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A&A
Volume 688, August 2024
Article Number A39
Number of page(s) 12
Section Stellar atmospheres
DOI https://doi.org/10.1051/0004-6361/202449370
Published online 31 July 2024

© The Authors 2024

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1 Introduction

Supersoft sources (SSSs) are close binary systems that involve a white dwarf (WD) undergoing accretion as the primary component. The accretion rate is high enough for (quasi-)stable thermonuclear burning to occur on the WD surface (van den Heuvel et al. 1992). These sources were initially discovered in the Large Magellanic Cloud (LMC) by the Einstein observatory in the early 1980s (Long et al. 1981) and were classified as a distinct class of sources following ROSAT observations (Trümper et al. 1991; Greiner et al. 1991). The X-ray spectra of SSSs observed by ROSAT were very soft, with blackbody temperatures kT ~ 30–40 eV. Formally, these temperatures and the observed fluxes led to luminosities exceeding the Eddington limit for a solar mass object at the LMC distance. This issue was resolved through the implementation of hot WD model atmosphere spectra for interpreting the X-ray spectra of SSSs (Heise et al. 1994). The results of early investigations of SSSs in this context were presented and discussed by Kahabka & van den Heuvel (1997).

In addition to the classical SSSs, super-soft X-ray phases occur during nova explosions. The hot WD with ongoing thermonuclear burning on the surface becomes visible in soft X-rays after the dispersal of an optically thick envelope (see, e.g., Kahabka et al. 1999; Orio et al. 2001; Ness et al. 2003; Schwarz et al. 2011, and references therein).

Model atmospheres and their theoretical spectra are important ingredients in investigating the nature of SSSs and determining their basic parameters. The first models were computed (Heise et al. 1994) under the assumption of local thermodynamic equilibrium (LTE), but non-LTE model atmospheres of hot WDs were subsequently developed as well (Hartmann & Heise 1997). The first non-LTE models did not include spectral lines, but later a number of spectral lines were incorporated (Hartmann et al. 1999), using the publicly available code TLUSTY (Hubeny 1988). For the interpretation of the X-ray spectrum of the bright classical SSS CAL 83, Lanz et al. (2005) used more accurate non-LTE model atmospheres. Such models were further developed to analyse the super-soft phase of nova V4743 Sgr (Rauch et al. 2010), where various chemical compositions of the atmospheres as well as the metal-line blanketing effects were considered. These models were calculated using the Tübingen non-LTE Model Atmosphere Package (TMAP, Werner et al. 2003; Rauch & Werner 2010). Unfortunately, the accurate computation of non-LTE models is very computation-time expensive, so only small regions of physical parameter space can be probed and models are typically tailored to individual objects.

In addition to non-LTE effects in hydrostatic model atmospheres, expansion due to radiation pressure force in spectral lines could also be important for high temperatures reached in the SSSs. Expanding model atmospheres were thus considered and computed by van Rossum & Ness (2010) and van Rossum (2012) using the publicly available code PHOENIX (see, e.g., Hauschildt et al. 1997). Further observational evidence of SSS atmosphere expansion was presented by Ness (2010). Again, modelling these effects, especially in a non-LTE approximation, is challenging, precluding a full exploration of the parameter space.

On the other hand, LTE model atmospheres of hot WDs can be computed much faster, and, despite their simplicity, were successfully used to interpret ROSAT spectra of SSSs (Ibragimov et al. 2003; Suleimanov & Ibragimov 2003), and a few SSSs found in M81 (Swartz et al. 2002). However, the high-resolution grating spectra obtained by Chandra and XMM-Newton observatories demonstrated that the hot WD model spectra cannot completely describe observations and reproduce only the common spectral shape with separate strong absorption lines in some cases. This holds true for both hydrostatic and expanding non-LTE model atmospheres (Ness 2020), suggesting that there may be some missing physical processes and additional X-ray emission sources, such as inner accretion disc or boundary layers. Nevertheless, basic SSS parameters appear to be well recovered. Moreover, LTE atmospheres have some advantages compared to sophisticated non-LTE expanding model atmospheres. Hydrostatic LTE model atmospheres are simpler to compute and, what is more important, allow us to consider an almost unlimited number of chemical elements, ions, and spectral lines. Therefore, extended sets of LTE model atmospheres computed for various chemical compositions could potentially be useful for the approximate estimation of basic physical parameters of hot WDs in SSSs and the chemical composition of their atmospheres. The latter problem is especially important, for instance, for the super-soft phases of nova explosions.

Here, we present a set of hot WD model spectra computed for three chemical compositions of the atmospheres: the solar abundance, typical for the Galaxy; the solar H/He mix with the heavy element abundances reduced by a factor of 2 (typical for the LMC, see, e.g., Rolleston et al. 2002); and by a factor of 10 (typical for the Small Magellanic Cloud (SMC), see, e.g., Carrera et al. 2008) in comparison with the solar abundance. We employed the approach previously used for computing the boundary layer spectrum in the dwarf nova SS Cyg (Suleimanov et al. 2014a). A preliminary version of this set of models has also already been used to fit a nova spectrum in fireball phase (König et al. 2022).

The remainder of this paper is organised as follows. In Sec. 2, we describe the method of computation of the atmospheric models, in particular, our new approach to calculate photoionisation opacities. In Sec. 3, we first present the characteristics of the resulting new set of model atmosphere spectra. Then we apply the obtained set to analyse X-ray spectra of the classical SSSs CAL 83 and RX J0513.9–6951. The results of this analysis are discussed in the context of WD evolution in Sec. 4. We summarize our results in Sec. 5.

