Stable lifetime of compact, evenly-spaced planetary systems with non-equal masses (original) (raw)

The Stability of Multi-Planet Systems

Icarus, 1996

A system of two small planets orbiting the Sun on loweccentricity, low-inclination orbits is stable with respect to close encompassing each conjunction, and the authors derive encounters if the initial semi-major axis difference, ⌬, measured approximate analytic expressions for the corresponding in mutual Hill radii, R H , exceeds 2͙3 ළ, due to conservation of changes in the planets' orbits, in the planar problem, for energy and angular momentum. We investigate the stability of cases in which the difference, ⌬, in their semi-major axes systems of more than two planets using numerical integrations. is either small or large.

Dynamical instability and its implications for planetary system architecture

Monthly Notices of the Royal Astronomical Society

We examine the effects that dynamical instability has on shaping the orbital properties of exoplanetary systems. Using N-body simulations of non-EMS (Equal Mutual Separation), multiplanet systems we find that the lower limit of the instability timescale t is determined by the minimal mutual separation K min in units of the mutual Hill radius. Planetary systems showing instability generally include planet pairs with period ratio <1.33. Our final period ratio distribution of all adjacent planet pairs shows dip-peak structures near first-order mean motion resonances similar to those observed in the Kepler planetary data. Then we compare the probability density function (PDF) of the de-biased Kepler period ratios with those in our simulations and find a lack of planet pairs with period ratio >2.1 in the observationspossibly caused either by inward migration before the dissipation of the disc or by planet pairs not forming with period ratios >2.1 with the same frequency they do with smaller period ratios. By comparing the PDF of the period ratio between simulation and observation, we obtain an upper limit of 0.03 on the scale parameter of the Rayleigh distributed eccentricities when the gas disc dissipated. Finally, our results suggest that a viable definition for a 'packed' or 'compact' planetary system be one that has at least one planet pair with a period ratio less than 1.33. This criterion would imply that 4 per cent of the Kepler systems (or 6 per cent of the systems with more than two planets) are compact.

Study of stability of mean-motion resonances in multiexoplanetary systems

Journal of Physics: Conference Series, 2016

Many exoplanetary systems have been found to harbour more than one planet. Some of them have commensurability in orbital periods of the planets (resonant-planet pair). The aim of this work is to analyse the stability of resonant-planet pair configuration in two multiexoplanetary systems which have two planets in near mean-motion resonances, i.e. Kepler-9 and HD 10180 systems. This work considers numerical and comparative empiric-analytical studies. Numerical studies are performed using the integrator package SWIFT with an integration time of 10 Myr. Results from numerical integrations indicate that all orbital solution sets of the systems are stable. Further numerical explorations also demonstrate that the systems are stable for small perturbations in the orbital elements and mass variations. Analyses of stability based on comparative empiric-analytical are done by applying a known stability criterion to all systems. We find that all systems tend to be stable.

Stability Limits in Resonant Planetary Systems

The Astrophysical Journal, 2007

The relationship between the boundaries for Hill and Lagrange stability in orbital element space is modified in the case of resonantly interacting planets. Hill stability requires the ordering of the planets to remain constant while Lagrange stability also requires all planets to remain bound to the central star. The Hill stability boundary is defined analytically, but no equations exist to define the Lagrange boundary, so we perform numerical experiments to estimate the location of this boundary. To explore the effect of resonances, we consider orbital element space near the conditions in the HD 82943 and 55 Cnc systems. Previous studies have shown that, for non-resonant systems, the two stability boundaries are nearly coincident. However the Hill stability formula are not applicable to resonant systems, and our investigation shows how the two boundaries diverge in the presence of a mean-motion resonance, while confirming that the Hill and Lagrange boundaries are similar otherwise. In resonance the region of stability is larger than the domain defined by the analytic formula for Hill stability. We find that nearly all known resonant interactions currently lie in this extra stable region, i.e. where the orbits would be unstable according to the non-resonant Hill stability formula. This result bears on the dynamical packing of planetary systems, showing how quantifying planetary systems' dynamical interactions (such as proximity to the Hill-stability boundary) provides new constraints on planet formation models.

