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A&A
Volume 689, September 2024
Article Number A335
Number of page(s) 12
Section Stellar atmospheres
DOI https://doi.org/10.1051/0004-6361/202451195
Published online 23 September 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

A new X-ray source was discovered by eROSITA during the first part of its all-sky survey, whose position is consistent with the known classical nova (CN) AT 2018bej in the Large Magellanic Cloud (LMC, Ducci et al. 2020; Chomiuk et al. 2018). The X-ray spectrum was thermal with low temperature of ~30 eV and high luminosity >1037 erg s−1. These properties led to conclusion that the super-soft source (SSS) phase of CNe was observed for this nova (Ducci et al. 2020).

A CN may undergo the SSS phase during outbursts (Hachisu & Kato 2006). The ejected envelope becomes transparent some time after the outburst maximum and the hot nuclear burning shell, producing the soft X-ray emission, can be observed (Bode & Evans 2008). This picture is confirmed both by calculations (Starrfield et al. 2004; Soraisam et al. 2016) and observations (see e.g. Kahabka et al. 1999; Orio et al. 2001; Ness et al. 2003; Henze et al. 2014).

The SSS phase was named after the classical super-soft sources, which are a sub-class of cataclysmic variable (CV) stars with high mass-accretion rate of ~10−7 M⊙ yr−1 and (quasi-)steady state thermonuclear burning on the white dwarf(WD) surface (van den Heuvel et al. 1992; Rappaport et al. 1994). Furthermore, CVs are close binaries, consisting of a WD as a primary star and a main sequence or red giant companion that fills its Roche lobe. As a result, the matter flows from the companion to the WD (with the possible formation of an accretion disc) and CNe originate from thermonuclear explosions on WD surfaces in such objects (Starrfield et al. 2016). Namely, the thermonuclear explosion takes place in the accumulated hydrogen-rich material on the surface of the WD, which leads to the expansion of the envelope and to subsequent rise in the brightness of the system, as well as to mass ejection from the binary (see Bode & Evans 2008, for a review).

The first SSSs were discovered in the Magellanic Clouds by the Einstein observatory (Long et al. 1981; Seward & Mitchell 1981) and then were treated as a new class of X-ray objects during the ROSAT observations (Trümper et al. 1991). They are characterised by very soft thermal X-ray spectra with black-body temperatures of about 15–80 eV (150–900 kK) and high luminosities ~1036−1038 erg/s, comparable with the Eddington luminosity of a solar mass object (Kahabka & van den Heuvel 1997; Greiner 2000). Currently, more than 100 SSSs are known in ~20 external galaxies, the Magellanic Clouds, and our Galaxy (Greiner 2000; Kahabka 2006; Maitra & Haberl 2022).

The interpretation of SSS spectra is challenging. Low-resolution spectra can be often reproduced by a blackbody fit, but this model is not physically accurate. In general, blackbody fitting overestimates the interstellar neutral hydrogen column density, NH, and underestimates temperature, resulting in an overestimated luminosity (see e.g. Greiner et al. 1991; Heise et al. 1994). High-resolution spectra of SSSs are much more complex, with plenty of emission and absorption lines (with the clear role of orientation effects, see Ness et al. 2013).

Physically motivated models, which potentially can describe the observed SSS spectra are the theoretical spectra of hot WD atmospheres. Heise et al. (1994) constructed the first model atmospheres of hot WDs for explaining the SSS X-ray spectra. These models were computed under the assumption of local thermodynamic equilibrium (LTE). However, deviations from this assumption potentially can be significant at high temperature. Therefore, various authors tried to use the model spectra of hot WD atmospheres computed without LTE assumption (non-LTE, NLTE model atmospheres). In particular, the TLUSTY (Hubeny & Lanz 1995, 2011) code was used to model hot WD atmospheres and corresponding model spectra were used for fitting the BeppoSAX and ROSAT spectra of various SSSs (e.g. Hartmann & Heise 1996, 1997; Hartmann et al. 1999; Parmar et al. 1997, 1998). Lanz et al. (2005) performed a detailed NLTE model atmosphere analysis of Chandra and XMM-Newton spectroscopy of the SSS CAL 83 in the LMC.

The multi-purpose model-atmosphere NLTE code PHOENIX (Hauschildt & Baron 1999; Baron et al. 2010), modified to calculate model atmospheres for the late X-ray phase of CNe, was used to fit the Chandra observations of nova V4743 Sgr (Petz et al. 2005). The WD atmosphere was 1D spherical and was approximated by an expanding, but stationary in time, structure. The composition was assumed to be solar, leading to some uncertainties in the fit. Due to the strong evidence for expansion in the SSS spectra of novae (Ness 2010), a new version of PHOENIX was developed to account for this wind-type ejecta expansion (van Rossum & Ness 2010; van Rossum 2012). Also, a four-dimensional (4D) grid of wind-type model spectra was introduced for Teff = 450–750 kK, seven values of log g, four values of the wind asymptotic velocity, υ∞, and three values of mass-loss rate, M˙$\[\dot{M}\]$, while the chemical composition was set to solar (van Rossum 2012).

The Tübingen NLTE Model-Atmosphere Package (TMAP, Rauch 2003; Werner et al. 2003, 2012) model atmospheres were successfully used for spectral analysis of hot, compact stars (see Rauch & Werner 2010, for a brief summary). Using this code, Rauch et al. (2010) performed a NLTE spectral analysis of the nova V4743 Sgr in SSS phase. While the model was planeparallel and static (in contrast with Petz et al. 2005), the chemical composition was more realistic. Additionally, the grid of models was calculated for H-Ni with solar abundance ratios within Teff = 450–1050 kK and a fixed surface gravity of log g = 9 (see refs. in Rauch et al. 2010).

