arXiv subjects

Alessandra Celletti

Publications and source records attributed to Alessandra Celletti.

At least 37 records · Page 2Linked to original sources

arXiv2019

Existence of whiskered KAM tori of conformally symplectic systems

We study the existence of whiskered tori in a family $f_μ$ of conformally symplectic maps depending on parameters $μ$. Whiskered tori are tori on which the motion is a rotation, but they have as many expanding/contracting directions as allowed by the preservation of the geometric structure. Our main result is formulated in an "a-posteriori" format. We fix $ω$ satisfying Diophantine conditions. We assume that we are given 1) a value of the parameter $μ_0$, 2) an embedding of the torus $K_0$ into the phase space, approximately invariant under $f_{μ_0}$ in the sense that $f_{μ_0} \circ K_0 - K_0 \circ T_ω$ is small, 3) a splitting of the tangent space at the range of $K_0$, into three bundles which are approximately invariant under $D f_{μ_0}$ and such that the derivative satisfies "rate conditions" on each of the components. Then, if some non-degeneracy conditions are satisfied, we show that there is another parameter $μ_\infty$, an embedding $K_\infty$ and splittings close to the original ones which are invariant under $f_{μ_\infty}$. We also bound $|μ_\infty - μ_0|$, $\|K_\infty - K_0 \|$ and the distance of the initial and final splittings in terms of the initial error. The proof of the main theorem consists in describing an iterative process that takes advantage of cancellations coming from the geometry. Then, we show that the process converges to a true solution when started from an approximate enough solution. The iterative process leads to an efficient algorithm that is quite practical to implement. As an application, we study the singular problem of effects of small dissipation on whiskered tori. We develop formal expansions in the perturbative parameter and use them as input for the a-posteriori theorem. This allows to obtain lower bounds for the domain of analyticity of the tori as function of the perturbative parameter.

math.DS↗

arXiv2018

Hill four-body problem with oblate tertiary: an application to the Sun-Jupiter-Hektor-Skamandrios system

We consider a restricted four-body problem with a precise hierarchy between the bodies: two point-mass bigger bodies, a smaller one with oblate shape, and an infinitesimal body in the neighborhood of the oblate body. The three heavy bodies are assumed to move in a plane under their mutual gravity, and the fourth body moves under the gravitational influence of the three heavy bodies, but without affecting them. We start by finding the triangular central configurations of the three heavy bodies; since one body is oblate, the triangle is isosceles, rather than equilateral as in the point mass case. We assume that the three heavy bodies are in such a central configuration and we perform a Hill's approximation of the equations of motion describing the dynamics of the infinitesimal body in a neighborhood of the oblate body. Through the use of Hill's variables and a limiting procedure, this approximation amounts to sending the two other bodies to infinity. Finally, for the Hill approximation, we find the equilibrium points of the infinitesimal body and determine their stability. As a motivating example, we consider the dynamics of the moonlet Skamandrios of Jupiter's Trojan asteroid Hektor.

math.DS↗

arXiv2018

Element history of the Laplace resonance: a dynamical approach

We consider the three-body mean motion resonance defined by the Jovian moons Io, Europa, and Ganymede, which is commonly known as the Laplace resonance. In particular, we construct approximate models for the evolution of the librating argument over the period of 100 years, focusing on its principal amplitude and frequency, and on the observed mean motion combinations associated with the quasi-resonant interactions. First, we numerically propagated the Cartesian equations of motion of the Jovian system for the period under examination, and by comparing the results with a suitable set of ephemerides, we derived the main dynamical effects on the target quantities. Using these effects, we built an alternative Hamiltonian formulation and used the normal forms theory to locate the resonance and to compute its main amplitude and frequency. From the Cartesian model we observe that on the timescale considered and with ephemerides as initial conditions, both the librating argument and the diagnostics are well approximated by considering the mutual gravitational interactions of Jupiter and the Galilean moons (including Callisto), and the effect of Jupiter's J2 harmonic. Under the same initial conditions, the Hamiltonian formulation in which Callisto and J2 are reduced to their secular contributions achieves larger errors for the quantities above, particularly for the librating argument. By introducing appropriate resonant variables, we show that these errors can be reduced by moving in a certain action-angle phase plane, which in turn implies the necessity of a tradeoff in the selection of the initial conditions. In addition to being a good starting point for a deeper understanding of the Laplace resonance, the models and methods described are easily generalizable to different types of multi-body mean motion resonances. They are also prime tools for studying the dynamics of extrasolar systems.

astro-ph.EP↗

arXiv2017

The dynamics of the de Sitter resonance

We study the dynamics of the de Sitter resonance, namely the stable equilibrium configuration of the first three Galilean satellites. We clarify the relation between this family of configurations and the more general Laplace resonant states. In order to describe the dynamics around the de Sitter stable equilibrium, a one-degree of freedom Hamiltonian normal form is constructed and exploited to identify initial conditions leading to the two families. The normal form Hamiltonian is used to check the accuracy in the location of the equilibrium positions. Besides, it gives a measure of how sensitive it is with respect to the different perturbations acting on the system. By looking at the phase-plane of the normal form, we can identify a \sl Laplace-like \rm configuration, which highlights many substantial aspects of the observed one.

