Question

If a star \(A\) has radiant power three times that of the Sun and a temperature of \(6000\,K\) while the Sun has temperature \(2000\,K\), what is the ratio of their radii \((R_A : R_{Sun})\)?

• A. \(1:\sqrt{27}\)
• B. \(\sqrt{27}:1\)
• C. \(1:9\)
• D. \(1:27\)

Hint:

From Stefan–Boltzmann law \(P \propto R^2T^4\). After solving for \(R^2\), always take the square root to obtain the ratio of radii.

Answer 1

The Correct Option is A

Concept: The total radiant power emitted by a star is given by the Stefan–Boltzmann law: \[ P = \sigma A T^4 \] where \(P\) = radiant power, \(\sigma\) = Stefan–Boltzmann constant, \(A = 4\pi R^2\) = surface area of the star, \(T\) = absolute temperature. Substituting the surface area: \[ P = 4\pi \sigma R^2 T^4 \] Thus, \[ P \propto R^2 T^4 \]

Step 1: Write the ratio of powers.
\[ \frac{P_A}{P_{Sun}} = \frac{R_A^2 T_A^4}{R_{Sun}^2 T_{Sun}^4} \] Given: \[ \frac{P_A}{P_{Sun}} = 3 \] \[ T_A = 6000K, \quad T_{Sun} = 2000K \]

Step 2: Calculate the temperature ratio.
\[ \left(\frac{T_A}{T_{Sun}}\right)^4 = \left(\frac{6000}{2000}\right)^4 \] \[ = 3^4 = 81 \]

Step 3: Substitute into the power relation.
\[ 3 = \frac{R_A^2}{R_{Sun}^2} \times 81 \] \[ \frac{R_A^2}{R_{Sun}^2} = \frac{3}{81} \] \[ \frac{R_A^2}{R_{Sun}^2} = \frac{1}{27} \]

Step 4: Take square root to obtain radius ratio.
\[ \frac{R_A}{R_{Sun}} = \sqrt{\frac{1}{27}} \] \[ \frac{R_A}{R_{Sun}} = \frac{1}{\sqrt{27}} \]

Step 5: Final result.
\[ R_A : R_{Sun} = 1 : \sqrt{27} \]