The Rayleigh

Why is the sky blue? Because light scatters off molecules in the atmosphere, and shorter wavelengths scatter more. The math is beautiful. The explanation is simple. The physics underneath is QED.

John William Rayleigh figured it out in the 1870s, decades before quantum electrodynamics existed, using nothing more than classical wave theory and a thoughtful observation of the sky. But the full explanation — the real explanation, the one that connects the blue sky to virtual particles and the quantum vacuum — lives in QED.

Rayleigh's law states that the scattering cross-section of light by a particle much smaller than the wavelength is proportional to 1/λ⁴. The intensity of scattered light scales as the fourth power of the frequency. Blue light, at about 450 nm, scatters about ten times more strongly than red light at 650 nm. This is why the sky looks blue during the day — sunlight passes through the atmosphere, and the blue component is scattered in every direction by nitrogen and oxygen molecules. When you look up, you see the scattered blue light.

When you look at the sun at sunset, the light has passed through much more atmosphere. Most of the blue has been scattered away by the time it reaches you. What's left is red, orange, and yellow. The sky turns the color of a dying star. Rayleigh scattering is the reason sunsets are beautiful. The universe does not create beauty, but it does distribute it efficiently.

The 1/λ⁴ dependence comes from treating the molecule as an oscillating dipole. Light is an electromagnetic wave. When it hits a molecule, it oscillates the electron cloud. An oscillating dipole radiates. The power radiated by an oscillating dipole is proportional to the fourth power of the frequency. Therefore, shorter wavelengths scatter more. Simple. Elegant. Correct.

But wait. If the atmosphere scattered light according to Rayleigh's law alone, the sky should be violet, not blue. Violet light has an even shorter wavelength than blue, and 1/λ⁴ makes violet scatter even more. The reason the sky is blue and not violet is a coincidence of biology and physics: the sun's spectrum peaks in the green, so there's more green light than violet to scatter, and the human eye is more sensitive to blue than to violet. The sky is the color of where nature and biology intersect.

The quantum mechanical correction to Rayleigh scattering was computed by Heitler and by Placzek in the 1930s. They showed that the classical result is actually an approximation — a low-energy limit — of a more fundamental QED calculation. At optical frequencies, the photon energy is tiny compared to the binding energy of the electrons, so the classical result is essentially exact. But the reason the classical result works is that QED reduces to classical electromagnetism in the appropriate limit. The blue sky is a consequence of quantum field theory.

There's a subtlety here that even physicists sometimes miss. Rayleigh scattering is coherent scattering — the photon is absorbed and re-emitted by the molecule, but the process is elastic. The photon's energy doesn't change. This is different from Compton scattering, where the photon transfers energy to the electron and changes wavelength. In Rayleigh scattering, the molecule acts as a whole, not as individual electrons. The entire electron cloud oscillates.

The polarization of the scattered light is another signature of Rayleigh scattering. Light scattered at 90 degrees is completely linearly polarized. This is why polarized sunglasses work: they block the horizontally polarized light that has been scattered by the atmosphere (and by roads, water, and glass). Without Rayleigh scattering, polarized sunglasses would have nothing to do.

At higher altitudes, where the atmosphere is thinner, the sky looks darker — almost black, despite being "daytime." This is what astronauts see from the ISS: a black sky with a blue planet hanging in it. No atmosphere means no Rayleigh scattering, and without scattering, there is no blue. The blue sky is a surface phenomenon. It lives at the interface between atmosphere and vacuum.

Edgar Allan Poe, in his prose poem "Eureka," anticipated Rayleigh's result a century before Rayleigh published it. Poe wrote that the blue of the sky was caused by "a rarefied fluid ... whose refractive index ... is ... that of the solar spectrum." He had no mathematics, no experimental data, and no mechanism. But he was right. The blue of the sky is a property of the solar spectrum interacting with a rarefied medium. Poe's intuition was, in this case, more accurate than the physics of his time.

The blue sky is the most common example of physics in everyday life. Every human being who has ever looked up at the sky has witnessed Rayleigh scattering. It is the one piece of physics that the entire species shares. You don't need a textbook. You don't need a laboratory. You just need eyes and a clear day.

And behind it all — behind the blue, behind the white clouds, behind the red sunset — is quantum electrodynamics, computing the scattering amplitude of photons off molecules to an accuracy that would make Rayleigh weep.