Populations of super-soft X-ray sources in galaxies of different morphological types

We study populations of soft and super-soft X-ray sources in nearby galaxies of various morphological types with the special emphasis on characterizing populations of stable nuclear burning accreting WDs. Analysing the content of Chandra archive we assembled a sample of nearby galaxies suitable for studying populations of super-soft X-ray sources. Our sample includes 4 spiral galaxies, 2 lenticular galaxies and 3 ellipticals with stellar mass exceeding $10^{10}$ $M_\odot$ and X-ray sensitivity of the order of a ${\rm few}\times 10^{36}$ erg/s. We used combination of hardness ratio and median energy to pre-select X-ray sources with soft spectra, and temperature - X-ray luminosity diagram to identify super-soft X-ray sources - likely nuclear burning accreting white dwarfs. For spiral galaxies, there is a distinct and rare population of super-soft sources, largely detached from the rest of sources on the $kT_{bb}-L_X$ plane. The boundary between these sources and the much more numerous population of harder (but still soft) sources is consistent with the boundary of stable hydrogen burning on the white dwarf surface. Combined spectrum of soft sources located outside this boundary, shows clear emission lines of Mg and S, which equivalent width is similar to that in the combined spectrum of a large number of confirmed supernova remnants in M83. This confirms earlier suggestions that the vast majority of so called quasi-soft sources are supernova remnants. In early-type galaxies, populations of super-soft sources are about a factor of $\approx 8$ less abundant, in broad agreement with the population synthesis calculations. Specific frequencies of super-soft sources are: (2.08$\pm$0.46)$\times10^{-10}$ M$_{\odot}^{-1}$ in spiral galaxies and (2.47$\pm$1.34)$\times10^{-11}$ M$_{\odot}^{-1}$ in lenticular and elliptical galaxies, with the ratio of the latter to the former of $0.12\pm0.05$.


Introduction
CAL 83 and CAL 87 -the two prototypical super-soft Xray sources were discovered by Einstein observatory in the course of a survey of the Large Magellanic Clouds (Long et al. 1981). About a decade later, the first light observations of central region of LMC with ROSAT observatory led to a discovery of a similar source with a very soft spectrum, RX J0527.8-6954 (Trümper et al. 1991). With advent of Chandra and XMM-Newton, such sources were found in other, more distant, galaxies. Spectra of these sources are very soft and do not extend beyond ∼ 1 − 2 keV, their approximation with the black body model giving temperatures of the order of ∼ 10 − 100 eV and bolometric luminosities in the ∼ 10 36 − 10 38 erg/s range (Greiner et al. 1991;Kahabka et al. 1994). It is the extreme softness of their spectra that gave rise to their name "super-soft Xray sources" (SSS). Binary nature of some of these sources (CAL 83 and CAL 87 in the first place) was determined soon after their discovery (Pakull et al. 1988;Smale et al. 1988;Cowley et al. 1990) and, although they were initially thought to be low-mass X-ray binaries, it was soon realised that the compact object in these binaries is a white dwarf rather than a neutron star or a black hole (van den Heuvel et al. 1992). The proposition that the main source of energy in super-soft X-ray sources is thermonuclear burning of the accreted hydrogen naturally explained the unusual softness of their X-ray spectra, the combination of the energy output expected in hydrogen fusion and the surface area of a typical WD giving the effective temperatures in the correct range, close to the observed values (van den Heuvel et al. 1992).
Details of thermonuclear burning on the white dwarf surface are still a matter of debate. Calculations by Nomoto et al. (2007); Wolf et al. (2013) (and references therein) find that hydrogen fusion may proceed in a steady manner above some value of the mass accretion rate, in the so called stability strip stretching aroundṀ ∼ few × 10 −7 M yr −1 , its exact location depending on the WD mass. In this regime classical steady SSS are observed, while at lower mass accretion rates hydrogen fusion on the WD surface is unstable, giving rise to the explosions of Classical and recurrent novae. On the other hand, calculations by Prialnik & Kovetz (1995); Yaron et al. (2005); Starrfield et al. (2013), suggest that nuclear burning is unstable at any mass accretion rate. Although the nova explosion itself is primarily observed in the optical band, nuclear burning of the residual hydrogen on the WD surface after the explosion generates soft X-ray emission and the system is observed as a post-nova super-Article number, page 1 of 22 arXiv:2012.01344v1 [astro-ph.HE] 2 Dec 2020 A&A proofs: manuscript no. Galiullin Gilfanov soft X-ray source (Truran & Glasner 1995;Wolf et al. 2013;Soraisam et al. 2016). Thus, existence of super-soft X-ray sources, steady or not, is envisaged in either scenario.
Although accreting WDs are interesting objects on their own right, they became of broader astrophysical importance as possible progenitors of type Ia supernovae (e.g. Whelan & Iben 1973;Nomoto 1982). However, before exploding as a supernovae, a white dwarf needs to increase its mass from the initial value, typically in the ∼ 0.5 − 1.0 M range, to near the Chandrasekhar mass limit of about ∼ 1.4 M . For an accreting white dwarf, the only way to increase its mass is hydrogen and/or helium fusion on its surface. The mass growth is most efficient when nuclear burning is stable, as in all other regimes it is counteracted by various mass loss processes. However, growth of the white dwarf mass is also possible in certain domains of the parameter space, when the burning is unstable (Starrfield et al. 2013;Hillman et al. 2015). Nevertheless, it was demonstrated that the number of super-soft X-ray sources (Di Stefano 2010) and their total luminosity in the X-ray (Gilfanov & Bogdán 2010;Bogdán & Gilfanov 2011) and UV (Woods & Gilfanov 2013;Johansson et al. 2014; bands in nearby galaxies is by far insufficient to explain the observed SN Ia rates.
There is still no complete understanding of how population of nuclear burning accreting white dwarfs is related to the fundamental properties of their host galaxies, such as their stellar mass, star-formation rate and star-formation history and metallicity. On the theoretical side, there have been several attempts to address this problem by means of the population synthesis calculations (e.g. Yungelson & Livio 1998;Yungelson 2010). Recently, Chen et al. (2014Chen et al. ( , 2015Chen et al. ( , 2016 implemented a hybrid population synthesis calculations using the MESA code (Paxton et al. 2011 to compute the mass transfer phase of the binary evolution. They have produced a model which was in a reasonable agreement with a number of existing constrains on the soft X-ray luminosities and average intensity of ionising radiation field, as well as with statistics of classical novae in nearby galaxies. Observational studies of SSS populations are often impeded by the extreme softness of their spectra, making their search and characterisation difficult. Nevertheless, there have been a number of successful attempts, which intensified with the advent of Chandra and XMM-Newton (e.g. Swartz et al. 2002;Di Stefano & Kong 2003Pietsch et al. 2005;Henze et al. 2010Henze et al. , 2011Henze et al. , 2014. It was suggested that observed number of SSS on various types of galaxies are different (e.g. Di Stefano 2010). Bogdán & Gilfanov (2011) showed that specific frequencies of SSS with L X > 10 36 erg/s in disks of late-type galaxies by factor of ∼2 exceed that of bulges.
In this paper we continue the observational line of investigation of populations of accreting white dwarfs. To this end, we attempted to construct a suitable sample of galaxies from Chandra archive for search SSSs and implemented efficient source classification algorithms. Our goal is to identify steady nuclear burning white dwarfs and to estimate their specific frequencies in early-and late-type galaxies.
The paper is structured as follows: In Section 2 we introduce our sample of nearby galaxies observed by Chandra. Observations and data reduction procedures are described in Section 3. In Section 4 we introduce the initial source classification procedure based on combination of median energy and hardness ratio and in Section 5 we investigate properties of soft X-ray sources and separate nuclear burning white dwarfs from the rest of source population with the help of the temperature-luminosity diagram. The nature of soft and super-soft X-ray sources is described in Section 6 where we also estimate their specific frequencies and compare our results with previous work, and in Section 7 we summarise our findings.

