arXiv subjects

Elena Lega

Publications and source records attributed to Elena Lega.

At least 19 recordsLinked to original sources

arXiv2026

Dust dynamics in disk dust traps and late planetesimal formation

The streaming instability is the leading model for planetesimal formation in protoplanetary disks, but it typically operates within the first ~Myr. In the Solar System, however, some planetesimals (the chondrite parent bodies) formed 2-4 Myr after disk formation, implying that dust must have been retained for extended periods. Pressure bumps efficiently trap dust, but trapping alone does not guarantee planetesimal formation: even modest gas turbulence can inhibit vertical settling and radial concentration, preventing dust density from reaching Hill density. This motivates the study of alternative dust-gas instabilities, such as the Dusty Rossby Wave Instability (DRWI). We investigate the viability of such instabilities in global disk simulations using the multi-fluid code fargOCA. We first reproduce previous 2D shearing-box results in a global 2D viscous disk and characterize the dust clumping produced by the DRWI. We find that the instability is suppressed in fully 3D viscous disks by unperturbed high-z gas layers caused by dust settling near the midplane. We then explore the inviscid limit and find that multiple dust sub-rings form, concentrating solids into thin ring structures. These would appear observationally as a single radially broad, vertically thin ring, explaining observed protoplanetary disk rings without invoking anisotropic turbulence. Dust concentrations in the sub-rings may remain below the threshold for gravitational collapse, but gas photoevaporation enhances dust settling and radial concentration, eventually forming dense dust clumps in both viscous and inviscid cases. We conclude that planetesimal formation within dust-trapping pressure bumps is favored in very low-viscosity disks at late evolutionary stages, after sufficient gas removal by photoevaporation. This is consistent with the inferred late formation of chondrite parent bodies in the Solar System.

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arXiv2025

The reflex instability: exponential growth of a large-scale $m=1$ mode in astrophysical discs

We report the finding of a linear, non-axisymmetric, global instability in gas discs around stars, which may be relevant to other astrophysical discs. It takes the form of an $m=1$ mode that grows in the disc density distribution while the star-barycentre distance rises exponentially with a characteristic timescale that is orders of magnitude longer than the orbital period. We present results of hydrodynamical simulations with various codes and numerical methods, using either barycentric or stellocentric reference frames, with or without the disc's self gravity: all simulations consistently show an unstable mode growing exponentially. The instability disappears if, and only if, the reflex motion of the star due to the disc's asymmetry is not taken into account in the simulations. For this reason we refer to this instability as the reflex instability. We identify a feedback loop as a possible origin, whereby the acceleration of the star excites the eccentricity of the disc, yielding an $m=1$ mode in the density distribution which, in turn, pulls the star. The growth timescale of the instability decreases with increasing disc mass and is a few hundred orbits for disc-to-star mass ratios of a few percent. If truly physical, and not due to a numerical artifact that would be common to all the codes we have employed, the reflex instability could have a dramatic impact on protoplanetary discs evolution and planetary formation.

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arXiv2025

On inertial forces (indirect terms) in problems with a central body

Gravitational systems in astrophysics often comprise a body -- the primary -- that far outweights the others, and which is taken as the centre of the reference frame. A fictitious acceleration, also known as the indirect term, must therefore be added to all other bodies in the system to compensate for the absence of motion of the primary. In this paper, we first stress that there is not one indirect term but as many indirect terms as there are bodies in the system that exert a gravitational pull on the primary. For instance, in the case of a protoplanetary disc with two planets, there are three indirect terms: one arising from the whole disc, and one per planet. We also highlight that the direct and indirect gravitational accelerations should be treated in a balanced way: the indirect term from one body should be applied to the other bodies in the system that feel its direct gravitational acceleration, and only to them. We point to situations where one of those terms is usually neglected however, which may lead to spurious results. These ideas are developed here for star-disc-planets interactions, for which we propose a recipe for the force to be applied onto a migrating planet, but they can easily be generalized to other astrophysical systems.

