The likelihood of not detecting cavity-carving companions in transition discs : A statistical approach

Context. Protoplanetary discs with cavities, also known as transition discs, constitute ∼10% of the protoplanetary discs at submillimmeter wavelengths. As one of several explanations, one hypothesis suggests that these cavities are carved by undetected stellar or planetary companions. Aims. We present a novel approach for quantifying the likelihood that a companion that carves the cavity in a transition disc is not detected because it is too close to the central star (small projected separation) or too faint to be resolved. Methods. We generated two independent samples of stellar and planetary companions that were randomly oriented in the sky. We assumed a distribution of their eccentricity, mass ratio, and time-weighted orbital phases to study the statistical properties of the cavities they carve. We first calculated the likelihood that each companion in these samples appears at a certain projected separation d relative to its semi-major axis abin (d/abin). Then, we applied a disc truncation model to calculate the likelihood that each companion carves a cavity with a size acav relative to its semi-major axis abin and projected separation d, deriving distributions of abin/acav and d/acav. Results. We find that stellar companions carve cavities with sizes acav with a median about three times larger than their projected separation d (acav ∼ 3 d, and acav ∼ 1.7 d times for planets), but with a statistically significant tail (∼50%) towards higher values (acav ≫ 3 d). With this information, we estimated the likelihood that cavity-carving companions remain undetected because of projection effects when the system is observed with a spatial resolution R, P(d < R acav). Conclusions. Using observational constraints on companion masses, we applied this framework to 13 well-known transition discs. We conclude that an undetected stellar companion is unlikely in 8 out of the 13 systems we considered, with 5 notable exceptions: AB Aur, MWC 758, HD 135344B, CQ Tau, and HD 169142. A planet, on the other hand, may have remained undetected in any of the transition discs we considered.