ExoNAMD: Leveraging the spin–orbit angle to constrain the dynamics of multi-planet systems

¹ Dipartimento di Fisica, La Sapienza Università di Roma, Piazzale Aldo Moro 5, Roma 00185, Italy
² INAF – Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125 Firenze, Italy
³ Astronomical Institute of the Czech Academy of Sciences, Fričova 298, 25165 Ondřejov, Czech Republic
⁴ INAF – Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese, Italy

^★ Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 16 September 2025
Accepted: 3 December 2025

Abstract

Multi-planet systems are excellent laboratories for studying the formation and evolution of exoplanets inside the same stellar environment. The number of known multi-planet systems is expected to skyrocket with the advent of PLATO and the Roman space telescope. The spin–orbit angle is a key context information for the systems’ dynamical history, and in recent years a growing number of planets had their spin–orbit angles measured, revealing a large diversity in orbital configurations, from well-aligned to polar, and even retrograde, orbits. Still, observers lack a robust tool with which to compare the dynamical state of different systems and to select the most suitable ones for future avenues of exploration, such as investigating the evolutionary pathways and their links to the atmospheric composition. Here, we present ExoNAMD, an open source code aimed at evaluating the dynamical state of multi-planet systems via the Normalized Angular Momentum Deficit (NAMD) metric. The NAMD measures the deficit in angular momentum with respect to circular, co-planar orbits. It is normalized to compare systems with different architectures and provides a lower limit on the past dynamical excitation of the system. We find that using the spin–orbit angle parameter in the NAMD calculation (A-NAMD) improves the dynamical state’s description, compared to using only the relative inclinations (R-NAMD). Comparison of A-NAMD and R-NAMD also yields powerful insights into the interplay between eccentricity and spin–orbit angle. ExoNAMD is a timely tool for easy and fast comparison of the myriad of exoplanetary systems to be discovered by PLATO and Roman and to optimize the target selection and scientific output for future atmospheric characterization using ELTs, JWST, and Ariel.