Anne-Sophie Libert

Predicting links in ego-networks using temporal information

Link prediction appears as a central problem of network science, as it calls for unfolding the mechanisms that govern the micro-dynamics of the network. In this work, we are interested in ego-networks, that is the mere information of interactions of a node to its neighbors, in the context of social relationships. As the structural information is very poor, we rely on another source of information to predict links among egos’ neighbors: the timing of interactions. We define several features to capture different kinds of temporal information and apply machine learning methods to combine these various features and improve the quality of the prediction. We demonstrate the efficiency of this temporal approach on a cellphone interaction dataset, pointing out features which prove themselves to perform well in this context, in particular the temporal profile of interactions and elapsed time between contacts.

Centrality measures and thermodynamic formalism for complex networks.

In the study of small and large networks it is customary to perform a simple random walk where the random walker jumps from one node to one of its neighbors with uniform probability. The properties of this random walk are intimately related to the combinatorial properties of the network. In this paper we propose to use the Ruelle-Bowens random walk instead, whose probability transitions are chosen in order to maximize the entropy rate of the walk on an unweighted graph. If the graph is weighted, then a free energy is optimized instead of the entropy rate. Specifically, we introduce a centrality measure for large networks, which is the stationary distribution attained by the Ruelle-Bowens random walk; we name it entropy rank. We introduce a more general version, which is able to deal with disconnected networks, under the name of free-energy rank. We compare the properties of those centrality measures with the classic PageRank and hyperlink-induced topic search (HITS) on both toy and real-life examples, in particular their robustness to small modifications of the network. We show that our centrality measures are more discriminating than PageRank, since they are able to distinguish clearly pages that PageRank regards as almost equally interesting, and are more sensitive to the medium-scale details of the graph.

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 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 smaller time-scale. While the inclination of low planetary masses (<5M_Jup) is always damped, large planetary masses with large i can undergo a Kozai-cycle with the disc. These Kozai-cycles are damped in time. Eccentricity is generally damped, except for very massive planets (M_p = 5M_Jup) where eccentricity can increase for low inclinations. The dynamics tends to a final state: planets end up in midplane and can then, over time, increase their eccentricity as a result of interactions with the disc. The interactions with the disc lead to damping of i and e after a scattering event of high-mass planets. If i is sufficiently reduced, the eccentricity can be pumped up because of interactions with the disc. If the planet is scattered to high inclination, it can undergo a Kozai-cycle with the disc that makes it hard to predict the exact movement of the planet and its orbital parameters at the dispersal of the disc.

Analytical study of the proximity of exoplanetary systems to mean-motion resonances

Aims. In a previous paper (Libert & Henrard 2005, Celest. Mech. & Dyn. Astron., 93, 187) we used a twelfth-order expansion of the perturbative potential in powers of eccentricities to represent the secular effects of two coplanar planets. This expansion was applied successfully to non resonant exoplanetary systems. This study was based on a first order (in the masses of the planets) model and will fail for systems too close to a resonance. In this paper we test the effects of the proximity of a mean-motion resonance on the secular motion of the planets. Methods. We analyse the proximity of several exoplanetary systems to a mean-motion resonance zone by using a first-order (in the mass ratios) Lie algorithm on the perturbative potential expanded to the twelfth order in the eccentricities. This perturbation method evaluates the difference between osculating elements and averaged ones. It permits us to decide whether resonant contributions dominate the terms of this difference or not. Results. This study is applied to several exosystems. We find that HD 168443, HD 38529, HD 74156, HD 217107, and HD 190360 are far away from a mean-motion resonance zone. v Andromedae and HD 12661 are rather close to the 5/1 resonance, HD 169830 to the 9/1 one. Hence, a secular theory is enough to depict correctly the behaviour of all these systems. On the other hand, HD 108874 and HD 202206 suffer from large perturbations in their motion due to the closeness of the 4/1 and 5/1 resonances, respectively. We also perform a complete investigation of the proximity of the v Andromedae system to mean-motion resonances, by studying the changes in behaviour due to different values of the outer semi-major axis. The v Andromedae system begins to be really influenced by the 5/1 resonant terms when the value of the outer semi-major axis decreases from 2.53 to 2.445.

