Interferometric Survey of Stellar Parameters: Mass of the metallic A-type binary $\beta$ Aur
With the capabilities of the new visible CHARA/SPICA instrument and the multiple spectral band operation of CHARA, our goal is to resolve orbits of short-period binaries and develop a robust framework for combining interferometric, spectroscopic, and photometric observations into a single consistent model. For our target sample, we selected suitable binaries based on brightness, angular separation, and orbital properties based on the expected performance of the CHARA/SPICA instrument. As a case study, we analysed the bright eclipsing binary $\beta$ Aurigae, composed of two slightly evolved A1 stars. We combined new interferometric observations of $\beta$ Aur obtained with CHARA/SPICA, MIRC-X, and MYSTIC with archival MIRC data, radial velocities, and light curves. We first derived astrometric positions from interferometric observables and computed an orbital solution. Afterwards, we implemented a unified model, capable of tying interferometric modelling with the ellc code to simultaneously fit all observables using MCMC sampling. We performed a detailed analysis of the noise statistics of each data set and in the end we adopted a profile likelihood approach to account for underestimated noise and systematics. We derived a consistent orbital and physical solution for $\beta$ Aur through joint modelling. The inclusion of interferometric data tightly constrains the angular semi-major axis and inclination. Using profile likelihood to account for the different intrinsic levels of uncertainty of the fundamentally different observables, we derived the masses of the two stars, $M_1 = 2.359 \pm 0.005$ M$_S$ and $M_2 = 2.293 \pm 0.004$ M$_S$, their radii $R_1 = 2.752 \pm 0.002$ R$_S$ and $R_2 = 2.622 \pm 0.002$ R$_S$, and the distance to the binary, $d = 24.30\pm0.05$ pc.