First order resonance overlap and the stability of close two planet systems

Motivated by the population of multi-planet systems with orbital period ratios 1<P2/P1<2, we study the long-term stability of packed two planet systems. The Hamiltonian for two massive planets on nearly circular and nearly coplanar orbits near a first order mean motion resonance can be reduced to a one degree of freedom problem (Sessin & Ferraz Mello (1984), Wisdom (1986), Henrard et al. (1986)). Using this analytically tractable Hamiltonian, we apply the resonance overlap criterion to predict the onset of large scale chaotic motion in close two planet systems. The reduced Hamiltonian has only a weak dependence on the planetary mass ratio, and hence the overlap criterion is independent of the planetary mass ratio at lowest order. Numerical integrations confirm that the planetary mass ratio has little effect on the structure of the chaotic phase space for close orbits in the low eccentricity (e <~0.1) regime. We show numerically that orbits in the chaotic web produced primarily by first order resonance overlap eventually experience large scale erratic variation in semimajor axes and are Lagrange unstable. This is also true of the orbits in this overlap region which are Hill stable. As a result, we can use the first order resonance overlap criterion as an effective stability criterion for pairs of observed planets. We show that for low mass (<~10 M_Earth) planetary systems with initially circular orbits the period ratio at which complete overlap occurs and widespread chaos results lies in a region of parameter space which is Hill stable. Our work indicates that a resonance overlap criterion which would apply for initially eccentric orbits needs to take into account second order resonances. Finally, we address the connection found in previous work between the Hill stability criterion and numerically determined Lagrange instability boundaries in the context of resonance overlap.Comment: Accepted for publication in Ap

Transit timing variations for planets near eccentricity-type mean motion resonances

We derive the transit timing variations (TTVs) of two planets near a second-order mean motion resonance (MMR) on nearly circular orbits. We show that the TTVs of each planet are given by sinusoids with a frequency of jn_2 -(j-2){n_1, where j ≥ 3 is an integer characterizing the resonance and n2 and n1 are the mean motions of the outer and inner planets, respectively. The amplitude of the TTV depends on the mass of the perturbing planet, relative to the mass of the star, and on both the eccentricities and longitudes of pericenter of each planet. The TTVs of the two planets are approximated anti-correlated, with phases of φ and ≈φ + π, where the phase φ also depends on the eccentricities and longitudes of pericenter. Therefore, the TTVs caused by proximity to a second-order MMR do not in general uniquely determine both planet masses, eccentricities, and pericenters. This is completely analogous to the case of TTVs induced by two planets near a first-order MMR. We explore how other TTV signals, such as the short-period synodic TTV or a first-order resonant TTV, in combination with the second-order resonant TTV, can break degeneracies. Finally, we derive approximate formulae for the TTVs of planets near any order eccentricity-type MMR; this shows that the same basic sinusoidal TTV structure holds for all eccentricity-type resonances. Our general formula reduces to previously derived results near first-order MMRs

Measurement of planet masses with transit timing variations due to synodic "chopping" effects

Gravitational interactions between planets in transiting exoplanetary systems lead to variations in the times of transit that are diagnostic of the planetary masses and the dynamical state of the system. Here we show that synodic "chopping" contributions to these transit timing variations (TTVs) can be used to uniquely measure the masses of planets without full dynamical analyses involving direct integration of the equations of motion. We present simple analytic formulae for the chopping signal, which are valid (generally <10% error) for modest eccentricities e <~ 0.1. Importantly, these formulae primarily depend on the mass of the perturbing planet, and therefore the chopping signal can be used to break the mass/free-eccentricity degeneracy which can appear for systems near first order mean motion resonances. Using a harmonic analysis, we apply these TTV formulae to a number of Kepler systems which had been previously analyzed with full dynamical analyses. We show that when chopping is measured, the masses of both planets can be determined uniquely, in agreement with previous results, but without the need for numerical orbit integrations. This demonstrates how mass measurements from TTVs may primarily arise from an observable chopping signal. The formula for chopping can also be used to predict the number of transits and timing precision required for future observations, such as those made by TESS or PLATO, in order to infer planetary masses through analysis of TTVs.Comment: submitted to ApJ, comments appreciate

