Question

The intensity of a sunspot is found to be 3 times smaller than the intensity emitted by the solar surface. What is the approximate temperature of this sunspot if the temperature of the solar surfaceis 5800 K?
a. 4400 K
b. 470,000 K
c. 1900 K
d. 7600 K
e. 1400 K

Answer 1

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 $\frac{W}{m^{2}}$) of a black body is proportional to the fourth power of the temperature T of the body:

                                            $I=\sigma T^{4}$

with $\sigma=5.67x10^{-8} Wm^{-2} K^{-4}$ 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 $I_{sunspot}$ is found to be 3 times smaller than the intensity emitted by the solar surface $I_{surface}$, that means that:

                                   $I_{sunspot}=\frac{I_{surface}}{3}$

then using the expression of Stefan-Boltzmann law we get that

                                   $\sigma T_{sunspot} ^{4}=\sigma T_{surface} ^{4}$

we cross out $\sigma$ and use the fourth root in each side of the equation

                                   $\sqrt[4]{T_{sunspot} ^{4}}=\frac{\sqrt[4]{T_{surface} ^{4}}}{\sqrt[4]{3}}$

                                   $T_{sunspot}=\frac{T_{surface}}{\sqrt[4]{3} }$

then we use that

• $T_{surface}=5800 K$

• $\sqrt[4]{3}\approx1,316$

                                 

                                   $T_{sunspot}=\frac{5800 K}{1,316}$

                                   $T_{sunspot}=4407,3 K$

So finally we get that

                                   $T_{sunspot}\approx4400 K$