# Notes on the green house effect

### First calculation of the earth’s surface temperature

Calculate the surface temperature of the earth assuming no atmosphere.

The energy reaching the earth from the sun can be calculated from the blackbody radiation given off by the sun and is about . The surface of the earth is but only a circular cross section of faces the sun so if all the energy were absorbed the incoming power would be where  is the radius of the earth. In fact the earth’s albedo is about  which means the earth reflects about 30% of the incoming light, as mentioned in a previous chapter. The actual incoming solar energy absorbed by the earth per second is then  where .

The earth is also cooling by blackbody radiation given off into space and the amount, calculated from the Stephan-Boltzmann equation, is temperature dependent. The earth, then, will continue heating from incoming solar radiation until it reaches a temperature at which the outgoing blackbody radiation equals the incoming solar radiation. Only the side facing the sun absorbs energy but the entire surface area of   emits blackbody radiation (here we are assuming that the rotation of the earth distributes the average warming of the sun roughly equally over the entire surface). The energy leaving the earth is  where  is the earth’s surface temperature. Here we have assumed the earth without an atmosphere is a perfect blackbody with emissivity, , equal to one.

At equilibrium we can set the energy gain equal to the energy lost due to back body radiation which gives an approximate temperature of the earth’s surface for the case of no atmosphere. We have . So for the energy balance between incoming energy and outgoing energy we have

or  which gives .

The average surface temperature of the earth is actually around  which is  higher than value calculated.

### Second calculation of the earth’s surface temperature

Calculate the surface temperature of the earth assuming an atmosphere of a single layer, all at the same temperature. A schematic is shown in the figure below.

A simple radiative equilibrium model for the earth’s surface and a single atmospheric layer of uniform temperature.

We can get the approximate temperature for the atmosphere using the previous calculation for the balance between incoming solar energy and reradiated infrared energy. This calculation gave  which we will use here as the temperature of the atmospheric layer. This ignores convection, ocean evaporation and the fact that the atmosphere has a temperature profile which varies with height but is a useful first approximation.

The atmospheric layer allows visible light from the sun to pass through unimpeded but captures infrared emitted from the earth’s surface. This will cause the atmospheric layer to heat up until its blackbody emission (mostly in the infrared) is equal to the incoming infrared radiation from the earth. The atmosphere emits infrared in two directions so the energy balance between emission and absorption of radiation in the atmosphere is

where the emissivity of the atmosphere is used on both sides of the equation since this is the energy balance for the atmosphere. Simplifying gives .

The energy balance equation for the earth’s surface now has an additional term representing the extra infrared energy received from the radiating atmospheric layer. The surfaces of the earth and the atmospheric layer have the same area, , since they are in contact and we have, for an earth emissivity of one,

.

Substituting  into this equation gives

.

The emissivity of the atmosphere,  indicates how much energy it emits at a given temperature and depends on its exact chemical makeup. A value of   gives a surface temperature for the earth of  which is higher than the current measured average of . A value of   gives a surface temperature of , very close to the global average.

As noted above, the emissivity of the atmosphere is a key factor in determining the earth’s surface temperature. Emissivity is defined to be the ratio of the emitted radiation to that of a perfect blackbody at the same temperature. For objects in thermal equilibrium with their surroundings using blackbody radiation as the only means of heat transfer, the emissivity is equal to the absorptivity which is the ratio of energy absorbed to that of a perfect blackbody at the same temperature. An object which did not have this property could possibly absorb energy more efficiently than it emitted at a given temperature. This would break the second law of thermodynamics since the object could spontaneously heat itself at the expense of the surroundings without any input work. The absorption of radiation by the atmosphere is thus an important factor in determining the earth’s surface temperature since it equals the emissivity. In fact, about 20% of the incoming solar radiation is eventually absorbed by the atmosphere (from the earth and directly from incoming infrared from the sun) while 50% percent is eventually absorbed by the earth’s surface. In both cases the earth’s surface and the atmosphere will only heat up until they reach a given temperature, reradiating the absorbed heat as blackbody radiation to remain in equilibrium with incoming solar energy. About 4% of the solar radiation is reflected by the earth’s surface, 20% is reflected by clouds and 6% is reflected by the atmosphere, thus accounting for the 0.3 total albedo of the earth.

The figure below shows the solar spectrum at the top of the earth’s atmosphere and at the earth’s surface. The visible spectrum ranges from  to ; higher wavelengths are in the infrared range, lower wavelengths are ultraviolet. We found previously that for the sun with a surface temperature of  the peak in the solar spectrum is , squarely in the middle of the visible spectrum. A similar calculation for the surface of the earth, at , gives a peak wavelength of  which is in the far infrared.

