Many exo-solar systems discovered in the last decade consist of planets orbiting in resonant configurations and consequently, their evolution should show long-term stability. However, due to the mutual planetary interactions a multi-planet system shows complicated dynamics with mostly chaotic trajectories. We can determine possible stable configurations by computing resonant periodic trajectories of the general planar three body problem, which can be used for modeling a two-planet system. In this work, we review our model for both the planar and the spatial case. We present families of symmetric periodic trajectories in various resonances and study their linear horizontal and vertical stability. We show that around stable periodic orbits there exist regimes in phase space where regular evolution takes place. Unstable periodic orbits are associated with the existence of chaos and planetary destabilization.
The Kepler mission has discovered a plethora of multiple transiting planet candidate exosystems, many of which feature putative pairs of planets near mean motion resonance commensurabilities. Identifying potentially resonant systems could help guide future observations and enhance our understanding of planetary formation scenarios. We develop and apply an algebraic method to determine which Kepler 2-planet systems cannot be in a 1st-4th order resonance, given the current, publicly available data. This method identifies when any potentially resonant angle of a system must circulate. We identify…
Close-in planetary systems detected by the Kepler mission present an excess of periods ratio that are just slightly larger than some low order resonant values. This feature occurs naturally when resonant couples undergo dissipation that damps the eccentricities. However, the resonant angles appear to librate at the end of the migration process, which is often believed to be an evidence that the systems remain in resonance. Here we provide an analytical model for the dissipation in resonant planetary systems valid for low eccentricities. We confirm that dissipation accounts for an excess of pai…
Early dynamical evolution of close-in planetary systems is shaped by an intricate combination of planetary gravitational interactions, orbital migration, and dissipative effects. While the process of convergent orbital migration is expected to routinely yield resonant planetary systems, previous analyses have shown that the semi-major axes of initially resonant pairs of planets will gradually diverge under the influence of long-term energy damping, producing an overabundance of planetary period ratios in slight excess of exact commensurability. While this feature is clearly evident in the orbi…
A diversity of resonance configurations may be formed under different migration of two giant planets. And the researchers show that the HD 128311 and HD 73526 planetary systems are involved in a 2:1 mean motion resonance but not in apsidal corotation, because one of the resonance argument circulates over the dynamical evolution. In this paper, we investigate potential mechanisms to form the 2:1 librating-circulating resonance configuration. In the late stage of planetary formation, scattering or colliding among planetesimals and planetary embryos can frequently occur. Hence, in our model, we c…
We consider the dynamics of a small trojan companion of a hypothetical giant exoplanet under the secular perturbations of additional planets. By a suitable choice of action-angle variables, the problem is amenable to the study of the slow modulation, induced by secular perturbations, to the dynamics of an otherwise called `basic' Hamiltonian model of two degrees of freedom (planar case). We present this Hamiltonian decomposition, which implies that the slow chaotic diffusion at resonances is best described by the paradigm of modulational diffusion.
