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Appendix A: Temperaturegradient models: the poleon approximation
The assumption of the temperaturegradient model (Sect. 3) that disks are oriented poleon is incorrect, at least for some disks. In this appendix, we comment further on this approximation and its possible effect on the resulting halflight radii.
The poleon approximation is equivalent to assuming that the inclination i is zero, an orientation for which the disk’s position angle PA is not defined. For the interferometric observation, this orientation has the advantage that the model is independent of the baseline angle. Indeed, the relative angle between the disk’s position angle and the baseline angle is what generally plays the role in defining the model orientation. For this reason, the poleon approximation has generally been used for interferometric surveys with few observations per target (e.g., Monnier & MillanGabet 2002; Monnier et al. 2005).
Intrinsically, the poleon approximation is only justified for poleon or mildly inclined (e.g., i ≲ 20°) disks. However, a significant number of the disks in our sample will have a stronger inclination. To justify the use of a poleon disk geometry for determining the halflight radius of these disks, we perform the following simulation. We take two of the radiative transfer models of Sect. 6.3 with the same stellar/disk parameters^{13} but with two different inclinations: i = 10° (nearly poleon) and i = 60° (strongly inclined, close to the maximum for a nonobscured central object). For each of the two models, we calculated the halflight radius with the poleon temperaturegradient model, for a random set of five interferometric observations (i.e., five UV points). This experiment was repeated 500 times, and histograms of the determined halflight radii are shown in Fig. A.1. First, the Monte Carlo simulation shows that even for this strong inclination difference, the median halflight radius for both distributions differs by only 10%. Second, the fit of the strongly inclined disks is slightly biased toward underestimating the halflight radii found for the (almost) poleon disk, and the range of possible size estimates is 20–25% wider. These minor differences allow us to conclude that the midinfrared halflight radius of a poleon temperature gradient model is a robust parameter, even for disks that are strongly inclined. The conclusions based in Fig. 6 (for which the vertical axis is on a logarithmic scale) are thus unaffected by this approximation.
Two alternatives for this poleon approximation can be considered, for which we show below that they provide less robust or less confined results. First, it is obviously possible to extend the fit of the temperaturegradient model to include the disk inclination and position angle as fit parameters. We did this experiment for the above radiativetransfer model disk with i = 60°. In the first histogram in Fig. A.2, we see that the inferred halflight radius is much less constrained than under the poleon approximation in Fig. A.1. The two other histograms show the inferred inclination and position angle, neither of which are well constrained. It is clear that the originally robust size parameter (under the poleon approximation) is not robust when the disk orientation is assumed to be free.
Fig. A.1
Results of a Monte Carlo simulation for testing the influence of the poleon approximation of the temperaturegradient models on inclined disks. The blue and red histograms show the distribution of (normalized) halflight radius estimates for a (almost) poleon disk (i = 10°) and a strongly inclined disk (i = 60°), respectively. The median size estimates differ by 10%. 

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Fig. A.2
Results for the same Monte Carlo simulation as in Fig. A.1 (for the radiativetransfer disk with i = 60°), but with a temperaturegradient model that also includes the inclination i and the position angle PA as free parameters. Clearly, neither the halflight radius, nor i and PA are well constrained. The radiativetransfer disk has cosi = 0.5 (i.e., i = 60°) and PA = 0°/ 180°. 

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A second option is to fix a nonzero inclination for the temperaturegradient model and determine halflight radii with this inclined geometric model. To avoid biases related to the
unknown position angle, the applied model needs to be fit at the full range of position angles (PA = 0° to 180°/ 360°). The result of such a fit is a range of halflight radii (for the varying position angles) rather than a single value. Part of this size range will come from models that are oriented perpendicularly to the actual disk orientation. For strongly inclined disks, these halflight radius estimates will therefore be less precise than when a poleon model is taken. The result is a less confined size estimate than for the poleon approximation. The conceptually easier poleon approximation, which we have shown to be robust (even when disks are strongly inclined), was therefore the preferred approach in this work.
Appendix B: Overview of MIDI observations
Overview of observations.
© ESO, 2015