arXiv subjects

Christos Efthymiopoulos

Publications and source records attributed to Christos Efthymiopoulos.

At least 19 recordsLinked to original sources

arXiv2026

Arches of chaos, heteroclinic connections of first-order MMRs and the chaotic transport of small bodies in the Sun-Jupiter system

We investigate the heteroclinic connections between stable and unstable manifolds of unstable periodic orbits associated with the most important mean motion resonances (MMRs) in the Sun-Jupiter planar restricted three-body problem. We explicitly compute the stable and unstable manifolds of the unstable periodic orbits associated with the first order interior MMRs 2:1, 3:2, and the exterior MMR 2:3. We also compute short-time FLI maps showing the chaotic saddle structure created by the manifolds of several interior or exterior MMRs other than the 1:1 (co-orbital) resonance. Transits of particles from the exterior to the interior of Jupiter's orbit and vice versa are allowed for Tisserand parameter lesser than 3, and are shown to exist through a variety of heteroclinic channels. Besides the classical ones by Koon et al., we find heteroclinic connections between manifolds of short-period orbits around L3 and periodic orbits of interior or exterior first order MMRs, as well as direct connections between interior and exterior MMR manifolds not involving co-orbital periodic orbits. Through these manifolds and the corresponding FLI ridges, we explain the 'arches-of-chaos' in the asteroid orbital plane (a,e). Chaotic orbits shadowing heteroclinic trajectories exhibit resonance hopping, suggesting links to quasi-Hildas and Jupiter-family comets. Results are obtained in the circular RTBP but persist in the elliptic problem.

astro-ph.EP↗

arXiv2025

3D analytical theory of the perturbed single-synchronous state. Application to the post-impact Didymos-Dimorphos system

We develop the 3D generalization of the planar analytical theory presented in Gaitanas et. al., 2024, which deals with states slightly perturbed from the exact `single-synchronous equilibrium state' (SSES) of the full two-body problem. The SSES corresponds to two non-spherical gravitationally interacting bodies, settled in nearly circular relative orbit, with rotation axes normal to the orbital plane, rapid rotation of the primary and synchronous rotation of the secondary. In the present paper we remove all simplifying assumptions of our previous work Gaitanas et. al., 2024, and show how to compute analytical solutions describing a 3-dimensional perturbation of the system from the SSES in the framework of two distinct theories, called `linear' and `nonlinear'. Linear theory stems from averaging the equations of motion over the primary's rapid rotation angle. This maps the SSES to an equilibrium point of the averaged system, around which analytical solutions can be computed by linearization of the equations of motion. In nonlinear theory, instead, we compute a high order normal form for the Hamiltonian of motion through a sequence of canonical transformations in the form of series. Resonances between the basic system's frequencies appear in the nonlinear theory as small divisors. We show that, close to resonances, the nonlinear theory leads to a partially integrable model, sufficient to analytically describe the evolution of the relative orbit, but only of some of the Euler angles of the system. As a basic application, we compute analytical solutions representing various possible Didymos-Dimorphos post-impact orbital and rotational states. In this case, all analytical formulas here proposed are of direct utility in fitting algorithms exploiting available time series of post-impact observational data.}}

astro-ph.EP↗

arXiv2025

Bifurcation sequences in the secular 3D planetary 3-Body problem: a geometric approach

We implement the geometric method proposed in ([9], [3], [16]) to analytically predict the sequence of bifurcations leading to a change of stability and/or the appearance of new periodic orbits in the secular 3D planetary three body problem. Stemming from the analysis in [17], we examine various normal form models as regards the extent to which they lead to a phase space dynamics qualitatively similar as that in the complete system. For fixed total angular momentum, the phase space in Hopf variables is the 3D sphere, and the complete sequence of bifurcations of new periodic orbits can be recovered through formulas yielding the tangencies or degenerate intersections between the sphere and the surfaces of a constant second integral of motion of the normal form flow. In particular, we find the critical values of the second integral giving rise to pitchfork and saddle-node bifurcations of new periodic orbits in the system. This analysis renders possible to predict the most important structural changes in the phase space, as well as the emergence of new possible stable periodic planetary orbital configurations which can take place as the mutual inclination between the two planets is allowed to increase.

