T' = k·S1/4
S is the solar irradiance. The value of S for planet Earth is often quoted as 1367.6 W/m2.
It turns out that the temperature formula above is generally applicable to almost all the planets in our solar system. Figure 1 is a scatter chart of solar irradiance (to the 1/4th power) vs. the mean temperature of planets in our solar system.
Except for the planet Venus, it's clear we have a correlation so strong that it could only be the reflection of a straightforward law of Physics. Planet Earth actually falls a little bit outside the model. It's about 8 degrees Celsius warmer than the formula would predict. (I understand this is called the natural greenhouse effect.) If we remove Earth from the analysis, the correlation coefficient improves to 0.9995 and the slope of the linear trend becomes 46.163. Hence, the formula for the equilibrium temperature of any arbitrary planet, provided nothing unusual is going on with it, is the following.
T' = 46.163·S1/4
Per Figure 1, again, this appears to be quite precise. With this formula we can begin to estimate what might happen if solar irradiance were to increase from, say, 1363.4 W/m2 to 1366.7 W/m2. This is the range of variation reported in the Lean (2000) solar irradiance reconstruction between the years 1610 and 2000.
With an increase of that magnitude, I estimate that equilibrium temperature would rise by only 0.2 degrees. It can't possibly explain a 0.8- to 1.6-degree anomaly by itself.
When I first started to look at data and claims related to climate science, and noticed the concept of temperature sensitivity to CO2 doubling, my impression was that equilibrium temperature could be approximated by a formula such as this one:
T' = baseline + a·ln(C)
C is the concentration of CO2, most likely in ppmv, and a an unknown constant. Climate sensitivity would be given by
There are other types of formulas that model how CO2 would affect radiative forcing, as opposed to equilibrium temperature. These work in a different manner, but in general these formulas can only be approximations of the real world. They may be roughly applicable to our planetary reality, but they won't work in general.
The first problem with the conventional doubling model is that it can't work for really small concentrations of the greenhouse gas. Imagine there are only 100 molecules of CO2 in the entire atmosphere, and we subsequently double this concentration to 200 molecules. Should we expect equilibrium temperature to increase by about 3 degrees as a result?
The second problem is that greenhouse forcing is temperature dependent. In particular, it can't work for really low temperatures. It's clear from Figure 1 that a planet receiving zero solar irradiance would have a temperature of absolute zero. It would not irradiate out any energy due to its temperature, and no (theoretical) greenhouse gas would change this.
Let me propose a model that subsumes the previous models and addresses the two problems outlined.
T' = 46.163·(S·(1 + a·ln(b·C + 1)))1/4
When b·C is sufficiently large, the extra greenhouse forcing is proportional to the logarithm of C. When C is zero, the formula is reduced to the one derived from Figure 1. When solar irradiance is zero, temperature is still zero, regardless of C. When C increases from 100 to 200 molecules, the effect is negligible.
That's still an approximation in other ways, no doubt, but I expect to use it in the future.