How to calculate the temperature of Earth's surface

Question

From a dissertation, I read about the greenhouse gas's effect which make the surface warmer. It says, I quote:

The greenhouse effect results in an average surface temperature of approx. $14^\circ\ \rm C$, compared to $-19^\circ\ \rm C$ if the atmosphere would have been transparent to thermal radiation.

Is there some simple way to estimate the average temperature of Earth's surface using some parameters like sunlight intensity?

Answer 1

An easy calculation is to start with the solar constant, the power (energy per unit time) produced by solar radiation at a distance of one astronomical unit. This is 1.361 kilowatts per square meter. The surface area of the Earth is $4\pi R^2$, where $R$ is the radius of the Earth, while the cross section of the Earth to solar radiation is $\pi R^2$. Thus the Earth as a whole receives 1/4 of that solar constant.

Assume a planet with an atmosphere that is transparent in the thermal infrared, with the same albedo as that of the Earth (0.306), rotating rapidly like the Earth, and orbiting at the same distance from the Sun as the Earth. The effective temperature of this planet is given by the Stefan Boltzmann law: $$T = \left(\frac{(1-\alpha)\,I_\text{sc}}{4\sigma}\right)^{1/4}$$ where

• $\alpha$ is the albedo (0.306),
• $I_\text{sc}$ is the solar constant (1.361 kW/m²),
• $\sigma$ is the Stefan Boltzmann constant (5.6704×10⁻⁸ W/m²/K⁴), and
• the factor of 1/4 arises from the the fact that the Earth is a rapidly rotating spherical object.

The result is -19 °C.