12.2: Structure of the Atmosphere, Given the Radiation Field

At the outset of any atmosphere calculation we must decide on the particular atmosphere to be modeled. Choosing the parameters \(T_e\), \(g\), and \(\mu\) is analogous to choosing \(\mathrm{M}\), \(\mathrm{L}\), \(\mathrm{R}\), and \(\mu\) for the construction of a model stellar interior. Indeed, the relationship between them is straightforward: \[T_e=\left(\frac{L}{4 \pi \sigma R^2}\right)^{1 / 4} \quad g=\frac{G M}{R^2}\label{12.2.1}\]

a. Choice of the Independent Variable of Atmospheric Depth

Before beginning the model calculation itself, we must choose a depth parameter to serve as an independent variable. While the traditional choice has always been the optical depth, some more modern atmosphere codes use a "density depth" like \[h\left(x_0\right)=\int_0^{x_0} \rho(x) d x\label{12.2.2}\]

as the independent depth variable. As we see, this greatly simplifies the calculation of hydrostatic equilibrium, but introduces some difficulties in the solution of the equation of transfer. For the most part, we use the traditional optical depth as our independent depth variable.

Which optical depth should we use? Early investigators would pick one of the mean opacities to generate a mean optical depth, and this dimensionless, frequency-independent variable would provide an excellent parameter for describing the atmospheric structure. Unfortunately, the calculation of the mean opacity at numerous depths in the atmosphere is a nontrivial undertaking and is completely avoidable. Since there is no particular significance for any of the mean opacities, no optical depth scale is to be preferred over any other on the basis of the physical information contained there. Therefore, it makes good sense to pick a monochromatic optical depth at some frequency \(\tau(\nu_0)\) (or simply \(\tau_0\)) as the depth parameter, thereby avoiding the tedious calculation of the mean opacity and the associated mean optical depths. However, since the radiative transfer equation must be solved at each frequency, it will be necessary to interpolate the solution to the reference optical depth \(\tau_0\). So it would be wise to choose a frequency in the general vicinity of the maximum energy flow through the atmosphere and in a part of the spectrum where the opacity does not vary rapidly with frequency. This will tend to minimize interpolation errors when the solutions are transferred to the reference optical depth. For the majority of the development in this chapter, this is the choice that we make. The relevant optical depths are then given by \[\begin{aligned}
& d \tau_0=\left(\kappa_0+\sigma_0\right) \rho d x \\
& d \tau_\nu=\frac{\kappa_\nu+\sigma_\nu}{\kappa_0+\sigma_0} d \tau_0 \equiv k_\nu d \tau_0
\end{aligned}\label{12.2.3}\]

The parameter \(\kappa_\nu\) is just a normalized opacity which relates the differential monochromatic optical depth to its counterpart on the reference depth scale.

b. Assumption of Temperature Dependence with Depth

Having specified the nature of the atmosphere and chosen the depth parameter, we can begin the calculation of the atmospheric structure with depth. We have indicated that we will split the process of making the model into two parts by assuming the results of the radiative transfer solution in order to calculate the atmospheric structure. The form that this assumption takes is the dependence of the temperature with depth. That is, the end result for the solution of the radiative transfer calculation will be to generate the dependence of the temperature with depth in a manner that is consistent with radiative equilibrium. Thus, to begin our calculation, we must assume the existence of this temperature distribution. We may obtain this information either as a result of an earlier model calculation or from an initial approximation.

In Section 10.2, we spent considerable effort in solving the equation of transfer for the gray atmosphere. One of the results of this effort was the temperature distribution in the Eddington approximation [equation \ref{10.2.16}]. Remembering that for a gray atmosphere in radiative equilibrium \[S(\tau)=J(\tau)=B(\tau)\label{12.2.4}\]

we can write the more general result \[T^4(\tau)=\frac{3}{4} T_e^4[q(\tau)+\tau]\label{12.2.5}\]

where \(q(\tau)\) is the Hopf function specified in equation \ref{10.2.34}. Although the gray atmosphere does not specify a unique physical atmospheric structure, it does provide a temperature distribution that scales with the effective temperature and is consistent with radiative equilibrium. In addition, the opacity in a wide variety of stars is relatively independent of frequency over a large part of the spectrum, so that the gray atmosphere temperature distribution provides a good first approximation to the actual temperature distribution. However, the accuracy of this assumption does depend on the choice of reference optical depth being representative of the atmosphere as a whole, so we can only expect it to form an approximate first guess.

c. Solution of the Equation of Hydrostatic Equilibrium

The equation of hydrostatic equilibrium is a deceptively simple looking first-order differential equation. There are many sophisticated methods for obtaining the numerical solution to such an equation, but all such methods involve knowing at least one value (and usually several values) for the solution at and near the boundary. This poses both philosophical and practical problems. The boundary of a plane-parallel atmosphere, while well located in optical depth, is poorly placed in physical depth since it formally occurs where the density vanishes. In principle, this occurs at an infinite distance from the star where the plane-parallel approximation itself would no longer be valid. However, in practice, a small but finite optical depth is reasonably located with respect to the photosphere (i.e., near optical depth \(\text { τ.2/3 }\)) so that boundary conditions can be specified there without jeopardizing the accuracy of the solution. Since we have assumed a distribution of temperature with optical depth, there is no problem in determining the boundary temperature.

