ExoNAMD: Leveraging the spin-orbit angle to constrain the dynamics of multiplanetary systems

TL;DR

This work addresses the need for a robust, cross-system dynamical-state diagnostic for multiplanetary systems by introducing ExoNAMD, an open-source tool that computes both $R\text{-}NAMD$ and the newly proposed $A\text{-}NAMD$, which incorporates the spin-orbit angle $\\psi$ to capture true 3-D architectures. $NAMD$ is defined as $NAMD = (\sum_k m_k \sqrt{a_k} (1 - \sqrt{1 - e_k^2} \cos i_k)) / (\sum_k m_k \sqrt{a_k})$, and $A\text{-}NAMD$ as $(1/2)(\sum_k m_k \sqrt{a_k} (1 - \sqrt{1 - e_k^2} \cos \psi_k)) / (\sum_k m_k \sqrt{a_k})$, with uncertainties assessed via Monte Carlo sampling. The authors demonstrate the diagnostic power of combining $A\text{-}NAMD$ with $R\text{-}NAMD$ in a four-quadrants diagram, highlighting cases like K2-290 and Kepler-462 where the true dynamical history is clarified by including the 3-D spin-orbit information. The framework supports target selection and interpretation for upcoming atmospheric characterization efforts (JWST, Ariel) and paves the way for integrating dynamical context into large surveys from PLATO and Roman, ultimately linking dynamical histories to atmospheric composition across multiplanetary systems.

Abstract

Multiplanetary systems are excellent laboratories for studying the formation and evolution of exoplanets inside the same stellar environment. The number of known multiplanetary 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 angle measured, revealing a large diversity in orbital configurations, from well-aligned to polar, and even retrograde, orbits. Still, observers lack a robust tool 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 multiplanetary 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.

Figures (5)

• Figure 1: R-NAMD vs. multiplicity plot for the core sample described in the text. Same figure as Fig. 3 in T20 but with an expanded sample. The horizontal offsets around the multiplicity values are put arbitrarily for better visualization. The color scale, capped to 1, represents the relative uncertainty on the R-NAMD values, obtained from the arithmetic mean of the lower and upper errors from the Monte Carlo. Systems with relative uncertainty above unity are plotted with a white hollow circle. The linear fit is shown only as a visual aid as the slope is highly dependent on the (few) high multiplicity systems.
• Figure 2: A-NAMD vs. multiplicity plot. The shown systems are selected from the core sample on the basis of the availability of spin-orbit angle measurements, as described in the text. More systems are needed to confirm and quantify the sketched anti-correlation between A-NAMD and multiplicity. The linear fit is shown only as a visual aid.
• Figure 3: Comparison of NAMD values for the K2-290 system. The system consists of two planets on highly misaligned orbits. Left: R-NAMD as defined in T20 using relative inclinations. Middle: $\lambda$-NAMD, using projected spin-orbit angle ($\lambda$). Right: A-NAMD as defined in this work using the spin-orbit angle ($\psi$). The A-NAMD is the most suitable to capture the true architecture of the system as it is not affected by projection effects.
• Figure 4: Four-quadrants diagram. This figure illustrates the possible states in which planetary systems can be found depending on their dynamical history which is reflected in their eccentricities and spin-orbit angles. The Solar System is located at the intersection of the two gray dashed lines and separates the four quadrants, as discussed in the text. As an illustration, we include a value (red line) of a Solar System with planets misaligned by 15 degrees. The vertical highlighted area marks instead the R-NAMD values that can be associated to excited but stable systems that did not undergo large-scale instabilities in the simulations by rick23. In addition, we have included several synthetic systems based on real ones for which some parameters were missing (green points). These were inputted in order to better populate the quadrants.
• Figure 5: Comparison of NAMD values for the Kepler-462 system. The system consists of two planets on eccentric and misaligned orbits. Left: R-NAMD as defined in T20 using relative inclinations. Middle: $\lambda$-NAMD, using projected spin-orbit angle ($\lambda$). Right: A-NAMD as defined in this work using the spin-orbit angle ($\psi$). The A-NAMD metric is the most suitable to capture the true architecture of the system.