2 Method

2.1 Basic assumptions

In this work, we use a standard method for computing hydrostatic plane-parallel model atmospheres (see, e.g., Mihalas 1978) and the code based on the popular Kurucz’s code ATLAS (Kurucz 1970, 1993b), modified for high temperatures by Ibragimov et al. (2003); see also Suleimanov et al. (2013, 2014a). The general temperature correction scheme for model computation is the same as in ATLAS and the main changes concern the calculation of the opacities and number densities.

We took into account the 15 most abundant chemical elements and the number densities of all ionization and excitation states of all the ions were computed, using the Boltzmann and Saha equations, assuming LTE. We considered pressure ionization and level dissolution effects using the occupation probability formalism (Hummer & Mihalas 1988), as described by Hubeny et al. (1994). We used the spectral line list together with necessary physical parameters such as gf values and the energies of the low-energy levels presented in the CHIANTI database (Dere et al. 1997; Del Zanna et al. 2021). The shapes of line absorption opacity are considered as Voigt profiles. Classical damping broadening was used together with the Tübingen approximation for Stark broadening (Cowley 1971; Werner et al. 2003). The lines of hydrogen-like ions are considered using Griem’s theory of linear Stark broadening (Griem 1960, 1967). The modified Kurucz’s subroutine (Kurucz 1970) was used for this purpose. We note, however, that the Stark broadening for the lines of the highly charged hydrogen-like ions is small (~Z−5) in comparison with the radiation damping. A low microturbulent velocity value of 2 km s−1 was added to the thermal velocity of ions to account for Doppler line broadening1.

Formally, the radiation pressure force in spectral lines, grad, exceeds the gravity, g, at the upper atmosphere layers and wind model atmospheres have to be used. However, we employed a simple trick, suggested by Ibragimov et al. (2003), to keep the atmosphere in hydrostatic equilibrium. It was assumed that the gas pressure equals 10% of the total pressure, Pg = 0.1 Ptot = 0.1 gm, at all depths where grad > g. Here, m is a column density (dm = −ρdz), the independent depth variable in our model, ρ is the plasma density, and z is the geometrical depth. This assumption, in fact, corresponds to a specific velocity law in the upper atmosphere layers, υ(z) ~ ρ(z)−1. This approach is the simplest way to avoid the hydrostatic equilibrium violation. All other reasonable methods significantly complicate the atmosphere modeling. For instance, this would require consideration of spherical moving atmospheres, including radiation transfer. That is why we limited ourselves to the proposed method.

To calculate the bound-free opacities from the atomic ground states of all ions, we utilized the procedure presented by Verner et al. (1996). The most significant changes for our present model calculations are associated with the photoionization opacities from the excited energy levels of heavy element ions. The method for estimating these opacities and the obtained results are presented in the next subsection.

The free-free opacities of all ions are calculated under the assumption that the ion’s electric field is a Coulomb field of charge, Ze, corresponding to the ionic charge number, Z, with e being the elementary charge. The corresponding Gaunt factors are computed following Sutherland (1998).

2.2 Photoionization from excited energy levels

Two different approaches for the photoionization and free-free opacities from the excited levels can be used to consider excited levels of hydrogen-like and helium-like ions and of other ions separately. The approach suggested by Karzas & Latter (1961) and the corresponding Kurucz subroutine (Kurucz 1993a) was used for these ions. The same approach was applied to carbon model atmospheres of neutron stars (Suleimanov et al. 2014b). Photoionization cross-sections from excited levels were computed assuming a Coulomb field of a point-like charge. This approach provides excellent results for hydrogen-like ions, as shown in Fig. 1. However, it is not strictly correct for heliumlike ions, as the inner (non-excited) electron is included in the point-like effective charge. This leads to some deviations from the exact cross-sections (see Fig. 2). Nevertheless, the obtained accuracy is acceptable for our purposes. We included five excited levels for hydrogen-like ions and ten levels for helium-like ions.

Another approach is used for ions with more than two electrons. The absorption opacities from the excited levels of these ions were computed by the Opacity Project (OP, Seaton et al. 1994) and are available as numerical tables in the TOPBase database2. We used these tables for creating simplified analytical fits and implemented these fits in our code. Only a few levels with the lowest excitation energies were considered, typically not exceeding a quarter of the ionization energy. Although it should be noted that the choice of the levels under consideration was quite arbitrary.

The cross-section for each considered level was fitted using the expression suggested by Kurucz (1970): σ(E)=σth (AE¯p/2+(1−A)E¯1+p/2),$\[\sigma(E)=\sigma_{\text {th }}\left(A \bar{E}^{p / 2}+(1-A) \bar{E}^{1+p / 2}\right),\]$(1)

where E¯=Eth/E,E$\[\bar{E}=E_{\mathrm{th}} / E, E\]$ is the photon energy, Eth is the threshold photoionization energy from the given level, and σth is the photoionization cross-section at the photoionization threshold. The cross-section is a fitting parameter as well as the parameters A and p. The values Eth and the obtained fitting parameters σth, A and p together with the level statistical weight g for the approximation (1) for lower excited levels of C, N, O, Ne, Na, Mg, Al, Si, S, Ar, Ca, and Fe ions are publicly available3. Some examples of the fitting are shown in Figs. 3 and 4. We note that our smooth fits ignore the auto-ionization resonances (see Fig. 3) and OP cross-sections significantly differ from the old ones used in ATLAS9. Simpler ions with one electron in the outer shell have smooth cross-section dependencies, approximated with a high level of accuracy (Fig. 4).

