Blog - putting the Earth in a box

This page is a blog article in progress, written by Tim van Beek. To see the final polished article, go to the Azimuth Blog.

Is it possible to fly to Mercury in a spaceship without being fried?

If you think that it should be possible to do a simple back-on-the-envelope calculation taking into account the radiation of the sun, you’re right, of course! And NASA has already done that:

This is interesting for astronauts, but it is interesting for a first estimation of the climate of planets, too. In particular, for the Earth. In this post, I would like to talk about how this estimate is done and what it means for climate science.

How do physicists model a farm? They say: "First, let's assume that all cows are spherical with homogeneous milk distribution".- Anonymous

Theoretical physicists have a knack for creating the simplest possible model for very complicated systems and still have some measure of success with it. This is no different when the system is the whole climate of the earth:

The back-on-the-envelope calculation mentioned above has a name; people call it zero dimensional energy balance model.

Surprisingly, the story of energy balance models starts with a prominent figure in physics and one of the most important discoveries of 20th century physics: Max Planck and “black body radiation”.

Matter emits electromagnetic radiation, at least the matter that we know best. Physicists have also postulated the existence of matter out in space that does not radiate at all, called “dark matter”, but that doesn’t need to concern us here.

Around 1900, the German physicist Max Planck set out to solve an important problem of thermodynamics: To calculate the amount of radiation emitted by matter based on first principles.

To solve the problem, Planck made a couple of simplifying assumptions about the kind of matter he would think of. These assumptions characterize what is known in physics as a perfect ‘black body’.

A **black body** is an object that perfectly absorbs and therefore also perfectly emits all electromagnetic radiation at all frequencies. Real bodies don’t have this property; instead, they absorb radiation at certain frequencies better than others, and some not at all. But there are materials that do come rather close to a black body. Usually one adds another assumption to the characterization of an ideal black body: namely, that the *radiation is independent of the direction*.

When the black body has a certain temperature T, it will emit electromagnetic radiation, so it will send out a certain amount of energy per second for every square meter of surface area. We will call this the **energy flux** and denote this as $f$. The SI unit for $f$ is $W/{m}^{2}$: that is, watts per square meter. Here the **watt** is a unit of energy per time, or power.

This electromagnetic radiation comes in different wavelengths. So, can ask how much energy flux our black body emits per change in wavelength. We will call this the **monochromatic energy flux** ${f}_{\lambda}$. The SI unit for ${f}_{\lambda}$ is $W/({m}^{2}\phantom{\rule{thickmathspace}{0ex}}\mu m)$, where $\mu m$ stands for **micrometer**: a millionth of a meter, which is a unit of wavelength. We call ${f}_{\lambda}$ the ‘monochromatic’ energy flux because it gives a number for any fixed wavelength $\lambda $.

When we integrate the monochromatic energy flux over all wavelengths, we get the energy flux:

$$f={\int}_{0}^{\mathrm{\infty}}{f}_{\lambda}\phantom{\rule{thinmathspace}{0ex}}d\lambda $$

For the ideal black body, it turned out to be possible for Max Planck to calculate the monochromatic energy flux ${f}_{\lambda}$, but to his surprise, Planck had to introduce in addition the assumption that energy comes in quanta. This turned out to be the birth of quantum mechanics!

His result is called the $\mathrm{latex}{f}_{\lambda}(T)=\frac{{c}_{1}}{{\lambda}^{5}(\mathrm{exp}\frac{{c}_{2}}{\lambda T}-1)}$:

$${f}_{\lambda}(T)=\frac{{c}_{1}}{{\lambda}^{5}(\mathrm{exp}\frac{{c}_{2}}{\lambda T}-1)}$$

Here I have written ${c}_{1}$ and ${c}_{2}$ for two constants. These can be calculated in terms of fundamental constants of physics. But for us this does not matter now. What matters is what the function looks like as a function of the wavelength $\lambda $ for the temperature of the sun and of the earth.

As usual, Wikipedia has a great page about this:

• Black body radiation, Wikipedia.

The following picture shows the energy flux as a function of the wavelength, for different temperatures:

The earth radiates roughly like the 300 kelvin curve and the sun like the 5800 kelvin curve. You may notice that the maximum of the sun’s radiation is at the wavelengths that are visible to human eyes.

Real surfaces are a little bit different than the ideal black body:

As we can see, the real surface emits less radiation than the ideal black body. This is not a coincidence, the black body is by definition the body that generates the highest energy flux at a fixed temperature.

A simple way to take this into account is to talk about a **grey body**, which is a body that has the same monochromatic energy flux as the black body, but reduced by a constant factor, the **emissivity**.

It is possible to integrate the black body radiation over all wavelengths, to get the relation between temperature $T$ and energy flux $f$. The answer is surprisingly simple:

$$f=\sigma \phantom{\rule{thickmathspace}{0ex}}{T}^{4}$$

This is called the **Stefan-Boltzmann law**, and the constant $\sigma $ is called the **Stefan-Boltzmann constant**. Using this formula, we can assign to every energy flux $f$ a black body temperature $T$, which is the temperature that an ideal black body would need to have to emit $f$.

A planet like Earth gets energy from the Sun and loses energy by radiating to space. Since the Earth sits in empty space, these two processes are the only relevant ones that describe the energy flow.

