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To estimate the temperature of the Earth in radiation equilibrium with the Sun, we can use the Stefan-Boltzmann law. The power emitted by a black body is given by:
P = A \sigma T^4
where A is the surface area of the body, \sigma is the Stefan-Boltzmann constant (approximately 5.67 \times 10^{-8} W m^{-2} K^{-4}), and T is the temperature in Kelvin.

1. Power emitted by the Sun (P_s):
The total power emitted by the Sun is given by the formula:
P_s = A_s \sigma T_s^4
where A_s = 4\pi R_s^2 is the surface area of the Sun and T_s is the temperature of the Sun, which is given as 5,800 K. We substitute:
A_s = 4\pi (7 \times 10^8 m)^2
Therefore, the power emitted by the Sun becomes:
P_s = 4\pi (7 \times 10^8)^2 \times 5.67 \times 10^{-8} \times (5800)^4
2. Power received by the Earth (P_e):
The power received by the Earth is given by the area of the Earth intercepting the sunlight:
P_e = \frac{P_s}{4} (since the Earth only intercepts sunlight over a cross-sectional area). Therefore,
P_e = \frac{4\pi (7 \times 10^8)^2 \times 5.67 \times 10^{-8} \times (5800)^4}{4}
3. Estimate the temperature (T_E) of the Earth:
Using the same formula, set P_e = A_e \sigma T_E^4, where A_e = 4\pi R_e^2.
Assume the radius of the Earth to be around 6.4 \times 10^6 m:
T_E = (\frac{P_e}{\sigma A_e})^{1/4} = (\frac{P_s/4}{\sigma (4\pi (6.4 \times 10^6)^2)})^{1/4}
Substituting the values will yield an estimate of the temperature of the Earth. This results in approximately 255 K, adjusting for the Earth's albedo gives a final average temperature of about 288 K (15 °C).
Thus, this calculation illustrates how using the radiative balance informs the temperature of Earth based on solar properties.