The challenging factor in describing spectra of SSSs is the unknown composition of the WD atmosphere, which should reflect both the composition of the accreted material and the nucleosynthesis products. Therefore, a mixture of stellar and CNO-cycle processed material is expected. The resulting deficiency of hydrogen and enhancement of helium abundances, as well as the deficiency of carbon and the enhancement of nitrogen and oxygen abundances, have an impact on the strength of absorption lines and the general continuum shape (e.g. Rauch 2003; Rauch et al. 2010).

Nevertheless, LTE model atmospheres were used by Ibragimov et al. (2003) and Suleimanov & Ibragimov (2003) for the analysis of ROSAT spectra of 11 SSSs. They used a modified version of Kurucz’s ATLAS code. This models were also used to fit spectra of other SSSs (Burwitz et al. 2007; Swartz et al. 2002).

Recently Suleimanov et al. (2024) further refined the LTE approach, including a more extensive set of spectral lines and photoionisation opacities from the excited levels of heavy element ions. They successfully applied a computed set of model atmosphere spectra to the interpretation of grating Chandra spectra of two classical SSS sources, CAL 83 and RX J0513.9 − 6951. It was shown that the new LTE model atmospheres enable us to represent the grating Chandra spectrum of CAL 83 with the same accuracy as the non-LTE atmospheres computed by Lanz et al. (2005) and provide similar results. However, the LTE approach has several advantages over NLTE, with the main advantage being faster computational speed, allowing us to compute much more models with various parameters. Additionally, it is easier to take into account a large number of ions and their excited levels as well as a larger number of spectral lines compared to the NLTE case.

In this article, we present a new grid of high-gravity hot LTE model atmospheres with various chemical compositions, computed using the code developed by Suleimanov et al. (2024). This grid has been used to fit eROSITA and XMM-Newton spectral observations of the supersoft phase of the CN AT 2018bej and we present the obtained results here.

The X-ray observations we used are described in Sec. 2. The model atmospheres and the corresponding grid of emergent spectra computed for fitting the observed spectra are presented in Sec. 3. The results of spectral fitting are presented in Sec. 4 and in Sec. 5 we discuss them in the context of modern models of thermonuclear-burned envelopes of WDs. We summarise our obtained results in Sec. 6.

2 Observations

2.1 eROSITA

As part of the eRASS1 all-sky survey, eROSITA onboard the Spectrum-Roentgen-Gamma (SRG) observatory (Predehl et al. 2021) started scanning over the position of AT 2018bej on February 12, 2020, with visits every 4 hours of 50 seconds in duration. A new X-ray source was found at RA(J2000)=06h26m21⋅s58 Dec(J2000)=−69∘45′49⋅′′5$\[\mathrm{R A(J 2000)=06}^{\mathrm{h}} 26^{\mathrm{m}} 21^{\mathrm{s}}_\cdot{58} ~\mathrm{Dec}(\mathrm{J} 2000)=-69^{\circ} 45^{\prime} 49^{\prime\prime}_\cdot5\]$, with an estimated uncertainty of 10″ radius (including systematics). This position is consistent with the known CV AT 2018bej/ASAS-SN 18jj (Ducci et al. 2020; Chomiuk et al. 2018). Later, this region was scanned during eRASS2–eRASS4 surveys in 2020 and 2021 (see the observation log in Table 1).

The data were reduced with the task srctool from the eROSITA Standard Analysis Software System (eSASS, Brunner et al. 2022) version 211214, pipeline configuration c020 (Merloni et al. 2024, see Table E.1). We defined an annulus region with 20″ and 100″ inner and outer radius, respectively, to extract the source from eRASS1–eRASS2 observations and an inner circle with 31″ radius in the case of eRASS3, but the source was barely visible during the eRASS3 survey and it was not visible in eRASS4, so the eRASS4 observation is not shown in Table 1. For background extraction we used a 130″ circle from a nearby source-free region.

We combined the data from the five cameras with an on-chip filter: TM1–TM4 and TM6. We did not use data from TM5 and TM7 as they have calibration problems due to a light leak (Predehl et al. 2021). A cut in the fractional exposure of 0.15 (FRACEXP > 0.15) was applied to light curves to exclude data from the edge of the detectors. The obtained light curves are shown in Fig. 1, while the spectrum (eRASS1 epoch) is shown in Fig. 2. The eRASS1 light curve is constant on long timescales, indicating that the supersoft state started prior to the beginning of the eROSITA observations.

Table 1

X-ray observation log of AT 2018bej.

thumbnail Fig. 1

eROSITA light curve of AT 2018bej in the energy band 0.2–2 keV, combining data from TM1–TM4, and TM6. eROSITA scanned the source position during four epochs eRASS1–4 until the middle of 2021. The source was barely visible during eRASS3 (right panel) and was not detected during eRASS4. The XMM-Newton observation started ~10 days after the end of eRASS1 scanning. The Gehrels (1986) approximation is used for bins with unknown count rate errors.
thumbnail Fig. 2

eROSITA eRASS1 spectrum, combining data from TM1–TM4, and TM6. The absorbed blackbody model is also shown. The obtained blackbody parameters are presented in Table 2.

