15.1: Introduction

Thus far we have made considerable use of the concepts of equilibrium. In the stellar interior, the departures from a steady equilibrium distribution for the photons and gas particles were so small that it was safe to assume that all the constituents of the gas behaved as if they were in STE. However, near the surface of the star, photons escape in such a manner that their energy distribution departs from that expected for thermodynamic equilibrium, producing all the complexities that are seen in stellar spectra. However, the mean free path for collisions between the particles that make up the gas remained short compared to that of the photons, and so the collisions could be regarded as random. More importantly, the majority of the collisions between photons and the gas particles could be viewed as occurring between particles in thermodynamic equilibrium. Therefore, while the radiation field departs from that of a black body, the interactions determining the state of the gas continue to lead to the establishment of an energy distribution for the gas particles characteristic of thermodynamic equilibrium. This happy state allowed the complex properties of the gas to be determined by the local temperature alone and is known as LTE.

However, in the upper reaches of the atmosphere, the density declines to such a point that collisions between gas particles and the remaining "equilibrium" photons will be insufficient for the establishment of LTE. When this occurs, the energy level populations of the excited atoms are no longer governed by the Saha-Boltzmann ionization-excitation formula, but are specified by the specific properties of the atoms and their interactions.

Although the state of the gas is still given by a time-independent distribution function and can be said to be in steady or statistical equilibrium, that equilibrium distribution is no longer the maximal one determined by random collisions. We have seen that the duration of an atom in any given state of excitation is determined by the properties of that atomic state. Thus, any collection of similar atoms will attempt to rearrange their states of excitation in accordance with the atomic properties of their species. Only when the interactions with randomly moving particles are sufficient to overwhelm this tendency will the conditions of LTE prevail. When these interactions fail to dominate, a new equilibrium condition will be established that is different from LTE. Unfortunately, to find this distribution, we have to calculate the rates at which excitation and de-excitation occur for each atomic level in each species and to determine the population levels that are stationary in time. We must include collisions that take place with other constituents of the gas as well as with the radiation field while including the propensity of atoms to spontaneously change their state of excitation. To do this completely and correctly for all atoms is a task of monumental proportions and currently is beyond the capability of even the fastest computers. Thus we will have to make some approximations. In order for the approximations to be appropriate, we first consider the state of the gas that prevails when LTE first begins to fail.

A vast volume of literature exists relating to the failure of LTE and it would be impossible to cover it all. Although the absorption of some photon produced by bound-bound transitions occurs in that part of the spectrum through which the majority of the stellar flux flows, only occasionally is the absorption by specific lines large enough to actually influence the structure of the atmosphere itself. However, in these instances, departures from LTE can affect changes in the atmosphere's structure as well as in the line itself. In the case of hydrogen, departures in the population of the excited levels will also change the "continuous" opacity coefficient and produce further changes in the upper atmosphere structure. To a lesser extent, this may also be true of helium. Therefore, any careful modeling of a stellar atmosphere must include these effects at a very basic level. However, the understanding of the physics of non-LTE is most easily obtained through its effects on specific atomic transitions. In addition, since departures from LTE primarily occur in the upper layers of the atmosphere and therefore affect the formation of the stellar spectra, we concentrate on this aspect of the subject.