# Multiwavelength modeling the SED of Luminous Supersoft X-ray Sources in
Large Magellanic Cloud and Small Magellanic Cloud

Augustin Skopal  Astronomical Institute, Slovak Academy of Sciences,
059 60 Tatranská Lomnica, Slovakia

###### Abstract

Classical supersoft X-ray sources (SSSs) are understood as close binary
systems in which a massive white dwarf (WD) accretes from its companion at
rates sustaining a steady hydrogen burning on its surface generating
bolometric luminosities of $10^{36}-2\times 10^{38}$$\rm\,erg\,s^{-1}$. Here,
we perform for the first time the global supersoft X-rays to near-infrared
(NIR) spectral energy distribution (SED) for the brightest SSSs in LMC and
SMC. We test a model in which the ultraviolet–NIR is dominated by emission
from a compact (unresolved) circumstellar nebula represented by the ionized
gas out-flowing from the SSS. The SED models correspond to luminosities of
SSSs of a few times $10^{38}-10^{39}$$\rm\,erg\,s^{-1}$, radiating at
blackbody temperatures of $\approx 3\times 10^{5}$ K, and indicate nebular
continuum, whose emission measure of $\gtrsim 2\times 10^{60}$ cm-3
corresponds to a wind mass-loss at rates $\gtrsim 2\times 10^{-6}$
$M_{\odot}\,{\rm yr}^{-1}$. Such the extreme parameters suggest that the
brightest SSSs could be unidentified optical novae in a post-nova SSS state
sustained at a high long-lasting luminosity by resumed accretion, possibly at
super-Eddington rates. New observations and theoretical multiwavelength
modeling of the global SED of SSSs are needed to reliably determine their
parameters, and thus to understand their proper stage in stellar evolution.

Fundamental parameters of stars — X-rays: binaries — X-rays: individual: (RX
J0513.9-6951, RX J0058.6-7135, RX J0543.6-6822, RX J0527.8-6954)

††facilities: ROSAT(PSPC), XMM-Newton, Chandra, FUSE, HST(GHRS, FOS), IUE(SWP,
LWP), USNO-B Catalog, NOMAD Catalog, 2MASS All-Sky Catalog, IRSF Magellanic
Clouds Point Source Catalog, 2MASS 6X Point Source Working Database/Catalog,
The SMC Stellar Catalog and Extinction Map, The LMC Stellar Catalog and
Extinction Map, Catalogue of Stellar Spectral Classifications, the 3rd release
of the DENIS database

##  1 Introduction

Luminous supersoft X-ray sources (SSSs) were discovered in the Large
Magellanic Cloud (LMC) and in the Small Magellanic Cloud (SMC) by Long et al.
(1981) and Seward & Mitchell (1981). They are characterized by very soft
thermal spectra, emitting predominantly at energies $\lesssim$0.5 keV (Greiner
et al., 1991). Their extremely soft spectra correspond to blackbody
temperatures of $\approx$15–80 eV and bolometric luminosities of
$10^{36}-2\times 10^{38}$$\rm\,erg\,s^{-1}$. These objects are commonly
accepted as binaries that consist of a massive white dwarf (WD) radiating
close to the Eddington luminosity due to a steady nuclear burning on its
surface, when the hydrogen-rich material is burned as fast as it is accreted,
at rates of the order of $10^{-7}$ $M_{\odot}\,{\rm yr}^{-1}$ (van den Heuvel
et al., 1992; Kahabka & van den Heuvel, 1997). Following multiwavelength
observations have revealed a significant excess of the radiation in the
ultraviolet (UV) to near-infrared (NIR), well above the Rayleigh-Jeans tail of
the SSS radiation itself (see Sects. 1.1 – 1.4 and 2). To explain the high
luminosity of bright SSSs, and the longer wavelength excess, Popham & di
Stefano (1996) proposed a model, in which a significant fraction of the WD’s
radiation is converted to the UV–NIR by irradiating, (i) the flared accretion
disk and (ii) the donor star.

In this paper we selected four bright SSSs observed in the Magellanic Clouds,
RX J0513.9-6951, RX J0058.6-7135, RX J0543.6-6822 and RX J0527.8-6954, with
the aim to model their global X-ray—NIR spectral energy distribution (SED) by
the method of multiwavelength modeling described by Skopal (2015). The
motivation for this work is the absence of the SED model, which would
uniformly fit radiation of SSSs in both the supersoft X-ray and the UV–NIR
domains. Further, the extreme properties of SSSs and their spectral features
indicate the presence of the nebular radiation in the spectrum (see Appendix
A), which could explain the observed strong UV–NIR excess as it is currently
considered for the accreting nuclear-burning WDs in symbiotic binaries (Kenyon
& Webbink, 1984; Muerset et al., 1991; Skopal, 2005). Therefore, here we are
testing SED modeling, in which the UV–NIR is dominated by the nebular emission
instead of the irradiated disk/companion radiation.

Section 2 summarizes the observed supersoft X-rays to NIR SED of our targets,
while Sect. 3 introduces the method and results of our analysis. Their
discussion is found in Sect. 4, while Sect. 5 summarizes our findings, and
proposes tasks for future investigation. In the following subsections, we
first introduce our targets.

###  1.1 The LMC SSS RX J0513.9-6951

RX J0513.9-6951 is a transient binary supersoft X-ray source. Currently, it is
thought that the system contains a somewhat evolved star and a relatively
massive compact object at a 0.76-day orbit, viewed nearly pole-on (Crampton et
al., 1996; Southwell et al., 1996). The system is the brightens SSS emitting a
significant amount of the radiation also in the far-UV ($\gtrsim
10^{-13}$$\,\rm erg\,cm^{-2}\,s^{-1}\,\AA^{-1}$).

RX J0513.9-6951 was discovered during the ROSAT All-sky Survey (1990 July –
1991 January) in the LMC (Schaeidt et al., 1993). The authors found it to be
variable and matched its total average spectrum by a blackbody radiation (see
Table 1). Pakull et al. (1993) revealed the optical counterpart as a $B\approx
17$ mag blue star. Its spectrum was dominated by the hydrogen Balmer lines
with a signature of a broad P-Cyg absorption component indicating a mass-loss
from the system at $v_{\infty}\approx 3600$ km s-1. Absence of He I lines and
the presence of highly ionized ions (O VI 3811, 3834 doublet, He II) with
$I_{\rm{He\,\rm{\scriptsize I}\rm{\scriptsize I}}\,4686}\gg I_{\rm{H}\beta}$
reflect the presence of a very hot ionizing source in the binary. They also
pointed that the optical counterpart appears fainter during the X-ray
outburst.

Pronounced optical variability of RX J0513.9-6951 was revealed by optical
multicolor monitoring campaigns. Reinsch et al. (1996) measured an optical low
states during February to December, 1993, lasting around 40 days with a
decrease in the continuum by $\Delta V\sim 1$ mag and a factor of $\approx$10
decrease in emission line fluxes. The light curve obtained within the MACHO
project specified the occurrence of the low stages on a time-scale of $\sim
100-200$ days (Alcock et al., 1996; Southwell et al., 1996). In addition,
during the high-state, these authors revealed a small ($\Delta V\sim 0.05$)
light modulation with a period of 0.763 d, which is consistent with variations
in the radial velocities of He II $\lambda$4686 emission line, and thus
confirms a binary nature of the system (Crampton et al., 1996).

Broad wings of the strongest emission lines (He II and H$\beta$) accompanied
with satellite components indicated a bipolar outflow and thus the presence of
a disk in the binary. The bipolar outflow in the form of jets suggested that
the accretor is a WD (Southwell et al., 1996; Crampton et al., 1996).

Based on all previously published optical and X-ray data, Southwell et al.
(1996) pointed out the principal feature of the variability: the anti-
correlation between the supersoft X-ray and the optical fluxes. X-ray
monitoring of RX J0513.9-6951 showed that the flux anti-correlation is very
strict: There are cyclic changes between optical-low/X-ray-on states and
optical-high/X-ray-off states (e.g., Reinsch et al., 2000; Charles et al.,
2010).

Based on the HST ultraviolet spectroscopy, Gänsicke et al. (1998) derived the
neutral hydrogen density $N_{\rm H}=(5.5\pm 1.0)\times 10^{20}$ cm-3. The
authors found that this value is consistent with parameters determined by
model atmosphere fits to the ROSAT PSPC spectrum of July 1993 (see Table 1).

###  1.2 The SMC SSS RX J0058.6-7135 (LIN 333)

Henize (1956) labeled this object in his “Catalog of Emission Nebulae in the
Small Cloud” as LH$\alpha$-115 N67. Independently, Lindsay (1961) put it in
his catalog under the number 333, remarking features of a planetary nebula in
its spectrum. Therefore, the identifier LIN 333 is frequently used for this
object. Aller et al. (1987) analyzed its IUE spectra, and revealed a strong
nebular emission produced by the object, characterized with the emission
measure of $\sim 1.4\times 10^{60}$ cm-3 (see Appendix A) and a high electron
temperature of $\sim 30\,000$ K. They also measured the optical position of
the nebula, which helped Wang (1991) to identify LH$\alpha$-115 N67 with the
strong SSS 1E0056.8-7154, which is also named as SMP SMC 22 (Sanduleak et al.,
1978). Wang (1991) suggested that 1E0056.8-7154 is the single hot nucleus of
the planetary nebula LH$\alpha$-115 N67. The X-ray detection of this source
was first reported by Seward & Mitchell (1981) on the basis of observations
with the Einstein satellite. Following X-ray observations were carried out by
the ROSAT and XMM-Newton satellites, see Kahabka et al. (1994) and Mereghetti
et al. (2010), respectively. Modeling the X-ray SED of LIN 333 was performed
by Kahabka et al. (1994), Heise et al. (1994), and Mereghetti et al. (2010).
Corresponding parameters are introduced in Table 1. Based on their models,
Heise et al. (1994) found that accreting WD in a binary is consistent with the
ROSAT spectrum of 1E0056.8-7154, but also admitted a possibility that the SSS
is an exceptionally hot single central star of a planetary nebula. Comparing
all X-ray data and their models, spanning almost 30 years, Mereghetti et al.
(2010) did not find any apparent long-term variability in the luminosity or
spectrum of LIN 333. The authors pointed out that the lack of variability,
typical for other SSSs, and too low upper limit to the optical counterpart
($V\sim 21$, Villaver et al., 2004) do not favor a binary nature of LIN 333.
Therefore, they supported the interpretation that it is a single, very hot
star on its way to becoming a relatively massive ($\sim$1 $M_{\odot}$) WD.
Using the Spitzer Space Telescope infrared spectra, Bernard-Salas et al.
(2009) classified SMP SMC 22 as a high-excitation planetary nebula with high
excitation lines ([O IV] and [Ne V]). The absence of dust-related features in
the spectrum was ascribed to the high temperature and luminosity of SMP SMC
22.

In spite of these arguments, we interpret the results of our multiwavelength
modeling of the LIN 333 SED within a binary-star model, in which a WD accretes
from its companion at a high rate (Sect. 4.3). The main reason for this
assumption is the exceptionally high luminosity of LIN 333 compared to other
planetary nebulae, indication of a compact unresolved nebula in its spectrum –
common main features of all our targets (Fig. 1, Sect. 4.5, Appendix A), and
possible presence of the binary companion indicated by the optical fluxes (see
Appendix C).

