Carl Sagan proposed the faint young paradox, where liquid water could not exist on Earth with the faint young sun having an energy output of only 70% of what it is today. This is not only not a paradox, but an illustration of the lack of understanding of thermodynamic in astrophysics circles.
With the current average solar radiation at the top of Earth's atmosphere determined to be 1360.8Wm-2, the faint young sun energy would have been .7*1360.8=952.6Wm-2. Felt at the surface of the ocean, that 952.6Wm-2 is capable of raising the specific heat of water by 952.6 Joules per gram. With the specific heat of water being 4.2Joules/gram per degree C, 952.6/4.2=226.8 increase in temperature per gram of water at the surface skin layer. To illustrate, consider a cubic meter of water equal to 1 million layers of one gram of water each, i.e. there are 1000kilograms of water per cubic meter with the density of water being 1kilogram per liter. There are 1000 gram per liter or kilogram, one cubic meter of water has a mass of 1000 kilograms. Applying heat at a rate of 952.6Wm-2 per second for a period of one hour would increase the energy of the cubic meter of water by 952.6*60*60=3.429,360 Joules or 3.43Joules per gram of water. For a period of one hour at solar noon. For a equatorial location, one half of the 952.6Wm-2 would be the average energy applied for a 12 hour day. 952.6*4.2*60*60*12/2=86,419,872 Joules per cubic meter or 86.4 Joules per gram. The true question is if the energy gained in the day would be lost at night?
The specific heat of dry air is 1.006 Joules per gram more than 4 times lower than the specific heat of water at sea level. At the surface skin layer, the boundary between the liquid water and the dry air, the water would have to transfer energy to the air and cool. The paradox is that the surface skin layer would also radiate to space. Unlimited, the rate of cooling to both air and space would exceed the rate of warming.
The minimum surface temperature of liquid salt water is -1.9C which has a radiant energy of 307Wm-2. With an average solar energy applied of 952.6Wm-2/4 the average global energy, the energy in, 238.15Wm-2 which is less than the minimum radiant energy of a liquid water surface. However, for a tropical location, the average is actually, 952.2/2=476.1Wm-2 Allowing for the true orbital maximum insolation which is higher, the loss of energy for the mass of water gained during day only has to equal or less than the loss at night.
To set up or faint young sun scenario consider that as ice is formed in salt water, most of the salt is expelled as the ice forms resulting in fresh ice. The melting point of fresh ice is 1.9 C greater than the freezing point of salt water. As the salt water warms it would transfer heat to the surrounding fresh ice. The specific heat of fresh ice is 2.1Joules per gram. With all sides and the bottom of the cubic meter or liquid salt water enclosed by fresh ice, there would be 2.1*5=10.5 Watts per second transferred to the fresh ice. As the ice gains energy, it would approach the latent heat of fusion of 334Joules per gram at a temperature of 0 C degrees. Since the salt water can be at a lower temperature than the temperature of the latent heat of fusion, the energy flow from the salt water to fresh ice would approach zero, providing perfect insulation at this skin layer until either the salt water warmed above -1.9 C or froze, at which point it would have to release 334 Joules per gram. At equilibrium there would be no heat exchanged between the salt water and the fresh ice.
That leaves the surface exposed to the atmosphere. As the surface skin layer of the salt water released heat to the air, the specific heat capacity of the air would limit the heat flow. With a specific heat of 1.006, the conductive flow rate would be equal to 1.006 Watts per meter squared for still air. The radiant energy of the surface would be 307Wm-2 from the salt water but would have to increase to 316Wm-2 if the skin layer lost enough energy to form ice. There would again be an insulating effect of fresh ice now on all sides of the cubic meter of salt water. The rate of heat loss would be limited by the specific heat of fresh ice at -1.9 C degrees.
Our cubic meter of salt water now contains 86,419,872 Joules and on all sides is insulated by fresh ice. The rate of energy flow from the center of the cube is totally dependent on releasing 334 Watts per meter squared on six sides limited by a temperature difference of 1.9C, the difference between the temperature of the freezing salt and melting fresh ices. The surrounding ice insulation barrier limits the energy exchange to 2.1*6=12.6Wm-2 per second. over the course of the night, the salt water would lose, 12.6*60*60*12=544,322 Joules plus 6*334=2004 Joules for the surrounding heat of fusion of the initial skin layer of ice. At the skin ice layer exposed to the atmosphere, the initial radiant loss would be 316Wm-2 which would decrease as the skin layer cooled at a faster rate that the internal energy could provide. The radiant energy would decay to 2.01Wm-2. Assuming that the rate of decay to 2.01 resulted in half the initial emission energy of 316Wm-2, then 316*60*60*12/2=6,825,600 Joules. The final energy of the cubic meter of salt water would be 86,419,872 Joules minus 6,825,600 Joules radiated minus 544,322 Joules conducted plus 2004 Joules latent heat for the fusion of the initial skin layer resulting in 80,136,590 Joules of retained energy. So if salt water existed for any reason in the tropics, a faint young sun could maintain the liquid water and expand its area.
From a frozen surface the puzzle is different. If the albedo of the surface does not allow a minimum of 2 (for the full 24 hours) times 6,825,600 Joules to be absorbed per cubic meter then there would be no liquid water surviving until the next solar noon. The albedo of clear ice would allow nearly full absorption but for only solar noon plus and minus one to two hours. At 952.6Wm-2 per second the maximum total energy absorbed per hour would be 952.6*60*60=3,429,360 for full absorption. Not enough energy would be provided to ensure liquid water at the surface. Enough energy though to help melt a skin layer of ice form a liquid water starting point.
Depending on the geometry of the pool of liquid water, the volume would expand more rapidly than the surface area. The boundary of the liquid water would require some minimum energy. For the liquid water example, 6,825,600Joules is the approximate minimum required to maintain liquid salt water. That is only an average energy at the surface of 158Wm-2 either by solar or any other means of heat transfer. That means the faintest faint sun could be 632Wm-2 at the surface if any volume of liquid water existed.
Feel free to review the calculations and the logic, I will review this later myself.
Geothermal energy below the surface would be the key for the initial formation of liquid water if a snow ball Earth ever existed. With the large number of equatorial submarine volcanoes, it is unlikely that a snow ball Earth existed for long if it did at all.
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