E. Lega

Cyclical Growth and Primary Resource Constraint

The most important limit to growth, in traditional as well as more recent growth models, is the s... more The most important limit to growth, in traditional as well as more recent growth models, is the shortage of primary factors. This implies a lower growth rate unless technical progress makes it possible to partially release the primary resource constraint. However, this divertes the attention from the crucial aspects of dynamic processes : structural aspects and, paramount, the time structure of production. This paper is an attempt to construct a model of change, in a neo-austrian perspective, in which the sequential character of production appears as the truly essential limit to growt

Migration of Jupiter mass planets in discs with laminar accretion flows

Astronomy & Astrophysics, 2021

Context. Migration of giant planets in discs with low viscosity has been studied recently. Result... more Context. Migration of giant planets in discs with low viscosity has been studied recently. Results have shown that the proportionality between migration speed and the disc's viscosity is broken by the presence of vortices that appear at the edges of the planet-induced gap. Under some conditions, this 'vortex-driven' migration can be very slow and eventually stops. However, this result has been obtained for discs whose radial mass transport is too low (due to the small viscosity) to be compatible with the mass accretion rates that are typically observed for young stars. Aims. Our goal is to investigate vortex-driven migration in low-viscosity discs in the presence of radial advection of gas, as expected from angular momentum removal by magnetised disc winds. Methods. We performed three dimensional simulations using the grid-based code FARGOCA. We mimicked the effects of a disc wind by applying a synthetic torque on a surface layer of the disc characterised by a prescribed column density Σ A so that it results in a disc accretion rate ofṀ A = 10 −8 M yr −1. We have considered values of Σ A typical of the penetration depths of different ionising processes. Discs with this structure are called 'layered' and the layer where the torque is applied is denoted as 'active'. We also consider the case of accretion focussed near the disc midplane to mimic transport properties induced by a large Hall effect or by weak Ohmic diffusion. Results. We observe two migration phases: in the first phase, which is exhibited by all simulations, the migration of the planet is driven by the vortex and is directed inwards. This phase ends when the vortex disappears after having opened a secondary gap, as is typically observed in vortex-driven migration. Migration in the second phase depends on the ability of the torque from the planet to block the accretion flow. When the flow is fast and unimpeded, corresponding to small Σ A , migration is very slow, similar to when there is no accreting layer in the disc. When the accretion flow is completely blocked, migration is faster (typicallyṙ p ∼ 12 AU Myr −1 at 5 au) and the speed is controlled by the rate at which the accretion flow refills the gap behind the migrating planet. The transition between the two regimes, occurs at Σ A ∼ 0.2 g cm −2 and 0.65 g cm −2 for Jupiter or Saturn mass planets at 5.2 au, respectively. Conclusions. The migration speed of a giant planet in a layered protoplanetary disc depends on the thickness of the accreting layer. The lack of large-scale migration apparently experienced by the majority of giant exoplanets can be explained if the accreting layer is sufficiently thin to allow unimpeded accretion through the disc.

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Twist angles: a method for distinguishing islands, tori and weak chaotic orbits. Comparison with other methods of analysis

Astronomy & Astrophysics, 1998

Using the standard map as a model problem we have investigated the method of the twist angles (Co... more Using the standard map as a model problem we have investigated the method of the twist angles (Contopoulos and Voglis 1997) to distinguish islands tori and weak chaotic orbits. In the case of regular orbits, we have shown the relationship with the frequency studying in particular the case of second order resonances. We have tested the sensitivity of the method

format_quoteThe helicity angle method effectively distinguishes between islands and rotational tori, complementing existing chaos detection techniques.format_quote

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Weak Chaos and Diffusion in Hamiltonian Systems

The Dynamics of Small Bodies in the Solar System, 1999

Recent theoretical results have shown that the stability of a dynamical system is strictly relate... more Recent theoretical results have shown that the stability of a dynamical system is strictly related to the density of tori which foliate the phase space. Therefore, the numerical check of the dynamical behaviour of a given domain requires tools of analysis which should be as sensitive as the classical Lyapunov characteristic exponents but cheaper in computational time, in order to be applied to a large number of orbits. We define the different methods of analysis and we compare them using the two and four dimensional Standard map as a model problem. We apply such tools for a fine study of the vicinity of an invariant KAM torus. We then study standard map like mappings showing the relationship between the Lyapunov times and macroscopic instability times. The results obtained applied to the distribution of asteroids can help in solving some puzzling problems such that of “stable” chaos or that of the Kirkwood gaps.

