arXiv subjects

C. Beaugé

Publications and source records attributed to C. Beaugé.

At least 19 recordsLinked to original sources

arXiv2025

A formation mechanism for narrow rings around minor bodies

The recent discovery of narrow rings around minor bodies has raised many questions regarding their origin and current dynamics. Sharp ring boundaries seem indicative of shepherding moonlets, but none have been found. All rings lie close to spin-orbit resonances (SORs) with the central body, particularly the 1/3, even though it is not clear how these may be related. Furthermore, in at least one case the location of the ring is exterior to the Roche radius, adding to the striking differences with respect to giant planets. We study the dynamical evolution of a particle disk around a minor body, perturbed by the non-spherical component of the gravity field, particle collisions, and spin changes of the central mass linked to angular momentum conservation. By varying key parameters, we search for cases where the combined effects may lead to resonance capture and orbital configurations similar to those that have been observed. We performed N-body simulations of massless particles orbiting a central spherical body with a co-rotating mass anomaly. Collisions were modeled by adopting a simple radial damping force. Angular momentum conservation links the body spin to the disk orbital evolution. Since the gravitational effect of the test particles is neglected, this back-reaction is introduced externally assuming ad hoc spin-down rates and disk mass. Interaction between non-sphericity and collisions leads to the formation of a narrow ring that slowly recedes from the central mass. Spin-down of the minor planet shifts the SORs outward, enabling resonant capture. For suitable parameters, a portion -or all- of the initial disk can become trapped in the 1/3 SOR with the mass anomaly, in dynamically stable low-eccentricity orbits. Although the required disk mass is high ($\ge 1\%$ of the central body), long-term collisional erosion could reduce it to values that are consistent with observed ringlets.

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arXiv2023

A Six-Planet Resonance Chain in K2-138?

The K2-138 system hosts six planets and presents an interesting case study due to its distinctive dynamical structure. Its five inner planets are near a chain of 3/2 two-body mean-motion resonances, while the outermost body (planet {\it g}) is significantly detached, having a mean-motion ratio of $n_f/n_g \sim 3.3$ with its closest neighbor. We show that the orbit of $m_g$ is actually consistent with the first-order three-planet resonance (3P-MMR) characterized by the relation $2n_e - 4n_f + 3n_g = 0$ and is the first time a pure first-order 3P-MMR is found in a multi-planet system and tied to its current dynamical structure. Adequate values for the masses allow to trace the dynamical history of the system from an initial capture in a 6-planet chain (with $n_f/n_g$ in a 3/1 resonance), up to its current configuration due to tidal interactions over the age of the star. The increase in resonance offset with semi-major axis, as well as its large value for $n_f/n_g$ can be explained by the slopes of the pure three-planet resonances in the mean-motion ratio plane. The triplets slide outward over these curves when the innermost pair is pulled apart by tidal effects, in a \textit{pantograph-}like manner. The capture into the 3P-MMR is found to be surprisingly robust given similar masses for $m_g$ and $m_f$, and it is possible that the same effect may also be found in other compact planetary systems.

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arXiv2023

Tidal Evolution of Close-in Exoplanets and Host Stars

The evolution of exoplanetary systems with a close-in planet is ruled by the tides mutually raised on the two bodies and by the magnetic braking of the host star. This paper deals with consequences of this evolution and some features that can be observed in the distribution of the systems two main periods: the orbital periods and the stars rotational periods. The results of the simulations are compared to plots showing both periods as determined from the light curves of a large number of Kepler objects of interest. These plots show important irregularities as a dearth of systems in some regions and accumulations of hot Jupiters in others. It is shown that the accumulation of short-period hot Jupiters around stars with rotation periods close to 25 days results from the evolution of the systems under the joint action of tides and braking, and requires a relaxation factor for solar-type stars of around $10 \, s^{-1}$.

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arXiv2021

Creep Tide Model for the 3-Body Problem. The rotational evolution of a circumbinary planet