2.2 XMM-Newton

Following the eROSITA discovery, a proposal for highresolution spectral observations was made at the XMM-Newton telescope. The source was observed by XMM-Newton Reflection Grating Spectrometer (RGS, den Herder et al. 2001) instrument on 2020 March 14, 10 days after the last eRASS1 observation (for the observation log, see Table 1). The data were processed by the XMM-Newton Science Analysis Software (SAS 19.0.0) package; the first-order spectra were used. The obtained light curve is shown in Fig. 3. It also does not show any obvious periodicity.

thumbnail Fig. 3

XMM-Newton light curve of AT 2018bej in the energy band 0.3–3 keV from both RGS1 and RGS2 instruments.
Table 2

Preliminary fit of AT 2018bej with blackbody model.

2.3 Preliminary analysis

The Bayesian approach was used for analysing the X-ray spectra. We applied the analysis software Bayesian X-ray Analysis (BXA, Buchner et al. 2014; Buchner 2016a), which connects the nested sampling package UltraNest1 (Buchner 2021, 2016b) with XSPEC 2(Arnaud 1996; Arnaud et al. 1999). Posterior probability distributions and the Bayesian evidence were derived with the nested sampling Monte Carlo algorithm MLFriends (Buchner et al. 2014; Buchner 2019). To construct the corner plots, the corner3 package was used (Foreman-Mackey 2016, 2017).

It is known that for low-count spectra the usage of C-statistics (Cash 1979) is preferable instead of the χ2-statistics. We used its XSPEC implementation cstat as a likelihood in order to determine the best-fit parameters. The parameter uncertainties are derived from the 0.16 and 0.84 quantiles of the posterior distribution (which corresponds to 68% CL). As there is no convenient criterion of estimating the goodness of fit (as in the case of χν2$\[\chi_\nu^2\]$) for cstat, we give the statistics value and the number of degrees of freedom below.

At first, the eROSITA and RGS spectra were fitted by the simple blackbody model, while for the absorption, we used the Tübingen-Boulder ISM absorption model tbabs. The eROSITA spectrum with the blackbody fit is shown in Fig. 2. However, these results are considered non-physical, because the emitting radii of (1–6) · 105 km (assuming a LMC distance of 50 kpc, see Table 2) are inconsistent with a typical WD radius, and the resulting luminosity significantly exceeds the Eddington limit for typical WD masses. This is not surprising, as it is well known that blackbody fits tend to overestimate NH and underestimate the temperature (see e.g. Greiner et al. 1991; Heise et al. 1994). Therefore, hot WD model atmosphere spectra are necessary to reproduce SSS spectra.

Alternatively, Suleimanov et al. (2024) computed a new set of LTE model-atmosphere spectra and showed that these models allow us to represent the spectra of two classical SSS sources, CAL 83 and RX J0513.9 − 6951. In the cited article, three model grids were also calculated for three chemical compositions: the H/He mix was solar, and the abundance of heavy elements was set to solar, 0.1*solar, and 0.5*solar. The latter abundances correspond to the SMC and LMC composition. A preliminary analysis of all spectra with these models gives Teff ≈ 570–600 kK and log g ≈ 8.25, with the LMC composition preferred.

3 Model atmospheres

To describe the observed X-ray spectrum of the object, we computed our own model-grid spectra of hot WDs. We used our code, which is based on Kurucz’s code ATLAS (Kurucz 1970), and modified it for high temperatures and high densities (Ibragimov et al. 2003; Suleimanov et al. 2006, 2015). The 15 most abundant chemical elements from H to Ni were taken into account, along with about 20 000 lines from CHIANTI, Version 3.0, atomic database (Dere et al. 1997). The main model parameters are the effective temperature, Teff, and the gravity, Δ log g = log g − log gEdd. The latter indicates the distance of model from the Eddington limit: log⁡gEdd=log⁡(σeσSBT4/c)=4.818+4 log⁡(Teff/105 K),$\[\log g_{\mathrm{Edd}}=\log \left(\sigma_{\mathrm{e}} \sigma_{\mathrm{SB}} T^4 / c\right)=4.818+4 \log \left(T_{\mathrm{eff}} / 10^5 \mathrm{~K}\right),\]$(1)

as detailed in Suleimanov et al. (2024).

The chemical composition of the WD atmosphere during the SSS phase after the nova explosion is significantly different from that of the accreted matter (see e.g. Rauch et al. 2010) due to thermonuclear fusion products being brought to the surface. Therefore, it is necessary to calculate the model atmospheres with chemical compositions close to the expected one.

It was assumed that the initial chemical composition was LMC-like (solar H/He mix and half-solar heavy elements abundance). Then, some part of the hydrogen was transformed into helium via the CNO-cycle, resulting in an increase in the number fraction of CNO. We used the composition of Rauch’s NLTE models as a starting point. The ratios of the number fractions of hydrogen and helium in these models are equal to each other, the carbon number fraction is less than solar, and the number fractions of nitrogen and oxygen increase in comparison with the solar values. In particular, in Rauch’s model 0064, which we considered a fiducial model, AH/AHe = 0.48/0.48, and the number fraction of carbon is AC = 0.39 * solar. We considered various ratios of AH/AHe and ultimately adopted 0.46/0.54. Technically, from a statistical point of view, a ratio of AH/AHe = 0.0/1.0 is a bit more likely; however, we considered it less physical, as it is assumed that active hydrogen thermonuclear burning still occurs at the surface of the WD. The number fraction of carbon was considered as an additional free parameter of the model.

Thus, in addition to Teff and Δ log g, a new parameter has been introduced – the number fraction of carbon ‘chem’ in solar units. The number fraction of nitrogen is varied from 1.89 to 0.5 in solar units depending on ‘chem’ in assumption that (0.5 − chem) of carbon is transformed into nitrogen. Thus, the resulting abundance of carbon ‘chem’ changes from 0 (all C→N) to 0.5 (LMC composition). The abundance of other elements is assumed to be 0.5 * solar (we use solar abundances from Asplund et al. 2009).

