[Coda]

In your defense, Rayleigh scattering is usually taught very poorly at the high school level.

So before I start getting into the technicalities you're asking about... I'm just going to make one big statement: The sky is blue because the Sun is blue.

Now I'm going to roll it back to the beginning and build back up to that statement.

Though first, one quick definition, since I'm going to use it in my description and you were asking about it: A proportion is just a fraction -- a description of some part of a whole that has a certain property. If two things are proportional, then there's a relationship between them such that they go up and down together at related rates.

Part 1: Wavelengths of Light

As you know, what we perceive as "white" light is made up of a combination of colors. No individual photon of light is "white." (For that matter, no individual photon of light is magenta, either. Magenta doesn't exist except in your head.)

The color of light is determined by its wavelength, and the wavelength of light is directly determined by its energy. (For ANY wave, if you know two of wavelength, velocity, and energy, you can determine the third one. We know that light's velocity is c, so that means its wavelength and its energy have a directly inverse relationship with each other. When one goes up, the other goes down.) So whenever you see "color" or "wavelength" or "energy" you know it's all talking about the same thing.

The Sun is a fairly good approximation of a blackbody radiator. That means it emits photons in a continuous range of energies (this is called an "emission spectrum"), with the most of them at an energy level determined by the temperature of the blackbody (this is known as its "spectral peak") and then you get less of them the farther away from that energy level you go. The Sun burns at around 6000K, so it emits blue-green light the brightest, a little less violet light and yellow light, a little less near-ultraviolet light and red light, a little less far-ultraviolet light and infrared light, and so on. There's no gaps; every color in between is represented.

(Incandescent light bulbs are ALSO blackbody radiators, but since they're much cooler than the Sun, their spectral peak is in the infrared region of the spectrum, so most of the light coming from an incandescent bulb is heat, and most of the visible light is on the red end of the spectrum. Fluorescent and LED lights aren't blackbody radiators, so their emission spectra aren't smooth and continuous.)

Upshot: When your book talks about sunlight dropping off in the violet region, it's referring to the shape of this emission spectrum.

Part 2: Reflection

Throw a ball at a big flat surface. It bounces off at a predictable angle, right? Light does the same thing: shoot a beam of light at a big flat perfect surface (like a mirror) and it'll bounce off predictably.

Throw a ball at an uneven surface. Still bounces, but it isn't perfectly predictable; if you throw a bunch of balls, each one might bounce a different direction. You can estimate where they're likely to end up, but you can't predict any individual throw. Again, light does the same thing: shoot a beam of light at a sheet of paper, and while you know it's going to bounce, it's not going to bounce perfectly. It scatters, because each of the photons gets bounced in a little bit different direction.

Throw a ball at another ball. You can't even be sure if it's going to bounce, this time, because that's an awfully small target, and it's pretty likely that you're going to miss. You throw a bunch of balls, and most of them are going to go right on by without even so much as deflecting. And even if you hit, you can't be sure which way it's going to bounce because both balls are round, and you can't even be sure how hard it's going to bounce because the other ball is going to get knocked away too.

Now we're getting closer to the idea of Rayleigh scattering, but we're still not there yet.

Part 3: Crazy Physical Analogies That Actually Work!

Forgive me for that silly outburst of clickbait.

Blue light has a wavelength of 475 nanometers. A nitrogen molecule, like what the vast majority of our atmosphere is made up of, has a scattering diameter of a little under 1 femtometer -- around 600 million times smaller! That's like trying to hit a speck of dust with a beach ball, except the speck of dust weighs a trillion kilograms. (I actually calculated that number; I didn't just pick a big number arbitrarily -- I used E=mc^2 here!)

Now, fortunately for sanity, Newton's third law does in fact still apply at this scale. The principle of equal and opposite reaction still applies. (The first and second laws get fuzzy when you start approaching the speed of light. We already discussed the changes to the second law -- F = m*a is really an approximation of E^2 = m^2*c^4 + p^2*c^2 -- and the first law requires that you can construct a coordinate system with the object in question at relative rest, and you can't do this for photons because they're ALWAYS traveling at c no matter how you look at them.)

So given the third law, you can treat this situation like shooting said (indestructible, perfectly rigid) beach ball with said speck of dust traveling at the speed of light.

