Secular Dynamics of S-type Planetary Orbits in Binary Star Systems: Applicability Domains of First- and Second-Order Theories

TL;DR

This paper evaluates the accuracy and applicability of first- and second-order analytical models for the secular dynamics of S-type planets in tight binary systems, comparing them with full N-body simulations to guide model selection.

Contribution

It systematically assesses the validity domains of first- and second-order theories for planetary orbits in binary systems, providing practical guidelines for their use.

Findings

First-order models are simple but limited to small perturbations.

Second-order models have broader applicability but are more complex.

Neither model is sufficient in extreme cases, requiring higher-order theories or simulations.

Abstract

We analyse the secular dynamics of planets on S-type coplanar orbits in tight binary systems, based on first- and second-order analytical models, and compare their predictions with full N-body simulations. The perturbation parameter adopted for the development of these models depends on the masses of the stars and on the semimajor axis ratio between the planet and the binary. We show that each model has both advantages and limitations. While the first-order analytical model is algebraically simple and easy to implement, it is only applicable in regions of the parameter space where the perturbations are sufficiently small. The second-order model, although more complex, has a larger range of validity and must be taken into account for dynamical studies of some real exoplanetary systems such as $\gamma$-Cephei and HD 41004A. However, in some extreme cases, neither of these analytical models yields quantitatively correct results, requiring either higher-order theories or direct numerical simulations. Finally, we determine the limits of applicability of each analytical model in the parameter space of the system, giving an important visual aid to decode which secular theory should be adopted for any given planetary system in a close binary.