A sample of nearby galaxies
Galaxies for the analysis were selected from Chandra observatory archive for cycles 1-20. To this end we selected observations from the following categories: normal galaxies, active galaxies and quasars, clusters of galaxies, galactic diffuse emission and surveys. Only archival observations with ACIS detector without grating were selected. For each galaxy, we grouped observations made within same Chandra cycle and selected the group having largest total exposure time. We then cross-matched aim-point positions of selected Chandra observations with positions of galaxies in RC3 catalogue (de Vaucouleurs et al. 1991) to select potentially interesting observations. For each galaxy, we used the match radius of 0.5×D 25 . Parameters of galaxies required for the further selection were computed as follows. Stellar masses were computed from K s band luminosities using mass-to-light ratios computed following the prescription of Bell & de Jong (2001). Classification of galaxies, distances and K s -band magnitudes were taken from HyperLeda 1 catalogue (Makarov et al. 2014) and Updated Nearby Galaxy Catalog (Karachentsev et al. 2013). Absorption corrected (B-V) 0 colours were taken from RC3 catalogue.

Selection criteria
We built our selection procedure based on the following considerations.
i) Stellar mass of the host galaxy. Population synthesis calculations of Chen et al. (2014Chen et al. ( , 2015 predict specific frequencies of observable (apparent luminosity L X > 10 36 erg/s) stable burning accreting WDs in spiral galaxies after 10 Gyrs of evolution at the level of ∼ 4 × 10 −10 (∼ 2 × 10 −9 ) sources per M , assuming absorbing column density of 3 × 10 21 (3 × 10 20 ) cm −2 . For elliptical galaxies their prediction is ∼ 10 −10 sources per M for hydrogen column density 3 × 10 20 cm −2 . Based on these predictions we will be only considering sufficiently massive galaxies with stellar masses M * > ∼ 10 10 M , in which we can expect to find > ∼ 1 nuclear burning accreting WD.
ii) Distance. We excluded from our analysis a few large nearby galaxies which do not fit, by a large margin, in the Chandra field of view and require analysing mosaics of a large number of observations. These are: Magellanic Clouds, M31 and M33. We should also exclude too distant galaxies which will suffer from confusion. However, they are automatically excluded based on the sensitivity arguments (item iv below) iii) Inclination angle. Source populations in spiral galaxies which are viewed close to edge-on will be obscured by large intrinsic absorption by gas and dust in the disk, essentially reducing the numbers of expected super-soft sources to zero. Therefore in selecting late-type galaxies we applied  Fig. 1. The X-ray sensitivity-distance diagram of nearby galaxies observed by Chandra up to cycle 20. Each panel corresponds to a certain morphological type of galaxies. Filled (open) symbols correspond to galaxies with stellar mass M * ≥ 10 10 M (M * < 10 10 M ). See Section 2 for details. a criterion i < 70 degrees. For this reason, for example, Sculptor and NGC 4945 were not included in our final sample. No inclination cut was applied to early-type galaxies. iv) Sensitivity limit of available archival data. The sensitivity limit of archival data can be estimated as follows: where D is distance to the galaxy, ECF is counts-to-ergs conversion factor, S is the limiting number of source counts required to detect and characterise a source, t is total exposure time of archival data. We assumed black body model with temperature 60 eV and require S = 20 counts to detect and classify a source. For each galaxy we obtained ECF from black body model (kT = 60 eV) modified by the Galactic absorption column density in the direction of the galaxy. For the latter we used the HI4PI data (HI4PI Collaboration et al. 2016). For cycles 7-20, in computing ECF, we used the ACIS detector aim-point response files from Chandra Cali-bration Database 2 . For cycles 1-6 we generated aim-point response files ourselves. Circinus (5.3 × 10 21 cm −2 ) and IC 342 (3.6×10 21 cm −2 ) galaxies with high Galactic absorption column densities were excluded from our sample. We also excluded M87 and M82 due to the presence of bright and complex diffuse emission in these galaxies. We plot the distribution of galaxies of various morphological types on the distance -sensitivity limit plane in Fig.1. Based on these data we selected spiral galaxies M51, M81, M83, M101, lenticular galaxies Cen A and NGC3115 and elliptical galaxies NGC 3379, 4278 and 4697. For elliptical galaxies, sensitivity limits of the archival Chandra data exceed 10 37 erg/s, this is however compensated by zero intrinsic absorption in these galaxies, see Section 5.2.
We experimented with the selection criteria in order to check if our sample can be improved by adding a few galaxies lying on the boundary of our luminosity and stellar mass selection. We found, however, that relaxing the sensitiv-A&A proofs: manuscript no. Galiullin Gilfanov (Peletier et al. 1990). (5) -The total colour indexes corrected for galactic and internal extinction from RC3 catalogue. (6) -Stellar mass within D 25 of the galaxy obtained by mass-to-light ratio for K s band with colour index (B-V) 0 from RC3 using calibration from Bell & de Jong (2001). Total K s magnitudes were calculated using data from 2MASS Large Galaxy Atlas (Jarrett et al. 2003). Absolute K s magnitude of the Sun was assumed to be K s, =3.27 (Willmer 2018 ity threshold we need to raise the luminosity cut on the kT bb − L X plane (Section 5.2) which actually leads to the loss of super-soft sources of lower luminosity and degradation of the statistical accuracy of our analysis. Adding less massive galaxies, below the assumed cut of 10 10 M , on the other hand, does not lead to the quick improvement of the sensitivity to the populations of SSS as the total mass already included in the analysis exceeds ≈ 4 × 10 11 M . We therefore conclude that the list of galaxies assembled here is reasonably optimal for the current analysis. Basic parameters of these galaxies are summarised in Table 1.