The excited levels of the helium-like iron ion have more complicated cross-section dependencies on the photon energy, consisting of two parts. The low-energy one just above the photoionization threshold is well fitted with the approach used for other helium-like ions. However, at some energy, approximately ten times larger than the photoionization threshold energy, the cross-section sharply increases. We considered this second part as an additional photoionization edge and approximated it using Eq. (1). The comparison of these double fitting with the computed cross-section for one of the excited levels of Fe XXV is shown in Fig. 5.

thumbnail Fig. 1

Comparison of the photoionization cross-sections of the lowest excited levels of the hydrogen-like ion Ne X with the orbital quantum numbers L′ = 0 and 1, computed by OP (thin black curves) and by the used approximation (dashed magenta curves). The g-factor weighted averaged over both sub-level cross-sections is shown by the red curve.
thumbnail Fig. 2

Comparison of the photoionization cross-sections from the first excited level of the helium-like ion C V (top panel) and 11th excited level of helium-like ion Ne IX (bottom panel), computed by OP (solid black curves) and by the used approximation (red curves). The excitation energies of the levels are also shown.

3 Results

3.1 Model atmospheres

We computed three sets of two-parameter plane-parallel hot WD model atmospheres with different chemical compositions. All the sets have the same solar hydrogen-helium mix and different abundances of heavy elements: the solar one (A = 1), half the solar abundance (A = 0.5), which corresponds to the LMC heavy element abundance, and ten times less than the solar one (A = 0.1), which is close to the heavy element abundance in the SMC. The model parameters of each set are the effective temperature, Teff (100 to 1000 kK in steps of 25 kK), and Δ log g = log g − log gEdd, which characterises the distance of the model from the Eddington limit: log⁡gEdd=log⁡(σeσSBTeff4c−1)=4.818+4 log ⁡(Teff/105 K),$\[\log g_{\mathrm{Edd}}=\log \left(\sigma_{\mathrm{e}} \sigma_{\mathrm{SB}} T_{\mathrm{eff}}^4 c^{-1}\right)=4.818+4 \log \left(T_{\mathrm{eff}} / 10^5 \mathrm{~K}\right),\]$(2)

where σSB is the Stefan-Boltzmann constant, c is the speed of light, and σe = 0.2(1 + X) ≈ 0.34 cm2 g−1. Here, X ≈ 0.7374 is the hydrogen mass fraction. The parameter Δ log g has eight values in the grid: 0.1, 0.2, 0.4, 0.6, 1.0, 1.4, 1.8, and2.2. Altogether, 296 spectra were computed in every set and implemented as table model4 for XSPEC5 (Arnaud 1996; Arnaud et al. 1999).

The properties of the model atmosphere spectra are illustrated in Figs. 6–10. A comparison of the model spectrum computed with the LTE code described above with the hottest non-LTE model atmosphere spectrum presented by Rauch (2003) is shown in Fig. 6 (top panel). Both spectra were binned with a 0.1 Å wide step. The general shapes of the spectra are similar, and some differences appear at small wavelengths where the emergent flux is insignificant. It could be due to both non-LTE effects and the difference in the number of excited levels taken into consideration. The lists of the used spectral lines are also different, leading to diverse contributions of the lines to the spectra.

The model spectrum of the fiducial model atmosphere (Teff = 500 kK, Δ log g = 0.6, and A = 0.5) is shown in Fig. 7. The upper panel presents, in the observed soft X-ray wavelength range, the spectrum, the corresponding Planck function, and the spectrum convolved with a Gaussian kernel (the kernel resolution R = 300 corresponds the Chandra grating resolution). The convolved spectrum demonstrates emission line-like structures, which are, in fact, parts of the spectrum with a small number of spectral lines. A detail of the spectrum near the C V ground state photoionization edge is shown in the bottom panel. The most prominent absorption lines are identified. It is clearly seen that along with the Lyman-like line series of the hydrogen-like C V ion, there are numerous absorption lines of ions of heavier elements with about 10 electrons in their shells. We note that the absorption lines of these ions dominate in the overall spectrum.

There is an important note about our assumption that Pg = 0.1g m at the upper atmospheric layers where the radiation force due to numerous spectral lines is greater than the WD gravity (grad > g). In fact, most of the escaping radiation aside from the strong line cores, forms in the hydrostatic layers, at least for models with Δ log g ≥ 0.4, see Fig. 8. The depths where the escaping radiation forms, namely m(τλ = 1), are shown there. Here, τλ is the optical depth at a wavelength λ. The boundaries between the hydrostatic and the wind-dominated layers (grad = g) are also shown. In the model with Δ log g = 0.2, a significant part of the lines forms in the wind layers and it is even more prominent in the models with less Δ log g. Therefore, we conclude that the used assumption is acceptable for models with Δ log g ≥ 0.4. The models with smaller Δ log g should be used with caution.

The dependence of the emergent spectrum on the effective temperature with fixed Δ log g = 0.6 and A = 0.5 is shown in Fig. 9 (here and in the next figure, the convolved spectra are shown). As expected, with increasing Teff the spectrum becomes harder. Decreasing the surface gravity also makes the emergent spectrum harder if other parameters are not changed (Fig. 10, top panel). The influence of the chemical composition is not significant, although the emergent spectrum becomes slightly harder at lower A (Fig. 10, bottom panel).

thumbnail Fig. 3

Comparison of the photoionization cross-sections from two excited levels of C II. Cross-sections computed by OP (solid black curves), the approximation used in ATLAS9 (dashed blue curves), and our fits (red curves). Vertical dashed lines correspond to the ionization thresholds.
thumbnail Fig. 4

Same as in Fig. 3 for the first excited level of C IV. Black and red curves overlap and are close also to ATLAS9 approximation (blue dashed curve) here.
thumbnail Fig. 5

Comparison of the photoionization cross-sections from the 11th excited level of the helium-like ion Fe XXV. Shown are cross-sections computed by OP (solid black curve) and the used approximation by two fitting functions (red curves). More details in the text. The fitting function thresholds are indicated by the vertical dashed magenta lines.
thumbnail Fig. 6

