Question

Assume the Earth is in radiative equilibrium with an effective radiative temperature of 255 K. If the planetary albedo increases by 0.05, then the effective radiative temperature of the planet will be _________ \(\text{ K}\). (Round off to the nearest integer)

Hint:

An increase in planetary albedo reduces the amount of absorbed solar radiation, which lowers the effective temperature. Use the Stefan-Boltzmann law to relate changes in albedo to changes in temperature.

Answer 1

Correct Answer: 249

The effective radiative temperature of a planet is given by the Stefan-Boltzmann law:
\[ T_e = \left( \frac{(1 - \alpha) S}{4 \sigma} \right)^{1/4} \] Where:
- \( T_e \) is the effective radiative temperature
- \( \alpha \) is the planetary albedo
- \( S = 1370\ \text{W/m}^2 \) is the solar constant
- \( \sigma = 5.67 \times 10^{-8}\ \text{W/m}^2 \text{K}^4 \) is the Stefan-Boltzmann constant
Initially, the effective temperature \( T_{e1} \) is given by:
\[ T_{e1} = \left( \frac{(1 - \alpha_1) S}{4 \sigma} \right)^{1/4} = 255\ \text{K} \] Now, the albedo increases by 0.05, so the new albedo is \( \alpha_2 = \alpha_1 + 0.05 \). We can use the ratio of temperatures to solve for the new temperature \( T_{e2} \). Since the temperatures are related by the equation:
\[ \frac{T_{e2}}{T_{e1}} = \left( \frac{(1 - \alpha_2)}{(1 - \alpha_1)} \right)^{1/4} \] Substituting the values:
\[ \frac{T_{e2}}{255} = \left( \frac{(1 - \alpha_1 - 0.05)}{(1 - \alpha_1)} \right)^{1/4} \] Thus, the new temperature \( T_{e2} \) is approximately:
\[ T_{e2} = 249\ \text{K} \]