Model Atmosphere Spectrum Fit to the Soft X-Ray Outburst Spectrum of SS Cyg

The X-ray spectrum of SS Cyg in outburst has a very soft component that can be interpreted as the fast-rotating optically thick boundary layer on the white dwarf surface. This component was carefully investigated by Mauche (2004) using the Chandra LETG spectrum of this object in outburst. The spectrum shows broad (≈ 5 Å) spectral features that have been interpreted as a large number of absorption lines on a blackbody continuum with a temperature of ≈ 250 kK. Because the spectrum resembles the photospheric spectra of super-soft X-ray sources, we tried to fit it with high gravity hot LTE stellar model atmospheres with solar chemical composition, specially computed for this purpose. We obtained a reasonably good fit to the 60–125 Å spectrum with the following parameters: Teff = 190 kK, log g = 6.2, and NH = 8 · 10 cm−2, although at shorter wavelengths the observed spectrum has a much higher flux. The reasons for this are discussed. The hypothesis of a fast rotating boundary layer is supported by the derived low surface gravity.


Introduction
SS Cyg is one of the brightest cataclysmic variables (CVs), one of the best-studied dwarf nova stars (Warner 1995), and was the first CV discovered in X-ray radiation (Rappaport et al. 1974).The properties of the X-ray radiation of this close binary system have been extensively investigated, and are observed to be dramatically different in quiescence and in outburst.In quiescence, the X-ray spectrum is hard and can be described by an optically thin hot (kT ≈ 20 keV) plasma with an observed flux ≈ 2 • 10 −10 erg s −1 cm −2 .In outburst, this hard component decreases by a factor of ten, the plasma temperature is reduced to ∼ 6-8 keV, and an additional soft component appears with a blackbody temperature ≈ 200-300 kK (Córdova et al. 1980;Mc-Gowan et al. 2004;Ishida et al. 2009).It is commonly accepted that the X-ray radiation of non-magnetic CVs arises in the boundary layer (BL) between the white dwarf (WD) and the accretion disc (Pringle & Savonije 1979;Tylenda 1981;Patterson & Raymond 1985a, b;Kley 1991), which are optically thick at high accretion rates ( Ṁ > 10 16 g s −1 ) and optically thin at lower accretion rates.
The soft X-ray spectrum of SS Cyg in outburst was carefully investigated by Mauche (2004) using a high-resolution spectrum obtained with the Chandra LETG.He phenomenologically described the observed 40-130 Å spectrum by a blackbody with temperature T ≈ 250 kK and numerous broad absorption features of ions of cosmically abundant O, Ne, Mg, Si, S, and Fe.The BL luminosity and WD spin were also evaluated in this work.On the other hand, this spectrum looks like the photospheric spectra of super-soft X-ray sources (Lanz et al. 2005;Rauch et al. 2010; van Rossum 2012), so it probably could be described using the spectra of hot stellar model atmospheres.Boundary layers possibly rotate with almost Keplerian velocities and could have reduced (in comparison with WD) surface gravities close to the local Eddington limit.Therefore, we consider close to Eddington limit models in the present work.Here we present our attempt to fit the Chandra LETG spectrum of SS Cyg using such model spectra.We also make estimates of the BL parameters in the context of our model fits.

Model Atmospheres
The version of the LTE computer code ATLAS (Kurucz 1970;1993), modified by us to deal with high temperatures (Ibragimov et al. 2003;Suleimanov & Werner 2007), was used to model high temperature at-mospheres.In this code, local thermodynamic equilibrium (LTE) is assumed and the pressure ionization effects using the occupation probability formalism (Hummer & Mihalas 1988) as described by Hubeny et al. (1994) are taken into account.Coherent electron scattering together with the free-free and bound-free transitions of all ions of the 15 most abundant elements using cross-sections from Verner & Yakovlev (1995) were adopted for the continuum opacity.Line blanketing is also included using ∼ 25000 spectral lines from the CHI-ANTI, Version 3.0, atomic database (Dere et al. 1997).Twenty-two model atmospheres with solar chemical composition were computed using the described code.The effective temperatures of the models range between 150 kK and 250 kK with a step of 10 kK.Two values of the surface gravity for each effective temperature, namely log g = log g Edd + 0.2 and log g = log g Edd + 0.4, were used.Here log g Edd = log(σ e σ SB T 4 eff /c) = 4.88 + 4 log(T eff /10 5 K) is the surface gravity that has an equal radiation pressure force for a given T eff , and σ e ≈ 0.34 cm 2 g −1 is the electron scattering opacity for the assumed solar chemical composition.The positions of the computed models on the T eff -log g plane are shown in Fig. 1 (top panel).For the considered model atmospheres, a radiation pressure force g rad due to spectral lines becomes larger than the surface gravity at the upper atmospheric layers (see Fig. 1, bottom panel).To enforce hydrostatic equilibrium, we took a gas pressure equal to 10% of the total pressure (P gas = 0.1P tot ) at all atmospheric layers where g rad > g.
As shown in Fig. 2, the computed emergent spectra are dominated by a forest of absorption lines, and have to be convolved with the LETG spectral resolution and the interstellar gas transmission to be compared with the observed spectrum of SS Cyg (see Fig. 3).The spectra of models with different surface gravities are sufficiently different to discriminate them from a comparison with the observed spectrum (see Fig. 2, bottom panel).

