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Lagrangian descriptors (LDs) based on the arc length of orbits previously demonstrated their utility in delineating structures governing the dynamics. Recently, a chaos indicator based on the second derivatives of the LDs, referred to as $\vert\vert \Delta \rm{LD} \vert\vert$, has been introduced to distinguish regular and chaotic trajectories. Thus far, the derivatives are numerically approximated using finite differences on fine meshes of initial conditions. In this paper, we instead use the differential algebra (DA) framework as a form of automatic differentiation to introduce and compute $\vert\vert \Delta \rm{LD} \vert\vert$ up to machine precision. We discuss and exemplify benefits of this framework, such as the determination of reliable thresholds to distinguish ordered from chaotic trajectories. Our extensive parametric study quantitatively assesses the accuracy and sensitivity of both the finite differences and differential arithmetic approaches by focusing on paradigmatic discrete models of Hamiltonian chaos, namely the Chirikov's standard map and coupled $4$-dimensional variants. Our results demonstrate that finite difference techniques for $\vert\vert \Delta \rm{LD} \vert\vert$ might lead to significant misclassification rate, up to $20\%$ when the phase space supports thin resonant webs, due to the difficulty to determine appropriate thresholds. On the contrary, $\vert\vert \Delta \rm{LD} \vert\vert$ computed through DA arithmetic leads to clear bimodal distributions which in turn lead to robust thresholds. As a consequence, the DA framework reveals as sensitive as established first order tangent map based indicators, independently of the underlying dynamical regime. Finally, the benefits of the DA framework are also highlighted for non-uniform depleted meshes of initial conditions., Comment: Submitted to PhysicaD. 22 pages, 11 figures. Comments and feedback are welcome

Accurate modeling of tidal interactions is crucial for interpreting recent JWST observations of the thermal emissions of TRAPPIST-1~b and c and for characterizing the surface conditions and potential habitability of the other planets in the system. Indeed, the rotation state of the planets, driven by tidal forces, significantly influences the heat redistribution regime. Due to their proximity to their host star and the estimated age of the system, the TRAPPIST-1 planets are commonly assumed to be in a synchronization state. In this work, we present the recent implementation of the co-planar tidal torque and forces equations within the formalism of Kaula in the N-body code Posidonius. This enables us to explore the hypothesis of synchronization using a tidal model well suited to rocky planets. We studied the rotational state of each planet by taking into account their multi-layer internal structure computed with the code Burnman. Simulations show that the TRAPPIST-1 planets are not perfectly synchronized but oscillate around the synchronization state. Planet-planet interactions lead to strong variations on the mean motion and tides fail to keep the spin synchronized with respect to the mean motion. As a result, the sub-stellar point of each planet experiences short oscillations and long-timescale drifts that lead the planets to achieve a synodic day with periods varying from $55$~years to $290$~years depending on the planet., Comment: 7 pages, 6 figures

PLATO (PLAnetary Transits and Oscillations of stars) is ESA's M3 mission designed to detect and characterise extrasolar planets and perform asteroseismic monitoring of a large number of stars. PLATO will detect small planets (down to <2 R_(Earth)) around bright stars (<11 mag), including terrestrial planets in the habitable zone of solar-like stars. With the complement of radial velocity observations from the ground, planets will be characterised for their radius, mass, and age with high accuracy (5 %, 10 %, 10 % for an Earth-Sun combination respectively). PLATO will provide us with a large-scale catalogue of well-characterised small planets up to intermediate orbital periods, relevant for a meaningful comparison to planet formation theories and to better understand planet evolution. It will make possible comparative exoplanetology to place our Solar System planets in a broader context. In parallel, PLATO will study (host) stars using asteroseismology, allowing us to determine the stellar properties with high accuracy, substantially enhancing our knowledge of stellar structure and evolution. The payload instrument consists of 26 cameras with 12cm aperture each. For at least four years, the mission will perform high-precision photometric measurements. Here we review the science objectives, present PLATO's target samples and fields, provide an overview of expected core science performance as well as a description of the instrument and the mission profile at the beginning of the serial production of the flight cameras. PLATO is scheduled for a launch date end 2026. This overview therefore provides a summary of the mission to the community in preparation of the upcoming operational phases.