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arXiv2024

Hydrodynamical simulations of proto-Moon degassing

Similarities in the non-mass dependent isotopic composition of refractory elements with the bulk silicate Earth suggest that both the Earth and the Moon formed from the same material reservoir. On the other hand, the Moon's volatile depletion and isotopic composition of moderately volatile elements points to a global devolatilization processes, most likely during a magma ocean phase of the Moon. Here, we investigate the devolatilisation of the molten Moon due to a tidally-assisted hydrodynamic escape with a focus on the dynamics of the evaporated gas. Unlike the 1D steady-state approach of Charnoz et al. (2021), we use 2D time-dependent hydrodynamic simulations carried out with the FARGOCA code modified to take into account the magma ocean as a gas source. Near the Earth's Roche limit, where the proto-Moon likely formed, evaporated gases from the lunar magma ocean form a circum-Earth disk of volatiles, with less than 30% of material being re-accreted by the Moon. We find that the measured depletion of K and Na on the Moon can be achieved if the lunar magma-ocean had a surface temperature of about 1800-2000 K. After about 1000 years, a thermal boundary layer or a flotation crust forms a lid that inhibits volatile escape. Mapping the volatile velocity field reveals varying trends in the longitudes of volatile reaccretion on the Moon's surface: material is predominantly re-accreted on the trailing side when the Moon-Earth distance exceeds 3.5 Earth radii, suggesting a dichotomy in volatile abundances between the leading and trailing sides of the Moon. This dichotomy may provide insights on the tidal conditions of the early molten Earth. In conclusion, tidally-driven atmospheric escape effectively devolatilizes the Moon, matching the measured abundances of Na and K on timescales compatible with the formation of a thermal boundary layer or an anorthite flotation crust.

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arXiv2024

Gas dynamics around a Jupiter mass planet: II. Chemical evolution of circumplanetary material

In an ongoing effort to understand planet formation the link between the chemistry of the protoplanetary disk and the properties of resulting planets have long been a subject of interest. These connections have generally been made between mature planets and young protoplanetary disks through the carbon-to-oxygen (C/O) ratio. In a rare number of systems, young protoplanets have been found within their natal protoplanetary disks. These systems offer a unique opportunity to directly study the delivery of gas from the protoplanetary disk to the planet. In this work we post-process 3D numerical simulations of an embedded Jupiter-massed planet in its protoplanetary disk to explore the chemical evolution of gas as it flows from the disk to the planet. The relevant dust to this chemical evolution is assumed to be small, co-moving grains with a reduced dust-to-gas ratio indicative of the upper atmosphere of a protoplanetary disk. We find that as the gas enters deep into the planet's gravitational well, it warms significantly (up to $\sim 800$ K), releasing all of the volatile content from the ice phase. This change in phase can influence our understanding of the delivery of volatile species to the atmospheres of giant planets. The primary carbon, oxygen, and sulfur carrying ices: CO$_2$, H$_2$O, and H$_2$S are released into the gas phase and along with the warm gas temperatures near the embedded planets lead to the production of unique species like CS, SO, and SO$_2$ compared to the protoplanetary disk. We compute the column densities of SO, SO$_2$, CS, and H$_2$CS in our model and find that their values are consistent with previous observational studies.