Observations and Data Reduction
The observations were processed by following a standard CIAO (Fruscione et al. 2006) threads 3 (CIAO version 4.8 and CALDB version 4.7.1). We used tool chandra repro to reprocess data and create level 2 event file. To improve absolute astrometry we used wcs match and wcs update. Astrometry correction was done relative to the observation with the longest exposure time using positions of bright Xray sources to match frames. Event files have been merged by reproject obs with aim point position of observation with the longest exposure taken as the reference. Combined images and exposure maps were obtained running flux obs. In Table 2 we present short information about observations 3 http://cxc.harvard.edu/ciao/index.html used in analysis. Detail information available in Appendix A.

Source detection
Source detection was performed with wavdetect tool. We chose to use small √ 2 scale series with power of 0-3 of the wavelet functions to minimize the spurious detections associated with soft diffuse emission in galaxies. Scales of wavelets correspond to variation of the point spread function (PSF) within an aim point of single ACIS chip. We set up following parameters of wavdetect to create a background map from merged image: bkgsigthresh=0.01, maxiter = 10, iterstop = 10 −6 . The false source detection probability threshold was chosen at 10 −7 . Mask filtering were applied to all images of galaxies using exposure maps. Pixels were included to the source detection region where exposure map values were greater than 20% of the maximum (expthresh =0.2). False colour X-ray images of all galaxies with their D 25 and source detection regions are presented in Appendix B.
To maximise sensitivity of source detection, we used combination of the following energy bands: 0.3-1 keV; 1-2 keV; 0.3-2 keV; 0.5-8 keV; 2-8 keV bands. We ran wavdetect in each of these bands and merged their results into a single source list, excluding duplications. To this end we used STILTS package (Taylor 2006), with the match radius equal to the sum of their PSFs radii defined at the I. Galiullin and M. Gilfanov: Populations of super-soft X-ray sources in galaxies of different morphological types  80% encircled fraction. In the final merged list we chose coordinates of sources in the band where the significance of detection was maximal. In the very inner regions of some galaxies wavdetect found multiple overlapping sources. Visual inspection of the images showed that many of these sources are likely false detections caused by presence of substructures in the bright and complex diffuse emission of the nucleus. In many cases it appears impossible to separate true compact sources from such false detections. For this reason we decided to exclude in M51, M83 and Centaurus A galaxies their central regions of radius of 15 . In the case of Centaurus A we also excluded regions around bright Xray jets similar to Voss & Gilfanov (2006).
To define source and background counts extraction regions we produced PSF maps. In order to account for PSF variations across the combined image, we created PSF map for each observation with mkpsfmap tool using encircled energy of 80%, and combined individual maps weighting them with respective exposure maps. Combined source spectra used in the further analysis were extracted using CIAO tool specextract. For source regions we used circles with the radius equal to the 80% PSFs radius from the exposureweighted PSF map. The background regions were defined as annuli with the inner and outer radii equal to two and four times the PSFs radius respectively. The source count rates were corrected for the source counts leakage to the background region. When source or background regions of different sources overlapped, we excluded overlapping parts from the counts summation, correcting the PSF fractions accordingly.
The lists of detected sources were analysed to identify spurious detection due to fluctuations of local background. For each sources we tested the null hypothesis that observed counts in source and background regions are Poissonian realizations of local background with the same mean (Weisskopf et al. 2007, see Appendix A). We estimated probabilities using source and background counts from merged images in 0.3-2 keV band. All detections of statistical confidence of less than 3σ were excluded from the final source list. Additionally, an eye inspection of X-ray images has not revealed any potential spurious detections. In Table 2 we summarise numbers of X-ray sources detected in galaxies of our sample.

Stellar masses of the final sample of galaxies
Stellar masses of galaxies within D 25 and within the source detection regions listed in Tables 1 and 2 were determined following prescription of Zhang et al. (2012) as summarised below. We used background subtracted images from 2MASS Large Galaxy Atlas (Jarrett et al. 2003). Bright sources were visually removed from the images. Net counts of the images S were converted into magnitude using zero-point magnitude KMAGZP from a headers of fits files.
We obtained K s band absolute magnitudes using distances listed in Table 1. In computing luminosities, Solar absolute magnitude in K s band was assumed to be K s, =3.27 (Willmer 2018). Final stellar masses of the galaxies were obtained using K s band mass-to-light ratio (Bell & de Jong 2001).
where L K is K s band luminosity and (B-V) 0 is colour index corrected for galactic and internal extinction (see Table 1).

