Study of the stability of Kepler-90 with hypothetical planets in stable regions

The Kepler-90 (K90) extrasolar system is a system that has some characteristics similar to our Solar System: eight planets in a hierarchical arrangement, that is, the innermost planets are terrestrial and the outermost are gas giant planets. The main difference between K90 and the Solar System is that the former is a compact system and the outermost planet orbits at approximately one astronomical unit.

   In this work, we study the possibility of the existence of a planet not yet detected in the regions between planets c and i, and i and d, which are regions of stability of K90 as shown by Gaslac et al. (2024) through the frequency map technique. In the first part of the analysis, we consider the entire region between planets c and i. In the study, we analyze the influence of a hypothetical planet on the confirmed planets in the system. To this end, we considered a hypothetical planet with a radius equal to 0.1 Earth radii and a density of 4.91 g/cm^3 in different positions in the aforementioned region (between c and i), resulting in one hundred and fifteen different positions. In this first analysis, the eccentricities of the confirmed and hypothetical planets are equal to zero, and the hypothetical planets closest to c or i collided, totaling seven collisions. Another analysis performed is the variation in eccentricity that the hypothetical planets can cause on the confirmed planets b, c, and i, and we neglect cases where the hypothetical planets excite eccentricities greater than 10^-2. It is worth noting that the eccentricities of the planets in the system are not confirmed in the literature; thus, Gaslac, in his 2021 study, concluded that the system is stable when the planets assume eccentricities of the order up to 10^-2. We also considered the radii of the hypothetical planets to be equal to 0.5 and 0.7 Earth radii. In both cases, the results are similar: the hypothetical planets closest to c and i collide (resulting in 7 and 6 collisions in each case, respectively), and, in the case of 0.7 Earth radii, more hypothetical planets close to the confirmed ones can cause a greater variation in eccentricity in b, c, and i.
   
   We also initially considered that the eccentricities of the planets (confirmed and hypothetical) were equal to 0.001, a value for which the system is stable. We kept the values ​​of the density of the hypothetical planets and their positions. We considered the radii to be equal to 0.1 and 0.5 Earth radii. For the first radius, no hypothetical planet suffered an eccentricity variation greater than 10^-2, despite having random collisions. For the second radius, the hypothetical planets closest to c or i induced greater variations in eccentricities; however, in both cases, most hypothetical planets maintain the eccentricity variations within the stability limit. Thus, we conclude that, under the given conditions, it is possible that the existence of a hypothetical planet in the region proves the system in such a way that it does not induce a variation in eccentricity in the confirmed planets of the system that would make it unstable. In the next step, we will analyze the possibility of the existence of more massive planets in this region and the breaking of resonance between b and c due to the presence of hypothetical planets. Furthermore, we will investigate a region, also stable, between planets i and d.