Based on the preliminary fit analysis, altogether 288 models were calculated for Teff from 525 to 700 kK in steps of 25 kK, Δ log g = 0.1, 0.2, 0.4, 0.6, 1.0, 1.4, and chem = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5. We refer to this model as the chem-model hereafter. All the spectra were implemented as a tabular XSPEC-format model.

Examples of emergent spectra of the models are shown in Figs. 4 and 5 for different values of Teff and Δ log g, while chem = 0.3 is fixed. The absorption edges corresponding to photoionisation from the ground level of C VI (0.49 keV), N VI (0.55 keV), and N VII (0.66 keV) ions are clearly seen. These edges lead to significant deviation of the model atmosphere spectra from the corresponding blackbody flux (indicated by dashed-grey lines). The spectra are dominated by a forest of absorption lines (spectral series of some CNO ions are shown), but the main influence on the common shape of the spectra comes from the effective temperature. The spectra in Fig. 5 are convolved for clarity with a Gaussian kernel with XMM-Newton RGS resolution of R ≈ 650 in the observed soft X-ray energy range. The gravity mainly affects the 0.48–0.65 keV energy range, which is covered by RGS.

Figure 6 shows some examples of models for different values of the chem-parameter. The spectra were also convolved with a Gaussian kernel with XMM-Newton RGS resolution R ≈ 650 in the observed soft X-ray energy range. The chemical composition mainly affects the 0.48–0.65 keV energy range, which is covered by RGS. Although the changes resulting from the chemical composition are smaller, the spectral resolution of XMM-Newton allows for accurate determinations to be obtained (see the convolved spectrum in Fig. 6). In addition, the overall spectrum level in this region is clearly distinguishable in the eROSITA spectrum. Also, Fig. 7 shows the comparison between spectra convolved with the XMM response for chem = 0.27 and 0.5 (LMC composition), while temperature and gravity are fixed at Teff = 605 kK, Δ log g = 0.52. The notable deviation in the 0.48–0.55 keV region is clearly visible.

We also compared our LTE models with NLTE models computed by Rauch et al. (2010). Figure 8 shows the comparison between the emergent spectrum of Rauch’s NLTE model 006 and our LTE spectrum with the same chemical composition. Both spectra were binned to a 0.002 Å interval within the 10–60 Å wavelength range. The general shapes of the spectra are similar, however, there are some differences seen in the absorption-edge regions. Due to an over-ionisation of the ions (relative to the LTE case), the absorption jumps in the NLTE spectrum appear shallower. Additionally, a sharp jump occurs in the NLTE spectrum at 0.45 keV, associated with ionisation from an excited level of Si XI. This feature is much smoother in our spectrum. Our list of lines is also different, resulting in variations in the line distribution in the spectra. The line broadening of H-like ions seems to differ, as seen in the lower left panel in Fig. 8, where the N VI and N VII lines are depicted. The broadening of the resonance lines is primarily determined by radiative damping in addition to Doppler broadening. In contrast, the wide wings of the resonance line N VII in the NLTE spectrum are also affected by the Stark effect. Additionally, the level populations differ between LTE and NLTE models, causing the lines in the LTE spectrum to appear deeper. Namely, the upper layers of LTE models are obviously cooler than those of NLTE models and the corresponding ions are more populated. This indicates that a decreased abundance of some elements is needed compared to NLTE models to describe the same observed spectral line absorption.

We investigated the uncertainty that may arise from using LTE models instead of NLTE models. It is possible to demonstrate that a NLTE model spectrum with Teff = 550 kK reproduces almost the same general spectrum shape with the same absorption jump strengths as the LTE model spectrum with Teff = 600 kK, while the gravity log g is fixed, see Fig. A.1. Reducing log g while fixing the temperature leads to the same effect. Thus, using LTE atmospheres may lead to an overestimation of temperature by ~10% and/or to an underestimation of log g by ~0.5 dex (see Fig. A.2). We note that both of these effects occur simultaneously and we can only conclude that LTE model atmospheres slightly overestimate the effective temperature and slightly underestimate the surface gravity of a hot WD as a result of fitting its soft X-ray spectrum.

This occurs because the elements under NLTE conditions are over-ionised compared to the LTE case. Consequently, higher temperatures and/or lower plasma densities are needed in LTE atmospheres to compensate for this effect.

thumbnail Fig. 4

Emergent spectra of model atmospheres with different Teff values. The spectral series for some CNO ions are also indicated. The grey dashed lines represent the corresponding blackbody spectra.
thumbnail Fig. 5

Emergent model spectra with various surface gravity (three values of gravity Δ log g). The spectra are smoothed using a Gaussian kernel with resolution R ≈ 650. Notations are the same as in Fig. 4.
thumbnail Fig. 6

Emergent model spectra with different values of the chem parameter. The spectra are also smoothed (see Fig. 5). Notations are the same as in Fig. 4.
thumbnail Fig. 7

Comparison of the convolved spectra for different chemical compositions with the same values: Teff = 604 kK, Δ log g = 0.51, NH = 5.88 × 1020 cm−2. These are the cases for chem = 0.0 (i.e. AH/AHe = 0.46/0.54, Arest = 0.5 * solar, indicated in blue) and chem = 0.27 (the best-fit value of carbon abundance, in red). The 0.47–0.6 keV region is also shown in detail.
thumbnail Fig. 8

Comparison of the model atmospheres spectra computed by our code (in red) and the TMAP non-LTE code (Rauch et al. 2010, in blue). Both models have Teff = 600 kK, log g = 9. The chemical composition corresponds to model 006 from Rauch’s grid. The lower left panel shows in detail the comparison of absorption line profiles.
thumbnail Fig. 9

Corner plot of the parameters’ posterior distribution for the eRASS1 spectrum (left panel) and for the XMM-Newton RGS spectrum (right panel) of AT 2018bej. The WD mass, M, and radius, R, are considered as free parameters. The 2D contours correspond to 39.3, 86.5, and 98.9% confidence levels. The histograms show the normalised 1D distributions for a given parameter derived from the posterior samples. The hydrogen column density is fixed at NH = 5.88 × 1020 cm−2 for the RGS spectrum. The best-fit parameter values are presented in Table 3.

