Orbital Element Space Research Articles

Jacobian Spheroids, Shallow Encounters, and the Keyhole Map

The study of the dynamics of flybys is featured in several space-mission-related applications, spanning from the design of interplanetary orbits to the computation of impact probabilities with given celestial bodies in planetary protection and defense tasks. While patched-conics, two-body models suffice in accuracy for trajectory planning cases, collision probability assessment requires instead more insightful approaches. High-fidelity simulations have shown impact patterns to happen outside nominally colliding trajectories in terms of two-body analysis, all caused by distant or weak interactions between the given trajectory-celestial body pairs. This work proposes a novel concept of sphere of influence based on the Jacobian of the three-body dynamics, resulting in a wider definition of encounters that encompasses weak interactions up to an arbitrary threshold. The so-defined Jacobian spheroids are then used to construct the Keyhole map, a graphical tool that maps in the orbital elements space both nominal and off-nominal keyholes, i.e., collision paths accounting for three-body weak interactions. The proposed concepts arise from and are compared against the on-ground casualty risk assessment of the European Space Agency’s mission JUICE.

On the Co-orbital Motion of Any Inclination

The co-orbital motion occurs when a celestial body (e.g. an asteroid) shares the same semimajor axis with the perturbing object (e.g. a planet), and thus they are in a 1:1 mean motion resonance. Trojan asteroids in the tadpole orbits around planets in the Solar System are these co-orbital objects. The motion and origin of some Trojan asteroids, particularly those on high-inclination orbits, are still not fully understood. In this paper, a newly developed perturbation function expansion, which is applicable to the 1:1 resonance, is used to investigate the co-orbital motion in three-dimensional space. The position of the resonance center and the resonance width are calculated for different initial orbital elements, and the relationship between the orbital types and the initial orbital elements is analyzed. The results obtained by the analytical method are compared with and verified by the results from numerical simulations. A panorama of the co-orbital motion in the wide orbital elements space is obtained.

Astrodynamics-Informed Kinodynamic Sampling-Based Motion Planning for Relative Spacecraft Motion

This paper presents a sampling-based spacecraft relative motion planning algorithm to reconfigure a spacecraft from a given initial state to a desired final state, subject to kinodynamic (simultaneous kinematic and dynamics) constraints. We leverage theoretical advancement in the field of sampling-based motion planning in robotics and extend their application to achieve motion planning in astrodynamics. We present an astrodynamics-informed kinodynamic motion planning (AIKMP) algorithm that takes advantage of a spacecraft’s natural motion to compute a safe, fuel-efficient motion plan for spacecraft reconfiguration in a cluttered environment. This algorithm is developed in the relative orbital element space of the deputy spacecraft around a chief spacecraft that provides very useful geometric insight to relative spacecraft motion, as well as helps in generating an overall smooth final transfer trajectory without demanding additional postprocessing of the transfer solution. Through several example scenarios, we demonstrate that the AIKMP algorithm is regularly able to compute safe and fuel-efficient solutions to relative transfer problems. We also show that this algorithm iteratively improves on its computed solutions and thus holds the capability of finding near-optimal transfer solutions given a sufficient number of algorithm iterations—without requiring an initial guess of the solution.

Relegation-free closed-form perturbation theory and the domain of secular motions in the restricted three-body problem

We propose a closed-form (i.e., without expansion in the orbital eccentricities) scheme for computations in perturbation theory in the restricted three-body problem (R3BP) when the massless particle is in an orbit exterior to the one of the primary perturber. Starting with a multipole expansion of the barycentric (Jacobi-reduced) Hamiltonian, we carry out a sequence of normalizations in Delaunay variables by Lie series, leading to a secular Hamiltonian model without use of relegation. To this end, we introduce a book-keeping analogous to the one proposed in Cavallari and Efthymiopoulos (Celest Mech Dyn Astron 134(2):1–36, 2022) for test particle orbits interior to the one of the primary perturber, but here adapted, instead, to the case of exterior orbits. We give numerical examples of the performance of the method in both the planar circular and the spatial elliptic restricted three-body problem, for parameters pertinent to the Sun-Jupiter system. In particular, we demonstrate the method’s accuracy in terms of reproducibility of the orbital elements’ variations far from mean-motion resonances. As a basic outcome of the method, we show how, using as criterion the size of the series’ remainder, we reach to obtain an accurate semi-analytical estimate of the boundary (in the space of orbital elements) where the secular Hamiltonian model arrived at after eliminating the particle’s fast degree of freedom provides a valid approximation of the true dynamics.

