Prove: If atmospheric non-radiating O2 is exchanged for radiating (absorbing/emitting) CO2, emissivity, e, of planet to space must increase and corresponding global radiating temperature must decrease. More generally, if any “greenhouse” gas displaces a “non-greenhouse” gas, planet will cool.
The Stefan-Boltzmann Law of radiation intensity emitted by all matter in the universe is:
I = σ e (T/100)4
If e increases at constant I, T goes down, by algebra. Therefore if CO2 increases e at constant I, T goes down, causes global cooling.
I = intensity of any radiating body, w/m2 of its spherical surface, Earth emits and transfers radiant energy to outer space surroundings at average rate Io = 239. This is measured by satellite spectrophotometers.
T = temperature of radiating body, K
e = emissivity of radiating body, fraction 0 < e < 1. Perfect radiator black body e = 1, radiates a given intensity at lowest possible temperature. Perfect reflector e = 0.
σ = Stefan-Boltzmann radiation law constant, 5.67
NASA uses this relationship with an undisclosed estimate of e to measure (deduce) average global temperature.
GHGT Kiehl-Trenberth (http://chriscolose.files.wordpress.com/2010/03/kiehl4.jpg) atmosphere energy flows diagram says surface radiates with intensity 40 (at 15C) directly through atmosphere to space.
Is = 40 = 5.67 * 0.1025 * [(14.88 + 273.15)/100]**4
This says surface emissivity to space is about 0.1025.
Same GHGT K-T energy flows diagram says atmosphere radiates with intensity 199 (at -18C) directly to space.
Ia = 199 = 5.67 * 0.830 * [(-18.15 + 273.15)/100]**4
This says atmosphere emissivity to space is about 0.830.
If true, we have good estimates for each emissivity: es = 0.1025 and ea = 0.830.
Next we can find a combined weighted average global emissivity from atmosphere and surface.
e = (es * Is + ea * Ia ) / I = (0.1025 * 40 + 0.830 * 199) / 239 = 0.708243
Since absorptivity = 1.0 – reflectivity – transmissivity = 1.00 – 0.30 – 0.00 = 0.70; we confirm Earth’s global emissivity = 0.708 is very close to its absorptivity 0.70, in close agreement with Kirchhoff’s Law. Deviations are due to energy transfer within body by non-radiation processes, like photosynthesis.
Colorful Earth radiator e = 0.708 emits the measured intensity at average global T = 4.61C.
Io = Is + Ia = 40 + 199 = 239 = 5.67 * 0.7082 * (4.6068 + 273.15)**4
If Earth were a black body emitter, its global T would be -18.35C
239 = 5.67 * 1.0 * (-18.35 + 273.15)**4
Actually GHGT promoters say it is a colorful 0.612 emitter,
239 = 5.67 * 0.612 * (14.93 + 273.15)**4
at T = 14.93C, which is closer to surface T = 14.88.
The difference 14.93 – (-18.35) = +33.3C is the difference between colorful Earth’s radiating surface temperature and its theoretical black body equivalent when radiating at same intensity, 239. This 33C value is without physical meaning.
Since surface T = 15C, James Hansen, author of GHGT, invented a Green House Effect: GHE = surface T – black body global T = 15 – (-18) = +33C, which has no physical meaning. He erroneously attributed it to his newly invented greenhouse gases. This is the foundation of GHGT and subsequent research to reduce Earth’s CO2, mistakenly thinking it increases T when basic physics says it decreases T.
This S-B Law is valid for Earth’s radiating mixture of solids, liquids and gases; atoms, molecules and ions; blue, white, green & brown; thick equatorial atmosphere and thin polar atmosphere; hurricanes and tornadoes, floods and droughts, oceans and forests and deserts and ice, day & night; rotating and orbiting. The emissivity value 0.708 is an effective average physical property of atmosphere and surface that makes the measured intensity and temperature agree.
The difficulty is those e values are hard to estimate independently from the properties of Earth’s radiating mixtures. But they exist and can be inferred from I and T measurements with S-B Law. However average global surface and atmosphere T measurements are very hard to determine. Chemists can determine the effect of atmospheric CO2 on atmospheres emissivity ea = 0.830, but it is a complicated calculation. Most of the emissivity is due to liquid H2O with ew near 0.9.
[Emissivity of radiating gasses varies with T, P, composition and beam length. Most emissivity’s are determined for design of commercial furnaces, 100C < T < 2000C, rather than Earth’s low temperature, pressure and composition atmosphere. e of CO2 is about 0.18 from T = -18 to 100C at Pt = 1 atm and P*L = 3 ft-atm, where P is partial pressure of CO2 and L is beam length. e of H2O gas is about 0.48 at these conditions. Emissivity of CO2 and H2O decreases with lower P of atmosphere with altitude but may increase with lower T. Emissivity of O2 = 0. J Perry, “Chemical Engineer’s Handbook”, 1950, pg 490.]
