Computations and Results

Using our two independent methods, we investigated the effects of Compton scattering on the emergent spectra of three pure H white dwarf models. These were: models appropriate for the well-known DA white dwarfs Sirius B (assuming $T_{\rm eff}=24\,700$ K, $\log~g=8.6$) and HZ 43 ( $T_{\rm eff}=50\,000$ K, $\log~g=8.0$), together with a significantly hotter model with $T_{\rm eff}=1\cdot 10^5$ K and $\log~g = 6$ and $8$. These models cover the effective temperature range relevant to pure H DA atmospheres and specifically address the question of whether Compton scattering might be relevant to the X-ray bright DA white dwarfs Sirius B and HZ 43 that are central to the low-energy calibration of Chandra (Pease et al., 2003).

The results of our calculations from both methods are presented in Figures 2-5. In Fig. 2a the spectra of the model atmosphere for the DA white dwarf HZ 43 ( $T_{\rm eff}=50\,000$ K, $\log~g=8.0$) computed using Method 1 with (solid line) and without (dashed line) Compton effects are shown. We also calculated a non-LTE model atmosphere for HZ 43 using the code PRO2 (Werner et al., 2003) and computed the radiation transfer equation with Compton scattering (3) using this non-LTE model atmosphere structure. The corresponding spectra are shown in Fig. 2a by the dotted line and by open circles.

Figure 2: Spectra (a) of the DA white dwarf HZ43 model atmosphere with (solid line) and without (dashed line) Compton scattering. Also shown are the spectra of the non-LTE model atmosphere (dotted line) and the spectra of non-LTE model atmosphere with Compton scattering (open circles). Run (b) of Comptonisation parameter $Y_{\rm Compt}$ with wavelength (see Eqn. 19).

Figure 3: Emergent X-ray spectra for a pure hydrogen model atmosphere appropriate for the parameters of HZ43 computed for classical Thomson scattering (dashed) and Compton scattering (solid) using Method 2 (the method of Madej et al. 2004).

Calculations using Method 2 in LTE are illustrated in Fig. 3. The differences between Compton and Thomson models in these calculations are very similar to the differences found in Method 1 in Fig. 2a, and are significant only at wavelengths $< 25$ Å. The flux in the models with Compton scattering is smaller than the flux in the model without Compton scattering by 0.5 % at 50 Å, 1.3 % at 40 Å, 7% at 30 Å, and 40% at 20 Å (Method 1). Similarly, model computations performed with Method 2 yield the following differences for HZ 43: 0.5 % at 50 Å, 1.4 % at 40 Å, 5% at 30 Å, and 56 % at 20 Å.

These results can be understood more clearly if we consider the Comptonisation parameter $Y_{\rm Compt}$:

$$Y_{\rm Compt} = \frac{h\nu}{m_{\rm e}c^2} \max((\tau_{\rm e}^*)^2,\tau_{\rm e}^*),$$ (18)

where $h\nu/m_{\rm e}c^2$ is the relative photon energy lost during one scattering event off a cool electron, $\max((\tau_{\rm e}^*)^2,\tau_{\rm e}^*)$ is the number of scattering events the photon undergoes before escaping, $\tau_{\rm e}^*$ is the Thomson optical depth, corresponding to the depth where escaping photons of a given frequency are created. The Comptonisation parameter is, then, a representation of the influence of Compton scattering on the emergent spectrum: significant Compton effects are expected if the Comptonisation parameter approaches unity (Rybicki & Lightman, 1979). In Fig.2b the dependence of $Y_{\rm Compt}$ on the wavelength is shown. It is clear that the Comptonisation parameter is very small down to 25 Å, and this is indeed reflected in the emergent spectrum.

Similar results were obtained for the DA white dwarf Sirius B (assuming $T_{\rm eff}=24\,700$ K, $\log~g=8.6$). The corresponding spectra and the Comptonisation parameter are shown in Fig. 4. The effective temperature of this star is smaller, and the surface gravity is larger, therefore, Compton scattering is even less significant than for HZ 43. We also calculated a non-LTE model atmosphere for Sirius B using the PRO2 code and computed the radiation transfer equation with Compton scattering (3) using this non-LTE model atmosphere structure, as for HZ43. The corresponding spectra are shown in Fig. 4a by the dotted line and by open circles. Compton scattering changes only very slightly the emergent spectrum at wavelengths below 20 Å.

Figure 4: The same as Fig. 2 but here for the DA white dwarf Sirius B LTE model atmosphere.

In the Fig. 5 we present the spectra of the hot DA white dwarfs model atmospheres ( $T_{\rm eff} = 100 000$ K, $\log~g = 6$ and $8$) with and without Compton scattering. It is obvious that Compton effects are more significant for these hot DA models (especially with low surface gravity) than for Sirius B and HZ 43, but visible effects still do not occur for wavelengths $>50$ Å.

Figure 5: Spectra (a) of the DA white dwarf $T_{\rm eff} = 100 000$ K model atmospheres with and without Compton scattering. Solid lines correspond to model with $\log~g=8.0$, and dashed lines correspond to model with $\log~g = 6.0$. Softer high energy tails correspond to models with Compton effect. Run (b) of Comptonisation parameter $Y_{\rm Compt}$ with wavelength.