Using the formulas introduced in the previous section, you can determine both the flux and the luminosity produced by the specified surface.
To begin, calculate the flux:
$F\=\mathrm{\σ}\cdot {T}^{4}$
$F\=5.67\times {10}^{-8}\frac{W}{{\mathrm{K}}^{4}{\mathrm{m}}^{2}}{\left(1000K\right)}^{4}$
$F\=56700\mathrm{W}sol;{\mathrm{m}}^{2}$.
You can now use this result to determine the luminosity:
$L\=4\cdot \mathrm{\π}\cdot {R}^{2}\cdot F$
$L\=4\cdot \mathrm{\π}\cdot {\left(5\times {10}^{8}\mathrm{m}\right)}^{2}\cdot \left(56700\mathrm{W}sol;{\mathrm{m}}^{2}\right)$
$\mathrm{L}\=1.781\times {10}^{23}\mathrm{W}$.
Therefore, a surface with a radius of $5\times {10}^{8}\mathrm{m}$ and a temperature of $1000\mathrm{K}$ has a radiant intensity of $56700\mathrm{W}sol;{\mathrm{m}}^{2}$ and a luminosity of $1.781\times {10}^{23}\mathrm{W}$.