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Astron. Astrophys. 340, 617-625 (1998)

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2. Model atmosphere equations

2.1. The equation of radiative transfer

Equations of a model atmosphere with plane-parallel geometry can be written on the geometrical vertical depth z. Since both gas temperature and photon energies are small compared to the electron rest mass, [FORMULA] and [FORMULA], photon diffusion in frequency space due to repeated Compton scatterings can be written in the diffusion approximation. The corresponding equation of radiative transfer was given by Pomraning (1973, Eq. 8.620), and here it reduces to

[EQUATION]

which is valid for the isotropic distribution of scattered photons. [FORMULA] has the usual meaning of the mean intensity of monochromatic radiation. The absorption coefficient [FORMULA] (taken for 1 cm3) includes already stimulated emission. Scattering coefficient [FORMULA] is equal to the classical Thomson value, [FORMULA] cm-1, where [FORMULA] stands for the number of free electrons in unit volume. Stimulated scattering is represented by the last (quadratic) term in Eq. (2.1).

Eq. (2.1) clearly demonstrates, that the scattering emissivity is represented by a sum of coherent and noncoherent contributions (terms: [FORMULA] and that in square brackets, respectively). Noncoherent terms are proportional either to [FORMULA] or to [FORMULA], which are of the order [FORMULA] in atmospheres of hot white dwarfs and in their UV and X-ray spectra. Therefore they represent rather small perturbations in the equation of transfer.

The noncoherent scattering terms in Eq. (2.1) can also be obtained from the widely used Kompaneets diffusion equation (Rybicki & Lightman 1979), if we replace dimensionless variables in that equation by dimensional both frequencies and true/mean intensities of radiation. The equation of transfer, when written on the monochromatic optical depth scale [FORMULA] at some frequency [FORMULA], changes to

[EQUATION]

where [FORMULA]. At the effective temperatures corresponding to white dwarfs the Compton scattering term [FORMULA] is a small correction to the standard equation of transfer with coherent scattering, and it is nonlinear with respect to the radiation mean intensity [FORMULA]. In thermodynamical equilibrium ([FORMULA]), the noncoherent scattering term [FORMULA], which can be shown by direct computations.

Following the usual steps in the radiative transfer we can obtain the zeroth and first momenta of the equation of transfer

[EQUATION]

Variables [FORMULA], [FORMULA], and [FORMULA] denote mean intensity of radiation, monochromatic flux, and the second moment of true intensity [FORMULA], respectively. Both equations joined together form a single scalar equation of transfer of the second order

[EQUATION]

with the variable Eddington factors [FORMULA] closing the transfer problem, [FORMULA]. Eq. (2.5) can be solved by iterations (Mihalas 1978). All the following equations include factors [FORMULA] and therefore are exact. However, for numerical convenience, factors [FORMULA] were set to 0.333333 throughout this paper. Consequently, model computations presented in this paper are in fact computed in the Eddington approximation.

The equation of transfer (2.5) can be most conveniently written on the optical depth scale at some fixed (standard) frequency, which then becomes the unique independent variable. Such a standard optical depth will be denoted below by [FORMULA] with no subscript. The equation of transfer, Eq. (2.5), takes then the form

[EQUATION]

with the dimensionless variable [FORMULA].

2.2. Radiative equilibrium and linearization procedure

The equation of radiative equilibrium in a stellar atmosphere requires constancy of the bolometric flux with geometrical depth, [FORMULA]. Following Eq. (2.5) we obtain

[EQUATION]

An asterisk attached to the Planck function [FORMULA] and the Compton scattering term [FORMULA] denotes, that both of them are computed at the correct temperature distribution [FORMULA], which yet remains unknown.

We can expand both [FORMULA] and [FORMULA] in Taylor series at the actual [FORMULA], and neglect higher order terms

[EQUATION]

where [FORMULA]. Substituting to Eq. (2.7) one gets the linearized equation of radiative equilibrium

[EQUATION]

where

[EQUATION]

The correct equation of the radiative transfer is given by Eq. (2.7) with both thermal and Compton scattering terms including [FORMULA] and [FORMULA], in which unknown temperature corrections are already included in computing of the radiation field. Such a widely employed strategy with [FORMULA] term being ignored, ensured fast convergence of [FORMULA] to the required values at the radiative equilibrium in both LTE and NLTE model atmospheres (cf. Mihalas 1978). Eq. (2.6) changes then to

[EQUATION]

where

[EQUATION]

and

[EQUATION]

The right-hand side of the equation of transfer is a complex nonlinear expression with respect to the unknown variable [FORMULA].

2.3. Discretization and iterations

Computations of model stellar atmosphere are routinely performed with grids of discrete points, which replace integrals and differentials by quadrature sums and finite differences. The equation of transfer with continuum variables, Eq. (2.12), is then transformed to the following grids: standard optical depths [FORMULA], [FORMULA], and discrete frequencies [FORMULA], [FORMULA].

