An alternative stable solution for the Kepler-419 system, obtained with the use of a genetic algorithm
The mid-transit times of an exoplanet may be non-periodic. The variations in the timing of the transits with respect to a single period, that is, the transit timing variations (TTVs), can sometimes be attributed to perturbations by other exoplanets present in the system, which may or may not transit the star. Our aim is to compute the mass and the six orbital elements of an non-transiting exoplanet, given only the central times of transit of the transiting body. We also aim to recover the mass of the star and the mass and orbital elements of the transiting exoplanet, suitably modified in order to decrease the deviation between the observed and the computed transit times by as much as possible. We have applied our method, based on a genetic algorithm, to the Kepler-419 system. We were able to compute all fourteen free parameters of the system, which, when integrated in time, give transits within the observational errors. We also studied the dynamics and the long-term orbital evolution of the Kepler-419 planetary system as defined by the orbital elements computed by us, in order to determine its stability.