The radiation emitted by the Sun results at the distance of earth to an energy flux of about 1370 watts per square meter. We need to account for the fact, however, that the Earth receives energy from the Sun on one half of the globe only, on the area of a circle with the radius of the Earth, but radiates from the whole surface of the whole sphere. This means that the average outbound energy flux is actually $\frac{1}{4}$ of the inbound energy flux. (The question if there is some deeper reason for this simple relation was posed as a geometry puzzle here on Azimuth.)

So, now we are in a position to check if NASA got it right!

The Stefan-Boltzmann constant has a value of

$$\sigma =5.670400\times {10}^{-8}\frac{W}{{m}^{2}{K}^{4}}$$

which results in a black body temperature of about 279 kelvin, which is about 6 °C:

$$\frac{1370W{m}^{-2}}{4}\phantom{\rule{thickmathspace}{0ex}}\approx \phantom{\rule{thickmathspace}{0ex}}5.67\phantom{\rule{thinmathspace}{0ex}}\times \phantom{\rule{thinmathspace}{0ex}}{10}^{-8}\frac{W}{{m}^{2}{K}^{4}}\phantom{\rule{thinmathspace}{0ex}}\times \phantom{\rule{thinmathspace}{0ex}}(279K{)}^{4}$$

That is not bad for a first approximation! The next step is to take into account the ‘albedo’ of the Earth. The **albedo** is the fraction of radiation that is instantly reflected without being absorbed. The albedo of a surface does depend on the material of the surface, and in particular on the wavelength of the radiation, of course. But in a first approximation for the average albedo of earth we can take:

$${\mathrm{albedo}}_{\mathrm{earth}}=0.3$$

This means that 30% of the radiation is instantly reflected and only 70% contributes to heating earth. When we take this into account by multiplying the left side of the previous equation by 0.7, we get a black body temperature of 255 kelvin, which is -18 °C.

Note that the emissivity factor for grey bodies *does not change the equation*, because it works both ways: the absorption of the incoming radiation is reduced by the same factor as the emitted radiation.

The average temperature of earth is actually estimated to be some 33 kelvin higher, that is about +15 °C. This should not be a surprise, after all 70% of the planet is covered by liquid water! This is an indication that the average temperature is most probably not below the freezing point of water.

The albedo depends a lot on the material: for example, it is almost 1 for fresh snow. This is one reason people wear sunglasses for winter sports, even though the winter sun is considerably dimmer than the summer sun in high latitudes.

Since a higher albedo results in a lower temperature for the Earth, you may wonder what happens when there is more snow and ice? This results in a lower absorption, which leads to less heat, which results in even more snow and ice. This is an example of **positive feedback**, which is a reaction that strengthens the process that caused the reaction. There is a theory that something like this happened to the Earth about 600 million years ago. The scenario is aptly called Snowball Earth. This theory is based on geological evidence that at that time there was a glaciation that reached the equator! And it works the other way around, too.

Since a higher temperature leads to a higher radiation and therefore to cooling, and a lower temperature leads to a lower radiation, according to the Planck distribution, there is always a **negative feedback** present in the climate system of the earth. This is dubbed the **Planck feedback** and has already been mentioned in week 302 of “This Weeks Finds” here on Azimuth.

Now, the only variable that a zero dimensional energy balance model calculates is the average temperature of earth. But does it even make sense to talk about the “average” temperature of the whole planet?

It is always possible to “put a planet into a box”, calculate the inbound energy flux, and compute from this a black body temperature T—given that the inbound energy per second is equal to the outgoing energy per second, which is the condition of thermodynamic equilibrium for this system. We will always be able to calculate this temperature T, but of course there may be very strange things going on inside the box, that make it nonsense to talk about an average temperature. As far as we know, one side of the planet may be burning and the other side may be freezing, for example.

For planets with slow rotation and no atmosphere, this actually happens! This applies to Mercury and the moon of the Earth, for example. In the case of Earth itself, most of the heat energy is stored in the oceans and it spins rather fast. This means that it is not completely implausible to talk about a “mean surface air temperature”. But it could be interesting to take into account the different energy input at different latitudes! Models that do that are called “one dimensional” energy balance models. And we should of course take a closer look at the heat and mass transfer processes of the earth. But since this post is already rather long, I’ll skip that for now.

The simple back-of-the-envelope calculation of the simplest possible climate model shows that there is a difference of roughly 33 kelvin between the black body temperature and the mean surface temperature on earth.

There is an explanation for this difference; I bet that you have already heard of it! But I’ll postpone that one for another post.

If you would like to learn more about climate models, you should check out this book:

• Kendal McGuffie and Ann Henderson-Sellers, *A Climate Modelling Primer*, 3rd edition, Wiley, New York, 2005.

Whenever I wrote “NASA” I was actually referring to this paper:

• Albert J. Juhasz, An analysis and procedure for determining space environmental sink temperatures with selected computational results, NASA/TM—2001-210063, 2001.

The pictures of black body radiation are taken from the book

• Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, Adrienne S. Lavine, *Fundamentals of Heat and Mass Transfer*, 6th edition, Wiley, New York, 2006.

I want to paint it black.— The Rolling Stones

Last but not least: you can fly to Mercury without getting fried, but you have to paint your spaceship *white* in order to get a higher albedo.

Really? Well, it depends on the albedo of the whitest paint you can find: the one that reflects the Sun’s energy flux the most.

So, here’s a puzzle: what’s the whitest white paint you can find? What’s its albedo? And how hot would a spaceship with this paint get, if it were in Mercury’s orbit?

category: blog