit and the rectilinear relative orbit.     

High-order expansion of the solution of preliminary orbit determination problem

A method for high-order treatment of uncertainties in preliminary orbit determination is presented. The observations consist in three couples of topocentric right ascensions and declinations at three observation epochs. The goal of preliminary orbit determination is to compute a trajectory that fits with the observations in two-body dynamics. The uncertainties of the observations are usually mapped to the phase space only when additional observations are available and a least squares fitting problem is set up. A method based on Taylor differential algebra for the analytical treatment of observation uncertainties is implemented. Taylor differential algebra allows for the efficient computation of the arbitrary order Taylor expansion of a sufficiently continuous multivariate function. This enables the mapping of the uncertainties from the observation space to the phase space as high-order multivariate Taylor polynomials. These maps can then be propagated forward in time to predict the observable set at successive epochs. This method can be suitably used to recover newly discovered objects when a scarce number of measurements is available. Simulated topocentric observations of asteroids on realistic orbits are used to assess the performances of the method.     

Orbit computation for transneptunian objects

Using statistical orbital ranging, we systematically study the orbit computation problem for transneptunian objects (TNOs). We have automated orbit computation for large numbers of objects, and, more importantly, we are able to obtain orbits even for the most sparsely observed objects (observational arcs of a few days). For such objects, the resulting orbit distributions include a large number of high-eccentricity orbits, in which TNOs can be perturbed by close encounters with Neptune. The stability of bodies on the computed orbits has therefore been ascertained by performing a study of close encounters with the major planets. We classify TNO orbit distributions statistically, and we study the evolution of their ephemeris uncertainties. We find that the orbital element distributions for the most numerous single-apparition TNOs do not support the existence of a postulated sharp edge to the belt beyond 50 AU. The technique of statistical ranging provides ephemeris predictions more generally than previously possible also for poorly observed TNOs.     

任务轨道设计

本文首先说明太空任务与轨道设计的关系 ,接着介绍轨道的基本性质。从地球重力势的观点看各种常用的绕地轨道 ,包括地球和太阳同步轨道及Molniya轨道。从扰动的观点看常用的星际轨道 ,包括LISA、ASTROD、SOHO轨道。最后对星际轨道设计 ,说明二点边界值问题的数值解法、飞掠星体的应用、最佳化的考虑 ,并用以设计 2 0 1 5年发射的ASTROD初步任务轨道。     

Study on the Orbit Prediction Errors of Space Objects Based on Historical TLE Data

Two line element (TLE) released by the North American Aerospace Defense Command (NORAD) is widely used by aerospace workers, and the matched SGP4/SDP4 (Simplified General Perturbation Version 4/Simplified Deep-space Perturbation Version 4) model is used to propagate it. Nevertheless, no corresponding information about its accuracy and covariance is clearly given, thus the application of the TLE data is greatly restricted. In this paper, the determined and predicted orbits are compared to generate the orbit error data, based on the historical TLE data obtained from the Space-Track website and the SGP4/SDP4 model. By dividing different time bins, the fitting coefficients of the variation of orbit prediction error with time are given for each space object, and the characteristics of the error evolution are further discussed for the different types of orbits. The mean analytic model of the orbit prediction error evolution with time is given respectively for the four orbit types of space objects, which provides a valuable reference for extending the application of the TLE data.     

Distribution of lunar impacts by objects in parabolic low inclination orbits

This paper presents a computer investigation extending to the case of parabolic orbits, an earlier investigation conducted by Barricelli and Metcalfe (1969) on lunar impacts by external low eccentricity satellites as a means to interpret the asymmetric distribution of lunar maria. Parabolic orbits can be approximated by two kinds of objects:

1. High eccentricity external satellites may, near periapsis, approach the Moon with orbital velocity and other characteristics closely resembling those of a parabolic orbit.
2. Asteroids and meteoroids approaching the Earth-Moon system with a low velocity may have moved in a nearly parabolic orbit when they reached the lunar distance from the Earth at the time when the impacts which carved the lunar maria took place.

The investigation gives, therefore, not only additional information relevant to the interpretation of the distribution of lunar maria by the satellite impacts hypothesis (in this case high eccentricity ones), but also information about the alternative hypothesis (Wood, 1973) that asteroid impacts rather than satellite impacts were involved.     

