Question

The radiation emitted by a star A is 10000 times that of the Sun. If the surface temperature of the Sun and star A are 6000 K and 2000 K respectively, what is the ratio of the radii of the star A and the sun?
(A) $300:1$
(B) $600:1$
(C) $900:1$
(D) $1200:1$

Answer 1

Hint
The temperature of a star other than the sun can be determined by using a simple variation of the Stefan’s Law. According to the Stefan’s Law, the energy emitted by a star is directly proportional to the fourth power of its temperature.
$\Rightarrow E = \sigma A{T^4}$, where $E$ is the energy emitted, $A$is the area of the star, $T$is the effective temperature of the star, and $\sigma$is the Stefan-Boltzmann constant.

Complete step by step answer
We are provided with the energy or radiation emission of a star and our Sun. We are asked to find the ratio of their radii. We know that the energy of a star is given by:
$\Rightarrow E = \sigma A{T^4}$
We are given the following information about the two stars:
Temperature of star A: $T = 2000$K
Temperature of the Sun: ${T_S} = 6000$K
Energy of the star A: $E = 10000{E_S}$ where ${E_S}$is the energy of the Sun
When we take the two energy terms on one side:
$\Rightarrow \dfrac{E}{{{E_S}}} = 10000$
Substituting the formula for energy, we get:
$\Rightarrow \dfrac{{\sigma A{T^4}}}{{\sigma {A_S}{T_S}^4}} = 10000$
The area will consist of a squared radius term, and the constant$\sigma$ cancels out to give us:
$\Rightarrow \dfrac{{{R^2}{T^4}}}{{R_S^2{T_S}^4}} = 10000$
Now, to simplify this equation, we take the temperatures on the RHS, and leave the unknown radii on the LHS:
$\Rightarrow \dfrac{{{R^2}}}{{R_S^2}} = 10000\dfrac{{{T_S}^4}}{{{T^4}}} = {\left( {10\dfrac{{{T_S}}}{T}} \right)^4}$
We cancel the powers in the exponent, to get:
$\Rightarrow \dfrac{R}{{{R_S}}} = {\left( {10\dfrac{{{T_S}}}{T}} \right)^2}$
Putting the value of temperature in the above equation:
$\Rightarrow \dfrac{R}{{{R_S}}} = {\left( {10\dfrac{{6000}}{{2000}}} \right)^2} = {\left( {10 \times 3} \right)^2} = 900$
This gives the ratio as $900:1$. Hence, the answer is option (C).

Note
The original Stefan-Boltzmann Law gives a relationship between the energy emitted by a star per unit surface area and the temperature of the star. Since in this question we were not provided with the energy per unit surface area, we multiply an area term to the original equation to get our desired equation.