4 Analysis of the spectra

All the observed spectra, eRASS1, eRASS2, and XMM-Newton (see Table 1), were fitted with the chem-model described above. Using temperature, Teff, and gravity, Δ log g, as free parameters leads to a degeneracy in the gravity distribution and to a nonphysical estimation of the WD mass. Therefore, we fit the spectra using WD mass M and radius R as free parameters and set a strict upper limit of M = 1.4 M⊙. For both eROSITA and XMM spectra a uniform prior distribution was set for NH (in the range: (1–10) × 1020 cm−2), ‘chem’ parameter, effective temperature, Teff, WD mass, M (in the range: 0.3–1.4 M⊙) and radius R (in the range: (2–20) × 103 km). Also, the additional lower limit was set for the radius to obtain a physically adequate result; thus R should be greater than the radius of cold WD Rcold (Nauenberg 1972) with the same mass.

The example of the resulting corner plot for the eRASS1 spectrum is shown in the left panel of Fig. 9, while the obtained parameters for all the fits are presented in Table 3. Figure 10 shows the comparison between the observed and model eROSITA spectrum. The parameters’ posterior distributions for all spectra are narrow enough and have only one mode, resulting in smaller errors. However, the WD mass tends to its upper limit, especially for the XMM case.

It is important to emphasise that the obtained hydrogen column density NH is significantly different for the eROSITA and XMM spectra. Fitting the XMM spectrum provides the column density NH is about half of that found from fitting the eROSITA spectrum; namely, the analysis of the eRASS1 spectrum yields NH ≈ 5.88 × 1020 cm−2 (see Table 3 and Fig. 9, left panel). On the other side, the analysis of the XMM spectrum yields Nh ≈ 3.06 × 1020 cm−2. The last value is too small compared to the Galactic absorption column in the direction of the source. The average Galactic absorption in this direction is 5.6−0.8+0.7$\[5.6_{-0.8}^{+0.7}\]$ × 1020 cm−2 based on the extinction and absorption maps presented in Doroshenko (2024)5, who combined the results by Edenhofer et al. (2024); Yao et al. (2017) and Planck Collaboration XLVIII (2016). The HI survey (HI4PI Collaboration 2016) provides the close result 6.5−0.1+0.1$\[6.5_{-0.1}^{+0.1}\]$ × 1020 cm−2. Therefore, we consider the value of NH obtained from fitting the eROSITA observation to be more reliable and this is the value we fix when describing the XMM spectrum. Specifically, we fixed NH = 5.88 × 1020 cm−2. It should be also noted that we assume there is no extra NH local to the source, so we determined only the foreground absorption component, while the Tübingen-Boulder ISM absorption model tbabs was used.

The corner plot and obtained parameters for the XMM observation with the fixed NH are presented in Fig. 9 (right panel) and Table 3. The comparison between the observed and model RGS spectrum is shown in Fig. 11.

Interestingly, for both the XMM and eRASS1 spectra, regardless of the other parameters’ values, the carbon abundance ‘chem’ parameters are close to each other and coincide within the errors, namely, ≈0.27 and ≈0.24, respectively. They are derived quite accurately and the corresponding distributions are narrow and do not demonstrate multi-modal behaviour. Additionally, another spectrum from the next eROSITA survey, eRASS2, is available. Fitting this spectrum with the chem-model also yields quite a good fit with relatively small errors for ‘chem’, indicating its reliable determination. This ‘chem’ value is slightly lower, ≈ 0.20, than the value ≈0.27–0.24 obtained from eRASS1 and XMM spectra.

Given the availability of eROSITA spectra for two epochs, it is possible to trace the evolution of the system parameters, particularly the carbon abundance. To achieve this, a MCMC calculation was performed simultaneously for the eRASS1 and eRASS2 spectra using the chem-model, assuming that the WD mass is the same in both fits; therefore, it is a common free parameter for all spectra. The same uniform priors were set for the hydrogen column density, ‘chem’-parameter, effective temperature, WD mass, and radius. The corresponding posterior distribution is shown in Fig. 12 and the parameter values are presented in Table 3.

The posterior distributions of the two ‘chem’-parameters are quite narrow, and the corresponding errors are relatively small. We note that the hydrogen column density is a free parameter in this case and it converged to a plausible value of 5.83 × 1020 cm−2. The inferred WD mass, ≈1.07 M⊙, was also well determined.

We also compared our results with those obtained using non-LTE model spectra. A set of NLTE atmosphere models was calculated by Rauch et al. (2010) to describe the spectrum of nova V4743 Sgr during its SSS phase. The chemical composition of these models has been adjusted specifically for that nova. To compare our results with NLTE models we fit our observed spectra with Rauch’s model 006. The resulting WD radii (2.2–3.7) × 108 cm and masses of 0.4–1 M⊙ are significantly underestimated and the obtained radius is lower than the radius of cold WD with the similar mass. Therefore, we set a lower limit for the WD radius using Nauenberg’s M − R relation (see Nauenberg 1972) and the obtained results are presented in Table 4. The temperature goes down to ~660–690 kK, while the mass increases to ~1.2 M⊙. Again, we consider the NH value found by the eRASS1 analysis to be more reliable and performed an additional fitting of the XMM spectrum with the fixed Nh = 4.8 × 1020 cm−2. The fit quality deteriorates significantly, and the mass tends to its lower limit based on the radius constraint. While the low-resolution eROSITA spectrum is (more or less) well described by the NLTE model, the quality of the XMM fit is significantly worse, compared to our LTE fit.