Table 1: Physical parameters of our targets from the literature (designation
as in Sect. 3.1)

Object | $N_{\rm H}$ | $R_{\rm SSS}^{\rm eff}$ | $T_{\rm BB}/T_{\rm eff}$ | $\log(L_{\rm SSS})$ | Model | Ref.
---|---|---|---|---|---|---
| [$10^{20}\,{\rm cm^{-2}}$] | [ $R_{\odot}$ ] | [ K ] | [ ${\rm erg\,s^{-1}}$ ] | BB/Atm. |
RX J0513 | 9.4 |  $\sim$0.039 | 456000$\pm 78000$ |  $\sim$38.37 | BBa | 1
RX J0513 | 4–7 | 0.009–0.017 |  $\approx$560000 | 37.40–37.95 | Atm.b | 2
LIN 333 | 5.2 - 16 | 0.086 | 160000 - 400000 | 38.28 | BBl | 9
LIN 333 | 5 | 0.012 | 440000 | 37.30 | Atm.c | 3
LIN 333 | 8 | 0.072 | 360000 | 38.48 | BBc | 3
LIN 333 | 2.8–3.2 | 0.19–0.22 | 154000 | 37.81–37.93 | Atm.d | 4
LIN 333 | 5.2 | 0.049 | 313000 | 37.91 | BBd | 4
CAL 83 | 17 | $>0.06$ | 302000 | $L_{\rm SSS}>L_{\rm Edd}$ | BBe | 5
CAL 83 | 8 | 0.022 | 500000 | $L_{\rm SSS}\equiv L_{\rm Edd}$ | BBf | 5
CAL 83 | 8.87 | 0.014–0.016 | 516000 | 37.76 | Atm.g | 6
CAL 83 | 6.07 |  $\gtrsim$0.015 |  $\lesssim$403000 |  $\gtrsim$37.30 | Atm.g | 6
CAL 83 | 6.5 | 0.012$\pm 0.003$ | 539000$\pm 16000$ | 37.40$\pm 0.03$ | BBh | 7
CAL 83 | 6.5 | 0.018$\pm 0.001$ | 378000$\pm 9000$ | 37.37$\pm 0.01$ | Atm.h | 7
CAL 83 | 6.5$\pm 1.0$ | 0.01$\pm 0.001$ | 550000$\pm 25000$ | 37.54$\pm 0.13$ | Atm.i | 8
RX J0527 | 14 | $>0.16$ | 209000 | $L_{\rm SSS}>L_{\rm Edd}$ | BBj | 5
RX J0527 | 8.5 | 0.043 | 348000 | $L_{\rm SSS}\equiv L_{\rm Edd}$ | BBk | 5

a ROSAT, October 30, 1990; 1 - Schaeidt et al. (1993); b ROSAT, July 1993; 2 -
Gänsicke et al. (1998); c ROSAT, October 8, 1991; 3 - Heise et al. (1994); d
XMM-Newton, dates in Table 2; 4 - Mereghetti et al. (2010); e ROSAT, June 20,
1990 - best fit; 5 - Greiner et al. (1991); f as in e, but for the Eddington
luminosity ($L_{\rm Edd}$); g ROSAT, observation No. 500131p; 6 - Kahabka
(1998); h BeppoSAX, $N_{\rm H}$ fixed, March 7-8, 1997; 7 - Parmar et al.
(1998); i Chandra, XMM-Newton, $N_{\rm H}$ fixed, dates in Table 2; 8 - Lanz
et al. (2005); j ROSAT, June 16-25, 1990 - best fit; k as in j, but for
$L_{\rm Edd}$; l ROSAT, October 27-31, 1990; 9 - Kahabka et al. (1994).

###  1.3 The LMC SSS RX J0543.6-6822 (CAL 83)

The bright SSS CAL 83 was discovered in the Einstein survey of the LMC (Long
et al., 1981). Subsequent observations with EXOSAT from November 1983
confirmed its very soft X-ray spectrum with an upper limit of 0.3 count/s
(Pakull et al., 1985). The optical spectrum from May 1984 revealed its
similarity to that of the low mass X-ray binaries LMC X-2 and Sco X-1. From
radial velocity variations of the optical emission lines (H and He II
$\lambda$4686), Crampton et al. (1987) discovered that CAL 83 is a binary
system with a period of $\sim 1$ day. Orbital elements from radial velocities
of the He II $\lambda$4686 line were improved by Schmidtke et al. (2004).

Following X-ray observations of CAL 83 were carried out with the ROSAT,
BeppoSAX, Chandra and XMM-Newton satellites. Kahabka (1998) presented a
1.5-year X-ray light curve of CAL 83 as measured with the ROSAT/HRI from July
1995 to December 1996. The authors also indicated the first short-term X-ray-
off state of CAL 83 on JD 2 450 200 (April 26, 1996). The second and the third
X-ray-off state was reported by Greiner & Di Stefano (2002) and Lanz et al.
(2005) on the basis of Chandra observations from November 1999 and October
2001, respectively. This demonstrates recurrent nature for CAL 83. Modeling
the X-ray SED of CAL 83 was carried out by Greiner et al. (1991), Kahabka
(1998), Parmar et al. (1998), Paerels et al. (2001) and Lanz et al. (2005).
Corresponding parameters are found in Table 1.

The prototypical SSS CAL 83 was also intensively studied in the ultraviolet
and optical. It was identified with a variable, blue, $V\approx 17$, point
like source with an orbital period of 1.047 days (Cowley et al., 1984; Pakull
et al., 1985; Smale et al., 1988; Schmidtke et al., 2004; Rajoelimanana et
al., 2013). Optical 3900–6800 Å spectrum from May 1984 showed relatively weak
Balmer emission lines and a high He II $\lambda$4686/H$\beta$ ratio ($>5$).
The nebular [OIII] 5007 Å line was also present. The blue-shifted He II
$\lambda$4686 emission component suggested a mass outflow with velocities at
1500–2300 km s-1 (Pakull et al., 1985). CAL 83 is known to show irregular
brightness variations of more than a magnitude over timescales of months.
Using the MACHO $V$ and $R$ light curves (1993–1999), Greiner & Di Stefano
(2002) indicated that CAL 83 exhibits distinct and well-defined low,
intermediate, and high optical states, characterized with no color changes
during transition phases. They also found that X-ray-off states were observed
during optical high states. A variability has also been found in the
ultraviolet by Gänsicke et al. (1996), who observed CAL 83 during high and low
states. More recent optical and X-ray observations revealed a quasi-periodic
variation on a time-scale of 450 d with the optical low and high states
lasting for $\sim$200 and $\sim$250 d (Rajoelimanana et al., 2013). The
authors also clearly established the X-ray/optical flux anticorrelation. It
was found that the X-ray light curve exhibits significant modulations with
periodicities from 67 s to hours (Odendaal et al., 2014; Odendaal & Meintjes,
2017). It is generally assumed that variations in the optical flux are most
likely associated with changes in the accretion disk (e.g., Greiner & Di
Stefano, 2002).

Optical imaging of CAL 83 revealed the presence of an extended ionized nebula
surrounding CAL 83, first recognized by Pakull & Motch (1989) on the [O III]
$\lambda$5007 CCD image with a diameter of $\sim 50\arcsec$. Remillard et al.
(1995) reported that the nebula is caused by the illumination of the local ISM
by the central SSS CAL 83 with the time-averaged X-ray luminosity $>3\times
10^{37}$$\rm\,erg\,s^{-1}$. The authors inferred that the nebula has a bright
inner zone with a characteristic radius, particle density and mass of
$\sim$7.5 pc, 4–10 cm-3 and $\gtrsim$150 $M_{\odot}$, respectively, while the
diffuse outer nebula has a characteristic radius of $\sim$20 pc. Based on the
spectra taken over a $25\arcsec\times 25\arcsec$ field of view (FoV, i.e.,
$7.5\times 7.5$ pc2 for the distance of 55 kpc), Gruyters et al. (2012)
presented the morphology of the ionized gas within the inner nebula, for the
first time indicated there a He II $\lambda$4686 emission region, and
extracted the global spectrum of the ionization nebula within their FoV (see
Appendix B, in detail). Using the CLOUDY software and typical parameters of a
SSS, the authors modeled the CAL 83 nebula. However, none of their models was
able to fit the observed spectrum very well. It is of interest to note that
searching for ionization nebulae surrounding other strong SSSs in LMC and SMC
has been negative, which indicates that either the ISM densities in their
environments must be significantly lower than surrounding CAL 83, or the
average X-ray luminosities of these sources over the last $\lesssim$10000
years must be significantly lower than presently observed (Woods & Gilfanov,
2016; Farias et al., 2020).

###  1.4 The LMC SSS RX J0527.8-6954

This source was discovered during the calibration phase of ROSAT pointing the
LMC field between 16 and 25 June 1990 (Greiner et al., 1991). Profile of its
supersoft spectrum was very similar to that of CAL 83, but it was fainter by a
factor of $\sim 10$. The model SED was done by Greiner et al. (1991) (Table
1).

Repeated observations of this source exhibited a continuous decline in the
photon counts, from $\sim 0.2$ to $\sim 0.01$ cts s-1, during 1990 to 1995
(Greiner et al., 1996). The authors speculated that a likely mechanism causing
the X-ray variability could be a hydrogen flash on a massive WD ($\geq 1.1$
$M_{\odot}$) accreting at a high rate of $10^{-7}$ $M_{\odot}\,{\rm yr}^{-1}$.
This view is supported by the fact that the source was not detected by the
Einstein satellite 20 years ago, which thus appears that the source is
recurrent and was in a declining phase during the ROSAT observations. Also the
4-yr light curve revealed a steady fading by $\lesssim$0.5 mag and a 0.39-day
modulation with $\sim$0.05 mag semi-amplitude.

Based on the optical photometry obtained via the MACHO project, Alcock et al.
(1997) first suggested an optical counterpart to RX J0527.8-6954, and Cowley
et al. (1997) specified it as a B8 IV star with $V=17.3$ mag. Using a high-
resolution imaging, Oliveira et al. (2010) identified the X-ray source with a
B5e V star that is associated with a bipolar H$\alpha$ subarcsecond extended
emission. The authors pointed that for a massive WD and the Be V companion
with $M\sim 6$ $M_{\odot}$, the orbital period has to exceed 21 hours, and
thus the 0.39-day period cannot be associated with the orbit.

