Yes, the Sun is getting hotter as it evolves across the main sequence of the Hertzsprung-Russell diagram. However, this is happening on a time scale of hundreds of millions of years. You are right that it has absolutely no effect on our climate today, but it will over the next 200 million years or so. Water vapour does not escape into space, but it will become a larger part of the atmosphere as the Earth heats up, and water vapour is a very potent greenhouse gas. Eventually the water vapour will be lost. First, as the atmosphere heats up the random motion of water vapour molecules will increase, so more of them will end up in the high-velocity tail of the Maxwell-Boltzmann distribution, and thus have enough speed to escape. Second the changes in the chemistry of the atmosphere will make it easier for water molecules to chemically dissociate (and cosmic rays will contribute too). The net effect is that water vapour will be lost, but on time scales of hundreds of millions of years. Plate tectonics are driven by convection currents in the mantle, but they are lubricated by water. If there are no oceans it is much harder for plates to subduct. Once the oceans are gone plate tectonics become much more difficult. This is thought to be the reason that Venus, a planet with essentially the same mass and internal composition as the Earth) shows no evidence for having plate tectonics. My understanding is that this is still somewhat hypothetical, but that there is an emerging consensus in the the geological community that oceans play a major r?le. Finally, plate tectonics do not drive geological CO2 emission, vulcanism does. While it is true that plate tectonics does cause vulcanism volcanos can happen without it, as we see on Venus, and in Hawaii. So, when plate tectonics stop there will still be CO2 emission from vulcanism. However, the carbonate-sillicate weathering cycle will have stopped, and this is the primary geological way of removing CO2 from the atmosphere. The net effect will be CO2 being added by volcanos, but with nothing to remove it the concentration of CO2 in the atmosphere will increase. Unless if we buy some Puppeteer world-moving technology Earth is in for a hot time around its 5,500,000,000th birthday.
Source: http://rss.slashdot.org/~r/Slashdot/slashdotScience/~3/sRTBjeJBm70/story01.htm
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