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Celebrating Einstein
"Einstein and the Daytime Sky"

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B.  Effects of an uneven random distribution

Einstein's work involved giving some greater mathematical precision to some recent ideas of the physicist Marian von Smoluchowski.  Smoluchowski considered the significant effect that variations in a fluid's density can have on how light waves travel through the fluid.

When a fluid, either gas or liquid, is compressed, its molecules get crowded into a smaller space, so the fluid becomes more dense.  Since molecules constantly move about in random directions, there is always some chance that the fluid will spontaneously compress itself as its molecules move closer together at a given time.  This chance is extremely small if very many molecules are involved.  But it's a less unlikely coincidence for just a few independently-moving molecules to happen to move towards each other, so the chance that a small mass of fluid will compress spontaneously is much greater.  Similarly, the chance that a portion of fluid will spontaneously expand and decrease its density depends on the number of molecules that have to move apart:  small sets of molecules will happen to move apart fairly often, while larger sets of molecules are less likely to move apart at any given time.

The likelihood of a spontaneous density change depends on the temperature.  Higher temperature means higher average speed for the molecules-or, viewed another way, higher temperature means higher average energy per molecule.  The molecules' energy is relevant, because it turns out that the chance of a given-size density change is practically zero unless the energy needed to compress or expand the fluid by the right amount is roughly equal to the average energy of motion of a single molecule.

How will this affect light?  Light waves travel more slowly through matter than through empty space-even more slowly, the greater the density of the material.  If a transparent fluid has different densities at different places, light waves will speed up or slow down as they travel through.  Furthermore, whenever light changes its speed, the light waves will scatter; where the density changes a great deal within a short distance, the light will scatter strongly.

While Smoluchowski had worked out this much, he had not calculated the intensity of the scattered light waves.  In taking up the problem, Einstein considered two points.  The likelihood of a given-sized change in a fluid's density depends on the temperature; Einstein calculated this dependence.  Using the laws of electromagnetic force fields, Einstein also calculated how strongly light waves would scatter when they ran into any given density variation.  Then, accounting for the likelihood of each possible variation at a given temperature, Einstein combined these two results to see how the light would be affected on the average-much scattering due to the most likely density variations, plus a little scattering due to the less likely density variations.

Einstein's result showed that the scattered light's intensity depends on several factors.

The last factor is particularly significant for why the earth's atmosphere-a gaseous fluid itself-looks blue in the daytime.  If fluids scattered all light waves the same, independently of their frequency, the light from the sky and the direct light from the sun would have the same frequency mix and would thus both be the same color.  But the atmosphere, like any fluid, scatters high-frequency waves much more than low-frequency waves.  In fact, if two light waves are scattered by the air, one wave having twice the frequency of the other, the higher-frequency scattered wave will be about 16 times as intense as the lower-frequency scattered wave.  (The ratio would be exactly 16 if air slowed the speed of all light waves equally, but the air slows higher-frequency light slightly more.)  Unlike direct sunlight, scattered sunlight has so much less of the low-frequency waves like red and yellow that it looks blue.     (.....continued)

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Last Modified: 2/26/2007

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