Stability Of Planetary Systems Research Articles

Encounters in star clusters and survival probabilities for planets

The stability of young planetary systems is strongly influenced by multiple factors, both internal and external. In this paper, we investigate the link between the environment in which young stars form and the possibility of having stable planetary systems around them. We analyze the robustness of such systems after an encounter with another star within the same stellar cluster. We employ a model for the star cluster to extract the encounter properties, such as the mass of the perturber star and its velocity. We perform numerical simulations on systems with a single planet perturbed by an external star, in order to calculate the emission probabilities of the planet. We also calculate analytically the stellar encounter rates in the cluster. We find that such probabilities are strongly dependent on the thermal velocity of the cluster. We also notice that these probabilities are generally quite small, below 3% for the systems tested by us.

Stability of planetary systems within the S-star cluster: the Solar system analogues

ABSTRACT A dynamically relaxed dense cluster comprised of about 40 stars (the so-called S-stars) inhabits the central region of our Galaxy. Their stars revolve around the Sgr A* massive object. To understand the dynamical evolution of planetary systems in a particular environment like that around Sgr A*, we carry out direct N-body simulations of planetary systems embedded in the S-star cluster. In this work, we investigate the short-term stability of the planets orbiting around S-stars after their close interactions with the central massive black hole of our Galaxy. We find that planetary systems go through encounters with the supermassive black hole (SMBH) and the nearby stars. We determine the frequency and the strength of planetary systems’ encounters with the nearby stars as these encounters remarkably increase for systems assigned to S-stars closer to the SMBH. The SMBH severely destabilizes the planetary systems, though we noted that the small oscillations in the mutual eccentricity and inclination of the planetary system could be caused by the planet–planet coupling and the near-resonance effect between the two planets. We obtain estimates of the fraction of survivor planets ($\sim 51{{\ \rm per\ cent}}$), and find that planets stripped from their hosting star are generally captured on close orbits around Sgr A*. We notify while gas giants are tidally disrupted, terrestrial planets do not. We estimate that Sgr A* flares can be due to the tidal disruption events of starless giant planets.

Long-term stability of planets in and around binary stars

ABSTRACT Planets are observed to orbit the component star(s) of stellar binary systems on so-called circumprimary or circumsecondary orbits, as well as around the entire binary system on so-called circumbinary orbits. Depending on the orbital parameters of the binary system, a planet will be dynamically stable if it orbits within some critical separation of the semimajor axis in the circumprimary case, or beyond some critical separation for the circumbinary case. We present N-body simulations of star-forming regions that contain populations of primordial binaries to determine the fraction of binary systems that can host stable planets at various semimajor axes, and how this fraction of stable systems evolves over time. Dynamical encounters in star-forming regions can alter the orbits of some binary systems, which can induce long-term dynamical instabilities in the planetary system and can even change the size of the habitable zone(s) of the component stars. However, the overall fraction of binaries that can host stable planetary systems is not greatly affected by either the assumed binary population or the density of the star-forming region. Instead, the critical factor in determining how many stable planetary systems exist in the Galaxy is the stellar binary fraction – the more stars that are born as singles in stellar nurseries, the higher the fraction of stable planetary systems.

An integrable model for first-order three-planet mean motion resonances

Recent works on three-planet mean motion resonances (MMRs) have highlighted their importance for understanding the details of the dynamics of planet formation and evolution. While the dynamics of two-planet MMRs are well understood and approximately described by a one-degree-of-freedom Hamiltonian, little is known of the exact dynamics of three-body resonances besides the cases of zeroth-order MMRs or when one of the bodies is a test particle. In this work, I propose the first general integrable model for first-order three-planet mean motion resonances. I show that one can generalize the strategy proposed in the two-planet case to obtain a one-degree-of-freedom Hamiltonian. The dynamics of these resonances are governed by the second fundamental model of resonance. The model is valid for any mass ratio between the planets and for every first-order resonance. I show the agreement of the analytical model with numerical simulations. As examples of application, I show how this model could improve our understanding of the capture into MMRs as well as their role in the stability of planetary systems.