Ultra-cool dwarf stars are abundant, long-lived, and uniquely suited to enable the atmospheric study of transiting terrestrial companions with JWST. Amongst them, the most prominent is the M8.5V star TRAPPIST-1 and its seven planets. While JWST Cycle 1 observations have started to yield preliminary insights into the planets, they have also revealed that their atmospheric exploration requires a better understanding of their host star. Here, we propose a roadmap to characterize the TRAPPIST-1 system -- and others like it -- in an efficient and robust manner. We notably recommend that -- although more challenging to schedule -- multi-transit windows be prioritized to mitigate the effects of stellar activity and gather up to twice more transits per JWST hour spent. We conclude that, for such systems, planets cannot be studied in isolation by small programs, but rather need large-scale, jointly space- and ground-based initiatives to fully exploit the capabilities of JWST for the exploration of terrestrial planets.

These last years several Systems with Tightly packed Inner Planets in the super-Earth mass regime have been discovered harboring chains of resonances. It is generally believed that planet pairs get trapped in MMR during the migration phase in the protoplanetary disk, while the tides raised by the host star provide a source of dissipation on very long timescales. In this work, we aim to study the departure from exact commensurabilities observed among the STIPs which harbor 3-planet resonances and analyze how tides play an important role in shaping the resonance offsets for the STIPs. We analyzed the resonance offsets between adjacent pairs for five multi-planetary systems, namely Kepler-80, Kepler-223, K2-138, TOI-178, and TRAPPIST-1, highlighting the existence of different trends in the offsets. On the one hand, we derived analytical estimates for the offsets, which confirm that the departure of the planetary pairs from the nominal MMRs are due to the 3-planet resonant dynamics. On the other hand, we performed N-body simulations including both orbital migration and tidal dissipation from the host star with simple prescriptions in order to test the effectiveness of this mechanism at shaping the observed trend in the offsets, focusing our study on the preservation of the resonant patterns in the different systems with the same general setup. We found that the trends in the offsets of the five detected systems can be produced by tidal damping effects, regardless of the considered value for the tidal factor. It is a robust mechanism that relaxes the system towards equilibrium while efficiently moving it along 3-planet resonances, which induces the observed resonance offset for each planet pair. In addition, we showed that for Kepler-80, K2-138, and TOI-178, the amplitudes of the resonant offsets can also be reproduced with appropriate tidal factor, for the estimated age of the systems., Comment: Accepted for publication in A&A

The aim of this work is to study the impact of a binary companion on the evolution of two-planet systems during both the type-II migration phase and their long-term evolution after the dissipation of the protoplanetary disk. We use the symplectic integrator SyMBA, modified to include a wide binary companion. We also include the Type-II migration of giant planets during the disk phase with suitable eccentricity and inclination damping as well as the disk gravitational potential acting on the planets and the nodal precession of the disk induced by the binary companion. We consider various inclinations, eccentricities, and separations of the binary companion. Disk migration allows the formation of planet pairs in mean-motion resonances (MMR) despite the presence of the binary companion. When the binary separation is wide (1000au), the timescale of the perturbations it raises on the planets is longer than the disk's lifetime and resonant pairs are routinely formed in the 2:1, 5:2 and 3:1 MMR. Provided the planet-planet interaction timescale is smaller than the binary perturbations timescale, these systems can remain in resonance long after the disk has dissipated. When the binary separation is smaller (250au), only planets in the 2:1 resonance tend to remain in a resonant state and more chaotic evolutions are observed, as well as more ejections. After those ejections, the remaining planet can become eccentric due to the perturbations from the binary companion and for strongly inclined binary companions captures in the von Ziepel-Lidov-Kozai resonance can occur, while in systems with two planets this mechanism is quenched by planet-planet interactions. Our simulations reveal that the interplay between planet-disk, planet-planet and planet-binary interactions can lead to the formation of resonant pairs of planets which remain stable over timescales much longer than the disk's lifetime., Comment: 15 pages, 16 figures

