Method 2

Our second approach adopts the equation of transfer for absorption and scattering presented by Sampson (1959) and Pomraning (1973, see Eqn. (2.167)). The equation of transfer can be expressed in the form

$${\partial ^2 f_\nu J_\nu \over {\partial \tau_\nu^2 }} = {k_\nu \over {k_\nu + \sigma_\nu} } \, (J_\nu - B_\nu) +$$

$$+ {k_\nu \over {k_\nu + \sigma_\nu} } \, J_\nu \, \int \limits _{0}... ...eft(1+ {c^2 \over {2h{\nu ^\prime }^3 }} J_{\nu ^\prime} \right)\, d\nu ^\prime$$

$$+ {k_\nu \over {k_\nu + \sigma_\nu} } \, \left( 1+ {c^2 \over {2h\nu ^3}} J_\nu \right)\, \times$$

$$\int \limits _{0}^{\infty} \Phi (\nu ,\nu ^\prime) J_{\nu ^\prime... ...3 \exp \left[ -{{h(\nu - {\nu ^\prime }) }\over {kT}} \right] \, d\nu ^\prime .$$ (14)

This transfer equation is written on the monochromatic optical depth scale $d\tau_\nu=-(k_\nu+\sigma_\nu) \, \rho \, dz$. The variable $I_\nu$ denotes the energy-dependent specific intensity, and $J_\nu$ is the mean intensity of radiation.

Transformation of the equation of transfer by Pomraning (1973) to the Eq. 14 and the definition of required angular approximations for Compton scattering in a stellar atmosphere was outlined by Madej (1994); Madej (1991) and Madej et al. (2004).

Our equations and theoretical models of Method 2 use detailed differential cross-sections for Compton scattering, $\sigma(\nu \to \nu^\prime, \vec{n} \cdot \vec{n^\prime })$, which were taken from Guilbert (1981). Cross-sections correspond to scattering in a gas of free electrons with relativistic thermal velocities, and they are also completely valid at low temperatures. Differential cross-sections were then integrated numerically to obtain large grids of Compton scattering opacity coefficients $\sigma_\nu$

$$\sigma_\nu = \oint_{\omega^{\prime}} \frac{d\omega^\prime}{4... ...\sigma(\nu \to \nu^\prime, \vec{n} \cdot \vec{n^\prime }) \, ,$$ (15)

and grids of angle-averaged Compton scattering redistribution functions $\Phi (\nu, \nu^\prime)$

$$\Phi(\nu,\nu^\prime) = {1 \over {\sigma_\nu}} \oint_{\omega... ...\sigma(\nu \to \nu^\prime, \vec{n} \cdot \vec{n^\prime }) \, .$$ (16)

The function $\Phi$ denotes the zeroth angular moment of the angle-dependent differential cross-section, normalized to unity.

The scattering frequency redistribution function was introduced by Pomraning (1973)--see Eqn. (7.95) of his book--and it represents the normalized probability density of scattering from a given frequency $\nu$ to the outgoing frequency $\nu^\prime$.

Note that Eq. 14 also includes stimulated scattering terms, $1+c^2/2h \nu^3 \, J_\nu$, which ensure the physically correct description of Compton scattering. The actual equations and calculations used here strongly differ from those in Madej (1998). The physics of Compton scattering used here is also fundamentally different to Method 1. Note, that the latter method and that of Madej (1998) employ the well-known Kompaneets diffusion approximation to Compton scattering kernels. The apparent difference between Methods 1 and 2 is that they use either the differential Kompaneets kernel (Method 1) or kernels given by integrals over the detailed Compton scattering profiles (Method 2).

The equation of radiative equilibrium (the energy balance equation) requires that

$$\int^\infty _0 H_\nu \, d\nu = \frac{\sigma_R T_{\rm eff}^4}{4\pi} \, .$$ (17)

The above condition is fulfilled in a hot stellar atmosphere, where energy transport by convective motions can be neglected.

Computing derivatives of both sides of Eqn. 17 and using the equation of transfer, Eqn. 14, one can obtain the alternative energy balance equation

$$\int\limits_{0}^{\infty} k_\nu \, (J_\nu\, - B_\nu) \, d\nu + \int\limits_{0}^{\infty} \sigma_\nu J_\nu d\nu \int\limits_{0}^{\... ...eft(1+ {c^2 \over {2h{\nu ^\prime }^3 }} J_{\nu^\prime} \right)\, d\nu^\prime\,$$

$$\hskip-30mm - \int\limits_{0}^{\infty} \sigma_\nu \, d\nu \int\li... ...xp \left[ -{{h(\nu - {\nu^\prime }) }\over {kT}} \right] \, d\nu^\prime = 0\, ,$$

which can be compared with its analog, the Eqn. 5 of Method 1.

The computer code ATM21 used for the model calculations was described in detail in Madej & Rózanska (2000) and Madej et al. (2004). The structure of the code is based on the partial linearization scheme by Mihalas (1978), in which corrections of temperature $\Delta T$ and the function $\Delta \Phi$ are built into the equation of transfer. The high numerical accuracy and very good convergence properties of the ATM21 code are vital for the present paper, and allowed us to compute accurate spectra for atmospheric parameters appropriate for the white dwarfs HZ43 and Sirius B using Method 2, outlined above. These calculations supersede less accurate X-ray spectra of HZ43 which were presented in the earlier paper, Madej (1998).

For the present research, the ATM21 code included numerous bound-free LTE opacities of neutral hydrogen and free-free opacity of ionized hydrogen, which always remains in LTE. No hydrogen lines were included in the actual computations. In each temperature iteration the code solves the equation of hydrostatic equilibrium to obtain stratifications of gas pressure $P_g$ and density $\rho$ in the model atmosphere. After that the ATM21 code solves the set of coupled equations of radiative transfer with implicit temperature corrections and finds the stratification of $\Delta T$ in the model atmosphere. The equation of radiative transfer was solved using the Feautrier method and the technique of variable Eddington factors (Mihalas, 1978). Boundary conditions along the $\tau_\nu$ axis were the same as in Method 1. Explicit expressions for the temperature corrections, $\Delta T$, can be found e.g. in Madej et al. (2004).