Determining the number of integrals of motion by an adapted correlation dimension method.

An adapted correlation dimension algorithm is used to numerically determine the number of integrals of motion in a variety of conservative classical systems. The method is demonstrated on three sample systems that display various degrees of integrable and chaotic behavior: the Hénon-Heiles Hamiltonian, an asymmetric top molecule in an electric field, and the planetary system HD128311. Two additional applications of the method emerge: using the adapted correlation dimension algorithm (a) to study partial barriers and turnstiles in phase space and (b) to predict the long-time stability of planetary systems using short-time data.

Stability of planetary systems: A numerical didactic approach

We discuss the numerical methods needed to study the dynamics of interacting planetary systems. We argue that the most appropriate method for studying the many-body dynamics is Runge-Kutta with an adaptive step size. We demonstrate that planetary systems are highly susceptible to catastrophic events in which collisions between the planets are almost unavoidable. We then discuss a recently proposed mechanism that explains how planetary systems may have spontaneously evolved into a self-organized periodic state in which catastrophic events are avoided.

Comparison between Laplace-Lagrange secular theory and numerical simulation: the case of $$\upsilon $$ υ Andromedae planetary system

The large increase in exoplanet discoveries in the last two decades showed a variety of systems whose stability is not clear. In this work we chose the $\upsilon$ Andromedae system as the basis of our studies in dynamical stability. This system has a range of possible masses, as a result of detection by radial velocity method, so we adopted a range of masses for the planets $c$ and $d$ and applied the secular theory. We also performed a numerical integration of the 3-body problem for the system over a time span of 30 thousand years. The results exposed similarities between the secular perturbation theory and the numerical integration, as well as the limits where the secular theory did not present good results. The analysis of the results provided hints for the maximum values of masses and eccentricities for stable planetary systems similar to $\upsilon$ Andromedae.

Stability and self-organization of planetary systems.

We show that stability of planetary systems is intimately connected with their internal order. An arbitrary initial distribution of planets is susceptible to catastrophic events in which planets either collide or are ejected from the planetary system. These instabilities are a fundamental consequence of chaotic dynamics and of Arnold diffusion characteristic of many body gravitational interactions. To ensure stability over astronomical time scale of a realistic planetary system-in which planets have masses comparable to those of planets in the solar system-the motion must be quasiperiodic. A dynamical mechanism is proposed which naturally evolves a planetary system to a quasiperiodic state from an arbitrary initial condition. A planetary self-organization predicted by the theory is similar to the one found in our solar system.

The architecture and formation of the Kepler-30 planetary system

We study the orbital architecture, physical characteristics of planets, formation and long-term evolution of the Kepler-30 planetary system, detected and announced in 2012 by the KEPLER team. We show that the Kepler-30 system belongs to a particular class of very compact and quasi-resonant, yet long-term stable planetary systems. We re-analyse the light curves of the host star spanning Q1-Q17 quarters of the KEPLER mission. A huge variability of the Transit Timing Variations (TTV) exceeding 2 days is induced by a massive Jovian planet located between two Neptune-like companions. The innermost pair is near to the 2:1 mean motion resonance (MMR), and the outermost pair is close to higher order MMRs, such as 17:7 and 7:3. Our re-analysis of photometric data allows us to constrain, better than before, the orbital elements, planets' radii and masses, which are $9.2 \pm 0.1$, $536 \pm 5$, and $23.7 \pm 1.3$ Earth masses for Kepler-30b, Kepler-30c and Kepler-30d, respectively. The masses of the inner planets are determined within $\sim 1\%$ uncertainty. We infer the internal structures of the Kepler-30 planets and their bulk densities in a wide range from $ (0.19 \pm 0.01)$ g$\cdot$cm$^{-3}$ for Kepler-30d, $(0.96\pm0.15)$ g$\cdot$cm$^{-3}$ for Kepler-30b, to $(1.71 \pm 0.13)$ g$\cdot$cm$^{-3}$ for the Jovian planet Kepler-30c. We attempt to explain the origin of this unique planetary system and a deviation of the orbits from exact MMRs through the planetary migration scenario. We anticipate that the Jupiter-like planet plays an important role in determining the present dynamical state of this system.

