. . . And I have felt
A presence that disturbs me with the joy
Of elevated thoughts; a sense sublime
Of something far more deeply interfused,
Whose dwelling is the light of setting suns,
And the round ocean and the living air
And the blue sky, and in the mind of man,
A motion and a spirit, that impels
All thinking things, all objects of all thought,
And rolls through all things. . . .

--William Wordsworth, Tintern Abbey

There's something about easy questions that makes them difficult to answer. That sounds like a contradiction in terms, so let me clarify: The questions that are the easiest to state are often the hardest to answer. An easy example is the children's question, "Why?" Why do bees buzz? Why do we sleep? Why do people get angry? And so on. Sometimes, the questions are hard to answer because we have to sanitize our response, but other times the questions are hard because they touch on fundamental problems.

Things are no different in the sciences. Probably the most famous mathematical problem in history, and one that eluded solution for over three centuries, is known as Fermat's Last Theorem. In his copy of Arithmetica, a treatise on mathematical problems with integer solutions by the Greek scholar Diophantus (c. 210-290), the French polymath Pierre de Fermat (1601-1665) wrote that there were no solutions to the equation

xⁿ + yⁿ = zⁿ

for x, y, z, and n all positive integers, with n > 2. He added that he had a "marvelous proof" of this claim, which he was not able to fit into the margin.

That sounds a little irresponsible, but Fermat was a very sharp guy, and every other time he said that he had a proof of a statement, he did. After some time, all of his other problems had been solved, and only this last one remained, hence its name (although it really should have been called "Fermat's Last Conjecture," since it hadn't been proved yet). Not until 1995, when the English-born mathematician Andrew Wiles (b. 1953) published a proof totalling over 200 pages, did Fermat's Last Theorem finally fall.

When it did, it was to 20th century and not 17th century mathematics, and in the process, a lot of pioneering work was done. Fermat's Last Theorem is one of the easiest of Fermat's conjectures and problems to state, and yet it took the combined effort of over 300 years of the best mathematicians in the world to crack it. -- Well, not quite all of the best. The Swiss mathematician Leonhard Euler (1707-1783) was said to have refused to work on Fermat's Last Theorem because he didn't want to waste years on a probable failure.

Here's a question that's both a children's "Why?" and a scientific puzzler: Why is the sky blue?

That sounds like it ought to be an easy question, but it isn't. It took scientists a long time before they figured it out, and when they did, it was because they had also learned a lot about other things: how light works, and what the air around us is like.

We can begin to get a handle on the problem if we observe that there are plenty of times when the sky is not blue. It's certainly not blue on a cloudy day, for example -- it's white, or grey. And at night, the sky is black. You might counter that the white clouds just block your view of the sky, which is ultimately blue; that the night is black only because the blue of the sky is unilluminated. If one could only shine a powerful enough flashlight on the night sky, then its blue splendor would become apparent.

You can tell from the way we talk about these things that the usual conception of the sky is a big dome that surrounds the Earth in all directions. This dome sits above the clouds (if there are any), it can be lit up by the Sun during the day, or it can remain in shadow at night.

However, there are some simple observations that upset this simple idea of the sky as a dome. For example, if you look at the Moon right after the new Moon, when it's a thin crescent in the western evening sky, you can often see the faint dark grey portion of the Moon. This is often called "the old Moon in the new Moon's arms," and it's the result of earthshine -- sunlight reflected off the Earth that lights the Moon. This obviously means that the Moon is beneath the sky, even the part you ordinarily don't see.

And yet, as many people are unaware, I find, the Moon can also be seen during the day, especially near first or last quarter, when the Moon is close enough to the Sun to be in the sky at the same time, yet not so close that the Sun's glare drowns it out. If you look closely at those times, you'll see that the part of the Moon that is not lit up by the Sun is not the faint dark grey of the old Moon, nor the bright grey of the new Moon. Instead, it is blue -- the same blue as the rest of the sky. If you view the Moon through a telescope, it's even more apparent that the blue of the sky passes in front of the Moon. But if the sky is a dome, how can it possibly be both in front of and in back of the Moon?