Planetary formation theories and, more specifically, migration models predict that planets can be captured in mean-motion resonances (MMRs) during the disc phase. The distribution of period ratios between adjacent planets shows an accumulation in the vicinity of the resonance, which is not centred on the nominal resonance but instead presents an offset slightly exterior to it. Here we extend on previous works by thoroughly exploring the effect of different disc and planet parameters on the resonance offset during the disc migration phase. The dynamical study is carried out for several first-order MMRs and for both low-mass Earth-like planets undergoing type-I migration and giant planets evolving under type-II migration. We find that the offset varies with time during the migration of the two-planet system along the apsidal corotation resonance family. The departure from the nominal resonance increases for higher planetary masses and stronger eccentricity damping. In the Earth to super-Earth regime, we find offset values in agreement with the observations when using a sophisticated modelling for the planet-disc interactions, where the damping timescale depends on the eccentricity. This dependence causes a feedback which induces an increase of the resonance offsets. Regarding giant planets, the offsets of detected planet pairs are well reproduced with a classical $K$-factor prescription for the planet-disc interactions when the eccentricity damping rate remains low to moderate. In both regimes, eccentricities are in agreement with the observations too. As a result, planet-disc interactions provide a generic channel to generate the offsets found in the observations., Comment: 13 pages, 12 figures, article accepted for publication in MNRAS

We analyze solutions drawn from the recently published posterior distribution of the TRAPPIST-1 system, which consists of seven Earth-size planets appearing to be in a resonant chain around a red dwarf. We show that all the planets are simultaneously in two-planet and three-planet resonances, apart from the innermost pair for which the two-planet resonant angles circulate. By means of a frequency analysis, we highlight that the transit-timing variation (TTV) signals possess a series of common periods varying from days to decades, which are also present in the variations of the dynamical variables of the system. Shorter periods (e.g., the TTVs characteristic timescale of 1.3 yr) are associated with two-planet mean-motion resonances, while longer periods arise from three-planet resonances. By use of $N$-body simulations with migration forces, we explore the origin of the resonant chain of TRAPPIST-1 and find that for particular disc conditions, a chain of resonances - similar to the observed one - can be formed which accurately reproduces the observed TTVs. Our analysis suggests that while the 4-yr collected data of observations hold key information on the two-planet resonant dynamics, further monitoring of TRAPPIST-1 will soon provide signatures of three-body resonances, in particular the 3.3 and 5.1 yr periodicities expected for the current best-fit solution. dditional observations would help to assess whether the innermost pair of planets is indeed resonant (its proximity to the 8:5 resonance being challenging to explain), and therefore give additional constraints on formation scenarios., Comment: 11 pages, 14 figures, accepted for publication in A&A

Previous works on the divergence of first-order mean-motion resonances (MMRs) have studied in detail the extent of the pericentric and apocentric libration zones of adjacent first-order MMRs, highlighting possible bridges between them in the low-eccentricity circular restricted three-body problem. Here, we describe the previous results in the context of periodic orbits and show that the so-called circular family of periodic orbits is the path that can drive the passage between neighbouring resonances under dissipative effects. We illustrate that the circular family can bridge first and higher order resonances while its gaps at first-order MMRs can serve as boundaries that stop transitions between resonances. In particular, for the Sun-asteroid-Jupiter problem, we show that, during the migration of Jupiter in the protoplanetary disc, a system initially evolving below the apocentric branch of a first-order MMR follows the circular family and can either be captured into the pericentric branch of an adjacent first-order MMR if the orbital migration is rapid or in a higher order MMR in case of slow migration. Radial transport via the circular family can be extended to many small body and planetary system configurations undergoing dissipative effects (e.g., tidal dissipation, solar mass-loss and gas drag)., Comment: Accepted for publication in MNRAS