math-ph↗

arXiv2025

Fully analytical propagator for lunar satellite orbits in closed form

We present a fully analytical propagator for the orbits of lunar artificial satellites in a lunar gravity and third-body model sufficiently precise for a wide range of practical applications. The gravity model includes the twelve most important lunar gravity harmonics as well as the Earth's quadrupole tidal terms with a precise representation of the Earth's lunicentric ephemeris, and it gives an accuracy comparable to the way more extended semi-analytical propagator SELENA [6] for satellite orbits at altitudes from 300 to 3000 km. Extra terms of a more complete gravity model are straightforward to include using the formulas of the presently discussed analytical theory. The theory is based on deriving an approximate analytical solution of the secular part of the equations of motion using a Hamiltonian normal form in closed form. In total, we have two types of element transformations: from osculating to mean elements (as in [6]), and from mean to proper elements. The solution of the problem in proper elements is trivial, and, through the inverses of the above transformations, it allows to recover the position and velocity of a satellite analytically at any time t given initial conditions of the osculating elements at time $t_0$ without any intermediate numerical propagation. The propagator model is valid in time spans of several decades, and for every initial condition leading to no-fall on the Moon's surface, except for identified thin zones around a set of secular resonances corresponding to commensurabilities between the satellite's secular frequencies and the secular frequencies of the lunicentric Earth's orbit. Open software python and symbolic routines implementing our propagator are provided in the repository [14]. Precision tests with respect to fully numerical orbital propagation in Cartesian coordinates are reported.

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arXiv2024

Analytical theory of the spin-orbit state of a binary asteroid deflected by a kinetic impactor

We study the perturbed-from-synchronous librational state of a double asteroid, modeled by the Full Two Rigid Body Problem (F2RBP), with primary emphasis on deriving analytical formulas which describe the system's evolution after deflection by a kinetic impactor. To this end, both a linear and nonlinear (canonical) theory are developed. We make the simplifying approximations (to be relaxed in a forthcoming paper) of planar binary orbit and axisymmetric shape of the primary body. To study the effect of a DART-like hit on the secondary body, the momentum transfer enhancement parameter $β$ is introduced and retained as a symbolic variable throughout all formulas derived, either by linear or nonlinear theory. Our approach can be of use in the context of the analysis of the post impact data from kinetic impactor missions, by providing a precise modeling of the impactor's effect on the seconadry's librational state as a function of $β$.

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arXiv2024

Secular dynamics and the lifetimes of lunar artificial satellites under natural force-driven orbital evolution

In this paper, we study the long-term (time scale of several years) orbital evolution of lunar satellites under the sole action of natural forces. In particular, we focus on secular resonances, caused either by the influence of the multipole moments of the lunar potential and/or by the Earth's and Sun's third-body effect on the satellite's long-term orbital evolution. Our study is based on a simplified secular model obtained in `closed form' using the same methodology proposed in the recently published report on the semi-analytical propagator of lunar satellite orbits, SELENA. Contrary to the case of artificial Earth satellites, in which many secular resonances compete in dynamical impact, we give numerical evidence that for lunar satellites only the 2 g resonance affects significantly the orbits at secular timescales. We interpret this as a consequence of the strong effect of lunar mascons. We show that the lifetime of lunar satellites is, in particular, nearly exclusively dictated by the 2 g resonance. By deriving a simple analytic model, we propose a theoretical framework which allows for both qualitative and quantitative interpretation of the structures seen in numerically obtained lifetime maps. This involves explaining the main mechanisms driving eccentricity growth in the orbits of lunar satellites. In fact, we argue that the re-entry process for lunar satellites is not necessarily a chaotic process (as is the case for Earth satellites), but rather due to a sequence of bifurcations leading to a dramatic variation in the structure of the separatrices in the 2 g resonance's phase portrait, as we move from the lowest to the highest limit in inclination (at each altitude) where the 2 g resonance is manifested.

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arXiv2023

SELENA: Semi-analytical Integrator for Lunar Artificial Satellites

The present report summarizes the main theory and implementation steps associated with SELENA (SEmi-anaLytical intEgrator for a luNar Artificial satellite), i.e. the semi-analytical propagator for lunar satellite orbits developed in the framework of the the R&T R-S20/BS-0005-062 CNES research activity in collaboration between the University of Padova (UniPd), and the Aristotle University of Thessaloniki (AUTH), both acting as contractors with CNES. A detailed account of the method, algorithms and symbolic manipulations employed in the derivation of the final theory are described in detail in this report: they invoke the use of canonical perturbation theory in the form of Lie series computed in `closed form', i.e., without expansions in the satellite's orbital eccentricity. These algorithms are provided in the form of a symbolic package accompanying the present report. The package contains symbolic algebra programs, as well as explicit data files containing the final Hamiltonian, equations of motion and transformations (i.e. the coefficients and exponents of each variable in each term) leading to the averaging of the short-periodic terms in the satellite's equations of motion.

