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1. It is believed that in the Archeaneon (2.5-4 billion years ago) the sun’s radiative output was 30% less than it is today.
(i) What would the temperature of the sun have been at that time?
(ii) At what wavelength would the peak emission from the sun have been?
(iii) Ignoring the effects of the atmosphere, what would the temperature of the Earth have been at this time (i.e. the effective radiative temperature)? Assume that the Earth’s albedo was 0.3, the same as today.
i) Use the Stefan-Boltzmann formula, and assume an emissivity of 1. You will need the sun's current output.
You can use a simpler method if you know the sun's current surface temperature is 5800 K. You reduce the output 30%, T^4 has to be reduced by a factor 0.7.
(Told/Tnew)^4 = 0.7
Told/Tnew = 0.915
Told = 0.915*5800 = 5300 K
ii) Use the Wien displacement law. If you haven't learned it yet, learn it now. (Lamdamax)*T = 0.29 cm* K
iii) Write a radiation balance equation that says the received radiation from the sun (which is proportional to Tsun^4) equals the radiation emitted by the earth which is proportional to Tearth^4. Solve for Tearth. You will need the solid angle subtended by the sun. You can find online references of how this is done for the sun's actual temperature.
The earth's temperature should also end up being about 0.915 of its present average value (which is about 290K), making it about 265 K.