The discovery of numerous circumprimary planets in the last few years has brought to the fore the question of planet formation in binary systems. The significant dynamical influence, during the protoplanetary disk phase, of a binary companion on a giant planet has previously been highlighted for wide binary stars. In particular, highly inclined binary companion can induce perturbations on the disk and the planets, through the Lidov-Kozai resonance, which could inhibit the formation process. In this work, we aim to study how the disk gravitational potential acting on the planet and the nodal precession \textbf{induced by the wide binary companion with separation of 1000 AU} on the disk act to suppress the Lidov-Kozai perturbations on a migrating giant planet. We derive new approximate formulas for the evolution of the disk's inclination and longitude of the ascending node, in the case of a rigidly precessing disk with a decreasing mass and perturbed by a wide binary companion, which are suitable for $N$-body simulations. We carry out 3200 simulations with several eccentricity and inclination values for the binary companion. The gravitational and damping forces exerted by the disk on the planet tend to keep the latter in the midplane of the former, and suppress the effect of the binary companion by preventing the planet from getting locked in the Lidov-Kozai resonance during the disk phase. We also confirm that because of nodal precession induced by the binary, a primordial spin-orbit misalignment could be generated for circumprimary planets with an inclined binary companion., Comment: 11 pages, 10 figures, to be published in MNRAS

We investigate the inclination-growth mechanisms for two-planet systems during the late protoplanetary disc phase. In previous works, much attention has been directed to the inclination-type resonance, and it has been shown that it asks for high eccentricities to be acquired during the migration of the giant planets. By adopting eccentricity and inclination damping formulae based on hydrodynamical simulations (instead of the K-prescription), we have carried out 20 000 numerical simulations, where we vary the initial planetary eccentricities, the migration rate, and the dispersal time of the gas disc. Our results confirm that highly mutually inclined systems are unlikely to be produced by an inclination-type resonance of two migrating giant planets. However, in ~1 per cent of the simulations, inclination-type resonance is observed, and a dynamical study of the evolutions reveals that the inclination-type resonance mechanism operates in three cases: (i) when the inner planet reaches the inner cavity of the disc, (ii) at moderate to high eccentricities for faster migration rates, and (iii) at low to moderate eccentricities after a phase of orbital destabilization and re-arrangement., Comment: Published in MNRAS

Aims. There are a growing number of giant planets discovered moving around one stellar component of a binary star, most of which have very diverse eccentricity. These discoveries raise the question of their formation and long-term evolution because the stellar companion can strongly affect the planet formation process. We aim to study the dynamical influence of a wide binary companion on the evolution of a single giant planet migrating in a protoplanetary disk. Methods. Using a symplectic n-body integrator adapted for binary star systems and modeling the dissipation due to the disk by appropriate formulae emerging from hydrodynamical simulations, we carried out 3600 simulations with different orbital parameters for the planet and different eccentricity and inclination values for the binary companion. The long-term evolution of the planets was followed for 100 Myr and the different dynamical behaviors were unveiled using a quadrupolar Hamiltonian approach. Results. We highlight the high probability for planets to experience, during the disk phase, a scattering event or an ejection due to the presence of a binary companion. We show that a capture in a Lidov-Kozai resonant state is far from automatic when the binary companion star is highly inclined, since only $11 \%$ of the systems end up locked in the resonance at the end of the simulations. Nevertheless, in the presence of a highly inclined binary companion, all the planetary evolutions are strongly influenced by the Lidov-Kozai resonance and the nonresonant evolutions present high eccentricity and inclination variations associated with circulation around the Lidov-Kozai islands., Comment: Accepted for publication in A&A

