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Astron. Astrophys. 363, 1055-1064 (2000)

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3. The equation of transfer

The key element and starting point in our research is a realistic expression for the differential Compton scattering cross section [FORMULA], describing scattering by electrons in relativistic thermal motion. In this research we adopted formulae from Guilbert (1981). The differential cross section can be decomposed according to

[EQUATION]

where variable [FORMULA] denotes the total Compton scattering cross section (in cm2). Variable [FORMULA] denotes the scattering redistribution function (scattering kernel), normalized to unity

[EQUATION]

In the following equations we shall use the function [FORMULA], which is defined as the zeroth angular moment of [FORMULA] (Pomraning 1973).

We introduce auxiliary functions [FORMULA] and [FORMULA], which are defined by

[EQUATION]

[EQUATION]

Both functions [FORMULA] and [FORMULA] contain stimulated scattering correction factors, and fulfil strictly the detailed balancing condition in thermodynamic equilibrium (cf. Paper I).

We assume the following time-independent equation of radiative transfer, suitable for planar geometry

[EQUATION]

where we included both thermal and scattering emission. We assume here nongrey true absorption [FORMULA], and noncoherent Compton scattering opacity, [FORMULA], both taken for 1 gram. Absorption coefficient [FORMULA] included already LTE stimulated emission. Dimensionless absorption [FORMULA].

Variable [FORMULA] denotes the mean intensity of external irradiation at the level [FORMULA], with [FORMULA]

[EQUATION]

and variable [FORMULA] denotes the flux

[EQUATION]

Following the standard approach we compute zeroth and first momenta of the equation of transfer

[EQUATION]

[EQUATION]

3.1. Radiative equilibrium

The constraint of radiative equilibrium implies on each depth level [FORMULA] that

[EQUATION]

where [FORMULA] (cgs units). Taking derivative [FORMULA] of the above equation, we obtain the alternative form of the equation of radiative equilibrium

[EQUATION]

where [FORMULA]. Functions [FORMULA], [FORMULA], and [FORMULA] are computed at the unknown temperature [FORMULA], in which they fulfil strictly the constraint of radiative equilibrium.

3.2. Hydrostatic equilibrium

The total scalar pressure [FORMULA], and the gradient of gas pressure in hydrostatic equilibrium is given by

[EQUATION]

see also Paper II for more details.

3.3. Temperature corrections

We assume linear Taylor expansions

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA], variables T and [FORMULA] denoting the actual and final values of temperature, respectively. We follow here the standard strategy of model stellar atmosphere computations, where changes of various physical variables are represented by linear perturbations, and nonlinear (quadratic) terms are neglected. This method was proven by many numerical experiments to be efficient, general, and stable (Mihalas 1978, p. 180).

Then, we immediately obtain linear temperature corrections

[EQUATION]

The sum [FORMULA] equals the total mean intensity of radiation on the level [FORMULA]. New functions, [FORMULA] and [FORMULA], are defined by

[EQUATION]

[EQUATION]

Variables [FORMULA], [FORMULA] or 2, denote partial derivatives of the redistribution functions with respect to temperature

[EQUATION]

and

[EQUATION]

where [FORMULA]. In thermodynamic equilibrium ([FORMULA], and [FORMULA]) we must have [FORMULA].

3.4. The equation of transfer and boundary conditions

Following the standard procedure we combine Eqs. 8 - 9 and get

[EQUATION]

Such an equation implicitly contains unknown temperature corrections [FORMULA]. Eddington factors [FORMULA]. The final form of the equation of transfer is

[EQUATION]

The above linearized equation of transfer was solved in the actual model stellar atmosphere computations.

a. At the upper boundary we apply the equation which directly results from the equation for momenta, Eq. 9

[EQUATION]

The surface factor [FORMULA], which is equal to [FORMULA] in the initial Eddington approximation. The above upper boundary condition is just a relation between the surface mean intensity and the flux. Factors [FORMULA] can be iterated to exact, frequency-dependent values simultaneously with variable Eddington factors [FORMULA] (Mihalas 1978).

Note, that the upper boundary condition, Eq. 23, is formally the same as for a nonilluminated atmosphere. This is because the external illumination was explicitly separated in the equation of transfer, Eq. 6.

b. At the lower boundary we assume the diffusion approximation

[EQUATION]

Useful form of that condition and its derivation is given in the Appendix.

3.5. Computational details

Full set of model atmosphere equations solved in this paper consists of discretized radiative transfer Eqs. (22), written on frequencies [FORMULA], supplemented by the equation of hydrostatic equilibrium (12), and the equation for new temperatures [FORMULA], with [FORMULA] given by Eq. (16). The LTE equation of state for ideal gas, and expressions for monochromatic opacities close the whole system.

Actual model computations were performed with the two numerical assumptions:

- For simplicity, electron scattering cross section per one particle has been set to Thomson value, [FORMULA] cm2. However, the normalized redistribution functions [FORMULA] were always precisely reconstructed for each discrete frequency and temperature.

- To avoid convergence problems, two Compton scattering terms in the equation of transfer,

[EQUATION]

were always computed with the mean intensity [FORMULA] on each depth level taken from the initial Thomson scattering model atmosphere, as described in Paper II. This approximation was absolutely necessary to avoid divergence of temperature corrections. However, such a choice certainly introduces some systematic errors.

During a given iteration, tables of both integrals were carefully recomputed for each optical depth [FORMULA] and running frequency [FORMULA], with new temperatures determined at the end of previous iteration. Therefore Compton scattering contribution to the equation of transfer was a scalar inhomogeneity changing from one iteration to other, which was just added to the thermal term (Planck function). The same applies to scalars L and [FORMULA], and all derivatives of Compton scattering functions in the temperature correction term.

Method of solution and elimination scheme were adopted from Madej (1998), and Paper II, with the modifications described above. Convergence of our code was very rapid: in large part the redistribution of external hard X-ray flux to soft X-rays or EUV was computed in the first iteration. Relative temperature correction between neighbouring iterations were very large, and we had to artificially restrict them to max. 3-4, in order to avoid numerical instabilities.

Computer code ATM21 represents implementation and extension of the partial linearization approach, described by Mihalas (1978, pp. 179-180). According to this, we did not linearize neither the variable [FORMULA], nor dimensionless absorption [FORMULA].

We have neglected the effects of angular distribution of radiation in Compton scattering integrals just for simplicity. Also we have made other simplification by averaging of nonisotropic Compton scattering differential cross section [FORMULA] over cosine of scattering angle, [FORMULA]. However, it is necessary to point out here that Compton scattering of X-rays can be highly anisotropic, and then the significance of angular effects gets important. Hubeny (2000) expresses opinion, that for photon frequencies [FORMULA] Hz (energies less than 40 keV) the angular effects may easily be more important than errors introduced by the Kompaneets equation approximation (which was rejected in this research).

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© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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