(b) Analytical solution
Using the formula of the radiative exchange between gray diffuse surfaces (i to j):
\(\Sigma_{j=1}^N\left[\frac{\delta_{ij}}{\epsilon_j}-\left(\frac{1-\epsilon_j}{\epsilon_j}\right)F_{i-j}\right]q'_j+H_{oi}=\Sigma_{j=1}^N\left(\delta_{ij}-F_{i-j}\right)E_{bj},\ i=1...N\)
with \(\delta_{ij}\) the kroneker delta, \(\epsilon\) the emissivity, \(F\) the view factor, \(E_b\) the emissivity
Taking into advantage that there is no reflection (\(H=0\)), and that there is only heat flux between the convex sphere (1) and the plane (2); the surrounding (3) beeing non-paricipating the two equation can be solved for the heat fluxes:
\(q'_1=\frac{\frac{1}{\epsilon_2}\left(\sigma T_1^4-F_{1-2}\sigma T_2^4\right)+F_{1-2}\ \frac{1-\epsilon_2}{\epsilon_2}\left(-F_{2-1}\sigma T_1^4+\sigma T_2^4\right)}{\frac{1}{\epsilon_1\epsilon_2}-F_{2-1}\ \frac{1-\epsilon_1}{\epsilon_1}F_{1-2}\frac{1-\epsilon_2}{\epsilon_2}}\)
\(q'_2=\frac{\frac{1}{\epsilon_1}\left(\sigma T_1^4-F_{2-1}\sigma T_1^4\right)+F_{2-1}\ \frac{1-\epsilon_1}{\epsilon_1}\left(-F_{1-2}\sigma T_2^4+\sigma T_1^4\right)}{\frac{1}{\epsilon_1\epsilon_2}-F_{2-1}\ \frac{1-\epsilon_1}{\epsilon_1}F_{1-2}\frac{1-\epsilon_2}{\epsilon_2}}\)