Planetary Temperature and Climate Change

I. Radiative equilibrium
A planet like the Earth has its surface T set by radiative equilibrium, where the average T of the surface is set by the need to radiate all the light that impinges on it from the Sun. We saw the equation for this in class:
F_S = L / 4 pi D^2
is the energy flux (per time per unit area) impinging on the Earth (for L the luminosity of the Sun and D the distance to the Earth), and
F_S = sigma*T_s^4 * (4 pi R_E^2)*(1 - A)*(1 + tau)
sets the average surface T_s of the Earth by radiative equilibrium. Here I use a better expression than the 1-A+tau I gave in class, because it works better when A and tau are not necessarily small. The 1-A factor corrects for scattering from the Earth's surface (which reduces the absorbed flux), and the 1+tau factor corrects for the excess optical depth tau encountered in the infrared (compared to the visible) by the heat attempting to escape Earth (as that heat gets returned to the Earth an average of tau times due to greenhouse gases in the atmosphere, where tau is approximately the optical depth of those gases).

If one ignores A and tau, one gets an average surface T of about 280 K, which is a little cool (the actual average T is more like 290 K). Thus the combined tau-A must raise T by about 3 percent, or T^4 by about 4 times that, or some 12 percent. Hence we infer (1-A)*(1+tau) is about 1.12. Measurements show the Earth's albedo A is about 0.3 (and much of this is from clouds), but there are other effects as well (such as the latent heat of water evaporation when sunlight hits the ocean) that reduce the amount of heat the surface must emit in the form of infrared light. If we lump all these effects into the effective albedo, it comes out more like A=0.6. Thus the tau needs to be about 1.7. This is a good thing, because otherwise the Earth's oceans would freeze, but global warming is occurring due to too much of this good thing.

II. The Greenhouse Effect on Earth and Venus
In our solar system, the hottest planet is Venus, even though it is not closest to the Sun. That is because a runaway greenhouse effect in its thick CO2 atmosphere produces surface T of about 470 K on average. Due to being closer to the Sun, we only expect Venus to be about 8 percent hotter than Earth (since it is 30 percent closer to the Sun), but actually it is over 60 percent hotter. Hence the greenhouse effect on Venus increases the flux from the Sun by a factor of about 6. On Venus the atmosphere is optically thick even in the visible spectrum, so this factor 6 correction to the incident flux looks like (1+tau_IR)/(1+tau_V), where tau_IR is the average optical depth in the IR (so tau_IR is like the number of times the energy is returned to the surface), and tau_V is the optical depth in the visible (so 1/(1+tau_V) is the fraction of sunlight that makes it to the surface in the first place). Venus' atmosphere has about 75 times the column depth of Earth's, so should have a tau_V that is in the 10-20 range, which implies its tau_IR should be of order 100. Infrared eyes on Venus could not see the Sun at all!

A complication with the concept of optical depth in the infrared is that actually it varies a great deal with wavelength, because the molecular bands responsible for it are stronger at some wavelengths than others. Any effective optical depth tau allows about 1/(1+tau) fraction of the light to get through (a more precise analysis of the radiative transfer is needed to be more accurate here), so if you have windows in the infrared opacity where the atmosphere is transparent (as on Earth), then 1/(1+tau) gives the fractional size of the wavelength space that corresponds to those windows (since that's the fraction of the infrared light that will get trhough). So on Earth, increasing the effective tau in the infrared means partially filling in the windows in the infrared spectrum, whereas on Venus all the windows are already full (it has a CO2 atmosphere that has almost 200,000 times more CO2 than Earth's atmosphere). The bottom line is, to calculate the greenhouse effect requires not only an accurate treatment of the radiative transfer, but also an accurate accounting of the wavelength dependence of the opacity, both of which are outside our current scope but go into the calculations that climate scientists must carry out.

