Question

The sun has a radiated power of 3.9 · 1026 W, a surface temperature of 5800 K and a radius of 6.96 · 108 m. Find out if the sun is radiating like a black body?

Answer 1

According to the Stefan–Boltzmann law, the radiant emittance (energy per unit surface per unit time) of a black body is given by:

$j_{black} = \sigma T^4$ ,

where $\sigma = 5.67\cdot 10^{-8} \dfrac{W}{m^2 K^4}$ is the Stefan–Boltzmann constant and $T$ is the temperature of the black body. For the black body with the temperature 5800 K the radiant emittance will be:

$j_{black} = \sigma T^4 =5.67\cdot 10^{-8} \dfrac{W}{m^2 K^4}\cdot 5800^4K^4 \approx 64.2 \dfrac{MW}{m^2 }$ .

On the other hand, by definition, the radiant emittance of the sun is:

$j_{sun} = \dfrac{P}{4\pi r^2} = \dfrac{3.9\cdot 10^{26}}{4\pi\cdot 6.96^2\cdot 10^{16}} \approx 64.1 \dfrac{MW}{m^2 }$.

Answer. Comparing $j_{black}$ and $j_{sun}$ one can conclude, that the Sun is radiating like a black body with rather good accuracy.