Stefan-Boltzmann law and the earth’s temperature

The amount of solar radiation determines the surface temperature of the earth. The calculation is based on Stefan-Boltzmann law. We will carry out the calculation briefly.

The amount of solar radiation reaching the surface above the earth’s atmosphere is about 1360 watts per square meter. The intersection of the earth is pi*r^2 while the earth’s surface area is 4*pi*r^2. Hence the average energy density from the sun to the earth’s surface is 1360/4 =340 watts per square meter. The Stefan-Boltzmann law is I = sigma* T^4. Hence 

T = (I/sigma)^(1/4) =( 340/(5.67*10^(-8)))^(1/4) = 279 K

From the calculation based on Stefan-Boltzmann law, the temperature at the earth’s surface would be 279K if all solar energy reached the surface of the earth. 

The albedo of the atmosphere is 30%, or 70% of the solar energy actually reaches the earth’s surface. This is 

340*70%= 238 watts. With this adjustment, the theoretical temperature at the surface of the earth would be 

T = (I/sigma)^(1/4) =( 238/(5.67*10^(-8)))^(1/4) = 255.

The measured average temperature at the surface of the earth is about 288K, 33K above the theoretical calculation. This 33K increase is attributed to the greenhouse effect exerted by carbon dioxide and other gases. This is the official theory. 

There are several questions about the official theory. First, the earth’s atmosphere reduces the amount of earth’s radiation energy leaving the earth as well. Currently, the official theory only counts the reduction of the solar radiation energy reaching the earth. Atmosphere is like a house. A house can block some sunlight. At the same time, heat loss is reduced due to the house. Overall, it is often warmer inside the house than outside. With this consideration, the theoretical value of the earth’s surface temperature will be adjusted upward. Currently, higher temperature on the earth is mainly attributed to the existence of carbon dioxide by the official narrative.

The inward solar radiation has shorter wavelength than the earth’s outward radiation. How radiations with different wavelengths interact with the atmosphere? 

Second, the earth is not a blackbody. Not being a blackbody will reduce the amount of radiation energy to the outer space. The average emissivity of the earth is about 0.96. With this adjustment, the theoretical value of the earth’s surface temperature will be a little bit higher. With these two adjustments, the theoretical value of the earth’s surface temperature would be higher than 255K. 

Third, the earth is not in thermal equilibrium. Stefan-Boltzmann law applies to blackbodies in thermal equilibrium. The earth is not in thermal equilibrium. It is undergoing daily fluctuations and other fluctuations. Since Stefan-Boltzmann law is non-linear, an averaging process will not capture the overall effect. 

Does averaging cause significant error in estimation? Moon doesn’t have an atmosphere. Its temperature fluctuation is more dramatic. We may have a look at the moon for an answer.

The temperature on the moon can reach 120° Celsius or 400 Kelvin during lunar daytime at the moon's equator, and plummet to -130° C, 140 K at night. 

In certain spots near the moon’s poles temperatures can drop even further, reaching - 253°C or 20 K. Without solar radiation, the temperature approaches to 0 K, as expected. 

Since the distances of the moon and the earth to the sun are similar, we would expect the average temperature of the moon and the sun to be similar. Take the average of high and low temperatures at the moon’s equator, we have

           ½*( 400 + 140) = 270 K

This is similar to 279 K, the theoretical average of the earth’s temperature based on Stefan-Boltzmann law. However, the above calculation is the average of the equator. The overall average temperature of the moon must be much lower. 

The above calculation shows that the theoretical average temperature of the earth would be much lower than 255 K if the nonlinearity of Stefan-Boltzmann law is considered.

The answers to these questions, and many other questions on climate research, may already be available in literature. If not, the questions can be answered by empirical investigations and further theoretical study.  However, there has been little discussion on fundamental questions in climate science in mainstream media and academia. For the ruling class, the science for climate change has “settled”. For them, science is never about seeking truth, science is merely a tool for political control and economic exploitation.