The dynamics of the planetary system of Kepler-90

The planetary system of Kepler-90 has eight planets, b, c, i, d, e, f, g and h, in increasing distance from the star. The outer planet has an orbital distance equals to 1 AU. Planets g and h are similar to gas giants, planets d, e, and f are similar to super-Earths and planets b and c are about to 2 Earth radii. Small planets are closer to the star while the larger ones are distant from the star. Numerical simulations performed by Cabrera et al (2014) and Granados et at (2018) showed that some of these planets are in mean motion resonances(MMR) between them. For example, planets b and c are in 4:5 MMR. Through the frequency map analysis and long term evolution, we analyse their stability and the region surrounding them for a sample of parameters of the planets, such as their masses, semi-major axes, and eccentricities. We analyse three different intervals of eccentricity: the first interval is from 0 to 10^-3, the second interval from 10^-3 to 10^-2 and the third interval is from 10^-2 to 10^-1. These values of eccentricity were randomly chosen. Our results showed that the planets which have the eccentricities derived from the first two intervals are stable, while most of the planets with larger eccentricity, ranging from 10^-2to 10^-1, are ejected from the system. We also generated a set of test particles in a digram a versus e disturbed by the planets. This system was numerically simulated for 10⁴ T_h (T_h the orbital period of planet h). We identified three stable regions: stable region 1, between the planets c and i, is located from 0.10AU to 0.194AU for values of the eccentricity from 0 to 0.3; region 2, between the planets i and d, located from 0.215AU to 0.281AU for 0 < e < 0.1; and the region 3, between planets d and e, from 0.32AU to 0.39AU and 0 < e < 0.06. Numerical simulations also showed that some test particles are close to 4:5, 5:6, 7:8, and 10:11 MMR with planet i.