Table 2: Log of the used observations from the space

Date | Julian date | Region | Observing | $T_{\rm exp}$
---|---|---|---|---
yyyy/mm/dd | JD 2 4… | [nm] | satellite | [ks]
RX J0513.9-6951
1990/10/301 | 48195.5 | 1.92–6.2 | ROSAT2 | 1.9
2001/10/31 | 52214.5 | 94–118 | FUSE3 | 6.6
2003/07/05 | 52827.0 | 94–118 | FUSE3 | 33.0
1996/11/13 | 50401.7 | 115–144 | HST3 | 2.8
1992/04/24 | 48736.6 | 115–198 | IUE3 | 24.0
1992/04/25 | 48737.6 | 186–335 | IUE3 | 23.0
1995/06/29 | 49898.4 | 115–198 | IUE4 | 22.8
RX J0058.6-7135 (LIN 333)
1991/10/08 | 48537.5 | 2.5–6.2 | ROSAT | 17
2007/06/04 | 54256.8 | 2.5–6.3 | XMM-Newton | 7.7
2009/05/14 | 54966.3 | 2.5–6.3 | XMM-Newton | 16.54
2009/09/25 | 55100.3 | 2.5–6.3 | XMM-Newton | 32.3
2002/07/25 | 52480.5 | 92–119 | FUSE | 26.4
1984/04/16 | 45806.5 | 150 - 190 | IUE | 14
1984/04/17 | 45807.5 | 190 - 333 | IUE | 16
1995/03/20 | 49796.5 | 150 - 540 | HST | 2.5
RX J0543.6-6822 (CAL 83)
1990/06/20 | 48063.5 | 2.1–5.9 | ROSAT | 3.6
2000/04/23 | 51658.0 | 2.4–5.4 | XMM-Newton | 45.1
2001/08/15 | 52137.3 | 2.4–5.4 | Chandra | 35.4
2001/09/22 | 52175.3 | 93.5–118 | FUSE | 45.6
1996/11/10 | 50398.0 | 115–144 | HST | 2.8
1984/11/24 | 46029.1 | 115–198 | IUE | 15.1
1987/12/12 | 47142.0 | 115–198 | IUE | 14.4
1990/10/29 | 48194.1 | 115–198 | IUE | 24.4
1990/11/07 | 48203.4 | 115–198 | IUE | 22.8
1987/12/11 | 47141.0 | 186–332 | IUE | 16.1
1988/08/12 | 47397.3 | 197–333 | IUE | 12.9
RX J0527.8-6954
1990/06/20 | 48063.5 | 2.6–5.9 | ROSAT | 18.8
1992/09/01 | 48867.2 | 115–198 | IUE | 16.7
1992/09/01 | 48867.3 | 240–318 | IUE | 6.0

1 avearage of the second part, 2 the total spectrum (see Fig. 2 of Schaeidt et
al., 1993), 3 high state, 4 low state.

##  2 Observations

This section summarizes observations of our targets we used to model their
global SED in the continuum. Spectroscopic observations with Far Ultraviolet
Spectroscopic Explorer (FUSE), Hubble Space Telescope (HST) and International
Ultraviolet Explorer (IUE) were obtained from the satellite archives with the
aid of the MAST. Other observations were taken from the literature and
catalogs as referred in the following subsections. Data obtained from the
space are collected in Table 2. Stellar magnitudes in the optical–NIR were
converted to fluxes according to the calibration of Henden & Kaitchuck (1982)
and Bessell (1979). To correct the observed X-ray fluxes for absorptions we
used the tbabs absorption model for the interstellar matter composition with
abundances given by Wilms et al. (2000) (e.g., $\log(A_{\rm OI})+12=8.69$).
Observations for $\lambda>912$ Å were dereddened with $E_{\rm B-V}$ = 0.06 and
0.08 (unless otherwise specified), and physical parameters were scaled with
the distance $d$ = 49 and 60 kpc to objects in the LMC and SMC, respectively
(Mateo, 1998).

###  2.1 RX J0513.9-6951

The observed SED of RX J0513.9-6951, used in this contribution, covers the
spectral range from 1.92 nm to 2200 nm. It consists of the ROSAT PSPC
observations (1.92 – 6.20 nm) as published by Schaeidt et al. (1993). The data
were adopted from their figure 2. The far-UV spectra with FUSE (B0160104,
D0060102) were taken during the high state of RX J0513.9-6951 as found by
Hutchings et al. (2002) for the 2001 observation, and according to the light
curve of Burwitz et al. (2008) for the 2003 spectrum. The average continuum
level of both spectra is comparable, with $F(2001)/F(2003)\sim 1.2$. According
to the light curve of Cowley et al. (2002), the high resolution HST/GHRS
spectrum (Z3HS0205T) was also taken during a high state. Further spectroscopic
observation in the ultraviolet was carried out with the IUE satellite. The
1992 spectrum (SWP44467, LWP22883) corresponds to a high state, because its
short-wavelength part fits well the HST spectrum. However, the 1995 spectrum
(SWP55170) was taken at the optical minimum, i.e., during a low state (Cowley
et al., 2002).

Observations from the space were supplemented with the ground-based optical
spectroscopy obtained during both the high state (Pakull et al., 1993) and the
low state (Reinsch et al., 1996). The former was taken nearly simultaneously
with the IUE high-state spectra from 1992. In addition, optical $BVR$ broad-
band photometry was estimated from Fig. 1 of Reinsch et al. (1996). Further
$RI$ and infrared $JHK$ photometric measurements were found in the USNO-B1.0
Catalog (R = 16.54, I = 15.75) (Monet et al., 2003), the NOMAD Catalog (V =
16.640, R = 16.540) (Zacharias et al., 2005), and the 2MASS All-Sky Catalog of
Point Sources ($J=16.492\pm 0.157,H\sim 15.517,K\sim 15.728$) (Skrutskie et
al., 2006).

###  2.2 RX J0058.6-7135 (LIN 333)

Multiwavelength observations, we used to model the SED of this SSS, cover the
spectral range from the supersoft X-rays to the optical ($\sim 2.4-360$ nm).
The supersoft X-ray fluxes made by the ROSAT satellite were reconstructed with
the aid of Fig. 2 in Heise et al. (1994), and those carried out with the XMM-
Newton satellite were estimated according to the blackbody model of Mereghetti
et al. (2010). The ultraviolet spectra were obtained from the satellite
archives of FUSE (C0560301000), IUE (SWP22766, LWP3168) and HST FOS
(Y2N30104T, Y2N30105T, Y2N30106T). Spectroscopic observations were
supplemented with the $UBVR$ broad-band photometry, taken from the catalogs of
Massey (2002) (U = 16.16, B = 16.86, V = 16.37, R = 16.09), Zaritsky et al.
(2002) (U = 16.643, B = 17.240, V = 16.924), Skiff (2010) (V = 16.90), and the
3rd release of the DENIS catalog (I = 17.662, J = 16.156).

###  2.3 RX J0543.6-6822 (CAL 83)

The observed SED of CAL 83, used in this paper, covers the spectral range from
2.07 nm to 790 nm. The calibrated supersoft X-ray fluxes were reconstructed
from Fig. 1 of Greiner et al. (1991) (ROSAT) and Fig. 1 of Lanz et al. (2005)
(Chandra, XMM-Newton). In both cases the source was in the X-ray-on state (see
Kahabka, 1998, for the ROSAT observation).

In the ultraviolet, eleven far-UV spectra of CAL 83 were carried out by FUSE
(B0150101) on September 22, 2001 (Schmidtke et al., 2004). The continuum
fluxes were estimated within the range of 93.5 – 118 nm. The high resolution
HST/GHRS spectrum (Z3HS0105T) covers the region from $\sim 114.9$ to $\sim
143.5$ nm. Further spectroscopic observation in the ultraviolet was carried
out with the IUE satellite. Continuum of the SWP spectra No. 24554, 32511,
39995, 40075 and the LWP spectra No. 12257, 13902 were consistent best with
those of the HST and FUSE spectra. Therefore, we used their average fluxes
weighted with exposure times to model the SED. Other IUE spectra had a higher
continuum, in particular the SWP30024 spectrum was a factor of $>2$ above the
used minimum values. According to the measured variations in both the X-ray
and the UV–optical fluxes and their anticorrelation (see Greiner & Di Stefano,
2002), it is probable that the minimum level of the UV continuum, which we
used to model the SED, corresponds to the X-ray-on phase of CAL 83.

Spectroscopic observations were supplemented with the optical $UBVRI$ broad-
band photometry, taken from catalogs of Massey (2002) ($U=16.25$, $B=17.18$,
$V=17.44$), Zaritsky et al. (2004) ($U=16.022$, $B=16.864$, $V=16.976$) and
the 3rd release of the DENIS database ($R=17.0$, $I=16.538$). Compared are
values reported by Crampton et al. (1987) ($B=17.19-17.34$, $V=17.22-17.35$),
and fluxes from the optical, 400–680 nm, spectrum published by Pakull et al.
(1985) scaled with their average photometric measurements ($U-B=-1.04\pm
0.06$, $B-V=0.02\pm 0.06$, $B=17.2\pm 0.3$). In the near-IR Kato et al. (2007)
measured $J=16.98$, $H=16.81$ and $K=16.77$, and 2MASS 6X Point Source Working
Database/Catalog reports $J=17.433$, $H=17.016$ and $K=16.079$.

Table 3: Physical parameters of bright SSSs in LMC and SMC from
multiwavelength SED models (see Sect. 3.1)

Object | Supersoft X-ray source | Nebula |
---|---|---|---
| $N_{\rm H}$ | $R_{\rm SSS}^{\rm eff}$ | $T_{\rm BB}$ | $\log(L_{\rm SSS})$ | $\log(L_{\rm X})^{7}$ | $T_{\rm e}$ | $EM$ |  $\chi^{2}_{\rm red}$/d.o.f.
|  [$10^{20}\,{\rm cm^{-2}}$] |  [$R_{\odot}$] | [K] |  [${\rm erg\,s^{-1}}$] |  [${\rm erg\,s^{-1}}$] | [K] |  [$10^{60}\,{\rm cm^{-3}}$] |
RX J05131 |  12$\pm 2$ |  0.19$\pm 0.02$ |  $350000\pm 10000$ |  39.28$\pm 0.14$ |  38.25$\pm 0.13$ |  30000$\pm 10000$ |  2.3$\pm 0.2$ | 2.8 / 40
RX J05132 | – |  0.26$\pm 0.03$ | $350000^{3}$ |  39.54$\pm 0.15$ |  38.52$\pm 0.14$ |  30000$\pm 10000$ |  8.6$\pm 0.5$ | 3.6 / 38
LIN 3334 |  9.0$\pm 0.2$ |  0.17$\pm 0.02$ | $305000\pm 10000$ |  38.96$\pm 0.13$ |  37.66$\pm 0.12$ |  37000$\pm 5000$ |  1.6$\pm 0.2$ | 2.6 / 39
LIN 3335 |  7.1$\pm 0.2$ |  0.18$\pm 0.02$ | $267000\pm 7000$ |  38.75$\pm 0.12$ |  37.45$\pm 0.11$ | – | – | 4.4 / 22
CAL 834 |  10.3$\pm 0.5$ |  0.15$\pm 0.02$ | $345000\pm 15000$ |  39.04$\pm 0.16$ |  37.99$\pm 0.15$ |  30000$\pm 15000$ |  2.9$\pm 0.3$ | 3.8 / 61
CAL 836 |  12.6$\pm 0.5$ |  0.16$\pm 0.02$ | $357000\pm 15000$ |  39.14$\pm 0.16$ |  38.15$\pm 0.15$ | – | – | 2.9 / 35
RX J0527 |  7.1$\pm 0.2$ |  0.16$\pm 0.02$ | $255000\pm 20000$ |  38.57$\pm 0.06$ |  36.83$\pm 0.05$ |  9000$\pm 2000$ |  4.3$\pm 0.3$ | 3.3 / 34

1 low state, 2 high state: the $(1-9)\times 10^{3}$ Å model SED for $T_{\rm
h}$ adapted from the low state (Fig. 2), 3 fixed value, 4 model SED with the
ROSAT data, 5 model SED with the XMM-Newton data, 6 model SED with the
Chandra/XMM-Newton data, 7 the luminosity in the 0.2–1.0 keV range.

