The effective black body radiation temperature, T, of the Earth was assumed to vary with age (t in Ga) as follows (see Kasting and Grinspoon 1990):
This expression assumes a constant planetary albedo.
Abiotic temperatures were computed from an assumed B = 100 (see
Schwartzman and Volk 1991a); the biotic enhancement of weathering on the present Earth is 100.
Equilibrium temperatures were computed from thermodynamic data for the Urey reaction:
K = e-(4992/r) = pCO2, the partial pressure of carbon dioxide (in atm) in the atmosphere/ocean system.
With a greenhouse function that takes into account the variation in solar luminosity, both the temperature and carbon dioxide level are soluble at any time. The equilibrium temperatures and corresponding atmospheric carbon dioxide levels represent hypothetical states at low temperatures because of the very slow kinetics of the abiotic solid/gas reaction at these conditions. The computed equilibrium temperatures are a first approximation only of what on a real Earth would be more complex equilibria involving CaMgFe silicates and carbonates.
We used the greenhouse function provided by Caldeira and Kasting (1992b) because it gives more plausible temperatures for the low pCO2 range than the Walker et al. (1981) function (Kasting, personal communication):
T = Te + AT, where T is the mean global surface temperature (oK).
- 6.7084y-2 + 73.221y-1 - 30,882 T~xy-1, where y = log pCO2 (in bars).
We assumed a constant planetary albedo, using the above expression for the variation of Te (Caldeira and Kasting assumed a dependence of albedo on temperature). Our computed temperatures are only roughly comparable with those of Caldeira and Kasting owing to model assumptions; for example, Caldeira and Kasting's biotic temperatures were deliberately maximized with atmospheric pCO2 held at 10~6 bars at times greater than 1 billion years in the future.
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