The answer is, it can't. We have to let go of the idea that the sky is a dome. Whatever creates the blue of the sky has to be between us and the Moon; otherwise, the Moon would appear in front of the blue sky. And what is between us and the Moon? Air, of course: our own atmosphere.

But what is it about air that creates the blue sky? In 1859, the Irish physicist John Tyndall (1820-1893) passed a beam of white light, which is a mixture of light of all the different colors of the spectrum, through a clear container of water in which small, invisible particles were suspended. When he looked at the beam of light from the side, it looked blue, but when he looked directly into the beam from the other side, it looked orange, an effect now called the Tyndall Effect.

Tyndall correctly deduced that the particles in the water were scattering the light, and were furthermore scattering blue light to a greater extent than light of other colors. When he looked at the beam from the side, he was seeing scattered blue light, so the beam looked blue. On the other hand, when he looked into the beam, he was seeing light that had the blue scattered out of it, so it looked orange. He suggested that similar particles in the air were responsible for the blue color of the sky. When you look away from the Sun, he proposed, you see scattered blue light, but if you were to look into the Sun (not a good idea for safety reasons), you would see slightly orange light.

The reason that the Sun doesn't actually look orange, Tyndall added, is that the atmosphere is too thin to scatter away enough of the blue light. The sky is only as blue as it is because the Sun has so much light of all colors to begin with that the tiny bit of the blue light that is scattered is sufficient to give our sky its cerulean hue.

It was a remarkably insightful guess, but it was lacking details. First of all, if Tyndall was correct, then shouldn't the sky get bluer as the air got dustier? That didn't seem to happen, though; if anything, extra dust seemed to make the sky redder, rather than bluer. Secondly, what caused the particles to scatter more blue light than red? Tyndall didn't know. Also, he didn't quantify the effect. That made it difficult to test whether the right amount of blue was scattered via the Tyndall effect to account for the sky color.

Since the color of the sky is remarkably consistent on clear days, dusty or otherwise, it has to be caused by substances that are in the air everyday, and not present only on dusty days, or rainy days, or whatever. The principal constituent gases of the atmosphere are nitrogen and oxygen, so those are likely the main causes.

In 1871, the English physicist John William Strutt, Lord Rayleigh (1842-1919) took up the problem. Like Tyndall, he also began with the idea that particulate scattering of light was responsible for the color of the sky. But he went further than Tyndall had. In particular, he quantified the effect, deriving a formula that predicted just how much light would be scattered, and how that scattering would vary by color. This had two consequences: first, it meant that the explanation could be tested directly, by experiment; secondly, the formula and its derivation were still valid, even when Rayleigh and others later decided that it was gas molecules and not particles in the air that were responsible.

To be sure, it took people a long time before they realized that there was anything at all around us, that the empty space between you and me wasn't in fact totally empty. One of the first to recognize this was the Greek philosopher Empedocles of Acragas (c. 492-432 B.C.). He performed experiments to show that something in the emptiness around us was capable of exerting pressure and plugging holes; this pressure is what keeps soda in the straw if you cover the top with your finger. To do this, it couldn't really be empty nothingness; there had to be some substance exerting the pressure, which Empedocles declared to be air.

He then made the daring suggestion that air was actually matter that had been divided so finely that it became invisible, a suggestion further developed by Democritus of Abdera (c. 470-380 B.C.). Democritus suggested that there was no such thing as a true fluid, that things that appeared fluid and continuous only looked that way because their component parts were too small to see individually. Substances such as air and water could, in principle, be separated into small, indivisible pieces, which Democritus called atoms, from a Greek word meaning "without cuts."

This all sounds enormously modern, and in a way it is, but Democritus came to his conclusions because they matched his pattern of thought. For example, he also thought that the Milky Way, which looks like a cloud of gauzy light to the unaided eye, could also be broken up into individual stars, which would therefore play the same role as atoms in the ordinary matter around us. He was forever seeing the same pattern of indivisibility, which probably led many of his contemporaries to think of him as a crank. What's more, Democritus got the crazy idea that atoms were shaped like regular polyhedra, so that earth atoms were shaped like cubes, fire atoms were shaped like tetrahedra, and so forth. The upshot is that very few Greeks believed Democritus and his atomic theory, in Democritus' own time and for centuries afterward.

