During my holiday I could enjoy many beautiful views of sunsets just above the see. It is a natural and, I think, very common question why all the colours of sky suddenly shift to the red/orange hues.

You probably know the answer to that question. Light is being scattered on various molecules suspended in the air with blue-coloured photons are more likely to enjoy that phenomenon. This is why sky is blue and when the Sun is setting photons have much longer way to get into your eyes than in the midday. More blue light is scattered leaving a majority of reddish colours in the light ray. Hence, the proximity of the setting Sun is red.

The first approximation of the quantitative answer to that question is Rayleigh Scattering that is, a result stating that the intensity of scattered light is proportional to the negative fourth power of the wavelength. This diversifies blue from red very much! Bluish photons have higher energy hence shorter wavelength. Red light - just the opposite. To obtain Rayleigh's famous result, instead of going into very details of the scattering theory, we will use dimensional analysis.

Dimensional analysis is an extremely useful technique in obtaining various formulas without knowing all about the phenomenon. It uses symmetry and we certainly will come back to its theoretical and philosophical aspects in the future. Here, we sketch only an intuitive approach to it. Generally speaking, in order to obtain a formula for anything you would like, you have to collect all relevant quantities and multiply their respective powers. The result of the multiplication has to have the dimensionality of the quantity you are looking for. As you can guess, there is no unique answer to that problem and but this procedure always works and rigorous details are stated in the famous Buckingham Pi Theorem. We will surely come back to this fascinating topic.

Let us see how the dimensional analysis can help us in finding the law of Rayleigh Scattering. Consider a particle of volume $V$ which scatters a light beam of wavelength $\lambda$ sent by a source distant by $d$. Denote the initial intensity of light by $I_0$ and after scattering by $I$. Then,

$\displaystyle \frac{I}{I_0} = C V^a d^\beta \lambda^c$,

where $C$ is an unknown constant and $a,b,c$ are to be found. Of course, the right-hand side can depend on some other variables and we never do not know the exact number of them. Fortunately, dimensional analysis is a self-correcting method meaning that if we do not include sufficient number of quantities we will arrive at a contradiction (either formal or experimental). On the other hand, if we take too many variables they will cancel leaving only the necessary ones. Now, the left-hand side of the above equation is nondimensional while $[V]=L^3$, $[d]=L$ and $[\lambda]=L$, where $L$ is dimension of length (such as meter). Because the dimensionalities have to agree we must have

$\displaystyle 3a+b+c = 0$,

or

$\displaystyle c = -b - 3a$.

As we can see, there is a two-dimensional subspace of this homogeneous equation and without physical insight we cannot proceed any further. First, since the source emits energy in every direction the intensity of light decays as $r^{-2}$, which gives us $b=-2$. This can be visualized by noticing that the energy of photons at some particular time is constant on a sphere with certain radius. If time passes, photons propagate and sphere becomes larger but the total energy of light stays the same over its surface.

Next, the amplitude of scattered light is proportional to the number of molecules which, in turn, is proportional to their volume $V$. Moreover, we know that the energy of a wave is proportional to the square of the amplitude hence we have $a=2$. Therefore $c=2-3\times 2 = -4$. That is the famous Rayleigh's $\lambda^{-4}$ law!

As a quick application of our formula we calculate how much blue photons are scattered with comparison with the red ones. The appropriate wavelengths are $\lambda_r \approx 700 nm$ while $\lambda_b \approx 475 nm$ for red and blue light rescpectively. Then,

$\displaystyle \frac{I_b}{I_r} = \left(\frac{\lambda_r}{\lambda_b}\right)^4 = 4.75$,

which says that almost five times more blue light is scattered than the red. If we had chosen the extremes of the visible spectrum we would obtain almost ten times more scattering of violet light than red. We can see that this phenomenon is indeed very sensitive on the frequency.

Rayleigh scattering is a first approximation of the so-called Mie theory and it is valid where scattering particles are much larger than wavelength. This is the case of setting sun but for other phenomena such as rainbow Rayleigh approximation would not give the right answer.