Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 315 / 2 / b / ii / Solution

At $20\mu \mathrm{m}$,The normalized flux weights of the $400$, $600$, and $800\mathrm{K}$ layers are thereforeUsing the Planck law at the band centre and $R_p/R_{\ast }=0.0354$ givesThis is only slightly larger than the $71.7\mathrm{ppm}$ isothermal $600\mathrm{K}$ value because the cool upper-layer suppression and hot deep-layer enhancement nearly cancel in this broad weighting.