ner_tags: O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Tag O O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Observatory O B-Telescope O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O B-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O
tokens: x -axis in all the panels are the observation wavelength. The horizontal dashed line in the lower right panel shows where τ obs = 0.6. On the other hand, we caution that the DSHARP opacity adopted here is for compact spheres without porosity, and that the mixture of different compositions are handled with the Bruggeman rule. Changing composition or porosity can change the opacity and albedo dramatically, as alluded to in Birnstiel et al. ( 2018 ). For example, the dust becomes more reflective with more water ice. Porosity reduces the resonant opacity features, making the opacity curve smoother. Carbonaceous materials have a substantial effect on the dust opacity. The maximum absorption feature can be shifted to much longer wavelengths than 2 πs max with some choices of carbonaceous materials. On the other hand, the peak of albedo is still at 2 πs max with different choices of carbonaceous materials (Birnstiel et al. 2018 ), making both χ and α measurements (Section 3.3 ) less sensitive to the choices of carbonaceous materials. More detailed calculations exploring different dust compositions, porosities, and size distributions are needed in future. 3.3. Dust Properties and the Spectral Index <italic>α</italic> Just based on the DSHARP opacity, we want to explore how we can constrain the dust size distributions using the measured χ and the spectral index α . The spectral index α iswhere the approximation on the right is with the Rayleigh–Jeans approximation. Clearly, the measured α depends on totally different quantities in the optically thick and thin regimes. In the more familiar optically thin regime, it depends on how the absorption coefficient changes with the frequency. With κ abs ∝ ν β , α is β + 2. In the optically thick regime, α depends on χ . Assuming that χ ∝ ν γ , α is γ + 2. As shown in Figure 1 , ω changes monotonically with 1/ χ that changes as λ γ . Thus, if the albedo ω increases with wavelength ( γ > 0), α = γ + 2 measured at this wavelength span will be larger than 2, and vice versa. To be more specific, if the disk is optically thick, we can use the measured α from observations to derive γ . Then we can constrain the relationship between ω ν 1 and usingwhich is based on Equation ( 13 ). Figure 9 shows how κ abs , χ and the spectral index α change with different dust populations for different ALMA and VLA bands, using the DSHARP opacity. The panels in the second row show χ in the optically thick limit. The third row shows α in the optically thin limit (thus denoted as α thin ), while the bottom row is α in the optically thick limit ( α thick ). Here, we simply use β + 2 or γ + 2 to calculate α , and thus do not consider the deviation of the blackbody radiation from the Rayleigh–Jeans limit (Huang et al. 2018a ). χ and both α s have sharp transitions around s max ∼ λ /2 π . When s ≪ λ /2 π , scattering is not important and χ = 1, α thick = 2, α thin ∼ 4. When s ≳ λ /2 π , χ becomes less than 1, α thick becomes larger than 2,