Question

due to the sun-earth distance the average temperature of earth is maintained at 255k. Derive this temperature of earth and state any assumptions made and state the variables used.

Answer 1

due to the sun-earth distance the average temperature of earth is maintained at 255k. Derive this temperature of earth and state any assumptions made and state the variables used.

Solution

Using the Stefan-Boltzmann law and energy conservation law for Earth's radiation balance

$$\alpha \pi R _ {\text {earth}} ^ {2} L = 4 \pi \varepsilon R _ {\text {earth}} ^ {2} \sigma T ^ {4}$$

Where $1 - \alpha$ is reflection coefficient, $\varepsilon$ is emissivity, $T$ is temperature of earth, $R_{\text{earth}}$ is radius of earth, $\sigma$ is Stefan-Boltzmann constant, $L$ is solar constant.

$$\alpha L = 4 \varepsilon \sigma T ^ {4}$$ $$T = \sqrt [ 4 ]{\frac {\alpha L}{4 \varepsilon \sigma}}$$

I defined constant as

$$\alpha \approx 0.6$$ $$\varepsilon \approx 0.9$$ $$L = 136 \, \mathrm{W} / \mathrm{m} ^ {2}$$

Answer

$$T = \sqrt [ 4 ]{\frac {0.7 * 136 \, \mathrm{W} / \mathrm{m} ^ {2}}{4 * 5.67 * 10 ^ {- 0}}} K \approx 252K$$