Calculate the size of a star if its surface temperature is 6000 K and has a power output of 5.6 x 1026 W.
Calculate the size of a star if its surface temperature is 6000 K and has a power output of 5.6 x 1026 W.
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![**Question:**
Calculate the size of a star if its surface temperature is 6000 K and it has a power output of \( 5.6 \times 10^{26} \) W.
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**Explanation:**
This question involves determining the radius of a star based on its surface temperature and power output, also known as luminosity. The relationship between these quantities is given by the Stefan-Boltzmann Law, which states:
\[ L = 4 \pi R^2 \sigma T^4 \]
Where:
- \( L \) is the luminosity (or power output) of the star.
- \( R \) is the radius of the star.
- \( \sigma \) is the Stefan-Boltzmann constant (\( \sigma = 5.67 \times 10^{-8} \, \text{W m}^{-2} \, \text{K}^{-4} \)).
- \( T \) is the surface temperature of the star in Kelvin.
Given:
- \( L = 5.6 \times 10^{26} \) W
- \( T = 6000 \) K
Rearranging the Stefan-Boltzmann equation to solve for \( R \):
\[ R = \sqrt{\frac{L}{4 \pi \sigma T^4}} \]
By substituting the given values into the equation, you can find the radius \( R \) of the star.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3fe8677b-d2b4-4cf1-b1ed-08820154fcb5%2F98190ee8-6392-4702-8aa2-0482d060ce39%2Fpq9rpw9_processed.png&w=3840&q=75)
Transcribed Image Text:**Question:**
Calculate the size of a star if its surface temperature is 6000 K and it has a power output of \( 5.6 \times 10^{26} \) W.
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**Explanation:**
This question involves determining the radius of a star based on its surface temperature and power output, also known as luminosity. The relationship between these quantities is given by the Stefan-Boltzmann Law, which states:
\[ L = 4 \pi R^2 \sigma T^4 \]
Where:
- \( L \) is the luminosity (or power output) of the star.
- \( R \) is the radius of the star.
- \( \sigma \) is the Stefan-Boltzmann constant (\( \sigma = 5.67 \times 10^{-8} \, \text{W m}^{-2} \, \text{K}^{-4} \)).
- \( T \) is the surface temperature of the star in Kelvin.
Given:
- \( L = 5.6 \times 10^{26} \) W
- \( T = 6000 \) K
Rearranging the Stefan-Boltzmann equation to solve for \( R \):
\[ R = \sqrt{\frac{L}{4 \pi \sigma T^4}} \]
By substituting the given values into the equation, you can find the radius \( R \) of the star.
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