Asteroid spin-states of a 4 Gyr collisional family

Context. Families of asteroids generated by the collisional fragmentation of a common parent body have been identified using clustering methods of asteroids in their proper orbital element space. However, there is growing evidence that some of the real families are larger than the corresponding cluster of objects in orbital elements, and there are families that escaped identification by clustering methods. An alternative method has been developed in order to identify collisional families from the correlation between the asteroid fragment sizes and their proper semi-major axis distance from the family centre (V-shape). This method has been shown to be effective in the cases of the very diffuse families that formed billions of years ago. Aims. Here we use multiple techniques for observing asteroids to provide corroborating evidence that one of the groups of asteroids identified as a family from the correlation between size and proper semi-major axis of asteroids are real fragments of a common parent body, and thus form a collisional family. Methods. We obtained photometric observations of asteroids in order to construct their rotational light curves; we combine them with the literature light curves and sparse-in-time photometry; we input these data in the light curve inversion methods, which allow us to determine a convex approximation to the 3D shape of the asteroids and their orientation in space, from which we extract the latitude (or obliquity) of the spin pole in order to assess whether an object is prograde or retrograde. We included in the analysis spin pole solutions already published in the literature aiming to increase the statistical significance of our results. The ultimate goal is to assess whether we find an excess of retrograde asteroids on the inward side of the V-shape of a 4 Gyr asteroid family identified via the V-shape method. This excess of retrograde rotators is predicted by the theory of asteroid family evolution. Results. We obtained the latitude of the spin poles for 55 asteroids claimed to belong to a 4 Gyr collisional family of the inner main belt that consists of low-albedo asteroids. After re-evaluating the albedo and spectroscopic information, we found that nine of these asteroids are interlopers in the 4 Gyr family. Of the 46 remaining asteroids, 31 are found to be retrograde and 15 prograde. We also found that these retrograde rotators have a very low probability (1.29%) of being due to random sampling from an underlying uniform distribution of spin poles. Conclusions. Our results constitute corroborating evidence that the asteroids identified as members of a 4 Gyr collisional family have a common origin, thus strengthening their family membership.

Orbital Stability Around the Primary of a Binary Asteroid System

Orbiting the primary of a binary asteroid system is extremely challenging due to the perturbative effects of the primary’s nonspherical gravity and the secondary’s close-proximity third-body gravity. In this work, the stability of perturbed Keplerian orbits around the primary is investigated from the point of view of the long-term eccentricity oscillation. Numerical investigations indicate that the eccentricity undergoes a large-amplitude oscillation, caused by the secular perturbation of the secondary’s gravity and may cause impact. A two-degree-of-freedom dynamic model is established, based on the doubly averaged, semi-analytical orbital dynamics incorporating effects of the primary’s oblateness and the secondary’s nonspherical third-body gravity. The oscillation of eccentricity, including its phase and amplitude, and its dependence on initial orbital geometry, is investigated through the phase space structure. The results can reveal the origin of the instability, predict stable and unstable regions in the space of orbital elements, and determine the initial orbital geometry that can ensure the secular stability. The binary asteroid system 2003 YT1 is used as an example to present our verifications and analyses, and the results can also be applied to other binary asteroid systems and even planetary systems with the central body’s oblateness and the third-body gravity dominating the perturbative environments.

The young Adelaide family: Possible sibling to Datura?

