The GRAVITY young stellar object survey VII. The inner dusty disks of T Tauri stars

Context. T Tauri stars are surrounded by dust and gas disks. As material reservoirs from which matter is accreted onto the central star and planets are built, these protoplanetary disks play a central role in star and planet formation. Aims. We aim at spatially resolving at sub-astronomical unit (sub-au) scales the innermost regions of the protoplanetary disks around a sample of T Tauri stars to better understand their morphology and composition. Methods. Thanks to the sensitivity and the better spatial frequency coverage of the GRAVITY instrument of the Very Large Telescope Interferometer, we extended our homogeneous data set of 27 Herbig stars and collected near-infrared K-band interferometric observations of 17 T Tauri stars, spanning effective temperatures and luminosities in the ranges of similar to 4000-6000 K and similar to 0.4-10 L-circle dot, respectively. We focus on the continuum emission and develop semi-physical geometrical models to fit the interferometric data and search for trends between the properties of the disk and the central star. Results. As for those of their more massive counterparts, the Herbig Ae/Be stars, the best-fit models of the inner rim of the T Tauri disks correspond to wide rings. The GRAVITY measurements extend the radius-luminosity relation toward the smallest luminosities (0.4-10 L-circle dot). As observed previously, in this range of luminosities, the R proportional to L-1/2 trend line is no longer valid, and the K-band sizes measured with GRAVITY appear to be larger than the predicted sizes derived from sublimation radius computation. We do not see a clear correlation between the K-band half-flux radius and the mass accretion rate onto the central star. Besides, having magnetic truncation radii in agreement with the K-band GRAVITY sizes would require magnetic fields as strong as a few kG, which should have been detected, suggesting that accretion is not the main process governing the location of the half-flux radius of the inner dusty disk. The GRAVITY measurements agree with models that take into account the scattered light, which could be as important as thermal emission in the K band for these cool stars. The N-to-K band size ratio may be a proxy for disentangling disks with silicate features in emission from disks with weak and/or in absorption silicate features (i.e., disks with depleted inner regions and/or with large gaps). The GRAVITY data also provide inclinations and position angles of the inner disks. When compared to those of the outer disks derived from ALMA images of nine objects of our sample, we detect clear misalignments between both disks for four objects. Conclusions. The combination of improved data quality with a significant and homogeneous sample of young stellar objects allows us to revisit the pioneering works done on the protoplanetary disks by K-band interferometry and to test inner disk physics such as the inner rim morphology and location