Secular dynamics of multiplanetary circumbinary systems: stationary solutions and binary-planet secular resonance

We present an analytical formalism to study the secular dynamics of a system consisting of $$N-2$$ planets orbiting a binary star in outer orbits. We introduce a canonical coordinate system and expand the disturbing function in terms of canonical elliptic elements, combining both Legendre polynomials and Laplace coefficients, to obtain a general formalism for the secular description of this type of configuration. With a quadratic approximation of the development, we present a simplified analytical solution for the planetary orbits for both the single planet and the two-planet cases. From the two-planet model, we show that the inner planet accelerates the precession rate of the binary pericenter, which, in turn, may enter in resonance with the secular frequency of the outer planet, characterizing a secular resonance. We calculate an analytical expression for the approximate location of this resonance and apply it to known circumbinary systems, where we show that it can occur at relatively close orbits, for example at 2.4 au for the Kepler-38 system. With a more refined model, we analyse the dynamics of this secular resonance and we show that a bifurcation of the corresponding fixed points can affect the long- term evolution and stability of planetary systems. By comparing our results with complete integrations of the exact equations of motion, we verified the accuracy of our analytical model.

Galactic perturbations on the population of wide binary stars with exoplanets

Aims. The aim of this work is to study the dynamical effects of the Galaxy on binary star systems with physical and orbital charac- teristics similar to those of the population of known wide binary stars with exoplanets. As secondary goal we analyse the possible consequences on the stability of a hypothetical planetary system orbiting one of the stellar components. Methods. We numerically reproduced the temporal evolution of a sample of 3 * 10 5 binary star systems disturbed by the Galactic potential and passing stars in an environment similar to the solar neighbourhood. Results. Our results show that the dynamical evolution of the population of wide binary stars with exoplanets in the solar neigh- bourhood is modelled by the process of disruption of binary star systems induced by the Galaxy. We found that this process depends mainly on the separation between both stars, whereas it is almost independent of the initial orbital configuration. Moreover, our calcu- lations are in agreement with the results of previous works regarding the indirect influence of the Galaxy on the stability of planetary systems in wide binary stars. However, the effects on the planetary region show a dependence on the initial configuration of binary stars. Finally, we obtain an indirect test of the impulse approximation model for dynamical studies of binary star systems.

AMD-stability in the presence of first-order mean motion resonances

The AMD-stability criterion allows to discriminate between a-priori stable planetary systems and systems for which the stability is not granted and needs further investigations. AMD-stability is based on the conservation of the Angular Momentum Deficit (AMD) in the averaged system at all orders of averaging. While the AMD criterion is rigorous, the conservation of the AMD is only granted in absence of mean-motion resonances (MMR). Here we extend the AMD-stability criterion to take into account mean-motion resonances, and more specifically the overlap of first order MMR. If the MMR islands overlap, the system will experience generalized chaos leading to instability. The Hamiltonian of two massive planets on coplanar quasi-circular orbits can be reduced to an integrable one degree of freedom problem for period ratios close to a first order MMR. We use the reduced Hamiltonian to derive a new overlap criterion for first order MMR. This stability criterion unifies the previous criteria proposed in the literature and admits the criteria obtained for initially circular and eccentric orbits as limit cases. We then improve the definition of AMD-stability to take into account the short term chaos generated by MMR overlap. We analyze the outcome of this improved definition of AMD-stability on selected multi-planet systems from the Extrasolar Planets Encyclopeadia.