The convergent migration of two planets in a gaseous disc can lead to capture in mean motion resonance (MMR). In addition, pairs of planets in or near MMRs are known to produce strong transit timing variations (TTVs). In this paper we study the impact of disc-induced migrations on the TTV signal of pairs of planets that enter a resonant configuration. We show that disc-induced migration creates a correlation between the amplitude and the period of the TTVs. We study the case of K2-24, a system of two planets whose period ratio indicates that they are in or near the 2:1 MMR, with non-zero eccentricities and large-amplitude TTVs. We show that a simple disc-induced migration cannot reproduce the observed TTVs, and we propose a formation scenario in which the capture in resonance occurring during migration in a disc with strong eccentricity damping is followed by eccentricity excitation during the dispersal of the disc, assisted by a third planet whose presence has been suggested by radial velocity observations. This scenario accounts for the eccentricities of the two planets and their period ratio, and accurately reproduces the amplitude and period of the TTVs. It allows for a unified view of the formation and evolution history of K2-24, from disc-induced migration to its currently observed properties., Comment: 9 pages, 7 figures. Accepted for publication in Astronomy and Astrophysics

Many extrasolar systems possessing planets in mean-motion resonance or resonant chain have been discovered to date. The transit method coupled with transit timing variation analysis provides an insight into the physical and orbital parameters of the systems, but suffers from observational limitations. When a (near-)resonant planetary system resides in the dynamical neighbourhood of a stable periodic orbit, its long-term stability, and thus survival, can be guaranteed. We use the intrinsic property of the periodic orbits, namely their linear horizontal and vertical stability, to validate or further constrain the orbital elements of detected two-planet systems. We computed the families of periodic orbits in the general three-body problem for several two-planet Kepler and K2 systems. The dynamical neighbourhood of the systems is unveiled with maps of dynamical stability. Additional validations or constraints on the orbital elements of K2-21, K2-24, Kepler-9, and (non-coplanar) Kepler-108 near-resonant systems were achieved. While a mean-motion resonance locking protects the long-term evolution of the systems K2-21 and K2-24, such a resonant evolution is not possible for the Kepler-9 system, whose stability is maintained through an apsidal anti-alignment. For the Kepler-108 system, we find that the stability of its mutually inclined planets could be justified either solely by a mean-motion resonance, or in tandem with an inclination-type resonance. Going forward, dynamical analyses based on periodic orbits could yield better constrained orbital elements of near-resonant extrasolar systems when performed in parallel to the fitting of the observational data., Comment: Accepted for publication in A&A

Extrasolar systems with planets on eccentric orbits close to or in mean-motion resonances are common. The classical low-order resonant Hamiltonian expansion is unfit to describe the long-term evolution of these systems. We extend the Laplace-Lagrange secular approximation for coplanar systems with two planets by including (near-)resonant harmonics, and realize an expansion at high order in the eccentricities of the resonant Hamiltonian both at orders one and two in the masses. We show that the expansion at first order in the masses gives a qualitative good approximation of the dynamics of resonant extrasolar systems with moderate eccentricities, while the second order is needed to reproduce more accurately their orbital evolutions. The resonant approach is also required to correct the secular frequencies of the motion given by the Laplace-Lagrange secular theory in the vicinity of a mean-motion resonance. The dynamical evolutions of four (near-)resonant extrasolar systems are discussed, namely GJ 876 (2:1 resonance), HD 60532 (3:1), HD 108874 and GJ 3293 (close to 4:1)., Comment: 21 pages, 7 figures