Fossil fuel combustion converts non-radiating O2 molecules to radiating CO2 molecules. Since absorptivity and emissivity of CO2 > O2, the emissivity of Earth’s atmosphere must increase with CO2. Therefore, if Io does not change with CO2, T decreases, cooling.
If you accept exchanging non-radiating O2 molecules for radiating CO2 molecules by fossil fuel combustion, you can answer the first question: does that increase or decrease ea and then T? It increases ea and decreases T. If you wish to quantify the effect of CO2 on e and T, you must justify the work to find the effect of CO2 on ea, which will not be very accurate.
To check climate sensitivity, CS, assume doubling CO2 from 400 to 800 ppmv increases ea by 0.001, from 0.830 to 0.831.
e = (es * Is + ea * Ia ) / I = (0.1025 * 40 + 0.831 * 199) / 239 = 0.709075, an increase of 0.000833
Io = Is + Ia = 40 + 199 = 239 = 5.67 * 0.7091 * (4.5211 + 273.15)**4
So T drops from 4.6068 to 4.5211; CS = -0.08576C/doubling.
It is easy to conclude this is vanishingly small, no matter what the effect of CO2 on ea might be.
So global sensitivity to doubling CO2 is CS = -0.08576C, global cooling. Controversy resolved by physics and elementary algebra. No need for $1 billion/day research since 1996 to prove the impossible, global warming. If you disagree with Stefan-Boltzmann radiation law of physics, used successfully since 1884, take it up with them, not me.
Turning to effect of CO2 on intensity Io = 239.
Earth receives and absorbs radiant energy from outer space, sun, at rate
Isu = (1 – Albedo – transmissivity) S/4,
S = solar radiation intensity intercepted by Earth; 1365 to 1370 w/m2 incident disk or 1365/4 to 1370/4 w/m2 of incident sphere.
Albedo = reflectivity, fraction reflected back to space, mostly by clouds, estimate 0.3. Assume independent of CO2. If not and the relationship is provided, it can be accounted for easily.
Transmissivity = 0.0
Isu = (1 – 0.3) 1368/4 = 239.4
Input rate = Isu + sum Qi from volcanoes, core, combustion, other effects
sum Qi is small, estimate 0.1, and not affected by CO2.
Output rate = I0 + sum Qo
sum Qo = 0.5, mostly by photosynthesis, a cooling effect which increases with CO2, The known rate of reaction and deforestation are easily accommodated.
Conservation of energy, First Law of Thermodynamics, FLoT, says energy is neither created nor destroyed: output rate = input rate.
Io + sum Qo = Isu + sum Qi
239.0 + 0.5 = 239.4 + 0.1
Substituting: I0 = (1 – Alb) S/4 + sum (Qi – Qo) = σ e (T/100)4
Dividing by σ e: (T/100)4 = [(1 – Alb) S/4 + sum (Qi – Qo)] / σ e = Io / σ e
[4.601 + 273.15)/100]4 = [(1 – 0.3) 1368/4 + sum (0.1 – 0.5)]/5.67 * 0.7082 = 239/5.67 * 0.7082
This basic, rigorous equation gives T as function of all input variables and physical properties affecting T. It includes only two laws of physics, S-B Law of radiation intensity and conservation of energy for Earth. There is no need for statistical fitting of data to empirical correlations; particularly inaccurate T and CO2 data. The only terms that might be influenced by CO2 are e and Qo (and perhaps Alb), both cooling.
If S or sum Qi increases, T increases. If Alb, sum Qo, or e increases, T decreases.
An aside. Since heat capacity, Cp, of CO2 is greater than the heat capacity of the O2 it displaced by the oxidation reaction, increasing CO2 increases heat capacity of the atmosphere. This rotates the temperature vs altitude profile counterclockwise about its centroid, at about 5 km and -18C, since its slope for any planet is –g/Cp, easily derived from conservation of energy, FLoT, as kinetic energy is converted to potential energy with altitude, cooling. While bulk average global atmospheric temperature is unaffected, lower altitude air warms and upper altitude cools. Surface would warm accordingly.
There are several mechanisms for CO2 to affect temperatures; I have identified two warming and four cooling and described one surface warming and two global cooling’s here. My best guess net effect is within -0.2C < CS < +0.1C. No wonder data regression models can’t find it.