The spectral energy distribution of the sun has maxima at `4753overset@A` What is the temperature of a star for which spectral distribution has maxima at `10350overset@A` [temperature of sun is 6000 K]

Options

A. 2700 K
B. 2755.4 K
C. 3700 K
D. 37554 K

Answer

To find the temperature of a star based on the wavelength at which its spectral energy distribution has a maximum, we can use Wien's Displacement Law. This law states that the wavelength of the peak emission (λ_max) is inversely proportional to the temperature (T) of the black body. The formula is given by:

\[ \lambda_{max} \cdot T = b \]

where \( b \) is Wien's displacement constant, approximately equal to \( 2898 \, \mu m \cdot K \) or \( 2898 \times 10^{-10} \, m \cdot K \).

### Step-by-step Solution:

1. **Identify the known values**:
- Wavelength of the sun's maximum emission, \( \lambda_{sun} = 4753 \, \text{Å} = 4753 \times 10^{-10} \, \text{m} \)
- Temperature of the sun, \( T_{sun} = 6000 \, K \)
- Wavelength of the star's maximum emission, \( \lambda_{star} = 10350 \, \text{Å} = 10350 \times 10^{-10} \, \text{m} \)

2. **Use Wien's Displacement Law for the sun**:
\[ \lambda_{sun} \cdot T_{sun} = b \]
Substituting the known values:
\[ 4753 \times 10^{-10} \cdot 6000 = b \]
Calculate \( b \):
\[ b = 4753 \times 10^{-10} \cdot 6000 = 2.8518 \times 10^{-6} \, m \cdot K \]

3. **Use Wien's Displacement Law for the star**:
\[ \lambda_{star} \cdot T_{star} = b \]
Substituting the known values:
\[ 10350 \times 10^{-10} \cdot T_{star} = 2.8518 \times 10^{-6} \]

4. **Solve for \( T_{star} \)**:
\[ T_{star} = \frac{2.8518 \times 10^{-6}}{10350 \times 10^{-10}} \]
Calculate \( T_{star} \):
\[ T_{star} = \frac{2.8518 \times 10^{-6}}{1.035 \times 10^{-6}} \approx 2750 \, K \]

### Final Answer:
The temperature of the star is approximately \( 2750 \, K \).