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Relationship between optical thickness and spectral gradient for all multi-thermal numerical models is shown in black. The simple, isothermal expression from Eq. (4), without the correction for α factor is shown in red. The orange region shows the corrected relationship from Eq. (7), where α is evaluated for ALMA Band 3 at temperatures between 103 and 106 K. The dashed-blue line shows τ = 1.
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