14.1: Introduction

If we take the classical picture of the atom as the definitive view of the formation of spectral lines, we would conclude that these lines should be delta functions of frequency and appear as infinitely sharp black lines on the stellar spectra. However, many processes tend to broaden these lines so that the lines develop a characteristic shape or profile. Some of these effects originate in the quantum mechanical description of the atom itself. Others result from perturbations introduced by the neighboring particles in the gas. Still others are generated by the motions of the atoms giving rise to the line. These motions consist of the random thermal motion of the atoms themselves which are superimposed on whatever large scale motions may be present. The macroscopic motions may be highly ordered, as in the case of stellar rotation, or show a high degree of randomness such as is characteristic of turbulent flow.

In practice, all these effects are present and give the line its characteristic shape. The correct representation of these effects allows for the calculation of the observed line profile and in the process reveals a great deal about the conditions in the star that give rise to the spectrum. Of course the photons that give rise to the absorption lines in the stellar spectrum have their origins at different locations in the atmosphere. So the conditions giving rise to a spectral line are really an average of a range of conditions. Thus, when we talk of the excitation temperature or the kinetic temperature appropriate for a specific spectral line, it must be clear that we are referring to some sort of average temperature appropriate for that portion of the atmosphere in which most of the line photons originate. For strong lines with optical depths much greater than the optical depth of the adjacent continuum, the physical depth of the line-forming region is quite small, and the approximation of the physical conditions by their average value is a good one. Unfortunately, for very strong lines, the optical depths can range to such large values that the line-forming region is located in the chromosphere, where most of the assumptions that we have made concerning the structure of the stellar atmosphere break down. A discussion of such lines will have to wait until we are ready to relax the condition of LTE.

In describing the shape or profile of a spectral line, we introduce the notion of the atomic line absorption coefficient. This is a probability density function that describes the probability that a given atom in a particular state of ionization and excitation will absorb a photon of frequency ν in the interval between \(ν\) and \(v+\mathrm{d}v\). We then assume that an ensemble of atoms will follow the probability distribution function of the single atom and produce the line. In order to make the connection between the behavior of a single atom and that of a collection of atoms, we shall make use of the Einstein coefficients that were introduced in Section 11.3.