TTVFast: An efficient and accurate code for transit timing inversion problems

Transit timing variations (TTVs) have proven to be a powerful technique for confirming Kepler planet candidates, for detecting non-transiting planets, and for constraining the masses and orbital elements of multi-planet systems. These TTV applications often require the numerical integration of orbits for computation of transit times (as well as impact parameters and durations); frequently tens of millions to billions of simulations are required when running statistical analyses of the planetary system properties. We have created a fast code for transit timing computation, TTVFast, which uses a symplectic integrator with a Keplerian interpolator for the calculation of transit times (Nesvorny et al. 2013). The speed comes at the expense of accuracy in the calculated times, but the accuracy lost is largely unnecessary, as transit times do not need to be calculated to accuracies significantly smaller than the measurement uncertainties on the times. The time step can be tuned to give sufficient precision for any particular system. We find a speed-up of at least an order of magnitude relative to dynamical integrations with high precision using a Bulirsch-Stoer integrator.Comment: Submitted to ApJ. Our code is available in both C and Fortran at: http://github.com/kdeck/TTVFast . If you download this version, please check back after the referee process for a possibly updated versio

Stability of Satellites in Closely Packed Planetary Systems

We perform numerical integrations of four-body (star, planet, planet, satellite) systems to investigate the stability of satellites in planetary Systems with Tightly-packed Inner Planets (STIPs). We find that the majority of closely-spaced stable two-planet systems can stably support satellites across a range of parameter-space which is only slightly decreased compared to that seen for the single-planet case. In particular, circular prograde satellites remain stable out to $\sim 0.4 R_H$ (where $R_H$ is the Hill Radius) as opposed to $\sim 0.5 R_H$ in the single-planet case. A similarly small restriction in the stable parameter-space for retrograde satellites is observed, where planetary close approaches in the range 2.5 to 4.5 mutual Hill radii destabilize most satellites orbits only if $a\sim 0.65 R_H$. In very close planetary pairs (e.g. the 12:11 resonance) the addition of a satellite frequently destabilizes the entire system, causing extreme close-approaches and the loss of satellites over a range of circumplanetary semi-major axes. The majority of systems investigated stably harbored satellites over a wide parameter-space, suggesting that STIPs can generally offer a dynamically stable home for satellites, albeit with a slightly smaller stable parameter-space than the single-planet case. As we demonstrate that multi-planet systems are not a priori poor candidates for hosting satellites, future measurements of satellite occurrence rates in multi-planet systems versus single-planet systems could be used to constrain either satellite formation or past periods of strong dynamical interaction between planets.Comment: 11 pages, 5 figures. Accepted for publication, ApJ

Stellar Spin-Orbit Misalignment in a Multiplanet System

Stars hosting hot Jupiters are often observed to have high obliquities, whereas stars with multiple coplanar planets have been seen to have low obliquities. This has been interpreted as evidence that hot-Jupiter formation is linked to dynamical disruption, as opposed to planet migration through a protoplanetary disk. We used asteroseismology to measure a large obliquity for Kepler-56, a red giant star hosting two transiting coplanar planets. These observations show that spin-orbit misalignments are not confined to hot-Jupiter systems. Misalignments in a broader class of systems had been predicted as a consequence of torques from wide-orbiting companions, and indeed radial velocity measurements revealed a third companion in a wide orbit in the Kepler-56 system.United States. National Aeronautics and Space Administration (Science Mission Directorate)United States. National Aeronautics and Space Administration (NASA Postdoctoral Program at Ames Research Center)National Science Foundation (U.S.) (NSF Graduate Research Fellowship)National Science Foundation (U.S.) (NSF Graduate Research Fellowship, grant DGE1144469)Netherlands Organization for Scientific ResearchBelgian Federal Science Policy Office (BELSPO, contract PRODEX COROT)United States. National Aeronautics and Space Administration (NASA Kepler Participating Scientist program)National Science Foundation (U.S.) (NSF grant AST-1105930)David & Lucile Packard FoundationAlfred P. Sloan FoundationHarvard-Smithsonian Center for Astrophysics (Hubble Fellow