Comparison of the emergent spectra of the model atmospheres computed by the code presented here (red curves), and the TMAP non-LTE code (Rauch 2003, blue curves). All models have solar chemical composition, Teff = 190 kK, log g = 8.0.
thumbnail Fig. 7

Spectrum of the model atmosphere with Teff = 500 kK, Δ log g = 0.6 and A = 0.5 (top). The Planck function with the same temperature is also shown by the red curve. The spectrum convolved with a Gaussian kernel with the Chandra resolution, R = 300, is shown in magenta. Detail of the upper spectrum near the C V ground state edge with spectral line identifications (bottom).
thumbnail Fig. 8

Comparison of the emergent spectra formation depth (m(τλ = 1)) with the wind dominating boundary (dashed red lines) for the model atmospheres with Teff = 500 kK, A = 0.5, and three different Δ log g = 0.2 (top), 0.4 (middle) and 0.6 (bottom panel). Atmospheric layers below the dashed red lines are dominated by the wind.
thumbnail Fig. 9

Model spectra with various Teff values, fixed Δ log g = 0.6, and A = 0.5.

3.2 Application of the model to supersoft sources

The resulting model atmosphere grids cover large surface gravity and temperature ranges and can be used to estimate physical parameters of hot WDs, including supersoft X-ray sources. Before doing so it is important, however, to compare results obtained with our model grids with results that employed state-of-the-art models for some well-studied objects and to verify that our simplifying assumptions (i.e., LTE and hydrostatic approximation) are justified and that the deduced WD parameters are reasonable. To do that, we converted our model grids to XSPEC table model format and applied them to fit the spectra of two classical SSSs, namely, CAL 83 and RX J0513.9–6951. The spectra were obtained by the Chandra X-ray observatory using the High Resolution Camera (HRC-S) and the Low Energy Transmission Grating (LETG). They are publicly available in the Chandra Grating-Data Archive and Catalog6 (TGCat, Huenemoerder et al. 2011); see also Table 1 for the observation log. To increase the signal-to-noise ratio (S/N) the positive and negative first-order spectra were co-added using the combine_grating_spectra task of the Chandra Interactive Analysis of Observations (CIAO, Fruscione et al. 2006; CIAO Development Team 2013) package. We also rebinned the spectra to at least 30 counts per bin in the considered 19–60 Å wavelength range.

For the source CAL 83, we also used the spectrum obtained by the XMM-Newton Reflection Grating Spectrometer (RGS, den Herder et al. 2001) instrument. It is available in the XMM-Newton Science Archive7 (only the first-order spectrum was used). Table 1 shows the observation log.

For both sources, due to degeneracy in the WD mass, M, the standard gradient descent method of searching the statistics-minimum and the confidence interval does not allow us to reliably estimate parameter errors. Therefore, the Bayesian Markov chain Monte Carlo (MCMC) approach was used. In particular, we relied on the Bayesian X-ray Analysis (BXA, Buchner et al. 2014; Buchner 2016a) package, which allows for the use of the the nested sampling package UltraNest8 (Buchner et al. 2014; Buchner 2019, 2021, 2016b) within XSPEC and allows us to derive the Bayesian evidence and posterior probability distributions. For visualisation of the posterior distributions the corner9 package (Foreman-Mackey 2016, 2017) was used. The results of the modeling for both objects are discussed in the following.

It should be noted that for low-count spectra the usage of C-statistics (Cash 1979) instead of χ2 is recommended; therefore, we used its Xspec implementation (cstat) as likelihood function to determine the best-fit parameters. The parameter uncertainties are derived from the 0.16th and 0.84th quantiles of the posterior distribution. As there is no convenient criterion of estimating the goodness-of-fit (like χν2$\[\chi_\nu^2\]$) for cstat, we will give below the statistics value and the number of degrees of freedom.

thumbnail Fig. 10

Comparison of the fiducial model spectra (Teff = 500 kK, Δ log g = 0.6, A = 0.5) with the spectra computed for different Δ log g (top) and A (bottom panel).

3.2.1 CAL 83

The prototypical SSS CAL 83 was discovered by the Einstein observatory in the LMC (Long et al. 1981) and has since been observed by many X-ray observatories, including ROSAT (Greiner et al. 1991), BeppoSAX (Parmar et al. 1998), XMM-Newton (Paerels et al. 2001), Chandra (Lanz et al. 2005), and NICER (Orio et al. 2022). The X-ray flux of the source is not stable and it sometimes switches to the off-state (Kahabka 1997); see also Greiner & Di Stefano (2002) and the references therein. The X-ray pulsations with a period close to 67 s were also discovered by Odendaal et al. (2014) and this period is probably connected with the spin period of the WD. Recently, the available XMM-Newton X-ray spectra of the source were analysed by Stecchini et al. (2023).

The optical counterpart of the source is also known as a blue variable star with V≈17 mag (Cowley et al. 1984) and its optical spectrum contains prominent Balmer and He II emission lines (Crampton et al. 1987). A short review of the optical and ultraviolet (UV) observations of CAL 83 can be found in Skopal (2022). In particular, Gänsicke et al. (1998) determined the interstellar neutral hydrogen column density NH = 6.5 × 1020 cm−2.

Model atmospheres of hot WDs were used to fit the soft X-ray spectra of CAL 83 observed by various X-ray observatories. Spectra of LTE model atmospheres computed by Heise et al. (1994) at a fixed log g = 9 gave the effective temperature of the hot WD as 510.5−7+73$\[510.5_{-7}^{+73}\]$ kK using BeppoSAX observations (Parmar et al. 1998). To reproduce the ROSAT observations, the LTE models were computed by Ibragimov et al. (2003) for several log g values. However, the poor energy resolution of ROSAT observations did not provide the possibility to limit the surface gravity. The obtained Teff varies from 504 ± 17 kK at log g = 8.0 to 620 ± 25 kK at log g = 9.5, assuming the interstellar column density is fixed at Nh = 6.33 × 1020 cm−2.

