Answer:
If the temperature of the solar surface is 5800 K then the approximate temperature of the sunspot is a) 4400 K.
Explanation:
The most straightforward way to solve this is using Stefan-Boltzmann law that states that I the energy radiated per unit surface area per unit time (watt per unit area [tex]\frac{W}{m^{2}}[/tex]) of a black body is proportional to the fourth power of the temperature T of the body:
[tex]I=\sigma T^{4}[/tex]
with [tex]\sigma=5.67x10^{-8} Wm^{-2} K^{-4}[/tex] being the Stefan constant.
A black body is an idealized physical body that is a perfect absorber because it absorbs all incident electromagnetic radiation and is also an ideal emitter. The Sun is considered to be a black body at different layers and different temperatures.
We are told that the intensity of a sunspot [tex]I_{sunspot}[/tex] is found to be 3 times smaller than the intensity emitted by the solar surface [tex]I_{surface}[/tex], that means that:
[tex]I_{sunspot}=\frac{I_{surface}}{3}[/tex]
then using the expression of Stefan-Boltzmann law we get that
[tex]\sigma T_{sunspot} ^{4}=\sigma T_{surface} ^{4}[/tex]
we cross out [tex]\sigma[/tex] and use the fourth root in each side of the equation
[tex]\sqrt[4]{T_{sunspot} ^{4}}=\frac{\sqrt[4]{T_{surface} ^{4}}}{\sqrt[4]{3}}[/tex]
[tex]T_{sunspot}=\frac{T_{surface}}{\sqrt[4]{3} }[/tex]
then we use that
[tex]T_{sunspot}=\frac{5800 K}{1,316}[/tex]
[tex]T_{sunspot}=4407,3 K[/tex]
So finally we get that
[tex]T_{sunspot}\approx4400 K[/tex]