The solar spectrum at the top of the atmosphere (upper curve) and at the earth’s surface (lower curve). Places where the spectrum has been absorbed or blocked by a specific atmospheric gas and does not reach the surface are marked by the respective gas (water vapor, carbon dioxide, oxygen and ozone) [12].

Wavelengths which are absorbed by the atmosphere show up as dips in the solar spectrum as measured at the earth’s surface. These dips are clearly evident in the figure (a more detailed picture for longer wavelengths is shown in the figure below). The atmosphere will absorb outgoing blackbody radiation from the earth at these same wavelengths, heating up until the net blackbody radiation emitted by the atmosphere equals the absorbed radiation. The main constituents of the atmosphere, nitrogen and oxygen, do not block reradiated infrared energy from the earth although they do block short wavelength ultraviolet (UV) radiation from reaching the earth. As mentioned in Chapter One, the UV blocking property of  and  is important in protecting life on earth from these harmful wavelengths.

Water vapor, although a very small percentage of the atmosphere, clearly plays a significant role in absorbing several parts of the solar spectrum as shown in the figure. In fact water vapor and water droplets in clouds together are thought to be responsible for approximately 65% of the  increase in surface temperature of the earth. Other gases, now referred to as greenhouse gases, which also play a role in absorbing infrared radiation are , , , ,  and the various chlorofluorocarbons. Carbon dioxide, much of which occurs naturally in the atmosphere, contributes as much as 25% of the increase in the earth’s surface temperature above the no-atmosphere prediction. Other greenhouse gases are thought to contribute the remaining measured warming effect of . As mentioned previously, without this  warming the earth would be too cold for most life as we know it. The concerns of climate scientists today are related to changes in these natural forcings and feedbacks.

A more detailed view of the absorption spectra of the atmosphere is shown in the figure below. The electrons associated with individual atoms and molecules have quantized energy levels. This means that only photons with precisely the right energy can be absorbed by changing an electron from one energy level to another. This is the basis for discrete absorption and emission spectra specific to a given element or molecule and is used by scientists to identify the particular elements or molecules in a sample of gas. In addition to the electron energy levels, complex molecules can have vibrational and rotational energies associated with the relative motion if their components. These energies are also quantized which means that the molecule can only change to a different frequency of vibration or rotation by the absorption of a discrete quantity of energy.

Figure 7.5 Absorption spectra of the atmosphere with the particular absorption of certain atmospheric constituents indicated.

As mentioned previously, the energy of a photon is given by  where  is Planck’s constant and  is the photon wavelength. For isolated molecules at low temperatures the particular quantized energies of a given molecule can only absorb photons of certain wavelengths (there is actually a small range of wavelengths due to Heisenberg’s uncertainty principle but that does not concern us here). For groups of molecules at temperatures above absolute zero this strict condition for absorption is relaxed a little because of the thermal motion of the molecules and collisions with other molecules. A molecule in motion already has a little extra kinetic energy so that when it meets a photon which doesn’t quite have the right energy it may still be able to absorb that photon by using some of its own kinetic energy to make up the difference. A molecule may also absorb a photon with a little too much energy if the extra energy can be transferred into kinetic energy of the molecule. Collisions of molecules at high pressure also have a similar effect on the emission and absorption of photons, allowing for a wider range of wavelengths to be absorbed. For these reasons the discrete absorption spectra spreads into broad bands as shown in the figures above. The location of the wavelengths of the blackbody spectra of the sun and of the earth are also shown in figure. Note that the atmosphere allows much of the suns energy to pass through but the reradiated infrared spectrum of the earth is mostly blocked.

 Gas Percent of the Earth’s Atmosphere Radiative Efficiency () Radiative Forcing  () Nitrogen () 78.1 Oxygen () 20.9 Argon () 0.93 Water vapor 0.48 Carbon Dioxide ( ) 0.035 ( anthropomorphic) Liquid and Solid Water 0.002 A feedback mechanism (see text). Neon () 0.0018 Helium () 0.00052 Methane () 0.00017 (anthropomorphic) Krypton () 0.00010 Nitrous Oxide () 0.00003 (anthropomorphic) Ozone () 0.000007 (anthropomorphic) Chlorofluorocarbons (CFC) 0.00000014 to (anthropomorphic) Aerosols 0.00000002 (anthropomorphic)

Major components of the earth’s atmosphere by percentage of molecules present and their radiative efficiencies [5], [14], [15]. Figures listed as anthropomorphic indicate forcings due to changes in the average background amounts of these substances.

Anthropogenic caused changes to climate forcing occurring over the past 250 years [5]. The lower graph shows the confidence in the data as estimated by the IPCC [5].