A star has a radius of 2RO and has a temperature of 3TO. Calculate the luminosity in solar units.
A star has a radius of 2RO and has a temperature of 3TO. Calculate the luminosity in solar units.
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![**Stellar Luminosity Calculation**
A star has a radius of \(2R_{\odot}\) and has a temperature of \(3T_{\odot}\). Calculate the luminosity in solar units.
In this problem, we need to calculate the luminosity of a star compared to the Sun. Here, \(R_{\odot}\) represents the radius of the Sun and \(T_{\odot}\) represents the temperature of the Sun.
To find the luminosity, we can use the formula for luminosity in terms of radius and temperature:
\[ L = 4 \pi R^2 \sigma T^4 \]
where:
- \(L\) is the luminosity,
- \(R\) is the radius of the star,
- \(\sigma\) is the Stefan-Boltzmann constant,
- \(T\) is the temperature of the star.
Given that the star's radius is \(2R_{\odot}\) and its temperature is \(3T_{\odot}\), we substitute these values into the equation:
\[ L = 4 \pi (2R_{\odot})^2 \sigma (3T_{\odot})^4 \]
Since we are calculating luminosity in solar units and knowing the formula for solar luminosity \(L_{\odot} = 4 \pi R_{\odot}^2 \sigma T_{\odot}^4\), we can relate the star's luminosity to the solar luminosity.
Rewriting the equation:
\[ L = L_{\odot} \left( \frac{2R_{\odot}}{R_{\odot}} \right)^2 \left( \frac{3T_{\odot}}{T_{\odot}} \right)^4 \]
Simplifying the terms:
\[ L = L_{\odot} \times 2^2 \times 3^4 \]
\[ L = L_{\odot} \times 4 \times 81 \]
\[ L = 324L_{\odot} \]
Thus, the luminosity of the star is \(324\) times the solar luminosity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadbb6fa3-6b43-4484-8eb2-c92bfed12f92%2Fea036851-544f-4aa5-999b-4658041cd007%2F2vswqms_processed.png&w=3840&q=75)
Transcribed Image Text:**Stellar Luminosity Calculation**
A star has a radius of \(2R_{\odot}\) and has a temperature of \(3T_{\odot}\). Calculate the luminosity in solar units.
In this problem, we need to calculate the luminosity of a star compared to the Sun. Here, \(R_{\odot}\) represents the radius of the Sun and \(T_{\odot}\) represents the temperature of the Sun.
To find the luminosity, we can use the formula for luminosity in terms of radius and temperature:
\[ L = 4 \pi R^2 \sigma T^4 \]
where:
- \(L\) is the luminosity,
- \(R\) is the radius of the star,
- \(\sigma\) is the Stefan-Boltzmann constant,
- \(T\) is the temperature of the star.
Given that the star's radius is \(2R_{\odot}\) and its temperature is \(3T_{\odot}\), we substitute these values into the equation:
\[ L = 4 \pi (2R_{\odot})^2 \sigma (3T_{\odot})^4 \]
Since we are calculating luminosity in solar units and knowing the formula for solar luminosity \(L_{\odot} = 4 \pi R_{\odot}^2 \sigma T_{\odot}^4\), we can relate the star's luminosity to the solar luminosity.
Rewriting the equation:
\[ L = L_{\odot} \left( \frac{2R_{\odot}}{R_{\odot}} \right)^2 \left( \frac{3T_{\odot}}{T_{\odot}} \right)^4 \]
Simplifying the terms:
\[ L = L_{\odot} \times 2^2 \times 3^4 \]
\[ L = L_{\odot} \times 4 \times 81 \]
\[ L = 324L_{\odot} \]
Thus, the luminosity of the star is \(324\) times the solar luminosity.
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