Constraints on the height of the inner disk rim in pre-main-sequence stars

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The structure of inner region of protoplanetary disks around young pre-main-sequence stars is still poorly understood. This part of the disk is shaped by various forces that influence dust and gas dynamics, and by dust sublimation, which creates abrupt drops in the dust density. This region also emits strong near-infrared excess that cannot be explained by classical accretion disk models, which suggests the existence of some unusual dust distribution or disk shape. The most prevalent explanation to date is the puffed-up inner disk rim model, where the disk exhibits an optically thin cavity around the star up to the distance of dust sublimation. The critical parameter in this model is the inner disk rim height $z_{\rm max}$ relative to the rim distance from the star $R_{\rm in}$. Observations often require $z_{\rm max}/R_{\rm in}\gtrsim0.2$ to reproduce the near-infrared excess in the spectra. We compile a comprehensive list of processes that can shape the inner disk rim and combine them into a self-consistent model. Two of them, radiation pressure force and the gas velocity profile, have never been applied in this context before. The aim was to find the most plausible theoretical values of $z_{\rm max}/R_{\rm in}$. The results show that this value is $\lesssim$0.13 for Herbig Ae stars, $\lesssim$0.11 for T Tau stars, and $\lesssim$0.10 for young brown dwarfs. This is lower than the observational requirements for Herbig Ae stars. We argue that the same problem exists in T Tau stars as well. We conclude that the puffed-up inner rim model cannot be the sole explanation for the near-infrared excess in young pre-main-sequence stars.