It should be noted that since the source luminosity is near the Eddington limit (L ~ 0.12–0.49 LEdd for different LTE/NLTE fits), NLTE effects are not negligible. However, the spectral resolution of eROSITA observations is insufficient to reveal these effects. While the XMM resolution, in principle, allows us to distinguish NLTE effects, they would only moderately affect the result (e.g. by ~10% in the temperature; see the end of Sect. 3). In addition, the available NLTE models were computed only with fixed log g = 9. Possibly, NLTE models with smaller log g would be able to fit the observed spectra with better accuracy. Nevertheless, Suleimanov et al. (2024) compared the results of fitting the Chandra and XMM spectra of the classical SSS CAL 83 using LTE models and NLTE models by Lanz et al. (2005), finding that both approaches yield similar results. This supports the use of LTE models for analysing the spectra of SSSs.

Table 3

Spectral parameters of the LTE model fit for AT 2018bej.

thumbnail Fig. 10

Model atmosphere best-fit of the eROSITA eRASS1 (in red) and eRASS2 (in blue) spectra. The obtained parameters are presented in Table 3.
thumbnail Fig. 11

XMM-Newton RGS spectrum obtained with the best-fit LTE model. The spectra of RGS 1 (left panel) and RGS 2 (right panel) instruments are shown. The best-fit parameters can be found in Table 3. The hydrogen column density is fixed at NH = 5.88 × 1020 cm−2.
Table 4

Fit of AT 2018bej spectra with NLTE(a) model.

5 Discussion

Our analysis of the joint eRASS1 and eRASS2 fit allows us to investigate the evolution of the source on a half-year scale. Although the effective temperature of the hot WD slightly decreases and the photospheric radius slightly increases, the changes in the parameter values are marginal and remain within the errors. Formally, the carbon abundance decreases from 0.21 to 0.16 of the solar value, but the error of ‘chem’ determination is comparable with the difference between these two values, as seen in Fig. 13. In conclusion, even the low-resolution spectral analysis shows hints of the evolution of the source parameters, including the WD’s chemical composition, but observations more separated in time are necessary for a more reliable study of evolution.

It should be noted that the mass obtained from fitting different spectra varies, with a tendency to approach its upper limit. Therefore, for a more reliable mass determination, we compare the obtained results with theoretical predictions. Models of WD envelopes with hydrogen thermonuclear burning have been computed by several groups (see e.g. Wolf et al. 2013; Nomoto et al. 2007). In particular, Nomoto et al. (2007) calculated the R − Teff dependencies for different masses of WDs with hydrogen-rich envelope. We converted these dependencies into Teff − log g dependencies, and put all the obtained data on the corresponding plane (see Fig. 14). We included all our fits on the figure, including the different eRASS observations, the XMM observation with fixed NH, and the joint eROSITA fit with a common M parameter.

All the positions of the source on the plane Teff − log g lie at the model dependencies corresponding to 1.1–1.15 M⊙, although it should be noted that the errors are large for the eRASS2 fit. This value of the WD mass is in a good agreement with the WD mass derived from the joint eRASS1 and eRASS2 fit. It means that the estimation of the WD mass M = 1.05–1.15 M⊙ could be correct. We also emphasise that all fit positions found lie above the steady-burning limit for WDs (red dashed line in Fig. 14). This confirms our assumption that the source undergoes steady hydrogen burning on its surface. Moreover, the eRASS2 source position moves further away from the lower boundary of the stable burning strip, indicating an increase in the intensity of the thermonuclear burning from eRASS1 to eRASS2 observations. This fact is supported by the increased bolometric luminosity.

It should be noted that setting NH as a free parameter for the XMM fit leads to a significant increase in gravity, causing the model mass to approach the limit mass of 1.4 M⊙. As mentioned earlier in this paper, we do not consider such a fit to be satisfactory (and the corresponding point is not shown in Fig. 14).

thumbnail Fig. 12

Corner plot of the parameters’ posterior distribution for eRASS1 and eRASS2 spectra fitted simultaneously with a common WD mass parameter. The 2D contours correspond to 39.3, 86.5, and 98.9% confidence levels. The histograms show the normalised 1D distributions for a given parameter derived from the posterior samples. The best-fit parameter values are presented in Table 3.
thumbnail Fig. 13

1D distribution for eRASS1 (in blue) and eRASS2 (in orange) ‘chem’-parameter. The spectra were fitted simultaneously using the Bayesian approach, while the WD mass parameter was common for both spectra. Vertical lines show the 16th, 50th, and 84th percentiles for the ‘chem’ distribution.
thumbnail Fig. 14

Positions of the source in the Teff − log g plane according to different eRASS observations (red circles with the number 1 and 2), and according to joint fit (red empty circles), see Table 3. The XMM spectrum fit with fixed NH is shown by blue circle. Model dependencies for various WD masses, taken from Nomoto et al. (2007), are shown by black dashed curves. The numbers at the curves indicate WD masses (in solar masses). The lower boundary of the stable thermonuclear burning band is shown by the dashed red line. The Eddington limit for solar H/He abundances is shown by the blue dashed line.

