1,720,983 research outputs found
Closed-form perturbation theory in the restricted three-body problem without relegation
Full text linkStages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral
No full textWe investigate the long term evolution of trajectories in the Fermi-Pasta-Ulam (FPU) system, using as a probe the first non-trivial integral J in the hierarchy of integrals of the corresponding Toda lattice model. To this end we perform simulations of FPU-trajectories for various classes of initial conditions produced by the excitation of isolated modes, packets, as well as 'generic' (random) initial data. For initial conditions corresponding to localized energy excitations, J exhibits variations yielding 'sigmoid' curves similar to observables used in literature, e.g., the 'spectral entropy' or various types of 'correlation functions'. However, J(t) is free of fluctuations inherent in such observables, hence it constitutes an ideal observable for probing the timescales involved in the stages of FPU dynamics. We observe two fundamental timescales: i) the 'time of stability' (in which, roughly, FPU trajectories behave like Toda), and ii) the 'time to equilibrium' (beyond which energy equipartition is reached). Below a specific energy crossover, both times are found to scale exponentially as an inverse power of the specific energy. However, this crossover goes to zero with increasing the degrees of freedom N as epsilon(c) similar to N-b, with b is an element of [1.5, 2.5]. For 'generic data' initial conditions, instead, J(t) allows to quantify the continuous in time slow diffusion of the FPU trajectories in a direction transverse to the Toda tori
A deep dive into the 2g+h resonance: separatrices, manifolds and phase space structure of navigation satellites
No full textDespite 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 2 g+ h lunisolar resonance. In particular, (i) we compute the correct forms of the separatrices of the resonance in the inclination-eccentricity (i, e) space for fixed semi-major axis a. This allows to compute the change in the width of the 2 g+ h resonance as the altitude increases. (ii) We discuss the crucial role played by the value of the inclination of the Laplace plane, iL. Since iL is comparable to the resonance’s separatrix width, the parametrization of all resonance bifurcations has to be done in terms of the proper inclination ip, instead of the mean one. (iii) The subset of circular orbits constitutes an invariant subspace embedded in the full phase space, the center manifold C, where actual navigation satellites lie. Using ip as a label, we compute its range of values for which C becomes a normally hyperbolic invariant manifold (NHIM). The structure of invariant tori in 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 2 g+ h resonance by the [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.] 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 [InlineEquation not available: see fulltext.] 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
Secular dynamics and the lifetimes of lunar artificial satellites under natural force-driven orbital evolution
No full textIn 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’, i.e., without expansions in the satellite's orbital eccentricity. 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 2g−resonance (ω̇=0) 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 2g 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 2g resonance's phase portrait, as we move from the lowest to the highest limit in inclination (at each altitude) where the 2g resonance is manifested
A detailed dynamical model for inclination-only dependent lunisolar resonances. Effect on the “eccentricity growth” mechanism
No full textThe 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 Daquin et al. (2022), 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 ΩL. In addition, on the basis of our model, we discuss the role played by the Ω-ΩL and the 2Ω-ΩL 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
Numerical construction of precise distribution functions for self-gravitating disks of given surface density profile
Full text linkreservedThis thesis presents an algorithm to numerically produce distribution function (DF) models which represent exact collisionless equilibria for a thin and truncated self-gravitating disk with a fixed-in-advance surface density profile ΣD(ρ). Firstly a Shu-type DF is modified in such a way that the self-consistency condition leads to an integral equation for an unknown function S(ρ) whose specification completes the determination of the DF. Then a discretized form of the integral equation is solved to obtain numerically S(ρ). Families of different DFs yielding the same initial density profile ΣD(ρ) can be produced by choosing different input radial velocity dispersion profiles σρ(ρ). The algorithm leads to DFs which, while constrained in principle only to exactly reproduce the imposed surface density profile ΣD(ρ), in practice reproduce also to a good accuracy the imposed velocity