A certain star has a temperature twice that of the Sun and a luminosity 64 times greater than the solar value. What is its radius, in solar units?
A certain star has a temperature twice that of the Sun and a luminosity 64 times greater than the solar value. What is its radius, in solar units?
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![### Problem Statement:
A certain star has a temperature twice that of the Sun and a luminosity 64 times greater than the solar value. What is its radius, in solar units?
### Explanation:
In this problem, we are given:
- The temperature of the star is twice that of the Sun, denoted as \( T = 2T_{\odot} \),
- The luminosity of the star is 64 times the luminosity of the Sun, denoted as \( L = 64L_{\odot} \).
We need to find the radius of the star in solar units (i.e., in terms of the Sun's radius).
### Key Formula to Use:
The relationship between luminosity, radius, and temperature of a star is given by the Stefan–Boltzmann law:
\[ L = 4\pi R^2 \sigma T^4 \]
where:
- \( L \) is the luminosity,
- \( R \) is the radius,
- \( T \) is the temperature,
- \( \sigma \) is the Stefan–Boltzmann constant.
For simplicity when comparing to the Sun, we can use ratios:
\[ \frac{L}{L_{\odot}} = \left( \frac{R}{R_{\odot}} \right)^2 \left( \frac{T}{T_{\odot}} \right)^4 \]
Substituting the given values:
\[ 64 = \left( \frac{R}{R_{\odot}} \right)^2 \left( 2 \right)^4 \]
\[ 64 = \left( \frac{R}{R_{\odot}} \right)^2 \times 16 \]
To isolate \( \left( \frac{R}{R_{\odot}} \right)^2 \):
\[ \left( \frac{R}{R_{\odot}} \right)^2 = \frac{64}{16} \]
\[ \left( \frac{R}{R_{\odot}} \right)^2 = 4 \]
Taking the square root of both sides:
\[ \frac{R}{R_{\odot}} = 2 \]
### Conclusion:
The radius of the star is 2 solar units. This means the star's radius is twice that of the Sun.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbdf4ad1c-8d6e-4c6e-ab0d-20fae6a9f061%2Fd455a0bf-ff2d-4706-a450-86f90129295c%2Fnb111rm_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
A certain star has a temperature twice that of the Sun and a luminosity 64 times greater than the solar value. What is its radius, in solar units?
### Explanation:
In this problem, we are given:
- The temperature of the star is twice that of the Sun, denoted as \( T = 2T_{\odot} \),
- The luminosity of the star is 64 times the luminosity of the Sun, denoted as \( L = 64L_{\odot} \).
We need to find the radius of the star in solar units (i.e., in terms of the Sun's radius).
### Key Formula to Use:
The relationship between luminosity, radius, and temperature of a star is given by the Stefan–Boltzmann law:
\[ L = 4\pi R^2 \sigma T^4 \]
where:
- \( L \) is the luminosity,
- \( R \) is the radius,
- \( T \) is the temperature,
- \( \sigma \) is the Stefan–Boltzmann constant.
For simplicity when comparing to the Sun, we can use ratios:
\[ \frac{L}{L_{\odot}} = \left( \frac{R}{R_{\odot}} \right)^2 \left( \frac{T}{T_{\odot}} \right)^4 \]
Substituting the given values:
\[ 64 = \left( \frac{R}{R_{\odot}} \right)^2 \left( 2 \right)^4 \]
\[ 64 = \left( \frac{R}{R_{\odot}} \right)^2 \times 16 \]
To isolate \( \left( \frac{R}{R_{\odot}} \right)^2 \):
\[ \left( \frac{R}{R_{\odot}} \right)^2 = \frac{64}{16} \]
\[ \left( \frac{R}{R_{\odot}} \right)^2 = 4 \]
Taking the square root of both sides:
\[ \frac{R}{R_{\odot}} = 2 \]
### Conclusion:
The radius of the star is 2 solar units. This means the star's radius is twice that of the Sun.
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