Originally Posted by Bill Illis
I think it is better to think of it as the Solar Irradiance reaching the Earth was 30% lower 4.55 billion years ago and it has increased in very close to a straight line over time.
So, Solar Irradiance 520 million years ago was = 1366 (0.7*520/4550) = 1319 = 252.8K
or -2.2K change 520 million years ago from lower solar irradiance.
Go back to 4.55 billion years ago and the Te was 233K.
This comes from D. Gough 1981, Kasting 1988 and outlined a little better in a more recent paper by Kasting.
http://geosc.psu.edu/~kasting/Person...annurev_03.pdf
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Thanks for the link. The calculation you present is better founded on the power law relating temperature to energy; but over the time scales of GeoCarb, this is not actually going to make much difference. A linear approximation works okay.
The real problem is that this calculation ignores all feedback effects, which are actually pretty crucial to the final temperature. Going back 520 Mya, the fractional decrease in insolation is 0.3*520/4550 = 0.0343. Insolation, after allowing for albedo, now about 240 W/m
2. The reduction is 0.0343*240 = 8.3 W/m
2. Assuming a climate sensitivity of about 0.75 K per W/m
2 you get a temperature change at the surface of 6.2K.
This is about what we get also from the numbers in the reference for the GeoCARB III reference, which is
The two papers are consistent with each other, and the factors used for temperature difference over time due to the dimmer Sun take into account both the numbers you have presented for how solar radiance changes, and also the climate feedbacks that affect how the planet responds to that change, which is not simply given by Stefan-Boltzman.
Your latest reference spells this out explicitly. You have cited
On page 442 (my bolding)
If one reduces the value of S by 30% in (1), holding A and ΔTg constant for simplicity, one finds that Te drops to 233 K and Ts = 266 K, well below the freezing point of water. If the calculation is repeated with a climate model that includes the positive feedback loop involving water vapor, the problem becomes even more severe. The dashed curves in Figure 4 show Te and Ts calculated using a one-dimensional, radiative-convective climate model, assuming constant CO2 concentrations and fixed relative humidity (Kasting, Toon & Pollack 1988). The results are remarkably similar to those predicted earlier by Sagan & Mullen: Ts drops below the freezing point of water prior to ~2 Ga. Combined with the snow/ice-albedo feedback loop, this temperature drop would almost certainly lead to a globally glaciated Earth. However, geologic evidence tells us that liquid water and life were both present as far back as 3.5 Ga and maybe longer. The oldest zircons, zirconium silicate minerals that must have formed in liquid water, are dated at more than 4.3 Ga and may indicate the presence of an ocean at that time (Catling & Kasting 2002, Mojzsis, Harrison & Pidgeon 2001, Wilde et al. 2001).
How can the faint young Sun problem be solved? A large decrease in cloudiness would do it (Rossow et al. 1982), but this seems unlikely for reasons mentioned in Section 3.1. Instead, the answer probably lies in increased concentrations of greenhouse gases. Both CO2 and CH4 are plausible candidates. ...
The numbers used in the GeoCARB III reference, which give a temperature difference of about 6 degrees 520 Mya, correspond to estimates that take water vapour and other factors into account, just as Kasting and Catling describe here.
Cheers -- sylas