The method
To classify and to separate soft and super-soft sources from sources with more usual spectra we used combination of hardness ratio HR and median energyẼ. The hardness ratio was defined as a ratio of the hard band count rate to the total broad band count rate: where R is the background subtracted count rate in the respective energy band. Statistical error of the hardness ratio was calculated using standard error propagation. Median energyẼ for each source was calculated from its observed counts spectrum considering the 0.3-8.0 keV energy band. The statistical error was computed using the standard formula for the variance of the sample median (Rider 1960) where the probability density at the median energy was approximately determined from observed counts distribution. In computing the median energy we chose not to subtract background from the source spectrum. Simulations have demonstrated that for the typical source and background spectra we are dealing with, this does not affect our source classification procedure (see Fig. 2 and its discussion below).
The motivation behind our choice of the parameters used for source classification is fairly obvious. The median energy characterizes overall softness of the spectrum. The Fig.2 (left panel) shows median energies of absorbed black body and power law models. Temperatures and photon index axes are located on the right and on the left side of Fig.2 (left panel). As one can see from the plot, in the parameter range relevant to nuclear-burning accreting white dwarfs (kT < ∼ 200 eV) and X-ray binaries/AGN/stars (Γ < ∼ 2), the two types of sources occupy different domains along theẼ axis with some gap in between 4 . Therefore one may expect a bi-modal distribution of sources overẼ (which may however be smeared by statistical uncertainties).
We used M51 to illustrate and verify classification of sources using their median energy (see corresponding panels in Fig.3). To this end we fit background subtracted spectra of all sources havingẼ ≤ 2 keV 5 , with an absorbed power law model. The absorption column density for each source was fixed as described in Section 5.1 below. Distributions of soft and hard X-ray sources over photon index are shown in the right panel of Fig.2. From these distributions it is clear, that indeed, the sources classified as soft and hard occupy different domains along the photon index axis. Small overlap of the two distributions is caused by statistical uncertainties in measuring the median energy and photon index.
This comparison shows that a classification scheme based on the median energy is efficient in identifying sources with soft spectra. The hardness ratio, on the other hand, characterizes the presence of source emission above 2 keV and helps to identify sources having lowẼ but also featuring hard emission components extending to higher energies. Such hard spectral components are not typical for nuclear burning white dwarfs and may be encountered, for example, in accreting black holes in the soft state and in some of supernova remnants.
Finally, we note that the median energy is not efficient in separating the bulk of SNRs from white dwarfs. This can be achieved with more detailed spectral fitting, which however produces meaningful results only for the brightest sources.
Use of the median energy and hardness ratio for characterization of the spectral shape is not entirely new in X-ray astronomy, for example Hong et al. (2004) proposed to use various quantiles and Prestwich et al. (2003) used colorcolor diagram for this purposes. Di Stefano & Kong (2003) used hardness ratio to classify super-soft X-ray sources in external galaxies. Color-color diagrams and hardness ratio techniques were used in numerous other work not mentioned here. However, to our knowledge, combination of the median energy and hardness ratio was employed for separating super-soft sources from the bulk of the population for the first time in this paper.

Distribution of sources in HR −Ẽ plane
The distribution of sources on the hardness ratio -median energy plane is shown in Fig.3. In producing these plots we applied to all data sets cuts of 20 net counts in order to exclude sources with too large uncertainties. For each galaxy we also plot distribution of sources over the median energy.

Late-type galaxies
Distributions of the sources over median energy in M51, M83 and M101 are clearly bimodal. Their HR-Ẽ diagrams have a distinct and relatively compact populations of soft sources atẼ < ∼ 1 keV and HR ∼ 0 along with the more extended tail of sources with more usual harder spectra. Based on their distributions over the median energy we chose the boundary ofẼ 0 = 1.2 keV to select soft sources in these galaxies. Hard sources were excluded from further consideration. We also excluded from further analysis sources having statistically significant flux (≥ 3σ) detected above 2 keV. Total numbers of so selected soft sources are presented in Table 2.
The median energy distribution of sources in M81 shows a single peak atẼ 0 ≈ 1.5 keV. with a possible shoulder towards lower values ofẼ. Lack of bi-modality may be a consequence of the star-formation history of this galaxy as discusses in section 6.4. Due to absence of obvious bimodality of the source distribution we applied selection criterion based on the absence of hard flux. In particular, we excluded from further analysis sources, having statistically significant flux (≥ 3σ) detected above 2 keV. Their numbers are listed in Table 2 .

Early-type galaxies
Distributions of sources over median energy for elliptical and lenticular galaxies is also uni-modal with the low energy shoulder present for some galaxies. Similar to M81, we kept only sources without statistically significant flux above 2 keV.

