31 research outputs found
Nonlinearly stable equilibria in the Sun-Jupiter-Trojan-Spacecraft four body problem
The Trojan asteroids have been highlighted as a main target for future discovery missions, which will enable key questions about the formation of our Solar system to be answered. Programs like the Japanese Jupiter and Trojan Asteroids Exploration Programme are already testing technology demonstrators like the IKAROS spacecraft to enable future interplanetary missions to Jupiter and the Trojans. In this paper an analytic analysis of the stability of the Low thrust Sun Jupiter Asteroid Spacecraft system, is presented, from a Hamiltonian point of view. Setting the three primaries in the stable Lagrangian equilateral triangle configuration, eight natural (i.e. with zero thrust) equilibrium points are identified, four of which are close to the asteroid, two stable and two unstable, when considering as primaries the Sun and any other two bodies of the Solar System. Artificial equilibria, which can be seen as low thrust perturbations of the natural ones, are then taken into account with the aim of identifying their linearly stable subset. The Lyapunov stability of these marginally stable points is then analysed using basic KAM (Kolmogorov Arnold Moser) theory and Arnoldâs stability theorem. In order to apply such theorem an iterative procedure to reduce the Hamiltonian into Birkhoffâs Normal Form is applied up to fourth order, explicitly defining, at each step, the generating function of a symplectic transformation. Despite the complexity of this process, Normal Forms are a fundamental, necessary step for any application of KAM theory; such theory, transforming a non-integrable system into a sum of perturbed integrable ones, enables the computation of a high order analytical approximation of the system dynamics, plus an estimation of the discrepancy from the initial model. As an application of KAM theory, a proof of the nonlinear stability for the low thrust generated equilibrium points under non resonant conditions is found using Arnoldâs stability theorem. Results show that Lyapunov stability is guaranteed along the linearly stable domain with the exception of a set of points with zero measure where the conditions to apply Arnoldâs theorem are not satisfied
Analytic perturbative theories in highly inhomogeneous gravitational fields
Orbital motion about irregular bodies is highly nonlinear due to inhomogeneities in the gravitational field. Classical theories of motion close to spheroidal bodies cannot be applied as for inhomogeneous bodies the Keplerian forces do not provide a good approximation of the system dynamics. In this paper a closed form, analytical method for developing the motion of a spacecraft around small bodies is presented, for the so called fast rotating case, which generalize previous results to second order, arbitrary degree, gravitational fields. Through the application of two different Lie transformations, suitable changes of coordinates are found, which reduce the initial non integrable Hamiltonian of the system into an integrable one plus a negligible, perturbative remainder of higher degree. In addition, an explicit analytical formulation for the relegated, first and second order, arbitrary degree Hamiltonian for relatively high altitude motion in any inhomogeneous gravitational field is derived in closed-form. Applications of this algorithm include a method for determining initial conditions for frozen orbits around any irregular body by simply prescribing the desired inclination and eccentricity of the orbit. This method essentially reduces the problem of computing frozen orbits to a problem of solving a 2-D algebraic equation. Results are shown for the asteroid 433-Eros
Halo orbits around the collinear points of the restricted three-body problem
We perform an analytical study of the bifurcation of the halo orbits around
the collinear points , , for the circular, spatial, restricted
three--body problem. Following a standard procedure, we reduce to the center
manifold constructing a normal form adapted to the synchronous resonance.
Introducing a detuning, which measures the displacement from the resonance and
expanding the energy in series of the detuning, we are able to evaluate the
energy level at which the bifurcation takes place for arbitrary values of the
mass ratio. In most cases, the analytical results thus obtained are in very
good agreement with the numerical expectations, providing the bifurcation
threshold with good accuracy. Care must be taken when dealing with for
small values of the mass-ratio between the primaries; in that case, the model
of the system is a singular perturbation problem and the normal form method is
not particularly suited to evaluate the bifurcation threshold.Comment: 35 pages, 3 figures, updated version accepted for publication on
Physica
Analytical perturbative theories of motion in highly inhomogeneous gravitational fields : Ariadna AO/1-6790/11/NL/CBI
In this report we show that modern computer performances and state-of-the-art algebraic manipulator software are sufficiently developed to carry out our generalised analytical perturbative theory. This report addresses three technical aspects to develop a general perturbative theory and illustrates its power by applying it to investigate the inhomogeneous gravitational fields of asteroids
Qualitative and analytical results of the bifurcation thresholds to halo orbits
We study the dynamics in the neighborhood of the collinear Lagrangian points
in the spatial, circular, restricted three--body problem. We consider the case
in which one of the primaries is a radiating body and the other is oblate
(although the latter is a minor effect). Beside having an intrinsic
mathematical interest, this model is particularly suited for the description of
a mission of a spacecraft (e.g., a solar sail) to an asteroid.
The aim of our study is to investigate the occurrence of bifurcations to halo
orbits, which take place as the energy level is varied. The estimate of the
bifurcation thresholds is performed by analytical and numerical methods: we
find a remarkable agreement between the two approaches. As a side result, we
also evaluate the influence of the different parameters, most notably the solar
radiation pressure coefficient, on the dynamical behavior of the model.