###  2.4 RX J0527.8-6954

In this case, the available observed SED in the continuum covers the spectral
range from 2.65 nm to 550 nm. The calibrated supersoft X-ray fluxes were
obtained by the ROSAT satellite during the pointing observations of the LMC
field in 1990 (Greiner et al., 1991). According to Greiner et al. (1996) the
source was at the brighter stage.

Ultraviolet spectra of RX J0527.8-6954 were carried out with the IUE satellite
on September 1, 1992 (SWP45499, LWP23826). The LWP23826 spectrum was
underexposed, therefore its part from the short-wavelength edge to $\sim 240$
nm was not used in the modeling (dotted line in Fig. 1d).

The optical brightness of RX J0527.8-6954 was measured by Cowley et al. (1997)
in $UBV$ passbands at the Cerro Tololo Inter-American Observatory during
December 1992, December 1993 and November 1994. They determined $V=17.3$,
$B=17.4$ and $U=17.2$. On September 2004, Oliveira et al. (2010) observed RX
J0527.8-6954 using the Gemini South Telescope with the Gemini Multi Object
Spectrograph and distinguished the original star into two components. They
determined the flux ratio in the V band between the components to 1.43, and
corrected the $V$ magnitude of the RX J0527.8-695 optical counterpart to 17.9.

Figure 1:  A comparison of the measured (in magenta, Sect. 2, Table 2) and
model SEDs (heavy black line) of our targets with corresponding data-to-model
ratios (d/m). Filled/open squares are the unabsorbed/observed X-ray fluxes.
The blue and green lines denote components of radiation from the SSS and the
nebula, respectively (see Sect. 3, Eq. (2)). a – RX J0513.9-6951. b – RX
J0058.6-7135 (LIN 333): Two models fitting the ROSAT—UV and XMM-Newton—UV
fluxes were performed. $U,B,V,R$ fluxes and the optical part of the HST
spectrum suggest the presence of an additional source of radiation (marked by
?, see Appendix C). c – RX J0543.6-6822 (CAL 83): Models performed for the
ROSAT and Chandra/XMM-Newton data (each complemented with the same UV and
optical fluxes), are very similar (Table 3). The $J,H,K$ fluxes were not used
for the modeling. d – RX J0527.8-6954: Dotted line is the underexposed part of
the LWP23826 spectrum.

##  3 Analysis and results

###  3.1 Modeling the SED

The primary aim of this paper is to reconstruct the SED of bright SSSs in LMC
and SMC from supersoft X-rays to the optical or NIR. In this way to determine
physical parameters of the main components of radiation in the observed
continuum. To achieve this aim we use the method of multiwavelength modeling
the composite continuum of SSSs described by Skopal (2015). Here we introduce
its basic assumptions and relationships.

In our SED modeling the observed continuum, $F(\lambda)$, is assumed to
consist of a stellar component of radiation, $F_{\rm SSS}(\lambda)$, from the
WD pseudo-photosphere, and nebular continuum, $F_{\rm N}(\lambda)$, from the
ionized out-flowing gas (see Appendix A and Sect. 4.4), i.e.,

$F(\lambda)=F_{\rm SSS}(\lambda)+F_{\rm N}(\lambda).$ (1)

In the model we compare the stellar continuum with the blackbody radiation at
a temperature $T_{\rm BB}$, while the nebular continuum is approximated by
contributions from f–b and f–f transitions in hydrogen plasma radiating at the
electron temperature $T_{\rm e}$. In cases, where observations suggest a
discontinuity in the continuum around $\lambda 2050$ Å, contributions from
doubly ionized helium are also included, and the He++ abundance is set to 0.1.
According to Eqs. (6) and (8) of Skopal (2015), Eq. (1) can be expressed in
the form,

$F(\lambda)=\theta_{\rm SSS}^{2}\pi B_{\lambda}(T_{\rm BB})e^{-\sigma_{\rm
X}(\lambda)\,N_{\rm H}}+k_{\rm N}\times\varepsilon_{\lambda}({\rm
H,He^{+}},T_{\rm e}),$ (2)

where $\theta_{\rm SSS}=R_{\rm SSS}^{\rm eff}/d$ is the angular radius of the
SSS, given by its effective radius (i.e. the radius of a sphere with the same
luminosity) and the distance $d$. Attenuation of the light in the X-ray domain
is given by the total cross-section for photoelectric absorption per hydrogen
atom, $\sigma_{\rm X}(\lambda)$ (Cruddace et al., 1974), and the total column
density of neutral atoms of hydrogen, $N_{\rm H}$ (i.e., given by both the
interstellar and circumstellar matter component). The second term at the right
is the nebular continuum expressed by its volume emission coefficient
$\varepsilon_{\lambda}({\rm H,He^{+}},T_{\rm e})$ scaled with the factor
$k_{\rm N}=EM/4\pi d^{2}$, where $EM=\int_{V}n_{\rm e}n^{+}{\rm d}V$ is the
emission measure of the nebula given by the electron and a ion concentration,
$n_{\rm e}$ and $n^{+}$, in the volume $V$ of the ionized element under
consideration. The assumption that the nebular radiation in the continuum can
be characterized by a single $T_{\rm e}$ is supported by the velocity
distribution of free electrons in nebulae that tends to be Maxwellian, because
the electrostatic encounters of free electrons are much more likely than any
other inelastic scatterings (see Bohm & Aller, 1947). Therefore, using a
single-temperature nebula in the SED modeling expresses the measured UV–NIR
fluxes quite well (see the SED models in Skopal, 2005, 2015, 2019, and Fig. 1
here). In the modeling procedure we simultaneously fit absorbed X-ray
continuum fluxes and dereddened UV to NIR fluxes searching for parameters
$\theta_{\rm SSS}$, $T_{\rm BB}$, $N_{\rm H}$, $T_{\rm e}$ and $k_{\rm N}$
(see Skopal, 2015, in detail)111The corresponding software and application
example are available at https://doi.org/10.5281/zenodo.6985175. The
luminosity of the SSS is calculated as

$L_{\rm SSS}=4\pi d^{2}\theta_{\rm SSS}^{2}\sigma T_{\rm BB}^{4}.$ (3)

Physical parameters corresponding to our SED models are collected in Table 3.

###  3.2 $N_{\rm H}$ from Rayleigh scattering

The high-resolution HST/GHRS spectra ($\lambda$115–144 nm) show a pronounced
attenuation of the continuum around the Ly-$\alpha$ line that is caused by
Rayleigh scattering on atomic hydrogen. This effect can be used to
independently determine $N_{\rm H}$ in the direction of the SSS (see Sect.
2.3.1 of Skopal, 2015). For this purpose we compare the function,

$F(\lambda)=\theta_{\rm SSS}^{2}\pi B_{\lambda}(T_{\rm BB})\,e^{-\sigma_{\rm
Ray}(\lambda)\,N_{\rm H}}+k_{\rm N}\times\varepsilon_{\lambda}({\rm H},T_{\rm
e}),$ (4)

to the observed spectrum. The Rayleigh cross-section for scattering by
hydrogen in its ground state, $\sigma_{\rm Ray}(\lambda)$, is approximated by
Eq. (5) of Nussbaumer et al. (1989). Other parameters have the same meaning as
in Eq. (2), and are fixed according to the solution of the global SED.

###  3.3 Multiwavelength model SED of RX J0513.9-6951

Owing to the strict soft X-ray/optical flux anticorrelation, it was possible
to perform the X-ray—NIR model SED only for the low state of RX J0513.9-6951
(Fig. 1a), while during the high state, only the far-UV–NIR observations were
available to model the corresponding SED (Fig. 2). The best fitting X-ray—NIR
model SED parameters, $N_{\rm H}=(1.2\pm 0.2)\times 10^{21}$ cm-2,
$\theta_{\rm SSS}=8.8\times 10^{-14}$ and $T_{\rm BB}=(350000\pm 10000)$ K,
correspond to a very high luminosity of the SSS source, $L_{\rm SSS}=(1.9\pm
0.7)\times 10^{39}$$\rm\,erg\,s^{-1}$, with the effective radius, $R_{\rm
SSS}^{\rm eff}=(0.19\pm 0.02)$ $R_{\odot}$. The model SED shows also a
significant contribution from the thermal nebula with the emission measure,
$EM=(2.3\pm 0.2)\times 10^{60}$ cm-3 that dominates the spectrum for
$\lambda\gtrsim 2000$ Å.

During the optical-high state, the nebular emission increased by a factor of
$\sim$3.5 (Table 3), and the WD’s effective radius inflated to $(0.26\pm
0.03)$ $R_{\odot}$ for the same $T_{\rm BB}$ as during the optical-low state.
The former suggests that the variable nebular emission is responsible for the
optical variability with $\Delta V\approx 1$ mag, while the latter reflects
the increase of the far-UV fluxes of the stellar component of radiation (the
blue lines in Fig. 2, see Sect. 4.6 in detail).

Finally, fitting the far-UV HST/GHRS spectrum by Eq. (4) we found that the
hollow around Ly-$\alpha$ corresponds to $N_{\rm H}=(8\pm 2)\times 10^{20}$
cm-2. A comparison of the model SED and the HST spectrum constrains the
reddening $E_{\rm B-V}=0.11\pm 0.02$. The spectrum and model (4) are shown in
Fig. 3.

###  3.4 Multiwavelength model SED of RX J0058.6-7135

Observations of LIN 333 span the wavelength range from supersoft X-rays to the
optical. Fluxes in the near-UV and $UBV$ bands ($\sim 1\times 10^{-15}$$\,\rm
erg\,cm^{-2}\,s^{-1}\,\AA^{-1}$, see Fig. 1b), suggest a contribution from the
nebula and, possibly, from a companion to the SSS in a binary (see Appendix
C). In modeling the supersoft X-ray—optical SED we fitted 13 flux-points
between 24 and 62 Å and 31 flux-points between 1500 and 3600 Å by Eq. (2).
Observations with FUSE were not used directly in the SED-fitting procedure,
because their continuum level is a factor of 1.1–1.5 above that given by the
IUE and HST spectra. Nevertheless, the steep slope of their dereddened fluxes
follows the short-wavelength ends of the IUE and HST spectra, which documents
dominant contribution from the SSS to the far-UV (compared are selected fluxes
from the LiF2A channel spectrum (1087 – 1181 Å)).

