If the temperature at Saturn's cloud tops is 97 K and the planet radiates three times more energy than it receives from the Sun, use Stefan's law to calculate what the temperature would be in the absence of any internal heat source.
If the temperature at Saturn's cloud tops is 97 K and the planet radiates three times more energy than it receives from the Sun, use Stefan's law to calculate what the temperature would be in the absence of any internal heat source.
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![### Calculating Saturn's Temperature using Stefan's Law
**Problem Statement:**
If the temperature at Saturn’s cloud tops is 97 K and the planet radiates three times more energy than it receives from the Sun, use Stefan’s law to calculate what the temperature would be in the absence of any internal heat source.
**Explanation:**
- **Stefan’s Law** (also known as the Stefan-Boltzmann law) relates the power radiated per unit area of a black body to its temperature. The formula is:
\[
P = \sigma T^4
\]
where \(P\) is the power per unit area, \(\sigma\) is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8} \, \text{Wm}^{-2}\text{K}^{-4}\)), and \(T\) is the temperature in Kelvin.
- Given Data:
- Present temperature of Saturn’s cloud tops, \(T\): 97 K
- Saturn radiates three times more energy than it receives from the Sun.
**Calculation:**
1. Let \(P_s\) be the power per unit area received from the Sun.
2. The power radiated by Saturn, \(P_r\), is three times \(P_s\):
\[
P_r = 3P_s
\]
3. According to Stefan’s law:
\[
P_r = \sigma T^4
\]
4. To find the temperature \(T_{sun}\) that Saturn would have in the absence of internal heat (when \(P_r = P_s\)):
\[
P_s = \sigma T_{sun}^4
\]
5. Using the relationship:
\[
P_r = 3P_s
\]
and substituting,
\[
\sigma T^4 = 3\sigma T_{sun}^4
\]
6. Simplifying, we get:
\[
T^4 = 3 T_{sun}^4
\]
7. Solving for \(T_{sun}\):
\[
T_{sun} = \left(\frac{T^4}{3}\right)^\frac{1}{4}
\]
8. Plugging in the known temperature \(T = 97 \, K](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc44715e-e997-4081-aac6-58f75fd336c7%2Fbaa5aedf-9263-4f9b-91db-3f935c22a670%2F4ygzv7_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating Saturn's Temperature using Stefan's Law
**Problem Statement:**
If the temperature at Saturn’s cloud tops is 97 K and the planet radiates three times more energy than it receives from the Sun, use Stefan’s law to calculate what the temperature would be in the absence of any internal heat source.
**Explanation:**
- **Stefan’s Law** (also known as the Stefan-Boltzmann law) relates the power radiated per unit area of a black body to its temperature. The formula is:
\[
P = \sigma T^4
\]
where \(P\) is the power per unit area, \(\sigma\) is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8} \, \text{Wm}^{-2}\text{K}^{-4}\)), and \(T\) is the temperature in Kelvin.
- Given Data:
- Present temperature of Saturn’s cloud tops, \(T\): 97 K
- Saturn radiates three times more energy than it receives from the Sun.
**Calculation:**
1. Let \(P_s\) be the power per unit area received from the Sun.
2. The power radiated by Saturn, \(P_r\), is three times \(P_s\):
\[
P_r = 3P_s
\]
3. According to Stefan’s law:
\[
P_r = \sigma T^4
\]
4. To find the temperature \(T_{sun}\) that Saturn would have in the absence of internal heat (when \(P_r = P_s\)):
\[
P_s = \sigma T_{sun}^4
\]
5. Using the relationship:
\[
P_r = 3P_s
\]
and substituting,
\[
\sigma T^4 = 3\sigma T_{sun}^4
\]
6. Simplifying, we get:
\[
T^4 = 3 T_{sun}^4
\]
7. Solving for \(T_{sun}\):
\[
T_{sun} = \left(\frac{T^4}{3}\right)^\frac{1}{4}
\]
8. Plugging in the known temperature \(T = 97 \, K
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