# Temperature as function of $CO_2$

The carbon stored in fossil fuels must have been taken out of the earth atmosphere from CO2 in past periods by plants. Thereby reducing the amount of CO2 drastically and increasing the amount of O2 to current levels. But this change in the composition of the atmosphere did not cause an enormous cooling down of the planet. So how can it be that a small increase of CO2 now would significantly increase temperature. These processes should be proportional to the percentage of CO2 ?

• I deleted some comments that were not aimed at improving the question. Please remember that comments are supposed to critique or improve the post being commented on, not to give partial answers or to question the motivations of other users. – ACuriousMind Mar 13 at 18:52

The relationship is nonlinear, not even logarithmic. To oversimplify a very complicated subject ...

The key issue is the altitude of the tropopause, where convective transport through the troposphere gives way to radiative transport through the stratosphere. Convective instability ends where the adiabatic temperature gradient (a.k.a. lapse rate) exceeds the temperature gradient needed to drive radiative transport through blocked spectral bands, the latter being proportional to the opacity of the air in the blocked bands. (A wavelength is considered blocked from a given altitude if the air above is optically thick.) Increasing the opacity would raise altitude of the tropopause. Given the exponential thinning of the stratosphere, doubling the opacity would raise the tropopause about 4.5 km. Each extra kilometer of altitude could raise the surface temperature as much as 6.5 degC, according to the effective lapse rate of the Standard Atmosphere, and the theoretical adiabatic lapse rate in dry air is even higher, 9.8 degC/km.

Doubling CO2 would not even double the opacity of the dry air in the stratosphere, since CO2 only blocks a band from ~13 to ~17 microns. Moreover, much of this band is said to be “saturated” since absorption cannot exceed 100%, no matter how much CO2 there might be. Wikipedia has a nice figure of the stratospheric transmission spectra, which shows that additional CO2 would merely broaden the shoulders of the absorption band.

• On the website you refer to I found the formula relating change in temperature to the relative concentration of CO2: $\Delta T_2=\lambda 5.35 \ln \frac{C}{C_0}$. Back to my original question: how can it be that a severe reduction in CO2 during the great oxygenation event did not freeze the planet ? Suppose that we assume the CO2 levels dropped to 1% then the temperature would fall with about 20 K which explains why the earth didn't freeze. So indeed there is a great range of CO2 levels possible which sustain life on earth. – Jack Mar 13 at 14:22
• "Thereby... and increasing the amount of O2 to current levels" This assumption is wrong. Oxygenation is thought to have resulted from the reduction of FeO2 in the oceans by cyanobacteria. See en.m.wikipedia.org/wiki/Great_Oxygenation_Event. You should study a whole lot more if you want sustain your ambition level. – my2cts Mar 13 at 15:03
• "this change in the composition of the atmosphere did not cause an enormous cooling down of the planet." What is this assertion based on? – my2cts Mar 13 at 15:05
• Yes the oxygenation started with the production of oxygen by bacteria, but the result was that ozone was formed in the atmosphere and this blocked UV radiation from the sun, so life could further develop on the planet, it was photosynthesis that ultimately reduced the levels of CO2 – Jack Mar 13 at 15:39
• I learned from the answer that the relation is logarithmic. The concentration amount of CO2 can fall -100% or rise 10000% with a temperature change from -20 K to 20 K – Jack Mar 13 at 15:49

It's worth noting that the amount of 'available' $$\mathrm{CO_2}$$ in the system is not constant over very long geological times. $$\mathrm{CO_2}$$ gets dumped into the atmosphere by vulcanism & other geological processes, and this then gets scavenged from it by life. So, although all the $$\mathrm{CO_2}$$ now stashed as coal & oil came from the atmosphere, there was no point at which the atmosphere had all the $$\mathrm{CO_2}$$ that is currently stashed in biomass & its products in it, because other processes are adding $$\mathrm{CO_2}$$ to the system over time.

That being said I think that it is indeed the case that before life started scavenging $$\mathrm{CO_2}$$ from the atmosphere it was indeed a very major component of it, and there was a really pretty large greenhouse effect. That's a good thing as the Sun was significantly dimmer, and that greenhouse effect was enough to keep the planet warm enough to stop the oceans freezing, which would have at least severely hindered the origin of life.

It's quite easy to see how large the effect must have been in fact. The Sun has got about $$30\%$$ more luminous over its lifetime: we know it now has a power of about $$3.9\times 10^{26}\,\mathrm{W}$$, so in the early Solar System its power output was about $$3\times 10^{26}\,\mathrm{W}$$. And you can then work out how hot a simple-minded black-body Earth would be in both cases (so, ignoring any greeenhouse effect or albedo, and also ignoring changes in orbital radius -- I think changes in orbital radius have been fairly small although I may be wrong).

Today, such a black-body Earth would have a temperature of about $$279\,\mathrm{K}$$, or about $$6\,\mathrm{{}^\circ C}$$. This is about $$7\,\mathrm{K}$$ below the actual average surface temperature.

At the start of the solar system, a black-body Earth would have a temperature of about $$261\,\mathrm{K}$$ or about $$-11\,\mathrm{{}^\circ C}$$.

So without a really significant greenhouse effect the oceans would have frozen and life probably would not have got started.

It's also worth noting that relatively large changes in surface temperature have indeed happened over the life of the Earth: life is perfectly possible with a surface temperature significantly warmer than it is now, and significantly colder (I'm not going to quote figures because I don't know them but probably ten degrees in either direction is perfectly fine, fifty isn't). When temperature changes very rapidly with time, as we are now experiencing, it's not good news, but even then it's not actually a threat to the existence of life as a whole.

An example of what happens when there is no life to scavenge $$\mathrm{CO_2}$$ dumped into the atmosphere over long periods by geological processes is Venus, of course.