# The blue sky — is the simple Rayleigh explanation wrong?

Reading various answers here ([1], [2], [3]), Wikipedia, nasa.gov, and other places, the common explanation of our blue sky is Rayleigh Scattering due to gases and particles in the air, maybe with some Mie Scattering thrown in, too. But then I found this discussion on a physics forum, and it reminded me of a discussion I had with a knowledgeable person a long time ago:

The basic premise is that it's not the individual molecules (mainly nitrogen, oxygen) that are responsible for the scattering, but rather the probabilistic fluctuations of the spaces between them.

Quoting from the end of that physicsforum.com discussion (and ignoring the somewhat spicy back-and-forth between the participants):

Condense air into liquid. It will have about thousand times MORE molecules per same volume. According to Rayley, etc there shall be thousand times MORE scattered light.

In reality it is way LESS.

The reason that Rayley formula (derived on the assumption that each molecule radiates incoherently from others) was originally considered as a satisfactory explanation, is that for an ideal gas mean of SQUARES of deviations (fluctuations) of number of molecules from the average for any given volume by chance coincides with the number of molecules in that volume. In the absense of those deviations scattering is zero (like in solids and to some degree in liquids).

I think first this was cleared by L.I. Mandelstamm in his paper "About optically homogenious and inhomogenious media" published in 1907.

This reasoning makes sense to me. If the N₂/O₂ molecules were the "particles" involved in Rayleigh scattering, then with twice as many molecules, we should have twice as much scattering. See other examples given in that discussion re: crystals and solids, too.

Indeed, even Einsein and Smoluchowski use density fluctuations as the explanation for blue sky.

Einstein's key insight in his paper was that the phenomena of critical opalescence and the blue color of the sky, which are not obviously related to each other, are both due to density fluctuations caused by the molecular constitution of matter.

Even after the paper's publication, however, the relationship between the two phenomena remained unclear to Smoluchowski. In 1911 he published a paper in which he claimed that the blue color of the sky has two causes: Rayleigh scattering by the molecules of the air, and Smoluchowski-Einstein scattering by density fluctuations.[16] Einstein immediately responded to this paper, pointing out that "a 'molecular opalescence' in addition to the fluctuation opalescence does not exist."[17] Smoluchowski readily accepted Einstein's criticism.

Another source: https://www.ias.ac.in/article/fulltext/reso/005/04/0037-0045

He [Einstein] realized that the fluctuations in density in the critical region would lead to the corresponding fluctuations in the refractive index of the medium. These refractive index variations would behave like atomic size scatterers and give rise to the scattering of light. If the fluctuations are large, the light scattering also becomes large

And one more, with more rigor: http://users.df.uba.ar/bragas/Web%20roberto/Papers/sobelman%20light%20scattering.pdf

Later on, when the concept of fluctuations was realized (Smoluchowski, Einstein), it became clear that the scattering in rarefied gases is determined by the fluctuations of density or the number of particles, i.e. by the quantity $$\overline{\Delta N^2}$$. But for an ideal gas one has $$\overline{\Delta N^2} = N$$, i.e. the result arrived at is precisely the same as in the consideration of the light scattering by individual particles.

So, as far as I can tell, this is the correct explanation. The Rayleigh scattering formula is not wrong per-se, but the "particles" involved are not air molecules, but rather thermal density fluctuations, which just so happen to have matching mathematical characteristics for an ideal gas to make the Rayleigh formula work as-is.

Yet on Wikipedia, we have (emphasis added): "The daytime sky appears blue because air molecules scatter shorter wavelengths of sunlight more than longer ones (redder light)...most of the light in the daytime sky is caused by scattering, which is dominated by ... Rayleigh scattering. The scattering due to molecule-sized particles (as in air) is greater in the directions ..."

So which is it?

Help me, physics.stackexchange — you're my only hope.

EDIT I also found this in a textbook scanned online:

This equation shows that the blue color of the sky is due to fluctuations in the density of the atmosphere...

Yet at the same time, the same text refers to fluctuation-based scattering as "Rayleigh scattering," which in most other sources I've found is a term associated with particle interactions, not fluctuations.

