Secular resonances as a source of dynamic chaoticity in the long-term orbital evolution of uncontrolled satellites

We present the results of MEGNO analysis of the long-term orbital evolution of uncontrolled objects of satellite radio navigation systems in medium Earth orbits and geostationary orbits. It is shown that secular resonances affecting these objects may give rise to dynamic randomness in their long-term orbital evolution.     

Dynamic Structure of the GLONASS and GPS Orbital Space: Problem of Disposal of Retired Objects

The paper presents the results of a study of the dynamic structure of the orbital space of the navigation systems GLONASS and GPS. It is shown that the dynamic structure of the GLONASS region is determined by the action of one stable Lidov–Kozai secular resonance. The motion of almost all the retired objects of the GLONASS system is stable throughout the 100-year study period. In the GPS region, there is an orbital resonance and a large number of secular resonances. Their combined influence leads to a rapid increase in the eccentricity of the orbits of the retired objects of the system. Features of the dynamic structure of the orbital space are used to find the graveyard (parking) orbits of the retired objects of navigation systems.     

Geostationary secular dynamics revisited: application to high area-to-mass ratio objects

The long-term dynamics of the geostationary Earth orbits (GEO) is revisited through the application of canonical perturbation theory. We consider a Hamiltonian model accounting for all major perturbations: geopotential at order and degree two, lunisolar perturbations with a realistic model for the Sun and Moon orbits, and solar radiation pressure. The long-term dynamics of the GEO region has been studied both numerically and analytically, in view of the relevance of such studies to the issue of space debris or to the disposal of GEO satellites. Past studies focused on the orbital evolution of objects around a nominal solution, hereafter called the forced equilibrium solution, which shows a particularly strong dependence on the area-to-mass ratio. Here, we (i) give theoretical estimates for the long-term behavior of such orbits, and (ii) we examine the nature of the forced equilibrium itself. In the lowest approximation, the forced equilibrium implies motion with a constant non-zero average ‘forced eccentricity’, as well as a constant non-zero average inclination, otherwise known in satellite dynamics as the inclination of the invariant ‘Laplace plane’. Using a higher order normal form, we demonstrate that this equilibrium actually represents not a point in phase space, but a trajectory taking place on a lower-dimensional torus. We give analytical expressions for this special trajectory, and we compare our results to those found by numerical orbit propagation. We finally discuss the use of proper elements, i.e., approximate integrals of motion for the GEO orbits.     

Resonant dynamics of Medium Earth Orbits: space debris issues

The Medium Earth Orbit region is the home of the navigation constellations. It is shown how the orbits of these constellations of satellites are strongly affected by the so-called inclination dependent luni-solar resonances. The analytical theory of these resonances is recalled and a large set of numerical integrations is used to investigate the stability of the orbits of the constellations over very long time spans. The stability issue is important in the definition of possible disposal strategies for the constellation spacecraft, after their end-of-life. Two possible disposal strategies are envisaged involving either stable or unstable orbits (from the eccentricity growth point of view). In particular it is shown how, to have disposal orbits with moderately short lifetime (of the orders of 40–50 years), a very large disposal maneuver would be required to rise the inital eccentricity to values above ~ 0.3.     

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.     

An analysis of the region of the Phocaea dynamical family

In this work, I conduct a preliminary analysis of the Phocaea family region. I obtain families and clumps in the space of proper elements and proper frequencies, study the taxonomy of the asteroids for which this information is available, analyse the albedo and absolute magnitude distribution of objects in the area, obtain a preliminary estimate of the possible family age, study the cumulative size distribution and collision probabilities of asteroids in the region, the rotation rate distribution and obtain dynamical map of averaged elements and Lyapunov times for grids of objects in the area.
Among my results, I identified the first clump visible only in the frequency domain, the (6246) Komurotoru clump, obtained a higher limit for the possible age of the Phocaea family of 2.2 Byr, identified a class of Phocaea members on Mars-crossing orbits characterized by high Lyapunov times and showed that an apparently stable region on time-scales of 20 Myr near the  ν₆  secular resonance is chaotic, possibly because of the overlapping of secular resonances in the region. The Phocaea dynamical group seems to be a real S-type collisional family, formed up to 2.2 Byr ago, whose members with a large semimajor axis have been dynamically eroded by the interaction with the local web of mean-motion and secular resonances. Studying the long-term stability of orbits in the chaotic regions and the stability of family and clumps identified in this work remain challenges for future works.     