Equivalent hydrogen column density
Due to the softness of sources under consideration, the value of equivalent hydrogen column density N H is important for accurate spectral analysis. This problem is of relevance for late-type galaxies.
Because of degeneracy with other parameters N H is difficult to determine from fitting X-ray spectra, especially for fainter sources with smaller number of counts, which constitute the majority of our sample. Therefore we chose to fix N H at values determined in radio band. To this end we combined CO 2-1 and 21 cm maps from the HI Nearby Galaxy Survey (THINGS, Walter et al. 2008) and the HERA CO-Line Extragalactic Survey (HERACLES, Leroy et al. 2009). For each source we computed absorption column density as where I HI is velocity-integrated intensities of THINGS maps in units of Jy beam −1 km s −1 . FWHM maj and FWHM min are the major and minor axes of the beam in arsec (Walter et al. 2008, see Eq. 5). For N H 2 we used: where W CO is HERACLES velocity-integrated CO J = 2 → 1 map intensity in units of K km s −1 , X CO is the CO J = 1 → 0 to H 2 conversion factor and R 21 is CO J = 2 → 1 to CO J = 1 → 0 ratio. We assumed Galactic value of X CO = 2×10 20 cm −2 (K km s −1 ) −1 (Strong & Mattox 1996;Dame et al. 2001) for majority of galaxies and R 21 =0.8 (see Leroy et al. 2009, Eq.3 and references therein). The X CO factor is known to vary between galaxies, in particular for M51 it is ∼ 4 times smaller than the Galactic value (Garcia-Burillo et al. 1993;Guelin et al. 1995). For M81, no CO maps of sufficient quality are available and we accounted only for absorption by neutral hydrogen. For M83 we used CO 1-0 intensity map from NED 6 where galaxy was observed by Crosthwaite et al. (2002) with NRAO 12m telescope. Table 3 shows median values of absorption column densities N H of soft sources in late-type of galaxies for two cases: f = 0.5 and f = 1. We adopted f = 0.5 i.e. that sources are subjected to a half of the absorption intrinsic to galaxy. Naively, this corresponds to the assumption that all sources are located close to the mid-plane of the host galaxy.
We analysed spectra of bright sources with ≥200 net counts and compared results of X-ray spectral fits with fixed N H with spectral fits where N H was a free parameter of the fit. The results are shown in Fig.4. As one can see, although  (1) -Galaxy name. (2) and (3)  there is a considerable scatter in the N H values, values of temperature and, especially, X-ray luminosity are in a reasonable overall agreement. Unfortunately, such analysis is possible only for a small fraction of sources. We therefore chose to use fixed absorption column density, computed, for each source, from Eq.5. Elliptical and lenticular galaxies in our sample have much lower intrinsic neutral hydrogen column density, below ∼ few × 10 19 cm −2 (Serra et al. 2012). We therefore can safely assume N H = N H,Gal for early-type galaxies in our sample, with a caveat regarding Centaurus A. In the dust lanes of this galaxy, absorption column density is at the nearly constant level of ≈ 5 × 10 21 cm −2 and drops rather sharply outside the dust lanes (Struve et al. 2010). Therefore, for the sources located in the dust lanes (about ∼ 1/3 of all soft sources in Centaurus A) we accounted for the intrinsic absorption in the same way as it was done for spiral galaxies, but assuming a constant value of intrinsic absorption of 5 × 10 21 cm −2 , while for the sources outside dust lanes we used Galactic value. We found that particu-Article number, page 9 of 22 lar details of this calculation are unimportant. It should be also noted that Centaurus A galaxy also stands out in the early-type galaxy sample due to its ongoing star-formation on the level of ∼ 0.8 M /yr (Espada et al. 2019). It will be further discussed in Section 6.2.

X-ray spectral analysis and kT bb − L X diagram
Spectra of X-ray sources, classified as soft, were approximated with an absorbed black body model, with the absorption column density being fixed at the values computed as described in Section 5.1. For spectra with small number of counts absorption correction of X-ray luminosity often bears large uncertainties and may lead to physically meaningless values. Therefore we chose to use absorbed Xray luminosities of sources in further analysis and apply absorption to models, when comparing them to the data. We used Tuebingen-Boulder ISM absorption model (Wilms et al. 2000) (tbvarabs model in XSPEC). Abundances of late-type galaxies were fixed at values listed on Table 3. For early-type galaxies Wilms et al. (2000) solar elemental abundance was assumed. The spectral fit was performed with XSPEC (Arnaud 1996) in the 0.3-8 keV energy band using C -statistics (Cash 1979). Background spectrum was included in the spectral fitting. Spectra were binned to have at least one count per bin, following recommendations of the XSPEC manual 7 . Confidence intervals for the parameters were obtained using error command in XSPEC. Distribution of sources on the X-ray luminosity -temperature (hereafter kT bb − L X ) plane is shown in Fig. 5. In plotting kT bb −L X diagram we applied a cut of 20 net counts, in order to exclude faint sources with too large error bars. The X-ray luminosities were computed for the 0.3-2 keV energy band. The thin solid line near the bottom of each panel shows X-ray luminosity corresponding to 20 net counts, as a function of the black body temperature. In plotting these curves we converted counts to rate using the maximal exposure time of the observation. The absorbed X-ray luminos-ity was computed using count rate -flux conversion for the black body spectrum with the given temperature and absorption column density. The luminosity corresponding to 20 counts exceeds by a factor of ∼3 luminosities of faintest sources, therefore samples of sources above these curves should not be subject to significant incompleteness effects. 8 On each panel, the blue dashed line shows the luminosity limit for M51 data corrected for the difference between absorption column densities in M51 and the galaxy under consideration. The correction factor is close to unity at temperatures exceeding a few hundred eV. However, it may be considerable at low temperatures, for soft spectra. These lines permit to compare sensitivities of different datasets independently of the absorption. In particular, this comparison demonstrates that despite the rather significant scatter in Chandra exposure and distances to galaxies, selected galaxies have comparable detection sensitivity to sources with soft spectra.
In Fig.5, solid red curves show the lower bound of the stability strip for the nuclear burning of hydrogen on the WD surface (Wolf et al. 2013). In plotting these curves, we used the Stefan-Boltzmann law to compute the effective temperature and the mass-radius relation for zerotemperature white dwarfs from Panei et al. (2000). The WD mass changes along the curve from 0.51M to 1.34M . We then used the black body model in XSPEC, apply ISM absorption and compute 0.3-2 keV band luminosity, corresponding to the given WD mass and effective temperature. For late-type galaxies, absorption column densities were fixed at corresponding median values from Table 3 while for early-type galaxies we used the Galactic value N H,Gal , as explained above. For Centaurus A we also show location of this boundary for sources located in the dust lanes. Green lines in Fig.5 show tracks of the spectral model of relativistic accretion disk around a Schwarzschild black hole (Ebisawa et al. 1991). Along these lines, the mass accretion rate changes from 0.02Ṁ Edd toṀ Edd , different lines correspond to the different black hole mass as marked by the number near the curves. To compute these tracks, we approximated the accretion disk emission spectrum by an absorbed black body model. The absorption was applied similar to other models plotted in the figure.
All sources plotted in Fig.5 have no significant (≥ 3σ) flux above 2 keV. For three late-type galaxies with bi-modal median energy distributions (M83, M51 and M101) we plot only sources with median energyẼ < ∼ 1.2 keV. These sources show rather compact distributions in the kT bb − L X plane with temperatures 20∼300 eV and X-ray luminosities in 10 36 ∼10 38 ergs/s range. For other galaxies, we only used the criterion of lack of statistically significant (≥ 3σ) emission above 2 keV. For these galaxies, distributions of sources extend significantly towards higher temperatures (cross-hair symbols in Fig.5), which is not surprising as such a selection includes many sources with regular spectra. These sources are dominated by X-ray binaries and background AGN and their discussion is beyond the scope of this paper. Sources with low median energyẼ < ∼ 1.2 keV (solid orange symbols) tend to have lower temperatures than the bulk of sources in these galaxies and will be discussed in the following section, along with soft sources in M83, M51 and M101.