And another example: https://www.sciencedirect.com/topics/chemistry/rayleigh-scattering

Here, they refer separately to "the key role of fluctuations in RS [Rayleigh Scattering] from condensed media" and "particle RS," as if to indicate that the "Rayleigh" term applies to both, even though (as I understand it) Rayleigh's original model was just about particles, even if it accidentally also yielded the correct formula for the special case of ideal gases.

Bit of a mess of terminology...

• Wikipedia is not a reliable source on physics. You can find common misconceptions, personal views and incomplete coverage of known theories there. Omission of index of refraction fluctuations in explanation of the light scattering in atmosphere is one example of this. Commented Jul 6, 2023 at 20:06
• @JánLalinský Not just Wikipedia, but apparently also nasa.gov, universities, and other science-oriented resources (including this site, and others like it).
– jwd
Commented Jul 6, 2023 at 20:11
• Can you explain why you (or your sources) claim water scatters "way less" light than air? When we look at the blue sky we're looking through something like 50km of air. About 200m down in the water it's essentially pitch black. So it would seem that water vs air actually perfectly satisfies the "by that argument there should be 1000X the scattering" issue that you're posing. Commented Jul 6, 2023 at 22:58
• @AXensen : I think you're broadly right, but I'd point out the "pitch black" actually is from absorption, not scattering. Scattering doesn't reduce light intensity, just destroys its directional coherence. Loss of intensity, thus, is absorption. However, water does have a blue tint at intermediate depths between a few m to the 200 m where it becomes too dark to see. Commented Jul 6, 2023 at 23:09
• @The_Sympathizer Scattering without absorption can reduce the intensity of light through a thick barrier, because some of the light can leak back out the incident surface. This isn't what happens in seawater, but it's what happens in clouds and snowbanks, which are bright on their sunny sides and dark underneath.
– rob
Commented Jul 6, 2023 at 23:55

## 2 Answers

The daytime sky appears blue because air molecules scatter

Well, this is not actually wrong. Molecules do scatter light, each molecule has a particular cross section. And this cross section is very small, so the dependence on wave frequency and scattering angle also follows the Rayleigh formula.

The light scattered by each molecule is subsequently scattered by the neighboring molecules, and then again and again; these fields all add up to yield the final scattered field. This is the microscopic description.

The total scattered field is the result of interference of all the $$n$$-times-scattered fields. It so happens, that when the medium is made of an ordered (periodic or quasi-periodic) arrangement of scattering centers (atoms, molecules, identical glass spheres etc.), we get only a few directions of scattering due to destructive interference in all other directions. Often these directions are the directions of the refracted and (if the light first meets an interface like vacuum-medium) reflected waves, sometimes diffraction can happen (again, on interface).

In a gas, since the positions of the scattering centers are uncorrelated, the $$n$$-times-scattered fields are also uncorrelated, and thus the addition of the $$n$$-times-scattered far fields can be viewed as incoherent, i.e. we can simply add intensities (rather than fields) and get the same resulting intensity.

This can also be described in a macroscopic way, by considering the macroscopic dielectric permittivity of the medium and its fluctuations. This will yield the result in terms of refractive index, rather than in terms of molecular cross sections. The end result will be the same.

I'm not so sure of very dense gases though. Maybe the assumption of total uncorrelation between the scattered fields of different orders (numbers of re-scatterings) would be wrong. Then, if singly-scattered amplitude appears to be comparable to that of the incident light (can such a gas even exist?), the result of macroscopic vs the naive microscopic computation would be different.

Finally, there seems to be some doubt in the OP as to whether Lord Rayleigh actually was naive enough not to understand the limitation of this approach. From his paper (ref. 1) we can see that he was fully aware of the requirement of total disorder for the addition of intensities to work:

We have no reason to suppose that the purest water is any more transparent than (14) would indicate; but it is more than doubtful whether the calculations are applicable to such a case, where the fundamental supposition, that the phases are entirely at random, is violated.