To perform the analytical and numerical computations, we start by
implementing a center manifold reduction. Next, we estimate the bifurcation
values using qualitative techniques (e.g. Poincar\'e surfaces, frequency
analysis, FLIs). Concerning the analytical approach, following \cite{CPS} we
implement a resonant normal form, we transform to suitable action-angle
variables and we introduce a detuning parameter measuring the displacement from
the synchronous resonance. The bifurcation thresholds are then determined as
series expansions in the detuning. Three concrete examples are considered and
we find in all cases a very good agreement between the analytical and numerical
results
Addressing imminent impactors threat from distant retrograde orbits (DRO)
Planetary Defence is gaining momentum after the launching toward the Didymos binary system of NASA DART, the first asteroid deflection mission, foreseeing also the deployment of ASIâs LICIAcube. Moreover, the ESA Hera spacecraft, which will contribute to assessing the DART impact momentum transfer, is in full realization phase. After the well established US planetary defence activities, the European Union has recently included the NEO hazard in its own Space Programme in order to extend and complement the ESA initiatives i.e. the establishment of NEO Coordination centre at ESRIN and the realization of the wide-field high-sensitivity Flyeye telescope. Both NASA and ESA also plan to improve the efficiency of their observational networks by launching space mission devoted to NEO observations from space. Finally, the ever-growing NEO discovery rate moves toward increasingly smaller objects passing close to our planet, thus posing new challenges in performing follow- up observations for determining their orbital and physical properties. Within this framework, addressing the so called âimminent impactorsâ threat, posed by objects in the 10-40 m range in route of collision with the Earth (the Tunguska-class objects), has become a key issue for planetary defence. Deflection capabilities are useless if a celestial body large enough to produce significant damage can sneak up on the Earth undetected, as could asteroids hiding by the Sun, lurking in the well-known blind spot that ground-based observations can never peer into. In this respect the advantages of placing a telescope on a stable Distant Retrograde Obit (DRO) around the Earth when compared with other orbital configurations have already been proven, and they are now well established in the literature. In this work the feasibility of a mission scenario foreseeing a constellation of four spacecraft in DRO is invesitgated in detail, comparing several target orbits and different transfer strategies, including lunar swing-bys. The more efficient orbital configurations in terms of accessibility and detection capabilities are investigated and validated using case studies of historic asteroid undetected close encounters. Results prove that a DRO constellation would be able to detect and refine the trajectory of a Tunguska-size object with a warning time exceeding the requirement set for natural disasters. The possibility of contributing to the physical characterization of an imminent impactor is also discussed, which is essential for building up an efficient rapid response system for civil protection purposes.European Union funding: 87040
Assessment of flyby methods as applied to close encounters among asteroids
Orbital flybys have been extensively studied for spacecraft missions, resulting in effective mathematical and physical models. However, these modelsâ applicability to natural encounters involving asteroids has not been explored. This paper examines the applicability of two such theories, patched conics (PC) and the Keplerian map (KM), to asteroid encounters. A review of the two methods will be provided, highlighting their assumptions and range of applicability. Simulations of asteroidâasteroid encounters will then be performed to evaluate their effectiveness in these scenarios. The simulation parameters are set by collecting data on actual asteroidâasteroid encounters, hereby presented, generally characterised by high close approach distances and small masses of the perturbing bodies, if compared to those used to build the flyby theories. Results show that the PC theoryâs effectiveness diminishes with increasing approach distances, aligning with its assumptions. Moreover, the prediction of the model is better in the geometric configurations where the flyby has major effects on the orbital energy change. The KM theory has shown good effectiveness for encounters occurring outside the sphere of influence of the perturbing body, even for very high distances. This research investigates flyby modelsâ strengths and weaknesses in asteroid encounters, offering practical insights and future directions.European Space Agency (ESA) Open Space Innovation Platform (OSIP) campaign and by Cranfield University (ESA Contract no. 4000134762/21/NL/MH/hm-Asteroid Collisions).Aerospac
The evolution of the Line of Variations at close encounters: an analytic approach
We study the post-encounter evolution of fictitious small bodies belonging to the so-called Line of Variations (LoV) in the framework of the analytic theory of close encounters. We show the consequences of the encounter on the local minimum of the distance between the orbit of the planet and that of the small body and get a global picture of the way in which the planetocentric velocity vector is affected by the encounter. The analytical results are compared with those of numerical integrations of the restricted three-body proble
Influence of the dynamic classification of asteroids on observation astrometric errors: a statistical analysis
The ephemerides of minor planets are computed on the basis of astrometric observations. The asteroid orbit determination process requires these observations to be properly weighted to take into account the expected accuracy of the data. If not directly provided by the observers, the weights are, in general, computed after a station-specific statistical analysis on the observation residuals, where the influence of external factors such as epoch of observation, magnitude, and employed catalogue has been proven. In this paper, we perform a statistical analysis on observation residuals of the major surveys taking into account a new factor, i.e. the dynamical classification of asteroids, to understand whether the observation quality may have a dependence on the different type of observed object. If an influence is actually found, then it will be possible to develop a new weighting system based on these results. The weights will be easily applicable once one knows the asteroid orbit. In particular, four stations have been found having different qualities depending on whether they are observing near-Earth asteroids or main-belt asteroids. Moreover, the cross-correlation between the dynamic classification and epoch, magnitude, and catalogue is investigated, as well as the influence of these factors on observationsâ quality