We investigate the dynamical stability of the Kepler-60 planetary system with three super-Earths. We determine their orbital elements and masses by Transit Timing Variation (TTV) data spanning quarters Q1-Q16 of the KEPLER mission. The system is dynamically active but the TTV data constrain masses to ∼ 4 m ⊕ and orbits in safely wide stable zones. The observations prefer two types of solutions. The true three-body Laplace MMR exhibits the critical angle librating around 45 • and aligned apsides of the inner and outer pair of planets. In the Laplace MMR formed through a chain of two-planet 5:4 and 4:3 MMRs, all critical angles librate with small amplitudes ∼ 30 • and apsidal lines in planet's pairs are anti-aligned. The system is simultaneously locked in a three-body MMR with librations amplitude 10 o . The true Laplace MMR can evolve towards a chain of two-body MMRs in the presence of planetary migration. Therefore the three-body MMR formed in this way seems to be more likely state of the system. However, the true three-body MMR cannot be disregarded a priori and it remains a puzzling configuration that may challenge the planet formation theory.

We describe the Reversibility Error Method (REM) and its applications to planetary dynamics. REM is based on the time-reversibility analysis of the phase-space trajectories of conservative Hamiltonian systems. The round-off errors break the time reversibility and the displacement from the initial condition, occurring when we integrate it forward and backward for the same time interval, is related to the dynamical character of the trajectory. If the motion is chaotic, in the sense of non-zero maximal Characteristic Lyapunov Exponent (mLCE), then REM increases exponentially with time, as exp λt, while when the motion is regular (quasi-periodic) then REM increases as a power law in time, as t α , where α and λ are real coefficients. We compare the REM with a variant of mLCE, the Mean Exponential Growth factor of Nearby Orbits (MEGNO). The test set includes the restricted three body problem and five resonant planetary systems: HD 37124, Kepler-60, Kepler-36, Kepler-29 and Kepler-26. We found a very good agreement between the outcomes of these algorithms. Moreover, the numerical implementation of REM is astonishing simple, and is based on solid theoretical background. The REM requires only a symplectic and time-reversible (symmetric) integrator of the equations of motion. This method is also CPU efficient. It may be particularly useful for the dynamical analysis of multiple planetary systems in the KEPLER sample, characterized by loweccentricity orbits and relatively weak mutual interactions. As an interesting side-result, we found a possible stable chaos occurrence in the Kepler-29 planetary system.

We analyse the Transit Timing Variation (TTV) measurements of a system of two super-Earths detected as Kepler-29, in order to constrain the planets' masses and orbital parameters. A dynamical analysis of the best-fitting configurations constrains the masses to be ∼ 6 and ∼ 5 Earth masses for the inner and the outer planets, respectively. The analysis also reveals that the system is likely locked in the 9:7 mean motion resonance. However, a variety of orbital architectures regarding eccentricities and the relative orientation of orbits is permitted by the observations as well as by stability constraints. We attempt to find configurations preferred by the planet formation scenarios as an additional, physical constraint. We show that configurations with low eccentricities and anti-aligned apsidal lines of the orbits are a natural and most likely outcome of the convergent migration. However, we show that librations of the critical angles are not necessary for the Kepler-29 system to be dynamically resonant, and such configurations may be formed on the way of migration as well. We argue, on the other hand, that aligned configurations with e 0.03 may be not consistent with the migration scenario.