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arXiv2023

Action-Angle formalism for extreme mass ratio inspirals in Kerr spacetime

We introduce an action-angle formalism for bounded geodesic motion in Kerr black hole spacetime using canonical perturbation theory. Namely, we employ a Lie series technique to produce a series of canonical transformations on a Hamiltonian function describing geodesic motion in Kerr background written in Boyer-Lindquist coordinates to a Hamiltonian system written in action-angle variables. This technique allows us to produce a closed-form invertible relation between the Boyer-Lindquist variables and the action-angle ones, while it generates in analytical closed form all the characteristic functions of the system as well. The expressed in the action-angle variable Hamiltonian system is employed to model an extreme mass ratio inspiral (EMRI), i.e. a binary system where a stellar compact object inspirals into a supermassive black hole due to gravitational radiation reaction. We consider the adiabatic evolution of an EMRI, for which the energy and angular momentum fluxes are computed by solving the Teukolsky equation in the frequency domain. To achieve this a new Teukolsky equation solver code was developed.

gr-qc↗

arXiv2023

Semi-analytical estimates for the chaotic diffusion in the Second Fundamental Model of Resonance. Application to Earth's navigation satellites

We discuss the applicability of the Melnikov and Landau-Teller theories in obtaining semi-analytical estimates of the speed of chaotic diffusion in systems driven by the separatrix-like stochastic layers of a resonance belonging to the `second fundamental model' (SFM)\cite{henrard1983second}. Stemming from the analytic solution for the SFM in terms of Weierstrass elliptic functions, we introduce stochastic Melnikov and Landau-Teller models allowing to locally approximate chaotic diffusion as a sequence of uncorrelated `jumps' observed in the time series yielding the slow evolution of an ensemble of trajectories in the space of the adiabatic actions of the system. Such jumps occur in steps of one per homoclinic loop. We show how a semi-analytical determination of the probability distribution of the size of the jumps can be arrived at by the Melnikov and Landau-Teller approximate theories. Computing also the mean time required per homoclinic loop, we arrive at estimates of the chaotic diffusion coefficient in such systems. As a concrete example, we refer to the long-term diffusion of a small object (e.g. Earth navigation satellite or space debris) within the chaotic layers of the so-called $2g+h$ lunisolar resonance, which is of the SFM type. After a suitable normal form reduction of the Hamiltonian, we compute estimates of the speed of diffusion of these objects, which compare well with the results of numerical experiments.

astro-ph.EP↗

arXiv2023

Relegation-free closed-form perturbation theory and the domain of secular motions in the Restricted 3-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 (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.

math-ph↗

arXiv2022

The phase-space architecture in extrasolar systems with two planets in orbits of high mutual inclination

We revisit the secular 3D planetary three-body problem aiming to provide a unified formalism for studying the structure of the phase space for progressively higher values of the mutual inclination $i_{mut}$ between the two planets' orbits. We propose a `book-keeping' technique yielding (after Jacobi reduction) a clear decomposition of the secular Hamiltonian as $H_{sec}=H_{planar} +H_{space}$, where $H_{space}$ contains all terms depending on $i_{mut}$. We numerically compare several models obtained via expansion in the orbital eccentricities or via multipole expansion. We find the mimimum required truncation orders to accurately represent the dynamics. We explore the transition, as $i_{mut}$ increases, from a `planar-like' to a `Lidov-Kozai' regime. Using a typical (non-hierarchical) example, we show how the structure of the phase portraits of the integrable secular dynamics of the planar case is reproduced to a large extent also in the 3D case. We estimate semi-analytically the level of $i_{mut}$ up to which the dynamics remains nearly-integrable. In this regime, we propose a normal form method by which the basic periodic orbits of the nearly-integrable regime (apsidal corotation resonances) can be computed semi-analytically. On the other hand, as the energy increases the system gradually moves to the `Lidov-Kozai' regime. The latter is dominated by two different families of inclined periodic orbits ($C_1$ and $C_2$), of which $C_2$ becomes unstable via the usual Lidov-Kozai mechanism. We discuss the connection between the above families of periodic orbits. Finally, we study numerically the form of the phase portraits for different mass and semi-major axis ratios of the two planets, aiming to establish how generic are the phenomena reported above as the systems parameters are chosen close to one or more hierarchical limits.