Atomic theory as we recognize it was introduced by the English meteorologist and chemist, John Dalton (1766-1844). He was led to atomic theory in part because experiments showed that substances such as hydrogen, carbon, nitrogen, and oxygen always combine in amounts involving the same ratios.

For example, in any reaction producing carbon dioxide, which we now know contains one carbon atom and two oxygen atoms, 6 units (of mass) of carbon would always go with 16 units of oxygen. In water, which contains two hydrogen atoms and a single oxygen atom, 2 units of hydrogen always went with 16 units of oxygen. If these substances were really fluids, then it seemed that there should be some variability in the way they combined. By comparing the ratios in which atoms combined in chemical reactions, Dalton concluded that these substances were made of indivisible pieces, which he also called atoms.

Although there was now some experimental evidence for the existence of atoms, it was still circumstantial; there was still a certain element of faith involved. It wasn't until Einstein developed his theory of Brownian motion in 1905 that we saw direct evidence for the atomic world. That doesn't mean that all work on atoms stood still. For instance, it was first believed that the atoms themselves were little round balls of homogeneous matter, and that the only thing that distinguished atoms from one another was their mass.

Eventually, though, experiments involving the conductance of electricity through wires and through a vacuum showed that atoms could not be entirely homogeneous. In particular, it must be possible for different parts of the atom to have different electric charges. As it turned out, these parts were all first discovered at Cambridge University. The electron was discovered by J. J. Thomson (1856-1940) in 1897, the proton was discovered by Ernest Rutherford (1871-1937) in 1919, and the neutron was discovered by James Chadwick (1891-1974) in 1932. In the classical model of the atom, the electrons circle the nucleus, the impossibly dense core of the atom, which is made up of protons and neutrons. [1]

Electrons are negatively charged, while the nucleus is positively charged, due to its protons; the neutrons are neutral and uncharged. Unlike charges attract, so there is an attraction between the electrons and the nucleus. If we stick to the classical model of electrons orbiting the central nucleus like planets orbiting the Sun, then electrons must orbit at certain speeds and distances to keep from falling into the nucleus. (Even though the electrons and nucleus have equal and opposite charges, the nucleus is a few thousand times more massive than all the electrons put together, and is much harder to move around. That's why we can speak of just the electrons moving around the nucleus, without losing too much accuracy.)

These orbits are, however, free to move up and down -- that is, in a direction perpendicular to the orbital plane. If an electron's orbit is, for whatever reason, above the nucleus, the attraction between electron and nucleus tends to move the orbit downward; if the orbit is below the nucleus, the attraction tends to move the orbit upward. The force is directly proportional to the distance between the orbital plane and the nucleus. So, if something happens to shove the orbit, say, upward, then when it's released, it will start back down toward the nucleus. It doesn't stop when it reaches the nucleus, but instead continues downward.

Eventually, the attraction of the nucleus stops the electron's progress, and it heads back up, where the process repeats. It's almost as if the electron were suspended on a spring. This up-and-down bobbing has a natural or resonant frequency that depends on the strength of the attraction between the nucleus and the electrons, and since it (like other similar motions) resembles the back-and-forth of a vibrating musical instrument, it is often called harmonic motion.

And what would cause the orbit to move up or down? The electrons and the nucleus both carry charges, so one possibility is an electromagnetic field. Because they have opposite charges, the same field will move the electrons in one direction and the nucleus (very slightly) in the other.

The Scottish physicist James Clerk Maxwell (1831-1879) had shown that light is an electromagnetic field in oscillation -- in other words, that it's an electromagnetic wave. Rayleigh applied this finding to the Tyndall effect. He proposed that the scattering particles were composed of positive and negative charges, and that an electromagnetic interaction between the light and the particles produced the scattered light. Maxwell also showed that a moving electric charge will generate light, so that the energy the incoming light loses in stimulating the particles is not truly lost; it is merely re-emitted in a different direction by the bobbing electrons, and it is this re-emission that we call scattering. (Although the idea that the scattered light is re-emitted was superseded when quantum mechanics was developed.)

