EDP Sciences
Free Access
Volume 520, September-October 2010
Article Number A39
Number of page(s) 23
Section Interstellar and circumstellar matter
DOI https://doi.org/10.1051/0004-6361/200913909
Published online 27 September 2010

Online Material

Appendix A: Validation

A.1 Procedure validation

We evaluate the robustness of our procedure by fitting synthetic spectra with known abundances, temperatures and continua. This allows to check whether these quantities could be recovered when the spectra are processed with the B2C procedure. To this end, we used the best fit, synthetic spectra for the 58 objects analyzed in this paper (and presented in Sect. 3), as fake, but representative observations for serving as inputs to our B2C procedure. We reconstructed these synthetic spectra using the outputs of the fitting process on the original data. Namely, the same continuum emission, the same relative abundances for the dust species and their corresponding temperatures. The uncertainties chosen for the synthetic spectra are those of the original observed spectra.

Figure A.1 shows the ratio between the output and input crystallinity, as a function of the ratio between the output and input grain sizes, for both the warm and cold components. The plot shows ratios of about 1 on both axis, especially for the cold component (blue squares on the figure), suggesting some best fits obtained with the procedure may not be unique. But from a statistical point view, the B2C procedure produces reproducible results as the mean ratios between output and input quantities remain close to 1 (vertical and horizontal bars on the figure).

More precisely, the warm component crystallinity is slightly overestimated by 22% with a dispersion (i.e. the standard deviation) about this value of 30%. The inferred cold component crystallinity is satisfactorily reproduced at the 2% level with respect to the input value, and the dispersion, 35%, is rather similar to that for the crystallinity of the warm grains. The mean mass-averaged size of the warm grains is slightly underestimated by $7 \pm 11$% with respect to the input value, while the mean mass-averaged size of the cold grains is well reproduced but with a larger dispersion ($1 \pm 32$%).

A closer look at the dispersion of the calculated crystalline fraction as a function of the input crystallinity is shown in Fig. A.2 for the 58 fits to synthetic spectra, for both the warm (red open circles) and cold (blue open squares) components. The over-estimation of the warm component crystalline fractions seen in Fig. A.1 is mostly visible for objects with low-crystalline fractions (below 20%). For the cold component, it actually tends to be rather underestimated for large crystalline fractions (above $\sim$40%). The overall trend for both the warm and cold components is that the dispersion raises with the decreasing crystallinity. This finds an explanation in the fact that objects showing high crystalline fractions display strong, high-contrast crystalline features that are less ambiguously matched by theoretical opacities than objects with lower crystalline fractions.

\end{figure} Figure A.1:

Results for the fits to the 58 synthetic spectra to test our ``B2C'' procedure. The x-axis shows the dispersion in grain size (ratio of inferred over input mean mass-averaged grain sizes) and the y-axis the ratio between inferred and input crystallinity. Red open circles correspond to the warm component, and blue open squares to the cold component. The filled circles are mean values, with error bars corresponding to standard deviations.

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\end{figure} Figure A.2:

Ratios between output over input crystallinity fractions as a function of the input crystallinity fractions, for the fits to the 58 synthetic spectra. Red open circles are results for the warm component and blue open squares are for the cold component.

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A.2 Influence of the continuum on the cold component

\par\mbox{\includegraphics[angle=0,width=9cm,origin=bl]{13909_f26.ps}\includegraphics[angle=0,width=9cm,origin=bl]{13909_f27.ps} }
\end{figure} Figure A.3:

Left panel: crystalline fraction for the cold component as a function of the integrated continuum substracted flux in the range 22-35 $\mu $m (see text for details). Right panel: same for the cold mean mass-averaged grain size.

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The continuum estimation is a critical step when modeling spectra, especially for the cold component, and may contribute to the larger uncertainties on the inferred cold component crystallinities and sizes discussed in previous subsection. The continuum may affect the model outputs in the following way: a high-flux continuum in the 20-30$~\mu$m spectral range leaves very little flux to be fitted under the spectrum, possibly leading to a composition with few large and/or featureless grains. A low-flux continuum could, on the other hand, lean toward large/featureless grains to fill the flux left. We therefore examine if (whether) such trends are present in the results of our B2C procedure (or not) for the cold component.

