# In which altitude does Rayleigh scattering produce the blue color of the sky and are density fluctuations required?

In my Optics textbook, Rayleigh scattering is introduced as producing the blue color of the sky by predominantly scattering short wavelengths because of the well-known $$\lambda^{-4}$$-dependency. It is further explained that in dense optical media (e.g. glass, but also the lower atmosphere), destructive interference occurs in all except the forward direction, thus no scattering sideways occurs. Only in less dense media in which the mean free photon path length exceeds one wavelength (or period), light is incoherently scattered to the sides because the net interference cancels out.

It is concluded, that Rayleigh scattering occurs only high up in the atmosphere. This is supported by the fact that horizontally far away objects close to the surface, e.g. mountains, are not red (like the sunset as blue photons are scattered out of the line of sight). So obviously there is no or only minor Rayleigh scattering close to the surface.

So far so good. But now is said that in the lower and middle atmosphere the air is too dense to allow Rayleigh-scattering, so that the blue color must have another reason, and that's density fluctuations (found by Einstein and Smoluchowski, but not further discussed). I don't understand this since the previous finding that blue Rayleigh-scattering sideways occurs only high up in atmosphere seems to match observations (no red mountains).

So, in which altitude is the blue light due to Rayleigh scattering produced? And what is the role of density fluctuations? Are they needed, or is the precondition (low density, so that mean free path length $$> 2\pi$$ ) sufficient?

Note, there is a similar question here Where in the atmosphere is the blue light scattered? (and I found also some others, but none was solving my problem). 1) It's still unclear to me in which altitude the blue photons are scattered (sideways), and 2) either high altitudes (thin air) is enough, which fits to the explanation of no red mountains, or density fluctuations are required in thicker air, but then sideways Rayleigh-scattering occurs in low altitudes and mountains would be red...

in dense optical media (e.g. glass, but also the lower atmosphere), destructive interference occurs in all except the forward direction, thus no scattering sideways occurs

This is wrong. Destructive interference that prevents sideways scattering only occurs in crystals (and maybe also in quasicrystals). In disordered systems like gases there's no such interference: the scattering centers move chaotically, so even if they form a crystal at some point in time (which itself is extremely unlikely), this crystal symmetry is immediately broken a tiny time later.

So, the scattering that produces blue color happens everywhere in the atmosphere. You can easily notice this if you fly in a plane, or climb a mountain, and then look down. The aerial perspective you'll see is exactly the result of Rayleigh scattering, and you can easily note that at a sunny day it's visible even from relatively small altitudes like $$1\,\mathrm{km}$$.

but then sideways Rayleigh-scattering occurs in low altitudes and mountains would be red

Solar light, that at the horizon we can see as orange, travels through the whole atmosphere horizontally until it gets to the observer. Its brightness is so high that it overwhelms any aerial perspective, so the latter doesn't noticeably influence the color.

On the other hand, mountains are much closer than that. They do become somewhat yellower with distance, but then aerial perspective adds blue tint to the air gap between the observer and the mountain. Since extinction not only changes hue, but also decreases brightness, the result is that the yellowest mountains "dissolve" in the aerial perspective, while the color of those that remain visible is either hard to discern due to aerial perspective, or it's almost unextincted because they are too close.

• This is in contradiction to my textbook, E. Hecht Optics (but an older version...). Sect 4.2 states that lateral scattering doesn't occur in the lower atmosphere and thus density fluctuations are required (their role is my question). Two pages later in Sect. 4.2.1 it says scattering of blue light happens mainly in 150 km altitude. 2) the destructive interference sideways in dense media is in Sect. 4.2.2 not restricted to crystals, but dense gases and liquids are mentioned as well... It only says the more homogeneous the medium, the better the lateral destructive interference. Jul 18, 2021 at 16:47
• @CharlesTucker3 Hecht's book looks more like an encyclopedia to me than a textbook. In the chapters 4.2.1 and 4.2.2 there's not even any maths except vague references to phasors. And, unfortunately, these chapters are very wrong regarding density vs scattering. I suggest to try an actual physics textbook like Landau&Lifshitz "Electrodynamics of Continuous Media", where Rayleigh scattering is discussed in chapter 120, and the formula for Rayleigh scattering coefficient in gases is derived as equation $(120.4)$. Jul 18, 2021 at 17:32
• @CharlesTucker3 BTW, if we follow Hecht's expectations, we'd get the Rayleigh-scattered shine, as seen from the ISS, in the form of a sphere, simiar to airglow, but blue rather than green. But, in practice, we see that it's bright down to the surface. Moreover, the closer to the surface, the brighter. See e.g. this photo. Jul 18, 2021 at 18:12
• Many thanks for the clarification! I thought Hecht is kind of a standard textbook for optics, but following your suggestion I will move to Landau Lifshitz! Jul 18, 2021 at 18:37