Resonant chains are groups of planets for which each pair is in resonance, with an orbital period ratio locked at a rational value (2/1, 3/2, etc.). Such chains naturally form as a result of convergent migration of the planets in the proto-planetary disk. In this article, I present an analytical model of resonant chains of any number of planets. Using this model, I show that a system captured in a resonant chain can librate around several possible equilibrium configurations. The probability of capture around each equilibrium depends on how the chain formed, and especially on the order in which…
We consider the general spatial three body problem and study the dynamics of planetary systems consisting of a star and two planets which evolve into 2/1 mean motion resonance and into inclined orbits. Our study is focused on the periodic orbits of the system given in a suitable rotating frame. The stability of periodic orbits characterize the evolution of any planetary system with initial conditions in their vicinity. Stable periodic orbits are associated with long term regular evolution, while unstable periodic orbits are surrounded by regions of chaotic motion. We compute many families of s…
We study the dynamics of a two-planet system, which evolves being in a $1/1$ mean motion resonance (co-orbital motion) with non-zero mutual inclination. In particular, we examine the existence of bifurcations of periodic orbits from the planar to the spatial case. We find that such bifurcations exist only for planetary mass ratios $ρ=\frac{m_2}{m_1}<0.0205$. For $ρ$ in the interval $0<ρ<0.0205$, we compute the generated families of spatial periodic orbits and their linear stability. These spatial families form bridges, which start and end at the same planar family. Along them the mutual planet…
We present a new mechanism of generating large planetary eccentricities. This mechanism applies to planets within the inner cavities of their companion protoplanetary disks. A massive disk with an inner truncation may become eccentric due to non-adiabatic effects associated with gas cooling, and can retain its eccentricity in long-lived coherently-precessing eccentric modes; as the disk disperses, the inner planet will encounter a secular resonance with the eccentric disk when the planet and the disk have the same apsidal precession rates; the eccentricity of the planet is then excited to a la…
We present six years of new radial-velocity data from the Anglo-Australian and Magellan Telescopes on the HD 73526 2:1 resonant planetary system. We investigate both Keplerian and dynamical (interacting) fits to these data, yielding four possible configurations for the system. The new data now show that both resonance angles are librating, with amplitudes of 40 degrees and 60 degrees, respectively. We then perform long-term dynamical stability tests to differentiate these solutions, which only differ significantly in the masses of the planets. We show that while there is no clearly preferred s…
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…
At least 30\% of main sequence stars host planets with sizes of between 1 and 4 Earth radii and orbital periods of less than 100 days. We use N-body simulations including a model for gas-assisted pebble accretion and disk--planet tidal interaction to study the formation of super-Earth systems. We show that the integrated pebble mass reservoir creates a bifurcation between hot super-Earths or hot-Neptunes ($\lesssim15M_{\oplus}$) and super-massive planetary cores potentially able to become gas giant planets ($\gtrsim15M_{\oplus}$). Simulations with moderate pebble fluxes grow multiple super-Ear…
The planetary dynamics of $4/3$, $3/2$, $5/2$, $3/1$ and $4/1$ mean motion resonances is studied by using the model of the general three body problem in a rotating frame and by determining families of periodic orbits for each resonance. Both planar and spatial cases are examined. In the spatial problem, families of periodic orbits are obtained after analytical continuation of vertical critical orbits. The linear stability of orbits is also examined. Concerning initial conditions nearby stable periodic orbits, we obtain long-term planetary stability, while unstable orbits are associated with ch…
The most accurate method for modelling planetary migration and hence the formation of resonant systems is using hydrodynamical simulations. Usually, the force (torque) acting on a planet is calculated using the forces from the gas disc and the star, while the gas accelerations are computed using the pressure gradient, the star, and the planet's gravity, ignoring its own gravity. For the non-migrating the neglect of the disc gravity results in a consistent torque calculation while for the migrating case it is inconsistent. We aim to study how much this inconsistent torque calculation can affect…
