Question

Sunspots appear dark, although their temperatures are typical
5000 K, because the rest of the suns surface is even horer,
about 5800 K. Compare the radiation rates of surfaces of the
same emissivity whose temperatures are respectively 5000 usd
5800 K​

Answer 1

According to the Stefan–Boltzmann law, the radiation rates are given by the eqation:

$$j = \sigma\varepsilon T^4$$

where $\sigma$ is the constant, $\varepsilon$ is the emissivity, $T$ is a temperature. The ration of the radiation rates for the different temperatures will be:

$$\dfrac{j_1}{j_2} = \dfrac{\sigma\varepsilon T_1^4 }{\sigma\varepsilon T_2^4 }= \dfrac{ T_1^4 }{ T_2^4 } = \dfrac{5000^4 }{5800^4 }=0.55$$

Thus, the radiation rates of the sunspot is 0.55 less then the radiation rate of the rest of the Sun's surface.