Aims. To date, more than 600 multi-planetary systems have been discovered. Due to the limitations of the detection methods, our knowledge of the systems is usually far from complete. In particular, for planetary systems discovered with the radial velocity (RV) technique, the inclinations of the orbital planes, and thus the mutual inclinations and planetary masses, are unknown. Our work aims to constrain the spatial configuration of several RV-detected extrasolar systems that are not in a mean-motion resonance. Methods. Through an analytical study based on a first-order secular Hamiltonian expansion and numerical explorations performed with a chaos detector, we identified ranges of values for the orbital inclinations and the mutual inclinations, which ensure the long-term stability of the system. Our results were validated by comparison with n-body simulations, showing the accuracy of our analytical approach up to high mutual inclinations (approx. 70{\deg}-80{\deg}). Results. We find that, given the current estimations for the parameters of the selected systems, long-term regular evolution of the spatial configurations is observed, for all the systems, i) at low mutual inclinations (typically less than 35{\deg}) and ii) at higher mutual inclinations, preferentially if the system is in a Lidov-Kozai resonance. Indeed, a rapid destabilisation of highly mutually inclined orbits is commonly observed, due to the significant chaos that develops around the stability islands of the Lidov-Kozai resonance. The extent of the Lidov-Kozai resonant region is discussed for ten planetary systems (HD 11506, HD 12661, HD 134987, HD 142, HD 154857, HD 164922, HD 169830, HD 207832, HD 4732, and HD 74156)., Comment: Accepted for publication in A&A

Many triple-star systems have an inner pair with an orbital period of a few days only. A common mechanism to explain the short-period pile-up present in the observations is the migration through Lidov-Kozai cycles combined with tidal friction. Here, we revisit this mechanism and aim to determine the initial orbital configurations leading to this process. We show that the mutual inclination of the triple-star system is not the only critical parameter, since the eccentricity as well as the argument of the pericenter of the inner orbit also play an important role in the establishment of the Lidov-Kozai migration. Our framework is the secular hierarchical three-body problem (octupole order approximation) with general relativity corrections, including the effects of tides, stellar oblateness and magnetic spin-down braking. Both the orbital and the spin evolutions are considered. Extensive numerical simulations with uniform and non-uniform distributions of the initial orbital parameters are carried out, and unbiased initial conditions leading to Lidov-Kozai migration are revealed. Finally, we highlight the importance of the initial "Kozai constant" $h=\sqrt{(1-e^2)}\cos{i}$ in the dynamical evolution of triple-star systems, by showing that phase portraits at given $h$-values unveil different evolution paths., Comment: Accepted for publication in MNRAS

We consider a planetary system consisting of two primaries, namely a star and a giant planet, and a massless secondary, say a terrestrial planet or an asteroid, which moves under their gravitational attraction. We study the dynamics of this system in the framework of the circular and elliptic restricted TBP, when the motion of the giant planet describes circular and elliptic orbits, respectively. Originating from the circular family, families of symmetric periodic orbits in the 3/2, 5/2, 3/1, 4/1 and 5/1 mean-motion resonances are continued in the circular and the elliptic problems. New bifurcation points from the circular to the elliptic problem are found for each of the above resonances and thus, new families, continued from these points are herein presented. Stable segments of periodic orbits were found at high eccentricity values of the already known families considered as whole unstable previously. Moreover, new isolated (not continued from bifurcation points) families are computed in the elliptic restricted problem. The majority of the new families mainly consist of stable periodic orbits at high eccentricities. The families of the 5/1 resonance are investigated for the first time in the restricted three-body problems. We highlight the effect of stable periodic orbits on the formation of stable regions in their vicinity and unveil the boundaries of such domains in phase space by computing maps of dynamical stability. The long-term stable evolution of the terrestrial planets or asteroids is dependent on the existence of regular domains in their dynamical neighbourhood in phase space, which could host them for long time spans. This study, besides other celestial architectures that can be efficiently modelled by the circular and elliptic restricted problems, is particularly appropriate for the discovery of terrestrial companions among the single-giant planet systems discovered so far., Comment: Accepted for publication in Celestial Mechanics and Dynamical Astronomy

