## Why Is the Sky Blue?

Apr 01, 2013
Volume 28, Issue 4

There are many ways in which spectroscopy is relevant to matters celestial. Here, we'll tackle one of the classic ones, one to which many know the simple explanation, but few know the details. In this column, we'll go over the details.

About 16 years ago (gee, have I been doing this column THAT long?), I wrote a column titled "Spectroscopy in the Sky," which was a discussion about rainbows. Well, there are other ways in which spectroscopy is relevant to matters celestial. Here we'll tackle one of the classic ones, one to which many know the simple explanation, but few know the details. In this column, we'll go over the details. I am honored to share this installment's by-line with my colleague from our Physics Department, Jim Lock, whose expertise is in particle scattering (that's a foreshadowing of the real reason the sky is blue!).

Why the Sky Is Blue — The Easy Answer

Because of Rayleigh scattering (1).

Why the Sky Is Blue — The Details

Consider an electromagnetic wave of light of angular frequency ω and wavelength λ. This wave passes through the earth's atmosphere as it goes from space to the surface of the earth. Let us pretend that air consists of molecules that have an effective radius a. The incoming wave at the position of the molecule has an electric field E inc given by the expression

where E 0 is the maximum field amplitude for the frequency ω, i is the square root of -1, t is time, and Û is the unit vector in the direction of the wave's polarization. As the light interacts with the air molecule, it imposes a time-dependent induced electric dipole, p(t), on the molecule whose value is given by

The proportionality constant a in equation 2 is called the polarizability of the molecule. Technically, its units are C·m2 /V (coulombs times meters squared per volt), but because it is often combined with the 4πε0 term found in most electrostatic formulas, it is usually expressed in units of (length)3 , typically cm3 or Å3. Because the induced electric dipole moment is time-dependent, it represents an oscillation of electronic charge within the molecule. The molecule thus acts as a small antenna, producing electric dipole radiation. We can consider the behavior of this radiation from three different points of view of increasing sophistication.