Population synthesis of accreting white dwarfs: II. X-ray and UV emission

Figure 2 shows the evolution of the mass-normalized X-ray luminosity of the population of accreting WDs in the starburst and constant SFR cases. For our predicted observable X-ray luminosity, we apply absorption with a column density $N_{\rm H}=3.0\times 10^{20}\rm{cm}^{-2}$, which is a typical value for the extinction within the Galaxy at high Galactic latitudes (Dickey & Lockman, 1990). In addition, for early-type galaxies we also show the observed X-ray luminosity per unit mass with column density $1.8\times 10^{20}\rm{cm}^{-2}<N_{\rm H}<6.7\times 10^{20}\rm{cm}^{-2}$(the shaded region in the plot). This range of $N_{\rm H}$ corresponds to that observed for the six elliptical galaxies in Table 1. In the calculation, we only considered Galactic absorption, since the intrinsic absorption of these elliptical galaxies is small.

Our calculations predict that in the starburst case, the X-ray luminosity sharply decreases after 1 Gyr (Fig. 2). At early times (stellar age $t<1.0$ Gyr), the X-ray luminosity in the starburst case is comparable with that in the constant SFR case for galaxies with the same mass. However, it is smaller for $t>1.0$ Gyr. In order to compare our results with observations we use X-ray data from Bogdán & Gilfanov (2010) and Zhang, Gilfanov & Bogdán (2012) for nearby elliptical galaxies. The data and our predictions are summarised in Table 1 where we list the observed X-ray luminosities along with the predicted X-ray luminosities and SSS numbers for six elliptical galaxies (Bogdán & Gilfanov, 2010; Zhang, Gilfanov & Bogdán, 2012). Note that observed luminosities are total soft X-ray luminosities of unresolved emission and soft compact sources and therefore should be regarded as upper limits on the soft X-ray luminosity of nuclear burning white dwarfs (see Bogdán & Gilfanov (2010) for details). In the calculation of the predicted X-ray luminosities and SSS numbers, we have taken absorption into consideration for different galaxies with different column density values.

One can see in Fig. 2 that, at the ages exceeding 6 Gyr, our predicted X-ray luminosities are comparable to that which are observed in nearby ellipticals, and is roughly two orders of magnitude below the soft X-ray flux expected if all SNe Ia are produced via the SD scenario (see Gilfanov & Bogdán (2010)). This is consistent with our prediction that the SD channel does not significantly contribute to the SN Ia rate at late delay times (see Fig. 8 in Paper I). However, for the two youngest galaxies in the Bogdán & Gilfanov (2010) sample, NGC3585 and NGC3377, our standard calculations predict up to an order of magnitude larger soft X-ray luminosities than observed. The strong dependence of the X-ray luminosity on the age of the stellar population predicted by our calculation seems to contradict observations. In interpreting this result one should keep in mind that there is some controversy regarding the ages of the two youngest galaxies. Idiart, Silk & de Freitas Pacheco (2007) found that the age of NGC3377 is around 7.8 Gyr and Georgiev, Goudfrooij & Puzia (2012) listed literature values of the age of NGC3377, varying from 3.5 Gyr to 8.9 Gyr. In the case of NGC3585, Michard (2006) found a single stellar population age of 1.7 Gyr, while Hempel et al. (2007) suggested that its age should be larger than 3 Gyr. On the other hand, metallicity may be important to explain the discrepancy between model predictions and observations. O’Sullivan & Ponman (2004) and Georgiev, Goudfrooij & Puzia (2012) found that the metallicities of the two youngest galaxies, NGC3585 and NGC3377, are likely to be lower than solar metallicity. However, solar abundance is adopted in our calculation. The metallicity plays a complicated role in the evolution of accreting WDs. The metallicity influences the initial-final mass relation for stars, the stable burning limits of accreting WDs, and the evolution of mass transfer rates (Umeda et al., 1999; Meng, Chen & Han, 2008; de Mink, Pols & Yoon, 2008; Kistler et al., 2013). Therefore, any strong conclusion based on the example of these two galaxies may be premature. On the other hand, a similar discrepancy was also found in comparing predicted populations of low mass X-ray binaries with observations, based on a larger and different sample of galaxies and age determinations (Zhang, Gilfanov & Bogdán, 2012; Fragos et al., 2013).