Foreground stars
To identify foreground stars we cross-correlated the lists of soft X-ray sources with the GAIA (Gaia Collaboration et al. 2016) catalog, Data Release 2 (Gaia Collaboration et al. 2018). With the search radii of 2 we typically found ∼2 matches per galaxy except for Centaurus A galaxy where we found 12 matches. Large number of matches for Centaurus A is caused by its relative proximity to the Galactic plane. Some of the GAIA matches have statistically significant measurements of parallax or proper motion, with S/N>3. These sources are classified as stars in the Galaxy and excluded from further analysis. We mark them with open diamonds in Fig. 5.

Nature of soft and super-soft X-ray sources
Based on first Chandra studies of nearby galaxies it was proposed to divide soft sources in external galaxies into supersoft and quasi-soft (Di Stefano & Kong 2003. The former are characterised by color temperatures of kT < ∼ 100 eV and are believed to be mostly associated with accreting white dwarfs with nuclear burning of hydrogen on their surface, while the latter have temperatures in the range kT ∼ 100 − 300 eV and, possibly, are unrelated to white dwarfs.

Super-soft X-ray sources and accreting white dwarfs
In the commonly used albeit (over)simplified picture, it is assumed that the spectral energy distribution of nuclear burning WDs can be roughly described by a black body spectrum (cf. Woods & Gilfanov 2013) with the color temperature close to the effective temperature. Observations of well-known super-soft X-rays sources in the Milky Way, Magellanic Clouds (e.g. Greiner 1996;Kahabka & Trumper 1996) seem to broadly support this assumption. The effective temperature of an accreting white dwarf powered by nuclear burning of hydrogen on its surface is given by: where H = 6 × 10 18 erg/g is the energy release in hydrogen fusion per unit mass of hydrogen, X H ≈ 0.72 is hydrogen mass fraction of the accreting material,Ṁ is the mass accretion rate and R phot is the photospheric radius of the hydrogen burning layer. Numerical calculations show that at the bottom of the stability strip R phot ∼ R WD and photospheric radius increases with increase ofṀ (Wolf et al. 2013). The red solid line in Fig.5 shows the temperatureluminosity relation computed at the lower bound of the stability strip and assuming R phot = R WD . For the definition of the (white dwarf mass dependent) lower boundary of stable nuclear burning we used results of Wolf et al. (2013) and for the WD radius we used zero temperature radius (depending on the WD mass) from Panei et al. (2000). The region on the kT bb − L X diagramm above and to the left of this line is permitted for accreting white dwarfs stably burning hydrogen on their surface, while in the part of the diagram to the bottom-right stable nuclear burning of hydrogen is not possible and it should be populated with other types of soft X-ray sources. Indeed, in Fig.5 one can see that in latetype galaxies (M83, M51, M101 and M81), there is a distinct group of rare soft and luminous sources located above these line. These sources have temperatures of ∼several tens of eV and seem to clearly stand out in the populations of soft sources in these galaxies. In interpreting Fig.5 one should bear in mind that it shows absorbed luminosities in 0.3−2.0 keV band and that absorption and bolometric corrections increase ∼exponentially with decrease of the color temperature, therefore the softest sources are truly luminous.
Thus, we classify sources to the left of the red line in Fig.5 as classical super-soft X-ray sources. We also include those sources which are located outside the stable nuclear burning boundary but their error bars cross this boundary. In addition, we added to this list two sources in M101, located outside but close to the stable nuclear burning boundary (the two sources in Fig.5 upper right panel with kT > ∼ 100 eV and L X > ∼ 10 37 erg/s.). The catalog of super-soft X-ray sources in presented in Table C.1, and their numbers in galaxies from our sample are listed in Table 2.

Specific frequency of super-soft X-ray sources across morphological types
From Fig.5, it is evident that super-soft X-ray sources are considerably more abundant in late-type galaxies than in early-type galaxies. However, for a statistically sound conclusion one needs to take into consideration completeness limits of various data sets. To this end we used the sensitivity limit for M51 galaxy, corrected for the difference in the absorption column density as described in Section 5.2, to select super-soft X-ray sources. In each galaxy, we selected sources to the left of the thick red line and above the dashed blue line in Fig. 5. This selection is within the 20 counts flux limit (solid black line in Fig. 5.) in all galaxies, except for M83, where at the low temperature end, the 20 counts limit for this galaxy is by a factor of about ∼ 2 higher. This may result to slight underestimation of the specific frequency of super-soft sources in late-type galaxies. In counting supersoft sources, we included all the sources which error bars crossed the stable nuclear burning boundary. We also included two sources in M101, located outside but close to the stable nuclear burning boundary (the two sources in Fig.5 upper right panel with kT > ∼ 100 eV and L X > ∼ 10 37 erg/s). In Centaurus A galaxy, we used different criteria for sources locate in and outside the dust lanes (c.f. dotted line in Fig. 5), although this does not change the final result. In order to estimate specific frequencies of super-soft X-ray sources, we group the data separately for early-and late-type galaxies. There are 29 super-soft sources in latetype galaxies (M83, M51, M81, M101), which, for the total stellar mass of 13.9×10 10 M yield the specific frequency of f S SSS ≈ (2.08 ± 0.46) × 10 −10 M −1 In computing statistical errors we used Gehrels approximation (Gehrels 1986). Counting lenticular and elliptical galaxies together (Centaurus A, NGC3115, NGC3379, NGC4278, NGC4697), we find 7 super-soft X-ray sources sources for the total stellar mass of 28.3 × 10 10 M yielding specific frequency of Centaurus A galaxy, although classified as S0, shows some star-formation activity within its dust lanes, at the the level of ∼ 0.8 M /yr (Espada et al. 2019). However, its SSS specific frequency is closer to early-type galaxies than to latetype galaxies. In order to check how its inclusion in our sample affects our results, we compute SSS specific frequency for elliptical galaxies only: This number is ∼ 20% smaller but compatible with the value obtained including Centaurus A.
The lower limit for the SSS specific frequency in earlytype galaxies at the 3σ confidence level is f E SSS > 8.2 × 10 −12 M −1 .
The ratio of specific frequencies of SSS in late-type to early-type galaxies is i.e. late-type galaxies contain about ≈ 8 times more supersoft X-ray sources than early-type galaxies. Using Bayesian approach, we estimate that their specific frequencies are different at the confidence level corresponding to about ≈ 5.8σ. We searched for variations of the SSS specific frequency between galaxies of the same morphological type. To this end we compared observed numbers of super-soft X-ray sources in each galaxy with their expected values computed from the corresponding SSS specific frequency and stellar mass of the galaxy. For early-type galaxies we found no statistically significant deviations of observed numbers of super-soft X-ray sources from their expected values. This is not surprising, given small numbers of SSS in early-type galaxies. For the late-type galaxies, however, we observe statistically significant deviations in M83 and M81 where the observed numbers of super-soft sources differ from the expected values at the statistically significant levels exceeding ∼ 3σ. This can be understood considering that populations of accreting nuclear burning white dwarfs are determined by the long-term star-formation history of the galaxy, rather than by its mass, as discussed in the following section.
For late-type galaxies, we also compute the SSS specific frequency per unit star formation rate. Using SFR values, collected from the literature (Table 1 and references there) we obtain: Overall, the numbers of super-soft X-ray sources in latetype galaxies appear to somewhat better correlate with their star-formation rate than with the stellar mass.

