12.1: Statement of the Basic Problem

We have now acquired all the material necessary to construct a model of the atmosphere of a star. This material includes not only the dependence of the state variables P, T and ρ with depth in the atmosphere but also an approximation to the emergent spectrum. That predicted spectrum will not contain the details of the stellar absorption lines, but will show the departures from the Planck function of a radiation field in thermodynamic equilibrium. The major departures are caused by the absorption edges corresponding to the ionization limits for the elements included in the calculation. Even if there were no such discontinuities in the frequency dependence of the absorption coefficient, the emergent spectra would still differ from those of a blackbody. Since the photons emerge from different depths in the atmosphere, having different temperatures, even a gray atmosphere spectrum will depart from the Planck function. The more sophisticated spectrum results from the solution of the equation of radiative transfer, the calculation of which represents a major part of the construction of a model atmosphere.

In developing this material, we have used the same conservation laws that yielded the equations of structure for the stellar interior. However, the resulting formulation is somewhat different. The conservation of momentum yielded the expression for hydrostatic equilibrium, as it did in the stellar interior. However, the assumption of a plane-parallel structure for the atmosphere and the use of a different depth coordinate have caused the expression of hydrostatic equilibrium to take a somewhat different form than for the stellar interior. The conservation of energy is at the root of radiative equilibrium. This condition is imposed on the Boltzmann transport equation itself which was used to produce the equation of radiative transfer. However, because of the departure of the radiation field from STE, the flow of radiation was described by an integral equation, implying that the solution at any point depends on the solution at all points. As a result, we no longer have the simple differential equation for the radiative gradient that was appropriate for the stellar interior. Even the equation of state, which results from saying that the local velocity field of the particles is largely isotropic and dominates any macroscopic flow velocity, is present in basically the form used in the stellar interior. Although the calculation of the mass absorption coefficient appears to present a greater problem for stellar atmospheres, this is largely an illusion. The construction of an accurate model interior requires careful calculation of the frequency-dependent absorption coefficient, and the range of atomic phenomena that must be included is actually greater than that of an atmosphere because of the greater range of possible ionization states. However, in the stellar atmosphere, the frequency dependence of the absorption coefficient enters far more directly into the solution and plays a greater role. The use of the Rosseland Mean opacity for stellar interiors tends to average out the "mistakes" in the opacity calculations whereas those mistakes in a stellar atmosphere are directly visible in the emergent spectrum. The presence of molecules is an added complication for the theory of stellar atmospheres that does not plague the theory of stellar interiors.

Much has been concealed by writing the opacity as a function of the state variables. But while the details are messy and LTE has been assumed, the process is straightforward. The major difference between the calculation of a stellar interior and the construction of a model stellar atmosphere can be seen in the last of equations \ref{12.1.1}. No longer do we have a situation that can be mathematically described as a linear two-point boundary value problem. Because of the assumption of plane parallelism the "eigenvalues" of the problem have been reduced to two, \(T_e\) and \(g\). In addition, the four nonlinear differential equations of the interior structure have been replaced by one first order differential equation and an integral equation for the source function from which all physically relevant moments of the radiation field can be calculated. The global nature of this integral equation forces a rather different approach to the construction of a model stellar atmosphere from that adopted for the stellar interior.

We can summarize the equations of atmospheric structure, obtained from these conservation laws and assumptions: \[\begin{array}{ll}
\frac{d P}{d \tau_\nu}=\frac{g}{\kappa_\nu+\sigma_\nu} & \text {hydrostatic equilibrium } \\
d \tau_\nu=-\left(\kappa_\nu+\sigma_\nu\right) \rho d x & \text {definition of } \tau_\nu \\
P=\frac{\rho k T}{\mu m_h}+P_r & \text {equation of state } \\
\mathbf{F}=\frac{\sigma T_e^4}{\pi} & \text {definition of } T_e \\
\int_0^{\infty} \kappa_\nu\left\{J_\nu\left(\tau_\nu\right)-B_\nu\left[T\left(\tau_\nu\right)\right]\right\} d \nu=0 & \text {radiative equilibrium } \\
S_\nu\left(\tau_\nu\right)=\epsilon_\nu\left(\tau_\nu\right) B_\nu\left[T\left(\tau_\nu\right)\right]+\frac{1}{2}\left[1-\epsilon_\nu\left(\tau_\nu\right)\right] \int_0^{\infty} S_\nu(t) E_1\left|\tau_\nu-t\right| d t&\text{integral equation of radiative transfer}
\end{array}\label{12.1.1}\]

In general, we split the problem into two parts, each of which can be solved with the knowledge of the other. After making as shrewd a guess as possible for the approximate solution of one, we solve the other and use those results to improve the initial guess for the first. We can then proceed to solve these two halves of the problem alternately until we obtain an answer that is self-consistent and satisfies the conditions of radiative equilibrium throughout the atmosphere. The basic division of the problem is to calculate the depth dependence of the state variables under the assumption of the radiation field and then to use this atmospheric structure to improve the radiation field. Since the initial guess for the radiation field is not likely to be correct, we cannot expect that radiative equilibrium will be satisfied throughout the atmosphere. Thus we try to use the calculated departures from radiative equilibrium to modify the physical structure so as to produce a radiation field that more nearly satisfies radiative equilibrium. Since we have already dealt with the solution of the equation of radiative transfer, most of this chapter involves the iterative aspect of the problem. Proper formulation of such a correction scheme will provide the basis for forming a rigorous iterative algorithm that will converge to a fully self-consistent model atmosphere with a structure that yields a radiation field satisfying radiative equilibrium throughout the entire atmosphere. However, we must begin with some comments on how to find the dependence of state variables on depth in the atmosphere, given the radiation field. This involves the solution of the differential equation for hydrostatic equilibrium.