11.3: Continuous Opacity

In Section 4.1b we discussed the way in which a gas can absorb photons, and we calculated the continuous opacity due to an atom of hydrogen. The calculation of the opacity of other individual species of atoms follows the same type of argument, and so we do not deal with them in detail here since these details can be found in other elsewhere1. However, in the stellar interior, we were able to characterize the opacity of the stellar material by a single parameter known as the Rosseland mean opacity. This resulted from the fact that the radiation field was itself in thermodynamic equilibrium and therefore depended on the temperature alone. Therefore, all parameters that arise from the interaction of the radiation field and the gas, which is also in thermodynamic equilibrium, must be described in terms of the state variables alone. Unfortunately, in the stellar atmosphere, although the gas can still be considered to be in thermodynamic equilibrium (LTE), the radiation field is not. Thus, the opacity must be determined for each frequency for which a significant amount of radiant energy is flowing through the gas.

Traditionally, the dominant source of opacity has been considered to be that arising from bound-free atomic transitions which are called "continuous" opacity sources. However, contributions to the total opacity that result from bound-bound atomic transitions have been found to play an important role in forming the structure of the atmosphere in a wide variety of stars. Since we will deal with the formation of spectral lines arising from bound-bound transitions in considerable detail later, we defer the discussion of bound-bound opacity until then. It is sufficient to know that the total opacity can be calculated for each frequency of importance and to explicitly consider some of the important sources.

a. Hydrogenlike Opacity

Any atom which has a single electron in its outer shell will absorb photons in a manner similar to that for hydrogen, so we should expect the opacity to have a form similar to that of equation \ref{4.1.17}. Indeed the expression differs from that of hydrogen by only a factor involving the atomic weight and the atomic constants appropriate for the particular atom. Thus, the opacity per gram of the ionized species is \[\kappa_\nu(\mathrm{H} \text {-like })=\frac{32 \pi e^6 \mathrm{R} Z^4 e^{-\chi_{\mathrm{H} 1} /(k T)}}{3 \sqrt{3} h^3 c m_h \nu^3}\left(\sum_{n, \nu_n<\nu}^{\infty} \frac{g_n}{n^3} e^{-\epsilon_n /(k T)}+\frac{g_{\mathrm{ff}} k T}{2 \chi_{\mathrm{H} 1}}\right)\label{11.2.1}\]

where R is the Rydberg constant. For ionized helium, Z = 2. Since hydrogen and helium are by far the most abundant elements in stars, neutral hydrogen and ionized helium must be considered major sources of opacity.

b. Neutral Helium

Again, because of the large abundance of neutral helium it long been regarded as an important source of stellar opacity. However, since neutral helium has two electrons in the outer shell, the atomic absorption coefficient is much more difficult to calculate. Approximate values for the opacity of neutral helium were first given by Ueno et al.² in 1954. Later Stewart and Webb³ calculated the opacity arising from the ground state, and in general the contributions from the first two excited states must be treated separately. For states of excitation greater than 2, the approximate solutions will generally suffice.

c. Quasi-atomic and Molecular States

Although molecular opacity plays an important role in the later-type stars (and will be dealt with later), one might think that molecular opacity is unimportant at temperatures greater than those corresponding to the disassociation energies of the molecules. However, some molecules and atomic states may form for a short time and absorb photons before they disassociate. If the abundance of the atomic species that give rise to these quasi-states is great, they may provide a significant source of opacity. The prototypical example of a short lived or quasi-state is the H-minus ion.

Classically, the existence of the H-minus ion can be inferred from the incomplete screening of the proton by the orbiting electron of Hydrogen (see figure 11.1). Although the orbiting electron is "on one side" of the proton, it is possible to bind an additional electron to the atom for a short time. During this time, the additional electron can undergo bound-free transitions. Quantum mechanically, there exists a single weakly bound state for an electron near a neutral hydrogen atom. This state is weakly bound since the dipole moment of the hydrogen atom is small. The negatively charged configuration is called the H-minus ion and is important in stars only because of the great abundance of hydrogen and electrons at certain temperatures and densities. The Saha equation for the abundance of such ions will be somewhat different from that for normal atoms as the existence of the ion will depend on the availability of electrons as well as hydrogen atoms. Thus the Saha equation for the H-minus ion would have the form \[\frac{N(\mathrm{H}^{-})}{N(\mathrm{H})}=\Phi(T) P_e\label{11.2.2}\]

The parameter \(\boldsymbol{\phi}(T)\) involves partition functions and the like, but depends on only the temperature. Since the Saha equation for neutral hydrogen implies that the ratio of ionized to neutral hydrogen will also be proportional to the electron pressure, the abundance of the H-minus ion will depend quadratically on the electron pressure. Thus, the relative importance of H-minus to hydrogen opacity will decrease with decreasing pressure. So H-minus is less important than hydrogen as an opacity source for giants than for main sequence dwarfs of the same spectral type.

Figure 11.1 shows a classical representation of a Bohr Hydrogen atom with an additional electron bound temporarily so as to form an H-minus ion.

The opacity of a single H-minus ion has been calculated by a number of people over the past 40 years and is not a simple quantum mechanical calculation. The weak binding of the additional electron allows for the existence of a single bound state from which bound-free transitions make contributions to the continuous opacity. The contribution from the bound-free transitions has a peak at about 0.8 micrometers (µm) and is roughly bell-shaped in frequency with a half width of about 1 µm. The free-free contribution rises steadily into the infrared, becoming equal to the bound-free absorption at about 1.45 µm.

d. Important Sources of Continuous Opacity for Main Sequence Stars

OB Stars For the hottest stars on the main sequence (the O stars), most of the hydrogen is ionized along with a substantial amount of the helium. The largest source of continuous opacity is therefore due to electron scattering. From spectral type B2 to O, neutral helium joins electron scattering as an important opacity source. In the early-type stars later than B2, hydrogen becomes a significant opacity source which increases in importance as one considers later B-type stars. In these stars, the opacity due to bound-bound (see Section 15.4) and bound-free transitions in the metals becomes increasingly important for the ultraviolet part of the spectrum.

Stars of Spectral Type A0 to F5 At spectral type A0, neutral hydrogen is the dominant source of continuous opacity. As one moves to later spectral types, H-minus opacity increases in importance and dominates the opacity in the late A type and F stars. For stars of spectral type F, another molecule emerges as an important opacity source. Although the H₂ molecule cannot exist for long at these temperatures, enough does exist at any instant so that the once ionized form, H₂⁺, provides up to 10 percent of the continuous opacity. The H-minus opacity continues to grow in importance throughout this range of spectral type.

Stars of Spectral Type F5 to G In this range H-minus continues to provide more than 60 percent of the continuous opacity. However, other molecules are not of particular importance until one gets to spectral types later than G. The continuous opacity of metals is particularly important in the ultraviolet range below 3000Å. The opacity due to atomic hydrogen diminishes steadily into the G stars, but is still of major importance.

Late Spectral-Type K to M Stars Very little of the spectral energy distribution of these stars can be considered to result from continuum processes - particularly in the later spectral types. The absorption from the myriads of discrete transitions of molecules so dominates the spectrum of the late-type stars that little is apparent except the large molecular absorption bands. However, insofar as continuum processes still take place, Rayleigh scattering from H₂ molecules is the most important source. Absorption arising from the disassociation of molecules also provides an important source of continuous opacity, particularly in the infrared region of the spectrum.