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arXiv2024

Dust rings trap protoplanets on eccentric orbits and get consumed by them

We study the orbital evolution and mass growth of protoplanets with masses $M \in [0.1-8]$~M$_\oplus$ in the vicinity of a dusty ring, using three-dimensional numerical simulations with a two-fluid model and nested-meshes. We find two stable, eccentric orbits that lock the planet in the ring vicinity, thereby inhibiting its migration and allowing it to accrete dust from the ring. One of these orbits has an eccentricity comparable to the aspect ratio of the gaseous disc and has its periastron within the ring, enabling intermittent accretion during each pass. The other orbit has a smaller eccentricity and an apoastron slightly inside the ring. A planet locked at the outer orbit efficiently accretes from the ring and can reach the critical mass for runaway gas accretion on timescales $\gtrsim 10^5$ yr (for a 10~M$_\oplus$ dust ring at 10~au) while a planet locked at the inner orbit has a slower growth and might not supersede the super-Earth stage over the disc lifetime. While in our runs a low-mass embryo forming within the ring eventually joins the outer orbit, it is likely that the path taken depends on the specific details of the ring. The trapping on the outer orbit arises from an intermittent, strong thermal force at each passage through the ring, where the accretion rate spikes. It is insensitive to uncertainties that plague models considering planets trapped on circular orbits in rings. It is highly robust and could allow a growing planet to follow an expanding ring over large distances.

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arXiv2024

The Solar System could have formed in a low-viscosity disc: A dynamical study from giant planet migration to the Nice model

In the context of low-viscosity protoplanetary discs (PPDs), the formation scenarios of the Solar System should be revisited. In particular, the Jupiter-Saturn pair has been shown to lock in the 2:1 mean motion resonance while migrating generally inwards, making the Grand Tack scenario impossible. We explore what resonant chains of multiple giant planets can form in a low-viscosity disc, and whether these configurations can evolve into forming the Solar System in the post gas disc phase. We used hydrodynamical simulations to study the migration of the giant planets in a disc with viscosity $\alpha=10^{-4}$. After a transition phase to a gas-less configuration, we studied the stability of the obtained resonant chains through their interactions with a disc of leftover planetesimals by performing N-body simulations using rebound. The gaps open by giant planets are wider and deeper for lower viscosity, reducing the damping effect of the disc and thus weakening resonant chains. Exploring numerous configurations, we found five stable resonant chains of four or five planets. In a thin PPD, the four giant planets revert their migration and migrate outwards. After disc dispersal, under the influence of a belt of planetesimals, some resonant chains undergo an instability phase while others migrate smoothly over a billion years. For three of our resonant chains, about 1% of the final configurations pass the four criteria to fit the Solar System. The most successful runs are obtained for systems formed in a cold PPD with a massive planetesimal disc. This work provides a fully consistent study of the dynamical history of the Solar System's giant planets, from the protoplanetary disc phase up to the giant planet instability. Although building resonant configurations is difficult in low-viscosity discs, we find it possible to reproduce the Solar System from a cold, low-viscosity protoplanetary disc.

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arXiv2024

A recipe for eccentricity and inclination damping for partial gap opening planets in 3D disks

In a previous paper we showed that, like the migration speed, the eccentricity damping efficiency is modulated linearly by the depth of the partial gap a planet carves in the disk surface density profile, resulting in less efficient $e$-damping compared to the prescription commonly used in population synthesis works. Here, we extend our analysis to 3D, refining our $e$-damping formula and studying how the inclination damping efficiency is also affected. We perform high resolution 3D locally isothermal hydrodynamical simulations of planets with varying masses embedded in disks with varying aspect ratios and viscosities. We extract the gap profile and orbital damping timescales for fixed eccentricities and inclinations up to the disk scale height. The limit in gap depths below which vortices appear, in the low-viscosity case, happens roughly at the transition between classical type-I and type-II migration regimes. The orbital damping timescales can be described by two linear trends with a break around gap depths $\sim80\%$ and with slopes and intercepts depending on the eccentricity and inclination. These trends are understood on physical grounds and are reproduced by simple fitting formulas whose error is within the typically uncertainty of type-I torque formulas. Thus, our recipes for the gap depth and orbital damping efficiencies yield a simple description for planet-disk interactions to use in N-body codes in the case of partial gap opening planets that is consistent with high-resolution 3D hydro-simulations. Finally, we show examples of how our novel orbital damping prescription can affect the outcome of population synthesis experiments.