Round-about production, co-ordination failure, technological change, and the wage–employment dilemma

Journal of Economic Behavior & Organization, 2001

The contrasting experiences of European and the US labour markets suggest to most analysts the co... more The contrasting experiences of European and the US labour markets suggest to most analysts the conclusion that technological shocks would be the cause of the appearance of unemployment, and institutional settings, that generate real wage rigidity, would be the cause of its persistence. In this paper we intend to invalidate this conjecture. In order to do so we shall turn to an analytical framework which differs from the standard neo-classical one in that it allows to enlighten what happens step by step along a way that begins with the breaking of the existing steady state due to an attempt to bring about a change in technology.

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Apprentissage, croissance et cycles : la politique monétaire face aux chocs d'offre

Le Centre pour la Communication Scientifique Directe - HAL - SHS, 2000

Reduced gas accretion on super-Earths and ice giants

A large fraction of giant planets have gaseous envelopes that are limited to about 10 % of their ... more A large fraction of giant planets have gaseous envelopes that are limited to about 10 % of their total mass budget. Such planets are present in the Solar System (Uranus, Neptune) and are frequently observed in short periods around other stars (the so-called Super-Earths). In contrast to these observations, theoretical calculations based on the evolution of hydrostatic envelopes argue that such low mass envelopes cannot be maintained around cores exceeding five Earth masses. Instead, under nominal disc conditions, these planets would acquire massive envelopes through runaway gas accretion within the lifetime of the protoplanetary disc. In this work, we show that planetary envelopes are not in hydrostatic balance, which slows down envelope growth. A series of 3-dimensional, global, radiative hydrodynamical simulations reveal a steady state gas flow, which enters through the poles and exits in the disc midplane. Gas is pushed through the outer envelope in about 10 orbital timescales. I...

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Highly inclined and eccentric massive planets I: Planet-disc interactions

In the Solar System, planets have a small inclination with respect to the equatorial plane of the... more In the Solar System, planets have a small inclination with respect to the equatorial plane of the Sun, but there is evidence that in extrasolar systems the inclination can be very high. This spin-orbit misalignment is unexpected, as planets form in a protoplanetary disc supposedly aligned with the stellar spin. Planet-planet interactions are supposed to lead to a mutual inclination, but the effects of the protoplanetary disc are still unknown. We investigate therefore planet-disc interactions for planets above 1M_Jup. We check the influence of the inclination i, eccentricity e, and mass M_p of the planet. We perform 3D numerical simulations of protoplanetary discs with embedded high-mass planets. We provide damping formulae for i and e as a function of i, e, and M_p that fit the numerical data. For highly inclined massive planets, the gap opening is reduced, and the damping of i occurs on time-scales of the order of 10^-4 deg/yr M_disc/(0.01 M_star) with the damping of e on a smalle...

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Evolution of the Tangent Vectors and Localization of the Stable and Unstable Manifolds of Hyperbolic Orbits by Fast Lyapunov Indicators

SIAM Journal on Applied Mathematics, 2014

The Fast Lyapunov Indicators are functions defined on the tangent fiber of the phase-space of a d... more The Fast Lyapunov Indicators are functions defined on the tangent fiber of the phase-space of a discrete (or continuous) dynamical system, by using a finite number of iterations of the dynamics. In the last decade, they have been largely used in numerical computations to localize the resonances in the phase-space and, more recently, also the stable and unstable manifolds of normally hyperbolic invariant manifolds. In this paper, we provide an analytic description of the growth of tangent vectors for orbits with initial conditions which are close to the stable-unstable manifolds of a hyperbolic saddle point of an area-preserving map. The representation explains why the Fast Lyapunov Indicator detects the stable-unstable manifolds of all fixed points which satisfy a certain condition. If the condition is not satisfied, a suitably modified Fast Lyapunov Indicator can be still used to detect the stable-unstable manifolds. The new method allows for a detection of the manifolds with a number of precision digits which increases linearly with respect to the integration time. We illustrate the method on the critical problem of detection of the so-called tube manifolds of the Lyapunov orbits of L1, L2 in the circular restricted three-body problem.