Since nuclear-burning white dwarfs are characteristically high temperature sources, the emission from this population will ionize the ISM and will have an important influence on the structure of the ionized gas and emission lines of a galaxy. In particular, compared with the single stellar population, accreting WDs are likely to emit predominantly beyond the He II photoionizing limit (see Fig.1 in Woods & Gilfanov 2013). So in this section, we will investigate the H ionizing and He II ionizing luminosity of accreting, nuclear-burning WDs as a binary population. In addition, we will compute the predicted ratio of two recombination lines produced in the ISM, He II $\lambda$4686Å/H$\beta$, and compare this with observations.

In our calculation, we assumed the binary fraction to be 50%. It has been suggested that the binary fraction may depend on the binary parameters (Kouwenhoven et al., 2009; Kraus & Hillenbrand, 2009; Sana et al., 2012). We neglect this effect, and therefore may slightly underestimate the binary fraction in our calculations. According to its standard definition, the binary fraction includes all binary stars, irrespective of their separation and evolutionary state. Our population synthesis calculations keep track of only those binaries in which one of the components is a mass accreting white dwarf. Such mass-transferring binaries are in fact a very small minority in the population (in terms of the number of systems and total mass). In accounting stars in wide binaries we assume that their emission is similar to the emission of single stars of the same spectral type. To compute emission of single stars and stars in wide binaries we use the stellar population synthesis code of Bruzual & Charlot (2003), as described below. In computing the normalisation of the stellar emission we use the total mass of stars, ignoring the small fraction ($0.7\%$ by mass ) of stars in mass transferring close binaries. Hereafter, we refer to the stars in wide binaries and single stars as single population (SP) and accreting, nuclear burning WDs as binary population (BP). The total emission of the entire population is computed as a sum of emission of the SP and BP. The SP-only model represents the hypothetical case of absence of mass-accreting white dwarfs in the stellar population.

In Fig. 3, we show the evolution of the H-ionizing and He-ionizing luminosities as a function of stellar age in the starburst case. For the binary population, we only show the results for model a025, since we do not find a significant difference between models a025 and a050. To compute emission from the SP, we use the Bruzual & Charlot (2003) stellar population synthesis model. In running their model, we assumed solar metallicity and Chabrier (2003) IMF, as this model is not available for Kroupa IMF. This introduces only small inconsistency with our BP calculations (conducted for the Kroupa IMF). The Chabrier IMF has the same exponent as the Kroupa IMF for stellar masses above 1.0 $M_{\odot}$ and a more complicated shape at smaller masses. If the two IMFs are normalised to the same total mass, their high mass ($>1.0M_{\odot}$) parts have the same shape and only small ($\sim 7\%$) difference in normalisation. This will result in a similarly small difference ($\sim 7\%$) in the predicted ionizing luminosity between the two IMFs, since stars with initial mass below 1.0 $M_{\odot}$ do not contribute to the ionising UV radiation. Such a small difference is insignificant given the large dynamical range of the ionising luminosities plotted in Fig. 3. In Fig. 3, we do not show the case of a constant SFR, as in this case the H-ionizing and He-ionizing emission from accreting white dwarfs are significantly smaller than emission from the SP. This is due to the fact that in the constant SFR case there are many luminous young stars with high effective temperatures, which dominate the ionizing emission.