The modeling solution with the ROSAT data has the reduced $\chi^{2}=2.6$, and
corresponds to the fitting parameters of the SSS, $\theta_{\rm SSS}=(6.6\pm
0.7)\times 10^{-14}$, $T_{\rm BB}=305\,000\pm 10\,000$ K and $N_{\rm
H}=(9.0\pm 0.2)\times 10^{20}$ cm-2. These parameters imply the effective
radius, $R_{\rm SSS}^{\rm eff}=0.17\pm 0.02(d/60\,\,{\rm kpc})$ $R_{\odot}$,
and the luminosity, $L_{\rm SSS}=9.1\pm 3.0\times 10^{38}(d/60\,\,{\rm
kpc})^{2}$$\rm\,erg\,s^{-1}$, which translates into the unabsorbed X-ray
luminosity $L_{\rm X}(0.2-1.0\,{\rm keV})=(4.6\pm 1.6)\times
10^{37}(d/60\,\,{\rm kpc})^{2}$$\rm\,erg\,s^{-1}$. Using the average fluxes of
the three XMM-Newton spectra (see Table 2, Sect. 2.2, Mereghetti et al., 2010)
and the same UV–optical fluxes as above, yield the following parameters of the
SSS: $N_{\rm H}=(7.1\pm 0.2)\times 10^{20}$ cm-2, $T_{\rm BB}=267\,000\pm
7\,000$ K, $R_{\rm SSS}^{\rm eff}=0.18\pm 0.02(d/60\,\,{\rm kpc})$ $R_{\odot}$
and $L_{\rm SSS}=5.7\pm 1.8\times 10^{38}(d/60\,\,{\rm
kpc})^{2}$$\rm\,erg\,s^{-1}$.

The nebular component of radiation dominates the spectrum from around 2500 Å
to longer wavelengths (Fig. 1b). Its amount is scaled with $k_{\rm N}=3.8\pm
0.4\times 10^{12}\,{\rm cm^{-5}}$, which corresponds to $EM=(1.6\pm 0.2)\times
10^{60}(d/60\,\,{\rm kpc})^{2}$ cm-3. The nebula radiates at a high $T_{\rm
e}=37\,000\pm 5000$ K. Both, $EM$ and $T_{\rm e}$ are in a good agreement with
those derived by Aller et al. (1987) from emission lines: $EM\sim 1.8\times
10^{60}(d/63.1\,\,{\rm kpc})^{2}$ cm-3, $T_{\rm e}\sim 25\,300-31\,600$ K (see
their Tables 5 and 7, Appendix A here).

Finally, it is of interest to note that the long time-scale of observations of
LIN 333 (1984 – 2009, Table 2) with comparable X-ray and UV fluxes
demonstrates a stability of this bright SSS (Mereghetti et al., 2010).

Figure 2:  The UV–optical SED during the optical-high (a) and low (b) state of
RX J0513.9-6951. This demonstrates that the pronounced optical variability
with $\Delta V\approx 1$ mag can be caused by the variable nebular emission
(Sect. 3.3).

Figure 3:  The HST/GHRS spectrum of RX J0513.9-6951 dereddened with $E_{\rm
B-V}=0.11$ (in magenta). A strong depression of the continuum around the
Ly-$\alpha$ line (grey area) is caused by Rayleigh scattering of the SSS
radiation on hydrogen atoms with a column density of $(8\pm 2)\times 10^{20}$
cm-2 (black and dotted lines). Meaning of blue and green lines as in Fig. 1.
Geocoronal Ly-$\alpha$ line is denoted by $\oplus$.

Figure 4:  As in Fig. 3, but for CAL 83. The spectrum is dereddened with
$E_{\rm B-V}=0.11$.

###  3.5 Multiwavelength model SED of CAL 83

As in the case of RX J0513.9-6951, the high values of the near-UV–optical
fluxes (a few $\times 10^{-15}$$\,\rm erg\,cm^{-2}\,s^{-1}\,\AA^{-1}$), their
steepness (e.g., $U-B\sim-1$ mag) and the presence of emission lines of the
hydrogen Balmer series, He II $\lambda$4686, NIII 4640 Å and [OIII] 5007 Å in
the optical spectrum (e.g., Pakull et al., 1985) indicate a strong nebular
continuum that dominates the spectrum for $\lambda>1\,800$ Å (see Fig. 1c).

We modeled the SED of CAL 83 between 20.7 and 9 000 Å by Eq. (2). The best
model with ROSAT data has the reduced $\chi^{2}=3.8$ (for 61 d.o.f.) and
determines the variables of the SSS, $\theta_{\rm SSS}=(6.9\pm 0.6)\times
10^{-14}$, $T_{\rm BB}=345\,000\pm 15\,000$ K and $N_{\rm H}=(1.03\pm
0.05)\times 10^{21}$ cm-2, which yields $R_{\rm SSS}^{\rm eff}=0.15\pm
0.02(d/49\,{\rm kpc})$ $R_{\odot}$ and $L_{\rm SSS}=(1.1\pm 0.5)\times
10^{39}(d/49\,{\rm kpc})^{2}$$\rm\,erg\,s^{-1}$. Model variables of the
nebular component of radiation, $k_{\rm N}=(1.0\pm 0.1)\times 10^{13}$ cm-5
and $T_{\rm e}=30\,000+20\,000/-10\,000$ K correspond to $EM=(2.9\pm
0.3)\times 10^{60}(d/49\,{\rm kpc})^{2}$ cm-3 (Table 3).

The model SED with Chandra/XMM-Newton fluxes is very similar to that with the
ROSAT data (see Fig. 1c, Table 3). The corresponding parameters of the SSS
are, $N_{\rm H}=(1.3\pm 0.05)\times 10^{21}$ cm-2, $T_{\rm BB}=357\,000\pm
15\,000$ K, $R_{\rm SSS}^{\rm eff}=0.16\pm 0.02(d/49\,\,{\rm kpc})$
$R_{\odot}$ and $L_{\rm SSS}=(1.4\pm 1.1)\times 10^{39}(d/49\,\,{\rm
kpc})^{2}$$\rm\,erg\,s^{-1}$. As in the case of LIN 333, the long time-scale
of observations demonstrate that CAL 83 is a stable SSS during its X-ray-on
states.

Finally, the attenuation of the continuum around Ly-$\alpha$, observed on the
HST/GHRS spectrum, corresponds to $N_{\rm H}=(8\pm 2)\times 10^{20}$ cm-2 (see
Fig. 4).

###  3.6 Multiwavelength model SED of RX J0527.8-6954

The steepness of the SWP45499 spectrum suggests that the SSS radiation
dominates also a wide range of the far-UV continuum (115–190 nm), while the
LWP23826 spectrum has an opposite slope, whose profile suggests a dominant
contribution from a low temperature nebula (see Fig. 1d). Available
observations allow us to model the continuum fluxes between 26.5 and 5 500 Å.
The best fit to the X-ray and UV data (39 fluxes) has the reduced
$\chi^{2}=3.3$ (for 34 d.o.f.) and determines the model variables,
$\theta_{\rm SSS}=(7.3\pm 0.7)\times 10^{-14}$, $T_{\rm BB}=255\,000\pm
20\,000$ K and $N_{\rm H}=(7.1\pm 0.2)\times 10^{20}$ cm-2, which yield
$R_{\rm SSS}^{\rm eff}=0.16\pm 0.02(d/49\,{\rm kpc})$ $R_{\odot}$ and $L_{\rm
SSS}=(3.7\pm 0.5)\times 10^{38}(d/49\,{\rm kpc})^{2}$$\rm\,erg\,s^{-1}$. The
nebular component of radiation is determined by $k_{\rm N}=(1.5\pm 0.1)\times
10^{13}$ cm-5 and $T_{\rm e}=9\,000\pm 2\,000$ K that gives $EM=(4.3\pm
0.3)\times 10^{60}(d/49\,{\rm kpc})^{2}$ cm-3 (Table 3).

##  4 Discussion

###  4.1 Amount of absorption to the SSSs

Independently of the SED modeling, we estimated the value of $N_{\rm H}$ also
from Rayleigh scattering of the SSS radiation on H atoms that creates a strong
depression of the continuum around the Ly$\alpha$ line (see Figs. 3 and 4).
The larger inaccuracy of this method is given by a weaker sensitivity of the
Rayleigh scattering process to the amount of scatterers in the Ly$\alpha$
wings, and by the complexity of the depression profile due to superposition of
rich absorption-line spectrum to the continuum. On the other hand,
determination of $N_{\rm H}$ from the SED modeling suffer by rather uncertain
X-ray fluxes taken from already published figures and their extreme
sensitivity to the amount of absorption within the supersoft X-ray
wavelengths. Another source of uncertainties in determining values of $N_{\rm
H}$ is given by using non-simultaneous X-ray and UV observations, the temporal
variability of the SSS radiation, and/or the circumstellar component of
$N_{\rm H}$ originating in the surrounding nebula (Sect. 4.6, Appendix A).
Nevertheless, $N_{\rm H}$ values obtained by these two different methods are
comparable (see Table 3, and Figs. 3 and 4 for RX J0513.9-6951 and CAL 83). In
the case of LIN 333 and CAL 83 we modeled different X-ray data together with
the same UV–optical observations (Table 3, Figs. 1b and 1c). Despite this
shortcoming, the $N_{\rm H}$ values we found for LIN 333 are in good agreement
with the total column density $N_{\rm H}\approx(7\pm 4)\times 10^{20}$ cm-2
determined by Wang (1991), which includes contributions in the Galaxy, in the
SMC, and inside the nebula. For RXJ0527.8-6954 we used X-ray fluxes from June
1990 and probably fainter far-UV fluxes from September 1992 as suggested by
the X-ray and optical light curves (see Greiner et al., 1996; Alcock et al.,
1997). If this is so, the true value of $N_{\rm H}$ should be little higher
than that given in Table 3.

Figure 5:  Comparison of the multiwavelength SED models of this paper (dark
thick lines, Table 3) and the corresponding X-ray-data blackbody models from
the literature (see legends, Table 1, Sect. 4.2).

###  4.2 Comparison with previous models

Figure 5 compares our multiwavelength models and the X-ray-data models from
the literature described in Sects. 1.1 to 1.4, and summarized in Table 1. In
both cases, the SED models match satisfactorily the absorbed X-ray fluxes.
However, the unabsorbed models differ, often significantly, from our
multiwavelength models. Mostly, they are a factor of $\sim 10-100$ below the
far-UV data. Such the discrepancy between the multiwavelength modeling and the
modeling only the X-ray data can be ascribed to the mutual dependence between
$N_{\rm H}$, $L_{\rm SSS}$, and $T_{\rm BB}$ parameters, as described in Sect.
4.1 of Skopal (2015).

RX J0513.9-6951: The X-ray blackbody solution of Schaeidt et al. (1993)
corresponds to $L_{\rm SSS}=2.3\times 10^{38}$$\rm\,erg\,s^{-1}$, which is a
factor of $\sim 15$ below the far-UV fluxes. The model atmosphere fit by
Gänsicke et al. (1998) yields even lower luminosity of $(2.5-9)\times
10^{37}$$\rm\,erg\,s^{-1}$ (see Sect. 1.1).

RX J0058.6-7135 (LIN 333): Blackbody and atmospheric X-ray SED models
performed by Heise et al. (1994) are a factor of $\sim$9 and $\sim$300 below
the FUSE data, respectively. They were already discussed by Skopal (2015) as
an illustrative example of the mutual dependence of the model parameters,
$N_{\rm H}$, $L_{\rm SSS}$ and $T_{\rm BB}$ (see Sect. 4.1 and Fig. 7 there).
Similar results were obtained also by Mereghetti et al. (2010) using the XMM-
Newton observations (see Table 1). The authors questioned their atmospheric
model when they found that the flux predicted in the visible band is lower by
$\sim$30% than the upper limit ($V\sim 21$) obtained from the HST imaging of
the central star by Villaver et al. (2004).

