At the moment, the Earth is warming because the solar energy entering the atmosphere exceeds the infrared energy that is radiated back into space. If the concentration of greenhouse gases can be stabilized, a certain equilibrium temperature will eventually be reached where the radiated infrared energy will balance the incoming solar radiation. Just as your home air conditioner takes time to readjust the temperature of your home after you adjust the thermostat, the Earth takes a considerable period to rise to a new equilibrium temperature after its thermostat has been turned up by increasing emissions of greenhouse gases. The change in the equilibrium temperature due to a given change in the concentration of greenhouse gases is known as the "climate sensitivity," and you may find it useful to search for this term when doing your own research.
There are two approaches scientists take when attempting to calculate climate sensitivity. The first uses computer modeling using the known properties of CO2 and other greenhouse gases, combined with estimates of the feedback loops which amplify the temperature rise. Because of the difficulty in estimating the latter, there are large variations in the projections. For a doubling in CO2, the projected rise in temperature varies from 1.5 to 5.2°C. These are only medium-term models, however, and the long-term model, on which I focus, projects a rise in temperature of 4.7°C (8.5°F).
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