Self-calibration is essential to measuring the intrinsic source size, since the original phase noise on the longest baselines is relatively high. To illustrate this problem, we also used a data set in which only the most compact configurations (for which the correlated flux is well above the noise) have been self-calibrated. The most affected disk parameters are given in Table A.1 for fully and partially self-calibrated data. The corresponding (derotated and deprojected) visibilities are shown in Fig. A.1 (to be compared with Fig. 3). The coherence loss for uv distances above 300 m is quite significant, allowing only a lower limit to the disk sizes to be derived for the more compact ones. Self-calibration has no strong impact on the derived disk orientations, and only a small one (due to seeing effects) on the inclination. As shown in Table A.1, the recovered flux is slightly larger, and the sources are more compact, requiring higher surface densities to reproduce the observations.
Deprojected visibility profiles for the sources on a common relative scale, with best-fit profiles superimposed. Partially self-calibrated version. Red is for the power-law fit, blue for the viscous profile.
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Effect of phase self-calibration.
The use of the covariance matrix to estimate the errorbars is only well suited to the most resolved sources. For the more compact ones, errorbars on several parameters are necessarily asymmetric. The upper bound on Rout is determined by the visibility curves, while the lower bound is set by the necessity of providing a sufficient total flux. The value of Rout is thus well constrained even for these very compact sources. The inclination of the very compact sources is, of course, loosely defined, as it can only be constrained by measuring different sizes for different orientations of the baselines. However, this only has a moderate impact on Rout: as the contribution of an optically thick core is in general dominant, is constant to first order, unless the disk becomes nearly edge-on.
To avoid any substantial bias on the derivation of Rout and its errorbars for small sources, we explored a two-parameter χ2 surface as a function of (Rout,i), fitting position, orientation, surface density and emissivity index. The results are presented in Fig. B.1. These figures show the expected behavior for Rout. The errors quoted in Table 2 are in general conservative: the lower bound on Rout is sharply constrained due to the flux requirement (see Eq. (6)).
In practically all cases, the optically thick solution (Σ > 50 g cm-2) is within 2σ of the best fit value. We also point out that given the unavoidable phase errors even after self-calibration, the residual seeing effects increase the best fit radius. The quoted best fit surface densities are thus lower limit in this respect, since only noise is included in the errorbars.
χ2 contours as a function of Rout and i for all compact sources (red contours, 1 to 5σ by increasing width), overlaid on the surface density values in gray scale. The surface density contours are 5, 10, 20, 40, 80 and 160 g cm-2 (of dust+gas, assuming a gas-to-dust ratio of 100). The source name is indicated in each panel.
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© ESO, 2014