A stars wavelength is observed to be 200 nm. Calculate the approximate surface temperature of the star
A stars wavelength is observed to be 200 nm. Calculate the approximate surface temperature of the star
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![### Astronomy Education: Calculating Star Temperatures
**Question:**
A star's wavelength is observed to be 200 nm. Calculate the approximate surface temperature of the star.
**Explanation:**
To calculate the approximate surface temperature of a star based on its observed wavelength, we can use Wien's Displacement Law. Wien's Displacement Law is given by the formula:
\[ \lambda_{\text{max}} \times T = b \]
where:
- \(\lambda_{\text{max}}\) is the wavelength at which the star emits the most light (in meters).
- \(T\) is the surface temperature of the star (in Kelvin).
- \(b\) is Wien's displacement constant, approximately equal to \(2.897 \times 10^{-3} \, \text{m} \cdot \text{K}\).
**Step-by-Step Solution:**
1. **Convert Wavelength to Meters:**
Given \(\lambda_{\text{max}} = 200 \, \text{nm}\).
Convert nanometers to meters: \( 200 \, \text{nm} = 200 \times 10^{-9} \, \text{m}\).
2. **Apply Wien's Displacement Law:**
\[ T = \frac{b}{\lambda_{\text{max}}} \]
Using \( b = 2.897 \times 10^{-3} \, \text{m} \cdot \text{K} \) and \(\lambda_{\text{max}} = 200 \times 10^{-9} \, \text{m}\),
\[ T = \frac{2.897 \times 10^{-3}}{200 \times 10^{-9}} \]
3. **Calculate the Temperature:**
\[ T = \frac{2.897 \times 10^{-3}}{200 \times 10^{-9}} \]
\[ T = \frac{2.897 \times 10^{-3}}{2 \times 10^{-7}} \]
\[ T = 1.4485 \times 10^{4} \, \text{K} \]
Therefore, the approximate surface temperature of the star is \( 14,485 \, \text{K} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3fe8677b-d2b4-4cf1-b1ed-08820154fcb5%2Fdded2bb3-243a-4389-a1d6-ecc8afb4ab18%2Fij5leix_processed.png&w=3840&q=75)
Transcribed Image Text:### Astronomy Education: Calculating Star Temperatures
**Question:**
A star's wavelength is observed to be 200 nm. Calculate the approximate surface temperature of the star.
**Explanation:**
To calculate the approximate surface temperature of a star based on its observed wavelength, we can use Wien's Displacement Law. Wien's Displacement Law is given by the formula:
\[ \lambda_{\text{max}} \times T = b \]
where:
- \(\lambda_{\text{max}}\) is the wavelength at which the star emits the most light (in meters).
- \(T\) is the surface temperature of the star (in Kelvin).
- \(b\) is Wien's displacement constant, approximately equal to \(2.897 \times 10^{-3} \, \text{m} \cdot \text{K}\).
**Step-by-Step Solution:**
1. **Convert Wavelength to Meters:**
Given \(\lambda_{\text{max}} = 200 \, \text{nm}\).
Convert nanometers to meters: \( 200 \, \text{nm} = 200 \times 10^{-9} \, \text{m}\).
2. **Apply Wien's Displacement Law:**
\[ T = \frac{b}{\lambda_{\text{max}}} \]
Using \( b = 2.897 \times 10^{-3} \, \text{m} \cdot \text{K} \) and \(\lambda_{\text{max}} = 200 \times 10^{-9} \, \text{m}\),
\[ T = \frac{2.897 \times 10^{-3}}{200 \times 10^{-9}} \]
3. **Calculate the Temperature:**
\[ T = \frac{2.897 \times 10^{-3}}{200 \times 10^{-9}} \]
\[ T = \frac{2.897 \times 10^{-3}}{2 \times 10^{-7}} \]
\[ T = 1.4485 \times 10^{4} \, \text{K} \]
Therefore, the approximate surface temperature of the star is \( 14,485 \, \text{K} \).
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