I consider the dynamics of mean motion resonances between pairs of co-planar planets and derive a new integrable Hamiltonian model for planets' resonant motion. The new model generalizes previously-derived integrable Hamiltonians for first-order resonances to treat higher-order resonances by exploiting a surprising near-symmetry of the full, non-integrable Hamiltonians of higher-order resonances. Whereas past works have frequently relied on truncated disturbing function expansions to derive integrable approximations to resonant motion, I show that no such expansion is necessary, thus enabling …
The present work studies the secular resonance associated with the critical argument $σ= \varpi$ ($\varpi$ is the longitude of pericentre) for inner test particles moving in low-eccentricity region with inclination $i$ smaller than $39^{\circ}$. To formulate the dynamical model, the double-averaged Hamiltonian is formulated up to an arbitrary order in the semimajor axis ratio, and then those high-order periodic terms are removed from the double-averaged Hamiltonian by means of Hori--Deprit transformation technique. The resulting Hamiltonian determines a resonant model with a single degree of f…
HR 8799 is a star accompanied by four massive planets on wide orbits. The observed planetary configuration has been shown to be unstable on a timescale much shorter than the estimated age of the system (~ 30 Myr) unless the planets are locked into mean motion resonances. This condition is characterised by small-amplitude libration of one or more resonant angles that stabilise the system by preventing close encounters. We simulate planetary systems similar to the HR 8799 planetary system, exploring the parameter space in separation between the orbits, planetary masses and distance from the Sun …
In this paper, we use a semi-analytical approach to analyze the global structure of the phase space of the planar planetary 3/1 mean-motion resonance, in cases where the outer planet is more massive than its inner companion. We show that the resonant dynamics can be described using only two fundamental parameters, the total angular momentum and the scaling parameter. The topology of the Hamiltonian function describing the resonant behaviour is studied on the representative planes that allows us to investigate a large domain of the phase space of the three-body problem without time-expensive nu…
Resonant Hamiltonian Dynamics in the CR3BP: Bistability and Stochastic Resonance in Binary Planetary Systems
Sapienza, Universitá di Roma（罗马第一大学）；INFN, Sezione di Roma, Sapienza, Universitá di Roma（罗马大学罗马分部意大利国家核物理研究所）；INAF, Observatory of Roma at Monteporzio Catone（蒙波尔齐奥卡托内罗马天文台）；Centro Ricerche Enrico Fermi（恩里科·费米研究中心）
Jupiter and Saturn play host to an impressive array of satellites, making it reasonable to suspect that similar systems of moons might exist around giant extrasolar planets. Furthermore, a significant population of such planets is known to reside at distances of several Astronomical Units (AU), leading to speculation that some moons thereof might support liquid water on their surfaces. However, giant planets are thought to undergo inward migration within their natal protoplanetary disks, suggesting that gas giants currently occupying their host star's habitable zone formed further out. Here we…
Aim. The eccentric von Zeipel-Lidov-Kozai (ZLK) effect is widely used to explain dynamical phenomena in varieties of astrophysical systems. The purpose of this work is to make clear the dynamical essence of the eccentric ZLK effect by constructing an inherent connection between such an effect and dynamics of secular resonance in restricted hierarchical planetary systems. Methods. Dynamical structures of apsidal resonance are analytically studied by means of perturbative treatments. The resonant model is formulated by averaging the Hamiltonian (up to octupole order) over rotating ZLK cycles, pr…
We present a numerical study on the stability of all fourth- and fifth-order retrograde mean motion resonances (1/3, 3/1, 1/4, 4/1, 2/3, and 3/2) in the 3-body problem composed of a solar mass star, a Jupiter mass planet, and an additional body with zero mass (elliptic restricted problem) or masses corresponding to either Neptune, Saturn, or Jupiter (planetary problem). The fixed point families exist in all cases and are identified through libration of all resonant angles simultaneously. In addition, configurations with libration of a single resonant angle were also observed. Our results for t…
Inspired by the close-proximity pair of planets in the Kepler-36 system, we consider two effects that may have important ramifications for the development of life in similar systems where a pair of planets may reside entirely in the habitable zone of the hosting star. Specifically, we run numerical simulations to determine whether strong, resonant (or non-resonant) planet-planet interactions can cause large variations in planet obliquity---thereby inducing large variations in climate. We also determine whether or not resonant interactions affect the rate of lithopanspermia between the planet p…