astro-ph.EP↗

arXiv2022

A detailed dynamical model for inclination-only dependent lunisolar resonances. Effect on the "eccentricity growth" mechanism

The focus of this paper is on inclination-only dependent lunisolar resonances, which shape the dynamics of a MEO (Medium Earth Orbit) object over secular time scales (i.e. several decades). Following the formalism of arXiv:2107.14507, we discuss an analytical model yielding the correct form of the separatrices of each one of the major lunisolar resonances in the "action" space $(i, e)$ (inclination, eccentricity) for any given semi-major axis $a$. We then highlight how our method is able to predict and explain the main structures found numerically in Fast Lyapunov Indicator (FLI) cartography. We focus on explaining the dependence of the FLI maps from the initial phase of the argument of perigee $ω$ and of the longitude of the ascending node $Ω$ of the object and of the moon $Ω_L$. In addition, on the basis of our model, we discuss the role played by the $Ω-Ω_L$ and the $2 Ω-Ω_L$ resonances, which overlap with the inclination-only dependent ones as they sweep the region for increasing values of $a$, generating large domains of chaotic motion. Our results provide a framework useful in designing low-cost satellite deployment or space debris mitigation strategies, exploiting the natural dynamics of lunisolar resonances that increase an object's eccentricity up until it reaches a domain where friction leads to atmospheric re-entry.

astro-ph.EP↗

arXiv2022

Kolmogorov algorithm for isochronous Hamiltonian systems

We present a Kolmogorov-like algorithm for the computation of a normal form in the neighborhood of an invariant torus in `isochronous' Hamiltonian systems, i.e., systems with Hamiltonians of the form $\mathcal{H}=\mathcal{H}_0+\varepsilon \mathcal{H}_1$ where $\mathcal{H}_0$ is the Hamiltonian of $N$ linear oscillators, and $\mathcal{H}_1$ is expandable as a polynomial series in the oscillators' canonical variables. This method can be regarded as a normal form analogue of a corresponding Lindstedt method for coupled oscillators. We comment on the possible use of the Lindstedt method itself under two distinct schemes, i.e., one producing series analogous to those of the Birkhoff normal form scheme, and another, analogous to the Kolomogorov normal form scheme in which we fix in advance the frequency of the torus.

math-ph↗

arXiv2021

Nekhoroshev estimates for the orbital stability of Earth's satellites

We provide stability estimates, obtained by implementing the Nekhoroshev theorem, in reference to the orbital motion of a small body (satellite or space debris) around the Earth. We consider a Hamiltonian model, averaged over fast angles, including the $J_2$ geopotential term as well as third-body perturbations due to Sun and Moon. We discuss how to bring the Hamiltonian into a form suitable for the implementation of the Nekhoroshev theorem in the version given by Pöschel(1993) for the `non-resonant' regime. The manipulation of the Hamiltonian includes i) averaging over fast angles, ii) a suitable expansion around reference values for the orbit's eccentricity and inclination, and iii) a preliminary normalization allowing to eliminate particular terms whose existence is due to the non-zero inclination of the invariant plane of secular motions known as the `Laplace plane'. After bringing the Hamiltonian to a suitable form, we examine the domain of applicability of the theorem in the action space, translating the result in the space of physical elements. We find that the necessary conditions for the theorem to hold are fulfilled in some non-zero measure domains in the eccentricity and inclination plane (e, i) for a body's orbital altitude (semi-major axis) up to about 20000 km. For altitudes around 11000 km we obtain stability times of the order of several thousands of years in domains covering nearly all eccentricities and inclinations of interest in applications of the satellite problem, except for narrow zones around some so-called `inclination-dependent' resonances. On the other hand, the domains of Nekhoroshev stability recovered by the present method shrink in size as the semi-major axis a increases (and the corresponding Nekhoroshev times reduce to hundreds of years), while the stability domains practically all vanish for a > 20000 km.

astro-ph.EP↗

arXiv2021

Closed form perturbation theory in the restricted three-body problem without relegation