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arXiv2017

Dynamics of resonances and equilibria of Low Earth Objects

The nearby space surrounding the Earth is densely populated by artificial satellites and instruments, whose orbits are distributed within the Low-Earth-Orbit region (LEO), ranging between 90 and 2 000 $km$ of altitude. As a consequence of collisions and fragmentations, many space debris of different sizes are left in the LEO region. Given the threat raised by the possible damages which a collision of debris can provoke with operational or manned satellites, the study of their dynamics is nowadays mandatory. This work is focused on the existence of equilibria and the dynamics of resonances in LEO. We base our results on a simplified model which includes the geopotential and the atmospheric drag. Using such model, we make a qualitative study of the resonances and the equilibrium positions, including their location and stability. The dissipative effect due to the atmosphere provokes a tidal decay, but we give examples of different behaviors, precisely a straightforward passage through the resonance or rather a temporary capture. We also investigate the effect of the solar cycle which is responsible of fluctuations of the atmospheric density and we analyze the influence of Sun and Moon on LEO objects.

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arXiv2017

Hamiltonian formulation of the spin-orbit model with time-varying non-conservative forces

In a realistic scenario, the evolution of the rotational dynamics of a celestial or artificial body is subject to dissipative effects. Time-varying non-conservative forces can be due to, for example, a variation of the moments of inertia or to tidal interactions. In this work, we consider a simplified model describing the rotational dynamics, known as the spin-orbit problem, where we assume that the orbital motion is provided by a fixed Keplerian ellipse. We consider different examples in which a non-conservative force acts on the model and we propose an analytical method, which reduces the system to a Hamiltonian framework. In particular, we compute a time parametrisation in a series form, which allows us to transform the original system into a Hamiltonian one. We also provide applications of our method to study the rotational motion of a body with time-varying moments of inertia, e.g. an artificial satellite with flexible components, as well as subject to a tidal torque depending linearly on the velocity.

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arXiv2016

Dynamical models and the onset of chaos in space debris

The increasing threat raised by space debris led to the development of different mathematical models and approaches to investigate the dynamics of small particles orbiting around the Earth. Such models and methods strongly depend on the altitude of the objects above Earth's surface, since the strength of the different forces acting on an Earth orbiting object (geopotential, atmospheric drag, lunar and solar attractions, solar radiation pressure, etc.) varies with the altitude of the debris. In this review, our focus is on presenting different analytical and numerical approaches employed in modern studies of the space debris problem. We start by considering a model including the geopotential, solar and lunar gravitational forces and the solar radiation pressure. We summarize the equations of motion using different formalisms: Cartesian coordinates, Hamiltonian formulation using Delaunay and epicyclic variables, Milankovitch elements. Some of these methods lead in a straightforward way to the analysis of resonant motions. In particular, we review results found recently about the dynamics near tesseral, secular and semi-secular resonances. As an application of the above methods, we proceed to analyze a timely subject namely the possible causes for the onset of chaos in space debris dynamics. Precisely, we discuss the phenomenon of overlapping of resonances, the effect of a large area-to-mass ratio, the influence of lunisolar secular resonances. We conclude with a short discussion about the effect of the dissipation due to the atmospheric drag and we provide a list of minor effects, which could influence the dynamics of space debris.

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arXiv2016

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.

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arXiv2016

Analytical development of the lunisolar disturbing function and the critical inclination secular resonance

We provide a detailed derivation of the analytical expansion of the lunar and solar disturbing functions. Although there exist several papers on this topic, many derivations contain mistakes in the final expansion or rather (just) in the proof, thereby necessitating a recasting and correction of the original derivation. In this work, we provide a self-consistent and definite form of the lunisolar expansion. We start with Kaula's expansion of the disturbing function in terms of the equatorial elements of both the perturbed and perturbing bodies. Then we give a detailed proof of Lane's expansion, in which the elements of the Moon are referred to the ecliptic plane. Using this approach the inclination of the Moon becomes nearly constant, while the argument of perihelion, the longitude of the ascending node, and the mean anomaly vary linearly with time. We make a comparison between the different expansions and we profit from such discussion to point out some mistakes in the existing literature, which might compromise the correctness of the results. As an application, we analyze the long--term motion of the highly elliptical and critically--inclined Molniya orbits subject to quadrupolar gravitational interactions. The analytical expansions presented herein are very powerful with respect to dynamical studies based on Cartesian equations, because they quickly allow for a more holistic and intuitively understandable picture of the dynamics.