The late-stage formation of giant planetary systems is rich in interesting dynamical mechanisms. Previous simulations of three giant planets initially on quasi-circular and quasi-coplanar orbits in the gas disc have shown that highly mutually inclined configurations can be formed, despite the strong eccentricity and inclination damping exerted by the disc. Much attention has been directed to inclination-type resonance, asking for large eccentricities to be acquired during the migration of the planets. Here we show that inclination excitation is also present at small to moderate eccentricities in two-planet systems that have previously experienced an ejection or a merging and are close to resonant commensurabilities at the end of the gas phase. We perform a dynamical analysis of these planetary systems, guided by the computation of planar families of periodic orbits and the bifurcation of families of spatial periodic orbits. We show that inclination excitation at small to moderate eccentricities can be produced by (temporary) capture in inclination-type resonance and the possible proximity of the non-coplanar systems to spatial periodic orbits contributes to maintaining their mutual inclination over long periods of time., Comment: 13 pages, 7 figures, Published in Celestial Mechanics and Dynamical Astronomy, 02/2018

Hundreds of giant planets have been discovered so far and the quest of exo-Earths in giant planet systems has become intriguing. In this work, we aim to address the question of the possible long-term coexistence of a terrestrial companion on an orbit interior to a giant planet, and explore the extent of the stability regions for both non-resonant and resonant configurations. Our study focuses on the restricted three-body problem, where an inner terrestrial planet (massless body) moves under the gravitational attraction of a star and an outer massive planet on a circular or elliptic orbit. Using the Detrended Fast Lyapunov Indicator as a chaotic indicator, we constructed maps of dynamical stability by varying both the eccentricity of the outer giant planet and the semi-major axis of the inner terrestrial planet, and identify the boundaries of the stability domains. Guided by the computation of families of periodic orbits, the phase space is unravelled by meticulously chosen stable periodic orbits, which buttress the stability domains. We provide all possible stability domains for coplanar symmetric configurations and show that a terrestrial planet, either in mean-motion resonance or not, can coexist with a giant planet, when the latter moves on either a circular or an (even highly) eccentric orbit. New families of symmetric and asymmetric periodic orbits are presented for the 2/1 resonance. It is shown that an inner terrestrial planet can survive long time spans with a giant eccentric outer planet on resonant symmetric orbits, even when both orbits are highly eccentric. For 22 detected single-planet systems consisting of a giant planet with high eccentricity, we discuss the possible existence of a terrestrial planet. This study is particularly suitable for the research of companions among the detected systems with giant planets, and could assist with refining observational data., Comment: Accepted for publication in A&A

Evidence of mutually inclined planetary orbits has been reported for giant planets these last years. Here we aim to study the impact of eccentric and inclined massive giant planets on the terrestrial planet formation process, and investigate whether it can possibly lead to the existence of inclined terrestrial planets. We have performed 126 simulations of the late-stage planetary accretion in eccentric and inclined giant planet systems. The physical and orbital parameters of the giant planet systems result from n-body simulations of three giant planets in the late stage of the gas disc, under the combined action of Type II migration and planet-planet scattering. Fourteen two- and three-planet configurations have been selected, with diversified masses, semi-major axes (resonant configurations or not), eccentricities and inclinations (including coplanar systems) at the dispersal of the gas disc. We have then followed the gravitational interactions of these systems with an inner disc of planetesimals and embryos (9 runs per system), studying in detail the final configurations of the formed terrestrial planets. While coplanar giant systems harbour several terrestrial planets, generally as massive as the Earth and mainly on low eccentric and low inclined orbits, terrestrial planets formed in systems with mutually inclined giant planets are usually fewer, less massive (<0.5 M_{\Earth}) and with larger eccentricities and inclinations. This work shows that terrestrial planets can form on stable inclined orbits through the classical accretion theory, even in coplanar giant planet systems emerging from the disc phase.