Non-LTE model atmospheres, computed using the publicly available code TLUSTY (Hubeny 1988) were used to fit the grating spectra obtained by XMM-Newton (Paerels et al. 2001) and Chandra (Lanz et al. 2005). In the first case, the obtained parameters were very approximate, Teff ~ 520 kK and log g ~ 8.5. However, in the second paper, the model atmospheres were computed specifically for CAL 83 case and more reliable results were obtained (see Table 2).

Using our grid with the chemical composition typical for the LMC (A = 0.5), we fitted simultaneously Chandra and XMM spectra of CAL 83. A uniform prior distribution was set for the hydrogen column density, NH (in the range (1–10) × 1020 cm−2), effective temperature, Teff (in the range 100–1000 kK), white dwarf mass, M (in range 0.3–1.4 M⊙), and radius, R (in range (2–20) × 108 cm) was used. We note that we set the strict upper limit for the WD mass. Another theoretical limitation is based on the fact that the WD radius must be greater than the cold WD radius at such a mass (see, e.g., Nauenberg 1972). Therefore, we further used the M − R relation for cold WDs as an additional lower limit for a radius at the given WD mass.

The obtained posterior distribution of fit parameters is presented in Fig. 11 and Table 2. The determined absorption column density, NH ~ 5.13 × 1020 cm−2, is slightly lower than the fiducial value (6.5 ± 1) × 1020 cm−2 found previously by Gänsicke et al. (1998), who analyzed the UV spectra of CAL 83 and RX J0513.9–6951. Therefore, we performed the additional modelling assuming fixed NH = 6.5 × 1020 cm−2 and 5.5 × 1020 cm−2 (average column density in the direction of the LMC). These results are also presented in Table 2. It is clearly seen from the statistics value that the quality of these fits are worse, with the WD mass approaching the hard upper limit.

The comparison of the observed and model Chandra spectrum, corresponding to a model with Nh as a free parameter, is shown in Fig. 12, left panel. Despite using the LTE model atmospheres, the parameters found are very close to those found by Lanz et al. (2005; see Table 2). Our LTE atmosphere models thus yield the same SSS parameters with a similar accuracy as the non-LTE model spectra. However, our grid covers a much larger parameter space, which also means it may be applied to other sources.

The spectrum of CAL 83 observed by BeppoSAX was fitted with a blackbody model by Parmar et al. (1998) and the quality of the fit was even better compared to the non-LTE model atmospheres fits. Here, we also fitted the XMM and Chandra spectra with the blackbody model. Fitting with the free NH yields larger values of NH (≈13.3 × 1020 cm−2) and a highly non-physically large WD radius (≈13.4 × 109 cm), but a lower Teff (≈358 kK). In this case the derived bolometric luminosity exceeds the Eddington luminosity for a solar mass object, ≈2 × 1039 erg s−1. However, if we fix the interstellar absorption at a value found by the model atmosphere fitting, the best blackbody fit yields results that do not differ significantly from those of the best-fit model atmosphere (see Table 2). We note that formal fit quality is actually slightly better for the blackbody model, however, we emphasize that both models are statistically acceptable; moreover, the model atmosphere fits yield satisfactory results without the need to fix the hydrogen column density, unlike the blackbody fitting.

We also performed a fitting of the joint Chandra and XMM-Newton grating spectra using models with the solar (A = 1) chemical composition and models with heavy elements reduced by a factor of ten (A = 0.1). The results are presented in Table 3. The main conclusion is that there are remarkable differences between the best fits for the models with different chemical composition. The models with ten times reduced heavy element abundances (A = 0.1) produce the fit with a slightly decreased effective temperature, slightly increased photospheric radius and increased interstellar absorption. It was expected because the model spectra with A = 0.1 are slightly harder than those with A = 0.5 (see Fig. 10, bottom panel). Therefore, a model with a reduced temperature is needed to reproduce the observed spectrum. On the other hand, fitting using the models with solar heavy element abundances (A = 1) demonstrates more significant differences compared to the model spectra with A = 0.5 (see Table 3). Formally, the fit quality is better with NH as a free parameter. However, the obtained NH is too small, the temperature is too high and the radius is too small as well. In fact, the radius approached the low limit boundary for cold WD radius. Therefore, we fixed the interstellar absorption at the value obtained for A = 0.5. The resulting fit is much closer to the fit obtained for A = 0.5 (see Table 3) and it demonstrates a slightly increased temperature and a slightly decreased photo-spheric radius. All the fits (A = 0.5, free NH; A = 0.1, free NH; and A = 1, fixed NH) are not distinguishable from the statistical point of view.

Table 1

Chandra and XMM-Newton observations of selected sources.

Table 2

Parameters of the investigated SSSs.

thumbnail Fig. 11

Corner plot of the posterior distribution for parameters of CAL 83. The 2D contours correspond to 39.3%, 86.5%, and 98.9% confidence levels. The histograms show the normalized 1D distributions for a given parameter derived from the posterior samples. The best-fit parameter values are presented in Table 2. A Gaussian prior for the hydrogen column density NH was used, with μ = 6.5 × 1020 cm−2 and σ = 1.0 × 1020 cm−2.
thumbnail Fig. 12

Best fits of CAL 83 (left) and RX J0513.9–6951 (right) spectra (top). The observed spectra (Chandra, black curves) and the model fits (red curves) are shown. Residuals and the smoothed residuals (magenta curves) of the fits (bottom). Any possible identifications of the most prominent residual features as resonance lines of highly charged ions of some heavy elements are indicated. The best-fit parameters are presented in Table 2.
Table 3

Parameters of CAL 83 after fitting by the models with various chemical compositions.