6 Summary

We performed a spectral analysis of the supersoft X-ray phase of the classical nova AT 2018bej, which was observed for the first time in X-rays by the eROSITA and XMM-Newton telescopes. To describe the spectrum, we calculated high-gravity hot LTE model atmospheres with chemical compositions typical for nova SSS phases. In particular, we assumed that the number fractions of hydrogen and helium are almost equal, namely, AH/AHe = 0.46/0.54. The heavy element abundances are taken equal to the half of the solar abundances for all the elements, excluding carbon and nitrogen. We considered five different values of AC from 0.0 to 0.5 of the solar value. The nitrogen abundance is connected with AC under the assumption that all the carbon deficiency transforms to the nitrogen excess above 0.5 of the solar value. The grid covers Teff = 525–700 kK in steps of 25 kK, Δ log g = log g − log gEdd = 0.1, 0.2, 0.4, 0.6, 1.0, 1.4, and five values of carbon abundance (‘chem’-parameter). The code developed by Suleimanov et al. (2024) was used for this aim.

Separate fits of the individual eRASS1 and eRASS2 observations reveal remarkable changes in the physical properties of the source on a half-year timescale. The effective temperature decreased, while the photospheric radius increased. Although the observed soft X-ray flux decreased, the derived bolometric luminosity slightly increased. The most potentially interesting change is the marginal decrease in carbon abundance between these two observations, from ≈0.24 to ≈0.20 of the solar one. The fit of the XMM/RGS spectrum yields results close to those obtained from the eRASS1 spectrum analysis if we fix the interstellar absorption.

In addition to the individual fits, we performed a simultaneous fitting of the eROSITA spectra for two epochs (eRASS1 and eRASS2) with a common WD mass parameter. This assumption is strongly physically motivated and allows us to trace the evolution of the source more reliably. However, the analysis demonstrates only marginal evolution of parameters. Although the effective temperature slightly decreases, accompanied by a slight increase in the WD radius, the bolometric luminosity is constant. A slight carbon abundance decrease was also confirmed.

We compared the WD mass obtained from the joint eROSITA spectra fit of 1.07−0.11+0.08 M⊙$\[1.07_{-0.11}^{+0.08} ~M_{\odot}\]$ with the theoretical predictions. The predictions were derived from the positions of the best-fit parameters of the spectra on the Teff − log g plane, where model curves for different WD masses are also displayed. The positions of the fits correspond to WD masses of 1.1–1.15 M⊙. Thus, we concluded that the mass of the WD in the investigated nova falls within the range of 1.05–1.15 M⊙.

To summarise, our simplified LTE approach provides a satisfactory description of the X-ray spectra of the supersoft X-ray stage of a classical nova. It may potentially help to explain the spectra of other similar objects.

Acknowledgements

This work was supported by the Deutsche Forschungsgemeinschaft under grants WE1312/56-1 (AT) and WE1312/59-1 (VFS). This work is partially supported by the Bundesministerium für Wirtschaft und Energie through the Deutsches Zentrum für Luft- und Raumfahrt e. V. (DLR) under the grant number FKZ 50 QR 2102 (LD). This work is based on data from eROSITA, the soft X-ray instrument aboard SRG, a joint Russian-German science mission supported by the Russian Space Agency (Roskosmos), in the interests of the Russian Academy of Sciences represented by its Space Research Institute (IKI), and the Deutsches Zentrum für Luft- und Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL) and its subcontractors, and is operated by NPOL with support from the Max Planck Institute for Extraterrestrial Physics (MPE). The development and construction of the eROSITA X-ray instrument was led by MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and Astrophysics of the University of Tübingen, with the support of DLR and the Max Planck Society. The Argelander Institute for Astronomy of the University of Bonn and the Ludwig Maximilians Universität Munich also participated in the science preparation for eROSITA. The eROSITA data shown here were processed using the eSASS software system developed by the German eROSITA consortium.

Appendix A Additional figures

Here, we present the results that have allowed us to estimate the uncertainties arising due to using LTE model atmospheres instead of NLTE models. The main NLTE effect important for us is over-ionisation, which leads to a decrease in the photoionisation jumps. To compensate for this effect under an LTE assumption, we have to increase the plasma temperature or decrease the plasma density. We demonstrate here that a NLTE model atmosphere with Teff = 550 kK reproduces the general spectral shape of the LTE model atmosphere with Teff = 600 kK at the same surface gravity log g = 9.0 (see Fig. A.1). Rauch’s NLTE model 006 was used for comparison. The chemical composition of the LTE and NLTE models is identical. The spectra are binned to a 0.002Å interval within the 10 − 60 Å wavelength range, and the NLTE spectrum is scaled to align directly below the LTE one for the purpose of comparison. Is is clearly seen that the general spectrum shape, including the absorption jumps, is reproduced by the LTE model with a temperature that is ~ 10% higher than the NLTE one. The same effect is seen when we decrease the gravity of the LTE model, as shown in Fig. A.2. Both spectra have the same Teff = 600 kK but the LTE model has a lower surface gravity, log g = 8.5, instead of log g = 9 in the NLTE model.

thumbnail Fig. A.1

Comparison of the model atmospheres spectra computed by our code (in red) and the TMAP non-LTE code (Rauch et al. 2010, in blue). Shown are LTE model with Teff = 600 kK and scaled NLTE model with Teff = 550 kK. The gravity log g = 9 is fixed. The chemical composition corresponds to model 006 from Rauch’s grid. The lower left panel shows in detail the comparison of absorption line profiles.
thumbnail Fig. A.2

Same as in Fig. A.1, but with a fixed Teff = 600 kK and different gravities: log g = 8.5 for LTE model and log g = 9 for the NLTE one.

