Scatter plot of and fitted dust temperature for the modified black-body fit. Two alternative estimates of AV are shown (see text), plus an average value. The horizontal grey lines show and the 15% uncertainty in the AV normalisation (Sect. 2.1). All points have similar relative uncertainty; the bars for the lowest Umin are shown. The uncertainties are systematic, so they affect each point in the same way; errors on the two axes are strongly anti-correlated. The Umin colour scheme is the same as Fig. 2; the filled symbols correspond to the SEDs shown in Fig. 1.
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Figure A.1 shows two different estimates of AV as a function of temperature, obtained from a modified black-body fit to the observed SEDs. Planck Collaboration XI (2014) provides two different empirical relations for computing the dust reddening E(B − V) from the emission. One relation considers E(B − V) to be proportional to the 850 μm dust optical depth (E(B − V) /τ0 = 1.49 × 104), the other relation considers E(B − V) proportional to the radiance (E(B − V) / ℛ = 5.4 × 105). We calculate E(B − V) in both ways, and then we convert the values to AV using the average diffuse ISM value for RV = E(B − V) /AV = 3.1 and compare the results.
The two estimates differ both by their average value and their trend with temperature: the AV obtained from τ0 is about 20% too low on average and decreases with temperature like the physical models (Fig. 2); the AV obtained from ℛ has a good average value but increases with temperature. By way of comparison we also show the geometric average of the two. This average matches the expected value better, despite having no physical justification. Interestingly, the two estimates implicitly make opposite assumptions: the AV obtained from τ0 assumes that the dust optical properties are fixed, or at least τ0/AV is fixed; and the AV from ℛ assumes a fixed G0, or at least a fixed absorbed power per grain10. This means that cold dust is more emissive than expected from models with fixed dust properties, but less emissive than expected from models where variable optical properties account for all observed variations.
The size distribution of dust grains has a strong effect on extinction, but it is not expected to affect the far-infrared opacity, which only depends on the total volume of the grains in the Rayleigh regime. It is natural, therefore, to consider variations in grain size distribution as a way of varying τFIR/AV. Assessing the effect of grain size variation in a physically realistic way is not straightforward: the physical processes that change grain sizes, e.g. shattering, sputtering, accretion, coagulation, also affect its structure and composition; also, the optical properties of the materials themselves may be size-dependent. Simply varying the grain size distribution in a model is therefore not likely to mimic the actual variations in the ISM, but can still provide interesting qualitative insights. In this Appendix we explore the effects of varying the size distribution using the C11 model. While this model does not fit the average Iλ/AV as well as J13, it is still close enough to be useful for a differential analysis. Its homogeneous grains and constant optical properties allow us to modify the grain size distribution independent of optical properties.
In C11, grains larger than ~10 nm are distributed according to a power law – n(a) ∝ aα, where a is the grain radius – with an exponential cutoff above ~150 nm. The parameter that mainly controls the size distribution is the exponent of the power law, α, which is −2.8 for carbonaceous grains and –3.4 for silicate grains. We repeat the procedure of Sect. 3.2 varying α by −0.5 and + 0.5 around its standard value. This changes RV by −0.7 and + 1.0, respectively; by comparison, Fitzpatrick & Massa (2007) give ~0.3 as the typical 1σ dispersion of RV in the diffuse ISM.
Effect of the grain size distribution on Iλ/ ℛ, as predicted by the C11 model. The SEDs shown are the coldest (blue dot-dashed line), the median (green solid line) and the warmest (red dotted line); normalisation is the same as per Fig. 4. The corresponding observations are plotted in grey behind the models. Larger symbols indicate larger average grain size, and smaller symbols indicate smaller average grain size (see text for details).
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The results are shown in Fig. B.1. Varying the size distribution has a small impact on the dust SED, and the range of temperatures reproduced is smaller than that observed despite the large, possibly overestimated, span in α. The figure also shows that models with smaller grains are, surprisingly, colder than models with larger grains, i.e. they have lower 100 μm emission and higher long-wavelength emission. This can be explained simply. In our modelling, the radiation field intensity is not fixed, but is rather derived from the observed radiance per unit extinction ℛ /AV. Changing the size distribution modifies our estimate for G0, so that models with smaller grains necessitate a weaker radiation field to satisfy those constraints. The decrease in G0 thus offsets the temperature increase due to size effects, and even reverses it in the case of the C11 model.
The details of the result presented in this appendix are likely to depend on the dust model and parametrization used. Still, varying the grain size distribution without its corresponding change in the dust optical properties is not likely to explain the observed variations of Iλ/AV. In Ysard et al. (2015) a similar study uses the J13 model, which is instead adapted to reproduce the interplay of grain size and optical properties.
© ESO, 2015