Kenton Kwok


Among the grandest and most ambitious questions that one can pose—the meaning of life, the existence of extra-terrestrial beings, or whether mathematics was discovered or invented, there is a place for the special questions we asked as a child. One of which was ‘why is the sky blue?’

To me, perhaps, it is often quite enlightening to hear how one aspect of physics can be used to open up new avenues for almost a completely different domain of research.

The Lord’s Answer

The answer of why the sky is blue has been known for more than a hundred years. The answer to the question came from one of the great English scientists John William Strutt, who is more well known as Lord Rayleigh, after a town in Essex [1]. Not only was he a nobleman but he was also a man who won the 1904 Nobel prize in physics after his work in discovering Argon—a noble gas. Other discoveries by him include the Rayleigh criterion for the resolution of light, the Rayleigh waves in acoustics and also Rayleigh scattering which concerns us today.

Picture of John William Strutt, 3rd Baron of Rayleigh [2]
What is Rayleigh Scattering?

As we know, light is an oscillation in the electromagnetic field and can interact with matter, such as the molecules in the atmosphere (nitrogen, oxygen etc.). When this interaction causes the removal of energy from the wave and a subsequent re-radiation, it is called scattering. If the scattering reemits light of the same wavelength, it is called elastic and otherwise, it is inelastic.

In air, the dominant form of scattering is Rayleigh scattering as the size of the air molecules (10-4 microns) is much smaller than visible light. If we take the model that the induced electric dipoles of the molecules act as a simple harmonic oscillator and solve Maxwell’s equations by performing some mathematics, we find that the scattered intensity is wavelength dependent. It turns out that the intensity of scattering is inversely proportional to the fourth power of the wavelength in the far field [3]. This is extremely strong selectiveness; for example as blue light has a wavelength of 400 nm and red light 650 nm, we can calculate that blue light is scattered (600/400)4 = 5 times more times than red light which is quite a large amount.

The form of the Rayleigh scattering equation is:

Where I0 is the initial intensity, N is the number of scatterers, alpha is the polarizability of the scatters, R is the distance between the observer and the scatterer and theta describes information about the angle scattered.

What gives us the colours in the sky?

If you are looking up at the sky and aren’t looking at the sun directly, you are actually observing scattered light. We just showed that blue light scatters, and hence deviates from its original path more than red light. Hence, it makes sense that the sky is blue because a large part of the light we observed is actually scattered light.

You might have also noticed (or like me, gazed in awe at) how the sky turns a flaming red during times of sunrise and sunset. Amazingly, the same theory can be applied to explain this phenomenon. Here, we are not observing the strongly scattered light but the mildly scattered red light. In combination with dramatic cloud formations, it results in the production of waves of stunningly instagrammable photos. But why doesn’t this happen when the sun is overhead and likewise takes a direct path to our eyes? The sun encounters a smaller number of scatterers when it is overhead in comparison to when it is at the horizon due to the difference in the amount of atmosphere the light passes through. This multiple scattering effect makes the overhead sun retain its white-hot colour rather than appearing red to our eyes [4].

Typical ‘Golden Hour’ time (Hong Kong 2020)
Other atmospheric phenomena

On the opposite spectrum of the level of instagrammability are hazy, pollution-induced days. It turns out that smog and aerosol particulates are roughly the size of the wavelength of visible light [5] and hence they do not obey the Rayleigh scattering regime. Another solution occurs which is the Mie scattering, which is not wavelength dependent, but it is what gives the hazy appearance of the sky [6].

Depiction of Rayleigh and Mie scattering. Image courtesy of Hyperphysics [7]

It is quite illuminating when we see scenarios that differ from Earth’s, especially if we know what Rayleigh scattering is dependent on. For example, white sunsets, rather than red sunsets exist on Mars, which gives some information about the size of the particulates in Mars. According to NASA, these images and the scattering effects allow scientists to understand the extent of its atmosphere. [8]

As a sidenote, not only scientists find this phenomenon interesting. Computer graphics artists also have to consider programming this into games and perhaps animated films to create a realistic look [9] [10]!

What Rayleigh did not envision

What I think is quite amazing is how same physics can be applied to whole new scenarios, often in applications, to give meaningful insights. Rayleigh scattering can also be used in thermometry, as if we consider our gas to be ideal, the concentration should be inversely proportional to the temperature of our species as the number density of scatterers decreases with increasing temperature [11]. By measuring the profile of scattered light, scientists can extract quantitative information about heat distribution and even visualise the flow and velocity of gases [12]. One application of this would be imaging what happens in engines, which would require filtered Rayleigh scattering.

2D Rayleigh thermometry of a flame. Image courtesy of Princeton University

Time and time again, we see elegant phenomena in physics manifest itself in interesting applications. Some may think science ruins the mysterious nature of beauty. But for those who enjoy science, knowing the mechanisms behind things we find interesting might illuminate a more vibrant, deep and meaningful understanding of the world.


References

[1] https://www.aps.org/publications/apsnews/201308/physicshistory.cfm

[2] https://en.wikipedia.org/wiki/John_William_Strutt,_3rd_Baron_Rayleigh#/media/File:John_William_Strutt.jpg

[3] http://irina.eas.gatech.edu/ATOC5235_2003/Lec9.pdf

[4] https://www.spc.noaa.gov/publications/corfidi/sunset/

[5] http://homework.uoregon.edu/pub/class/atm/scatter.html

[6] https://pages.mtu.edu/~scarn/teaching/GE4250/scattering_lecture_slides.pdf

[7] http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/blusky.html#c2

[8] https://solarsystem.nasa.gov/news/925/what-does-a-sunrise-sunset-look-like-on-mars/

[9] https://core.ac.uk/download/pdf/55631247.pdf

[10] https://developer.nvidia.com/gpugems/gpugems2/part-ii-shading-lighting-and-shadows/chapter-16-accurate-atmospheric-scattering

[11] https://cefrc.princeton.edu/sites/cefrc/files/Files/2011%20Lecture%20Notes/Alden/Lecture-7-Rayleigh.pdf

[12] https://iopscience.iop.org/article/10.1088/0957-0233/12/5/201/pdf