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All Tables

Table 1

X-ray observation log of AT 2018bej.

In the text
Table 2

Preliminary fit of AT 2018bej with blackbody model.

In the text
Table 3

Spectral parameters of the LTE model fit for AT 2018bej.

In the text
Table 4

Fit of AT 2018bej spectra with NLTE(a) model.

In the text

All Figures

thumbnail Fig. 1

eROSITA light curve of AT 2018bej in the energy band 0.2–2 keV, combining data from TM1–TM4, and TM6. eROSITA scanned the source position during four epochs eRASS1–4 until the middle of 2021. The source was barely visible during eRASS3 (right panel) and was not detected during eRASS4. The XMM-Newton observation started ~10 days after the end of eRASS1 scanning. The Gehrels (1986) approximation is used for bins with unknown count rate errors.
In the text
thumbnail Fig. 2

eROSITA eRASS1 spectrum, combining data from TM1–TM4, and TM6. The absorbed blackbody model is also shown. The obtained blackbody parameters are presented in Table 2.
In the text
thumbnail Fig. 3

XMM-Newton light curve of AT 2018bej in the energy band 0.3–3 keV from both RGS1 and RGS2 instruments.
In the text
thumbnail Fig. 4

Emergent spectra of model atmospheres with different Teff values. The spectral series for some CNO ions are also indicated. The grey dashed lines represent the corresponding blackbody spectra.
In the text
thumbnail Fig. 5

Emergent model spectra with various surface gravity (three values of gravity Δ log g). The spectra are smoothed using a Gaussian kernel with resolution R ≈ 650. Notations are the same as in Fig. 4.
In the text
thumbnail Fig. 6

Emergent model spectra with different values of the chem parameter. The spectra are also smoothed (see Fig. 5). Notations are the same as in Fig. 4.
In the text
thumbnail Fig. 7

Comparison of the convolved spectra for different chemical compositions with the same values: Teff = 604 kK, Δ log g = 0.51, NH = 5.88 × 1020 cm−2. These are the cases for chem = 0.0 (i.e. AH/AHe = 0.46/0.54, Arest = 0.5 * solar, indicated in blue) and chem = 0.27 (the best-fit value of carbon abundance, in red). The 0.47–0.6 keV region is also shown in detail.
In the text
thumbnail Fig. 8

Comparison of the model atmospheres spectra computed by our code (in red) and the TMAP non-LTE code (Rauch et al. 2010, in blue). Both models have Teff = 600 kK, log g = 9. The chemical composition corresponds to model 006 from Rauch’s grid. The lower left panel shows in detail the comparison of absorption line profiles.
In the text
thumbnail Fig. 9

Corner plot of the parameters’ posterior distribution for the eRASS1 spectrum (left panel) and for the XMM-Newton RGS spectrum (right panel) of AT 2018bej. The WD mass, M, and radius, R, are considered as free parameters. The 2D contours correspond to 39.3, 86.5, and 98.9% confidence levels. The histograms show the normalised 1D distributions for a given parameter derived from the posterior samples. The hydrogen column density is fixed at NH = 5.88 × 1020 cm−2 for the RGS spectrum. The best-fit parameter values are presented in Table 3.
In the text
thumbnail Fig. 10

Model atmosphere best-fit of the eROSITA eRASS1 (in red) and eRASS2 (in blue) spectra. The obtained parameters are presented in Table 3.
In the text
thumbnail Fig. 11

XMM-Newton RGS spectrum obtained with the best-fit LTE model. The spectra of RGS 1 (left panel) and RGS 2 (right panel) instruments are shown. The best-fit parameters can be found in Table 3. The hydrogen column density is fixed at NH = 5.88 × 1020 cm−2.
In the text
thumbnail Fig. 12

Corner plot of the parameters’ posterior distribution for eRASS1 and eRASS2 spectra fitted simultaneously with a common WD mass parameter. The 2D contours correspond to 39.3, 86.5, and 98.9% confidence levels. The histograms show the normalised 1D distributions for a given parameter derived from the posterior samples. The best-fit parameter values are presented in Table 3.
In the text
thumbnail Fig. 13

1D distribution for eRASS1 (in blue) and eRASS2 (in orange) ‘chem’-parameter. The spectra were fitted simultaneously using the Bayesian approach, while the WD mass parameter was common for both spectra. Vertical lines show the 16th, 50th, and 84th percentiles for the ‘chem’ distribution.
In the text
thumbnail Fig. 14

Positions of the source in the Teff − log g plane according to different eRASS observations (red circles with the number 1 and 2), and according to joint fit (red empty circles), see Table 3. The XMM spectrum fit with fixed NH is shown by blue circle. Model dependencies for various WD masses, taken from Nomoto et al. (2007), are shown by black dashed curves. The numbers at the curves indicate WD masses (in solar masses). The lower boundary of the stable thermonuclear burning band is shown by the dashed red line. The Eddington limit for solar H/He abundances is shown by the blue dashed line.
In the text
thumbnail Fig. A.1

Comparison of the model atmospheres spectra computed by our code (in red) and the TMAP non-LTE code (Rauch et al. 2010, in blue). Shown are LTE model with Teff = 600 kK and scaled NLTE model with Teff = 550 kK. The gravity log g = 9 is fixed. The chemical composition corresponds to model 006 from Rauch’s grid. The lower left panel shows in detail the comparison of absorption line profiles.
In the text
thumbnail Fig. A.2

Same as in Fig. A.1, but with a fixed Teff = 600 kK and different gravities: log g = 8.5 for LTE model and log g = 9 for the NLTE one.
In the text

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