To solve the problem, we will use the Stefan-Boltzmann law, which states that the energy radiated per unit surface area of a black body is proportional to the fourth power of its absolute temperature. The formula can be expressed as:
\[ E \propto R^2 T^4 \]
where:
- \( E \) is the energy emitted,
- \( R \) is the radius of the star,
- \( T \) is the surface temperature.
### Step-by-Step Solution:
1. **Identify Given Values**:
- Energy emitted by star A is \( E_2 = 10,000 \times E_1 \) (where \( E_1 \) is the energy emitted by the Sun).
- Surface temperature of the Sun, \( T_1 = 6000 \, K \).
- Surface temperature of star A, \( T_2 = 2000 \, K \).
2. **Set Up the Proportionality**:
Using the Stefan-Boltzmann law, we can write the ratio of the energies emitted by the two stars:
\[
\frac{E_1}{E_2} = \frac{R_1^2 T_1^4}{R_2^2 T_2^4}
\]
3. **Substituting Known Values**:
We know \( E_2 = 10,000 \times E_1 \), so:
\[
\frac{E_1}{10,000 \times E_1} = \frac{R_1^2 T_1^4}{R_2^2 T_2^4}
\]
This simplifies to:
\[
\frac{1}{10,000} = \frac{R_1^2 T_1^4}{R_2^2 T_2^4}
\]
4. **Rearranging the Equation**:
Rearranging gives:
\[
R_2^2 = R_1^2 \times 10,000 \times \frac{T_2^4}{T_1^4}
\]
5. **Substituting Temperatures**:
Substitute \( T_1 \) and \( T_2 \):
\[
R_2^2 = R_1^2 \times 10,000 \times \left(\frac{2000^4}{6000^4}\right)
\]
6. **Calculating the Temperature Ratio**:
Calculate \( \left(\frac{2000}{6000}\right)^4 \):
\[
\frac{2000}{6000} = \frac{1}{3} \implies \left(\frac{1}{3}\right)^4 = \frac{1}{81}
\]
7. **Substituting Back**:
Now substitute back into the equation:
\[
R_2^2 = R_1^2 \times 10,000 \times \frac{1}{81}
\]
\[
R_2^2 = R_1^2 \times \frac{10,000}{81}
\]
8. **Taking Square Root**:
Taking the square root of both sides gives:
\[
R_2 = R_1 \times \sqrt{\frac{10,000}{81}} = R_1 \times \frac{100}{9}
\]
9. **Finding the Ratio**:
The ratio of the radii \( \frac{R_2}{R_1} \) is:
\[
\frac{R_2}{R_1} = \frac{100}{9}
\]
10. **Final Ratio**:
Therefore, the ratio of the radius of star A to the radius of the Sun is:
\[
R_A : R_{\text{Sun}} = 100 : 9
\]
### Conclusion:
The ratio of the radii of star A to the Sun is \( \frac{100}{9} \) or approximately \( 11.11 : 1 \).
