Exponential distance relation (aka Titius–Bode rule) in extra solar planetary systems

ABSTRACT In this paper we present phenomenological evidence for the validity of an exponential distance relation (also known as generalized Titius–Bode law) in the 32 planetary systems (31 extra solar, plus our Solar System) containing at least five planets each (known up to July 2023). We produce the semi-log fittings of the data, and we check them against the statistical indicators of $R^2$ and ${\rm Median}$. Then we compare them with the data of 4000 artificial planetary systems created at random. In this way, a possible origin by chance of the Titius–Bode rule (TBR) is reasonably excluded. We also point out that in some systems the fittings can be definitely improved by the insertion of new planets into specific positions. We discuss the harmonic resonances method and fittings, and compare them with the Titius–Bode fittings. Moreover, for some specific systems, we compare the Titius–Bode fitting against a polynomial fitting ($r\sim n^2$). Further comparisons with previous relevant works are reported in the last section. It emerges that TBR describes 25 out of the 32 planetary systems ($78{{\ \rm per\ cent}}$) with a $R^2\ge 0.95$. Further, it results to be the most economical (in terms of free parameters) and best-fitting law for the description of spacing among planetary orbits. This analysis allows us to conclude that an exponential distance relation can reasonably be considered as ‘valid’, or strongly corroborated, also in extra solar planetary systems.