Revisiting the picture of circumbinary disc truncation

Circumbinary discs are observed to develop central cavities carved by the gravitational influence of the binary. Analytical estimates of cavity sizes predict truncation at $\sim 2 \textrm{--} 3$ binary separations, depending on the binary properties. However, numerical studies show only qualitative agreement with these predictions: cavity sizes often evolve on long timescales and can exceed substantially the analytically predicted values. In this work, we revise this paradigm, suggesting that tidal truncation in circumbinary discs responds to additional dynamical parameters that have so far been neglected. We analyse a suite of 80 numerical simulations of circumbinary discs to re-examine the physical mechanism responsible for cavity truncation and to provide a prescription for the cavity size independent of the state of evolution of the system. We find that truncation depends not only on the binary parameters $a_{\rm bin}$, $e_{\rm bin}$, and mass ratio $q$, but also on the instantaneous cavity eccentricity $e_{\rm cav}$ and the relative apsidal orientation $\varpi_{\rm bin}-\varpi_{\rm cav}$. These quantities jointly determine the pericentre of the innermost stable disc orbit $R_{\rm p}$, in a way that shares some similarities with orbital stability in the restricted three body problem. Hydrodynamical effects introduce secondary corrections, with the disc scale height $H$ and viscosity $\alpha$ mildly shifting the cavity edge relative to the purely gravitational prediction. We introduce a semi-analytical prescription that captures these dependences for $R_{\rm p}$ and cavity semi-major axis $a_{\rm cav}$. We conclude that cavity truncation for binaries with mass ratios $q>0.05$ is a process where the instantaneous orbital properties of the disc ($e_{\rm cav}$, $\varpi_{\rm cav}$) play a fundamental role and should be taken into account to accurately evaluate the truncation efficiency.