We present a tidal model for treating the rotational evolution in the general three-body problem with arbitrary viscosities, in which all the masses are considered to be extended and all the tidal interactions between pairs are taken into account. Based on the creep tide theory, we present the set of differential equations that describes the rotational evolution of each body, in a formalism that is easily extensible to the N tidally-interacting body problem. We apply our model to the case of a circumbinary planet and use a Kepler-38 like binary system as a working example. We find that, in this low planetary eccentricity case, the most likely final stationary rotation state is the 1:1 spin-orbit resonance, considering an arbitrary planetary viscosity inside the estimated range for the solar system planets. We derive analytical expressions for the mean rotational stationary state, based on high-order power series of the semimajor axes ratio a1 /a2 and low-order expansions of the eccentricities. These are found to reproduce very accurately the mean behaviour of the low-eccentric numerical integrations for arbitrary planetary relaxation factors, and up to a1/a2 \sim 0.4. Our analytical model is used to predict the stationary rotation of the Kepler circumbinary planets and find that most of them are probably rotating in a sub-synchronous state, although the synchrony shift is much less important than the one estimated in Zoppetti et al. (2019, 2020). We present a comparison of our results with those obtained with the Constant Time Lag and find that, unlike what we assumed in our previous works, the cross torques have a non-negligible net secular contribution, and must be taken into account when computing the tides over each body in an N-extended-body system from an arbitrary reference frame. These torques are naturally taken into account in the creep theory.

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arXiv2019

Tidal evolution of circumbinary systems with arbitrary eccentricities: applications for Kepler systems

We present an extended version of the Constant Time Lag analytical approach for the tidal evolution of circumbinary planets introduced in our previous work. The model is self-consistent, in the sense that all tidal interactions between pairs are computed, regardless of their size. We derive analytical expressions for the variational equations governing the spin and orbital evolution, which are expressed as high-order elliptical expansions in the semimajor axis ratio but retain closed form in terms of the binary and planetary eccentricities. These are found to reproduce the results of the numerical simulations with arbitrary eccentricities very well, as well as reducing to our previous results in the low-eccentric case. Our model is then applied to the well-characterised Kepler circumbinary systems by analysing the tidal timescales and unveiling the tidal flow around each different system. In all cases we find that the spins reach stationary values much faster than the characteristic timescale of the orbital evolution, indicating that all Kepler circumbinary planets are expected to be in a sub-synchronous state. On the other hand, all systems are located in a tidal flow leading to outward migration; thus the proximity of the planets to the orbital instability limit may have been even greater in the past. Additionally, Kepler systems may have suffered a significant tidally induced eccentricity damping, which may be related to their proximity to the capture eccentricity. To help understand the predictions of our model, we also offer a simple geometrical interpretation of our results.

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arXiv2019

Shannon Entropy Applied to the Planar Restricted Three-Body Problem

We present a numerical study of the application of the Shannon entropy technique to the planar restricted three-body problem in the vicinity of first-order interior mean-motion resonances with the perturber. We estimate the diffusion coefficient for a series of initial conditions and compare the results with calculations obtained from the time evolution of the variance in the semimajor-axis and eccentricity plane. Adopting adequate normalization factors, both methods yield comparable results, although much shorter integration times are required for entropy calculations. A second advantage of the use of entropy is that it is possible to obtain reliable results even without the use of ensembles or analysis restricted to surfaces of section or representative planes. This allows for a much more numerically efficient tool that may be incorporated into a working N-body code and applied to numerous dynamical problems in planetary dynamics. Finally, we estimate instability times for a series of initial conditions in the 2/1 and 3/2 mean-motion resonances and compare them with times of escape obtained from directed N-body simulations. We find very good agreement in all cases, not only with respect to average values but also in their dispersion for near-by trajectories

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arXiv2019

A self-consistent weak friction model for the tidal evolution of circumbinary planets

We present a self-consistent model for the tidal evolution of circumbinary planets. Based on the weak-friction model, we derive expressions of the resulting forces and torques considering complete tidal interactions between all the bodies of the system. Although the tidal deformation suffered by each extended mass must take into account the combined gravitational effects of the other two bodies, the only tidal forces that have a net effect on the dynamic are those that are applied on the same body that exerts the deformation, as long as no mean-motion resonance exists between the masses. We apply the model to the Kepler-38 binary system. The evolution of the spin equations shows that the planet reaches a stationary solution much faster than the stars, and the equilibrium spin frequency is sub-synchronous. The binary components evolve on a longer timescale, reaching a super-synchronous solution very close to that derived for the 2-body problem. After reaching spin stationarity, the eccentricity is damped in all bodies and for all the parameters analyzed here. A similar effect is noted for the binary separation. The semimajor axis of the planet, on the other hand, may migrate inwards or outwards, depending on the masses and orbital parameters. In some cases the secular evolution of the system may also exhibit an alignment of the pericenters, requiring to include additional terms in the tidal model. Finally, we derived analytical expressions for the variational equations of the orbital evolution and spin rates based on low-order elliptical expansions in the semimajor axis ratio and the eccentricities. These are found to reduce to the 2-body case when one of the masses is taken equal to zero. This model allow us to find a close and simple analytical expression for the stationary spin rates of all the bodies, as well as predicting the direction and magnitude of the orbital migration.

