On the extension of the Laplace-Lagrange secular theory to order two in the masses for extrasolar systems

TL;DR

This paper extends the Laplace-Lagrange secular theory to second order in masses for exoplanetary systems, improving accuracy near resonances and accounting for mean anomaly influences, with a practical classification criterion.

Contribution

The authors develop a second-order analytical approach to secular evolution, enhancing the accuracy of predictions especially near mean-motion resonances in exoplanetary systems.

Findings

Second-order theory outperforms first-order near resonances.

The approach accounts for mean anomaly effects on secular dynamics.

A classification criterion for planetary systems based on resonance proximity.

Abstract

We study the secular evolution of several exoplanetary systems by extending the Laplace-Lagrange theory to order two in the masses. Using an expansion of the Hamiltonian in the Poincar\'e canonical variables, we determine the fundamental frequencies of the motion and compute analytically the long-term evolution of the keplerian elements. Our study clearly shows that, for systems close to a mean-motion resonance, the second order approximation describes their secular evolution more accurately than the usually adopted first order one. Moreover, this approach takes into account the influence of the mean anomalies on the secular dynamics. Finally, we set up a simple criterion that is useful to discriminate between three different categories of planetary systems: (i) {\it secular} systems (HD 11964, HD 74156, HD 134987, HD 163607, HD 12661 and HD 147018); (ii) systems {\it near a mean-motion resonance} (HD 11506, HD 177830, HD 9446, HD 169830 and $\upsilon$ Andromedae); (iii) systems {\it really close to} or {\it in a mean-motion resonance} (HD 108874, HD 128311 and HD 183263).