Comparison with previous work and discussion
The fact that super-soft X-ray sources tend to be predominantly associated with young stellar populations has been noted previously (e.g. Di Stefano & Kong 2004). Studying populations of soft sources in nearby galaxies, Di Stefano (2010) mentioned the difference in the numbers of soft sources in early-and late-type galaxies, however they did not quantify these trends. Bogdán & Gilfanov (2011) using sample of 12 nearby late-type galaxies concluded that specific frequencies (per unit K-band luminosity) of luminous, L X > 10 36 erg/s, super-soft sources in disks of spiral galaxies by factor of ∼2 exceed that of bulges. Taking into account that the typical M * /L K ratio for bulges are approximately twice larger than for disks, this translates to about 4 times difference in specific frequency per unit mass, i.e. shows qualitatively same trend as our comparison of latetype and early-type galaxies, although the particular numbers are different. The difference is due to the fact, that Bogdán & Gilfanov (2011) used a selection procedure to identify super-soft X-ray sources, based on the hardness ratio with the threshold, corresponding to kT =200 eV. Population synthesis calculations of nuclear burning accreting WDs predict difference of specific frequencies of SSSs in spiral and elliptical galaxies (Chen et al. 2015). Using their model a025qc17 we estimate specific frequencies of super-soft X-ray sources with X-ray luminosities exceeding 10 36 erg/s. For spiral galaxies they obtained specific frequencies ranging from 4 × 10 −10 to 2 × 10 −9 sources per M , corresponding to absorption column densities of 3×10 21 cm −2 and 3×10 20 cm −2 . For elliptical galaxies they obtained ≈ 6×10 −11 sources per M (for column density 3×10 20 cm −2 ), i.e. > ∼ 7 times less frequent that in late-type galaxies. The contrast between late-and early-type galaxies is well consistent with our observations, especially taking into account dependence of these numbers on the age of the host galaxy (Chen et al. 2015). However, the actual numbers of specific frequencies are by at least a factor off ∼ 2 − 3 larger in simulations than in our data. Among other reasons, this can be understood as a result of an approximate account for absorption, as the numbers of observable super-soft X-ray sources are highly sensitive to the column density.
Another interesting result of this work is that within the morphological types, the number of super-soft X-ray sources does not seem to obey simple scaling relations with parameters of the host galaxy. In particular, there are large and statistically significant excursions in the specific frequency of SSS in spiral galaxies, normalised to their stellar mass. This can be expected in the view of the results of WD population modelling which predict that the population of super-soft sources peaks at about 1 Gyr after the star formation event (Chen et al. 2015). For this reason, numbers of super-soft sources in galaxies should not be expected to scale with stellar-mass or current star-formation rate. In the more general context, low specific frequency of super-soft X-ray sources, especially in early-type galaxies, support earlier conclusion that they are not the major class of type Ia supernova progenitors (Di Stefano 2010;Gilfanov & Bogdán 2010;Johansson et al. 2014;.

