The $\mu$ Herculis system solved after nearly three centuries

Abstract

$\mu$ Herculis is a bright, nearby quadruple system. Its brightest member, $\mu$ Her Aa, displays solar-like oscillations, establishing the system as a crucial benchmark for asteroseismology, provided that its mass can be determined independently of stellar models. We aim to resolve the full hierarchical architecture of the system and determine precise, model-independent dynamical masses for all four components (Aa, Ab, B, and C), along with a consistent astrometric solution for the system's centre of mass. We performed a joint fit of radial velocities, relative astrometry and absolute astrometry from \textit{Hipparcos}, \textit{Gaia} DR3, and ground-based catalogues, spanning nearly three centuries. Our forward-modelling framework simultaneously constrains the Keplerian orbits of the inner Aa--Ab and B--C subsystems, the wide A--BC orbit, and the sky motion and parallax of the total centre of mass. Leveraging several complementary datasets and the decisive 2023 periastron passage of the Aa--Ab pair, we precisely determine all orbital parameters and obtain sub-percent precision on the component masses: $M_{\rm Aa} = 1.134 \pm 0.007\,M_{\odot}$, $M_{\rm Ab} = 0.2286 \pm 0.0006\,M_{\odot}$, $M_{\rm C} = 0.445 \pm 0.005\,M_{\odot}$, and $M_{\rm B} = 0.417 \pm 0.005\,M_{\odot}$. We derive a system parallax of $\varpi_{\rm CM} = 120.069 \pm 0.089\,\mathrm{mas}$ that reconciles and improves upon the individual \textit{Hipparcos} and \textit{Gaia} DR3 values.