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Analysis of the Chaotic Behaviour of Orbits Diffusing along the Arnold Web

Celestial Mechanics and Dynamical Astronomy, 2006

In a previous work (Guzzo, Lega and Froeschlé, DCDS B Vol.5, 687-698, 2005) we have provided nume... more In a previous work (Guzzo, Lega and Froeschlé, DCDS B Vol.5, 687-698, 2005) we have provided numerical evidence of global diffusion occurring in slightly perturbed integrable Hamiltonian systems and symplectic maps. We have shown that even if a system is sufficiently close to be integrable, global diffusion occurs on a set with peculiar topology, the so-called Arnold web, and is qualitatively different from Chirikov diffusion, occurring in more perturbed systems. In the present work we study in more detail the chaotic behaviour of a set of 90 orbits which diffuse on the Arnold web. We find that the largest Lyapunov exponent does not seem to converge for the individual orbits while the mean Lyapunov exponent on the set of 90 orbits does converge. In other words, a kind of average mixing characterizes the diffusion. Moreover, the Local Lyapunov Characteristic Numbers (LLCNs), on individual orbits appear to reflect the different zones of the Arnold web revealed by the Fast Lyapunov Indicator. Finally, using the LLCNs we study the ergodicity of the chaotic part of the Arnold web.

format_quoteDetected slow local diffusion confined to the Arnold web, with diffusion coefficients decreasing faster than a power law of the perturbing parameter.format_quote

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Evolution of the tangent vectors and localization of the stable and unstable manifolds of hyperbolic orbits by Fast Lyapunov Indicators

The Fast Lyapunov Indicators are functions defined on the tangent fiber of the phase-space of a d... more The Fast Lyapunov Indicators are functions defined on the tangent fiber of the phase-space of a discrete (or continuous) dynamical system, by using a finite number of iterations of the dynamics. In the last decade, they have been largely used in numerical computations to localize the resonances in the phase-space and, more recently, also the stable and unstable manifolds of normally hyperbolic invariant manifolds. In this paper, we provide an analytic description of the growth of tangent vectors for orbits with initial conditions which are close to the stable-unstable manifolds of a hyperbolic saddle point of an area-preserving map. The representation explains why the Fast Lyapunov Indicator detects the stable-unstable manifolds of all fixed points which satisfy a certain condition. If the condition is not satisfied, a suitably modified Fast Lyapunov Indicator can be still used to detect the stable-unstable manifolds. The new method allows for a detection of the manifolds with a num...

format_quoteFLI successfully identifies stable and unstable manifolds as sharp local maxima, aiding in understanding complex dynamical systems like three-body problems.format_quote

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Parametres morphologiques associes a une analyse multi-echelle de la distribution des galaxies

Le probleme de la formation des grandes structures de l'univers est une des questions fondame... more Le probleme de la formation des grandes structures de l'univers est une des questions fondamentales de la cosmologie moderne. Plusieurs modeles ont ete developpes pour pouvoir reproduire la distribution observee des galaxies. Apres avoir resume les differents scenarios theoriques de formation des grandes structures, nous decrivons en particulier les scenarios cdm, hdm et mdm. Nous passons en revue les generalites sur les methodes de simulations numeriques n-corps utilises pour l'etude de la phase non-lineaire de l'instabilite gravitationnelle. On detaille le code numerique que nous avons utilise et sa mise en uvre sur une machine massivement parallele. Les methodes statistiques d'analyse des structures sont ensuite decrites. Leur capacite a distinguer les differents scenarios theoriques est discutee. Nous avons introduits une methode de detection des structures objective et independante des proprietes globales du scenario qu'on analyse. L'application de la tr...