In the starburst case, the H-ionizing emission from the population of accreting WDs is comparable to that from the SP in the $\sim 0.3-2.0$ Gyr age range; it is unimportant for much earlier ages and for old galaxies. The He-ionizing emission on the contrary is predicted to dominate the ionizing UV background at ages greater than $\sim 0.3$ Gyr. In the SP model, massive stars dominate the ionizing UV radiation at early times. As the galaxy age increases, the main sequence turn-off mass decreases and the effective temperatures of the hottest stars in the population drops. This leads to the decrease of the ionizing luminosity seen in the Fig.3 at $t\la 100$ Myr. Around $t\sim 100$ Myr, the first post-AGB stars appear, resulting in the sharp increase in the ionizing luminosity at $t\sim 100$ Myr, thereafter they dominate the UV emission. Note that in the SP, the He-ionizing luminosity provided by the massive stars drops very steeply and is outside the plotting range in the lower panel of Fig. 3. In the considered time range ($t\ga 30$ Myrs), the He-ionizing luminosity of the SP becomes significant only after the first post-AGB stars appear at $t\sim 100$ Myr. Note also, that the little “bump” around $1$ Gyr, which can be seen on the curves in Fig. 3, is due to the fact that $\sim 2M_{\odot}$ stars undergo a helium flash and their effective temperatures increase.

Woods & Gilfanov (2013, 2014) proposed that the UV emission from accreting WDs is capable of ionizing the ISM. They investigated the contribution of accreting, nuclear-burning WDs in the SD scenario (as SNe Ia progenitors) and pAGB stars to the ionizing background and their influence on the emission lines of warm ISM in early-type galaxies. They predicted that, if the SD scenario for SNe Ia is the dominant channel, SNe Ia progenitors should strongly ionize He, producing strong He recombination lines in the extended emission-line regions of early-type galaxies. Such lines are predicted to be much weaker if only single pAGB stars are considered. Using a similar line of reasoning, one implication of the large population of accreting WDs found in our results across a large range of delay-times is that there should be strong He II recombination lines (e.g. He II 4686$\AA$) detectable in stellar populations with ages larger than 0.3 Gyr but less than $\sim 5$ Gyr.

Among the recombination lines of He II, He II $\lambda$4686 is the strongest line in the optical band, and is not heavily absorbed by the ISM. In order to avoid the uncertainties in the covering factor, we directly compare the predicted ratio of He II $4686\AA /\rm{H}\beta$ with observations. In order to predict the emission line fluxes, we make use of the photoionization code MAPPINGS III (e.g. Kewley et al., 2001; Groves, Dopita & Sutherland, 2004). The procedure is similar to that in Woods & Gilfanov (2013). Here we briefly summarize the main points and assumptions in the calculation.

Serra et al. (2012) found that the common morphology of HI gas in elliptical galaxies is a regular HI disc or ring. The HI discs may extend to several kpc well beyond the nucleus, and may or may not be confined within the stellar body of the galaxy. Given that this gas will be ionized by the combined diffuse emission of many distant sources, we assume plane-parallel geometry in our calculation. We adopt a constant hydrogen density $n_{\rm H}=100~\rm{cm}^{-3}$, which agrees with the observed ratio of [SII] 6716Å/6731Å  in passively-evolving galaxies (Yan & Blanton, 2012). Regarding the metallicities of the gas, which can be estimated from the oxygen abundance, Athey & Bregman (2009) found an average value of $Z_{\rm oxygen}\simeq Z_{\rm oxygen,\odot}$. So we use solar abundances in our calculation. Although the hydrogen density may be varied as the galaxy evolves or the metallicity may be different, the line ratio calculated in this work is relatively insensitive to the density and the metallicity. Following Johansson et al. (2014), we assume that the ionization parameter $\rm{log}(U)\approx-3.5$, where $U=\frac{\dot{N}_{\rm ph}}{4\pi r^{2}n_{\rm H}c}.$

In Fig. 4, we present the computed He II 4686Å/H$\beta$ ratio powered by the emission from the SP together with the accreting WD population, and powered by the emission of the SP alone. At early times, the ratio begins to increase with the decrease in H-ionizing luminosity and increase of the He-ionizing luminosity. The ratio reaches its first maximum when the H-ionizing luminosity reaches its minimum. Around 0.15 Gyr, the H-ionizing luminosity from the SP increases and exceeds the emission from the accreting WDs by an order of magnitude, leading to the beginning of a decrease in this ratio. After that, the H-ionizing luminosity does not change significantly. Thereafter, the evolution of this ratio simply reflects the evolution of the He ionizing luminosity. The luminosity ratio at 10 Gyr is consistent with observations, but the computed ratio is larger than the observed one by a factor up to $\simeq 3$ in the age range $1-8$ Gyr. This comes about due to the predominance of accreting WDs in producing our predicted ionizing background at these delay times. Although the total luminosities of accreting WDs and post-AGB stars in this age range are comparable, the lack of a sharp cutoff at 54.4eV in the spectra of accreting WDs leads to a dramatically greater production of He II-ionizing photons. In particular, the He II-ionizing luminosity predicted in our a025 model exceeds that in the SP case by up to an order of magnitude at t$\sim$ 1Gyr (see Fig. 3).