astro-ph.EP↗

arXiv2016

Poynting-Robertson drag and solar wind in the space debris problem

We analyze the combined effect of Poynting-Robertson and solar wind drag on space debris. We derive a model within Cartesian, Gaussian and Hamiltonian frameworks. We focus on the geosynchronous resonance, although the results can be easily generalized to any resonance. By numerical and analytical techniques, we compute the drift in semi-major axis due to Poynting-Robertson and solar wind drag. After a linear stability analysis of the equilibria, we combine a careful investigation of the regular, resonant, chaotic behavior of the phase space with a long-term propagation of a sample of initial conditions. The results strongly depend on the value of the area-to-mass ratio of the debris, which might show different dynamical behaviors: temporary capture or escape from the geosynchronous resonance, as well as temporary capture or escape from secondary resonances involving the rate of variation of the longitude of the Sun. Such analysis shows that Poynting-Robertson and solar wind drag must be taken into account, when looking at the long-term behavior of space debris. Trapping or escape from the resonance can be used to place the debris in convenient regions of the phase space.

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arXiv2016

Bifurcations of lunisolar secular resonances for space debris orbits

Using bifurcation theory, we study the secular resonances induced by the Sun and Moon on space debris orbits around the Earth. In particular, we concentrate on a special class of secular resonances, which depend only on the debris' orbital inclination. This class is typically subdivided into three distinct types of secular resonances: those occurring at the critical inclination, those corresponding to polar orbits, and a third type resulting from a linear combination of the rates of variation of the argument of perigee and the longitude of the ascending node. The model describing the dynamics of space debris includes the effects of the geopotential, as well as the Sun's and Moon's attractions, and it is defined in terms of suitable action-angle variables. We consider the system averaged over both the mean anomaly of the debris and those of the Sun and Moon. Such multiply-averaged Hamiltonian is used to study the lunisolar resonances which depend just on the inclination. Borrowing the technique from the theory of bifurcations of Hamiltonian normal forms, we study the birth of periodic orbits and we determine the energy thresholds at which the bifurcations of lunisolar secular resonances take place. This approach gives us physically relevant information on the existence and location of the equilibria, which help us to identify stable and unstable regions in the phase space. Besides their physical interest, the study of inclination dependent resonances offers interesting insights from the dynamical point of view, since it sheds light on different phenomena related to bifurcation theory.

astro-ph.EP↗

arXiv2016

The theory of secondary resonances in the spin-orbit problem

We study the resonant dynamics in a simple one degree of freedom, time dependent Hamiltonian model describing spin-orbit interactions. The equations of motion admit periodic solutions associated with resonant motions, the most important being the synchronous one in which most evolved satellites of the Solar system, including the Moon, are observed. Such primary resonances can be surrounded by a chain of smaller islands which one refers to as secondary resonances. Here, we propose a novel canonical normalization procedure allowing to obtain a higher order normal form, by which we obtain analytical results on the stability of the primary resonances as well as on the bifurcation thresholds of the secondary resonances. The procedure makes use of the expansion in a parameter, called the detuning, measuring the shift from the exact secondary resonance. Also, we implement the so-called `book-keeping' method, i.e., the introduction of a suitable separation of the terms in orders of smallness in the normal form construction, which deals simultaneously with all the small parameters of the problem. Our analytical computation of the bifurcation curves is in excellent agreement with the results obtained by a numerical integration of the equations of motion, thus providing relevant information on the parameter regions where satellites can be found in a stable configuration.

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arXiv2016

A study of the lunisolar secular resonance $2\dotω+\dotΩ=0$

The dynamics of small bodies around the Earth has gained a renewed interest, since the awareness of the problems that space debris can cause in the nearby future. A relevant role in space debris is played by lunisolar secular resonances, which might contribute to an increase of the orbital elements, typically of the eccentricity. We concentrate our attention on the lunisolar secular resonance described by the relation $2\dotω+\dotΩ=0$, where $ω$ and $Ω$ denote the argument of perigee and the longitude of the ascending node of the space debris. We introduce three different models with increasing complexity. We show that the growth in eccentricity, as observed in space debris located in the MEO region at the inclination about equal to $56^\circ$, can be explained as a natural effect of the secular resonance $2\dotω+\dotΩ=0$, while the chaotic variations of the orbital parameters are the result of interaction and overlapping of nearby resonances.