The assessment of the semi-analytical method in the long-term orbit prediction of Earth satellites

To understand the long-term evolution and distribution of the space objects, it is necessary to predict their orbits. Compared with the short-term prediction of a few days, the priority concerns of the long-term orbit prediction are the calculation speed, and the accuracies of major orbital elements, including the semi-major axis and eccentricity which define the shape of the orbit, as well as the orbital inclination and the right ascension of ascending node which define the orientation of the orbit. Given these requirements, it is preferable to adopt the semi-analytical method, which averages the system over the orbital period, and integrates the averaged system using the numerical method. It is not new, however, in the available literature, we can hardly find a quantitative assessment regarding its accuracy and speed when it is applied to various types of orbits. In this paper, we would like to report our implementation and assessment of the semi-analytical method, expecting that it would help to estimate its feasibility in the long-term orbit prediction. The quantitative assessment covers the commonly used orbits for the Earth satellites. In some rare and special cases where the performance of our method appears abnormal, we discuss the reasons and possible solutions. We hope our results can provide some useful reference for the similar applications of the semi-analytical method since our method is a relatively common approach in terms of both accuracy and implementation.     

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.     

The Dynamics of Objects in the Inner Edgeworth–Kuiper Belt

Objects in 3:2 mean motion resonance with Neptune are protected from close encounters with Neptune by the resonance. Bodies in orbits with semi-major axis between 39.5 and about 42 AU are not protected by the resonance; indeed due to overlapping secular resonances, the eccentricities of orbits in this region are driven up so that a close encounter with Neptune becomes inevitable. It is thus expected that such orbits are unstable. The list of known Trans-Neptunian objects shows a deficiency in the number of objects in this gap compared to the 43–50 AU region, but the gap is not empty. We numerically integrate models for the initial population in the gap, and also all known objects over the age of the Solar System to determine what fraction can survive. We find that this fraction is significantly less than the ratio of the population in the gap to that in the main belt, suggesting that some mechanism must exist to introduce new members into the gap. By looking at the evolution of the test body orbits, we also determine the manner in which they are lost. Though all have close encounters with Neptune, in most cases this does not lead to ejection from the Solar System, but rather to a reduced perihelion distance causing close encounters with some or all of the other giant planets before being eventually lost from the system, with Saturn appearing to be the cause of the ejection of most of the objects.     

Capture of comets from the Oort cloud into Halley-type and Jupiter-family orbits

This paper analyzes the capture of comets into Halley-type and Jupiter-family orbits from the nearparabolic flux of the Oort cloud. Two types of capture into Halley-type orbits are found. The first type is the evolution of near-parabolic orbits into short-period orbits (with heliocentric orbital periods P < 200 years) as a result of close encounters with giant planets. This process is followed by a very slow drift of cometary orbits into the inner part of the Solar System. Only those comets may pass from short-period orbits into Halley-type and Jupiter-family orbits, which move in orbits with perihelion distances q < 13 au. In the second type of capture, the perihelion distances of cometary orbits become rather small (< 1.5 au) during the first stage of dynamic evolution under the action of perturbations from the Galaxy, and then their semimajor axes decrease as a result of diffusion. The capture takes place, on average, in 500 revolutions of the comet about the Sun, whereas in the first case, the comet is captured, on average, after 12500 revolutions. The region of initial orbital perihelion distances q > 4 au is found to be at least as important a source of Halley-type comets as the region of perihelion distances q < 4 au. More than half of the Halley-type comets are captured from the nearly parabolic flux with q > 4 au. The analysis of the dynamic evolution of objects moving in short-period orbits shows that the distribution of Centaurs orbits agrees well with the observed distribution corrected for observational selection effects. Hence, the hypothesis associating the origin of Centaurs with the Edgeworth-Kuiper belt and the trans-Neptunian region exclusively should be rejected.     

Optical surveys for space debris

Space debris—man-made non-functional objects of all sizes in near-Earth space—has been recognized as an increasing threat for current and future space operations. The debris population in near-Earth space has therefore been extensively studied during the last decade. Information on objects at altitudes higher than about 2,000 km is, however, still comparatively sparse. Debris in this region is best detected by surveys utilizing optical telescopes. Moreover, the instruments and the applied observation techniques, as well as the processing methods, have many similarities with those used in optical surveys for ‘astronomical’ objects like near-Earth objects (NEOs). The present article gives a general introduction to the problem of space debris, presents the used observation and processing techniques emphasizing the similarities and differences compared to optical surveys for NEOs, and reviews the results from optical surveys for space debris in high-altitude Earth orbits. Predictions on the influence of space debris on the future of space research and space astronomy in particular are reported as well.     