Results
The model spectra convolved with the Chandra LETG spectral resolution ∆λ = 0.05 Å were used to fit the observed soft X-ray spectrum of SS Cyg.The interstellar absorption (with the hydrogen column number den- The spin period of the WD in SS Cyg is 12 (9) s as inferred using the relation between the BL and accretion disk luminosities (Kluźniak 1987; Kley1991) where Ω K (R WD ) is the Kepler angular velocity at the WD radius.
The accepted model gravity on the WD surface in SS Cyg (log g WD = 8.46) is more than two orders of magnitude higher than the obtained BL effective surface gravity log g eff = 6.2.The surface gravity of the BL can be reduced only by fast rotation of the accreting matter Therefore, a relative BL angular velocity Ω BL /Ω K (R WD ) ≈ 0.98 was obtained using this relation.

Discussion and Conclusion
We present here the results of fitting the SS Cyg Chandra LETG spectrum in outburst with model atmosphere spectra.The obtained best fit model spectrum does not describe the observed spectrum at short wavelengths (< 60 Å) and in the 82-90 Å wavelength region and, therefore, it is not statistically acceptable (χ 2 /dof = 3.9).These deficiencies can be connected with model shortcomings: the most important ignored effect is atmosphere expansion due to a spectral linedriven stellar wind.This expansion can be significant because g rad > g at the outer layers of our model atmospheres (see also van Rossum 2012).In addition, non-LTE effects could be important (see, e.g., Rauch et al. 2010), the chemical composition may differ from solar, and, finally, the atomic data are almost certainly neither complete nor entirely accurate.
The second important shortcoming is connected with a likely complicated BL structure with a distribution of effective temperatures and surface gravities over its surface.Therefore, a simple one-zone BL model presented here is most probably insufficient and more sophisticated BL models have to be considered.
Nevertheless, the spectral modeling presented here supports a BL in SS Cyg that is, to first approximation, a hot (≈ 190 kK), fast rotating [Ω BL ≈ 0.98 Ω K (R WD )], narrow (H BL ≈ 0.063 R WD ) belt on the WD surface.

Figure 1 :
Figure 1: Top panel: Positions of the computed model atmospheres in the T eff -log g plane.The dashed curve demarcates the Eddington limit (log g = log g Edd ).Bottom panel: The relative radiation force vs. depth for various model atmospheres.

Figure 2 :
Figure 2: Top Panel: Emergent spectra of three model atmospheres with the same log g = log g Edd + 0.2 and various effective temperatures: 150 kK (solid curves), 200 kK (dashed curves), and 250 kK (dotted curves).Bottom panel: Emergent spectra of two model atmospheres with the same effective temperature (200 kK) and different log g.
sity N H as a fitting parameter) was also taken into account.The observed spectrum was fitted in the 60-125 Å wavelength range because at the shorter wavelengths our model spectra could be incorrect (see next section).The best-fit model parameters are T eff = 190 kK, N H = 8•10 19 cm −2 , and normalization K = 7.82•10 −26 , and correspond to the models with the lower surface gravity log g = log g Edd + 0.2.The reduced χ 2 = 3.9 is relatively large, hence the formal parameter errors are large, too, and we have not attempted to determine them.The best-fit model spectrum together with the observed spectrum is shown in Fig.3.The contours of χ 2 on the T eff -log N H parameter plane are shown in Fig.4.The normalization can be expressed asK = f R 2 WD /d 2 where d is the distance to SS Cyg and f is the WD fractional area occupied by the BL, which can be expressed as the relative BL extension along the WD surface f ≈ (2πR WD 2H BL )/(4πR 2 WD ) = H BL /R WD .The basic properties of the BL can be derived from the obtained fit parameters.Using the same system and WD parameters as used by Mauche (2004) -M WD = 1 M , R WD = 5.5 • 10 8 cm (therefore, model log g = 8.46), d = 160 pc, and an accretion disk bolometric luminosity in outburst L Disk = 10 35 erg s −1 -we obtain the fractional area of the BL f = 6.3 • 10 −2 (5 • 10 −3 ), the bolometric BL luminosity L BL = 1.8 • 10 34 (5 • 10 33 ) erg s −1 , and the relative BL luminosity L BL /L Disk = 0.18 (0.05), where the best-fit parameter values obtained by Mauche (2004) are shown in parentheses for comparison.

Figure 3 :
Figure 3: The Chandra LETG spectrum of SS Cyg in outburst (thick black curve) and the best-fit model atmosphere spectrum with T eff = 190 kK, log g = 6.2, and log N H = 19.9(thin red curve).The fitting was performed in the 60-125 Å wavelength range.The model spectrum at shorter wavelengths is shown by the dashed red curve.

Figure 4 :
Figure 4: Position of the best-fit model in the T eff -log N H parameter plane and contours of χ 2 = [1.5, 3, 6] χ 2 min .