In Fig. 5, we show the soft X-ray emission from SNBWDs in model a025 and in an additional model in which we assume that upon exceeding the maximum accretion rate for stable burning, accreting WDs enter a CE phase instead of becoming RAWDs (model NORAWD) and WDs cease to contribute to X-ray and UV output of the population. Similar to Fig. 5, we show the H-ionizing luminosity and He-ionizing luminosity in Fig. 6. For the young stellar populations with age $t<1$Gyr, the RAWDs dominate the H and He II ionizing emission. For old stellar populations ($t>1$ Gyr), the H and He ionizing luminosity is dominated by emission from the SNBWDs. This is due to the fact that, at early times, the binaries have massive donor stars and higher thermal timescale mass transfer rates. In contrast, at later times, the donors are less massive and the thermal mass transfer rate is lower, leading to more emission from the SNBWDs. In the model NORAWD, the soft X-ray luminosity, H-ionizing luminosity and He-ionizing luminosity are very different from that in model a025. In Paper I (see Fig. 2), we found that there are two stable burning phases for some accreting WD binaries. In model NORAWD, replacing the RAWD phase with a CE, only the first stably burning phase will contribute to the emission. So the H and He-ionizing luminosity in model NORAWD are comparable to the emission from SNBWDs at early and late times, when massive systems do not make a contribution. In Fig. 7, we show the computed HeII 4686Å/$\rm{H}\beta$ ratio in the model NORAWD and in model a025. The ratio reduces significantly, but it is still larger than the observed ones by a factor of 2. This corresponds to a remaining discrepancy of up to factors of 2 to 5 between our prediction for the He II-ionizing luminosity and that expected from the single stellar population alone (see Fig. 3).

In the section above, we found that there is a strong dependence of soft X-ray luminosity $L_{\rm 0.3-0.7keV}$ and the ratio He II $4686\AA /\rm{H}\beta$ on stellar age for non-star-forming early-type galaxies in our calculation, in contrast with observations (Fig. 2 and Fig. 4). In producing a population synthesis model, there exist several uncertainties in both the initial conditions and the treatment of stellar evolution which may influence the results. In the following, we discuss a (partial) list of several aspects which may significantly impact our model, and their viability in providing a solution. Some of the model parameters not considered here were either tested and found to be of minimal importance (e.g. IMF, mass ratio), or are simply unlikely to be of significant impact (e.g. treatment of convective overshooting, as this only matters for massive stars).

1) One important uncertainty is the retention efficiency of accreted matter, (see Bours, Toonen & Nelemans (2013) ²²2Note, in Yungelson (2010) He-retention efficiency is calculated differently from that quoted by Bours, Toonen & Nelemans (2013).. In our calculation, we assumed that the He burning retention efficiency is 100%, which is clearly an overestimate and results in an excess of massive WDs ³³3 The WD mass may also be overestimated because of the initial-final mass function used in BSE code (see Claeys et al., 2014)., which may be partially responsible for the high H and He ionizing emission of accreting WDs. On the other hand, Piersanti, Tornambé & Yungelson (2014) have shown that the conventional assumption that the retention efficiency of matter at the surface of a WD may be computed as a product of independently calculated retentions of H and He (i.e. $\eta=\eta_{\rm H}\times\eta_{\rm He}$), results in an underestimate of $\eta$. The reason is that H-burning heats the underlying He layer and the outbursts of He-burning are milder than that in the case of accretion of pure He.

