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## Overview

Fortunately, most of these formal variables are tightly coupled to a much smaller set of variables which we might, therefore, consider the fundamental'' variables of the model atmosphere problem. In our approach, these fundamental variables are the temperatures T, the gas pressures , and the population numbers ni at each radial point ri. The radiation field is considered a derived'' quantity and the problem is thus reduced to find a set of physical variables at each radial point i so that the system outlined in Fig. 2 is self-consistent. With this approach we have reduced the number of variables from several million to a few 100 thousand, which is still a daunting number.
Although it is possible to analytically bring the system into a form so that it could be solved by a Newton-Raphson approach [11], this idea is computationally prohibitive because of its enormous memory and time requirements (however, for smaller systems this approach has been used successfully). Furthermore, this approach is complex to implement and it is relatively hard to add more physics'' to the model atmosphere. We have thus developed a scheme of nested iterative solutions that considers the direct (or strong) couplings between important variables directly and iteratively accounts for the indirect coupling between sets of variables. With this approach the problem of constructing the model atmosphere can be separated into solving a large number of smaller problems with only a few 100 variables. The global requirement of a self-consistent solution is then reach by iteratively coupling these sets of variables to each other until a prescribed accuracy has been reached. This method works because the level of coupling between the variables is very different. For example, the temperature structure of the atmosphere depends mostly on the global constraint of energy conservation (represented by wavelength integrals over the whole spectrum) and on the ratios of several averaged opacities, but it does not depend strongly on the fine details of the radiation field or the individual population of the vast majority of the atomic levels. Therefore, correction to the temperature structure can be calculated approximately. The current errors of, e.g., the energy conservation equations, must be calculated exactly in order to this scheme to function, however, this is relatively simple. The general idea of calculating errors exactly but corrections to the variables approximately will work if the approximations are good enough for the scheme to converge at all. This method will require more iterations to reach convergence but this is more than offset by faster individual iterations and (very often) by better robustness. The latter is very important if many model atmospheres have to be constructed or if no good initial guesses for the the variables are known.