The Faint Sun paradox was first proposed by Carl Sagan who pointed out that the geological evidence  that liquid oceans existed on Earth 4 billion years ago appears incompatible with a solar output 30% dimmer than today. The sun is a main sequence star whose output is known to increase slowly with age. However the change in solar radiative forcing over this long period turns out to be huge ~ 87 w/m2.   It has been argued that an enhanced greenhouse effect due to very high CO2 and/or CH4 concentrations  could resolve this paradox [1]. However, recent geological evidence does not support CO2 as being responsible but instead the authors propose a greater ocean surface leading to a lower albedo as a likely solution [2]. Others have suggested that high cirrus clouds effectively warmed the Earth [3]. Although the atmosphere must have been very different before photosynthesis began, the presence of large liquid oceans still implies that clouds and water vapour played the same  role  in the Earth’s energy balance then as they do today.

All current IPCC models adopt positive net feedbacks of water vapour and clouds with increased radiative forcing. A doubling of CO2 leads to an increased radiative forcing of 3.6 watts/m2. Without any feedbacks the  surface temperature is then predicted to warm by about 1 degreeC restoring global energy balance through increased outgoing Infrared. IPCC models only predict larger temperature rises because they assume positive feedbacks. What do these positive feedbacks imply for the Faint Sun paradox?

Fig 1: Feedback (+) increase or (-) decrease forcing DS

Black body radiation from the Earth’s surface is the primary negative feedback on any temperature rise DT. The differential of Stefan Boltzman’s law gives the effective feedback parameter:

DS/DT = 4sigmaT^3 (= 3.75W/m2K-1 for T=288K)   This is essentially the term G0 in Figure 1.

If we define the sum of all other feedback effects as F, and an external increase in forcing as DS, then applying  energy balance gives

DT( 4sigmaT^3 -F) = DS    or  DT  =DS/(4sigmaT^3-F)     equation. 1.

The main feedback estimates used by IPCC models concern water vapour and clouds. The total feedbacks used  range from +1.6 to 2.5 giving an average  total positive feedback of ~ 2.0 W/m2K-1. Only with these feedbacks does a doubling of CO2 lead to larger temperature increases of 2 to 5 deg.C. If however it turns out that total feedbacks are actually negative or zero then global temperatures would rise only 0.3 to 1.1 degrees C.

The sun has brightened 30% over the last 4 billion years and current average solar radiation is 342 watts/m2. Assuming a slow linear increase of solar radiation with time gives a net forcing of 0.02 watts/m2 every 1 million years. The calculation for the temperature response to this forcing is shown below in Figure 2 for 3 different values of  F  -2, 0, +2

Fig 3: Temperature change per 0.02 watts/m2 (1 million years)

This can now be integrated over time to give predicted temperatures. This can be done two ways. Firstly we assume that the temperature 4 billion years ago was 280 degrees, since we know that liquid oceans were present. This result is shown in Figure 4.

Fig 4: Temperatures for different feedbacks starting with T=280K

It is clear that  positive feedback of +2 as used by IPCC leads to unreasonably high temperatures as solar radiance increases and appears to be wrong. Instead a negative value of -2 w/m2K-1  is more compatible with current temperatures. The second way of calculating  temperatures is instead to fix the current temperature at the observed 288K and work backwards by subtracting DT from solar forcing every million years. It is here that we see the basic problem of (linear) positive feedback models. If the temperature falls enough so that 4sigmaT^3 = F then we get a singularity. The results of this “hindcast” calculation are shown in Figure 5.

Figure 4: Predicted past temperatures fixing T=288 today

This result shows that the assumed feedbacks in IPCC models are incompatible with the Earth’s early history because of the fact that we know that the planet has been continuously covered in liquid water. I find it very difficult to accept that the net effect of water on climate can be a positive feedback.  While it is true that water vapor greenhouse effects  depend on surface temperatures though the Claussius-Claperyon equation, high and low level cloud change to the Earth’s  albedo seemingly must be more important. The evidence is that global surface temperatures have changed little over the Earth’s history. This is incompatible with a simple linear net  positive long term feedback from water vapour and clouds. During the early lifetime of the Earth the total feedback from water must have been negative to avoid run away surface heating as the sun brightened.

On the contrary,  it is proposed that the continuous 70% surface coverage of water  on Earth implies that the oceans self regulating global temperatures. A very simple possible model for this is described here

[1]Owen et al. (1979) Enhanced CO2 greenhouse to compensate for reduced solar luminosity on early Earth, Nature, 277, 640-642

[2] James Kasting & David Catling, Evolution of  a Habitable Planet, Annu. Rev. Astron. Astrophys. 2003. 41:429–63

[3] Roberto Rondanelli and Richard Lindzen (2010) Can thin cirrus clouds in the tropics provide a solution to the faint young Sun paradox, Journal Geophys Research Vol 115, D02108