Faint Young Sun Paradox Resolved

The faint young Sun paradox describes the apparent contradiction between observations of liquid water early in the Earth's history and the astrophysical expectation that the Sun's output would be only 70% as intense during that epoch as it is during the modern epoch. The issue was raised by astronomers Carl Sagan and George Mullen in 1972.

The analysis of the lapse rate in earlier posts suggests the following resolution of the paradox:

A reduction of 30% of the insolation could mean a reduction from 180 to 125 W/m2 absorbed by the Earth surface, and a reduction of 140 to 100 W/m2 to be radiated from the tropopause, assuming 40 W/m2 directly radiated through the atmospheric window in both cases. 

This would require a drop in the temperature of the tropopause from - 50 C to - 68 C (from 223 K to 205 K by Stefan-Boltzmann with 223 =(140/5.66)^0.25 x 100). The Earth surface temperature could then remain at + 15 C if the lapse rate increased from the present 6.5 C/km to 8.3 C/km (with the tropopause at 10 km altitude). 

The maximal lapse rate is 10 C/km and could be attained in an atmosphere without thermodynamics, in an atmosphere at rest without motion of the air, with heat transfer by conduction and radiation but no thermodynamics of convection and evaporation/condensation. 

An effect of thermodynamics in the present atmosphere can thus be viewed as a reduction of the lapse rate from 10 to 6.5 C/km with the difference increasing with the vigor of the thermodynamics. With a less vigorous atmosphere the lapse rate could thus increase from 6.5 to 8.3 C/km and thus sustain the same surface temperature with only 70% of the input from the Sun of today. 

In the extreme case of an atmosphere without motion with a lapse rate of 10 C/km,  a 50% Sun would thus be enough to sustain comfortable organic life at + 15 C, thus very early in the history of the solar system.  Organic life is supposed to be 4 billion years old, apparently ignited by a young 50% Sun. 

The argument supports the idea of the thermodynamics of the atmosphere as an air conditioner acting to reduce the lapse rate and thus cool the Earth surface as the inside, with the the tropopause as the outside with a temperature set by the input via Stefan-Boltzmann.

PS Several unsuccessful attempts to resolve the paradox have been presented recently:

• Faint Young Sun Paradox remains (Nature June 2011)
• Why early Earth was no snowball (Potsdam Institute Dec 2012)
• Hypothesis for Faint Young Sun Paradox (Purdue Univ May 2012) 
• Faint Young Sun Paradox not resolved (NASA May 2012)
• Paradox: Is there even life on Earth (Smithsonian Astrophysical Observatory Sept 2012)
• Faint Young Sun Problem (Geophysical Research Letters 2012).

Solutions attempts include:

• Early Earth Atmosphere: Right mix of greenhouse gases (Nathan Sheldon) 
• Geology: Much more geothermal energy 
• Biology: Life developed on a cold planet (John Priscu)
• Fundamental Physics: e.g., gravitational constant has varied 
• Astrophysical Solutions: Young Sun was not faint.