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I've read not long ago that the absorption line of the $\text{CO}_2$ becomes larger with temperature thanks to Doppler effect (if it's false it would be great to tell me). My question is the following: Does this imply, like I think, that more energy is "stored" on Earth (thanks to greenhouse effect) and so play a role in climate change?

If the answer is yes, how big is this role?

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    $\begingroup$ Your are asking about a very tiny (third-order?) feedback effect. If the absolute temperature rise is 1%, the doppler broadening will increase 0.5%. I think you will find that collision broadening is greater than doppler broadening throughout the troposphere. For example, the effective width of oxygen's microwave absorption lines due to collisions is about 0.3 GHz at sea level. $\endgroup$ – Bert Barrois Jun 4 '18 at 12:34
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No. Not in a significant way.

The full width at half maximum (FWHM) of a spectral line widened by thermal Doppler broadening is

$\Delta f_{\text{FWHM}}={\sqrt{\frac{8kT\ln(2)}{mc^{2}}}}f_{0}$

where $m$ is the mass of the emitting particle, $T$ is the temperature, $f_{0}$ is the frequency of the spectral line, and $k$ is the Boltzmann constant.

Let's look at the CO$_2$ absorption lines:

enter image description here

(Image from Wikipedia commons, source)

Let's take for example the band roughly between 12 and 20 $\mu m$. That corresponds to frequencies from 15 to 25 THz, with a mean of 20 THz.

If we solve the above equation for the mass of a CO$_2$ molecule ($44.01/6.02\times 10^{23}$ g), a frequency of 20 THz and an Earth's mean temperature of let's say 15°C (288.15 K) we get 1158 kHz broadening, and for an scenario 2°C warmer it would be 1162 kHz.

Therefore, the width of the absorption band would grow 8 kHz (4 kHz on either side). Which considering the original width of 10 THz corresponds to a 0.00008% increase. This could be roughly equated to an analogous increase in absorbed energy, which is absolutely negligible for all practical effects.

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