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 a bin ( d / a bin ). Then, we applied a disc truncation model to calculate the likelihood that each companion carves a cavity with a size a cav relative to its semi-major axis a bin and projected separation d , deriving distributions of a bin / a cav and d / a cav . Results . We find that stellar companions carve cavities with sizes a cav with a median about three times larger than their projected separation d ( a cav ∼ 3 d, and a cav ∼ 1.7 d times for planets), but with a statistically significant tail (∼50%) towards higher values ( a cav ≫ 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 a cav ). 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.