An ideal GB has uniform
emissivity across all wavelengths, from gamma through visible and infrared all the way to radio.
The above math shows that if the earth, without an atmosphere, is placed in a 1368W / m ^ 2 flux (which is the solar flux at the distance of 1 earth radius from the sun) then it will have a temperature of 278.6 K (or 5.4 C) if the earth was a BB or even an ideal GB (uniform
emissivity across all wavelengths).
GB's are bodies that have uniform
emissivity across all wavelengths.
The earth does not have uniform
emissivity across every wavelength.
Not exact matches
For other reasons, at LTE, the transmission (of a given type of photon) is the same in a pair of opposite directions, so in the absence of scattering,
emissivity and absorptivity must each be the same for opposite directions
across the same path of material, and thus they will be the same for absorption of photons from a direction and emission of photons into the opposite direction.
This is in addition to there being a much higher partial pressure of water vapor (up to 2.5 %) in the atmosphere than that of CO2 (400ppm which varies with height) It should also be noted that the absorptivity and
emissivity of liquid water is close to unity
across the full range of wavelength from UV to microwaves.
which is, indeed, the Stefan - Boltzmann formula for net power radiatively transferred to / from a hot reservoir at temperature to a cold reservoir at temperature
across a vacuum between to facing plates of area and
emissivity.
However, as an educational tool, we can calculate the results for a grey atmosphere — this means that the
emissivity is assumed to be constant
across all wavelengths.
Across the wavelengths where the earth is absorbing solar radiation (0.2μm - 4μm) the earth has approximately an
emissivity of 0.7.
The «flaw» was that the
emissivity of the earth was pegged at 0.7
across all wavelengths.
The 5.4 C result (line 20) is for an earth with uniform
emissivity of 0.7
across both spectrums.
B) How does»
Across the wavelengths where the earth is emitting its own radiation (4μm - 40μm) it has an
emissivity close to 1.»
I referred you back to my original calculation because the end result -LRB--18.3 C) was for an earth absorbing with absorptivity of 0.7
across solar wavelengths and an
emissivity of 1.0
across terrestrial wavelengths.
Five temperatures
across the globe averaged (272 +285 +271 +301 +297) / 5 = 282.2 K so radiation to space ignoring
emissivities is 5.67E - 08 • 282.2 ²² = 359.59 Wm - 2 right?