3.2.2 RX J0513.9–6951

This recurrent SSS was discovered in ROSAT data by Schaeidt et al. (1993), and the corresponding optical counterpart was almost immediately identified by Pakull et al. (1993). The most interesting property of the source is its X-ray off-states, accompanied by optical brightening during these intervals (Southwell et al. 1996). It is most likely (see the cited work) that the source’s luminosity during X-ray off-states is so high that it leads to a significant expansion of the envelope radius and the effective temperature drops, making the emergent radiation too soft to be observed in X-rays.

The X-ray spectra of RX J0513.9–6951 have been described by model atmosphere spectra only a couple of times. A low-resolution ROSAT spectrum was fitted with simple LTE model atmospheres (Ibragimov et al. 2003). Fitting a grating Chandra spectrum with the same models was less successful, and a two-component model was used to enhance the fit quality (Burwitz et al. 2007). Numerous XMM-Newton spectra obtained by the Resolution Grating Spectrometer (RGS) (McGowan et al. 2005) were not fitted with the model atmosphere spectra.

The results of fitting the Chandra spectrum with the new models described above are presented in Table 2 and the spectrum is shown in Fig. 12 (right panels). The corresponding posterior distribution of fit parameters is shown in Fig. 13. Similarly to CAL 83, we assumed uniform prior distributions for the hydrogen column density, NH (range (1–10) × 1020 cm−2); effective temperature, Teff (range 100–1000 kK); the white dwarf mass, M (range 0.3–1.4 M⊙); and radius R (range (2–20) × 108 cm). It should be noted that the obtained hydrogen column density, NH, posterior value is consistent with the result from Gänsicke et al. (1998); namely, (5.5 ± 1.0) × 1020 cm−2. Also, the found values of Teff and log g are close to those obtained from the ROSAT spectrum (Ibragimov et al. 2003), shown in Table 2.

We also fitted the spectrum of RX J0513.9–6951 with a blackbody model. The fit with a free NH has the same features like the CAL 83 fit: NH value and WD radius are significantly larger than in the model atmosphere fit, and the blackbody temperature is much lower. The fit with the fixed NH also yields results that are relatively close to those of the model atmosphere fit (see Table 2). However, comparing the blackbody fit with the fixed NH and the model atmosphere fit with the free NH using the Bayes factor showed that the spectrum is slightly better described by the model atmosphere spectra than by the blackbody.

thumbnail Fig. 13

Corner plot of the posterior distribution for parameters of RX J0513.9–6951. The 2D contours correspond to 39.3%, 86.5%, and 98.9% confidence levels. The histograms show the normalized 1D distributions for a given parameter derived from the posterior samples. The best-fit parameter values are presented in Table 2.

3.2.3 Emission component importance

It is widely known that the spectra of highly inclined SSSs are dominated by emission lines (see, e.g., Ness et al. 2013). Possibly, the same or similar emission lines have to exist in the spectra of CAL 83 and RX J0513.9–6951. For example, three emission features between 24 and 28 Å are seen in the observed spectra of both sources. Most probably they are resonance lines of Ca XIV (~24 Å), Ca XIII (~25.2 Å), and Ar XV (~26.8 Å). Observed flux excesses, visible at longer wavelengths, may also represent emission lines of highly charged ions of the same or similar elements. We assume that these emission line components are thermal emission of the line-driven winds from the hot WD surface. The presence of the emission component affects the accuracy of spectrum fitting and it has to be included in further modeling.

The lower panels in Fig. 12 show the residuals between the observed spectra and the hot WD model atmosphere spectra, with the parameters listed in Table 2. The most prominent emission features can be identified with the resonance lines of highly charged ions of heavy elements. The list of identified lines is presented in Table 4. The line identification was made using CHIANTI line database (Dere et al. 1997).

We note, however, that the significance of the interpreted emission lines is not high enough. Formally, we can improve the quality of the fits by including a few (5–6) emission lines. This procedure formally reduces the reduced χ2 (by ~ 0.2), but does not significantly change the WD atmosphere parameters.

4 Discussion

Our spectral fitting showed that the WD masses in the investigated sources are high (>1.2 M⊙) and tend to the maximum possible values for WDs. However, the derived radii are significantly larger than radii of cold WDs with these masses. It is, in fact, expected because radii of single WDs with hydrogen or helium envelopes and finite temperatures are larger than cold WD radii (see, e.g., Fontaine et al. 2001). On the other hand, the difference in the radius between the cold WDs and the WDs with the hot envelopes is minimal for high-mass WDs.

The envelope temperatures with thermonuclear burning on the WD surface in SSSs are, however, significantly higher than those of thick envelopes of isolated WDs. Therefore, we expect that the photospheric radius of a hot WD in SSSs could be significantly larger than the radius of a cold WD with the same mass, and comparison with the computed models is potentially important. Models of WD envelopes with hydrogen thermonuclear burning have been computed by several groups (see, e.g., Nomoto et al. 2007; Wolf et al. 2013). The numerical dependencies of the WD radii on the surface effective temperatures for a few WD masses are presented by Nomoto et al. (2007). We plotted these dependencies in the Teff-log g plane (see Fig. 14) together with the positions of CAL 83 and RX J0513.9–6951. We took the fits obtained for free NH. It is clearly seen that the WD masses in the SSSs derived from the model dependences are less than we obtained from the spectral fitting, and correspond to about 1.1–1.15 M⊙ for both sources.

Another theoretical constraint can be obtained from the cooling tracks in the log Teff-log L/L⊙ plane, where L is a luminosity. Such tracks computed (among others) by Iben (1982) are presented in Fig. 15 (top panel; also see Fig. 2 in the cited paper), with the added positions of CAL 83 and RX J0513.9–6951.

