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

Kepler-210: An active star with at least two planets

We report the detection and characterization of two short-period, Neptune-sized planets around the active host star Kepler-210. The host star's parameters derived from those planets are (a) mutually inconsistent and (b) do not conform to the expected host star parameters. We furthermore report the detection of transit timing variations (TTVs) in the O-C diagrams for both planets. We explore various scenarios that explain and resolve those discrepancies. A simple scenario consistent with all data appears to be one that attributes substantial eccentricities to the inner short-period planets and that interprets the TTVs as due to the action of another, somewhat longer period planet. To substantiate our suggestions, we present the results of N-body simulations that modeled the TTVs and that checked the stability of the Kepler-210 system.Comment: 8 pages, 8 Encapsulated Postscript figure

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

Cubulating hyperbolic free-by-cyclic groups: the general case

Let $\Phi:F\rightarrow F$ be an automorphism of the finite-rank free group $F$. Suppose that $G=F\rtimes_\Phi\mathbb Z$ is word-hyperbolic. Then $G$ acts freely and cocompactly on a CAT(0) cube complex.Comment: 36 pages, 11 figures. Version 2 contains minor corrections. Accepted to GAF

Congruence RFRS towers

We describe a criterion for a real or complex hyperbolic lattice to admit a RFRS tower that consists entirely of congruence subgroups. We use this to show that certain Bianchi groups PSL(Od) are virtually fibered on congruence subgroups, and also exhibit the first examples of RFRS Kähler groups that are not a subgroup of a product of surface groups and abelian groups

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