We have estimates of temperature and carbon dioxide levels that go back at least to the end of the Precambrian. It is generally agreed that the Cambrian era began 540 million years ago. Estimates of carbon dioxide levels for that time are generally agreed to show levels in the range of 6,000 ppm CO2. It is widely agreed that the level of CO2 in the atmosphere was 279 ppm before human activity began to increase this. It will be taken as a given that human activity increases CO2. If you disagree, please go elsewhere. It will also be taken as a given that increasing CO2 increases surface temperature. Again. Go argue with someone else if you disagree.
Equilibrium Climate Sensitivity
The �equilibrium climate sensitivity� (IPCC 1990, 1996) is defined as the change in global mean temperature, T2x, that results when the climate system, or a climate model, attains a new equilibrium with the forcing change F2x resulting from a doubling of the atmospheric CO2 concentration. (IPCC Tar, chapter 9 WG1)
Given this, one can estimate the number of doublings (N) from 279 ppm to 6,000 ppm. The equation for this is: 6,000=279×2^N. Solving, N=4.426625. In significant figures. N=4.4
The change in temperature is thus ΔT=ECS X N.
Increasing Solar Flux
The reason that the earth was the same temperature 540 million years ago as it is today is postulated to be because the incoming solar flux has increased over time. How much has the solar flux increased? This is a challenging number to find. Why has it increased? The primary cause of flux variation is changes in the distance from earth to the sun. One of the known variations is the Milankovic cycle. I could only find one reference (History Of Planetary And Geological Factors by I. I. Borzenkova) to how much the solar flux has decreased over the Phanerozoic. This reference asserts that the flux increases at a rate of about 5% per billion years or 3% for the last 600 million years. (This value is the weak link in this argument. If someone has a link to a more authoritative value, please let me know.) That is the sun is 3% hotter now than it was 600 million years ago. By incredible coincidence, the change in CO2 is exactly enough to exactly offset this change. Occam’s razor might come in handy here.
What is the impact on temperature of an increase in solar flux. For the real world, one must assume, calculate and estimate a great many things. For a black body however we can be exact. So let’s use a black body. The equation for radiant heat flux is q=εσT^4. In the equation, ε is emissivity and σ is the Stephan-Boltzmann constant = 5.6703E-8 W/(m^2 K^4) . Solving for T, we get T=(q/εσ)^(1/4). We know that incoming solar flux is 1,366 W/m². This impacts on an area equal to a flat disk with a radius equal to the radius of the earth. This is radiated out by an area equal to the surface area of a sphere (almost) with a radius equal to the radius of the earth. The area of the sphere is 4 times the (update, April 9, 2016: d’oh. should be area) radius of the disk. This means that the flux that must be radiated is 1/4 of 1366 W/m² or 341.5 W/m². Solving for T, this gives a black body equilibrium temperature for a sphere of the radius of the earth of 278.6 kelvin or 5.3°C. Lower values quoted on the internet reduce the solar constant to account for reflections by clouds, water, etc. This is referred to as “albedo”. It is generally given as 0.3, hence 70% of the solar flux must be re-radiated. If we accept that, then the equilibrium temperature of the earth is -18.2°C. The effective flux used here is 239.05 W/m². OK. So a 1% change in flux is 2.3905 W/m². Using the SB equation, at 239.05 W/m², a change of 2.3905 W/m² results in a change in temperature of 0.266 kelvin. Borzenkova asserts that the change in flux will result in a change in temperature of 1.4 kelvin. Let’s give him the benefit of the doubt and use 1.4 instead of 0.266. For a change in solar flux of 3%, we get a temperature change of 3 X 1.4 = 4.2 kelvin.
Getting back to forcings etc. We now have a ΔT of 4.2 kelvin. We have already established 4.4 doublings. So what is the ECS? ECS= ΔT/D or 4.2/4.4 = 0.95 kelvin / doubling of CO2. How much would the solar flux have to decrease to give an ECS of 3? We will take the 4.4 doublings as constant. This gives ΔT=13.2 kelvin. This means Δq=66.8W/m² {q(T=288k)=390.1, q(T=274.8k)=323.3}. This is a change of 66.8/341.5=19.56%.
If I haven’t screwed anything up, if solar flux is the reason we have a temperature now similar to when CO2 was 6,000 and the ECS for a doubling is 3, the solar flux must have been about 20% lower at the beginning of the Phanerozoic. If the change in solar flux over the Phanerozoic is 3%, then the ECS is 0.95 or lower.
If solar flux is constant, then ECS is 0.
Some references.
http://www.ipcc.ch/ipccreports/tar/wg1/345.htm
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