15.5: Line Blanketing and Its Inclusion in the Construction of Model Stellar Atmospheres and Its Inclusion in the Construction of Model Stellar Atmospheres

In Chapter 10, we indicated that the presence of myriads of weak spectral lines could add significantly to the total opacity in certain parts of the spectrum and virtually blanket the emerging flux forcing it to appear in other less opaque regions of the spectrum. This is particularly true for the early-type stars for which the major contribution from these lines occurs in the ultraviolet part of the spectrum, where most of the radiative flux flows from the atmosphere. Although it is not strictly a non-LTE effect, the existence of these lines generally formed high in the atmosphere can result in structural changes to the atmosphere not unlike those of non-LTE. The addition of opacity high up in the atmosphere tends to heat the layers immediately below and is sometimes called backwarming.

Because of their sheer number, the inclusion of these lines in the calculation of the opacity coefficient poses some significant problems. The simple approach of including sufficient frequency points to represent the presence of all these lines would simply make the computational problem unmanageable with even the largest of computing machines that exist or can be imagined. Since the early attempts of Chandrasekhar¹³, many efforts have been made to include these effects in the modeling of stellar atmospheres. These early efforts incorporated approximating the lines by a series of frequency "pickets". That is, the frequency dependence would be represented by a discontinuous series of opaque regions that alternate with transparent regions. One could then average over larger sections of the spectrum to obtain a mean line opacity for the entire region. However, this did not represent the effect on the photon flow through the alternatingly opaque and relatively transparent regions with any great accuracy. Others tried using harmonic mean line opacities to reduce this problem. Of these attempts, two have survived and are worthy of consideration.

a. Opacity Sampling

This conceptually simple method of including line blanketing takes advantage of the extremely large number of spectral lines. The basic approach is to represent the frequency-dependent opacity of all the lines as completely as possible. This requires tabulating a list of all the likely lines and their relative strengths. For an element like iron, this could mean the systematic listing of several million lines. In addition, the line shape for each line must be known. This is usually taken to be a Voigt function for it represents an excellent approximation for the vast majority of weak lines. However, its use requires that some estimate of the appropriate damping constant be obtained for each line. In many cases, the Voigt function has been approximated by the Doppler broadening function on the assumption that the damping wings of the line are relatively unimportant. At any frequency the total line absorption coefficient is simply the sum of the significant contributions of lines that contribute to the opacity at that frequency, weighted by the relative abundance of the absorbing species. These abundances are usually obtained by assuming that LTE prevails and so the Saha-Boltzmann ionization-excitation equation can be used.

If one were to pick a very large number of frequencies, this procedure would yield an accurate representation of the effects of metallic line blanketing. However, it would also require prodigious quantities of computing time for modeling the atmosphere. Sneden et al.¹⁴ have shown that sufficient accuracy can be obtained by choosing far fewer frequency points than would be required to represent each line accurately. Although the choice of randomly distributed frequency points which represent large chunks of the frequency domain means that the opacity will be seriously overestimated in some regions and underestimated in others, it is possible to obtain accurate structural results for the atmosphere if a large enough sample of frequency points is chosen. This sample need not be anywhere near as large as that required to represent the individual lines, for what is important for the structure is only the net flow of photons. Thus, if the frequency sampling is sufficiently large to describe the photon flow over reasonably large parts of the spectrum, the resulting structure and the contribution of millions of lines will be accurately represented. This procedure will begin to fail in the higher regions of the atmosphere where the lines become very sharp and non-LTE effects become increasingly important. In practice, this procedure may require the use of several thousand frequency points whereas the correct representation of several million spectral lines would require tens of millions of frequency points. For this reason (and others), this approach has been extremely successful as applied to the structure of late-type model atmospheres where the opacity is dominated by the literally millions of bound-bound transitions occurring in molecules. The larger the number of weak lines and the more uniform their distribution, the more accurate this procedure becomes. However, the longer the lists of spectral lines, the more computer time will be required to carry out the calculation. This entire procedure is generally known as opacity sampling and it possesses a great degree of flexibility in that all aspects of the stellar model that may affect the line broadening can be included ab initio for each model. This is not the case with the competing approach to line blanketing.

b. Opacity Distribution Functions

This approach to describing the absorption by large numbers of lines also involves a form of statistical sampling. However, here the statistical representation is carried out over even larger regions of the spectra than was the case for the opacity sampling scheme. This approach has its origins in the mean opacity concept alluded to earlier. However, instead of replacing the complicated variation of the line opacity over some region of the spectrum with its mean, consider the fraction of the spectral range that has a line opacity less than or equal to some given value. For small intervals of the range, this may be a fairly large number since small intervals correspond to the presence of line cores. If one considers larger fractions of the interval, the total opacity per unit frequency interval of this larger region will decrease, because the spaces between the lines will be included. Thus, an opacity distribution function represents the probability that a randomly chosen point in the interval will have an opacity less than or equal to the given value (see Figure 15.4). The proper name for this function should be the inverse cumulative opacity probability distribution function, but in astronomy it is usually referred to as just the opacity distribution function or (ODF). Carpenter¹⁵ gives a very complete description of the details of computing these functions while a somewhat less complete picture is given by Kurucz and Pettymann¹⁶ and by Mihalas ⁴(pp. 167-169).

The ODF gives the probability that the opacity is a particular fraction of a known value for any range of the frequency interval, and the ODF may be obtained from a graph that is fairly simple to characterize by simple functions. This approach allows the contribution to the total opacity due to spectral lines appropriate for that range of the interval to be calculated. Unfortunately, the magnitude of that given value will depend on the chemical composition and the details of the individual line-broadening mechanisms. Thus, any change in the chemical composition, turbulent broadening, etc., will require a recalculation of the ODF.

Figure 15.4 schematically shows the opacity of a region of the spectrum represented in terms of the actual line opacity (panel a) and the opacity distribution function (panel b).

In addition, ODFs must be calculated as a function of temperature and pressure (or

alternatively, electron density), and so their tabular representation can be extremely large. Their calculation also represents a significant computational effort. However, once ODF's exist, their inclusion in a stellar atmosphere code is fairly simple and the additional computational load for the construction of a model atmosphere is not great, particularly compared to the opacity sampling technique. This constitutes the primary advantage of this approach for the generation of model stellar atmospheres. For stars where the abundances and kinematics of the atmospheres are well known, ODF's provide by far the most efficient means of including the effects of line blanketing. This will become increasingly true as the number of spectral lines for which atomic parameters are known grows; although the task of calculating the opacity distribution functions will also increase.

Considerations such as these will enable the investigator to include the effects of line blanketing and thereby to create reasonably accurate models of the stellar atmosphere which will represent the structure correctly through the line forming region of a normal star. These, when combined with the model interiors discussed in the first six chapters of the book, will allow for the description of normal stars from the center to the surface. While this was the goal of the book, We cannot resist the temptation to demonstrate to the conservative student that the concepts developed so far will allow the models to be extended into the region above stars and to determine some properties of the stellar radiation field that go beyond what is usually considered to be part of the normal stellar model. So in the last chapter we will consider a few extensions of the ideas that have already been developed