We present families of symmetric and asymmetric periodic orbits at the 1/1 resonance, for a planetary system consisting of a star and two small bodies, in comparison to the star, moving in the same plane under their mutual gravitational attraction. The stable 1/1 resonant periodic orbits belong to a family which has a planetary branch, with the two planets moving in nearly Keplerian orbits with non zero eccentricities and a satellite branch, where the gravitational interaction between the two planets dominates the attraction from the star and the two planets form a close binary which revolves …
Many extrasolar planetary systems containing multiple super-Earths have been discovered. N-body simulations taking into account standard type-I planetary migration suggest that protoplanets are captured into mean-motion resonant orbits near the inner disk edge at which the migration is halted. Previous N-body simulations suggested that orbital stability of the resonant systems depends on number of the captured planets. In the unstable case, through close scattering and merging between planets, non-resonant multiple systems are finally formed. In this paper, we investigate the critical number o…
Dust grains migrating under Poynting-Robertson drag may be trapped in mean-motion resonances with planets. Such resonantly trapped grains are observed in the solar system. In extrasolar systems, the exozodiacal light produced by dust grains is expected to be a major obstacle to future missions attempting to directly image terrestrial planets. The patterns made by resonantly trapped dust, however, can be used to infer the presence of planets, and the properties of those planets, if the capture and evolution of the grains can be modelled. This has been done with N-body methods, but such methods …
Recent exoplanet observations reported a large number of multiple-planet systems, in which some of the planets are in a chain of resonances. The fraction of resonant systems to non-resonant systems provides clues about their formation history. We investigated the orbital stability of planets in resonant chains by considering the long-term evolution of planetary mass and stellar mass and using orbital calculations. We found that while resonant chains were stable, they can be destabilized by a change of $\sim$10% in planetary mass. Such a mass evolution can occur by atmospheric escape due to pho…
Convergent disk migration has long been suspected to be responsible for forming planetary systems with a chain of mean-motion resonances (MMR). Dynamical evolution over time could disrupt the delicate resonant configuration. We present TOI-1136, a 700-Myr-old G star hosting at least 6 transiting planets between $\sim$2 and 5 $R_\oplus$. The orbital period ratios deviate from exact commensurability by only $10^{-4}$, smaller than the $\sim$\,$10^{-2}$ deviations seen in typical Kepler near-resonant systems. A transit-timing analysis measured the masses of the planets (3-8$M_\oplus$) and demonst…
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 order to study the origin of the architectures of low mass planetary systems, we perform numerical surveys of the evolution of pairs of coplanar planets in the mass range $(1-4)\ \rmn{M}_{\oplus}.$ These evolve for up to $2\times10^7 \rmn{yr}$ under a range of orbital migration torques and circularization rates assumed to arise through interaction with a protoplanetary disc. Near the inner disc boundary, significant variations of viscosity, interaction with density waves or with the stellar magnetic field could occur and halt migration, but allow ircularization to continue. This was modelle…
Migration of planetary systems caused by the action of dissipative forces may lead the planets to be trapped in a resonance. In this work we study the conditions and the dynamics of such resonant trapping. Particularly, we are interested in finding out whether resonant capture ends up in a long-term stable planetary configuration. For two planet systems we associate the evolution of migration with the existence of families of periodic orbits in the phase space of the three-body problem. The family of circular periodic orbits exhibits a gap at the 2:1 resonance and an instability and bifurcatio…
Two Jovian-sized planets are orbiting the star HD155358 near exact mean motion resonance (MMR) commensurability. In this work we re-analyze the radial velocity (RV) data previously collected by Robertson et al. (2012). Using a Bayesian framework we construct two models - one that includes and one that excludes gravitational planet-planet interactions (PPI). We find that the orbital parameters from our PPI and noPPI models differ by up to $2σ$, with our noPPI model being statistically consistent with previous results. In addition, our new PPI model strongly favours the planets being in MMR whil…
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 r…
We study proximity of the Kepler-25 planetary system to a periodic configuration, which is known to be the final state of a system that undergoes smooth migration resulting from the planet-disc interaction. We show that the system is close to the periodic configuration of 2:1 mean motion resonance (MMR) what indicates that its past migration was neither disturbed significantly by turbulence in the disc nor the orbits were perturbed by planetesimals that left after the disc dispersal. We show that, because of the TTV model degeneracy, a periodic configuration is difficult to be found when the s…
