This is the fourth post in the series. The previous post connected radiation escaping to space with the atmosphere's temperature profile. Here is why adding CO2 still changes that radiation even when some absorption lines are already opaque.

1. What quantum mechanics actually says

A molecule absorbs infrared radiation through allowed transitions between its energy levels. The photon's energy must match the transition: energy difference = h × frequency, where h is Planck's constant. The transition must also satisfy the relevant selection rules; matching energy alone is not enough.

CO2's bending vibration produces an important absorption band near 15 micrometres. Rotation adds many closely spaced transitions. Molecular motion and collisions give the lines finite widths. These are measured spectral properties, not an assumption that CO2 absorbs every wavelength equally.

HITRAN's spectroscopy definitions and data parameters
MIT spectroscopy lecture: CO2 vibrations, lines and broadening

2. An opaque line centre is not an opaque spectrum

For an incident beam at a particular frequency, its transmitted fraction is:

transmitted fraction = exp(-optical depth)

Optical depth measures the absorbing material along the path, weighted by its absorption strength. Holding temperature, pressure and path fixed, doubling the CO2 amount doubles its contribution to optical depth.

For example, increasing optical depth from 1 to 2 reduces direct transmission from about 37% to 14%. Where optical depth is already 10, direct transmission is already tiny and adding more makes little difference to that beam. Both statements can be true at different frequencies.

The strong line centres can therefore be saturated while weaker parts of the band still respond. Adding CO2 primarily adds absorbers; it does not need to produce a large increase in total air pressure or individual line widths.

3. The atmosphere also emits

The transmission equation alone is not a complete atmosphere. Each layer emits infrared radiation as well as absorbing it, and collisions exchange energy among the gas molecules.

What reaches space includes radiation from the surface and from many atmospheric layers, attenuated along the way. More CO2 changes the layers and temperatures that contribute at each frequency. In much of the troposphere, temperature decreases with height, so shifting the contribution upward can reduce outgoing radiation at the existing temperatures.

That is not a claim that every wavelength comes from one altitude or that every atmospheric layer gets colder with height. The stratosphere matters too. The calculation integrates the whole spectrum and atmospheric profile. The surface and atmosphere then adjust until outgoing energy again balances absorbed sunlight.

Pierrehumbert: Infrared radiation and planetary temperature

4. Diminishing returns are still returns

Over a useful range around Earth's present conditions, CO2 radiative forcing is approximately logarithmic:

change in forcing = a × ln(C/C0)

C/C0 is the ratio of final to initial CO2 concentration. Equal percentage increases therefore give approximately equal forcing increments. Doubling adds a × ln(2); another doubling adds approximately the same again. That is different from adding the same number of ppm each time, and it is not a temperature formula.

The coefficient depends on the forcing definition and atmospheric conditions. A doubling produces a forcing of roughly 4 W/m². Understanding the logarithmic relationship requires the distribution of absorption strengths across the band; merely saying 'the lapse rate' does not derive it.

Romps et al.: Why the forcing from carbon dioxide scales as the logarithm of its concentration

5. There is an observational check

Feldman and colleagues analysed infrared spectra at sites in Oklahoma and Alaska over 2000-2010. Using those measurements, other atmospheric observations and radiative-transfer calculations, they isolated an increase in clear-sky downward surface radiation associated with the roughly 22 ppm CO2 increase: about 0.2 W/m² per decade at each site.

That is a local surface result, not the global top-of-atmosphere forcing or total greenhouse effect. It nevertheless checks the claim that additional CO2 has no radiative effect.

Feldman et al.: Observational determination of surface radiative forcing by CO2

So 'some lines already absorb nearly everything' is a valid observation. 'Therefore adding CO2 cannot change Earth's energy balance' does not follow. The question is what happens across the full spectrum in an atmosphere that both absorbs and emits.