9.2: Basic Assumptions for the Stellar Atmosphere

a. Breakdown of Strict Thermodynamic Equilibrium

The description of the energy distribution of the photons in the stellar interior was made particularly simple by the assumption that all constituents of the gas that made up the star were in their most probable macrostate, resulting from random or uncorrelated collisions. That is, they were in thermodynamic equilibrium. All aspects of such a gas can then be characterized by a single parameter, the temperature, which specifies the mean energy of the gas. All other aspects of the distribution function of the gas particles are described by the equilibrium distribution function for the respective kinds of particles.

The validity of this assumption relied on the fact that various components of the gas would undergo randomizing collisions within a volume where the state variables (specifically the temperature) could be considered constant. Within the deep interior of a star, these conditions are met as well as anywhere in the universe. However, any configuration must have a boundary, and it is there that we should expect this assumption to fail. Such is the case for stars. However, the manner of that failure has a peculiar characteristic in that the particles that make up a star are of two distinctly different types. The photons that make up such an important component of the gas behave quite differently from the particles that have a material rest mass. These photons follow different quantum statistics so that their equilibrium distribution functions are different − Bose-Einstein for the photons and generally maxwellian for everything else. In addition, the mean free path between collisions for material particles is very much less than that for photons. Thus, we would expect that the photons would be the first species of particles to be affected by the presence of a boundary, and this is indeed the case. As one moves outward through a star, the presence of the surface begins to affect the state of the gas when photons first begin to escape directly into space and fail to interact any longer with the material particles of the gas. Since the probability that a specific photon will escape depends on the atomic physics of the opacity corresponding to the photon's energy, we should not expect all photons to escape with equal facility. Thus, the photon distribution will depart progressively from that of the Planck's law as one approaches the boundary and our notion of STE will have broken down.

The increase in the photon mean free path brought about by the decreasing density introduces another problem not unrelated to that posed by the boundary. The variation of the state variables over a "typical" photon mean free path will become a significant fraction of the value of the variables themselves. Thus, the radiation field at any point near the boundary will be made up of photons originating in rather different physical environments. Thus, the characteristics of the radiation field will no longer be determined by the local values of the state variables, but will depend on the structure solution of the entire atmosphere. This global aspect of the properties of the local radiation field completely changes the mathematical formalism that describes the flow of radiation from that used in the interior.

b. Assumption of Local Thermodynamic Equilibrium

It is a happy consequence of the difference between photons and particles with material rest mass that the mean free path for photons is generally very much greater than that for other particles. Thus, while the photons may sense the boundary, there is a substantial region where the material particles do not. The material particles continue to undergo collisions with other material particles and photons, the majority of which still represent their thermodynamic equilibrium distribution. Thus, the material particles of the gas will continue to behave as if they were in thermodynamic equilibrium as one approaches the boundary. Certainly, the point will be reached when collisions between material particles and other constituents of the gas will become sufficiently infrequent that the nonequilibrium photons of the gas will force departures from the Maxwell-Boltzmann energy distribution expected for particles in thermodynamic equilibrium. But by this point in the atmosphere (in many stars), the majority of the photons will have escaped, much of the stellar spectrum will have been established, and the atmospheric structure below this point will be determined. Thus, the notion that the distribution function for the material particles remains that obtained from the local values of the state variables in thermodynamic equilibrium, while the photon distribution does not, is a useful notion. It is called local thermodynamic equilibrium (LTE) and it is one of the central assumptions for much of the remainder of this book. To understand the physical situation that prevails when LTE fails, one must first understand the solution to the problems for which LTE is valid.