The continuation of resonant periodic orbits from the restricted to the general three body problem is studied in a systematic way. Starting from the Keplerian unperturbed system we obtain the resonant families of the circular restricted problem. Then we find all the families of the resonant elliptic restricted three body problem which bifurcate from the circular model. All these families are continued to the general three body problem, and in this way we can obtain a global picture of all the families of periodic orbits of a two-planet resonant system. We consider planar motion only. We show t…
Among multi-planet planetary systems there are a large fraction of resonant systems. Studying the dynamics and formation of these systems can provide valuable informations on processes taking place in protoplanetary disks where the planets are thought have been formed. The recently discovered resonant system HD 60532 is the only confirmed case, in which the central star hosts a pair of giant planets in 3:1 mean motion resonance. We intend to provide a physical scenario for the formation of HD 60532, which is consistent with the orbital solutions derived from the radial velocity measurements. O…
In this study, a new expansion of planetary disturbing function is developed for describing the resonant dynamics of minor bodies with arbitrary inclinations and semimajor axis ratios. In practice, the disturbing function is expanded around circular orbits in the first step and then, in the second step, the resulting mutual interaction between circular orbits is expanded around a reference point. As usual, the resulting expansion is presented in the Fourier series form, where the force amplitudes are dependent on the semimajor axis, eccentricity and inclination and the harmonic arguments are l…
The multi-planetary system HD128311 hosts at least two planets. Its dynamical formation history has been studied extensively in the literature. We reanalyse the latest radial velocity data for this system with the affine-invariant Markov chain Monte Carlo sampler EMCEE. Using the high order integrator IAS15, we perform a fully dynamical fit, allowing the planets to interact during the sampling process. A stability analysis using the MEGNO indicator reveals that the system is located in a stable island of the parameter space. In contrast to a previous study, we find that the system is locked in…
Planetary-scale waves at the Venusian cloud-top cause periodic variations in both winds and ultraviolet (UV) brightness. While the wave candidates are the 4-day Kelvin wave and 5-day Rossby wave with zonal wavenumber 1, their temporal evolutions are poorly understood. Here we conducted a time series analysis of the 365-nm brightness and cloud-tracking wind variations, obtained by the UV Imager onboard the Japanese Venus Climate Orbiter Akatsuki from June to October 2017, revealing a dramatic evolution of planetary-scale waves and corresponding changes in planetary-scale UV features. We identif…
Multi-planetary systems detected by the Kepler mission present an excess of planets close to first-order mean-motion resonances (2:1 and 3:2) but with a period ratio slightly higher than the resonant value. Several mechanisms have been proposed to explain this observation. Here we provide some clues that these near-resonant systems were initially in resonance and reached their current configuration through tidal dissipation. The argument that has been opposed to this scenario is that it only applies to the close-in systems and not to the farthest ones for which the tidal effect is too weak. Us…
An Adolescent and Near-Resonant Planetary System Near the End of Photoevaporation
Nanjing University（南京大学）；University of Hawai‘i（夏威夷大学）；Florida Institute of Technology（佛罗里达理工学院）；Tsinghua University（清华大学）；University of California Los Angeles（加州大学洛杉矶分校）；California Institute of Technology（加州理工学院）；University of California, San Diego（加州大学圣地亚哥分校）
Despite the discovery of thousands of exoplanets in recent years, the number of known exoplanets in star clusters remains tiny. This may be a consequence of close stellar encounters perturbing the dynamical evolution of planetary systems in these clusters. Here, we present the results from direct $N$-body simulations of multiplanetary systems embedded in star clusters containing $N = 8k, 16k, 32k$, and $64k$ stars. The planetary systems, which consist of the four Solar system giant planets Jupiter, Saturn, Uranus, and Neptune, are initialized in different orbital configurations, to study the e…
We present a numerical study on the stability of the 1/2, 2/1 and 1/1 retrograde mean motion resonances in the 3-body problem composed of a solar mass star, a Jupiter mass planet and an additional body with zero mass (elliptic restricted 3-body problem) or masses corresponding to either Neptune, Saturn or Jupiter (planetary 3-body problem). For each system we obtain stability maps using the n-body numerical integrator REBOUND and computing the chaos indicator mean exponential growth factor of nearby orbits (MEGNO). We show that families of periodic orbits exist in all configurations and they c…