Later, when attention was focused on the atoms and molecules of gas in the atmosphere as the scattering agents, the electrons and the nucleus played the roles of the opposite charges in Rayleigh's original particles. When the oscillating electromagnetic field of a beam of light strikes an atom, most of the field continues unaffected -- meaning that the beam of light passes through -- but some of it stimulates the electrons and nucleus of the atom, causing the electrons to bob up and down.

The question that Rayleigh asked himself was: Just how much of the incoming light is scattered away? Rayleigh's analysis involved a significant amount of math, but we can get a flavor of it if we look at a similar system.

Consider, for example, a simple spring. It resists being either stretched or compressed, and the force it exerts in returning to its natural length is proportional to the amount of stretching or compression, so it too exhibits harmonic motion. Also, like electrons in an atom, it has a resonant frequency. The resonant frequency is determined by the spring's stiffness, which is expressed in a parameter called the spring constant.The higher the spring constant, the stiffer the spring, and the higher the resonant frequency. (That's why the tighter you wind a guitar string, the higher the pitch of the note it sounds.)

Since the spring behaves much as an atom does in its natural motion, it stands to reason that it will also respond to applied forces in the same way. Let's lay out the spring horizontally on a frictionless table, so that no complicating issues arise. Mark the left end of the spring A, and the right end of the spring B. Now, so long as no force is applied to the spring, nothing happens to it.

But suppose we now wave the end marked A in an oscillating motion. If you wave it fairly slowly, almost all of that oscillation gets transmitted to B. This corresponds to a situation in which most of the light doesn't interact with the atom, and is merely transmitted through as though the atom weren't there. On the other hand, if you wave it quickly, quite a bit of the oscillation gets absorbed by the spring, and somewhat less of it is passed on to B. In fact, if you wave it quickly enough, all of the oscillation gets absorbed by the spring, and B just sits there. The frequency at which this happens turns out to be the resonant frequency of the spring. The amount of the oscillation that gets absorbed by the spring varies with the frequency of the oscillation in a way that gets complicated as we approach the resonant frequency. So long as we stick to the low-frequency end, though, we can characterize that behavior in a simple way.

In oscillating back and forth, B is constantly changing its velocity. For that to happen, force has to be applied to B; this force is applied by A, through the spring. Now, if the spring were perfectly stiff, like a solid rod, all of the force starting at A would be transmitted to B. When A is pushed all the way to the right or pulled all the way to the left, B would also be pushed all the way to the right or pulled all the way to the left. But the spring is not perfectly stiff, so it deforms, alternately stretching and compressing. The amount of deformation for a given spring is proportional to the force applied on it at A, which in turn is proportional to the acceleration at A.

This acceleration, in turn, depends on the frequency of the oscillation. Suppose you double the frequency with which you wave A back and forth. That end of the spring now takes half the time to go from all the way to the left to all the way to the right, so its velocity at any point must be double what it was before. What's more, the acceleration is the change in velocity, divided by the time taken to make that change. In a single oscillation, the velocity goes from maximum speed to the left to maximum speed to the right, back to maximum speed to the left again. At double the frequency, the change in velocity is twice what it was before, and that change is performed in half the time, so the total acceleration is actually quadrupled.

Similarly, if you triple the frequency, the total acceleration is three squared, or nine times what it was before; in general, if you multiply the frequency by n times, the acceleration is multiplied by n squared. This acceleration is expressed as an n-squared-fold increase in the amount of compression and stretching out of the spring.

Electrons in atoms react to the electromagnetic influence of a light wave in much the same way as a spring does to a mechanical force. If you double the frequency of the incident light, then the electrons oscillate four times as much, and at the new frequency, too. This means that the light they emit also oscillates four times as much. Another way of saying that is that the amplitude of the scattered light is multiplied by four. The energy of harmonic motion at any given frequency is proportional to the amplitude squared, so if you double the frequency, the energy of the scattered light is multiplied by 16. In general, if you multiply the frequency by n, the amplitude gets multiplied by n squared, and the energy gets multiplied by n squared twice, or n raised to the fourth power. [2]

As noted above, this only works if the frequency of the light striking the atoms is well below the resonant frequency. Fortunately, for the atoms in our atmosphere, the resonant frequency is well into the ultraviolet range (which is why the atmosphere is good at blocking most of the Sun's ultraviolet light), and for visible light frequencies, the fourth-power law works pretty well. This was all worked out in fairly rigorous detail by Rayleigh, so the resulting scattering is commonly known as Rayleigh scattering. Later, when quantum mechanics was developed, Rayleigh scattering could be explained in terms of a new paradigm, but that didn't invalidate Rayleigh's analysis.

