Question

A star 'A' has radiant power equal to 3 times that of the Sun. The temperature of star 'A' is \( 6000 \, \mathrm{K} \) and that of the Sun is \( 2000 \, \mathrm{K} \). What is the ratio of their radii?

• A. \( 900:1 \)
• B. \( 81:1 \)
• C. \( 729:1 \)
• D. \( 27:1 \)

Hint:

N/A

Answer 1

The Correct Option is A

The radiant energy \( E \) emitted by a star is given by: \[ E = \sigma \epsilon A T^4. \] For spherical stars: \[ A \propto R^2, \] where \( R \) is the radius. Hence: \[ E \propto R^2 T^4. \] Step 1: Write the energy ratio.
For two stars, the ratio of radiant energies is: \[ \frac{E_1}{E_2} = \frac{R_1^2 T_1^4}{R_2^2 T_2^4}. \] Given: \[ E_1 = 10000 \times E_2. \] Substitute: \[ \frac{10000 \cdot E_2}{E_2} = \frac{R_1^2 \cdot T_1^4}{R_2^2 \cdot T_2^4}. \] Simplify: \[ 10000 = \frac{R_1^2 (2000)^4}{R_2^2 (6000)^4}. \] Step 2: Simplify the temperature ratio.
The ratio of temperatures is: \[ \frac{T_1}{T_2} = \frac{2000}{6000} = \frac{1}{3}. \] Raise to the power of 4: \[ \left( \frac{1}{3} \right)^4 = \frac{1}{81}. \] Substitute back: \[ 10000 = \frac{R_1^2}{R_2^2} \cdot \frac{1}{81}. \] Multiply through by 81: \[ \frac{R_1^2}{R_2^2} = 10000 \cdot 81 = 900^2. \] Step 3: Take the square root.
\[ \frac{R_1}{R_2} = \sqrt{900} = 900:1. \] Thus, the ratio of their radii is: \[ R_1 : R_2 = 900:1. \]