The Effect of Atmospheric Drag on Satellite Orbits During the Bastille Day Event

Geomagnetic storms driven by solar eruptions are known to have significant effects on the total density of the upper atmosphere in the altitude range 250–1000 km. This in turn causes a measurable effect on the orbits of resident space objects in this altitude range. We analyzed a sample of these orbits, both from sensor data and from orbital element sets, during the period surrounding the 14 July 2000 solar activity. We present information concerning the effects of this event on the orbits of resident space objects and how well accepted atmospheric models were able to represent it. As part of this analysis, we describe a technique for extracting atmospheric density information from orbital element sets. On daily time scales, the effect of geomagnetic activity appears to be more important than that of prompt radiation. However, the limitations in time and amplitude quantization of the accepted solar indices are evident. A limited comparison is also made with previous solar storm events.     

Circular and zero-inclination solutions for optical observations of Earth-orbiting objects

Situational awareness of Earth-orbiting particles is important for human extraterrestrial activities. Given an optical observation, an admissible region can be defined over the topocentric range/range-rate space, with each point representing a possible orbit for the object. However, based on our understanding of Earth orbiting objects, we expect that certain orbits in that distribution, such as circular or zero-inclination orbits, would be more likely than others. In this research, we present an analytical approach for describing the existence of such special orbits for a given observation pass, and investigate topological features of the range/range-rate space by means of singularities in orbital elements.     

A symplectic mapping model for the 3/4 exterior

We present a symplectic mapping model to study the evolution of a small body at the 3/4 exterior resonance with Neptune, for planar and for three dimensional motion. The mapping is based on the averaged Hamiltonian close to this resonance and is constructed in such a way that the topology of its phase space is similar to that of the Poincaré map of the elliptic restricted three-body problem. Using this model we study the evolution of a small object near the 3/4 resonance. Both chaotic and regular motions are found, and it is shown that the initial phase of the object plays an important role on the appearance of chaos. In the planar case, objects that are phase-protected from close encounters with Neptune have regular orbits even at eccentricities up to 0.44. On the other hand objects that are not phase protected show chaotic behaviour even at low eccentricities. The introduction of the inclination to our model affects the stable areas around the 3/4 mean motion resonance, which now become thinner and thinner and finally at is=10° the whole resonant region becomes chaotic. This may justify the absence of a large population of objects at this resonance.     

GLISSE: A GPU-optimized planetary system integrator with application to orbital stability calculations.

We present a GPU-accelerated numerical integrator specifically optimized for stability calculations of small bodies in planetary systems. Specifically, the integrator is designed for cases when large numbers of test particles (tens or hundreds of thousands) need to be followed for long durations (millions of orbits) to assess the orbital stability of their initially “close-encounter free” orbits. The GLISSE (Gpu Long-term Integrator for Solar System Evolution) code implements several optimizations to achieve a roughly factor of 100 speed increase over running the same code on a CPU. We explain how various hardware speed bottlenecks can be avoided by the careful code design, although some of the choices restrict the usage to specific types of application.As a first application, we study the long-term stability of small bodies initially on orbits between Uranus and Neptune. We map out in detail the small portion of the phase space in which small bodies can survive for 4.5 billion years of evolution; the ability to integrate large numbers of particles allow us to identify for the first time how instability-inducing mean-motion resonance overlaps sharply define the stable regions. As a second application, we map the boundaries of 4 Gyr stability for transneptunian objects in the 5:2 and 3:1 mean-motion resonances, demonstrating that long-term perturbations remove the initially stable Neptune-crossing members.     

Migration of Trans-Neptunian Objects to the Terrestrial Planets

The orbital evolution of more than 22000 Jupiter-crossing objects under thegravitational influence of planets was investigated. We found that the meancollision probabilities of Jupiter-crossing objects (from initial orbits close tothe orbit of a comet) with the terrestrial planets can differ by more than twoorders of magnitude for different comets. For initial orbital elements close tothose of some comets (e.g., 2P and 10P), about 0.1% of objects got Earth-crossingorbits with semi-major axes a < 2 AU and moved in such orbits for more than a Myr (up to tens or even hundreds of Myrs).Results of our runs testify in favor of at least one of these conclusions: (1) the portionof 1-km former trans-Neptunian objects (TNOs) among near-Earth objects (NEOs)can exceed several tens of percent, (2) the number of TNOs migrating inside the solarsystem could be smaller by a factor of several than it was earlier considered, (3) mostof 1-km former TNOs that had got NEO orbits disintegrated into mini-comets and dustduring a smaller part of their dynamical lifetimes if these lifetimes are not small.     