This fourth-power dependence of Rayleigh scattering is why blue light is scattered so much more than red light. Blue light has a frequency about seven-fourths that of red light (depending on where you say the red and blue frequencies precisely begin and end). Since seven-fourths raised to the fourth power is almost 10, blue light is scattered nearly 10 times as efficiently as red light is.

Even so, as Tyndall guessed, most of the light from the Sun is not scattered; that's why the Sun appears as bright as it does -- because most of the light by which we see the Sun travels in a straight line to us. What little light is scattered reaches us from all directions, and is responsible for the appearance of the sky. Since blue light is scattered much more than light of other colors, the sky appears blue. The air also scatters starlight just as it does sunlight, but since starlight is almost immeasurably dimmer than sunlight, the night sky mostly just appears black.

But -- there is a color in the visible spectrum beyond blue: violet. Violet light has a frequency that is somewhat higher than blue light, and should therefore be more scattered than blue light. So why doesn't the sky look violet? Part of the reason is that the Sun doesn't emit as much violet light as blue light. So even though the violet light is scattered more, there's less of it to begin with. Even more importantly, our eyes are not equally sensitive to all colors. For example, they're most sensitive to green light, and their sensitivity drops significantly for blue light, and quite a bit further still for violet light. It's this, primarily, that explains why our sky doesn't look violet.

Clouds look white for a different reason. When water droplets initially form and are still small, they also scatter blue light preferentially, because the electrons in all the water molecules of a small droplet oscillate in phase. Eventually, though, the droplets get large enough (about the size of a wavelength of the incoming light) that the electrons are no longer in phase -- their synchronization is limited by the speed of light -- and the wavelength dependency largely goes away. Since all colors are scattered efficiently by the clouds, they appear white.

In fact, Rayleigh scattering is rarely the sole reason for the color of the sky. From first principles, the sky on Mars should be fairly dark, because its thin atmosphere is fairly inefficient at scattering light, and the predominant influence should be the black of outer space. Blue light is still scattered more efficiently, so the sky on Mars should be a very dark blue.

However, when the Viking landers first arrived on Mars in 1976, the first pictures they returned showed a bright, Earthlike sky blue. This turned out to be due to color tuning by technicians who weren't planetary scientists and therefore adjusted the receptors to replicate what they thought the sky should look like. But even when this was corrected, the sky still wasn't dark blue. Instead, it was salmon-colored. What was the explanation for that?

Even though Mars's atmosphere is thin, it's a very windy place -- its high winds can approach velocities of 200 m/s (about 450 mph). It's also very dusty, and the winds kick dust up into the air, where it may take days to settle. Before it does, however, another burst of wind typically kicks it back up again, and the semi-permanent dust in the air gives the sky of Mars its characteristically salmon color. Future residents of Mars, it seems, may never see its "pristine" sky for all the dust in its air.

Brian Tung is a computer scientist by day and avid amateur astronomer by night. He is an active member of the Los Angeles Astronomical Society and runs his own astronomy website. His previous publications in Strange Horizons can be found in our Archive.

Notes

[1] This can't really be what happens. Why not? Because if the electrons really orbited the nucleus, the electromagnetic field within the atom would fluctuate, even in an atom "at rest." In that case, the atom would radiate energy, slowly, and the atom would collapse. For such reasons, the classical model of the atom has long since been discarded, except for a few lingering terminologies. For example, in quantum mechanics, the electrons don't orbit the nucleus, but can instead be found in random locations throughout the atom. However, any given electron is more likely to be found in some places than in others, and the probability distribution of the electron's location is still called its orbital.

[2] But the energy of a light wave is also proportional to its frequency, I can hear you saying. That's true. What we're interested in here, however, is the fraction of the light's energy that gets scattered away, and that fraction isn't directly affected by this dependence.