The effect of the boundary upon particles that lie within a mean free path of the boundary extends to convective blobs. In the stellar interior, we were able to make do with the crude mixing-length theory because the differences between the adiabatic gradient and that predicted by the mixing-length theory were so small that large errors in this difference became rather small errors in the actual gradient. This was due to the large size of the mixing length, which implied great efficiency for convective transport. This will no longer be the case in the stellar atmosphere, for it is not possible to have a mixing length greater than the local distance to the boundary, and that is the order of a photon mean free path. Thus, convection, should it even occur in the deeper sections of the atmosphere, will be nowhere as efficient as it was in the interior. The mixing-length theory, while crude, can be used to estimate the impact of convection on the atmospheric structure. Fortunately, radiation dominates, by definition, in the outer sections of the atmosphere, and so convection will not be a major concern.

c. Continuum and Spectral Lines

In describing the spectral energy distribution of the photons emerging from a star, it is traditional to distinguish between the smooth distribution of photons and the dark interruptions, or lack of photons, called spectral absorption lines. These features arise because the opacity of atomic bound-bound transitions is so large compared to that of bound-free and free-free processes that photons with energies corresponding to those bound-bound transitions do not sense the boundary until they are relatively near it. At this point in the atmosphere, the temperature has declined to the point where the emitted radiation is less intense than that originating deeper in the atmosphere. Thus, there will be fewer photons at the frequencies corresponding to the bound-bound transitions, giving rise to the absorption lines of stellar spectra.

Remember that the distinction between continuum and line is largely artificial, and often the continuum is shot through with myriads of weak lines. The utility of the concept persists, and we are careful to explain exactly what is meant by the distinction. Since a large section of this book is be devoted to the processes that give rise to spectral lines (and throughout that section we assume that the structure of the atmosphere is known), we assume that continuum processes and photons involved in those processes are the photons that determine the structure of the atmosphere.

d. Additional Assumptions of Normal Stellar Atmospheres

Although some of the development of the theory of stellar atmospheres is presented in great generality, the basic focus of this book is on the theory of "normal" stars. This development is appropriate for most of the stars on the main sequence and some others. We indicate where the assumptions fail in the description of the atmospheres of other stars and what can be done about them, but for now we adopt the traditional assumptions of stellar atmospheres.

In addition to the assumption of LTE, we assume that the thickness of the atmosphere is small compared to the radius of the star. Under these conditions, the surface geometry may be assumed to be that of a plane- parallel slab of infinite thickness possessing a surface extending to infinity in all directions (see Figure 9.1). Since most of the stellar mass will reside inside the atmosphere, it is consistent with the plane-parallel atmosphere approximation to assume that the surface gravity is constant. Thus, the notion of hydrostatic equilibrium given in equation \ref{2.1.6} simplifies to \[\frac{d P(r)}{d r}=-\frac{G M(r) \rho}{r^2}=-g \rho\label{9.1.1}\]

Furthermore, since no sources of energy are likely to be present in the stellar atmosphere and we need not worry about time dependent entropy terms, the conservation of energy [equation \ref{7.1.1}] becomes \[\nabla \cdot \vec{F}=0=\frac{d F}{d x} \quad F=\text { const }=\sigma T_e^4\label{9.1.2}\]

If all the energy is to be carried by radiation, equation \ref{9.1.2} ensures that the radiant flux in the atmosphere will be constant.

Figure 9.1 shows the semi-infinite plane that is appropriate for describing the local conditions for stars with thin atmospheres.

Thus these are the fundamental assumptions for the theory of normal stellar atmospheres:

1. LTE prevails. All properties of the material gas can be specified in terms of the local thermodynamic variables.
2. The atmospheric structure is affected by the continuum opacity only.
3. The local geometry is that of a plane-parallel slab.
4. The local surface gravity can be regarded as constant throughout the atmosphere.
5. All energy is carried by radiation, and there are no sources of energy within the atmosphere.

Under these conditions, in addition to the chemical composition, only two parameters are required to specify the structure of the atmosphere: they are \[g=\frac{G M(r)}{R^2} \quad T_e^4=\frac{L}{4 \pi \sigma R^2}\label{9.1.3}\]

Since R^-2 appears in both the expressions for \(T_e\) and \(g\); it is no longer an independent parameter required for specifying the atmospheric structure. This is a result of the plane-parallel approximation and does not represent a fundamental difference between the theory of stellar atmospheres and the theory of stellar interiors. If that approximation were to be relaxed, R would be required, indicating that the same parameters (M, L, and R) are necessary for the specification of the model's structure as were required for stellar interiors.