Applying the method of analytical continuation of periodic orbits, we study quasi-satellite motion in the framework of the three-body problem. In the simplest, yet not trivial model, namely the planar circular restricted problem, it is known that quasi-satellite motion is associated with a family of periodic solutions, called family $f$, which consists of 1:1 resonant retrograde orbits. In our study, we determine the critical orbits of family $f$ that are continued both in the elliptic and in the spatial model and compute the corresponding families that are generated and consist the backbone o…
It has been suggested that HD 200964 is the first exoplanetary system with two Jovian planets evolving in the 4/3 mean- motion resonance. Previous scenarios to simulate the formation of two giant planets in the stable 4/3 resonance configuration have failed. Moreover, the orbital parameters available in the literature point out an unstable configuration of the planetary pair. The purpose of this paper is i) to determine the orbits of the planets from the RV measurements and update the value of the stellar mass (1.57 M), ii) to analyse the stability of the planetary evolution in the vicinity an…
Gaussian process regression is a widespread tool used to mitigate stellar correlated noise in radial velocity time series. It is particularly useful to search for and determine the properties of signals induced by small-size, low-mass planets ($R_p<4R_{\rm \oplus}$, $m_p<10M_{\rm \oplus}$). By using extensive simulations based on a quasi-periodic representation of the stellar activity component, we investigate the ability in retrieving the planetary parameters in 16 different realistic scenarios. We analyse systems composed by one planet and host stars having different levels of activity, focu…
Planets around binary stars and those in multiplanet systems may experience resonant eccentricity excitation and disruption due to perturbations from a distant stellar companion. This "evection resonance" occurs when the apsidal precession frequency of the planet, driven by the quadrupole associated with the inner binary or the other planets, matches the orbital frequency of the external companion. We develop an analytic theory to study the effects of evection resonance on circumbinary planets and multiplanet systems. We derive the general conditions for effective eccentricity excitation or re…
How might an advanced alien civilization manipulate the orbits within a planetary system to create a durable signpost that communicates its existence? While it is still debated whether such a purposeful advertisement would be prudent and wise, we propose that mean-motion resonances between neighboring planets -- with orbital periods that form integer ratios -- could in principle be used to encode simple sequences that one would not expect to form in nature. In this Letter we build four multi-resonant planetary systems and test their long-term orbital stability. The four systems each contain 6 …
Orbital mean motion resonances in planetary systems originate from dissipative processes in disk-planet interactions that lead to orbital migration. In multi-planet systems that host giant planets, the perturbation of the protoplanetary disk strongly affects the migration of companion planets. By studying the well-characterized resonant planetary system around GJ 876 we aim to explore which effects shape disk-driven migration in such a multi-planet system to form resonant chains. We modelled the orbital migration of three planets embedded in a protoplanetary disk using two-dimensional locally …
We consider a two-planet system, which migrates under the influence of dissipative forces that mimic the effects of gas-driven (Type II) migration. It has been shown that, in the planar case, migration leads to resonant capture after an evolution that forces the system to follow families of periodic orbits. Starting with planets that differ slightly from a coplanar configuration, capture can, also, occur and, additionally, excitation of planetary inclinations has been observed in some cases. We show that excitation of inclinations occurs, when the planar families of periodic orbits, which are …
Resonant planetary systems contain at least one planet pair with orbital periods librating at a near-integer ratio (2/1, 3/2, 4/3, etc.) and are a natural outcome of standard planetary formation theories. Systems with multiple adjacent resonant pairs are known as resonant chains and can exhibit three-body resonances -- characterized by a critical three-body angle. Here we study three-body angles as a diagnostic of resonant chains through tidally-damped N-body integrations. For each combination of the 2:1, 3:2, 4:3, and 5:4 mean motion resonances (the most common resonances in the known resonan…