Low-mass gas envelopes around accreting cores embedded in radiative 3D discs

A Fossilized Snowline in the Solar System Protoplanetary Disk

Global 3D radiation-hydrodynamic simulations of gas accretion: Opacity-dependent growth of Saturn-mass planets

Astronomy & Astrophysics

The full spatial structure and temporal evolution of the accretion flow into the envelopes of gro... more The full spatial structure and temporal evolution of the accretion flow into the envelopes of growing gas giants in their nascent discs is only accessible in simulations. Such simulations are constrained in their approach of computing the formation of gas giants by dimensionality, resolution, consideration of self-gravity, energy treatment and the adopted opacity law. Our study explores how a number of these parameters affect the measured accretion rate of a Saturn-mass planet. We present a global 3D radiative hydrodynamics framework using the FARGOCA-code. The planet is represented by a gravitational potential with a smoothing length at the location of the planet. No mass or energy sink is used; instead luminosity and gas accretion rates are self-consistently computed. We find that the gravitational smoothing length must be resolved by at least ten grid cells to obtain converged measurements of the gas accretion rates. Secondly, we find gas accretion rates into planetary envelopes ...

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Quasi-static contraction during runaway gas accretion onto giant planets

Astronomy & Astrophysics

Gas-giant planets, like Jupiter and Saturn, acquire massive gaseous envelopes during the approxim... more Gas-giant planets, like Jupiter and Saturn, acquire massive gaseous envelopes during the approximately 3 Myr-long lifetimes of protoplanetary discs. In the core accretion scenario, the formation of a solid core of around ten Earth masses triggers a phase of rapid gas accretion. Previous 3D grid-based hydrodynamical simulations found that runaway gas accretion rates correspond to approximately 10 to 100 Jupiter masses per Myr. Such high accretion rates would result in all planets with larger than ten Earth-mass cores to form Jupiter-like planets, which is in clear contrast to the ice giants in the Solar System and the observed exoplanet population. In this work, we used 3D hydrodynamical simulations, that include radiative transfer, to model the growth of the envelope on planets with different masses. We find that gas flows rapidly through the outer part of the envelope, but this flow does not drive accretion. Instead, gas accretion is the result of quasi-static contraction of the in...

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Three-dimensional representations of the tube manifolds of the planar restricted three-body problem

Physica D: Nonlinear Phenomena, 2016

Abstract The stable and unstable manifolds of the Lyapunov orbits of the Lagrangian equilibrium p... more Abstract The stable and unstable manifolds of the Lyapunov orbits of the Lagrangian equilibrium points L1, L2 play a key role in the understanding of the complicated dynamics of the circular restricted three-body problem. By developing a recent technique of computation of the stable and unstable manifolds, based on the use of Fast Lyapunov Indicators modified by the introduction of a filtering window function, we compute sample three-dimensional representations of the manifolds which show an original vista about their complicated development in the phase-space.

A study of the past dynamics of comet 67P/Churyumov-Gerasimenko with fast Lyapunov indicators

Astronomy & Astrophysics, 2015

Comet 67P/Churyumov-Gerasimenko is the target of the Rosetta mission. On the basis of backward nu... more Comet 67P/Churyumov-Gerasimenko is the target of the Rosetta mission. On the basis of backward numerical integrations of a large set of simulated comets whose initial conditions were obtained from small variations of the orbital parameters of comet 67P, and using the analysis of suitable chaos indicators, we detect the phase-space structure of the past close encounters of the comet with Jupiter. On the basis of these computations we find that the probability that the comet could have been injected into the inner solar system from distances larger than 100 AU from the Sun in the past 150 000 years is about 60 percent.

format_quoteThe phase-space structure shows initial conditions with close encounters in a specified time interval, aiding chaos theory applications.format_quote

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On the Relationship between Local Lyapunov Characteristic Numbers, Largest Eigenvalues and Maximum Stretching Parameters

The Dynamics of Small Bodies in the Solar System, 1999

The computation of the Lyapunov Characteristic Indicators (LCI hereafter) remains the only tool w... more

On the Diffusion Along Resonant Lines in Continuous and Discrete Dynamical Systems

International Journal of Modern Physics B, 2003

We detect and measure diffusion along resonances in a discrete symplectic map for different value... more We detect and measure diffusion along resonances in a discrete symplectic map for different values of the coupling parameter. Qualitatively and quantitatively the results are very similar to those obtained for a quasi-integrable Hamiltonian system, i.e. in agreement with Nekhoroshev predictions, although the discrete mapping does not fulfill completely, a priori, the conditions of the Nekhoroshev theorem.

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