The luminosity of the Rigel star is 17000 times that of the sun. Assume both to be perfectly back bodies. If the surface temperature of the sun 6000 K, then the temperature of the star is around `("Take "17000^(1//4)=11.4)`

Options

A. 68400 K
B. 58400 K
C. 38600 K
D. 32600 K

Answer

To solve the problem of finding the temperature of the Rigel star given its luminosity in relation to the Sun, we can use the Stefan-Boltzmann Law. Here are the steps to derive the temperature of the Rigel star:

### Step-by-Step Solution:

1. **Understand the relationship between luminosity and temperature**:
The luminosity (E) of a black body is related to its temperature (T) by the Stefan-Boltzmann Law:
\[ E = \sigma T^4 \]
where \(\sigma\) is the Stefan-Boltzmann constant.

2. **Set up the equation for the Sun and the Rigel star**:
Let \(E_1\) be the luminosity of the Rigel star and \(E_2\) be the luminosity of the Sun. We know:
\[ E_1 = 17000 \times E_2 \]
Let \(T_1\) be the temperature of the Rigel star and \(T_2\) be the temperature of the Sun. The temperature of the Sun is given as \(T_2 = 6000 \, K\).

3. **Write the equations for luminosity**:
For the Sun:
\[ E_2 = \sigma T_2^4 \]
For the Rigel star:
\[ E_1 = \sigma T_1^4 \]

4. **Take the ratio of the luminosities**:
\[ \frac{E_1}{E_2} = \frac{\sigma T_1^4}{\sigma T_2^4} \]
The \(\sigma\) cancels out:
\[ \frac{E_1}{E_2} = \frac{T_1^4}{T_2^4} \]

5. **Substitute the known values**:
Using the relationship \(E_1 = 17000 \times E_2\):
\[ 17000 = \frac{T_1^4}{(6000)^4} \]

6. **Rearranging the equation**:
\[ T_1^4 = 17000 \times (6000)^4 \]

7. **Taking the fourth root**:
\[ T_1 = (17000)^{1/4} \times 6000 \]

8. **Calculate \((17000)^{1/4}\)**:
Given that \((17000)^{1/4} \approx 11.4\):
\[ T_1 \approx 11.4 \times 6000 \]

9. **Final calculation**:
\[ T_1 \approx 68400 \, K \]

### Conclusion:
The temperature of the Rigel star is approximately **68400 K**.