Formation of ‘3D’ multiplanet systems by dynamical disruption of multiple-resonance configurations

Assuming that giant planets are formed in thin protoplanetary discs, a “3-D” system ‐ i.e. a planetary system composed of two (or more) planets, whose orbital planes have large values of mutual inclination ‐ can form, provided that the mutual inclination is excited by some dynamical mechanism. Resonant interactions and close planetary encounters are thought to be the primary inclination-excitation mechanisms, resulting in a resonant or non-resonant system, respectively. If by the end of planet formation the system is dynamically “hot”, then a phase of planet-planet scattering can be expected; however, this need not be the case in every system. Here we propose an alternative formation scenario, starting from a system composed of three giant planets in a nearly co-planar configuration. A s was recently shown for the case of the solar system, planetary migration in the gas disc (Type II migration) can force the planets to become trapped in a multiply-resonant state (similar to the Laplace resonance in the Galilean satellites). We simulate this process, assumi ng different values for the planetary masses and mass ratios. We show that, such a triple resonance generally becomes unstable, as the resonance excites the eccentricities of all planets, an d planet-planet scattering sets in. One of the three planets is typically ejected from the system, le aving behind a dynamically “hot” (but stable) two-planets configuration. The resulting two- planet systems typically have large values of semi-major axes ratio (α = a1/a2< 0.3), while the mutual inclination can be as high as 70 ◦ , with a median of∼ 30 ◦ . These values are quite close to the ones recently obtained for theυ-Andromedae system. A small fraction of our two-planet systems (∼ 5%) ends up in the stability zone of the Kozai resonance. In a few cases, the tri ple resonance can remain stable for long times and a “3-D” system can form by resonant excitation of the orbital inclinations; such a three-planet system could be stable if enough eccentricity damping is exerted on the planets. Finally, in the single-planet resulting systems, which are formed when two planets are ejected from the system, the inclination of the planet’s orbital plane with respect to the initial invariant plane ‐ presumably the plane perpendicular to the star’s spin axis ‐ can be as large as∼ 40 ◦ .

Habitability of planets on eccentric orbits: Limits of the mean flux approximation

Unlike the Earth, which has a small orbital eccentricity, some exoplanets discovered in the insolation habitable zone (HZ) have high orbital eccentricities (e.g., up to an eccentricity of ~0.97 for HD 20782 b). This raises the question of whether these planets have surface conditions favorable to liquid water. In order to assess the habitability of an eccentric planet, the mean flux approximation is often used. It states that a planet on an eccentric orbit is called habitable if it receives on average a flux compatible with the presence of surface liquid water. However, because the planets experience important insolation variations over one orbit and even spend some time outside the HZ for high eccentricities, the question of their habitability might not be as straightforward. We performed a set of simulations using the global climate model LMDZ to explore the limits of the mean flux approximation when varying the luminosity of the host star and the eccentricity of the planet. We computed the climate of tidally locked ocean covered planets with orbital eccentricity from 0 to 0.9 receiving a mean flux equal to Earth’s. These planets are found around stars of luminosity ranging from 1 L ⊙ to 10 -4 L ⊙ . We use a definition of habitability based on the presence of surface liquid water, and find that most of the planets considered can sustain surface liquid water on the dayside with an ice cap on the nightside. However, for high eccentricity and high luminosity, planets cannot sustain surface liquid water during the whole orbital period. They completely freeze at apoastron and when approaching periastron an ocean appears around the substellar point. We conclude that the higher the eccentricity and the higher the luminosity of the star, the less reliable the mean flux approximation.

On the influence of the Kozai mechanism in habitable zones of extrasolar planetary systems

Aims. We investigate the long-term evolution of inclined test particles representing a small Earth-like body with negligible gravitational effects (hereafter called massless test-planets) in the restricted three-body problem, and consisting of a star, a gas giant, and a massless test-planet. The test-planet is initially on a circular orbit and moves around the star at distances closer than the gas giant. The aim is to show the influences of the eccentricity and the mass of the gas giant on the dynamics, for various inclinations of the test-planet, and to investigate in more detail the Kozai mechanism in the elliptic problem. Methods. We performed a parametric study, integrating the orbital evolution of test particles whose initial conditions were distributed on the semi-major axis - inclination plane. The gas giants initial eccentricity was varied. For the calculations, we used the Lie integration method and in some cases the Bulirsch-Stoer algorithm. To analyze the results, the maximum eccentricity and the Lyapunov characteristic indicator were used. All integrations were performed for 10 5 periods of the gas giant. Results. Our calculations show that inclined massless test-planets can be in stable configurations with gas giants on either circular or elliptic orbits. The higher the eccentricity of the gas giant, the smaller the possible range in semi-major axis for the test-planet. For gas giants on circular orbits, our results illustrate the well-known results associated with the Kozai mechanism, which do not allow stable orbits above a critical inclination of approximately 40°. For gas giants on eccentric orbits, the dynamics is quite similar, and the massless companion exhibits limited variations in eccentricity. In addition, we identify a region around 35° consisting of long-time stable, low eccentric orbits. We show that these results are also valid for Earth-mass companions, therefore they can be applied to extrasolar systems: for instance, the extrasolar planetary system HD 154345 can possess a 35° degree inclined, nearly circular, Earth-mass companion in the habitable zone.