Context.Very young asteroid families may record processes that accompanied their formation in the most pristine way. This makes analysis of this special class particularly interesting.Aims.We studied the very young Adelaide family in the inner part of the main belt. This cluster is extremely close to the previously known Datura family in the space of proper orbital elements and their ages overlap. As a result, we investigated the possibility of a causal relationship between the two families.Methods.We identified Adelaide family members in the up-to-date catalogue of asteroids. By computing their proper orbital elements we inferred the family structure. Backward orbital integration of selected members allowed us to determine the age of the family.Results.The largest fragment (525) Adelaide, an S-type asteroid about 10 km in size, is accompanied by 50 sub-kilometre fragments. This family is a typical example of a cratering event. The very tiny extent in the semi-major axis minimises chances that some significant mean motion resonances influence the dynamics of its members, though we recognise that part of the Adelaide family is affected by weak, three-body resonances. Weak chaos is also produced by distant encounters with Mars. Simultaneous convergence of longitude of node for the orbits of six selected members to that of (525) Adelaide constrains the Adelaide family age to 536 ± 12 kyr (formal solution). While suspiciously overlapping with the age of the Datura family, we find it unlikely that the formation events of the two families are causally linked. In all likelihood, the similarity of their ages is just a coincidence.

Обобщенный алгоритм определения параметров орбиты ИСЗ на основе квадратичных функционалов.

The questions of development a generalized algorithm for determining the parameters of the low circular orbit (LCO) of an Earth satellite (ES) based on the use of quadratic functionals are in the focus of this paper. The functionals represent the square of the differences between the measured values of the ES sighting angles and the frequency of the received signal with the values of the same parameters obtained for the assumed values of the Keplerian orbital elements in accordance with the adopted model of the ES motion. Estimates of the orbit parameters are formed from the condition of the minimum of the proposed quality functionals. The proposed algorithm is aimed at the developing two equations for the relationship between the measured values of the azimuth and elevation angles, as well as the frequency of the received satellite signal and the parameters of the satellite orbit. The use of the indicated constraint equations makes it possible to pass from the six-dimensional space of the Keplerian orbital elements to the four-dimensional space of the Keplerian orbital elements when constructing the algorithm and choosing the initial approximations of the orbit parameters. Such a reduction in the dimension of space makes it possible to significantly reduce the amount of computational expenditure, which ensures the stability of the algorithm and expands the possibilities of its practical use with limited resources (computing power and restrictions on the permissible processing time). The following Keplerian orbital elements are proposed as four basic parameters: eccentricity, ascending node longitude, orbital inclination, and perigee argument. The other two elements, the semi-major axis of the orbit and the mean anomaly, are expressed as functions of four basic parameters. This choice is determined by the fact that, in the case of LCO, the pivoting of the initial values of the eccentricity and the argument of perigee is quite simple, which makes it possible to ensure convergence to the exact values of the orbit parameters in a wide value of the initial approximations. Within the Keplerian approximation of the satellite's orbital motion, mathematical relations are presented that determine the operations performed within the framework of the considered algorithm. However, a more complete consideration of the factors influencing the motion of the satellite only leads to more volumetric relations, but does not fundamentally affect the construction of the algorithm itself.

Neural-aided GNC reconfiguration algorithm for distributed space system: development and PIL test

This paper presents a neural-aided Guidance, Navigation & Control algorithm for reconfiguration of distributed space systems. The guidance algorithm is based on Artificial Potential Fields (APF) in the Relative Orbital Elements (ROE) space. Since the relative orbit determination measurements are typically referred to the Cartesian metrics (e.g. range or range rate), a linear mapping between the set of ROE and the Cartesian coordinates expressed in the Local-Vertical-Local-Horizontal (LVLH) reference frame is derived. The navigation and control algorithms rely on the relative dynamics expressed in the same ROE set of coordinates. To cope with uncertainties and nonlinearities of the system, a Radial Basis Function Neural Network (RBFNN) is employed to reconstruct the perturbed dynamics. The Artificial Neural Network (ANN) is coupled with an adaptive Extended Kalman Filter for state estimation. A feedback control is designed to track the desired state, whose stability is analyzed using Lyapunov theory. The guidance, navigation and control algorithms are tested in a high-fidelity numerical orbit propagator. Moreover, the algorithm is tested in relevant Processor-In-the-Loop (PIL) simulations using a TI C2000-Delfino MCU F28379D. The results demonstrate the effectiveness of the algorithm for relative reconfiguration maneuvers involving relative distances ~102 m with limited fuel consumption and constrained available thrust (⩽1mN). In particular along-track maneuvers, relative plane change and formation enlargement are analysed in the paper, showing the comparison between the proposed algorithm and the legacy one without the neural network. The benefit of implementing a neural network is particularly highlighted when the nonlinearities or unmodelled terms in the on-board dynamics become prominent.