RX J0543.6-6822 (CAL 83): Blackbody solution at the Eddington limit ($N_{\rm
H}=8\times 10^{20}$ cm-2, $kT_{\rm BB}=43$ eV, $R_{\rm BB}=1.5\times 10^{9}$
cm, Greiner et al., 1991) is a factor of $\sim 20$ below the far-UV continuum,
while their softer model ($N_{\rm H}=1.7\times 10^{21}$ cm-2, $kT_{\rm BB}=26$
eV) fits the absorbed X-ray data for $\theta_{\rm SSS}\sim 4\times 10^{-13}$
(i.e., $R_{\rm BB}=6.0\times 10^{10}$ cm, $L_{\rm SSS}\sim 2\times
10^{40}$$\rm\,erg\,s^{-1}$) that is a factor of $\sim 30$ above the far-UV
fluxes. Independent modeling of the BeppoSAX spectrum of CAL 83 by Parmar et
al. (1998) also matches well the observed X-ray data, but the unabsorbed model
is by a factor of $\sim 80$ below the far-UV continuum.

RX J0527.8-6954: Similar results were obtained also for this SSS by Greiner et
al. (1991).

Figure 6 compares our SED model for RX J0513.9-6951 and that of Popham & di
Stefano (1996), who calculated models of irradiated flared accretion disks
around luminous steady-burning WDs in bright LMC SSSs (CAL 83, CAL 87, and RX
J0513.9-6951) and compared the resulting disk fluxes to optical and UV
observations. To increase the radiation from the disk at these wavelengths,
the disk has to be flared to reprocess sufficiently large amount (more than
1/4) of the WD radiation into the optical and UV (see their Fig. 2). Figure 6
shows their resulting model (gray line) that consists of three components of
radiation: (i) From a 1.2 $M_{\odot}$ steady-burning WD ($\dot{M}_{\rm
acc}=5\times 10^{-7}$ $M_{\odot}\,{\rm yr}^{-1}$, $L_{\rm WD}=1.5\times
10^{38}$$\rm\,erg\,s^{-1}$, $T_{\rm BB}=5\times 10^{5}$ K and $R_{\rm
WD}=1.84\times 10^{9}$ cm), (ii) from the irradiated disk, whose heated
surface increases the optical and UV fluxes by a factor of 3–5, while the
heated star-disk boundary makes much brighter the high-energy spectrum, and
(iii) from the irradiated fraction of the donor star that contributes mainly
in the optical spectrum. For a comparison, the disk spectrum with no external
heating corresponding to the above-mentioned parameters (i.e., the accretion
luminosity of $2.7\times 10^{36}$$\rm\,erg\,s^{-1}$) and the outer disk edge
of $1.35\times 10^{11}$ cm (Popham & di Stefano, 1996) is shown by the dashed
line222For the sake of simplicity, we adopted optically thick disk that
radiates locally like a blackbody (Warner, 1995).. Note that a more
sophisticated calculation performed by Popham & di Stefano (1996) provides
similar profile of the disk spectrum, but it is just somewhat shifted to
shorter wavelengths (see their Figs. 1 and 2). The Popham & Di Stefano’s model
is calculated for the disk inclination of 60∘. If viewing the disk face-on,
its flux increases twice, and will match better the measured UV and optical
fluxes. However, it is rather below the NIR fluxes and does not reproduce
their slope. The model does not match the X-ray fluxes, at all.

Our multiwavelength SED models of the bright SSSs correspond to super-
Eddington luminosity even for a high-mass WD accretor. The SED from the near-
UV to longer wavelengths indicates a dominant contribution from thermal
nebula. Both results are fairly startling with respect to the present picture
of SSSs, whose luminosities can reach in maximum the Eddington limit for a WD
accretor, and the UV–optical spectrum is thought to be dominated by the light
from accretion disk irradiated by the central steady-burning WD (Popham & di
Stefano, 1996). Below, we discuss these properties in detail.

Figure 6:  Comparison of our SED model for RX J0513.9-6951 and that including
the steady-burning WD at the Eddington limit irradiating the flared accretion
disk and the donor star (gray line, adopted from Fig. 4 of Popham & di
Stefano, 1996). The model is aimed to explain the strong UV and optical
fluxes. Compared is a spectrum of accretion disk with no external heating
(dashed line, Sect. 4.2).

###  4.3 Problem of the high luminosity

####  4.3.1 Accreting white dwarf ?

According to van den Heuvel et al. (1992), the radiation of SSSs is generated
by a stable nuclear burning of hydrogen accreted onto massive WDs. This idea
is based on a theoretical conclusion that there is a small range of accretion
rates, $\dot{M}_{\rm acc}$, at which the hydrogen-rich material is quasi-
steady burning on the WD’s surface as it is accreting (Paczynski & Zytkow,
1978; Fujimoto, 1982). The material is transported onto the accretor trough an
accretion disk. Then, the maximum luminosity of a SSS is given by that
generated by the nuclear burning on the WD surface and the gravitational
potential energy of the accretor, converted into the radiation by the disk and
its boundary layer, i.e.,

$L_{\rm SSS}=\eta X\dot{M}_{\rm acc}+\frac{GM_{\rm WD}\dot{M}_{\rm
acc}}{R_{\rm WD}},$ (5)

where $\eta\sim 0.007\times c^{2}\sim 6.3\times 10^{18}$ erg g-1 is the energy
production of 1 gram of hydrogen due to the nuclear fusion of 4 protons, and
$X\equiv 0.7$ is the hydrogen mass fraction in the accreted matter. $M_{\rm
WD}$ and $R_{\rm WD}$ are the WD mass and radius. However, even the extreme
parameters of the WD accretor, $M_{\rm WD}\sim 1.4$ $M_{\odot}$, $R_{\rm
WD}\sim 0.003$ $R_{\odot}$ (Panei et al., 2000) and the mass transferred
through the disk at a high rate of a few times $10^{-7}$ $M_{\odot}\,{\rm
yr}^{-1}$ that sustains the stable H-burning (see e.g., Fig. 2 of Shen &
Bildsten, 2007) generates the total luminosity of $\gtrsim
10^{38}$$\rm\,erg\,s^{-1}$, which is one order of magnitude below that
suggested by our multiwavelength models SED (Table 3). The observed luminosity
would require a high accretion rate of $\dot{M}_{\rm acc}\sim 10\,\dot{M}_{\rm
Edd}$, where $\dot{M}_{\rm Edd}$ is the Eddington accretion rate. This result
is not consistent with the model of steady burning WD for the brightest SSSs
in LMC and SMC, unless the radiation of the source is not isotropic. Here, the
emission from steady nuclear burning onto a WD should be beamed with a factor,
$L/L_{\rm sph}\lesssim 0.1$, where $L_{\rm sph}$ is the inferred isotropic
luminosity and $L$ is the true source luminosity.

####  4.3.2 Accreting neutron star ?

A neutron star (NS) as the accretor in the brightest SSSs provides a somewhat
better possibility to explain their high luminosities, because of a larger
efficiency in converting gravitational energy into the heat and radiation. The
efficiency of radiative emission in units of the total mass energy accreted on
a compact object is

$\xi=\frac{L_{\rm SSS}}{\dot{M}_{\rm acc}\,c^{2}}=\frac{GM_{\rm acc}}{R_{\rm
acc}\,c^{2}}.$ (6)

For a 1.4 $M_{\odot}$ NS with a radius of 10–15 km, $\xi\sim 0.2-0.14$, while
for a WD (1 $M_{\odot}$, 7000 km), $\xi\sim 2\times 10^{-4}$. Considering that
the efficiency of turning mass energy into radiation via the nuclear fusion is
of $\sim 7\times 10^{-3}$ (Eq. (5)), the nuclear fusion reactions can be a
vital source of energy for accreting WDs, but are negligible in converting the
mass energy accreted by a NS.

To generate the luminosity of $\sim 10^{39}$$\rm\,erg\,s^{-1}$, the above
mentioned parameters of a NS require the accretion rate $\dot{M}_{\rm
acc}=0.85-1.3\times 10^{-7}$ $M_{\odot}\,{\rm yr}^{-1}$. These values are more
realistic than those required by the WD accretor. Such the material can be
supplied by mass transfer from a close Roche-lobe filling companion. However,
these accretion rates are still above the critical values even for a high mass
NS.

The presence of a NS as the accretor in the luminous SSSs could be evidenced
by e.g., detection of strictly periodic short-term light variation on the
time-scale of milliseconds to seconds, or by direct measuring of the size of
the X-ray emission region from eclipse analysis (Mukai, 2017). However, no
such observations indicating these properties of SSSs have been reported yet.
In the cases, where a NS probably accretes from the wind of its companion, we
measure a significantly harder X-ray spectrum dominating 1–10 keV (Masetti et
al., 2007b, a) than it is generated by our brightest SSSs. Therefore, the
presence of accreting NSs in the brightest LMC and SMC SSSs seems unlikely.

####  4.3.3 Accreting black hole ?

The high X-ray luminosities $L_{\rm X}\gtrsim 10^{39}$$\rm\,erg\,s^{-1}$ can
be interpreted by a sub-Eddington accretion onto intermediate-mass black hole
(BH, Freeland et al., 2006). For example, Kong & Di Stefano (2003) applied
this possibility to the luminous SSS XMMU J005510.7–373855 in the spiral
galaxy NGC 300 ($L_{\rm bol}=0.2-2\times 10^{39}$$\rm\,erg\,s^{-1}$), and
found that accretion disk around a $\approx$2800 $M_{\odot}$ BH radiating at
$kT=60$ eV is required to generate the luminosity of
$10^{39}$$\rm\,erg\,s^{-1}$. However, such the massive BHs, that are usually
connected with the star-forming regions (Irwin et al., 2004), are unlikely to
be present in LMC and SMC.

On the other hand, King (2008) considered that the high luminosity of
ultraluminous X-ray sources (ULXs) is generated by a stellar-mass BH accreting
material from a close companion star at super-Eddington rates ($\dot{M}_{\rm
acc}>\dot{M}_{\rm Edd}$) with some degree of geometric beaming. According to
Gladstone et al. (2009), a relatively cool SSS photosphere for the super-
Eddington accretion flows could be created by a massive wind that completely
envelopes the inner-disc regions. They also concluded that all spectral
features in their sample of ULXs can be explained by stellar mass BHs at high
accretion rates.

However, there are two main observational arguments contradicting the presence
of BHs in the brightest SSSs in LMC and SMC.

  * •

First, it is the profile of the X-ray spectrum that is considerably harder
than that produced by our brightest SSSs (Jin et al., 2010; Soria & Kong,
2016). In addition, the spectral shape of ULXs can show either a single peak
at 2-3 keV or two peaks (one below 1 keV and one above 3 keV, see Gladstone et
al., 2009), or otherwise structured shape of the spectrum (Motch et al., 2014;
Urquhart & Soria, 2016) – all totally different from those, here studied,
brightest SSSs.

  * •

Second, if the jets are observed, their velocities are expected to be
comparable to the escape velocity of the central compact object (Livio, 1997).
Accordingly, jets observed from objects containing a BH accretor are
accelerated to relativistic speed (Mirabel & Rodríguez, 1999; Fender et al.,
2006; Takeuchi et al., 2010; Liu et al., 2015), while those launched by a WD
accretor reach speeds of the order of thousands of km s-1 only (Southwell et
al., 1996; Becker et al., 1998; Skopal et al., 2009b).

