Question

The radiation emitted by a star A is 10,000 times that of the sun. If the surface temperature of the sun and star A are 6000K and 2000K, respectively, the ratio of the radii of the star A and the sun is
A) 300:1
B) 600:1
C) 900:1
D) 1200:1

Answer 1

In this solution, we will use the Stefan-Boltzmann law which tells us that the power radiated by a black body will be proportional to the fourth power of the temperature for the black body. The stars can be considered as a black body.

Formula used: In this question, we will use the following formula:
$P = \sigma A{T^4}$ where $P$ is the power radiated by a blackbody at temperature $T$ and $\sigma$ is the Stefan-Boltzmann constant and $A$ is the area of the radiating area.

Complete step by step answer
We’ve been told that the radiation emitted by a star A is 10,000 times that of the sun and the temperature of the sun and star A are 6000K and 2000K. According to the Stefan-Boltzmann law,
$P = \sigma A{T^4}$
Now assuming the shape of the star as a sphere, the area of the star will be $A = 4\pi {R^2}$. Taking the ratio of the power emitted for star A and the sun, we get
$\dfrac{{{P_A}}}{{{P_S}}} = \dfrac{{{r_A}^2{T_A}^4}}{{{r_S}^2{T_S}^4}}$
Now, we’ve been told $\dfrac{{{P_A}}}{{{P_S}}} = \dfrac{{10000}}{1}$ and the temperature of the star A is ${T_A} = 2000\,K$ and the temperature of the sun is ${T_S} = 6000\,K$
On substituting the values we get,
$\dfrac{{10000}}{1} = \dfrac{{{r_A}^2}}{{{r_S}^2}} \times \dfrac{{{{\left( {2000} \right)}^2}}}{{{{\left( {6000} \right)}^2}}}$
$\Rightarrow \dfrac{{{r_A}^2}}{{{r_S}^2}} = \dfrac{{270000}}{1}$
Which gives us,

$\dfrac{{{r_A}}}{{{r_S}}} = \dfrac{{900}}{1}$ which corresponds to option (C).

Note
We have assumed that the stars have a perfect blackbody which means that they do not absorb any light and emit completely. However, stars are not perfect blackbodies as the gases around the stars absorb a lot of radiation from the star resulting in different spectral lines observed by us. The colour of the star is determined by the temperature of the star. Hotter stars have a smaller wavelength while cooler stars have a larger wavelength.