Optical-depth rescaling at fixed temperature ($\tilde{B}(\tilde{\tau })=B(\tilde{\tau }/a)$)

Multiplying an absorber abundance by $a$ with fixed cross sections and physical temperature profile multiplies optical depth by $a$. The Planck function as a function of the new coordinate must therefore be composed with division by $a$. A term proportional to ${\tau }^n$ acquires coefficient $a^{-n}$. Holding its old optical-depth coefficient fixed would instead change the physical temperature profile. This is the immediate radiative comparison before thermal adjustment.