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An article in yesterday's New York Times by Andrew Revkin dealing with rise in sea level from melting glaciers cites Dr. Eric Rignot of NASA as saying that unabated global warming could result in 3 feet of sea level rise from melting ice on Greenland and another 3 feet from Antarctica and about 18 inches from the melting of Alpine glaciers. He quotes Dr. Rignot as stating: " it is too early to reassure that all will stabilize and similarly there is no way to predict a catastrophic collapse," ....." But things are definitely far more serious than anyone would have thought five years ago." (Emphasis mine).
There's an old expression-"Stop the world, I want to get off." Well just in case the worst does happen, we may have think of having to go elsewhere(getting off in a sense). Where? I don't know, but here's the calculation on what upward speed we will need to get off.
We're going to determine the velocity of a particle projected outward in a radial direction from the earth and acted on by only one force, that of gravity.
According to Newton's theory of gravitation , the acceleration of the particle will be inversely proportional to the square of the distance of the particle from the center of the earth.
Let r be that distance and R be equal to the earth's radius, t is time, a represents acceleration and k is the proportionality constant in Newton's law.
Then a = dv/dt=k/r^2
Since the acceleration is negative because the velocity is decreasing as it travels upward, then k is negative. When r=R and a =-g, then g=-k/R^2. Substituting for k above, gives a= -gR^2/r^2.
Since a =dv/dt and v=dr/dt then:
a= dr/dt(dv/dr) =vdv/dr so that our diff equation is : vdv/dr= -gR^2 / r^2
The solution is v^2= 2gR^2 /r +c
When v=v0 and r=R then c = vo^2- 2gR so that a particle projected radially outward will go with a velocity
v^2=2gR^/r +v0^2- 2gR.
In order to escape from the earth the quantity v0^2-2gR > 0 (must be equal to or greater than zero).
So then the vo=the square root of 2gR is the escape velocity required . R=3,960 miles(approx.) and
g= 6.09x10^-3 miles/second^2, so that v0 =6.95 miles per second. Frictional air resistance in the lower atmosphere may not be negligble so that a slightly higher value may be required.
In other words to offset the projected of effects of global warming by quadrupling the atmospheric CO2 we would merely have to move the Earth about 3 million kilometers further from the Sun. (Don't try this at home) I say 'merely' but there are downsides to this solution. For one thing we could overshoot and one of the outcomes could be that we'd become a satellite of Jupiter! Wouldn't that be a hoot!? There would also be effects from the force caused by the initial acceleration in accord with Newton's second law. Who knows what these effects would be. Earthquakes? Seismic sea waves a hundred meters high? Perhaps this isn't such a good idea after all. We'd probably do well to doing things like switching to alternative fuels.
I believe I've figured out what we must do. Here it is the middle of the 21st century and our home planet is getting warmer and warmer. There's barely any ice left on Earth, except for central Antarctica. The east coast of the U.S. is just offshore of Pittsburgh, where summer daytime temperatures compete with what used to be average for Texas or Arizona back in the year 2007.
What's needed is a solution, which I'll get to later on, to undo the warming or we are undone as a civilization. The main problem is that the carbon dioxide levels are nearly at 600 parts per million by volume, more than twice the concentration than existed in the beginning of the Industrial Revolution. This doubling means that the rate of heat energy coming to the Earth's surface as a result of the enhanced Greenhouse Effect has increased by 4.5 watts per square meter. We must somehow undo this by whatever means we have available. I've done some prelimination calculations and find that we are receiving a total solar intensity of heat rate in the upper atmosphere of about 1400 watts per square meter. ( To be continued)