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Chapter 6: Problem 130

In about 5 billion years, the sun will evolve to a red giant. Assume that its surface temperature will decrease to about half its present value of \(6000 \mathrm{K},\) while its present radius of \(7.0 \times 10^{8} \mathrm{m}\) will increase to \(1.5 \times 10^{11} \mathrm{m}\) (which is the current Earth-sun distance). Calculate the ratio of the total power emitted by the sun in its red giant stage to its present power.

Short Answer

Expert verified
The ratio of the total power emitted by the sun in its red giant stage to its present power is approximately 18300.

Step by step solution

01

Determine the current and future values of temperature and radius

The given information is as follows: Current temperature: \(T_i = 6000 \, \mathrm{K}\), Current radius: \(R_i = 7.0 \times 10^8 \, \mathrm{m}\), Future temperature: \(T_f = \frac{1}{2} T_i\), Future radius: \(R_f = 1.5 \times 10^{11} \, \mathrm{m}\). Now, we need to find the total power emitted by the sun currently and in the red giant stage.
02

Using the Stefan-Boltzmann Law

To find the total radiant power at each stage, we will use the Stefan-Boltzmann Law, \(P = σ A T^{4}\), where \(P\) is the radiant power, \(A\) is the surface area of the sun, \(T\) is the temperature, and \(σ\) is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8} \, \mathrm{W} \cdot \mathrm{m}^{-2} \cdot \mathrm{K}^{-4}\)). We will need to find the surface areas of the sun in its current stage and red giant stage: \(A_i = 4 \pi R_i^2\), \(A_f = 4 \pi R_f^2\), Then, we find the total power emitted in each stage: \(P_i = σ A_i T_i^4\), \(P_f = σ A_f T_f^4\).
03

Calculate the ratio of total power emitted

To find the ratio between the power emitted in the red giant stage and currently, we simply take the ratio of \(P_f\) to \(P_i\): \(\text{Power Ratio} = \frac{P_f}{P_i}\). Substitute the values we found in the previous steps: \(\text{Power Ratio} = \frac{σ A_f T_f^4}{σ A_i T_i^4}\). Notice that the Stefan-Boltzmann constant \(σ\) can be cancelled out: \(\text{Power Ratio} = \frac{A_f T_f^4}{A_i T_i^4}\), Substitute the values for surface area and temperature, then solve for the power ratio.
04

Final Result

Substituting the expressions for \(A_f\), \(A_i\), \(T_f\), and \(T_i\) into the power ratio formula, we get: \(\text{Power Ratio} = \frac{(4 \pi R_f^2)(\frac{1}{2} T_i)^4}{(4 \pi R_i^2) T_i^4}\). Simplify the expression and substitute the values for \(R_i\) and \(R_f\) to compute the power ratio: \(\text{Power Ratio} = \frac{R_f^2}{4 R_i^2} \approx 18300\). So, the ratio of the total power emitted by the sun in its red giant stage to its present power is approximately 18300.

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