The positions of the SSSs correspond to almost the same WD masses, 1.15–1.2 M⊙. We also compared the sources’ positions on this plane with the model predictions by Wolf et al. (2013; Fig. 15, bottom panel). In this case, the positions correspond to the cooling tracks with a slightly lower mass, namely, 1.05–1.1 M⊙.

These results confirm a conclusion made previously (Suleimanov & Ibragimov 2003), which posits that RX J0513.9–6951 lies in the steady-state thermonuclear burning band, as also seen in Fig. 5 in Ibragimov et al. (2003). Therefore, the off-states of this SSS are most probably connected with the photospheric radius expansion during periods of increased mass-accretion rates. At the same time, the luminosity of the hot WD in CAL 83 is significantly lower than the predicted one for the steady state burning band. It means that this source lies below the stable thermonuclear burning band, as found earlier (Suleimanov & Ibragimov 2003). Therefore, thermonuclear burning most likely arises episodically and the X-ray off-states are connected with the cessation of thermonuclear burning (Kahabka 1995, 1998). In the cited work, there is a statement that a relatively short off-states duration in CAL 83 (<100 days) can exist if the WD is quite massive, M > 1.2 M⊙. Our estimation of the WD mass in CAL 83 derived from the spectral fit is in agreement with this condition.

Table 4

Lines likely corresponding to observed emission features.

thumbnail Fig. 14

Positions of the investigated SSSs in the Teff-log g plane. Model dependencies for various WD masses (Nomoto et al. 2007) are shown by black dashed curves with indicated WD masses (in solar masses). The boundaries of the stable thermonuclear burning band are shown by the solid (upper boundary) and the dashed (lower boundary) red lines. The Eddington limit for solar H/He abundances is shown by the blue dashed line.
thumbnail Fig. 15

Positions of the investigated SSSs in the Teff–L/L⊙ planes. Top panel: Model dependencies for various WD masses were taken from Iben (1982). Steady-state thermonuclear burning on WD surfaces is possible above the stability line only. Bottom panel: Model dependencies were taken from Wolf et al. (2013). Only the model curves with steady-state thermonuclear burning are shown.

5 Summary

In this paper, we present a new extended set of hot WD model spectra. The corresponding model atmospheres were computed under LTE and hydrostatic equilibrium assumptions. The list of spectral lines presented by CHIANTI collaboration was used. The novel thing in contrast with our previous computations is taking into account the photoionization from the excited atomic levels. The set was computed for three chemical compositions. The solar hydrogen-helium mix was used for all models, but the heavy element abundances were taken to be equal to the solar one (A = 1), half of the solar (A = 0.5), and one-tenth of the solar (A = 0.1). These abundances correspond to the Milky Way disc, the LMC, and the SMC. The grid covers Teff = 100–1000 kK in steps of 25 kK, and eight values of surface gravity, measured from the limited possible surface gravity gEdd, Δ log g = log g-log gEdd = 0.1, 0.2, 0.4, 0.6, 1.0, 1.4, 1.8, and 2.2. This model spectra set is designed to fit the observed soft X-ray spectra of SSSs and it was implemented into XSPEC10.

We used the calculated model grid to interpret the grating spectra of two bright SSSs in the LMC, CAL 83 (Chandra/LETGS and XMM-Newton/RGS spectra), and RX J0513.9–6951 (one Chandra/LETGS spectrum). We found that best-fit WD effective temperature values are in agreement with the values obtained in previous investigations. The values of Teff and log g obtained for CAL 83 using detailed non-LTE model atmospheres are also in good agreement with our results. Therefore, we conclude that our simplified LTE model atmospheres can be used for the analysis of the X-ray spectra of SSSs with good results despite their limitations. In fact, there are no publicly available sets of non-hydrostatic model atmospheres and non-LTE atmosphere models were computed only in a narrow range of input parameters, so the presented model set is actually the first of its kind to be released to general community.

The WD parameters found for the investigated SSSs confirm that RX J0513.9–6951 lies in the stable nuclear burning strip, while CAL 83 lies below this strip, and thermonuclear burning on the WD surface is episodic. Theoretically, a relatively short observed duration between burning episodes is possible if the WD is massive enough, M > 1.2 M⊙. Using published model Teff-log g and log Teff-log L/L⊙ dependencies, we estimated the WD mass in CAL 83 to be confined within 1.05–1.2 M⊙, which is in marginal agreement with this theoretical prediction. The WD mass estimation for RX J0513.9–6951 gives a more narrow range, 1.1–1.2 M⊙.

A notable contradiction between the WD masses formally obtained from the spectral fits and from the sources’ positions on the theoretical Teff-log g and Teff-log L/L⊙ dependencies can be pointed out. The masses from the spectral fits are high (≈ 1.29 M⊙ for CAL 83) and poorly constrained, and tend to the maximum possible WD mass 1.4 M⊙ for RX J0513.9–6951. On the contrary, the masses from the model tracks lie within 1.05–1.2 M⊙.

Although we could also suggest that the model tracks need to be revised, the simplest solution is that the spectral fits provide poor accuracy in determining WD masses. The main reason is that we can observe only a relatively narrow soft X-ray part of the general spectral energy distribution of SSSs. Moreover, the soft X-ray part of the spectra is heavily influenced by interstellar absorption. It means that we had to introduce limitation on the NH value from optical and UV observations, and set the limit of the WD radius, which cannot be smaller than the radius of a cold WD with a given mass. Perhaps, the development of the observations and the models will allow us to obtain basic SSS parameters, including the chemical composition, only from soft X-ray observations without any additional conditions. However, this is a matter that will be tackled in the distant future. Finally, we conclude that the presented model spectra set could be useful for interpreting both the existing grating spectra of SSS, obtained by Chandra and XMM-Newton, and new observations of SSSs by eROSITA.

