Millimetre spectral indices of transition disks and their relation to the cavity radius

Context. Transition disks are protoplanetary disks with inner depleted dust cavities that are excellent candidates for investigating the dust evolution when there is a pressure bump. A pressure bump at the outer edge of the cavity allows dust grains from the outer regions to stop their rapid inward migration towards the star and to efficiently grow to millimetre sizes. Dynamical interactions with planet(s) have been one of the most exciting theories to explain the clearing of the inner disk. Aims. We look for evidence of millimetre dust particles in transition disks by measuring their spectral index alpha(mm), with new and available photometric data. We investigate the influence of the size of the dust depleted cavity on the disk integrated millimetre spectral index. Methods. We present the 3-mm (100 GHz) photometric observations carried out with the Plateau de Bure Interferometer of four transition disks: LkH alpha 330, UX Tau A, LRLL 31, and LRLL 67. We used the available values of their fluxes at 345 GHz to calculate their spectral index, as well as the spectral index for a sample of twenty transition disks. We compared the observations with two kinds of models. In the first set of models, we considered coagulation and fragmentation of dust in a disk in which a cavity is formed by a massive planet located at different positions. The second set of models assumes disks with truncated inner parts at different radii and with power-law dust-size distributions, where the maximum size of grains is calculated considering turbulence as the source of destructive collisions. Results. We show that the integrated spectral index is higher for transition disks (TD) than for regular protoplanetary disks (PD) with mean values of (alpha) over bar (TD)(mm) = 2.70 +/- 0.13 and (alpha) over bar (PD)(mm) = 2.20 +/- 0.07 respectively. For transition disks, the probability that the measured spectral index is positively correlated with the cavity radius is 95%. High angular resolution imaging of transition disks is needed to distinguish between the dust trapping scenario and the truncated disk case.