Migration of Two Massive Planets into (and out of) First Order Mean Motion Resonances

We consider the dynamical evolution of two planets with nearly circular and nearly coplanar orbits undergoing eccentricity damping and convergent migration in the vicinity of a first order mean motion resonance. Following Goldreich & Schlichting, we include a coupling between the dissipative semimajor axis evolution and the damping of the eccentricities. In agreement with past studies, we find that this coupling can lead to instability of the resonance and that for a certain range of parameters capture into resonance is only temporary. Using a more general model, we show that whether escape from resonance can occur depends in a characteristic way on the mass ratio between the two planets as well as their relative eccentricity damping timescales. In particular, systems undergoing Type I migration with a more massive inner planet typically result in permanent capture. Additionally, we show that even when escape from resonance does occur, the timescale for escape is long enough such at any given time a pair of planets is more likely to be found in a low-order resonance rather than migrating between them. Thus, we argue that intrinsic instability of resonances cannot singlehandedly reconcile convergent migration with the observed lack of Kepler planet pairs found near resonances

Seven temperate terrestrial planets around the nearby ultracool dwarf star TRAPPIST-1

One aim of modern astronomy is to detect temperate, Earth-like exoplanets that are well suited for atmospheric characterization. Recently, three Earth-sized planets were detected that transit (that is, pass in front of) a star with a mass just eight per cent that of the Sun, located 12 parsecs away. The transiting configuration of these planets, combined with the Jupiter-like size of their host star—named TRAPPIST-1—makes possible in-depth studies of their atmospheric properties with present-day and future astronomical facilities. Here we report the results of a photometric monitoring campaign of that star from the ground and space. Our observations reveal that at least seven planets with sizes and masses similar to those of Earth revolve around TRAPPIST-1. The six inner planets form a near-resonant chain, such that their orbital periods (1.51, 2.42, 4.04, 6.06, 9.1 and 12.35 days) are near-ratios of small integers. This architecture suggests that the planets formed farther from the star and migrated inwards. Moreover, the seven planets have equilibrium temperatures low enough to make possible the presence of liquid water on their surfaces

An Empirically Derived Three-Dimensional Laplace Resonance in the Gliese 876 Planetary System

We report constraints on the three-dimensional orbital architecture for all four planets known to orbit the nearby M dwarf Gliese 876 based solely on Doppler measurements and demanding long-term orbital stability. Our dataset incorporates publicly available radial velocities taken with the ELODIE and CORALIE spectrographs, HARPS, and Keck HIRES as well as previously unpublished HIRES velocities. We first quantitatively assess the validity of the planets thought to orbit GJ 876 by computing the Bayes factors for a variety of different coplanar models using an importance sampling algorithm. We find that a four-planet model is preferred over a three-planet model. Next, we apply a Newtonian MCMC algorithm to perform a Bayesian analysis of the planet masses and orbits using an n-body model in three-dimensional space. Based on the radial velocities alone, we find that a 99% credible interval provides upper limits on the mutual inclinations for the three resonant planets ($\Phi_{cb}<6.20^\circ$ for the "c" and "b" pair and $\Phi_{be}<28.5^\circ$ for the "b" and "e" pair). Subsequent dynamical integrations of our posterior sample find that the GJ 876 planets must be roughly coplanar ($\Phi_{cb}<2.60^\circ$ and $\Phi_{be}<7.87^\circ$), suggesting the amount of planet-planet scattering in the system has been low. We investigate the distribution of the respective resonant arguments of each planet pair and find that at least one argument for each planet pair and the Laplace argument librate. The libration amplitudes in our three-dimensional orbital model supports the idea of the outer-three planets having undergone significant past disk migration.Comment: 19 pages, 11 figures, 8 tables. Accepted to MNRAS. Posterior samples available at https://github.com/benelson/GJ87