The stability of ultra-compact planetary systems (original) (raw)

A dynamical stability study of Kepler Circumbinary planetary systems with one planet

Monthly Notices of the Royal Astronomical Society, 2014

To date, 17 circumbinary planets have been discovered. In this paper, we focus our attention on the stability of the Kepler circumbinary planetary systems with only one planet, i.e. . In addition to their intrinsic interest, the study of such systems is an opportunity to test our understanding of planetary system formation and evolution around binaries. The investigation is done by means of numerical simulations. We perform numerical integrations of the full equations of motion of each system with the aim of checking the stability of the planetary orbit. The investigation of the stability of the above systems consists of three numerical experiments. In the first one, we perform a long-term (1 Gyr) numerical integration of the nominal solution of the six Kepler systems under investigation. In the second experiment, we look for the critical semimajor axis of the six planetary orbits, and finally, in the third experiment, we construct two-dimensional stability maps on the eccentricity-pericentre distance plane. Additionally, using numerical integrations of the nominal solutions we checked if these solutions were close to the exact resonance.

On the dynamical stability of planets in double stars

The importance of stability studies of planetary motion in binaries arises from the fact that double and multiple star systems are more numerous than single stars - at least in the solar neighborhood. Another impulse to carry out such dynamical studies was the discovery of planets in binaries, where we distinguish between two types of motion: P-type and S-type orbits. A dynamical stability study of two binary systems (γ Cephei and Gliese 86) is shown in this investigation, where we examined the region between the two stars in order to find stable zones where other planets might exist. For the determination of the stable zones we used two chaos indicators (1. the Fast Lyapunow Indicator - FLI and 2. the Mean Exponential Growth factor of Nearby Orbits - MEGNO) and additionally straight-forward numerical computations by applying the Lie integration method. In the general stability study of S-type motion we show the results for a double star with mass-ratio 0.2 which can be applied to t...

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.

Dynamics and stability of telluric planets within the habitable zone of extrasolar planetary systems

2008

Aims. We study gravitational perturbation effects of observed giant extrasolar planets on hypothetical Earth-like planets in the context of the three-body problem. This paper considers a large parameter survey of different orbital configuration of two extrasolar giant planets (HD 70642b and HD 4208b) and compares their dynamical effect on Earth-mass planetary orbits initially located within the respective habitable terrestrial region. We are interested in determining giant-planet orbit (and mass) parameters that favor the condition to render an Earth-mass planet to remain on a stable and bounded orbit within the continuous habitable zone. Methods. We applied symplectic numerical integration techniques to studying the short and long term time evolution of hypothetical Earth-mass planets that are treated as particles. In addition, we adopt the MEGNO technique to obtain a complete dynamical picture of the terrestrial phase space environment. Both multi-particle and single-particle simulations were performed to follow an Earth-mass planet in the habitable region and its subsequent long term evolution. Results. Our numerical simulations show that giant planets should be on circular orbits to minimize the perturbative effect on terrestrial orbits. The orbit eccentricity (and hence proximity) is the most important orbital parameter of dynamical significance. The most promising candidate for maintaining an Earth-mass planet on a stable and bounded orbit well-confined to the continuous habitable zone is HD 70642b. Even the large planetary mass of HD 70642b renders an Earth-mass planet habitable during the complete lifetime of the host star. The results allow us to extrapolate similar observed systems and points the necessity further constraining the uncertainty range in giant planet orbital eccentricity by future follow-up observations.

Stable lifetime of compact, evenly-spaced planetary systems with non-equal masses

arXiv (Cornell University), 2022

Compact planetary systems with more than two planets can undergo orbital crossings from planet-planet perturbations. The time which the system remains stable without orbital crossings has an exponential dependence on the initial orbital separations in units of mutual Hill radii. However when a multi-planet system has period ratios near mean-motion resonances, its stability time differs from the time determined by planet separation. This difference can be up to an order of magnitude when systems are set up with chains of equal period ratios. We use numerical simulations to describe the stability time relationship in non-resonant systems with equal separations but non-equal masses which breaks the chains of equal period ratios. We find a deviation of 30 per cent in the masses of Earth-mass planets creates a large enough deviation in the period ratios where the average stability time of a given spacing can be predicted by the stability time relationship. The mass deviation where structure from equal period ratios is erased increases with planet mass but does not depend on planet multiplicity. With a large enough mass deviation, the distribution of stability time at a given spacing is much wider than in equal-mass systems where the distribution narrows due to period commensurabilities. We find the stability time distribution is heteroscedastic with spacing-the deviation in stability time for a given spacing increases with said spacing.

Stability Limits in Extrasolar Planetary Systems

The Astrophysical Journal, 2006

Two types of stability boundaries exist for any planetary system consisting of one star and two planets. Lagrange stability requires that the planets remain bound to the star, conserves the ordering of the distance from the star, and limits the variations of orbital elements like semi-major axis and eccentricity. Hill stability only requires that the ordering of the planets remain constant; the outer planet may escape to infinity. A simple formula defines a region in orbital element space that is guaranteed to be Hill stable, although Hill stable orbits may lie outside the region as well. No analytic criteria describe Lagrange stability. We compare the results of 1000 numerical simulations of planetary systems similar to 47 UMa and HD 12661 with these two types of boundaries. All cases are consistent with the analytic criterion for Hill stability. Moreover, the numerically determined Lagrange boundary lies close to the analytic boundary for Hill stability. This result suggests an analytic formulation that may describe the criterion for Lagrange stability.

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.

Orbital Stability of Earth-Type Planets in Binary Systems

Arxiv preprint arXiv:0712.3266, 2007

About half of all known stellar systems with Sun-like stars consist of two or more stars, significantly affecting the orbital stability of any planet in these systems. Here we study the onset of instability for an Earth-type planet that is part of a binary system. Our investigation makes use of previous analytical work allowing to describe the permissible region of planetary motion. This allows us to establish a criterion for the orbital stability of planets that may be useful in the context of future observational and theoretical studies.

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.

Dynamical Stability of Terrestrial and Giant Planets in the HD 155358 Planetary System

2007

The results of a study of the dynamical evolution and the habitability of the planetary system of HD 155358 are presented. This system is unique in that it is one of the two low metallicity stars discovered to host a multiple planet system. HD 155358 is host to two Jupiter-sized planets, with minimum masses of 0.86 and 0.50 Jupiter-masses. The orbit of the lower mass planet of this system is located at the inner edge of the system's habitable zone. To determine whether this system can harbor terrestrial-type planets, the orbits of its planets and an Earth-like object were numerically integrated for different values of their masses and orbital eccentricities. Results indicate that this system could potentially host stable orbits for terrestrial-sized planets in its habitable zone, but the stability of these orbits is very sensitive to the precise characteristics of the giant planets of the system. The long-term stability of larger bodies (Neptuneand Saturn-mass) was also studied ...