Explanation:
In order to solve this problem, the Stefan-Boltzmann law will be useful. This law establishes that a black body (an ideal body that absorbs or emits all the radiation that incides on it) "emits thermal radiation with a total hemispheric emissive power proportional to the fourth power of its temperature":
[tex]P=\sigma A T^{4}[/tex] (1)
Where:
[tex]P[/tex] is the energy radiated by a blackbody radiator per second, per unit area (in Watts).
[tex]\sigma=5.6703(10)^{-18}\frac{W}{m^{2} K^{4}}[/tex] is the Stefan-Boltzmann's constant.
[tex]A[/tex] is the Surface of the body
[tex]T[/tex] is the effective temperature of the body (its surface absolute temperature) in Kelvin.
However, there is no ideal black body (although the radiation of stars like our Sun is quite close). Therefore, we will use the Stefan-Boltzmann law for real radiator bodies:
[tex]P=\sigma A \epsilon T^{4}[/tex] (2)
Where [tex]\epsilon[/tex] is the star's emissivity
Knowing this, let's start with the answer:
We have two stars where the emissivity [tex]\epsilon[/tex] of both is the same:
Star A with a radius [tex]r_{A}=200000km[/tex] and a surface temperature [tex]T_{A}=6000K[/tex].
Star B with a radius [tex]r_{B}=400000km[/tex] and a surface temperature [tex]T_{B}=3000K[/tex].
And we are asked to find the ratio of the rate of energy radiated by both stars:
[tex]\frac{P_{A}}{P_{B}}[/tex] (3)
Where [tex]P_{A}[/tex] is the rate of energy radiated by Star A and [tex]P_{B}[/tex] is the rate of energy radiated by Star B.
On the other hand, with the radius of each star we can calculate their surface area, using the formula for tha area of a sphere (assuming both stars have spherical shape):
[tex]A_{A}=4\pi r_{A}^{2}[/tex] (4)
[tex]A_{B}=4\pi r_{B}^{2}[/tex] (5)
Writting the Stefan-Boltzmann law for each star, taking into consideration their areas:
[tex]P_{A}=\sigma (4\pi r_{A}^{2}) \epsilon {T_{A}}^{4}[/tex] (6)
[tex]P_{B}=\sigma (4\pi r_{B}^{2}) \epsilon {T_{b}}^{4}[/tex] (7)
Substituting (6) and (7) in (3):
[tex]\frac{\sigma (4\pi r_{A}^{2}) \epsilon {T_{A}}^{4}}{\sigma (4\pi r_{B}^{2}) \epsilon {T_{B}}^{4}}[/tex] (8)
[tex]\frac{P_{A}}{P_{B}}=\frac{{r_{A}}^{2} {T_{A}}^{4}}{{r_{B}}^{2} {T_{B}}^{4}}[/tex] (9)
[tex]\frac{P_{A}}{P_{B}}=\frac{{(200000km)}^{2} {(6000K)}^{4}}{{(400000km)}^{2} {(3000K)}^{4}}[/tex] (10)
Finally:
[tex]\frac{P_{A}}{P_{B}}=4[/tex]