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., Comment: Published in Nature Astronomy, Comment, 02/2017

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. 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 evolution of $11000$ numerical experiments of three giant planets in the late stage of the gas disc, exploring different initial configurations, planetary mass ratios and disc masses. The dynamical evolutions of the planetary systems are studied along the simulations, with emphasis on the resonance captures and inclination-growth mechanisms. Most of the systems are found with small inclinations ($\le10^{\circ}$) 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 results are quasi-coplanar, $\sim5\%$ of them end up with high mutual inclinations ($\ge10^{\circ}$). Half of these highly mutually inclined systems result from two- or three-body MMR 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., Comment: 13 pages, 14 figures, Accepted for publication in A&A

Contrary to 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 $\sim0.97$ for HD~20782~b). This raises the question of the capacity of these planets to host surface 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, as the planets do 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, exploring 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, around stars of luminosity ranging from $L_\odot$ to $10^{-4}~L_\odot$. Using here a definition of habitability based on the presence of surface liquid water, we 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., Comment: Recommended for publication in A&A. 17 pages, 16 figures

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., Comment: submitted to EPJ Data Science

Uncovering unknown or missing links in social networks is a difficult task because of their sparsity and because links may represent different types of relationships, characterized by different structural patterns. In this paper, we define a simple yet efficient supervised learning-to-rank framework, called RankMerging, which aims at combining information provided by various unsupervised rankings. We illustrate our method on three different kinds of social networks and show that it substantially improves the performances of unsupervised metrics of ranking. We also compare it to other combination strategies based on standard methods. Finally, we explore various aspects of RankMerging, such as feature selection and parameter estimation and discuss its area of relevance: the prediction of an adjustable number of links on large networks., Comment: 43 pages, published in Machine Learning Journal

We study the secular evolution of several exoplanetary systems by extending the Laplace-Lagrange theory to order two in the masses. Using an expansion of the Hamiltonian in the Poincar\'e canonical variables, we determine the fundamental frequencies of the motion and compute analytically the long-term evolution of the keplerian elements. Our study clearly shows that, for systems close to a mean-motion resonance, the second order approximation describes their secular evolution more accurately than the usually adopted first order one. Moreover, this approach takes into account the influence of the mean anomalies on the secular dynamics. Finally, we set up a simple criterion that is useful to discriminate between three different categories of planetary systems: (i) {\it secular} systems (HD 11964, HD 74156, HD 134987, HD 163607, HD 12661 and HD 147018); (ii) systems {\it near a mean-motion resonance} (HD 11506, HD 177830, HD 9446, HD 169830 and $\upsilon$ Andromedae); (iii) systems {\it really close to} or {\it in a mean-motion resonance} (HD 108874, HD 128311 and HD 183263)., Comment: 22 pages, 4 figures

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., Comment: accepted for publication in Astronomy and Astrophysics

In this work we propose a new numerical approach to distinguish between regular and chaotic orbits in Hamiltonian systems, based on the simultaneous integration of both the orbit and the deviation vectors using a symplectic scheme, hereby called {\em global symplectic integrator}. In particular, the proposed method allows us to recover the correct orbits character with very large integration time steps, small energy losses and short CPU times. To illustrate the numerical performances of the global symplectic integrator we will apply it to two well-known and widely studied problems: the H\'enon-Heiles model and the restricted three-body problem.

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 neighbours 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 maximise the entropy rate of the walk on an unweighted graph. If the graph is weighted, then a free energy is optimised instead of entropy rate. Specifically, we introduce a centrality measure for large networks, which is the stationary distribution attained by the the Ruelle-Bowens random walk; we name it Entropy Rank. We introduce a more general version, 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 HITS on both toy and real-life examples, in particular their robustness to small modifications of the network. It is observed that our centrality measures have a better discriminating power than PageRank, being able to distinguish clearly pages that PageRank holds for almost equally interesting, and is more sensitive to the medium-scale details of the graph., Comment: 7 pages, 3 figures, submitted in 2010; change from v1: major (numerical experimentations, interpretation, comparison with other PageRank, etc.); change from v2: minor (typo)