The language of exoplanet ranking metrics needs to change

We have found many Earth-sized worlds but we have no way of determining if their surfaces are Earth-like. This makes it impossible to quantitatively compare habitability, and pretending we can risks damaging the field.

Highly inclined and eccentric massive planets - II. Planet-planet interactions during the disc phase

Context. Observational evidence indicates that the orbits of extrasolar planets are more various than the circular and coplanar ones of the solar system. Planet-planet interactions during migration in the protoplanetary disc have been invoked to explain the formation of these eccentric and inclined orbits. However, our companion paper (Paper I) on the planet-disc interactions of highly inclined and eccentric massive planets has shown that the damping induced by the disc is significant for a massive planet, leading the planet back to the midplane with its eccentricity possibly increasing over time. Aims. We aim to investigate the influence of the eccentricity and inclination damping due to planet-disc interactions on the final configurations of the systems, generalizing previous studies on the combined action of the gas disc and planet-planet scattering during the disc phase. Methods. Instead of the simplistic K-prescription, our N-body simulations adopt the damping formulae for eccentricity and inclination provided by the hydrodynamical simulations of our companion paper. We follow the orbital evolution of 11 000 numerical experiments of three giant planets in the late stage of the gas disc, exploring different initial configurations, planetary mass ratios and disc masses. Results. The dynamical evolutions of the planetary systems are studied along the simulations, with a particular emphasis on the resonance captures and inclination-growth mechanisms. Most of the systems are found with small inclinations (≤10°) at the dispersal of the disc. Even though many systems enter an inclination-type resonance during the migration, the disc usually damps the inclinations on a short timescale. Although the majority of the multiple systems in our simulations are quasi-coplanar, ∼5% of them end up with high mutual inclinations (=10°). Half of these highly mutually inclined systems result from two-or three-body mean-motion resonance captures, the other half being produced by orbital instability and/or planet-planet scattering. When considering the long-term evolution over 100 Myr, destabilization of the resonant systems is common, and the percentage of highly mutually inclined systems still evolving in resonance drops to 30%. Finally, the parameters of the final system configurations are in very good agreement with the semi-major axis and eccentricity distributions in the observations, showing that planet-planet interactions during the disc phase could have played an important role in sculpting planetary systems. (Less)

Interesting dynamics at high mutual inclination in the framework of the Kozai problem with an eccentric perturber

We study the dynamics of the 3D three-body problem of a small body moving under the attractions of a star and a giant planet which orbits the star on a much wider and elliptic orbit. In particular, we focus on the influence of an eccentric orbit of the outer perturber on the dynamics of a small highly inclined inner body. Our analytical study of the secular perturbations relies on the classical octupole Hamiltonian expansion (third-order theory in the ratio of the semimajor axes), as third-order terms are needed to consider the secular variations of the outer perturber and potential secular resonances between the arguments of the pericentre and/or longitudes of the node of both bodies. Short-period averaging and node reduction (by adoption of the Laplace plane reference frame) reduce the problem to two degrees of freedom. The 4D dynamics is analysed through representative planes which identify the main equilibria of the problem. As in the circular problem (i.e. perturber on a circular orbit), the ‘Kozai-bifurcated’ equilibria play a major role in the dynamics of an inner body on a quasi-circular orbit: its eccentricity variations are very limited for mutual inclination between the orbital planes smaller than ∼40°, while they become large and chaotic for higher mutual inclination. Particular attention is also given to a region around 35° of mutual inclination, detected numerically by Funk et al. and consisting of long-time stable and particularly low-eccentricity orbits of the small body. Using a 12th-order Hamiltonian expansion in eccentricities and inclinations, in particular its action-angle formulation obtained by Lie transforms from Libert & Henrard, we show that this region presents an equality of two fundamental frequencies and can be regarded as a secular resonance. Our results also apply to binary star systems where a planet is revolving around one of the two stars.