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arXiv2015

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 $L_1$, $L_2$, $L_3$ 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 $L_3$ 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.

math-ph↗

arXiv2015

Domains of analyticity of Lindstedt expansions of KAM tori in dissipative perturbations of Hamiltonian systems

Many problems in Physics are described by dynamical systems that are conformally symplectic (e.g., mechanical systems with a friction proportional to the velocity, variational problems with a small discount or thermostated systems). Conformally symplectic systems are characterized by the property that they transform a symplectic form into a multiple of itself. The limit of small dissipation, which is the object of the present study, is particularly interesting. We provide all details for maps, but we present also the modifications needed to obtain a direct proof for the case of differential equations. We consider a family of conformally symplectic maps $f_{μ, ε}$ defined on a $2d$-dimensional symplectic manifold $\mathcal M$ with exact symplectic form $Ω$; we assume that $f_{μ,ε}$ satisfies $f_{μ,ε}^*Ω=λ(ε) Ω$. We assume that the family depends on a $d$-dimensional parameter $μ$ (called drift) and also on a small scalar parameter $ε$. Furthermore, we assume that the conformal factor $λ$ depends on $ε$, in such a way that for $ε=0$ we have $λ(0)=1$ (the symplectic case). We study the domains of analyticity in $ε$ near $ε=0$ of perturbative expansions (Lindstedt series) of the parameterization of the quasi--periodic orbits of frequency $ω$ (assumed to be Diophantine) and of the parameter $μ$. Notice that this is a singular perturbation, since any friction (no matter how small) reduces the set of quasi-periodic solutions in the system. We prove that the Lindstedt series are analytic in a domain in the complex $ε$ plane, which is obtained by taking from a ball centered at zero a sequence of smaller balls with center along smooth lines going through the origin. The radii of the excluded balls decrease faster than any power of the distance of the center to the origin.

math.DS↗

arXiv2015

Dynamical investigation of minor resonances for space debris

We study the dynamics of the space debris in regions corresponding to minor resonances; precisely, we consider the resonances 3:1, 3:2, 4:1, 4:3, 5:1, 5:2, 5:3, 5:4, where a j:l resonance (with j, l integers) means that the periods of revolution of the debris and of rotation of the Earth are in the ratio j/l. We consider a Hamiltonian function describing the effect of the geopotential and we use suitable finite expansions of the Hamiltonian for the description of the different resonances. In particular, we determine the leading terms which dominate in a specific orbital region, thus limiting our computation to very few harmonics. Taking advantage from the pendulum-like structure associated to each term of the expansion, we are able to determine the amplitude of the islands corresponding to the different harmonics. By means of simple mathematical formulae, we can predict the occurrence of splitting or overlapping of the resonant islands for different values of the parameters. We also find several cases which exhibit a transcritical bifurcation as the inclination is varied. These results, which are based on a careful mathematical analysis of the Hamiltonian expansion, are confirmed by a numerical study of the dynamical behavior obtained by computing the so-called Fast Laypunov Indicators. Since the Hamiltonian approach includes just the effect of the geopotential, we validate our results by performing a numerical integration in Cartesian variables of a more complete model including the gravitational attraction of Sun and Moon, as well as the solar radiation pressure.

math-ph↗

arXiv2015

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. Poincaré 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.

math.DS↗

arXiv2015

A study of the main resonances outside the geostationary ring

We investigate the dynamics of satellites and space debris in external resonances, namely in the region outside the geostationary ring. Precisely, we focus on the 1:2, 1:3, 2:3 resonances, which are located at about 66 931.4 km, 87 705.0 km, 55 250.7 km, respectively. Some of these resonances have been already exploited in space missions, like XMM-Newton and Integral. Our study is mainly based on a Hamiltonian approach, which allows us to get fast and reliable information on the dynamics in the resonant regions. Significative results are obtained even by considering just the effect of the geopotential in the Hamiltonian formulation. For objects (typically space debris) with high area-to-mass ratio the Hamiltonian includes also the effect of the solar radiation pressure. In addition, we perform a comparison with the numerical integration in Cartesian variables, including the geopotential, the gravitational attraction of Sun and Moon, and the solar radiation pressure. We implement some simple mathematical tools that allows us to get information on the terms which are dominant in the Fourier series expansion of the Hamiltonian around a given resonance, on the amplitude of the resonant islands and on the location of the equilibrium points. We also compute the Fast Lyapunov Indicators, which provide a cartography of the resonant regions, yielding the main dynamical features associated to the external resonances. We apply these techniques to analyze the 1:2, 1:3, 2:3 resonances; we consider also the case of objects with large area-to-mass ratio and we provide an application to the case studies given by XMM-Newton and Integral.

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