We quantify the level of continuum by integrating the continuum-substracted spectra between 22 $\mu $m and 35 $\mu $m (x-axis in Fig. A.3). Low-flux continua have high integrated fluxes and are located on the right side of the plot in Fig. A.3, the high-flux continua being on the left side. The left panel of Fig. A.3 shows the crystalline fraction for the cold component, as a function of the integrated flux left once the continuum is substracted. The trend low continuum - low crystalline fraction is visible for the highest integrated flux values. But the large dispersion for low integrated flux values indicates that no strong bias is introduced as we obtain cases with very high-flux continua but still with very low crystalline fractions. It remains that for the low-flux continua cases, we cannot possibly obtain very high crystalline fractions as a lot of flux needs to be filled to match the spectra. Indeed, amorphous grains are the best choice to fulfill this requirement, therefore diminishing the cold component crystalline fraction.

The influence of the continuum on the estimated mass-averaged grain size is shown on the right panel of Fig. A.3. A very weak and dispersed anti-correlation is found between the two parameters. We obtain a $\tau$ value of -0.12 with a significance probability P = 0.17, meaning that the adopted shape for the continuum is not strongly influencing the inferred mean grain size. The trend is mostly caused by a few low-flux continua objects modeled with large grains.

To conclude, the continuum estimation is a challenging problem for the cold component and its adopted shape will always have an impact on the results for the crystallinity and grain size at the same time. Still, we have checked that, statistically speaking, we are not introducing a strong and systematic bias with our simple, two free parameter continuum.

A.3 Importance of silica and necessity for large grains

\vspace*{-3mm} \end{figure} Figure A.4:

Change in reduced $\chi ^2_{{\rm r}}$values for fits with and without silica ( right panel) and fits with two grain sizes (0.1 and 1.5 $\mu $m, left panel). The dashed line corresponds to y = x. The $\langle \Delta
\chi^2_{{\rm r}}\rangle$ values correspond to the mean difference of reduced $\chi ^2_{{\rm r}}$ between x-axis and y-axis simulations.

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Usually, Mg-rich silicates are considered for the dust mineralogy in proto-planetary disks (e.g. Henning & Meeus 2009). But both Olofsson et al. (2009) and Sargent et al. (2009) attribute some features in IRS spectra of young stars to silica (composition SiO2). To gauge the importance of silica in our B2C compositional approach, we run the B2C model with and without silica. Many fits were improved adding silica in the dust population (as shown for one example in Fig. A.5, the 20-22 $\mu $m range being the spectral range where the improvement is the more noticeable). As can be seen on the right panel of Fig. A.4, showing the reduced $\chi ^2_{{\rm r}}$ for simulations with and without silica, many fits are improved when silica is included, demonstrating the non-negligible importance of silica for our B2C model.

We have also critically examined the need for large, amorphous, 6.0 $\mu $m-sized grains in the B2C model as they are almost featureless contrary to the 0.1 and 1.5 $\mu $m-sized grains. We therefore run the B2C procedure on the 58 same objects with only 0.1 and 1.5 $\mu $m-sized grains. Left panel of Fig. A.4 shows that the mean value of the reduced $\chi ^2_{{\rm r}}$ for all the simulations is augmented by 2.50 when using only two grain sizes, showing that the majority of the fits are improved using three grain sizes instead of two.

\end{figure} Figure A.5:

Blowup on the fit to the ROX43A spectrum, performed with silica (red dot-dashed line) and without (blue dashed line).

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One can also worry about the influence of the offset $O_{\nu , 2}$values on the inferred grain sizes. As a large offset value leads to larger flux below the continuum-substracted spectrum, it may indeed favor larger grains. We therefore examined if there was any correlation between the fraction of large 6.0 $\mu $m-sized grains and the $O_{\nu , 2}$ values, and we quantified these relations with correlation coefficients. Considering the warm large grains and the $O_{\nu , 2}$ offsets, we find a $\tau$ value of 0.04 with a significance probability of P = 0.69. For the cold large grain fraction and the $O_{\nu , 2}$ values, we obtain $\tau = -0.03$ and P = 0.70. This means that there is no significant influence of the offsets $O_{\nu , 2}$ on the inferred grain sizes.

Table A.1:   Dust composition derived using the ``B2C'' procedure for 58 objects from our sample.

Table A.2:   Dust temperatures, continuum parameters (blackbody temperature and offset $O_{\nu , 2}$) derived using the ``B2C'' procedure for 58 objects from our sample.

...e_26.ps}\includegraphics[width=4cm,origin=bl]{spectra/spe_27.ps} }\end{figure} Figure A.6:

Fits to the 58 objects using the B2C procedure.

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...e_54.ps}\includegraphics[width=4cm,origin=bl]{spectra/spe_55.ps} }\end{figure} Figure A.6:


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\par\mbox{\includegraphics[width=4cm,origin=bl]{spectra/spe_56.ps}\includegraphics[width=4cm,origin=bl]{spectra/spe_57.ps} }\end{figure} Figure A.6:


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