The faint young Sun paradox describes the contradiction between the fact that there must have been liquid water 3.5 billion years ago for life to have begun then and the fact that the Sun's power output would only have been 70% as intense as it is now. The issue was raised by astronomers Carl Sagan and George Mullen in 1972. A recent explanation is that there must have been a lot of carbon dioxide (20,000 ppm) and methane (1000 ppm) in the early atmosphere to produce enough greenhouse effect at that early time to raise the Earth's temperature above water freezing temperature.
However, perhaps the solution lies in the fact that the Earth was nearer to the Sun 3.5 billion years ago and therefore received enough solar energy to warm its surface above water freezing temperature even though the Sun's power output was 70% of what it is today. There is evidence from radiometric measurements that the Earth-Sun distance is increasing by about 15 meters per year. This is in fact approximately the rate of increase one would expect if the solar system is expanding in the same manner as the Universe at large.
A rough calculation of the present temperature of the Earth
The solar flux density, $F$, going into the Earth's climate system is:
$$F = \frac{S}{4} (1 - A),$$
where $S$ is the solar constant and $A$ is the albedo. The solar constant $S$ is the power per unit area at the Earth's current mean distance from the Sun. The albedo $A$ is the fraction of solar power that gets reflected back into space. The factor of 4 comes from the fact that the Earth receives solar energy over its cross-sectional area but that energy is delivered to the climate over its surface area.
We take the value $S=1366\ $Watts/m$^2$ for the current solar constant at our present distance from the Sun. According to NASA the current albedo of the Earth is $A=0.306$.
Therefore we find that the current flux density $F$ is given by:
$$F = \frac{1366}{4}\left(1 - 0.306 \right) = 237\ \mbox{Watts/m}^2.$$
Now, by conservation of energy, the Earth must radiate out as much energy as it receives (in the long run) so that by inverting the Stefan-Boltzmann law we have:
$$T_e = \left(\frac{F}{\sigma}\right)^{1/4},$$
where the Stefan-Boltzmann constant $\sigma=5.6704\times 10^{-8}$W m$^{-2}$ K$^{-4}$.
Plugging in our value for $F$ we find the Earth's current temperature is predicted to be:
$$T_e = 254\ K = -19\ C.$$
In fact the current surface temperature of the Earth is about $288$ K or $15$ C. The extra $34$ K is due to the greenhouse effect created largely by water vapor and carbon dioxide in the Earth's atmosphere.
We have good evidence that the first life probably started in bodies of water about 3.5 billion years ago. The standard model of the Sun indicates that it was 30% less bright then as it is now. Let us calculate the temperature of the surface of the Earth at that time assuming no greenhouse effect.
The solar constant $S$ would then be $S=0.7 \times 1366=956$ Watts/m$^2.$
Assuming the same albedo $A$ as we have at present we calculate the solar flux density $F$ going into the Earth's climate as $F = 166$ Watts/m$^2.$
Using the Stefan-Boltzmann law we find the surface temperature of the Earth, without greenhouse effect, would have been:
$$T_e = 233\ K = -40\ C.$$
Thus under the assumption that the Earth-Sun distance does not change we would require a hefty greenhouse effect 3.5 billion years ago to ensure that there was an appreciable amount of liquid water on the surface of the Earth to allow life to develop.
Finally, let us assume an alternate hypothesis that the Earth was nearer to the Sun 3.5 billion years ago. As mention above there seems to be some experimental evidence that the Earth is currently receding from the Sun at a rate of $15$ meters per year.
At present the semi-major axis of the Earth's orbit is $a_{now}=1.49598\times 10^{11}$ meters.
If we extrapolate back using the current rate of Earth recession we find that 3.5 billion years ago the semi-major axis would have been $a_{past}=9.7098\times 10^{10}$ meters.
Therefore if the Earth was closer to the Sun the intensity of solar flux density, $S$, hitting the Earth would be increased over the present value by a factor $f$:
$$f = \left(\frac{1.49598\times 10^{11}}{9.7098\times 10^{10}}\right)^2 = 2.37.$$
Therefore applying this factor as well as the 30% reduction in the early Sun's intensity we find that 3.5 billion years ago the solar flux density would be:
$$S = 2.37 \times 0.7 \times 1366 = 2266\ \mbox{Watts/m}^2.$$
Assuming the same albedo $A$ as we have at present we calculate the solar flux density $F$ going into the Earth's climate as $F = 393$ Watts/m$^2.$
Using the Stefan-Boltzmann law we find the surface temperature of the Earth, without greenhouse effect, would have been:
$$T_e = 288\ K = 15\ C.$$
By coincidence that is the same as the present surface temperature of the Earth but without greenhouse warming. Certainly warm enough to sustain early life in liquid water.
