Breaking the Mass Inclination Degeneracy of Radial Velocity Measurements via Monitoring von Zeipel–Lidov–Kozai Cycles: Implications in the HD 41004 System

Abstract

We investigate the dynamical stability of the S-type planet in the compact binary HD 41004. Using N-body simulations, we find that the planet could be dynamically stable at a mutual inclination angle of up to ∼75°. The von Zeipel–Lidov–Kozai (vZLK) mechanism becomes active when the mutual inclination is greater than 39 2. High-inclination orbits exhibit coupled oscillations in eccentricity and inclination, along with apsidal precession. Synthetic radial velocity (RV) modeling shows that these secular variations produce measurable signatures across a broad range of timescales, from full vZLK cycles to observationally accessible decades. For instance, a high mutual inclination at 75° can induce RV drifts exceeding 5 m s−1 per planetary orbit (∼1.9 m s−1 yr−1) in circular binary configurations. The presence of eccentric vZLK further accelerates these drifts, enhancing the detectability. Long-term RV observations of this system offer a unique pathway to dynamically constrain the orbital inclination and thus determine the true mass of HD 41004 Ab. The degeneracy of mass inclination is well known when using RV measurements alone. Our results highlight that HD 41004 Ab and potentially other S-type planets in compact binaries are promising targets for breaking such a degeneracy by studying the dynamics induced by the vZLK mechanism through long-term high-precision RV monitoring.