Underestimation of the tidal force and apsidal motion in close binary systems by the perturbative approach: Comparisons with non-perturbative models

¹ STAR Institute, University of Liège, 19C Allée du Six-Août, 4000 Liège, Belgium
e-mail: lfellay@uliege.be
² Department of Physics, KTH Royal Institute of Technology, The Oskar Klein Centre, AlbaNova, 106 91 Stockholm, Sweden

Received: 2 October 2023
Accepted: 23 December 2023

Abstract

Context. Stellar deformations play a significant role in the dynamical evolution of stars in binary systems, impacting the tidal dissipation and the outcomes of mass transfer processes. The prevalent method for modelling the deformations and tidal interactions of celestial bodies solely relies on the perturbative approach, which assumes that stellar deformations are minor perturbations to the spherical symmetry. An observable consequence of stellar deformations is the apsidal motion in eccentric systems, which has be observationally determined across numerous binary systems.

Aims. Our objective is to assert the reliability of the perturbative approach when applied to close and strongly deformed binary systems.

Methods. We have developed a non-perturbative 3D modelling method designed to account for high stellar deformations. We focus on comparing the properties of perturbatively deformed stellar models with our 3D models, particularly in terms of apsidal motion.

Results. Our research highlights that the perturbative model becomes imprecise and underestimates the tidal force and rate of apsidal motion at a short orbital separation. This discrepancy primarily results from the first-order treatment in the perturbative approach, and cannot be rectified using straightforward mathematical corrections due to the strong non-linearity and numerous parameters of the problem. We have determined that our methodology affects the modelling of approximately 42% of observed binary systems with measured apsidal motion, introducing a discrepancy greater than 2% when the normalised orbital separation verifies q^−1/5a(1 − e²)/R₁ ≲ 6.5 (q is the mass ratio of the system, a is its semi-major axis, e is its orbital eccentricity and R₁ is the radius of the primary star).

Conclusions. The perturbative approach underestimates tidal interactions between bodies up to ∼40% for close low-mass binaries. All the subsequent modelling is impacted by our findings, in particular, the tidal dissipation is significantly underestimated. As a result, all binary stellar models are imprecise when applied to systems with a low orbital separation, and the outcomes of these models are also affected by these inaccuracies.