Optical depth in our models and comparison between observed and simulated spectral indices. Left and central panels: Azimuthally averaged radial profiles of the optical depth along the line of sight and the intensities from the full-resolution dust emission image of the simulation (i.e., not convoluted with the synthesized beam) with a final planet mass of 2.0 M_Jup after 4 × 10⁴ yr, in Band 7 and Band 4. The vertical dash-dotted brown line indicates the location of the planet, and the horizontal dotted gray line marks the optical depth τ = 1 level. The region where the emission is optically thick (optical depth τ ≥ 1) is shown in beige. Right panel: azimuthally averaged radial profiles of the spectral index from the ALMA observation and our simulation with a final planet mass of 2.0 M_Jup after 4 × 10⁴ yr. Values are obtained using Eq. (5) and the synthetic image intensity profiles in Fig. 5. Uncertainties include the original error in the intensity profiles and a 10% flux error due to calibration. As a reference, the distance of the planet from the central star is indicated by the vertical dash-dotted brown line, and the position of the peak intensity is displayed by the vertical dotted blue line. The beige-shaded area marks the radii where the full-resolution simulated emission is optically thick (τ ≥ 1) in both Band 7 and Band 4.