Therefore, considering the observed X-ray SEDs of our targets, and taking into
account that jets from RX J0513.9-6951 are ejected at $\pm$3800 km s-1
(Southwell et al., 1996), there is no other option than that a WD is present
also in the most luminous SSSs in LMC and SMC.

####  4.3.4 Luminous post-nova supersoft X-ray sources ?

Due to the above reasons, we assume that the accretor in the brightest SSSs,
we investigate in this paper, is a WD. A possibility how to keep a SSS at the
super-Eddington state for a long time is to fuel the burning WD after the
classical nova eruption, when it is in the post-nova SSS state, and is burning
at/above the nuclear Eddington limit, i.e., producing the super-Eddington
luminosity. Below we support this idea by observations of similar cases, and
theoretical considerations.

Examples from observations. A transient post-nova SSS phase occurs after the
optical outburst and is of varying duration. The shortest one is measured for
the 1 yr recurrent nova M31N 2008-12a in the Andromeda galaxy that lasts for
about two weeks (Henze et al., 2018), while in the cases of GQ Mus and LMC
1995, the SSS phase was detected for $\sim$10 yr (Oegelman et al., 1993;
Shanley et al., 1995) and 6–8 yr (Orio et al., 2003), respectively. Long-
lasting super-Eddington luminosity was also documented for some extraordinary
novae, e.g., FH Ser (Friedjung, 1987), LMC 1988 $\\#$1 (Schwarz et al., 1998),
LMC 1991 (Schwarz et al., 2001), SMCN 2016-10a (Aydi et al., 2018), and nova
V339 Del (Skopal, 2019). The unusually long phase of hydrogen burning was
indicated for the nova V723 Cas (1995) by detecting it as a bright SSS more
than 12 years after the outburst, with possible super-Eddington luminosity
(depending on the spectral model, see Ness et al., 2008). The transient SSS in
the nearby galaxy NGC 300, denoted as SSS1, has the bolometric luminosity of
$\approx 10^{39}$$\rm\,erg\,s^{-1}$ (Kong & Di Stefano, 2003). The source was
found in outburst in 1992, 2000, 2008 and 2016 suggesting a possible
recurrence period of about 8 years, and thus could be associated with a
recurrent nova (Carpano et al., 2019). Recently, Vasilopoulos et al. (2020)
discovered a 30-yr long-lived post-nova SSS in LMC undergoing residual surface
nuclear burning. A large number of post-nova SSSs (250–600) was identified in
M31 (Soraisam et al., 2016). The authors found that depending on the WD mass
distribution in novae, their unabsorbed X-ray luminosity distribution shows
significant steepening around $L_{\rm X}{\rm(0.2-1.0\,keV)}\approx
10^{38}$$\rm\,erg\,s^{-1}$ with a maximum at $\approx 2\times
10^{38}$$\rm\,erg\,s^{-1}$. According to our multiwavelength SED modeling,
$L_{\rm SSS}/L_{\rm X}{\rm(0.2-1.0\,keV)}\gtrapprox 10$ (see Table 3), which
would correspond to the bolometric luminosities of the brightest SSSs in M31
of $\approx 10^{39}$$\rm\,erg\,s^{-1}$. Note, however, that this ratio is a
strong function of the temperature: $L_{\rm SSS}/L_{\rm
X}{\rm(0.2-1.0\,keV)}\sim 21$ or $\sim$2.4 only for 300 000 or 600 000 K,
respectively. Finally, long-term accretion at/above the nuclear Eddington
limit is also indicated for symbiotic X-ray binaries RXJ0059.1-7505 (LIN 358)
in SMC (Skopal, 2015; Kuuttila et al., 2021; Washington et al., 2021) and
Draco C1 in the Draco dwarf spheroidal galaxy (Saeedi et al., 2018; Lewis et
al., 2020), where the donor star is an evolved red giant.

Theoretical considerations. According to Kovetz et al. (1988) the irradiation
of the donor star after the nova outburst can cause the red dwarf to expand
and induce a mass transfer rate enhanced by two orders of magnitude. If the
donor is a red giant (symbiotic binaries), the high wind-mass-transfer
efficiency is achieved by the wind focusing to the orbital plane (Skopal &
Cariková, 2015; Shagatova et al., 2016, 2021). In this case, the enhanced
accretion can be sustained also through the disk warping that leads to
ejection of transient jets (Skopal et al., 2018). If the accretion resumes
immediately at the beginning of the post-nova SSS phase, the SSS lifetime can
be substantially lengthened (Kato et al., 2017). Illustration of this effect
is in Fig. 15 of Henze et al. (2018). At a high accretion rates ($\gtrsim
5\times 10^{-7}$ $M_{\odot}\,{\rm yr}^{-1}$), so that a new outburst can occur
on the human lifetime, the SSS phase can be interrupted for a relatively short
time by the optical outburst being followed by the SSS phase at very high
bolometric luminosity. Such the type of variability has been observed for the
SSS SSS1 in NGC 300 (see above), and here for RX J0527.8-6954, where the
repeatability of its visibility is indicated on the timescale of $\approx 20$
years (Sect. 1.4).

According to the above-mentioned reasons we suggest that the brightest SSSs
could be unidentified optical novae that are in a post-nova SSS state, when
the residual surface nuclear burning is supported by rapid re-accretion from
the donor, which can help in sustaining the high luminosity of the burning WD
for a long time. The lifetime of the high luminosity, possibly super-
Eddington, will then depend on the energy output from both the residual
burning and the resumed accretion.

###  4.4 Mass-loss rate from the nebular emission

The high luminosities of SSSs, whose WDs probably accrete at a high rate, lead
to a mass-outflow in the form of a wind (e.g., Kato & Hachisu, 1994; Hachisu &
Kato, 2001, and Appendix A here). According to Kato & Hachisu (1994) the
optically thick/thin interface of the wind represents the SSS pseudo-
photosphere. In our cases, its average value is $R_{\rm SSS}^{\rm eff}\sim
0.17$ $R_{\odot}$ (Table 3). The optically thin wind above the hot and
luminous pseudo-photosphere is ionized, giving rise to the nebular radiation
(see Appendix A). Assuming that the wind flows out spherically symmetrically
at a constant velocity $v_{\infty}$, begins at the $R_{\rm SSS}^{\rm eff}$,
and its radial density distribution satisfies the mass continuity equation,
then the relationship between the mass-loss rate of the ionized wind,
$\dot{M}_{\rm SSS}$, and its emission measure, $EM$, can be expressed as,

$\dot{M}_{\rm SSS}=\left[4\pi\,(\mu m_{\rm H}v_{\infty})^{2}R_{\rm SSS}^{\rm
eff}EM\right]^{1/2}{\rm gs^{-1}},$ (7)

(see Eq. (7) of Skopal et al., 2014), where $\mu$ is the mean molecular weight
and $m_{\rm H}$ is the mass of the hydrogen atom. This relation assumes that
the outer radius of the ionized wind $\gg R_{\rm SSS}^{\rm eff}$. The average
value of $R_{\rm SSS}^{\rm eff}=0.17$ $R_{\odot}$ and $EM\gtrsim 10^{60}$ cm-3
(Table 3) yield $\dot{M}_{\rm SSS}\gtrsim 2\times 10^{-6}$ $M_{\odot}\,{\rm
yr}^{-1}$ for $v_{\infty}\equiv 1000$ km s-1.

Using the same assumptions as for the nebular continuum, similar values of
$\dot{M}_{\rm SSS}$ can be obtained also from hydrogen emission lines, because
they are formed in the same volume of the ionized hydrogen as the continuum.
For example, for the measured flux in the H$\beta$ line, $F_{{\rm H}\beta}$,
we can express its luminosity as,

$L_{{\rm H}\beta}=4\pi d^{2}F_{{\rm H}\beta}=h\nu_{\beta}\alpha({\rm
H}\beta,T_{\rm e})\int_{R_{\rm SSS}^{\rm eff}}^{\infty}\\!\\!n_{\rm
e}(r)n_{\rm p}(r)\,{\rm d}V,$ (8)

where $\alpha({\rm H}\beta,T_{\rm e})$ is the effective recombination
coefficient for the H$\beta$ transition, and $n_{\rm e}$ and $n_{\rm p}$ is
the concentration of electrons and protons in the spherical H+ zone with the
inner radius $R_{\rm SSS}^{\rm eff}$, from which the wind can become optically
thin in the H$\beta$ line. The integral in Eq. (8) represents the emission
measure of the H+ region (see Sect. 3.1). Using its expression from Eq. (7),
we obtain,

$\dot{M}_{\rm SSS}=\left[4\pi\,(\mu m_{\rm H}v_{\infty})^{2}R_{\rm SSS}^{\rm
eff}\frac{L_{{\rm H}\beta}}{h\nu_{\beta}\alpha({\rm H}\beta,T_{\rm
e})}\right]^{1/2}{\rm gs^{-1}},$ (9)

which corresponds to a lower limit of $\dot{M}_{\rm SSS}$, because in a line
transition the wind becomes optically thin above the photosphere, at distances
$>R_{\rm SSS}^{\rm eff}$. This problem was treated by Leitherer (1988), who
found that stellar winds of luminous hot stars become optically thin in the
H$\alpha$ line from a distance of $\sim$1.5 times the star’s radius.
Accordingly, $F_{{\rm H}\beta}=1.45\times 10^{-13}$$\rm\,erg\,cm^{-2}\,s^{-1}$
measured in the spectrum of LIN 333 by Aller et al. (1987) corresponds to
$\dot{M}_{\rm SSS}\sim 1.7\times 10^{-6}$ $M_{\odot}\,{\rm yr}^{-1}$ for
$\alpha({\rm H}\beta,30000)=1.088\times 10^{-14}$ cm3 s-1 (Hummer & Storey,
1987), $R_{\rm SSS}^{\rm eff}=0.17$ $R_{\odot}$ and $v_{\infty}\equiv 1000$ km
s-1. These observational values of $\dot{M}_{\rm SSS}$ are also consistent
with those suggested by the optically thick wind theory, for winds from the
nuclear-burning WDs of a high mass (see Eq. (23) of Kato & Hachisu, 1994). It
is of interest to note that wind mass-loss rate of the order of $10^{-6}$
$M_{\odot}\,{\rm yr}^{-1}$ was measured during the long-lasting SSS-phase of
the classical nova V339 Del, whose bolometric luminosity was kept at a high
level of $1-2\times 10^{39}$$\rm\,erg\,s^{-1}$ for $\sim$150 days (see Fig. 12
of Skopal, 2019).