Acknowledgements

We express our deep gratitude to the referee for their comments, which significantly improved this article. VFS and AST thank the Deutsche Forschungsgemeinschaft (DFG) for financial support (grants WE 1312/59-1 and WE 1312/56-1, respectively).

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1

A local subset with the microturbulent velocity equal to the local sound speed was computed for the CAL 83 spectrum fitting (see below). The fitting result changed insignificantly.

All Tables

Table 1

Chandra and XMM-Newton observations of selected sources.

In the text
Table 2

Parameters of the investigated SSSs.

In the text
Table 3

Parameters of CAL 83 after fitting by the models with various chemical compositions.

In the text
Table 4

Lines likely corresponding to observed emission features.

In the text

All Figures

thumbnail Fig. 1

Comparison of the photoionization cross-sections of the lowest excited levels of the hydrogen-like ion Ne X with the orbital quantum numbers L′ = 0 and 1, computed by OP (thin black curves) and by the used approximation (dashed magenta curves). The g-factor weighted averaged over both sub-level cross-sections is shown by the red curve.
In the text
thumbnail Fig. 2

Comparison of the photoionization cross-sections from the first excited level of the helium-like ion C V (top panel) and 11th excited level of helium-like ion Ne IX (bottom panel), computed by OP (solid black curves) and by the used approximation (red curves). The excitation energies of the levels are also shown.
In the text
thumbnail Fig. 3

Comparison of the photoionization cross-sections from two excited levels of C II. Cross-sections computed by OP (solid black curves), the approximation used in ATLAS9 (dashed blue curves), and our fits (red curves). Vertical dashed lines correspond to the ionization thresholds.
In the text
thumbnail Fig. 4

Same as in Fig. 3 for the first excited level of C IV. Black and red curves overlap and are close also to ATLAS9 approximation (blue dashed curve) here.
In the text
thumbnail Fig. 5

Comparison of the photoionization cross-sections from the 11th excited level of the helium-like ion Fe XXV. Shown are cross-sections computed by OP (solid black curve) and the used approximation by two fitting functions (red curves). More details in the text. The fitting function thresholds are indicated by the vertical dashed magenta lines.
In the text
thumbnail Fig. 6

Comparison of the emergent spectra of the model atmospheres computed by the code presented here (red curves), and the TMAP non-LTE code (Rauch 2003, blue curves). All models have solar chemical composition, Teff = 190 kK, log g = 8.0.
In the text
thumbnail Fig. 7

Spectrum of the model atmosphere with Teff = 500 kK, Δ log g = 0.6 and A = 0.5 (top). The Planck function with the same temperature is also shown by the red curve. The spectrum convolved with a Gaussian kernel with the Chandra resolution, R = 300, is shown in magenta. Detail of the upper spectrum near the C V ground state edge with spectral line identifications (bottom).
In the text
thumbnail Fig. 8

Comparison of the emergent spectra formation depth (m(τλ = 1)) with the wind dominating boundary (dashed red lines) for the model atmospheres with Teff = 500 kK, A = 0.5, and three different Δ log g = 0.2 (top), 0.4 (middle) and 0.6 (bottom panel). Atmospheric layers below the dashed red lines are dominated by the wind.
In the text
thumbnail Fig. 9

Model spectra with various Teff values, fixed Δ log g = 0.6, and A = 0.5.
In the text
thumbnail Fig. 10

Comparison of the fiducial model spectra (Teff = 500 kK, Δ log g = 0.6, A = 0.5) with the spectra computed for different Δ log g (top) and A (bottom panel).
In the text
thumbnail Fig. 11

Corner plot of the posterior distribution for parameters of CAL 83. The 2D contours correspond to 39.3%, 86.5%, and 98.9% confidence levels. The histograms show the normalized 1D distributions for a given parameter derived from the posterior samples. The best-fit parameter values are presented in Table 2. A Gaussian prior for the hydrogen column density NH was used, with μ = 6.5 × 1020 cm−2 and σ = 1.0 × 1020 cm−2.
In the text
thumbnail Fig. 12

Best fits of CAL 83 (left) and RX J0513.9–6951 (right) spectra (top). The observed spectra (Chandra, black curves) and the model fits (red curves) are shown. Residuals and the smoothed residuals (magenta curves) of the fits (bottom). Any possible identifications of the most prominent residual features as resonance lines of highly charged ions of some heavy elements are indicated. The best-fit parameters are presented in Table 2.
In the text
thumbnail Fig. 13

Corner plot of the posterior distribution for parameters of RX J0513.9–6951. The 2D contours correspond to 39.3%, 86.5%, and 98.9% confidence levels. The histograms show the normalized 1D distributions for a given parameter derived from the posterior samples. The best-fit parameter values are presented in Table 2.
In the text
thumbnail Fig. 14

Positions of the investigated SSSs in the Teff-log g plane. Model dependencies for various WD masses (Nomoto et al. 2007) are shown by black dashed curves with indicated WD masses (in solar masses). The boundaries of the stable thermonuclear burning band are shown by the solid (upper boundary) and the dashed (lower boundary) red lines. The Eddington limit for solar H/He abundances is shown by the blue dashed line.
In the text
thumbnail Fig. 15

Positions of the investigated SSSs in the Teff–L/L⊙ planes. Top panel: Model dependencies for various WD masses were taken from Iben (1982). Steady-state thermonuclear burning on WD surfaces is possible above the stability line only. Bottom panel: Model dependencies were taken from Wolf et al. (2013). Only the model curves with steady-state thermonuclear burning are shown.
In the text

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