Representation of the differential operator in Eq. (2.12) is a standard problem and will not be discussed below. Discretization of the right-hand side of the equation of transfer yields the following algebraic equation at the depth [FORMULA] and frequency [FORMULA]

[EQUATION]

where

[EQUATION]

Values of [FORMULA] denote the quadrature weights for frequency integration.

Eqs. (2.15) and (2.16) describe the radiation field on the intermediate optical depths, i.e. at the levels [FORMULA]. Compton scattering terms [FORMULA] are adorned here with tilde sign, which indicates that they are computed on the basis of the radiation field found in the previous iteration (see also an explanation below). Both boundary conditions at [FORMULA] and [FORMULA] can also be expressed by relevant algebraic expressions (see Madej 1994b).

At any single frequency [FORMULA] and optical depth [FORMULA], the equation of transfer (2.15) involves the whole spectrum of radiation hidden in temperature corrections [FORMULA]. In case if strictly coherent Thomson scattering terms are included ([FORMULA]), there exist a number of numerical methods well suited to handle problems of LTE and NLTE model atmosphere computations. A very useful and stable technique for model atmosphere computations (with step-by-step eliminations of numerous frequency points) can be derived from Rybicki's "grand matrix" formulation (Rybicki 1971, Mihalas 1978).

The presence of nonzero Compton terms involves frequency derivatives of the radiation mean intensity [FORMULA], and this very seriously complicates the overall elimination scheme. A numerical technique for model atmosphere computations with Compton scattering, the elimination scheme, and the set of sample models has been presented by Madej (1994b). However, accuracy and convergence properties of that method still require additional improvements. Moreover, that algorithm is a very specific construction and cannot be used to upgrade existing codes written by other authors using a variety of techniques.

This paper proposes a simple iterative approach to solve the above model atmosphere equations. At the first iteration one starts with a roughly guessed temperature stratification [FORMULA], where the computer solves the LTE equation of state and finds monochromatic opacities at each frequency and depth point, [FORMULA] and [FORMULA], respectively. Then the radiation field is found by a repeated solution of the radiative transfer equation, Eq. (2.15), in which [FORMULA], [FORMULA], and [FORMULA] were set to zero. I.e. this represents a standard coherent scattering problem. After that iteration, tables of both [FORMULA] and [FORMULA] were determined, yielding the initial temperature corrections [FORMULA] in the model, according to Eq. (2.16). The [FORMULA]'s include already Compton scattering terms.

The second and all the remaining iterations solve the full radiative transfer equation, Eq. (2.15) with a nonzero [FORMULA]. However, terms [FORMULA] and [FORMULA] are taken from the previous iteration. Therefore all subsequent iterations solve the standard model atmosphere problem, in which the new radiation field is computed as in the Thomson scattering case. The only difference is that new Compton scattering terms are computed on the basis of the radiation field from the previous iteration, and are appended either to the inhomogeneous thermal term ([FORMULA]), or to the [FORMULA] term in the current iteration.

In other words, such a scheme makes computing of the Comptonized model generally very similar to the classical computations with coherent Thomson scattering. Noncoherent scattering terms are iterated by a lambda-iteration, whereas the remaining part of the problem was iterated with the efficient Rybicki method. Such applied lambda-iterations yield radiative equilibrium in a Comptonized atmosphere very promptly. Fast convergence of temperature corrections [FORMULA] to the values reflecting Compton heating in the uppermost layers of a model is not surprising, since that heating is restricted to layers of very small optical depth (cf. the following sections). The presence of noncoherent scattering terms practically does not increase the computing time of the new algorithm, as compared with the original coherent scattering code.

2.4. Extending of Thomson scattering codes

The above Compton scattering algorithm represents a quite simple overlay, which can be joined with any particular Thomson scattering model atmosphere code, which includes already [FORMULA] corrections in the equation of transfer (Mihalas 1978). Addition of Compton scattering terms essentially does not influence the structure of the host code. The only two necessary interface adjustements extend the expression for [FORMULA], Eq. (2.16), and extend also the expression for the scaling coefficient

[EQUATION]

appearing on the right hand side of the equation of transfer, Eq. (2.15). The coefficient [FORMULA] appears normally in other model atmosphere codes (both LTE and NLTE), in which Thomson scattering was assumed and then thermal terms only contributed to the temperature corrections included in the equation of radiative transfer (Mihalas 1978).

The overall accuracy of the final computer code strongly depends on a very accurate representation of the partial derivatives [FORMULA] and [FORMULA], and consequently [FORMULA] and [FORMULA]. In this research a second-order difference representation is used for [FORMULA], and [FORMULA] is the first derivative of the Lagrange interpolation polynomial of the third degree connecting groups of neighbouring [FORMULA] points.

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© European Southern Observatory (ESO) 1998

Online publication: November 9, 1998
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