In order to check if He retention efficiency has a strong influence on our result, we recomputed our results with the He burning retention efficiency given in Hachisu, Kato & Nomoto (1999), and with the range in mass transfer rates corresponding to the stable burning regime as given in Wolf et al. (2013). In Fig. 8, we compare the soft X-ray luminosity from this new model with that from our standard model, and find the difference is not significant. In spite of this, we should emphasize that the retention efficiency suffers significant uncertainty, requiring further investigation. Therefore, without more precisely understood retention efficiency, we can not completely exclude the influence of this parameter.

2) In our calculations, solar abundance is adopted. As we discussed in Section 4.1, some galaxies in our X-ray comparison are likely to have low metallicities, which plays a complicated and still uncertain role in the evolution of accreting WDs. On the one hand, at very low metallicities the stable-burning regime may be extended to much lower mass transfer rates (Shen & Bildsten, 2007). However, at the same time low-Z WD binaries are expected to have much higher accretion rates (Langer et al., 2000), likely pushing many more binaries into dynamically unstable mass transfer. Either of these mechanisms require very low metallicities ($<$ 0.01$\rm{Z}_{\odot}$), unlikely to be typical of many of the relatively nearby early-type galaxies used in our study. Therefore, we leave consideration of the evolution of WD binaries at low-Z to a future study.

3) The X-ray comparison is based on a sample of six nearby galaxies (the only currently available set of measurements suitable for such a comparison). In addition, there is some controversy between different age estimates for the two youngest galaxies. However, while the incorrect age determination could in principle resolve the discrepancy in the X-ray band, it is unlikely to explain the He II 4686 $\AA /\rm{H}{\beta}$ line ratio, as demonstrated in Johansson et al. (2014).

4) It may be that our modeling of the emission spectra of nuclear burning white dwarfs is incorrect. This must involve a failure of the 1D theory, rather than any simple deviations from the black body spectral template used throughout the paper (see section 2). One obvious possibility is that the photospheric radii may be much larger than predicted by 1D models of the nuclear burning on the WD surface. This would shift the peak of their emission to the UV or optical bands where they would be remarkably unusual objects. However, Lepo & van Kerkwijk (2013) searched for such objects in the Small Magellanic Cloud and found no plausible candidates. Therefore this possibility seems to be unlikely.

5) One of the most important uncertainties in our calculations is the treatment of the CE phase. There are still many aspects of CE evolution which remain unclear, such as the available energy sources and the definition of the core-envelope boundary (see Ivanova et al. (2013) for a review,Hall & Tout (2014)). In addition, the criteria for the onset of a CE phase also suffer from great uncertainty. In the following subsection, we will discuss this possibility in depth and provide a possible solution to resolve the discrepancy between our computed results and observations.

In our calculations, for binaries harbouring giant donors, we adopted after Hjellming & Webbink (1987) and Webbink (1988) as the criterion for stable mass loss a critical value of mass ratio (hereafter, HW criterion):

In Fig. 9, we show the evolution of soft X-ray luminosity as a function of stellar age. Compared with our model using the HW criterion, the new criteria have larger critical mass ratios. With larger critical mass ratios, the predicted soft X-ray luminosity becomes lower. This is due to the fact that, with the requirement of a larger critical mass ratio, it will be relatively difficult for binaries to enter dynamically unstable mass transfer. Therefore, fewer binaries will enter a CE phase, and fewer accreting WDs will form. For old stellar populations, the predicted soft X-ray luminosities in model a025qc17 and model a025qc19 are lower than the observed values. Given that the observational soft X-ray luminosity is the upper limit for the emission of accreting WDs, our results are consistent with observations. Fig. 10 shows the evolution of the H-ionizing and He-ionizing luminosity for different models, respectively. These results are similar to model a025, but the H-ionizing and He-ionizing luminosity is lower, for the same reason as above. In Fig. 11, we present the dependence of line ratio HeII $\lambda 4686/\rm{H}\beta$ on stellar age for the new models. For old stellar populations in model a025qc17 and model a025qc19, the line ratio is much smaller than that in model a025 and consistent with observations.

The use of fixed unique critical mass ratios for the onset of rapid, unstable mass transfer is not realistic (Nelemans et al., 2000; Woods & Ivanova, 2011). Numerical experiments above show that stability criteria of mass loss for giant stars play a crucial role in defining the number of SNBWD. However, stability also depends on mass of the donor, specific momentum of matter lost from the system etc. Fine-tuning of all these parameters may bring even better agreement of model and observations, but it is computer-time prohibitive.