Physical and dynamical characterization of the Euphrosyne asteroid family

Aims. The Euphrosyne asteroid family occupies a unique zone in orbital element space around 3.15 au and may be an important source of the low-albedo near-Earth objects. The parent body of this family may have been one of the planetesimals that delivered water and organic materials onto the growing terrestrial planets. We aim to characterize the compositional properties as well as the dynamical properties of the family. Methods. We performed a systematic study to characterize the physical properties of the Euphrosyne family members via low-resolution spectroscopy using the NASA Infrared Telescope Facility. In addition, we performed smoothed-particle hydrodynamics (SPH) simulations and N-body simulations to investigate the collisional origin, determine a realistic velocity field, study the orbital evolution, and constrain the age of the Euphrosyne family. Results. Our spectroscopy survey shows that the family members exhibit a tight taxonomic distribution, suggesting a homogeneous composition of the parent body. Our SPH simulations are consistent with the Euphrosyne family having formed via a reaccumulation process instead of a cratering event. Finally, our N-body simulations indicate that the age of the family is 280−80+180 Myr, which is younger than previous estimates.

Inertial Navigation Method of Spacecraft Based on Geodesic Deviation Equation

A new method of spacecraft inertial navigation is proposed in paper. The method uses the earth gravitational field model and the on-board accelerometer to measure data, so the real-time deviation of the spacecraft relative to the geodetic orbit is calculated. Based on the post-Newtonian approximate gravity theory of general relativity, the high precision geodesic deviation equation around the earth is derived, and the equation of the spacecraft relative to the geodesic orbit is transformed into orbital element space. The position of the spacecraft is determined by calculating the deviation between the spacecraft and the nearest geodetic orbit. Because the reference points are selected different in the adjacent following inertia system during navigation, the accumulation error of the traditional inertial navigation is avoided, and the navigation precision can be effectively improved.

A Search for Young Asteroid Pairs with Close Orbits

We analyzed the dynamic evolution of a number of young asteroid pairs in close orbits in order to constrain their ages. Several methods of pair selection and estimation of their age are used: analysis of convergence of orbital elements; estimation of the Kholshevnikov metrics in the space of Keplerian orbital elements; estimation of relative distances and velocities at the moments of asteroid close approaches. Estimates of the age of asteroid pairs are obtained depending on the drift velocities of the semimajor axes of the orbits, due to the Yarkovsky effect.