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arXiv2022

A recipe for orbital eccentricity damping in the type-I regime for low viscosity 2D-discs

It is known that gap opening depends on the disc's viscosity; however, eccentricity damping formulas have only been derived at high viscosities, ignoring partial gap opening. We aim at obtaining a simple formula to model $e$-damping of the type-I regime in low viscosity discs, where even small planets may start opening partial. We perform high resolution 2D locally isothermal hydrodynamical simulations of planets with varying masses on fixed orbits in discs with varying aspect ratios and viscosities. We determine the torque and power felt by the planet to derive migration and eccentricity damping timescales. We first find a lower limit to the gap depths below which vortices appear; this happens roughly at the transition between type-I and type-II regimes. For the simulations that remain stable, we obtain a fit to the observed gap depth in the limit of vanishing eccentricities that is similar to the one currently used in the literature but is accurate down to $\alpha=3.16\times 10^{-5}$. We record the $e$-damping efficiency as a function of the observed gap depth and $e$: when the planet has opened a deep enough gap, a linear trend is observed independently of $e$; at shallower gaps this linear trend is preserved at low $e$, while it deviates to more efficient damping when $e$ is comparable to the disc's scale height. Both trends can be understood on theoretical grounds and are reproduced by a simple fitting formula. Our combined fits yield a simple recipe to implement type-I $e$-damping in $N$-body for partial gap opening planets that is consistent with high-resolution 2D hydro-simulations. The typical error of the fit is of the order of a few percent, and lower than the error of type-I torque formulas widely used in the literature. This will allow a more self-consistent treatment of planet-disc interactions of the type-I regime for population synthesis models at low viscosities.

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arXiv2020

On the structure and mass delivery towards circumplanetary discs

Circumplanetary discs (CPDs) form around young gas giants and are thought to be the sites of moon formation as well as an intermediate reservoir of gas that feeds the growth of the gas giant. How the physical properties of such CPDs are affected by the planetary mass and the overall opacity is relatively poorly understood. In order to clarify this, we use the global radiation hydrodynamics code FARGOCA, with a grid structure that allows resolving the planetary gravitational potential sufficiently well for a CPD to form. We then study the gas flows and density/temperature structures that emerge as a function of planet mass, opacity and potential depth. Our results indicate interesting structure formation for Jupiter-mass planets at low opacities, which we subsequently analyse in detail. Using an opacity level that is 100 times lower than that of ISM dust, our Jupiter-mass protoplanet features an envelope that is sufficiently cold for a CPD to form, and a free-fall region separating the CPD and the circumstellar disc emerges. Interestingly, this free-fall region appears to be a result of supersonic erosion of outer envelope material, as opposed to the static structure formation that one would expect at low opacities. Our analysis reveals that the planetary spiral arms seem to pose a significant pressure barrier that needs to be overcome through radiative cooling in order for gas to free-fall onto the CPD. The circulation inside the CPD is near-keplerian and modified by the presence of CPD spiral arms. For high opacities we recover results from the literature, finding an essentially featureless hot envelope. With this work, we demonstrate the first simulation and analysis of a complete detachment process of a protoplanet from its parent disc in a 3D radiation hydrodynamics setting.

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arXiv2019

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

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 that 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 10 grid cells to obtain converged measurements of the gas accretion rates. Secondly, we find gas accretion rates into planetary envelopes compatible with previous studies, and continue to explain those via the structure of our planetary envelopes and their luminosities. Our measured gas accretion rates are formally in the stage of Kelvin-Helmholtz contraction due to the modest entropy loss that can be obtained over the simulation time-scale, but our accretion rates are compatible with those expected during late run-away accretion. Our detailed simulations of the gas flow into the envelope of a Saturn-mass planet provides a framework for understanding the general problem of gas accretion during planet formation and highlights circulation features that develop inside the planetary envelopes. Those circulation features feedback into the envelope energetics and can have further implications for transporting dust into the inner regions of the envelope.