The optimization of satellite orbit for Space-VLBI observation

By sending one or more telescopes into space,Space-VLBI(SVLBI)is able to achieve even higher angular resolution and is therefore the trend of the VLBI technique.For the SVLBI program,the design of satellite orbits plays an important role for the success of planned observation.In this paper,we present our orbit optimization scheme,so as to facilitate the design of satellite orbits for SVLBI observation.To achieve that,we characterize the uv coverage with a measure index and minimize it by finding out the corresponding orbit configuration.In this way,the design of satellite orbit is converted to an optimization problem.We can prove that,with an appropriate global minimization method,the best orbit configuration can be found within the reasonable time.Besides that,we demonstrate that this scheme can be used for the scheduling of SVLBI observations.     

Coupled orbit-attitude dynamics of high area-to-mass ratio (HAMR) objects: influence of solar radiation pressure, Earth’s shadow and the visibility in light curves

The orbital and attitude dynamics of uncontrolled Earth orbiting objects are perturbed by a variety of sources. In research, emphasis has been put on operational space vehicles. Operational satellites typically have a relatively compact shape, and hence, a low area-to-mass ratio (AMR), and are in most cases actively or passively attitude stabilized. This enables one to treat the orbit and attitude propagation as decoupled problems, and in many cases the attitude dynamics can be neglected completely. The situation is different for space debris objects, which are in an uncontrolled attitude state. Furthermore, the assumption that a steady-state attitude motion can be averaged over data reduction intervals may no longer be valid. Additionally, a subset of the debris objects have significantly high area-to-mass ratio (HAMR) values, resulting in highly perturbed orbits, e.g. by solar radiation pressure, even if a stable AMR value is assumed. Note, this assumption implies a steady-state attitude such that the average cross-sectional area exposed to the sun is close to constant. Time-varying solar radiation pressure accelerations due to attitude variations will result in un-modeled errors in the state propagation. This work investigates the evolution of the coupled attitude and orbit motion of HAMR objects. Standardized pieces of multilayer insulation (MLI) are simulated in a near geosynchronous orbits. It is assumed that the objects are rigid bodies and are in uncontrolled attitude states. The integrated effects of the Earth gravitational field and solar radiation pressure on the attitude motion are investigated. The light curves that represent the observed brightness variations over time in a specific viewing direction are extracted. A sensor model is utilized to generate light curves with visibility constraints and magnitude uncertainties as observed by a standard ground based telescope. The photometric models will be needed when combining photometric and astrometric observations for estimation of orbit and attitude dynamics of non-resolved space objects.     

Evidence for the existence of groups of meteorite-producing asteroidal fragments

Abstract— The MORP camera network in western Canada observed 56 events which we associate with meteorites larger than 0.1 kg. An additional 33 Prairie Network (central USA) fireballs with published orbits were previously identified as the sources of meteorites of at least 0.25 kg. A comparison of the MORP orbits with each other and with the PN orbits, using the D′ criterion of orbital similarity, exhibits a surprising number of small values. This suggests there are groups of related objects among the 89 events. We evaluate the probability of small values of D′ arising by chance from a group of random orbits that has the distribution of orbital elements expected for meteorites. There is an excess of small values of D′ among the 89 meteoritic objects over the expectation for random orbits and a marked excess of very small values. Four groups comprising a total of 16 objects account for this excess. These groups exhibit a preference for the larger masses of the population and a very strong concentration of perihelia just slightly inside the Earth's orbit. Although it has been shown by others that gravitational perturbations will disperse Earth-crossing streams in times that are much less than cosmic-ray exposure ages, the properties of the four groups suggest they may be streams of fragments that crossed the Earth's orbit only recently. Such streams may include a considerable fraction of meteorites falling at a given time. Orbital evolution of these streams could alter the sample of meteorites arriving on Earth over time intervals that are less than the accumulation time of the Antarctic collections.     

Quasi-terminator orbits near primitive bodies

Quasi-terminator orbits are introduced as a class of quasi-periodic trajectories in the solar radiation pressure (SRP) perturbed Hill dynamics. These orbits offer significant displacements along the Sun-direction without the need for station-keeping maneuvers. Thus, quasi-terminator orbits have application to primitive-body mapping missions, where a variety of observation geometries relative to the Sun (or other directions) can be achieved. This paper describes the characteristics of these orbits as a function of normalized SRP strength and invariant torus frequencies and presents a discussion of mission design considerations for a global surface mapping orbit design.