The radiant energy \( E \) from a star is calculated using the formula: \[E = \sigma \epsilon A T^4.\] For spherical stars, the area \( A \) is proportional to the square of the radius \( R \): \[A \propto R^2.\] Therefore, the radiant energy is proportional to the square of the radius and the fourth power of the temperature: \[E \propto R^2 T^4.\]Step 1: Formulate the energy ratio.
For two stars, the ratio of their radiant energies is: \[\frac{E_1}{E_2} = \frac{R_1^2 T_1^4}{R_2^2 T_2^4}.\]Given the energy relationship: \[E_1 = 10000 \times E_2.\]Substituting this into the ratio equation: \[\frac{10000 \cdot E_2}{E_2} = \frac{R_1^2 \cdot T_1^4}{R_2^2 \cdot T_2^4}.\]This simplifies to: \[10000 = \frac{R_1^2 (2000)^4}{R_2^2 (6000)^4}.\]Step 2: Simplify the temperature ratio.
The ratio of the temperatures is: \[\frac{T_1}{T_2} = \frac{2000}{6000} = \frac{1}{3}.\]Raising this ratio to the fourth power: \[\left( \frac{1}{3} \right)^4 = \frac{1}{81}.\]Substituting this back into the equation from Step 1: \[10000 = \frac{R_1^2}{R_2^2} \cdot \frac{1}{81}.\]Multiplying both sides by 81: \[\frac{R_1^2}{R_2^2} = 10000 \cdot 81 = 900^2.\]Step 3: Calculate the radius ratio.
Taking the square root of both sides: \[\frac{R_1}{R_2} = \sqrt{900} = 900:1.\]Consequently, the ratio of their radii is: \[R_1 : R_2 = 900:1.\]