In Section 4.1a we discussed how to relate the mass fractions of hydrogen, helium, and "metals" to the mean molecular weight and thereby provide a connection between the number and mass density. The Saha equations for each element [equation \ref{11.1.16}] provide a relationship between the relative ionization fraction, the temperature, and the electron pressure. By remembering that the sum of all the various states of ionization for a particular element is simply equal to the number abundance for the element, it is possible to parameterize the opacity in terms of the gas pressure, temperature, and electron pressure. Thus, we may write the total pressure at any optical depth as \[P\left(\tau_i\right)=P_g\left[T\left(\tau_i\right), P_e\left(\tau_i\right)\right]+P_r\left[T\left(\tau_i\right)\right]\label{12.2.6}\]

In a similar manner it is possible to integrate the equation of hydrostatic equilibrium [as stated in equation \ref{11.5.3}] so that \[\begin{aligned}
P\left(\tau_i\right) & =\int_0^{\tau_i} g\left\{\kappa_0\left[T(\tau), P_e(\tau)\right]+\sigma_0\left[T(\tau), P_e(\tau)\right]\right\}^{-1} d \tau \\
& \equiv P\left[T\left(\tau_i\right), P_e\left(\tau_i\right)\right]
\end{aligned}\label{12.2.7}\]

We may look for a value of \(P_e\) that makes equations \ref{12.2.6} and \ref{12.2.7} self-consistent. Numerically this can be accomplished by creating tables of \(\kappa_0\) and \(\sigma_0\) as functions of \(P_e\) so that the integral in equation \ref{12.2.7} can be done directly by any efficient quadrature scheme and then a solution found by iteration with equation \ref{12.2.6}. Details of this procedure are given by Mihalas¹. In carrying out this procedure, one keeps the value of the optical depth \(\tau_\mathrm{i}\) sufficiently small that \(T(0).T(\tau_\mathrm{i})\). When one has found a self-consistent value of \(P_e(\tau_\mathrm{i})\) (and hence all the state variables), values of the state variables may be interpolated for all intermediate optical depths between 0 and \(\tau_\mathrm{i}\). This technique will provide all the required values of the pressure to initiate a general numerical integration procedure for the differential equation for hydrostatic equilibrium. Since all the other state variables are given in terms of algebraic expressions, the entire atmospheric structure may be obtained as a function of \(\tau_0\).

A note of caution should be interjected at this point concerning the numerical solution described by this procedure. The range in pressures to be expected from the solution is several powers of 10. For this reason, logarithmic variables are often used to improve the stability of the numerical solution. In any case, the method used to solve the equation of hydrostatic equilibrium should be reasonably sophisticated since the rapid initial growth of the pressure, if not carefully dealt with, can produce systematic errors that destroy the accuracy of the entire atmosphere as the integration proceeds. Several investigators have found it necessary to employ up to a seven-point predictor-corrector integration scheme to achieve the accuracy required. In addition, although the remaining equations are indeed algebraic, the Saha equations for the various elements and states of ionization represent a system of coupled nonlinear algebraic equations and must be solved by iteration. Furthermore, the equations for the opacity due to the different elements in their various states of ionization and excitation represent a significant amount of calculation. Thus, the calculation of \(\kappa_\nu(\tau_0)\) can be quite time-consuming and represents a significant time burden for the calculation of the model structure. This is particularly evident when one remembers that a multipoint numerical integration scheme requires multiple evaluations of the function \(\mathrm{g}/\kappa_\nu\) to carry out one step in the integration. The situation is further exacerbated by the realization that a rapidly varying numerical solution to a differential equation usually requires that the solution proceed with very small steps, and the range required for the independent variable will be of the order of 2 powers of 10. This is the reason that some modern atmosphere codes utilize a density depth as given in equation \ref{12.2.2} as the independent depth variable. With this choice, the calculation of the opacity is entirely avoided. However, as we see, this choice of an independent variable is not without its own set of problems.

The solution of the equation of hydrostatic equilibrium also provides us with the dependence on depth of all the state variables and the various states of ionization and excitation of the elements. With this information, it is possible to calculate the opacity and hence the radiation field at all points in the atmosphere.