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arXiv2018

Orbital migration and Resonance Offset of the Kepler-25 and K2-24 systems

Based on the model described in Ramos et al., 2017, we present an analytical+numerical study of the resonance capture under Type-I migration for the Kepler-25 (Marcy et al., 2014) and K2-24 (Petigura et al., 2016) Kepler systems, both close to a 2/1 mean-motion resonance. We find that, depending on the flare index and the proximity to the central star, the average value of the period-ratio between two consecutive planets show a significant deviation with respect to the resonant nominal value, up to values well in agreement with the observations.

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arXiv2018

Resonance Capture and Dynamics of 3-Planet Systems

We present a series of dynamical maps for fictitious 3-planets systems in initially circular coplanar orbits. These maps have unveiled a rich resonant structure involving two or three planets, as well as indicating possible migration routes from secular to double resonances or pure 3-planet commensurabilities. These structures are then compared to the present-day orbital architecture of observed resonant chains. In a second part of the paper we describe N-body simulations of type-I migration. Depending on the orbital decay timescale, we show that 3-planet systems may be trapped in different combinations of independent commensurabilities: (i) double resonances, (ii) intersection between a 2-planet and a first-order 3-planet resonance, and (iii) simultaneous libration in two first-order 3-planet resonances. These latter outcomes are found for slow migrations, while double resonances are almost always the final outcome in high-density disks. Finally, we discuss an application to the TRAPPIST-1 system. We find that, for low migration rates and planetary masses of the order of the estimated values, most 3-planet sub-systems are able to reach the observed double resonances after following evolutionary routes defined by pure 3-planet resonances. The final orbital configuration shows resonance offsets comparable with present-day values without the need of tidal dissipation. For the 8/5 resonance proposed to dominate the dynamics of the two inner planets, we find little evidence of its dynamical significance; instead, we propose that this relation between mean motions could be a consequence of the interaction between a pure 3-planet resonance and a 2-planet commensurability between planets c and d.

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arXiv2017

Planetary migration and the origin of the 2:1 and 3:2 (near)-resonant population of close-in exoplanets

We present an analytical and numerical study of the orbital migration and resonance capture of fictitious two-planet systems with masses in the super-Earth range undergoing Type-I migration. We find that, depending on the flare index and proximity to the central star, the average value of the period ratio, $P_2/P_1$, between both planets may show a significant deviation with respect to the nominal value. For planets trapped in the 2:1 commensurability, offsets may reach values on the order of $0.1$ for orbital periods on the order of $1$ day, while systems in the 3:2 mean-motion resonance (MMR) show much smaller offsets for all values of the semimajor axis. These properties are in good agreement with the observed distribution of near-resonant exoplanets, independent of their detection method. We show that 2:1-resonant systems far from the star, such as HD82943 and HR8799, are characterized by very small resonant offsets, while higher values are typical of systems discovered by Kepler with orbital periods approximately a few days. Conversely, planetary systems in the vicinity of the 3:2 MMR show little offset with no significant dependence on the orbital distance. In conclusion, our results indicate that the distribution of Kepler planetary systems around the 2:1 and 3:2 MMR are consistent with resonant configurations obtained as a consequence of a smooth migration in a laminar flared disk, and no external forces are required to induce the observed offset or its dependence with the commensurability or orbital distance from the star.

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arXiv2016

Chaotic Diffusion in the Gliese-876 Planetary System

Chaotic diffusion is supposed to be responsible for orbital instabilities in planetary systems after the dissipation of the protoplanetary disk, and a natural consequence of irregular motion. In this paper we show that resonant multi-planetary systems, despite being highly chaotic, not necessarily exhibit significant diffusion in phase space, and may still survive virtually unchanged over timescales comparable to their age.Using the GJ-876 system as an example, we analyze the chaotic diffusion of the outermost (and less massive) planet. We construct a set of stability maps in the surrounding regions of the Laplace resonance. We numerically integrate ensembles of close initial conditions, compute Poincaré maps and estimate the chaotic diffusion present in this system. Our results show that, the Laplace resonance contains two different regions: an inner domain characterized by low chaoticity and slow diffusion, and an outer one displaying larger values of dynamical indicators. In the outer resonant domain, the stochastic borders of the Laplace resonance seem to prevent the complete destruction of the system. We characterize the diffusion for small ensembles along the parameters of the outermost planet. Finally, we perform a stability analysis of the inherent chaotic, albeit stable Laplace resonance, by linking the behavior of the resonant variables of the configurations to the different sub-structures inside the three-body resonance.