Other soft X-ray sources
Along with rare super-soft X-ray sources, in all galaxies from our sample there are much more abundant populations of soft sources with somewhat harder, but still soft, spectra, located to the right of the stable nuclear burning boundary in Fig. 5. These sources were previously classified as quasi-soft X-ray sources (Di Stefano & Kong 2004;Orio 2006). Among other hypothesis, it was proposed that they are associated with supernova remnants or with accreting black holes in the soft spectral state (e.g. Di Stefano & Kong 2004;Patel et al. 2013;Long et al. 2014;Kuntz et al. 2016), although the contribution of the latter should be minimal (Section 5.2) The X-ray emitting SNRs will be largely unresolved by Chandra and appear as point sources. Indeed, the SNR emission shifts outside X-ray band when the shock slows down to about ∼ 200 km/sec and the post-shock temperature drops below ∼ 10 6 K (Vink 2017). This happens about ∼ 2 × 10 4 years after the explosion when the remnant size typically reaches ∼ 10 − 20 pc (Blondin et al. 1998). At the distances of spiral galaxies 3.6∼7.6 Mpc, linear dimension of 10 − 20 pc corresponds to the angular size of ∼ 0.4 − 0.9 , which can be barely resolved with Chandra ACIS for the brightest sources. Furthermore, the majority of supernova remnants will be classified by hardness ratio and median energy as soft sources.
To clarify the nature of soft sources, located to the right from the stable nuclear burning boundary, we used M83 as a test case. It is a galaxy with one of the best studied X-ray SNR population. Long et al. (2014) identified 87 X-ray detected SNR candidates based on their X-ray spectral properties, associations with optical emission nebulae and/or radio sources. From this sample, SNR nature of 67 sources was confirmed with optical spectroscopy using [SII]/Hα line ratio (Winkler et al. 2017). The list of 87 X-ray detected SNRs from Long et al. (2014) was cross-correlated with our catalog of X-ray sources detected in M83. With 1 matching radius, we found 66 matches, of which 6 sources have statistically significant emission beyond 2 keV and remaining 60 sources were classified as soft sources in our analysis. We replot kT bb − L X diagramm for M83 in Fig. 6 marking confirmed SNRs by cyan symbols. As one can see, all confirmed SNRs are located outside the stable hydrogen nuclear burning boundary on kT bb − L X diagram (i.e. to the right of the thick red line in Fig. 6) and their distribution on the kT bb − L X plane is morphologically quite similar to the distribution of the bulk of quasi-soft sources. We therefore conclude that the majority of quasi-soft sources must be SNRs. For many of the brightest quasi-soft sources in spiral galaxies likelihood values obtained in spectral fitting suggest that the spectrum is more complicated than predicted by a featureless black body model. A more detailed investigation of several such sources revealed that these deviations are mostly associated with the presence of emission lines in their spectra. In majority of cases the spectrum of thermal emission of optically thin plasma or a non-equilibrium ionization model significantly improved the quality of the fit, suggesting that these sources may be SNRs. However, such a detailed analysis is only possible for a handful of brightest sources.
In order to compare spectral properties of soft and super-soft sources, we combined their spectra in late-and early-type galaxies. Combined spectra are shown in Fig.7 separately for sources located inside and outside the stable hydrogen nuclear burning boundary on the kT bb − L X diagram (see Section 5.2). For comparison, we also show the combined spectrum of confirmed SNRs in M83. As one can see, combined spectrum of soft sources in spiral galaxies shows strong Mg and S lines and matches very well the combined spectrum of SNRs in M83. Similar values of the equivalent width of Mg and S lines support our conclusion that the vast majority of soft sources in these galaxies must be supernova remnants. In early-type galaxies, on the contrary, combined spectrum of soft sources is harder and does not show strong emission lines. As it was discussed in Section 5.2, by the method of selection, soft sources in early-type galaxies should include large numbers of X-ray binaries and background AGN. On the other hand, super-soft sources in all galaxies have much softer spectra, as expected. Their spectra are broadly similar in late-and early-type galaxies, although the quality of the latter does not permit to make a more quantitative conclusion.
We finally note that the numbers and specific frequencies of soft sources in galaxies from our sample seem to follow same trend as super-soft sources, decreasing from late-type to early-type galaxies. However a detailed investigation of dependence of the specific frequency of SNRs on the morphological type would require a more accurate selection of sources and is outside the scope of the present paper.

Summary
Based on public Chandra archival data, we studied populations of soft and super-soft X-ray sources with the goal to investigate populations of stable nuclear burning accreting WDs.
To this end, we searched Chandra data archive and constructed a sample of galaxies optimal for studying populations of steady nuclear burning accreting white dwarfs. We aimed at constructing the best possible sample of such galaxies. We thus selected four late-type and five early-type galaxies with stellar masses > ∼ 10 10 M and Chandra X-ray sensitivity limits comparable or better than ∼ 10 36 ergs/s. Our selection presents the nearly full sample of galaxies available in Chandra archive up to cycle 20, suitable for such a study (among galaxies fitting within the field of view of Chandra observation) We proposed a new approach to identify population of soft and super-soft X-ray sources based on combination of hardness ratio and median energy. Distributions of sources over median energy in late-type galaxies shows clear bimodality with a distinct populations of soft and super-soft sources. These sources show a rather compact distributions in the X-ray luminosity-temperature plane with temperatures of 20-300 eV and X-ray luminosities in the 10 36 ∼10 38 ergs/s range. The picture is more complex in lenticular and elliptical galaxies, where populations of soft sources smoothly connect to the populations of X-ray binaries and background AGN in their median energy distributions. For these galaxies we mainly used the lack of emission above 2 keV to select soft sources for detailed analysis.
We used X-ray luminosity-temperature diagram to further classify selected sources into super-soft and soft. To this end we used the boundary of stable hydrogen nuclear burning to separate super-soft X-ray sources -candidates to nuclear burning white dwarfs from the rest of the population of soft sources. In late-type galaxies, we found 29 super-soft source which yields the specific frequency of f S SSS ≈(2.08±0.46)×10 −10 M −1 . In lenticular and elliptical galaxies, there are 7 super-soft sources with the specific frequency of f E+S0 SSS ≈ (2.47±1.34)×10 −11 M −1 . Specific frequencies in late-type and early-type galaxies differ at the statistical confidence of ≈ 5.8σ. The ratio of specific frequencies of super-soft X-ray sources in late-type galaxies and early-type galaxies is f E+S0 SSS / f S SSS = 0.12 ± 0.05. Our results are broadly consistent with the population synthesis modelling of the populations of stable nuclear burning accreting WDs.

Acknowledgements
This research has made use of data obtained from the Chandra Data Archive and software provided by the Chandra X-ray Center (CXC) in the application packages CIAO. This publication makes use of data products from the Hy-perLeda and RC3 database. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation; the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The authors acknowledge partial support of this work by the Russian Government Program of Competitive Growth of Kazan Federal University (IG) and by the RSF grant 19-12-00369 (MG). The authors thank the anonymous referee for useful and inspiring suggestions which helped to improve the manuscript.  Soft sources in late-type galaxies SSS in late-type galaxies SSS in early-type galaxies Soft sources in early-type galaxies Optical and radio SNRs in M83 Fig. 7. Combined spectra of soft and super-soft X-ray sources in late-and early-type galaxies. Black and green symbols show integrated spectrum of all soft sources located outside the stable nuclear burning region for WDs (i.e. located to the right of the thick red line in Fig.5) in late-and early-type galaxies. Yellow symbols show integrated spectrum of sources, matching confirmed SNRs in M83. Blue and red symbols show the integrated spectra of super-soft X-ray sources, i.e. sources located to the left of and above the stable nuclear burning boundary in late-type and early-type galaxies respectively. The spectra are normalized to the same flux at 0.35 keV. See Section 6.1 for more detail.  Table 2 are computed for these regions. The inside areas of small green regions near the center of Centaurus A, M83 and M51 were excluded from analysis.