### References

1. Lord Rayleigh F.R.S. (1899) XXXIV. On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 47:287, 375-384, DOI: 10.1080/14786449908621276
• So how do you feel about this line (from a quote in the question)? "Einstein immediately responded to this paper, pointing out that "a 'molecular opalescence' in addition to the fluctuation opalescence does not exist."[17] Smoluchowski readily accepted Einstein's criticism." By my reading, Einstein says molecules do not cause the Rayleigh scattering you describe on their own. It is only density fluctuation that is responsible.
– jwd
Commented Jul 7, 2023 at 0:20
• "a 'molecular opalescence' in addition to the fluctuation opalescence does not exist." — the key phrase here is "in addition to". The descriptions via molecules or fluctuations are two different approaches to the same phenomenon. If you choose one, you can't say the other adds radiance to the result. "A molecule itself does not SCATTER" — this is surely incorrect. A sole molecule in a vacuum is a fluctuation of vacuum's optical properties, so even in this language it would obviously scatter. Commented Jul 7, 2023 at 7:48
• "results in scattering or is coherent" — here it seems to be a play on semantics. In my use of the word "scattering" both coherent and incoherent addition of output EM wave would be called scattering. Otherwise, by this logic, a Mie solution for a tiny sphere must not be called scattered field, because it corresponds to a single object (molecule or whatever else). Not a good choice of terminology IMO. Commented Jul 7, 2023 at 7:51
• "increasing the number of molecules per volume <...> we'd see a corresponding increase in the scattering effect" — we do, just look at Earth's atmosphere from space: you'll see that as your view crosses lower-density layers, radiance from there decreases, gradually fading to become black somewhere around 60 km in altitude. Similarly, lidar measurements find exactly the same: as you measure higher and higher layers, backscattering reduces in accordance with concentration (and with correction for aerosols). Commented Jul 7, 2023 at 11:04
• "But surely it's unlikely that the fundamental mechanism is changing?" — no, the mechanism is still the same: if there's some order, there's destructive interference that lowers the far field scattered intensity. If it's an ideal gas, there's no order at all. "Molecules of ideal gas don't magically behave differently than non-ideal-gas molecules." — well, they do behave differently due to interaction with each other, but this is not the main point: what matters most is that if they are arranged with some order, their scattered fields become aligned, leading to coherent addition. Commented Jul 7, 2023 at 19:59

Interesting question.

The mean free path of air molecules at sea level is of order $$\sim 10^2\rm\ nm$$ (which I got confirmed from a vague recollection, but couldn't instantly reproduce using an online calculator). Since this is substantially shorter than the wavelength of light, $$\sim 10^3\rm\ nm$$, the scattering will be a wave phenomenon which interacts with many molecules at once, and the structure matters. The local changes in index of refraction due to thermal fluctuations are a good way to think of it. (Thinking about fluctuations is how Einstein argued that Brownian motion proves the atomic hypothesis: his least appreciated discovery from the "annus mirabilis" trio in 1905.)

The argument about compressing air into a transparent liquid is silly, because it neglects that structure matters. Consider that glass is transparent, but a pile of shattered glass is white. Or ice is transparent, but a pile of snowflakes is white.

It's not instantly obvious to me why index-of-refraction fluctuations give rise to the chromatic dispersion that makes air blue instead of red ... but this remark got too long for the comment box.

• Note also that the mean free path depends on the pressure and density of the air. It is possible that an argument based on scattering from air molecules applies in the stratosphere, while a model based on fluctuating regions with different indices of diffraction explains why a mountain looks blue when seen from 40 miles away, through air near sea level.
– rob
Commented Jul 6, 2023 at 17:50
• Re: condensing air to a liquid: I think that commenter was indeed aware of structure issues (see their other comments about changes in scattering across crystal phase transitions, eg), just using that as a counterexample to the simplistic explanation. That is to say: if the naïve explanation were correct, then structure would not matter (only number of molecules), and condensing it down would increase scattering. But it doesn't. Therefore the naïve explanation is not sufficient.
– jwd
Commented Jul 6, 2023 at 18:14