Why are solar longwave and terrestrial shortwave radiations neglected in radiation balance models?

For an assignment I was given the question:

Explain why solar radiation is neglected in atmospheric “long wave” calculations and Earth radiation is neglected in atmospheric “short wave” calculations.

I realize that the wavelengths considered to be shortwave radiation are those smaller than or equal to 4$\mu m$. And I know that we say that as wavelength increases, radiation from the Earth increases while solar radiation increases.

As a result of this, Why are the sun's long-wave radiation and the Earth's short-wave radiation neglected?

Any explanation would be much appreciated!

• Not sure what you mean when you say "And I know that we say that as wavelength increases, radiation from the Earth increases while solar radiation increases." Did you perhaps mean a decreases in there somewhere instead?? – JeopardyTempest Mar 25 '18 at 1:00

It is because longwave (LW) correspond to a negligible part of the solar radiation and shortwave corresponds to a negligible part of solar radiation.

You can make an experiment yourself using NASA's Radiance calculator. By adjusting the parameters for Earth's and the Sun you will get the following plot of energy flux at different wavelengths (The red line is the radiation spectra of the Sun and the blue line that of the Earth)

The cool thing about this tool is that you can see the actual data (or download it as a spreadsheet) and estimate the total energy flux between any range of wavelength (the area under the curve).

In this case, for SW radiation (0.1 - 4 $\mu m$) and LW radiation (4 - 100 $\mu m$) you get the following numbers

Sun LW: $1.98 \times 10^7 W m^{-2} {sr}^{-1}$
Sun SW: $1.95 \times 10^5 W m^{-2} {sr}^{-1}$
Sun Total: $2.00 \times 10^7 W m^{-2} {sr}^{-1}$

Earth LW: $1.24 \times 10^2 W m^{-2} {sr}^{-1}$
Earth SW: $1.82 \times 10^{-1} W m^{-2} {sr}^{-1}$
Earth Total: $1.23 \times 10^2 W m^{-2} {sr}^{-1}$

So in terms of percentages:

Sun: 99.03% SW and 0.97% LW

Earth 0.15% SW and 99.85% LW

That illustrate my initial point: LW is negligible for the Sun and SW negligible for Earth. Therefore, in simple models you can ignore both solar LW and terrestrial SW.

And why it is that way?

Well the reason is because the peak of the emissions decrease inversely proportional to temperature (according to Wien's law), while the total power grows MUCH faster, proportional to the fourth power of the temperature (according to Stefan-Boltzmann law). So the Sun been much hotter, produce an enormously larger amount of energy per square meter, but most of that energy is delivered in much shorter wavelength.

Note: It is interesting to note that despite that it is OK to neglect solar LW it is still 1580 times stronger than terrestrial LW.

It is important to keep in mind the black-body approximated radiation for the Earth and the Sun. Notice how small the long-wave energy radiation is from the sun and how Earth's radiation peaks there.

I would speculate that this question has something to do with the fact that Earth's surface is heated by solar shortwave radiation, and that Earth's lower atmosphere is heated by the Earth's surface. It is a common misconception that the Earth feel's thermal "heat" (longwave) from the sun. The Earth is instead absorbing short-wave radiation and then re-radiating that energy to the atmosphere as long-wave radiation.