Surprisingly, the math actually does work out that way.

There's some fiddly electromagnetic fudge factors you have to take into account in order to define how big the balls are, but once you've established the effective scattering cross-section, Rayleigh scattering turns out to be a fairly clean formula that describes what you get when you average out all of the probabilities over a volume of space.

If you shoot the beach ball dead center, it's just going to go zipping directly in the opposite direction. If you hit it with a glancing shot on the side, both objects are going to ricochet in different directions.

Or, to drop the metaphor and go back to the actual question at hand: When a photon hits a nitrogen molecule, it'll bounce off in a direction based on how it hits and the relative sizes, and you can average out how far it's likely to deflect from its original straight-line path. And since different colors of light have different wavelengths and therefore different sizes, they have different statistics.

The Rayleigh scattering formula collects those statistics into a very convenient number: Given a beam of a given color of light shining through a unit volume of space, how intense is the light coming from the sides instead of passing straight through?

Part 4: Why Blue?

Now that we understand what Rayleigh scattering is, the next question is... Why blue?

Well... that's where those electromagnetic shenanigans I mentioned come into play. It turns out that the relevant cross-sectional radius has an inverse relationship with the wavelength, so that means the cross-sectional area has an inverse square relationship with the wavelength -- that is, counterintuitively, longer wavelengths mean a SMALLER target. (Actually, it's that higher frequencies make it less likely to have a collision, and higher frequencies mean lower wavelengths.) And then since the light can scatter in any direction, the amount of light coming in any one direction follows the inverse square law (brief description: a ray passing through the center of a sphere must pass through a point on the surface of that sphere, and surface area is 4pi*r^2), so Rayleigh scattering ends up being inversely proportional to the 4th power of the wavelength.

The important part of that last paragraph is that shorter wavelength -> more scattering.

For a ray of light that WOULDN'T otherwise pass in a straight line from the Sun to your eye, for you to be able to see it at all it would have to be scattered.

Remember that violet has the shortest wavelength of visible light, and red has the longest. So that means the red light from the sun is more likely to keep going instead of bouncing off of the air in order to come back and hit your eye.

Why isn't the sky purple, then? Because referring back to part 1, the Sun emits more blue light than it emits violet light, so even though a greater PROPORTION of purple light is getting scattered back to your eye, there's just not as much of it in the first place, so you see more blue.

Part 5: But You Said...!

Yep. I said the Sun is blue.

Sure doesn't look blue, does it? Looks yellow!

The light that IS coming directly from the Sun to your eye is ALSO getting scattered. The blue light is getting deflected off to the side more instead of reaching you. That leaves the light that gets scattered less -- the yellows and the oranges and the reds.

Why isn't it red, then, if red gets scattered least?

Same reason the sky isn't purple: the Sun emits more yellow light than red.

From outer space, the Sun looks white, and space looks black. There's no air in the way, so all of the colors of light mix together in our eye and we perceive white that might be ever so faintly tinted blue.

Part 6: Absorbing? Re-Emitting?

I didn't actually talk about this stuff above!

This is one of the reasons -- though by no means the ONLY reason -- I think Rayleigh scattering is taught poorly. The whole nitty-gritty details about absorption and re-emission is diving into too much detail on the WRONG PART of the phenomenon. It suggests that it's relevant to the macro-scale things we observe, when in fact you can understand how Rayleigh scattering works without ever needing to talk about it.

But to actually discuss it...

Photons don't actually BOUNCE off of things. Despite our fun little relativity discussion, they don't have mass; they're pure energy. So when a photon hits something, it transfers its energy into whatever it hits. But atoms can't hold arbitrary amounts of energy; they want to be at the lowest stable energy state they can, so if nothing happens to make the higher-energy state stable they quickly shed that extra energy in the form of another photon. And since the energy of the photon determines its wavelength, the photon that comes out is exactly the same color as the one that went in. (Obviously there are exceptions -- those exceptions are where chemistry happens.)

Photons also aren't just particles. They're also waves, and no macro-scale physical model can fully imitate ALL of the weirdness that implies. It's the wave nature of a photon that causes the counterintuitive behavior where longer wavelengths mean shorter cross-sections for collisions.

But you don't NEED to worry about that for a high-school level description.