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arXiv2015

The Resonance Overlap and Hill Stability Criteria Revisited

We review the orbital stability of the planar circular restricted three-body problem, in the case of massless particles initially located between both massive bodies. We present new estimates of the resonance overlap criterion and the Hill stability limit, and compare their predictions with detailed dynamical maps constructed with N-body simulations. We show that the boundary between (Hill) stable and unstable orbits is not smooth but characterized by a rich structure generated by the superposition of different mean-motion resonances which does not allow for a simple global expression for stability. We propose that, for a given perturbing mass $m_1$ and initial eccentricity $e$, there are actually two critical values of the semimajor axis. All values $a < a_{\rm Hill}$ are Hill-stable, while all values $a > a_{\rm unstable}$ are unstable in the Hill sense. The first limit is given by the Hill-stability criterion and is a function of the eccentricity. The second limit is virtually insensitive to the initial eccentricity, and closely resembles a new resonance overlap condition (for circular orbits) developed in terms of the intersection between first and second-order mean-motion resonances.

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arXiv2014

Capture Probability in the 3:1 Mean Motion Resonance with Jupiter

We study the capture and crossing probabilities into the 3:1 mean motion resonance with Jupiter for a small asteroid that migrates from the inner to the middle Main Belt under the action of the Yarkovsky effect. We use an algebraic mapping of the averaged planar restricted three-body problem based on the symplectic mapping of Hadjidemetriou (1993), adding the secular variations of the orbit of Jupiter and non-symplectic terms to simulate the migration. We found that, for fast migration rates, the captures occur at discrete windows of initial eccentricities whose specific locations depend on the initial resonant angles, indicating that the capture phenomenon is not probabilistic. For slow migration rates, these windows become narrower and start to accumulate at low eccentricities, generating a region of mutual overlap where the capture probability tends to 100%, in agreement with the theoretical predictions for the adiabatic regime. Our simulations allow to predict the capture probabilities in both the adiabatic and non-adiabatic cases, in good agreement with results of Gomes (1995) and Quillen (2006). We apply our model to the case of the Vesta asteroid family in the same context as Roig et al. (2008), and found results indicating that the high capture probability of Vesta family members into the 3:1 mean motion resonance is basically governed by the eccentricity of Jupiter and its secular variations.

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arXiv2014

Stellar Scattering and the Formation of Hot-Jupiters in Binary Systems

Hot Jupiters (HJs) are usually defined as giant Jovian-size planets with orbital periods $P \le 10$ days. Although they lie close to the star, several have finite eccentricities and significant misalignment angle with respect to the stellar equator. Two mechanisms have been proposed to explain the excited and misaligned sub-population of HJs: Lidov-Kozai migration and planet-planet scattering. Although both are based on completely different dynamical phenomena, they appear to be equally effective in generating hot planets. Nevertheless, there has been no detailed analysis comparing the predictions of both mechanisms. In this paper we present numerical simulations of Lidov-Kozai trapping of single planets in compact binary systems. Both the planet and the binary are initially placed in coplanar orbits, although the inclination of the impactor is assumed random. After the passage of the third star, we follow the orbital and spin evolution of the planet using analytical models based on the octupole expansion of the secular Hamiltonian. The present work aims at the comparison of the two mechanisms, as an explanation for the excited and inclined HJs in binary systems. We compare the results obtained through this paper with results in Beaugé & Nesvorný 2012, where the authors analyze how the planet-planet scattering mechanisms works. Several of the orbital characteristics of the simulated HJs are caused by tidal trapping from quasi-parabolic orbits, independent of the driving mechanism. These include both the 3-day pile-up and the distribution in the eccentricity vs semimajor axis plane. However, the distribution of the inclinations shows significant differences. While Lidov-Kozai trapping favors a more random distribution, planet-planet scattering shows a large portion of bodies nearly aligned with the equator of the central star.