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arXiv2019

Quasi-static contraction during runaway gas accretion onto giant planets

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 10 Earth masses triggers a phase of rapid gas accretion. Previous 3D grid-based hydrodynamical simulations found runaway gas accretion rates corresponding to approximately 10 to 100 Jupiter masses per Myr. Such high accretion rates would result in all planets with larger-than-10-Earth-mass cores forming Jupiter-like planets, in clear contrast to the ice giants in the Solar System and the observed exoplanet population. In this work, we use 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 inner envelope, which can be orders of magnitude smaller than the mass flow through the outer atmosphere. For planets smaller than Saturn, we measure moderate gas accretion rates that are below 1 Jupiter mass per Myr. Higher mass planets, however, accrete up to 10 times faster and do not reveal a self-driven mechanism that can halt gas accretion. Therefore, the reason for the final masses of Saturn and Jupiter remains difficult to understand, unless their completion coincided with the dissipation of the Solar Nebula.

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arXiv2018

Pebble isolation mass --- scaling law and implications for the formation of super-Earths and gas giants

The growth of a planetary core by pebble accretion stops at the so called pebble isolation mass, when the core generates a pressure bump that traps drifting pebbles outside its orbit. If the isolation mass is very small, then gas accretion is protracted and the planet remains at a few Earth masses with a mainly solid composition. For larger values of the pebble isolation mass, the planet might be able to accrete gas from the protoplanetary disc and grow into a gas giant. Previous works have determined a scaling of the pebble isolation mass with cube of the disc aspect ratio. Here we expand on previous measurements and explore the dependency of the pebble isolation mass on all relevant parameters of the protoplanetary disc. We use 3D hydrodynamical simulations to measure the pebble isolation mass and derive a simple scaling law that captures the dependence on the local disc structure and the turbulent viscosity parameter $\alpha$. We find that small pebbles, coupled to the gas, with Stokes number $\tau_{\rm f}<0.005$ can drift through the partial gap at pebble isolation mass. However, as the planetary mass increases, particles must be decreasingly smaller to penetrate through the pressure bump. Turbulent diffusion of particles, however, can lead to an increase of the pebble isolation mass by a factor of two, depending on the strength of the background viscosity and on the pebble size. We finally explore the implications of the new scaling law of the pebble isolation mass on the formation of planetary systems by numerically integrating the growth and migration pathways of planets in evolving protoplanetary discs. Compared to models neglecting the dependence of the pebble isolation mass on the $\alpha$-viscosity, our models including this effect result in larger core masses for giant planets. These larger core masses are more akin to the core masses of the giant planets in the Solar System.

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arXiv2017

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 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. In regions of the disc that are not significantly dust-depleted, envelope accretion onto cores of about five Earth masses can get stalled as the gas flow enters the deep interior. Accreted solids sublimate deep in the convective interior, but small opacity-providing grains are trapped in the flow and do not settle, which further prevents rapid envelope accretion. The transition to runaway gas accretion can however be reached when cores grow larger than typical Super-Earths, beyond 15 Earth masses, and preferably when disc opacities are below kappa=1 cm^2/g. These findings offer an explanation for the typical low-mass envelopes around the cores of Super-Earths.