We propose a closed-form normalization method suitable for the study of the secular dynamics of small bodies in heliocentric orbits perturbed by the tidal potential of a planet with orbit external to the orbit of the small body. The method makes no use of relegation, thus, circumventing all convergence issues related to that technique. The method is based on a convenient use of a book-keeping parameter keeping simultaneously track of all the small quantities in the problem. The book-keeping affects both the Lie series and the Poisson structure employed in successive perturbative steps. In particular, it affects the definition of the normal form remainder at every normalization step. We show the results obtained by assuming Jupiter as perturbing planet and we discuss the validity and limits of the method.

astro-ph.EP↗

arXiv2021

Arnold diffusion and Nekhoroshev theory

Starting with Arnold's pioneering work, the term "Arnold diffusion" has been used to describe the slow diffusion taking place in the space of the actions in Hamiltonian nonlinear dynamical systems with three or more degrees of freedom. The present text is an elaborated transcript of the introductory course given in the Milano I-CELMECH school on the topic of Arnold diffusion and its relation to Nekhoroshev theory. The course introduces basic concepts related to our current understanding of the mechanisms leading to Arnold diffusion. Emphasis is placed upon the identification of those invariant objects in phase space which drive chaotic diffusion, such as the stable and unstable manifolds emanating from (partially) hyperbolic invariant objects. Besides a qualitative understanding of the diffusion mechanisms, a precise quantification of the speed of Arnold diffusion can be achieved by methods based on canonical perturbation theory, i.e. by the construction of a suitable normal form at optimal order. As an example of such methods, we discuss the (quasi-)stationary-phase approximation for the selection of remainder terms acting as driving terms for the diffusion. Finally, we discuss the efficiency of such methods through numerical examples in which the optimal normal form is determined by a computer-algebraic implementation of a normalization algorithm.

math-ph↗

arXiv2021

Closed-form perturbation theory in the Sun-Jupiter restricted three body problem without relegation

We present a closed-form normalization method suitable for the study of the secular dynamics of small bodies inside the trajectory of Jupiter. The method is based on a convenient use of a book-keeping parameter introduced not only in the Lie series organization but also in the Poisson bracket structure employed in all perturbative steps. In particular, we show how the above scheme leads to a redefinition of the remainder of the normal form at every step of the formal solution of the homological equation. An application is given for the semi-analytical representation of the orbits of main-belt asteroids.

astro-ph.EP↗

arXiv2021

A deep dive into the $2g+h$ resonance: separatrices, manifolds and phase space structure of navigation satellites

Despite extended past studies, several questions regarding the resonant structure of the medium-Earth orbit (MEO) region remain hitherto unanswered. This work describes in depth the effects of the $2g+h$ lunisolar resonance. In particular, (i) we compute the correct forms of the separatrices of the resonance in the inclination-eccentricity space for fixed semi-major axis. This allows to compute the change in the width of the $2g+h$ resonance as the altitude increases. (ii) We discuss the crucial role played by the value of the inclination of the Laplace plane, $i_{L}$. Since $i_L$ is comparable to the resonance's separatrix width, the parametrization of all resonance bifurcations has to be done in terms of the proper inclination $i_{p}$, instead of the mean one. (iii) The subset of circular orbits constitutes an invariant subspace embedded in the full phase space, the center manifold $\mathcal{C}$. Using $i_p$ as a label, we compute its range of values for which $\mathcal{C}$ becomes a normally hyperbolic invariant manifold (NHIM). The structure of invariant tori in $\mathcal{C}$ allows to explain the role of the initial phase $h$ noticed in several works. (iv) Through Fast Lyapunov Indicator (FLI) cartography, we portray the stable and unstable manifolds of the NHIM as the altitude increases. Manifold oscillations dominate in phase space between $a=24,000$ km and $a=30,000$ km as a result of the sweeping of the $2g+h$ resonance by the $h-Ω_{\rm{Moon}}$ and $2h-Ω_{\rm{Moon}}$ resonances. The noticeable effects of the latter are explained as a consequence of the relative inclination of the Moon's orbit with respect to the ecliptic. The role of the phases $(h,Ω_{\rm{Moon}})$ in the structures observed in the FLI maps is also clarified. Finally,(v) we discuss how the understanding of the manifold dynamics could inspire end-of-life disposal strategies.

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