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arXiv2013

A new scenario for the origin of the 3/2 resonant system HD45364

In this paper we revise the model proposed by Rein et al. (2010) for the origin of the HD45364 exoplanetary system, currently known to host two planets close to the 3/2 mean-motion commensurability (MMR). We show that, due to high surface density of the protoplanetary disk needed for Type III migration, this model could only lead to planets in a quasi-resonant regime of motion, and thus not consistent with the resonant configuration obtained by Correia et al. (2009). Although both resonant and quasi-resonant solutions are statistically indistinguishable with respect to radial velocity measurements, their distinct dynamical behaviour is intriguing. We use the semi-analytical model to confirm the quantitative difference between two configurations. In order to form a system evolving inside the 3/2 resonance, we develop a different model. Our scenario includes an interaction between different (but slower) planetary migration types, planet growth, and gap formation in the protoplanetary disk. The choice of the described evolutionary path was due to a detailed analysis of the structure of the phase space in the vicinity of the 3/2 MMR employing dynamical mapping techniques. The outcomes of our simulations are able to reproduce very closely the 3/2 resonant dynamics obtained from the best-fit presented by Correia et al. (2009). In addition, varying the strength of the eccentricity damping, we can also simulate the quasi-resonant configuration similar to that in Rein et al. (2010). We furthermore show that our scenario is robust with respect to the physical parameters involved in the resonance trapping process. However, the confirmation of our scenario will be possible only with additional radial velocities measurements.

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arXiv2013

MAMA: An Algebraic Map for the Secular Dynamics of Planetesimals in Tight Binary Systems

We present an algebraic map (MAMA) for the dynamical and collisional evolution of a planetesimal swarm orbiting the main star of a tight binary system (TBS). The orbital evolution of each planetesimal is dictated by the secular perturbations of the secondary star and gas drag due to interactions with a protoplanetary disk. The gas disk is assumed eccentric with a constant precession rate. Gravitational interactions between the planetesimals are ignored. All bodies are assumed coplanar. A comparison with full N-body simulations shows that the map is of the order of 100 times faster, while preserving all the main characteristics of the full system. In a second part of the work, we apply MAMA to the γ-Cephei, searching for friendly scenarios that may explain the formation of the giant planet detected in this system. For low-mass protoplanetary disks, we find that a low-eccentricity static disk aligned with the binary yields impact velocities between planetesimals below the disruption threshold. All other scenarios appear hostile to planetary formation.

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arXiv2010

Dynamics of two planets in co-orbital motion

We study the stability regions and families of periodic orbits of two planets locked in a co-orbital configuration. We consider different ratios of planetary masses and orbital eccentricities, also we assume that both planets share the same orbital plane. Initially we perform numerical simulations over a grid of osculating initial conditions to map the regions of stable/chaotic motion and identify equilibrium solutions. These results are later analyzed in more detail using a semi-analytical model. Apart from the well known quasi-satellite (QS) orbits and the classical equilibrium Lagrangian points L4 and L5, we also find a new regime of asymmetric periodic solutions. For low eccentricities these are located at $(σ,Δω) = (\pm 60°, \mp 120°)$, where σis the difference in mean longitudes and Δωis the difference in longitudes of pericenter. The position of these Anti-Lagrangian solutions changes with the mass ratio and the orbital eccentricities, and are found for eccentricities as high as ~ 0.7. Finally, we also applied a slow mass variation to one of the planets, and analyzed its effect on an initially asymmetric periodic orbit. We found that the resonant solution is preserved as long as the mass variation is adiabatic, with practically no change in the equilibrium values of the angles.

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arXiv2009

Mapping the $ν_\odot$ Secular Resonance for Retrograde Irregular Satellites

Constructing dynamical maps from the filtered output of numerical integrations, we analyze the structure of the $ν_\odot$ secular resonance for fictitious irregular satellites in retrograde orbits. This commensurability is associated to the secular angle $θ= \varpi - \varpi_\odot$, where $\varpi$ is the longitude of pericenter of the satellite and $\varpi_\odot$ corresponds to the (fixed) planetocentric orbit of the Sun. Our study is performed in the restricted three-body problem, where the satellites are considered as massless particles around a massive planet and perturbed by the Sun. Depending on the initial conditions, the resonance presents a diversity of possible resonant modes, including librations of $θ$ around zero (as found for Sinope and Pasiphae) or 180 degrees, as well as asymmetric librations (e.g. Narvi). Symmetric modes are present in all giant planets, although each regime appears restricted to certain values of the satellite inclination. Asymmetric solutions, on the other hand, seem absent around Neptune due to its almost circular heliocentric orbit. Simulating the effects of a smooth orbital migration on the satellite, we find that the resonance lock is preserved as long as the induced change in semimajor axis is much slower compared to the period of the resonant angle (adiabatic limit). However, the librational mode may vary during the process, switching between symmetric and asymmetric oscillations. Finally, we present a simple scaling transformation that allows to estimate the resonant structure around any giant planet from the results calculated around a single primary mass.

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