III. Climate change
Partial filling in of the infrared windows is indeed what is currently happening on Earth, due to CO2 emissions. The fact that measured CO2 levels in the Earth's atmosphere have risen about 30 percent over the last 60 years has caused a general warming of the Earth's climate by about 1 Kelvin (which is about 2 degrees Fahrenheit). If this continues, the same (or larger) increase will occur in your lifetime, and a futher increase of some 3 degrees Fahrenheit could easily occur. That would have significantly unpleasant consequences that you do not want. It is unfortunate that the previous generation has largely ignored this issue, partly because it was not completely obvious that it was happening, and partly because one of the political parties has chosen to stick their head in the sand about it, but we are clearly at a point where the problem can be ignored no more. Sea levels are rising due to melting Greenland and Antarctic ice sheets, and the glaciers in "Glacier National Park" will all be gone in a few more decades. Indeed, the North pole (where Santa lives) is predicted to be open ocean in the summertime in just 10-20 more years, and when Santa's elves all drown, we will know that we waited too long to fix this.

IV. Why this will not be easy to fix
On the surface, we could just say, stop creating CO2 and we'll be fine. But this is easier said than done, because the reason we are creating CO2 is that the main way we release energy to function in the modern world is by replacing weak C-H bonds by strong C-O and H-O bonds. This is the inverse of what photosynthesis does, whereby plants use the energy from sunlight to break the strong C-O and H-O bonds they find in the CO2 and H2O in their environment, and replace them with C-H bonds, found in carbohydrates (that we eat for energy) and hydrocarbons (that we burn for heat and transportation). In the process of releasing the energy stored by photosynthesis, we return the weak C-H bonds to strong C-O and H-O bonds, releasing both CO2 and H2O, which are both greenhouse gases. The H2O doesn't much matter because there is a lot already, but the CO2 is being increased at a rate of about 0.5-1 percent per year, because there is not a lot of it already. In fact, the blue-green algaes that once populated our oceans have removed so much of the C from CO2, leaving O2 and bonding the C to H taken from H2O, that CO2 levels have dropped and O2 levels have risen-- to the point that bluegreen algaes are mostly gone. Yes, they polluted their own environment to the brink of extinction, but their loss is our gain-- we got both the O2 to breathe and the C-H bonds to eat, to support our energy-hungry aerobic metabolism. So we are in effect reversing several billion years of photosynthesis at a rate that would double the CO2 every century. Right now the CO2 is only 0.04 percent of the atmosphere, but it would only need to be doubled about 8 times to be a largely CO2 atmosphere, so we could in a sense reverse the effects on our atmosphere of billions of years of photosynthesis in less than a millennium, if we could avoid extinction for that long. Which of course we could not, so another way to get energy is going to be needed (or the carbon will need to be sequestered), and it all has to be done in an economically viable way. Right now the best prospects for this are the Sun and radioactive elements, where we can either directly absorb sunlight, or use the wind energy generated when the Earth absorbs it for us, or we can put lots of radioactive elements in a confined space to get them to accelerate their own decay rate. The technology to do these things already exists, what is lacking is only the economical motivation to do it. Perhaps it will eventually be possible to borrow from the source of energy the Sun itself uses-- fusion of hydrogen. We have been 20 years away from successful H fusion for at least 60 years now, but perhaps there can be a breakthrough there at some point.

Other options are to continue to use photosynthesis, in the form of biofuels, though at present these are not net contributors of energy because of all the energy requirements of the fertilizer and transportation costs. What will never be a net contributor of energy is "hydrogen fuel", say for cars, because although burning hydrogen is indeed a great source of energy (again because of those strong H-O bonds being created), one cannot "mine hydrogen"-- so one must first break the H away from something it is already bonded to, which inevitably requires as much energy as is later being released by the hydrogen burning. So we have to think in terms of the ultimate source of the energy, which pretty much always has to be nuclear energy from fusion (inside the Sun or in tokamaks) or fission (from radioactive elements that can be mined). Otherwise, it's back to breaking C-H bonds and replacing them with C-O bonds, which is indeed the cause of global warming.