dispersion profiles σρ(ρ), hence allowing to have control on the kinematic properties (e.g. the Q-profile) of the disk. Several properties of the obtained DFs are discussed and compared to the predictions of epicyclic and post-epicyclic theory for a disc with Schwarzchild’s DF. N-Body realizations of the algorithm are produced in an example of a galactic-type disc embedded in a live halo. The constancy in time and stability properties of the N-body system whose initial conditions are obtained through the computed DFs are tested via N-body simulations.This thesis presents an algorithm to numerically produce distribution function (DF) models which represent exact collisionless equilibria for a thin and truncated self-gravitating disk with a fixed-in-advance surface density profile ΣD(ρ). Firstly a Shu-type DF is modified in such a way that the self-consistency condition leads to an integral equation for an unknown function S(ρ) whose specification completes the determination of the DF. Then a discretized form of the integral equation is solved to obtain numerically S(ρ). Families of different DFs yielding the same initial density profile ΣD(ρ) can be produced by choosing different input radial velocity dispersion profiles σρ(ρ). The algorithm leads to DFs which, while constrained in principle only to exactly reproduce the imposed surface density profile ΣD(ρ), in practice reproduce also to a good accuracy the imposed velocity dispersion profiles σρ(ρ), hence allowing to have control on the kinematic properties (e.g. the Q-profile) of the disk. Several properties of the obtained DFs are discussed and compared to the predictions of epicyclic and post-epicyclic theory for a disc with Schwarzchild’s DF. N-Body realizations of the algorithm are produced in an example of a galactic-type disc embedded in a live halo. The constancy in time and stability properties of the N-body system whose initial conditions are obtained through the computed DFs are tested via N-body simulations
Semi-analytical estimates for the speed of diffusion in the second fundamental model of resonance: a Jeans-Landau-Teller approach
Full text linkWe consider the problem of slow chaotic diffusion in a near-integrable three-dimensional hamiltonian system, in the context of the second fundamental model of resonance. Performing analytical estimates for a class of Melnikov integrals, we provide an upper bound for the rate of diffusion. Analytical estimates are based on the so-called stationary phase approximation. We apply our analysis to a problem arising in celestial mechanics: the first order (3:2) asteroidal mean motion resonance in the restricted elliptic three-body problem.ope
Cosmology and dynamical systems: the relationship between the scale factor and cosmological fields.
Full text linkopenIn order to put a spotlight on the fundamental fields that shape the cosmic dynamics, this thesis
discusses in an introductive manner the connection between the realms of dynamical systems
theory and Cosmology. The primary objective is to discuss how mathematical models can be
employed to comprehend the complex dynamics governing the observable universe, by yielding
a nonlinear coupling between the scale factor, on one hand, and a possible cosmological field,
permeating the space at cosmological scale, on the other.
This thesis begins introducting the Lagrangian formalism applicable to cosmological fields,
processing towards an exploration of the Einstein Field Equations and Friedmann solutions, and
finally, delving into the dynamics of scalar-tensor theories and how they affect the scale factor’s
evolution. A fundamental reference for this exploration, especially in the final chapter, is the
ELSEVIER-published article titled ’Dynamical Systems Applied to Cosmology: dark energy
and modified gravity’ (2018). This article serves as a keystone, guiding the main considerations
on cosmological models influenced by dark energy.In order to put a spotlight on the fundamental fields that shape the cosmic dynamics, this thesis
discusses in an introductive manner the connection between the realms of dynamical systems
theory and Cosmology. The primary objective is to discuss how mathematical models can be
employed to comprehend the complex dynamics governing the observable universe, by yielding
a nonlinear coupling between the scale factor, on one hand, and a possible cosmological field,
permeating the space at cosmological scale, on the other.
This thesis begins introducting the Lagrangian formalism applicable to cosmological fields,
processing towards an exploration of the Einstein Field Equations and Friedmann solutions, and
finally, delving into the dynamics of scalar-tensor theories and how they affect the scale factor’s
evolution. A fundamental reference for this exploration, especially in the final chapter, is the
ELSEVIER-published article titled ’Dynamical Systems Applied to Cosmology: dark energy
and modified gravity’ (2018). This article serves as a keystone, guiding the main considerations
on cosmological models influenced by dark energy
- …
