Quantifying the similarity of planetary system architectures

The planetary systems detected so far exhibit a wide diversity of architectures, and various methods have been proposed to quantitatively study this diversity. Straightforward ways to quantify the difference between two systems, and more generally two sets of multi-planetary systems, are helpful for studying this diversity. In this work we present a novel approach, using a weighted extension of the energy distance (WED) metric, to quantify the difference between planetary systems on the logarithmic period-radius plane. We demonstrate the use of this metric and its relation to previously introduced descriptive measures to characterise the arrangements of Kepler planetary systems. By applying exploratory machine-learning tools, we attempt to find whether there is some order that can be ascribed to the set of multi-planet Kepler system architectures. Based on the WED, the 'Sequencer', which is such an automatic tool, identifies a progression from small and compact planetary systems to systems with distant giant planets. It is reassuring to see that a WED-based tool does indeed identify this progression. Next, we extend the WED to define the inter-catalogue energy distance-a distance metric between sets of multi-planetary systems. We have made the specific implementation presented in the paper available to the community through a public repository. We suggest using these metrics as complementary tools in attempts to compare different architectures of planetary systems and, in general, different catalogues of planetary systems.