Circumbinary and circumstellar discs around the eccentric binary IRAS 04158+2805 -- a testbed for binary-disc interaction

IRAS~04158+2805 has long been thought to be a very low mass T-Tauri star (VLMS) surrounded by a nearly edge-on, extremely large disc. Recent observations revealed that this source hosts a binary surrounded by an extended circumbinary disc with a central dust cavity. In this paper, we combine ALMA multi-wavelength observations of continuum and $^{12}$CO line emission, with H$\alpha$ imaging and Keck astrometric measures of the binary to develop a coherent dynamical model of this system. The system features an azimuthal asymmetry detected at the western edge of the cavity in Band~7 observations and a wiggling outflow. Dust emission in ALMA Band 4 from the proximity of the individual stars suggests the presence of marginally resolved circumstellar discs. We estimate the binary orbital parameters from the measured arc of the orbit from Keck and ALMA astrometry. We further constrain these estimates using considerations from binary-disc interaction theory. We finally perform three SPH gas + dust simulations based on the theoretical constraints; we post-process the hydrodynamic output using radiative transfer Monte Carlo methods and directly compare the models with observations. Our results suggest that a highly eccentric $e\sim 0.5\textrm{--}0.7$ equal mass binary, with a semi-major axis of $\sim 55$ au, and small/moderate orbital plane vs. circumbinary disc inclination $\theta\lesssim 30^\circ$ provides a good match with observations. A dust mass of $\sim 1.5\times 10^{-4} {\rm M_\odot}$ best reproduces the flux in Band 7 continuum observations. Synthetic CO line emission maps qualitatively capture both the emission from the central region and the non-Keplerian nature of the gas motion in the binary proximity.