TRADES: A new software to derive orbital parameters from observed transit times and radial velocities

Abstract

Aims. With the purpose of determining the orbital parameters of exoplanetary systems from observational data, we have developed a software, named TRADES (TRAnsits and Dynamics of Exoplanetary Systems), to simultaneously fit observed radial velocities and transit times data. Methods. We implemented a dynamical simulator for N-body systems, which also fits the available data during the orbital integration and determines the best combination of the orbital parameters using grid search, χ 2 minimization, genetic algorithms, particle swarm optimization, and bootstrap analysis. Results. To validate TRADES, we tested the code on a synthetic three-body system and on two real systems discovered by the Kepler mission: Kepler-9 and Kepler-11. These systems are good benchmarks to test multiple exoplanet systems showing transit time variations (TTVs) due to the gravitational interaction among planets. We have found that orbital parameters of Kepler-11 planets agree well with the values proposed in the discovery paper and with a a recent work from the same authors. We analyzed the first three quarters of Kepler-9 system and found parameters in partial agreement with discovery paper. Analyzing transit times (T 0 s), covering 12 quarters of Kepler data, that we have found a new best-fit solution. This solution outputs masses that are about 55% of the values proposed in the discovery paper; this leads to a reduced semi-amplitude of the radial velocities of about 12.80 ms −1 .

Mass determination of K2-19b and K2-19c from radial velocities and transit timing variations

“…Another case is the reanalysis of RV and TTVs data of the Kepler-9 planetary system by Borsato et al (2014) using the same analysis tool (TRADES) as in this paper, and who estimated planet masses ∼55 % lower than reported by in the original discovery paper (Holman et al 2010), which was based on the combination of RVs, transit times and durations. In this case, Borsato et al (2014) ascribe the discrepancy to the longer Kepler light-curve analysed by them, as well as different approaches in the interpretation of the TTVs.…”

“…In this case, Borsato et al (2014) ascribe the discrepancy to the longer Kepler light-curve analysed by them, as well as different approaches in the interpretation of the TTVs.…”

“…In a further analysis we derived masses and orbital parameters of K2-19b and K2-19c using the code TRADES (Borsato et al 2014) to simultaneously model RV measurements and TTV data. TRADES combines the Particle Swarm Optimization (PSO; Tada 2007) with the Levenberg-Marquardt algorithm (LM; Moré et al 1980).…”

Numerical and Analytical Modeling of Transit timing variations

“…8 We also show in Appendix A.4 that  is nearly independent of J, soif two nearby first-order MMRs both contribute comparable  dt values, the eccentricities still enter through the single quantity  . 9 Kepler-9 (Holman et al 2010;Borsato et al 2014;Dreizler & Ofir 2014), Kepler-18 (Cochran et al 2011), andKepler-30 (Sanchis-Ojeda et al 2012), for which relatively precise individual planet eccentricities measurements are reported, contain planets near the 2:1 MMR. 10 Three (or more) planets with mutual TTVs in principle yield three distinct  values, one for each pairwise interaction, which can be inverted to determine three individual z values.…”

The Kepler-19 System: A Thick-envelope Super-Earth with Two Neptune-mass Companions Characterized Using Radial Velocities and Transit Timing Variations

“…Dynamical analysis of the system was performed with TRADES 27 (TRAnsits and Dynamics of Exoplanetary Systems, Borsato et al 2014), an N-body integrator with the capability of fitting RVs and T 0 s simultaneously to determine the orbital parameters of the system through χ 2 minimization. Since only the innermost planet is transiting, it is extremely difficult to constrain the orbital parameters of all the planets in the system by TTV alone.…”