On the frequency of planetary systems around G dwarfs

We determine the fraction of G-dwarf stars that could host stable planetary systems based on the observed properties of binaries in the Galactic field, and in various postulated primordial binary populations, which assume that the primordial binary fraction is higher than that in the field. We first consider the frequency of Solar System analogues - planetary systems that form either around a single G-dwarf star, or a binary containing a G-dwarf where the binary separation exceeds 100-300au. If the primordial binary fraction and period distribution is similar to that in the field, then up to 63 per cent of G-dwarf systems could potentially host a Solar System analogue. However, if the primordial binary fraction is higher, the fraction of G-dwarf systems that could host a planetary system like our own is lowered to 38 per cent. We extend our analysis to consider the fraction of G-dwarf systems (both single and binary) that can host either circumprimary planets (orbiting the primary star of the binary) or circumbinary planets (orbiting both stars in the binary) for fiducial planetary separations between 1 - 100au. Depending on the assumed binary population, in the circumprimary case between 65 and 95 per cent of systems can host a planet at 1au, decreasing to between 20 and 65 per cent of systems that can host a planet at 100au. In the circumbinary case, between 5 and 59 per cent of systems can host a planet at 1au, increasing to between 34 and 75 per cent of systems that can host a planet at 100au. Our results suggest that the assumed binary fraction is the key parameter in determining the fraction of potentially stable planetary systems in G-dwarf systems and that using the present-day value may lead to significant overestimates if the binary fraction was initially higher.

Synthetic modelling of the light-travel time effect of circumbinary planets

AbstractIn recent years several multi-body, circumbinary planets have been proposed to orbit short-period eclipsing binaries. In light of the recent discoveries based on the Kepler data, the existence of such systems seems plausible. However, performing a detailed dynamical analysis reveals that the majority of the proposed planetary systems follow highly unstable orbits. In order to solve the origin of this problem, we have started to model synthetic light-travel time signals of stable planetary systems. In particular, we aim to study the response of the model in various circumstances (e.g red/white noise level, various sampling frequencies, in-homogeneous data sets, baseline dependency.) This work will significantly increase the confidence with which model work is carried out for future systems and help towards an understaning when models break down (e.g resulting in unstable systems).

Scattering of small bodies by planets: a potential origin for exozodiacal dust?

High levels of exozodiacal dust are observed around a growing number of main sequence stars. The origin of such dust is not clear, given that it has a short lifetime against both collisions and radiative forces. Even a collisional cascade with km-sized parent bodies, as suggested to explain outer debris discs, cannot survive sufficiently long. In this work we investigate whether the observed exozodiacal dust could originate from an outer planetesimal belt. We investigate the scattering processes in stable planetary systems in order to determine whether sufficient material could be scattered inwards in order to retain the exozodiacal dust at its currently observed levels. We use N-body simulations to investigate the efficiency of this scattering and its dependence on the architecture of the planetary system. The results of these simulations can be used to assess the ability of hypothetical chains of planets to produce exozodi in observed systems. We find that for older (>100Myr) stars with exozodiacal dust, a massive, large radii (>20AU) outer belt and a chain of tightly packed, low-mass planets would be required in order to retain the dust at its currently observed levels. This brings into question how many, if any, real systems possess such a contrived architecture and are therefore capable of scattering at sufficiently high rates to retain exozodi dust on long timescales.

From cosmos to intelligent life: the four ages of astrobiology

AbstractThe history of life on Earth and in other potential life-bearing planetary platforms is deeply linked to the history of the Universe. Since life, as we know, relies on chemical elements forged in dying heavy stars, the Universe needs to be old enough for stars to form and evolve. The current cosmological theory indicates that the Universe is 13.7 ± 0.13 billion years old and that the first stars formed hundreds of millions of years after the Big Bang. At least some stars formed with stable planetary systems wherein a set of biochemical reactions leading to life could have taken place. In this paper, I argue that we can divide cosmological history into four ages, from the Big Bang to intelligent life. The physical age describes the origin of the Universe, of matter, of cosmic nucleosynthesis, as well as the formation of the first stars and Galaxies. The chemical age began when heavy stars provided the raw ingredients for life through stellar nucleosynthesis and describes how heavier chemical elements collected in nascent planets and Moons gave rise to prebiotic biomolecules. The biological age describes the origin of early life, its evolution through Darwinian natural selection and the emergence of complex multicellular life forms. Finally, the cognitive age describes how complex life evolved into intelligent life capable of self-awareness and of developing technology through the directed manipulation of energy and materials. I conclude discussing whether we are the rule or the exception.