Chaos in the inert Oort cloud

Context. Distant trans-Neptunian objects are subject to planetary perturbations and galactic tides. The former decrease with the distance, while the latter increase. In the intermediate regime where they have the same order of magnitude (the “inert Oort cloud”), both are weak, resulting in very long evolution timescales. To date, three observed objects can be considered to belong to this category. Aims. We aim to provide a clear understanding of where this transition occurs, and to characterise the long-term dynamics of small bodies in the intermediate regime: relevant resonances, chaotic zones (if any), and timescales at play. Methods. The different regimes are explored analytically and numerically. We also monitored the behaviour of swarms of particles during 4.5 Gyrs in order to identify which of the dynamical features are discernible in a realistic amount of time. Results. There exists a tilted equilibrium plane (Laplace plane) about which orbits precess. The dynamics is integrable in the low and high semi-major axis regimes, but mostly chaotic in between. From about 800 to 1100 astronomical units (au), the chaos covers almost all the eccentricity range. The diffusion timescales are large, but not to the point of being indiscernible in a 4.5 Gyrs duration: the perihelion distance can actually vary from tens to hundreds of au. Orbital variations are damped near the ecliptic (where previous studies focussed), but favoured in specific ranges of inclination corresponding to well-defined resonances. Moreover, starting from uniform distributions, the orbital angles cluster after 4.5 Gyrs for semi-major axes larger than 500 au, because of a very slow differential precession. Conclusions. Even if it is characterised by very long timescales, the inert Oort cloud mostly features chaotic regions; it is therefore much less inert than it appears. Orbits can be considered inert over 4.5 Gyrs only in small portions of the space of orbital elements, which include (90377) Sedna and 2012VP113. Effects of the galactic tides are discernible down to semi-major axes of about 500 au. We advocate including the galactic tides in simulations of distant trans-Neptunian objects, especially when studying the formation of detached bodies or the clustering of orbital elements.

Neptune Trojans: a Revisit and a New Membertwo

Up to now, 17 Neptune Trojan asteroids have been detected with their orbits being well determined by continuous observations. This paper analyzes systematically their orbital dynamics. Our results show that except for two temporary members with relatively short lifespans on Trojan orbits, the vast majority of Neptune Trojans located within their orbital uncertainties may survive in the solar system age. The escaping probability of Neptune Trojans, through slow diffusion in the orbital element space in 4.5 billion years, is estimated to be ∼50%. The asteroid 2012 UW177 classified as a Centaur asteroid by the IAU Minor Planet Center currently is in fact a Neptune Trojan. Numerical simulations indicate that it is librating on the tadpole-shaped orbit around the Neptune's L4 point. It was captured into the current orbit approximately 0.23 million years ago, and will stay there for at least another 1.3 million years in the future. Its high inclination of i≈54∘ not only makes it the most inclined Neptune Trojan, but also makes it exhibit the complicated and interesting co-orbital transitions between the leading and trailing Trojans via the quasi-satellite orbit phase.

Ancient and primordial collisional families as the main sources of X-type asteroids of the inner main belt

Aims. The near-Earth asteroid population suggests the existence of an inner main belt source of asteroids that belongs to the spectroscopic X complex and has moderate albedos. The identification of such a source has been lacking so far. We argue that the most probable source is one or more collisional asteroid families that have escaped discovery up to now. Methods. We apply a novel method to search for asteroid families in the inner main-belt population of asteroids belonging to the X complex with moderate albedo. Instead of searching for asteroid clusters in orbital element space, which could be severely dispersed when older than some billions of years, our method looks for correlations between the orbital semimajor axis and the inverse size of asteroids. This correlation is the signature of members of collisional families that have drifted from a common centre under the effect of the Yarkovsky thermal effect. Results. We identify two previously unknown families in the inner main belt among the moderate-albedo X-complex asteroids. One of them, whose lowest numbered asteroid is (161) Athor, is ~3 Gyr old, whereas the second one, whose lowest numbered object is (689) Zita, could be as old as the solar system. Members of this latter family have orbital eccentricities and inclinations that spread them over the entire inner main belt, which is an indication that this family could be primordial, that is, it formed before the giant planet orbital instability. Conclusions. The vast majority of moderate-albedo X-complex asteroids of the inner main belt are genetically related, as they can be included into a few asteroid families. Only nine X-complex asteroids with moderate albedo of the inner main belt cannot be included in asteroid families. We suggest that these bodies formed by direct accretion of the solids in the protoplanetary disc, and are thus surviving planetesimals.

Parallel initial orbit determination using angles-only observation pairs

We propose a type of admissible-region analysis for track initiation in multi-satellite problems when angles are the primary observable. Pairs of optical observations are used to calculate candidate orbits via a Lambert solver by hypothesizing range values. The method is attractive because it allows multiple levels of parallelization of the track-initiation process. Orbital element partitions are introduced to divide the admissible region into smaller search spaces to be processed on individual computer nodes. For a specified rectangular partition in the space of orbital elements, constraints are developed to bound the values of range that will lead to initial orbit hypotheses (data association hypotheses) associated with that partition. These bounds allow us to parallelize the generation of candidate orbits, because each element-space partition can be handled independently of the others. Several constraints are developed and shown to limit the range pair hypotheses effectively to the constrained admissible region based on the orbital element partitions. Examples are provided to highlight the topology of the proposed constraints.

