# Contents

This page is a blog article in progress, written by Tim van Beek.

## Introducion

Last time, when we talked about putting earth into a box, we saw that a simple back-of-the-envelope calculation about the energy balance and the resulting average temperature of the earth is surprisingly close to the real world. But there is some gap, because the temperature predicted by a one dimensional energy balance model is lower than the actual temperature on earth.

In such a situation, as theoretical physicsists, we congratulate ourselves on a successful first approximation, , and look out for the next most important effect that we need to include in our model.

Most of you will of course heard about the effect that climate scientists talk about, which is often - but confusingly - called “greenhouse effect”, or “back radiation”. The term that is most accurate is downward longwave radiation (DLR), however, so I would like to use that instead.

Since some people have voiced concerns that there is something wrong with the whole theory of radiation, as it is applied to the atmosphere, I would like to explain this phenomenon starting with the simplest possible thought experiment: A single blackbody in empty space, kept at a fixed temperature by a “heat bath”. A heat bath is, like the perfect blackbody, an idealization of an infinite container of heat energy that can keep a system coupled to it at a fixed temperature.

We will use the heat bath in the example to simplify the reasoning about the effect that emitting a photon may have on the temperature of the blackbody: In our example, it does not have any effect, because we assume that the blackbody is kept at a constant temperature by the coupling to the heat bath, no matter what happens. In my thought experiment, I would like to have a blackbody of the shape of a cube that radiates from two surfaces only, like from the one with a 1 on it, which I call $A_1$, and from the one with a 6 on it, which I call $A_2$.

I would also like to ignore any radiation that is emitted at a different angle than perpendicular to the surface. This may seem odd, since the blackbody is supposed to be a diffusive emitter: So it will emitt into all directions in the same way. But this assumption does not alter anything about the thermodynamics of the thought experiment, I will use it to avoid fancy calculations involving “steradians” and other complicated 3D geometry stuff. You can do all the calculations with a more complex geometry as I do here, it does not change anything about the main point.

With all these assumptions, the energy flux is given given by the Stefan-Boltzmann law:

$E(T_A) [\frac{W}{m^2}] = \sigma T_A^4$

and the overall energy per second that the blackbody emits is simply

$\sigma T_A^4 (A_1 + A_2)$

Tim van Beek: I’m running out of letters because I used E for the flux, and would like to use it for energy per second also.

Now let’s assume that we bring in a similar blackbody $B$ with a lower Temperature $T_B$. Blackbody $A$ will keep radiating as before, blackbody $B$ will also radiate, but will emit a lower energy flux: Note that none of the two will stop radiating or change their radiation because the other one is present. Also, the photons emitted by blackbody B have no qualms to cross empty space and be absorbed by blackbody A, therefore there is an energy flux fom B to A. Of course, the net energy transfer per second, on a macroscopic level is from A to B:

$Energy_net = \sigma T_A^4 A_2 - \sigma T_B^4 B_1$

For simplicity, let’s say that $A_2 = B_1 = 1 m^2$, so that the difference becomes

$Energy_net = \sigma (T_A^4 - T_B^4)$

So, the second law of thermodynamics is not violated, because the net energy flow is from the hotter to the colder body. But individual photons are still allowed to tbe emitted by B (the colder body), travel through empty space and be absorbed by A (the hotter body).

In fact this has to happen if quantum statistical mechanics is correct.

## Explaining the 33 K Gap: IR-Backradiation

Tim van Beek: The following is just a random collection of material right now!

Here is a nice overview of the spectrum of electromagnetic radiation: BTW, if you doubt that a colder black body can emit low energy photons that are then absorbed by a hotter black body, increasing its energy in the process, you may ponder the question how a microwave oven works.

From the Planck density, we can determine that sun and earth, as black bodies, emit at different wavelenghts: Only some components of the atmosphere emit and absorb radiation in the IR part, the part where earth’s spectrum is. These are called - somewhat misleading - “greenhouse gases”. Two prominent ones are $H_2O$ and $CO_2$: The “atmospheric window” at 8 to 12μm is quite transparent, which means that this radiation passed from the surface to the atmosphere without much ado. Therefore, this window is used by satellites to estimate the surface temperature.

Tim van Beek: I would like to add radiation measurements, maybe some can be found here:

Devices to measure the infrared radiation of the planetary surface are called pyrgeometer, for pyr = fire and geo = earth.

Also have a look here.

Just to have a number, the flux of DLR (downwards longwave radiation) is about 300 $W m^{-2}$.

Several pictures have been taken from the book

• Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, Adrienne S. Lavine: “Fundamentals of Heat and Mass Transfer”, 6th edition

category: blog