The orbital stability of planets trapped in the first-order mean-motion resonances

Icarus, 2012

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 of the resonantly trapped planets beyond which orbital instability occurs after disk gas depletion. We find that when the total number of planets (N) is larger than the critical number (N crit), crossing time that is a timescale of initiation of the orbital instability is similar to non-resonant cases, while the orbital instability never occurs within the orbital calculation time (10 8 Kepler time) for N ≤ N crit. Thus, the transition of crossing time across the critical number is drastic. When all the planets are trapped in 7:6 resonance of adjacent pairs, N crit = 4. We examine the dependence of the critical number of 4:3, 6:5 and 8:7 resonance by changing the orbital separation in mutual Hill radii and planetary mass. The critical number increases with increasing the orbital separation in mutual Hill radii with fixed planetary mass and increases with increasing planetary mass with fixed the orbital separation in mutual Hill radii. We also calculate the case of a system which is not composed of

Exploring the boundary of dynamical stability of interacting two-planet system

Quantifying the proximity of planetary systems to dynamical stability may be useful in screening extrasolar systems that lie deep inside stability and thus may harbor additional terrestrial-sized planets. A preferred definition for dynamical stability is Lagrange stability which requires all planets to maintain their ordering while remaining bound to the system. Alas, there is yet no analytical expression for Lagrange stability. Hill stability is less strict and only requires all planets to maintain their ordering. However, an analytical criterion for constraining the Hill stability of an interacting two-planet coplanar system was derived. empirically noted that the boundary of Lagrange stability follows an approximately constant proximity to the Hill-stability criterion on the planetary stability maps of 47 UMa and HD 12661. Consequently, they proposed an empirical analytic expression for Lagrange stability based on the proximity of the planetary configuration to the Hill-stability criterion. This study is aimed at reexamining that proposal. 280 numerical simulations were used to generate a stability map of 47 UMa that extends in eccentricity phase space beyond the one derived by . Lagrange stability and Hill-stability criterion of each planetary configuration were compared. Results suggest that the proposed analytic expression for Lagrange stability remains valid, but only in its weak form. However, it was also found that the proximity of the boundary of Lagrange stability to the Hill-stability criterion does not remain constant, but is rather a function of eccentricities. Moreover, only a weak correlation was found between the time to develop major planetary perturbations and the proximity to the Hill-stability criterion. It is argued that these findings challenge the validity of universally using the proximity to the Hill-stability criterion as a single parameter for quantifying the distance to dynamical stability. A follow-up study with a larger sample of planetary configurations is advocated, aimed at studying in detail the dependence on orbital elements of the proximity of Lagrange stability to the Hill-stability criterion.

Resonant planetary dynamics: Periodic orbits and long-term stability

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.

Limits on orbit-crossing planetesimals in the resonant multiple planet system, KOI-730

Monthly Notices of the Royal Astronomical Society, 2013

A fraction of multiple planet candidate systems discovered from transits by the Kepler mission contain pairs of planet candidates that are in orbital resonance or are spaced slightly too far apart to be in resonance. We focus here on the four-planet system, KOI-730, that has planet periods satisfying the ratios 8:6:4:3. By numerically integrating four planets assumed to initially be in a resonant configuration in proximity to an initially exterior cold planetesimal disc, we find that of the order of a Mars mass of planet-orbit-crossing planetesimals is sufficient to pull this system out of resonance. Approximately one Earth mass of planet-orbit-crossing planetesimals increases the interplanetary spacings sufficiently to resemble the multiple planet candidate Kepler systems that lie just outside of resonance. This suggests that the closely spaced multiple planet Kepler systems, host only low-mass debris discs or their debris discs have been extremely stable. We find that the planetary inclinations increase as a function of the mass in planetesimals that have crossed the orbits of the planets. If systems are left at zero inclination and in resonant chains after depletion of the gas disc, then we would expect a correlation between distance to resonance and mutual planetary inclinations. This may make it possible to differentiate between dynamical mechanisms that account for the fraction of multiple planet systems just outside of resonance.

On the 2/1 resonant planetary dynamics - periodic orbits and dynamical stability

Monthly Notices of the Royal Astronomical Society, 2009

The 2/1 resonant dynamics of a two-planet planar system is studied within the framework of the three-body problem by computing families of periodic orbits and their linear stability. The continuation of resonant periodic orbits from the restricted to the general 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. The parametric continuation, within the framework of the general problem, takes place by varying the planetary mass ratio ρ. We obtain bifurcations which are caused either due to collisions of the families in the space of initial conditions or due to the vanishing of bifurcation points. Our study refers to the whole range of planetary mass ratio values (ρ ∈ (0, ∞)) and, therefore we include the passage from external to internal resonances. Thus we can obtain all possible stable configurations in a systematic way. Finally, we study whether the dynamics of the four known planetary systems, whose currently observed periods show a 2/1 resonance, are associated with a stable periodic orbit.

Resonant periodic orbits in the exoplanetary systems

Astrophysics and Space Science, 2014

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 chaotic evolution that destabilizes the planetary system. Stable periodic orbits are of particular importance in planetary dynamics, since they can host real planetary systems. We found stable orbits up to 60 • of mutual planetary inclination, but in most families, the stability does not exceed 20 •-30 • , depending on the planetary mass ratio. Most of these orbits are very eccentric. Stable inclined circular orbits or orbits of low eccentricity were found in the 4/3 and 5/2 resonance, respectively.