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arXiv2017

Planetary Torque in 3D Isentropic Disks

Planet migration is inherently a three-dimensional (3D) problem, because Earth-size planetary cores are deeply embedded in protoplanetary disks. Simulations of these 3D disks remain challenging due to the steep requirement in resolution. Using two different hydrodynamics code, FARGO3D and PEnGUIn, we simulate disk-planet interaction for a 1 to 5 Earth-mass planet embedded in an isentropic disk. We measure the torque on the planet and ensure that the measurements are converged both in resolution and between the two codes. We find that the torque is independent of the smoothing length of the planet's potential ($r_{\rm s}$), and that it has a weak dependence on the adiabatic index of the gaseous disk ($γ$). The torque values correspond to an inward migration rate qualitatively similar to previous linear calculations. We perform additional simulations with explicit radiative transfer using FARGOCA, and again find agreement between 3D simulations and existing torque formulae. We also present the flow pattern around the planets, and show that active flow is present within the planet's Hill sphere, and meridional vortices are shed downstream. The vertical flow speed near the planet is faster for a smaller $r_{\rm s}$ or $γ$, up to supersonic speeds for the smallest $r_{\rm s}$ and $γ$ in our study.

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arXiv2014

Stellar irradiated discs and implications on migration of embedded planets III: viscosity transitions

The migration strength and direction of embedded low-mass planets depends on the disc structure. In discs with an efficient radiative transport, the migration can be directed outwards for planets with more than 3-5 Earth masses. This is due to the entropy driven corotation torque, a process that extends the lifetimes of growing planetary embryos. We investigate the influence on the disc structure caused by a jump in the alpha parameter of the viscosity to model a dead-zone structure in the disc. We focus on M-dot discs, which have a constant net mass flux. Using the resulting disc structure, we investigate the consequences for the formation of planetesimals and determine the regions of outward migration for proto-planets. We performed numerical hydrosimulations of M-dot discs in the r-z-plane. We used the explicit/implicit hydrodynamical code FARGOCA that includes a full tensor viscosity and stellar irradiation as well as a two-temperature solver that includes radiation transport in the flux-limited diffusion approximation. Viscosity transitions inside the disc create transitions in density that stop inward migration for small planets through the so-called "planet trap" mechanism. This mechanism also works for planets down to M_P > 0.5M_E. Additionally, the viscosity transitions change the pressure gradient in the disc, which facilitates planetesimal formation via the streaming instability. However, a very steep transition in viscosity is needed to achieve in a pressure bump in the disc. The transition in viscosity facilitates planetesimal formation and can stop the migration of small-mass planets (M_P>0.5M_E), but still does not halt inward migration of smaller planets and planetesimals that are affected by gas drag. A very steep, probably unrealistic viscosity gradient is needed to trap planets of smaller masses and halt gas-drag-driven planetesimal migration at a pressure bump.

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arXiv2014

Stellar irradiated discs and implications on migration of embedded planets II: accreting-discs

The strength and direction of migration of embedded low mass planets depends on the disc's structure. It has been shown that, in discs with viscous heating and radiative transport, the migration can be directed outwards. In this paper we investigate the influence of a constant dM/dt-flux through the disc, as well as the influence of the disc's metallicity on the disc's thermodynamics. We focus on dM/dt discs, which have a net mass flux through them. Utilizing the resulting disc structure, we determine the regions of outward migration in the disc. We perform numerical hydrosimulations of dM/dt discs with viscous heating, radiative cooling and stellar irradiation in 2D in the r-z-plane. We use the explicit/implicit hydrodynamical code FARGOCA that includes a full tensor viscosity and stellar irradiation, as well as a two temperature solver that includes radiation transport in the flux-limited diffusion approximation. The migration of embedded planets is studied by using torque formulae. For a disc of gas surface density and viscosity, we find that the discs thermal structure depends on the product Sigma_G νand the amount of heavy elements, while the migration of planets additionally to the mentioned quantities, depends on the amount of viscosity itself. As a result of this, the disc structure can not be approximated by simple power laws. During the lifetime of the disc, the structure of the disc changes significantly in a non-linear way in the inner parts. In the late stages of the discs evolution, outward migration is only possible if the metallicity of the disc is high. For low metallicity, planets would migrate inwards and could potentially be lost. The presented disc structures and migration maps have important consequences on the formation of planets, as they can give hints on the different formation mechanisms for different types of planets as a function of metallicity.

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arXiv2013

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 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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