# On the 3D secular dynamics of radial-velocity-detected planetary systems

## TL;DR

This study analyzes the long-term stability and spatial configurations of RV-detected multi-planet systems using analytical and numerical methods, revealing stability conditions related to mutual inclinations and Lidov-Kozai resonance effects.

## Contribution

It introduces an analytical approach combined with chaos detection to constrain the inclinations and stability of non-resonant planetary systems, validated by n-body simulations.

## Key findings

- Long-term stability at low mutual inclinations (<35°).
- High mutual inclinations often lead to chaos and destabilization.
- Lidov-Kozai resonance influences the stability and extent of stable regions.

## Abstract

Aims. To date, more than 600 multi-planetary systems have been discovered. Due to the limitations of the detection methods, our knowledge of the systems is usually far from complete. In particular, for planetary systems discovered with the radial velocity (RV) technique, the inclinations of the orbital planes, and thus the mutual inclinations and planetary masses, are unknown. Our work aims to constrain the spatial configuration of several RV-detected extrasolar systems that are not in a mean-motion resonance. Methods. Through an analytical study based on a first-order secular Hamiltonian expansion and numerical explorations performed with a chaos detector, we identified ranges of values for the orbital inclinations and the mutual inclinations, which ensure the long-term stability of the system. Our results were validated by comparison with n-body simulations, showing the accuracy of our analytical approach up to high mutual inclinations (approx. 70{\deg}-80{\deg}). Results. We find that, given the current estimations for the parameters of the selected systems, long-term regular evolution of the spatial configurations is observed, for all the systems, i) at low mutual inclinations (typically less than 35{\deg}) and ii) at higher mutual inclinations, preferentially if the system is in a Lidov-Kozai resonance. Indeed, a rapid destabilisation of highly mutually inclined orbits is commonly observed, due to the significant chaos that develops around the stability islands of the Lidov-Kozai resonance. The extent of the Lidov-Kozai resonant region is discussed for ten planetary systems (HD 11506, HD 12661, HD 134987, HD 142, HD 154857, HD 164922, HD 169830, HD 207832, HD 4732, and HD 74156).