The common origin of family and non-family asteroids

All asteroids are currently classified as either family, originating from the disruption of known bodies, or non-family. An outstanding question is the origin of these non-family asteroids. Were they formed individually, or as members of known families but with chaotically evolving orbits, or are they members of old ghost families, that is, asteroids with a common parent body but with orbits that no longer cluster in orbital element space? Here, we show that the sizes of the non-family asteroids in the inner belt are correlated with their orbital eccentricities and anticorrelated with their inclinations, suggesting that both non-family and family asteroids originate from a small number of large primordial planetesimals. We estimate that ~85% of the asteroids in the inner main belt originate from the Flora, Vesta, Nysa, Polana and Eulalia families, with the remaining ~15% originating from either the same families or, more likely, a few ghost families. These new results imply that we must seek explanations for the differing characteristics of the various meteorite groups in the evolutionary histories of a few, large, precursor bodies. Our findings also support the model that asteroids formed big through the gravitational collapse of material in a protoplanetary disk.

Probabilistic Optical and Radar Initial Orbit Determination

Future space surveillance requires dealing with uncertainties directly in the initial orbit determination phase. An approach based on Taylor differential algebra is proposed to both solve the initial orbit determination problem and to map uncertainties from the observables space into the orbital elements space. This is achieved by approximating in Taylor series the general formula for probability density function mapping through nonlinear transformations. In this way, the mapping is obtained in an elegant and general fashion. The proposed approach is applied to both angles-only and two position vectors initial orbit determination for objects in low Earth orbit and geostationary equatorial orbit.

Statistics of impacts among orbiting bodies: a Monte Carlo approach

In this paper, we describe a method to investigate the statistics of collisions among bodies orbiting around a common central mass but not interacting with each other, like minor bodies of the Solar system. This method can be used to derive the frequency of collisions and the distribution of any dynamical parameter related to the collision circumstances. It is based on deriving the statistics of impacts from a random sampling of the orbital element space of the bodies under investigation. The fundamental approach is not new, but the mathematical framework is completely original and the final procedure is very simple and easily implementable. The motivation behind this work is to overcome all limitations due to the assumptions about the dynamical behaviour of orbits, which the methods developed so far are based on. We show all the theoretical details of the method and its practical usage, including the determination of error. We show a set of examples demonstrating the satisfactory agreement with independent approaches, when they are available. The limits and drawbacks of the methodology are also highlighted.

Mass transfer between debris discs during close stellar encounters

We study mass transfers between debris discs during stellar encounters. We carried out numerical simulations of close flybys of two stars, one of which has a disc of planetesimals represented by test particles. We explored the parameter space of the encounters, varying the mass ratio of the two stars, their pericentre and eccentricity of the encounter, and its geometry. We find that particles are transferred to the other star from a restricted radial range in the disc and the limiting radii of this transfer region depend on the parameters of the encounter. We derive an approximate analytic description of the inner radius of the region. The efficiency of the mass transfer generally decreases with increasing encounter pericentre and increasing mass of the star initially possessing the disc. Depending on the parameters of the encounter, the transfer particles have a specific distributions in the space of orbital elements (semimajor axis, eccentricity, inclination, and argument of pericentre) around their new host star. The population of the transferred particles can be used to constrain the encounter through which it was delivered. We expect that many stars experienced transfer among their debris discs and planetary systems in their birth environment. This mechanism presents a formation channel for objects on wide orbits of arbitrary inclinations, typically having high eccentricity but possibly also close-to-circular (eccentricities of about 0.1). Depending on the geometry, such orbital elements can be distinct from those of the objects formed around the star.