Observed spectral properties of near-Earth objects: results for population distribution, source regions, and space weathering processes

We present new visible and near-infrared spectroscopic measurements for 252 near-Earth (NEO) and Mars-crossing (MC) objects observed from 1994 through 2002 as a complement to the Small Main-Belt Asteroid Spectroscopic Survey (SMASS, http://smass.mit.edu/). Combined with previously published SMASS results, we have an internally consistent data set of more than 400 of these objects for investigating trends related to size, orbits, and dynamical history. These data also provide the basis for producing a bias-corrected estimate for the total NEO population (Stuart and Binzel, 2004, Icarus 170, 295-311). We find 25 of the 26 Bus (1999, PhD thesis) taxonomic types are represented, with nearly 90% of the objects falling within the broad S-, Q-, X-, and C-complexes. Rare A- and E-types are more common in the MC than NEO population (about 5% compared to <1%) and may be direct evidence of slow diffusion into MC orbits from the Flora and Hungaria regions, respectively. A possible family of MC objects (C-types) may reside at the edge of the 5:2 jovian resonance. Distinct signatures are revealed for the relative contributions of different taxonomic types to the NEO population through different source regions. E-types show an origin signature from the inner belt, C-types from the mid to outer belt, and P-types from the outer belt. S- and Q-types have effectively identical main-belt source region profiles, as would be expected if they have related origins. A lack of V-types among Mars-crossers suggests entry into NEO space via rapid transport through the ν6 and 3:1 resonances from low eccentricity main-belt orbits, consistent with a Vesta origin. D-types show the strongest signature from Jupiter family comets (JFC), with a strong JFC component also seen among the X-types. A distinct taxonomic difference is found with respect to the jovian Tisserand parameter T, where C-, D-, and X-type (most likely low albedo P-class) objects predominate for T?3. These objects, which may be extinct comets, comprise 4% of our observed sample, but their low albedos makes this magnitude limited fraction under-representative of the true value. With our taxonomy statistics providing a strong component to the diameter limited bias correction analysis of Stuart (2003, PhD thesis), we estimate 10-18% of the NEO population above any given diameter may be extinct comets, taking into account asteroids scattered into T<3 orbits and comets scattered into T>3 orbits. In terms of possible space weathering effects, we see a size-dependent transition from ordinary chondrite-like (Q-type) objects to S-type asteroids over the size range of 0.1 to 5 km, where the transition is effectively complete at 5 km. A match between the average surface age of 5 km asteroids and the rate of space weathering could constrain models for both processes. However, space weathering may proceed at a very rapid rate compared with collisional timescales. In this case, the presence or absence of a regolith may be the determining factor for whether or not an object appears “space weathered.” Thus 0.1 to 5 km appears to be a critical size range for understanding the processes, timescales, and conditions under which a regolith conducive to space weathering is generated, retained, and refreshed.     

Orbital correlation of space objects based on orbital elements

Orbital correlation of space objects is one of the most important elements in space object identification. Using the orbital elements, we provide correlation criteria to determine if objects are coplanar,co-orbital or the same. We analyze the prediction error of the correlation parameters for different orbital types and propose an orbital correlation method for space objects. The method is validated using two line elements and multisatellite launching data. The experimental results show that the proposed method is effective, especially for space objects in near-circular orbits.     

Dynamics,interrelations and evolution of the systems of asteroids and comets

The dynamics of larger interplanetary bodies is reviewed, with emphasis on evolutionary problems, interrelations, and open questions. Observational biases distinguishing the sample of known objects from the whole population are briefly discussed. A schematic division of the solar system into zones with different regimes of motion, and a rough taxonomy of orbit types are attempted. The role of individual major planets in controlling the dynamical evolution of interplanetary objects, in particular by stabilizing resonances and destabilizing close encounters, is compared. There are significant evolution asymmetries due to boundary conditions and preferential evolutionary paths; individual major displacements in the phase space of orbital elements conserve the Tisserand invariant with respect to the planet responsible, thus favouring certain evolutionary sequences against others. Very limited lifetimes of some orbit types imply a long-term balance between source and sink, and require a continuous supply of objects from other types of orbits. In this respect, the ultimate fate of extinct comets is of particular interest. Under very specific conditions, nongravitational effects of mass loss can result in stabilization of a formerly unstable orbit. Since the dividing line between the two basic interplanetary populations distinguished by origin and composition — the asteroids and the comets — is essentially that between stable and unstable motion, orbital data can be used to specify which of the known asteroid-like objects may be devolatilized cometary nuclei.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.