Four billion years ago the Sun’s radiative output was 30% less than it is today.
(i) If we assume the radius of the sun is the same, and that the Earth’s atmosphere was the same as it is now (that is, the atmosphere absorbs 10% of the incoming solar radiation and 80% of the outgoing terrestrial radiation), estimate the average surface temperature of the Earth four billion years ago using a single layer atmosphere.
(ii) In fact, the Earth’s atmosphere was drastically different 4 billion years ago (15-20% CO2, 0% O2). Suppose the atmosphere absorbed 5% of the incoming radiation but 95% of the outgoing radiation. What would be the temperature of the early Earth surface and its atmosphere?
The models that you need are described here:
http://www.scienceinschool.org/2008/issue9/climate
That is as far as I am prepared to go helping you with this. Perhaps another teacher will be able and willing to pursue it further.
To estimate the average surface temperature of the Earth four billion years ago, we will use a simplified model of Earth with a single layer atmosphere.
(i) Assuming the radius of the Sun and the Earth's atmosphere were the same 4 billion years ago, we can use the concept of the Earth's energy balance to estimate the average surface temperature.
The Earth's energy balance can be represented as follows:
Incoming solar radiation = Outgoing terrestrial radiation
Since the Sun’s radiative output was 30% less compared to today, we can calculate the remaining incoming solar radiation as follows:
Remaining incoming solar radiation = (100% - 30%) = 70% of the current solar radiation.
Given that the Earth's atmosphere absorbs 10% of the incoming solar radiation and 80% of the outgoing terrestrial radiation, we can calculate the effective radiation reaching the Earth's surface:
Effective incoming solar radiation = Remaining incoming solar radiation * (1 - Atmosphere absorption) = 70% * (1 - 10%) = 63% of the current solar radiation.
Next, we need to consider the outgoing terrestrial radiation. Since we assume the Earth's atmosphere is the same as it is now, 80% of the outgoing terrestrial radiation will be absorbed by the atmosphere:
Effective outgoing terrestrial radiation = Outgoing terrestrial radiation * (1 - Atmosphere absorption) = 100% * (1 - 80%) = 20% of the current outgoing terrestrial radiation.
With the Earth's energy balance equation, we have:
Effective incoming solar radiation = Effective outgoing terrestrial radiation
63% of the current solar radiation = 20% of the current outgoing terrestrial radiation
To determine the Earth's average surface temperature, we can use the Stefan-Boltzmann law, which relates the surface temperature and the outgoing radiation:
Outgoing terrestrial radiation = σ * (Surface temperature)^4
Where σ is the Stefan-Boltzmann constant. Rearranging the equation gives:
(Surface temperature)^4 = Effective outgoing terrestrial radiation / σ
Surface temperature = (Effective outgoing terrestrial radiation / σ)^(1/4)
Substituting the values, we have:
Surface temperature = (0.20 / σ)^(1/4)
By solving this equation, we can find the average surface temperature of the Earth four billion years ago using a single layer atmosphere.
(ii) Now, let's consider the scenario where the Earth's atmosphere was drastically different 4 billion years ago, which means the absorption properties are different.
In this case, we assume the atmosphere absorbed 5% of the incoming radiation and 95% of the outgoing radiation. The calculations follow a similar approach as in part (i):
Effective incoming solar radiation = Remaining incoming solar radiation * (1 - Atmosphere absorption) = 70% * (1 - 5%) = 66.5% of the current solar radiation.
Effective outgoing terrestrial radiation = Outgoing terrestrial radiation * (1 - Atmosphere absorption) = 100% * (1 - 95%) = 5% of the current outgoing terrestrial radiation.
Using the Earth's energy balance equation:
Effective incoming solar radiation = Effective outgoing terrestrial radiation
66.5% of the current solar radiation = 5% of the current outgoing terrestrial radiation
Applying the Stefan-Boltzmann law as before:
Surface temperature = (0.05 / σ)^(1/4)
By solving this equation, we can determine the average surface temperature of the early Earth with the different absorption properties in the atmosphere.
Keep in mind that these calculations make several simplifications, and the actual temperatures might have been influenced by various other factors during that period.