Free Access
Volume 581, September 2015
Article Number A107
Number of page(s) 25
Section Interstellar and circumstellar matter
Published online 16 September 2015

Online material

Appendix A: Temperature-gradient models: the pole-on approximation

The assumption of the temperature-gradient model (Sect. 3) that disks are oriented pole-on is incorrect, at least for some disks. In this appendix, we comment further on this approximation and its possible effect on the resulting half-light radii.

The pole-on 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 pole-on approximation has generally been used for interferometric surveys with few observations per target (e.g., Monnier & Millan-Gabet 2002; Monnier et al. 2005).

Intrinsically, the pole-on approximation is only justified for pole-on 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 pole-on disk geometry for determining the half-light 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 parameters13 but with two different inclinations: i = 10° (nearly pole-on) and i = 60° (strongly inclined, close to the maximum for a non-obscured central object). For each of the two models, we calculated the half-light radius with the pole-on temperature-gradient 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 half-light radii are shown in Fig. A.1. First, the Monte Carlo simulation shows that even for this strong inclination difference, the median half-light radius for both distributions differs by only 10%. Second, the fit of the strongly inclined disks is slightly biased toward underestimating the half-light radii found for the (almost) pole-on disk, and the range of possible size estimates is 20–25% wider. These minor differences allow us to conclude that the mid-infrared half-light radius of a pole-on 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 pole-on 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 temperature-gradient model to include the disk inclination and position angle as fit parameters. We did this experiment for the above radiative-transfer model disk with i = 60°. In the first histogram in Fig. A.2, we see that the inferred half-light radius is much less constrained than under the pole-on 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 pole-on approximation) is not robust when the disk orientation is assumed to be free.

thumbnail Fig. A.1

Results of a Monte Carlo simulation for testing the influence of the pole-on approximation of the temperature-gradient models on inclined disks. The blue and red histograms show the distribution of (normalized) half-light radius estimates for a (almost) pole-on disk (i = 10°) and a strongly inclined disk (i = 60°), respectively. The median size estimates differ by 10%.

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thumbnail Fig. A.2

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

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A second option is to fix a non-zero inclination for the temperature-gradient model and determine half-light 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 half-light 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 half-light radius estimates will therefore be less precise than when a pole-on model is taken. The result is a less confined size estimate than for the pole-on approximation. The conceptually easier pole-on 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

Table B.1

Overview of observations.

© ESO, 2015

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