Question

The Sun is approximately a sphere of radius \(6.963 \cdot 10^{5} \mathrm{~km},\) at a mean distance \(a=1.496 \cdot 10^{8} \mathrm{~km}\) from the Earth. The solar constant, the intensity of solar radiation at the outer edge of Earth's atmosphere, is \(1370 . \mathrm{W} / \mathrm{m}^{2}\). Assuming that the Sun radiates as a blackbody, calculate its surface temperature.

Short answer

Answer: The approximate surface temperature of the Sun is 5774.7 K.

Step-by-step solution

Convert the given values to standard units

We will convert the given values to SI units. That means we will convert the distance "a" from the Earth to the Sun and the radius "r" of the Sun from km to meters: \(a = 1.496 \cdot 10^8 \mathrm{~km} \cdot 10^3 \mathrm{~m/km} = 1.496 \cdot 10^{11} \mathrm{~m}\) \(r = 6.963 \cdot 10^5 \mathrm{~km} \cdot 10^3 \mathrm{~m/km} = 6.963 \cdot 10^8 \mathrm{~m}\)

Find the total power radiated by the Sun

Using the solar constant, we will find the total power radiated by the Sun. \(solar \thinspace constant = \frac{Total \thinspace power \thinspace radiated \thinspace by \thinspace Sun}{Area \thinspace of \thinspace sphere \thinspace centered \thinspace at \thinspace Sun}\) \(1370 \mathrm{~W/m^2} = \frac{P_\odot}{4\pi a^2}\) Now, we'll isolate \(P_\odot\) and get: \(P_\odot = 1370 \mathrm{~W/m^2} \cdot 4\pi a^2 \approx 3.828 \cdot 10^{26} \mathrm{~W}\)

Use Stefan-Boltzmann Law to find the surface temperature of the Sun

Stefan-Boltzmann Law states that: \(P_\odot = 4\pi r^2 \sigma T_s^4\) Here, \(\sigma\) is the Stefan-Boltzmann constant (\(5.67 \cdot 10^{-8} \mathrm{~W/m^2 K^4}\)) and \(T_s\) is the surface temperature of the sun that needs to be found. Now, we can solve for \(T_s\): \(T_s^4 = \frac{P_\odot}{4\pi r^2 \sigma} = \frac{3.828 \cdot 10^{26} \mathrm{~W}}{4\pi (6.963 \cdot 10^8 \mathrm{~m})^2 \cdot 5.67\cdot 10^{-8} \mathrm{~W/m^2 K^4}}\) \(T_s^4 \approx 2.341 \cdot 10^{17} \mathrm{K^4}\) Now, take the fourth root of both sides: \(T_s = \sqrt[4]{2.341 \cdot 10^{17} \mathrm{K^4}} \approx 5774.7\mathrm{~K}\) So, the surface temperature of the Sun is approximately \(5774.7 \mathrm{~K}\).