This is an introductory talk aimed at undergraduates with some familiarity with ordinary differential equations.
In 1801 the great astronomer Sir William Herschel suggested that our Sun, like other variable stars, might vary in brightness and that variability might be responsible for changing the climate on Earth. He noted that the price of wheat was high during periods when the dark spots were largely absent on the face of the Sun, presumably leading to cold weather and the scarcity of wheat. Note the role played by proxies---the sunspot number as a proxy for the sun's brightness, and the price of wheat as a proxy for the cold climate in Europe---since no reliable direct measurements were available of the Sun's irradiance at the top of Earth's atmosphere and of Earth's global temperature. Note also that no theory existed that quantitatively connected the variation in the solar irradiance to the climate change on Earth, and that it had not been shown that the correlation of the two was statistically different from random coincidence. Such unsatisfactory state of science persisted for two centuries. Despite thousands of scientific publications, the field of sun-weather connection fell into disrepute, according to some. A brief renaissance occurred when the astrophysicist John Eddy published his famous (and longest Science) paper in 1976 tying the lack of sunspots in the Sporer Minimum of the Sun in the 16th century and the Maunder Minimum in the 17th century to the Little Ice Age in Europe, supplemented by the Carbon-14 proxy in tree rings. Since 1979, NASA satellites have been able to directly measure some variability of the solar irradiance at the top of the atmosphere, covering almost four cycles of the 11-year sunspot cycle. We now also have reliable measurements of global temperature just long enough to statistically extract the solar signal from the other climate noise it is embedded in. So the time is ripe for completing a quantitative calculation of the Sun-Earth climate connection.
Recently the effect of the Sun on climate has become a hot topic of scientific debate again, as some researchers suggested that the recent trend of global warming, which most thought to be caused by man's activity, specifically the carbon dioxide emission, may instead be the result of a more active Sun. Since the secular trend of the Sun's irradiance increase is very small, it has been suggested by some that the climate may have a "hypersensitivity" to solar change as compared to the response to greenhouse gas change. So far no viable mechanism has been suggested that can explain the magnitude of the Earth's response to the small change in solar irradiance, and the debate continues. I will show, using the observed temperature data and a simple mathematical model, that the mechanism of climate's response to solar irradiance variability is not so different from that to the radiative forcing from greenhouse gases. In fact the observation of the Earth's response to the Sun's 11-year sunspot cycle can help us narrow the uncertainty in the prediction of our climate's global warming response to doubling carbon dioxide.