 Solution

Write ${\tau }_0={\tau }_{\infty }$, $n=2a$, and $t={\tau }_0(p/p_s{)}^2$. The integrated Planck function is $B(T)=\sigma T^4/\pi$ by the Stefan–Boltzmann law, andThe formal radiative transfer equation solution for upward radiance at the top isThe first term is transmitted surface emission; the second is emission from the atmosphere. Integrating radiance over the upward hemisphere gives the outgoing radiant flux per unit area,Here $E_j$ is an exponential integral. For fixed $a$ and ${\tau }_0\to \infty$, surface transmission is negligible and the upper limit can be extended to infinity. Positivity permits interchanging the integrals; putting $t=\mu x$ givesConsequently the exact hemispheric optically thick result isThis is a dimensional scale times a dimensionless definite integral depending only on $a$. In the commonly used vertical-ray approximation, define $F_{\mathrm{v}}=\pi I(0,1)$. ThenA vertical-ray approximation therefore has the same scaling but omits the angular factor $1/(a+1)$. Stating which quantity is used prevents a purely angular normalization discrepancy.