Simulating Rare Events in Dynamical Processes

Atypical, rare trajectories of dynamical systems are important: they are often the paths for chemical reactions, the haven of (relative) stability of planetary systems, the rogue waves that are detected in oil platforms, the structures that are responsible for intermittency in a turbulent liquid, the active regions that allow a supercooled liquid to flow... Simulating them in an efficient, accelerated way, is in fact quite simple. In this paper we review a computational technique to study such rare events in both stochastic and Hamiltonian systems. The method is based on the evolution of a family of copies of the system which are replicated or killed in such a way as to favor the realization of the atypical trajectories. We illustrate this with various examples.

The Global Symplectic Integrator: an efficient tool for stability studies of dynamical systems. Application to the Kozai resonance in the restricted three-body problem

Following the discovery of extrasolar systems, the study of long-term evolution and stability of planetary systems is enjoying a renewed interest. While non-symplectic integrators are very time-consuming because of the very long time-scales and the small integration steps required to have a good energy preservation, symplectic integrators are well suited for the study of such orbits on long time-spans. However, stability studies of dynamical systems generally rely on non-symplectic integrations of deviation vectors. In this work we propose a numerical approach to distinguish between regular and chaotic orbits in Hamiltonian systems, hereby called Global Symplectic Integrator. It consists of the simultaneous integration of the orbit and the deviation vectors using a symplectic scheme of any order. In particular, due to its symplectic properties, the proposed method allows us to recover the correct orbit characteristics using very large integration time-steps, fluctuations of energy around a constant value and short CPU times. It proves to be more efficient than non-symplectic schemes to correctly identify the behaviour of a given orbit, especially on dynamics acting on long time-scales. To illustrate the numerical performances of the global symplectic integrator, we will apply it to the well-known toy problem of Henon–Heiles and the challenging problem of the Kozai resonance in the restricted three-body problem, whose secular effects have periods of the order of 104–105 yr.

The planetary N-body problem: symplectic foliation, reductions and invariant tori

The 6n-dimensional phase space of the planetary (1+n)-body problem (after the classical reduction of the total linear momentum) is shown to be foliated by symplectic leaves of dimension (6n−2) invariant for the planetary Hamiltonian \({\mathcal{H}}\). Such foliation is described by means of a new global set of Darboux coordinates related to a symplectic (partial) reduction of rotations. On each symplectic leaf \({\mathcal{H}}\) has the same form and it is shown to preserve classical symmetries. Further sets of Darboux coordinates may be introduced on the symplectic leaves so as to achieve a complete (total) reduction of rotations. Next, by explicit computations, it is shown that, in the reduced settings, certain degeneracies are removed. In particular, full torsion is checked both in the partially and totally reduced settings. As a consequence, a new direct proof of Arnold’s theorem (Arnold in Russ. Math. Surv. 18(6):85–191, 1963) on the stability of planetary system (both in the partially and in the totally reduced setting) is easily deduced, producing Diophantine Lagrangian invariant tori of dimension (3n−1) and (3n−2). Finally, elliptic lower dimensional tori bifurcating from the secular equilibrium are easily obtained.

Stability of planetary systems with respect to masses

The stability in the sense of Lagrange of the Sun–Jupiter–Saturn system and 47 UMa system with respect to masses on a time scale of 106 years was studied using the method of averaging and numerical methods. When the masses of Jupiter and Saturn increase by 20 times (approximately, more accurate value depends on a time-scale of stable motion), these planets can have close approaches. Close approaches appear when analyzing osculating elements; they are absent in the mean elements. A similar situation takes place in the case of 47 UMa and other exoplanetary systems. The study of Lagrange stability with respect to masses allows us to obtain upper limits for masses of extrasolar planets.