As discussed in Paper I, not all SNBWDs can be detected as SSSs, since this is dependent on the luminosity and effective temperature of the source, and the column density of the gas along the line of sight. In this paper, we define SSSs as SNBWDs with soft X-ray (0.30-0.70keV) luminosity $L_{\rm x}>10^{36}$erg/s. With this definition, we computed the evolution of the number of SSSs, which is shown in Fig. 12. In the starburst case, the SSS population peaks around $\sim 1$ Gyr and then declines by $\sim 2$ orders of magnitude by the age of 10 Gyr. We note that the ionizing radiation from SNBWDs has a peak at the same age range (see the figures above). A similar peak was found in earlier studies of SNe Ia rates in the single degenerate scenario (e.g. Canal, Ruiz-Lapuente & Burkert, 1996; Han & Podsiadlowski, 2004; Ruiter, Belczynski & Fryer, 2009; Yungelson, 2010). The reason for this is that main-sequence stars with mass close to $\sim 2~M_{\odot}$ have lifetimes $\sim 1$ Gyr and radii and luminosities that enable a relatively long stage of thermal time-scale mass loss with the rate $\sim 10^{-6}M_{\odot}$/yr, corresponding to the stable hydrogen burning regime (see Figs. 2 and 10 in Paper I)⁴⁴4If mass loss is artificially fine-tuned, e.g., by the highly uncertain effect of mass-stripping effect of donors by an accretor wind (Hachisu, Kato & Nomoto, 1999), the peak may be shifted to younger ages, e.g., Mennekens et al. (2010). The peak is smeared by the scatter in the actual parameters of WD+MS pairs - masses, mass ratios of components, and initial separations.. The number of SSSs for a $10^{11}M_{\odot}$ galaxy at 10 Gyr is $10-20$ in the starburst case and $200-820$ in the constant SFR case assuming $N_{\rm H}=3\times 10^{20}\rm cm^{-2}$. Given the high intrinsic absorption of spiral galaxies, we also made a set of calculations assuming $N_{\rm H}=3\times 10^{21}\rm cm^{-2}$. In this case, the SSS number is $40-130$ for a spiral-like galaxy. Note, however, that the observable number of SSSs depends on the absorption column density, which in turn depends on the relative distribution of gas and SSSs in any given galaxy.

Given the hard emission of SSSs, it is suggested that they should be accompanied by ionized nebulae. Rappaport et al. (1994) modeled the ionization and temperature structure of such nebulae and predicted strong [OIII] $\lambda$5007 and He II $\lambda$4686 emission lines. Remillard, Rappaport & Macri (1995) searched for gaseous nebulae surrounding SSSs in the Large and Small Magellanic Clouds. They found only one SSS, Cal 83 (first identified as a high-excitation nebula by Pakull & Motch, 1989), with a detected nebula, with null detections among the 9 other known Magellanic SSSs. This means that either the time-averaged luminosity and/or temperature of these sources must be much lower than presently observed, or that the local ISM density around SSSs are typically too low. In fact, given that typical ISM densities are 1 – 2 orders of magnitude below that observed for the Cal 83 nebula, the latter appears more likely (see Woods & Gilfanov (submitted), who found a 70% probability that $\lesssim 1$ SSS in the LMC would have a detectable nebula). The situation is complicated further in the accretion-wind scenario – for SSSs in star-forming galaxies, a wind-blown bubble will excavate a cavity in the ISM surrounding the source which is 10 – 40 pc in radius (Badenes et al., 2007). Consequently, the ionization parameter and morphology of these nebulae will be much different.

Since the common morphology of HI gas in elliptical galaxies is a regular HI disk or ring, individual sources are not embedded in the ISM they ionize. With the above explanation and HI gas morphology in mind, we would expect that the individual emission line luminosity in elliptical galaxies should not be influenced by any complications introduced by the impact of the accreting WD on any surrounding ISM, as is the case in star-forming galaxies.