###  4.5 Dense unresolved circumstellar nebulae in SSSs

According to the mass continuity equation the particle density of the wind
$n(r)\propto r^{-2}$, where $r$ is the radial distance from the source of the
wind. For example, $n(1\,R_{\odot})\sim 9\times 10^{12}$ cm-3, while
$n(1\,AU)\sim 2\times 10^{8}$ cm-3 for wind parameters given in Sect. 4.4.
Further, the emitted flux by the wind at the distance $r$ is proportional to
$n^{2}$ (e.g., Eq. (8)), so the densities in our example correspond to the
flux ratio of the wind layers,
$F(1\,AU)/F(1\,R_{\odot})=4\pi(1\,AU)^{2}n(1\,AU)^{2}/4\pi(1\,R_{\odot})^{2}n(1\,R_{\odot})^{2}\sim
2\times 10^{-5}$. This implies that the vast majority of the nebular emission
is produced within a relatively small central part of the ionized wind – from
its optically thick/thin interface up to a few AU. Such dimensions of nebular
medium are similar to those generated by accreting nuclear-burning WDs in
symbiotic stars (Skopal, 2005), and were also considered by Nielsen & Gilfanov
(2015) in calculations the circumstellar mass-loss rates required for
obscuration of supersoft X-ray sources. For the purpose of our work, it is
important to note that such a small size of the circumstellar nebulae in SSSs
at the distance of LMC and SMC is not spatially resolvable by the current
observational technique. Further, the circumstellar nebulae generated by the
ionized wind in a binary are also very dense. In our cases, the emission
measure ($\sim n^{2}V$) of a few times $10^{60}$ cm-3 (Table 3) corresponds to
the mean particle density of a few times 109 cm-3. Such the dense nebulae are
characterized with a specific type of the emission line spectrum, whose fluxes
and their theoretical ratios significantly differ from those produced by the
low-density planetary nebulae and/or ISM nebulae predicted to be photoionized
by the central SSS (see suggestions for future work in Sect. 5). Here, for a
comparison with a dense circumstellar nebula in the symbiotic star AX Per,
Skopal et al. (2001) found the electron concentration in the H II and [O III]
zone surrounding its burning WD to be of a few times $10^{9}$ and $\approx
3\times 10^{7}$ cm-3, respectively. Further, they suggested that the extremely
steep Balmer decrement (e.g., H$\alpha$/H$\beta$ $\sim 7-10$) could be due to
electron collisions in a very dense H II region.

Finally, we note that the dense unresolvable circumstellar nebulae can be
indicated for a given object only by the method of disentangling its composite
spectrum (Sect. 3.1).

###  4.6 Origin of the X-ray/optical flux anticorrelation

The presence of a strong nebular radiation in the UV to NIR spectrum of our
targets ($EM\gtrsim 2\times 10^{60}$ cm-3, Table 3) and its variation (Fig. 2,
Sect. 3.3) allow us to explain the anticorrelation between their supersoft
X-ray and optical fluxes by a variable mass-outflow from the accreting burning
WD as in the case of the symbiotic binary AG Dra (see González-Riestra et al.,
2008; Skopal et al., 2009a). It is assumed that the variable mass-outflow
results from a variable mass-transfer. Charles et al. (2010) suggested a
similar view for the SSS RX J0513.9-6951, and also estimated the mass outflow
rate of $\approx 10^{-7}$ $M_{\odot}\,{\rm yr}^{-1}$ for the luminosity at the
Eddington limit.

An increase of the mass-loss increases the particle density above the hot WD’s
pseudo-photosphere, and thus the number of b–f absorptions. This leads to a
decrease of the supersoft X-ray photons, and consequently to an increase of
f–b and f–f transitions that causes an increase of the nebular emission, which
dominates the spectrum from the near-UV to longer wavelengths.

In this way the enhanced (wind) mass-outflow increases the optical depth for
the supersoft X-ray photons, and thus increases its optically thick/thin
interface, which is the WD’s pseudo-photosphere. That is why we indicate a
larger effective radius of the SSS during the X-ray-off/optical-high states.
For RX J0513.9-6951 the $R_{\rm SSS}^{\rm eff}$ radius inflated from
$\sim$0.19 $R_{\odot}$ during the optical-low state to $\sim$0.26 $R_{\odot}$
during the optical-high state (Sect. 3.3, Table 3). Independently, the
connection between the flux anticorrelation and the WD’s radius was
demonstrated for RX J0513.9-6951 by Charles et al. (2010) on the basis of the
XMM-Newton simultaneous soft X-ray, UV and optical observations (see their
Figs. 2 and 3). Finally, the inflated WD can also increase its radiation for
$\lambda>912$ Å (compare the blue lines in the panel a and b of Fig. 2).

In this way, the ionization/recombination process in the variable wind from
the SSS causes the simultaneous presence of the X-ray-on/off and optical-
low/high states. Corresponding changes in the nebular emission can explain
$\sim$1 mag changes in the optical brightness (Sect. 3.3).

##  5 Conclusions and future work

In this paper we performed the multiwavelength modeling of the supersoft
X-ray—NIR SED for the brightest Magellanic Cloud SSSs, RX J0513.9-6951, RX
J0543.6-6822, RX J0058.6-7135 and RX J0527.8-6954. The global SED models
satisfactorily fit the measured multi-band fluxes. The main results of the
modeling can be summarized as follows.

  1. 1.

The SED models revealed that their fundamental parameters, $L_{\rm SSS}$,
$T_{\rm BB}$ and $R_{\rm SSS}^{\rm eff}$ are $\gtrsim
10^{38}-10^{39}$$\rm\,erg\,s^{-1}$, $\approx 3\times 10^{5}$ K and $\sim 0.17$
$R_{\odot}$, respectively. Their supersoft radiation is attenuated with the
total $N_{\rm H}\sim(7-12)\times 10^{20}$ cm-2. The modeling identified the
nebular component of radiation with $EM\gtrsim 2\times 10^{60}$ cm-3 (Table
3). Luminosities of our targets differ significantly from those inferred
previously from modeling the X-ray data only (Sect. 4.2, Fig. 5).

  2. 2.

In spite of super-Eddington luminosities for a 1.4 $M_{\odot}$ compact object,
the accretor has to be a WD, because the X-ray spectrum profile and a low
speed of jets are not consistent with those of accreting NS or BH (Sects.
4.3.2 and 4.3.3).

  3. 3.

The high luminosities lead to the mass-loss at high rates. Assuming that the
nebular component of radiation is produced by the ionized wind from SSSs, its
emission measure corresponds to the mass-loss rate $\gtrsim 2\times 10^{-6}$
$M_{\odot}\,{\rm yr}^{-1}$ (Sect. 4.4). The ionized wind represents a dense
unresolved circumstellar nabula in SSSs (Sect. 4.5, Appendix A).

  4. 4.

Accordingly, we suggest that the brightest SSSs in LMC and SMC could be
unidentified optical novae in a post-nova SSS state, whose lifetime and
luminosity are enhanced by the resumed accretion, possibly at super-Eddington
rates (Sect. 4.3.4).

  5. 5.

The observed anticorrelation between the supersoft X-ray and optical fluxes
can be caused by a variable mass-outflow from the SSS due to variations in the
mass-transfer. A higher mass-loss absorbs more X-ray photons that are re-
emitted in the form of the nebular emission, and vice versa. As a result, we
observe simultaneously the X-ray-off/on and optical-high/low states (Sect.
4.6).

For future investigation, we suggest to use the multiwavelength modeling
including simultaneously obtained fluxes from both ascending and descending
part of the SSS spectrum. In this way to determine more reliable parameters
than can be obtained by modeling only one part of the spectrum. To test the
proposed model, and develop its more perfect version we suggest to acquire new
observations, and using better theoretical modeling. In particular the
following.

  * •

Flux calibrated low-resolution optical/NIR spectrum ($\lambda\gtrsim 350$ nm)
should verify the presence of the nebular continuum in the spectrum by its
specific profile and the presence of the Balmer jump in emission. Moreover,
for LIN 333, such the spectrum should confirm or refute its binary nature
proposed in Appendix C.

  * •

High-resolution spectra should directly indicate a high-velocity mass-outflow
by (presumably) broad wings of the strongest lines as H$\alpha$, H$\beta$ and
He II $\lambda$4686\. In addition, for RX J0513.9-6951, the presence of
satellite components to these lines would indicate a collimated bipolar
outflow resulting from a very high accretion rate onto the WD (see Southwell
et al., 1996).

  * •

In accordance with the study of Nielsen & Gilfanov (2015), using the
photoionization code CLOUDY, determine the critical value of the mass-loss
rate, $\dot{M}_{\rm crit}$, from the burning WD that causes the obscuration of
the central X-ray source for parameters determined by our modeling. The
corresponding $EM$ should be comparable with that obtained from the SED
modeling during the X-ray-off states. In this way to test the origin of the
X-ray/optical flux anticorrelation by a variable wind from the SSS proposed in
Sect. 4.6.

Furthermore, the calculated emission line spectrum produced by the ionized
wind from the X-ray source should provide additional constraints on the SSS
environment. By analogy with symbiotic stars, a dense (unresolved) nebula,
here suggested also for SSSs by our SED modeling (Sect. 4.5; Appendix A),
should be characterized by an anomalous Balmer decrement (e.g.,
H$\alpha$/H$\beta$$\gg 3$), and a very low rate of the nebular [O III] lines
to [O III] $\lambda$4363 – in strong contrast to values produced by the low-
density planetary nebulae (e.g., Gurzadyan, 1997), and/or nebulae predicted to
exist around SSSs imbedded in the low-density ISM (Nielsen & Gilfanov, 2015,
Sect. 4.5 here).

  * •

In modeling the global SED, consider also the component of radiation from the
irradiated accretion disk proposed by Popham & di Stefano (1996). Its possible
contribution in the far-UV will reduce the radiation from the burning WD here,
and thus its bolometric luminosity, and amount of the required absorption.

These tasks present new challenges for further observations and theoretical
modeling, which should help us in understanding the nature and evolutionary
status of the most luminous SSSs.

The author thanks the anonymous referee for constructive comments. This
research is based on observations made with the Far Ultraviolet Spectroscopic
Explorer, International Ultraviolet Explorer obtained from the MAST data
archive at the Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS 5–26555, and on observations made with the NASA/ESA Hubble Space
Telescope, and obtained from the Hubble Legacy Archive, which is a
collaboration between the Space Telescope Science Institute (STScI/NASA), the
Space Telescope European Coordinating Facility (ST-ECF/ESAC/ESA) and the
Canadian Astronomy Data Centre (CADC/NRC/CSA). DENIS is the result of a joint
effort involving human and financial contributions of several Institutes
mostly located in Europe. It has been supported financially mainly by the
French Institut National des Sciences de l’Univers, CNRS, and French Education
Ministry, the European Southern Observatory, the State of Baden-Wuerttemberg,
and the European Commission under networks of the SCIENCE and Human Capital
and Mobility programs, the Landessternwarte, Heidelberg and Institut
d’Astrophysique de Paris. 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. This research was
supported by the Slovak Research and Development Agency under the contracts
No. APVV-15-0458 and APVV-20-0148, and by the Slovak Academy of Sciences grant
VEGA No. 2/0030/21.

##  Appendix A Justification of nebular continuum in the spectrum of the
supersoft X-ray sources

The nebular component of radiation has not yet been considered in modeling the
SED of SSSs. We mean the spatially unresolved source of nebular emission, the
presence of which in the spectrum is indicated by multiwavelength modeling of
the SED (Sect. 3.1 and Fig. 1) as in the case of dense nebulae in symbiotic
binaries and/or classical novae (see Skopal, 2005, 2015, 2019, Sect. 4.5
here). Its presence also in the spectrum of SSSs can be justified as follows.

  1. 1.

It is probable that extremely luminous and hot SSSs will generate an outflow
of material driven by radiation pressure as it is observed and theoretically
justified for luminous hot stars (e.g., Castor et al., 1975; Klein & Castor,
1978), accreting WDs in symbiotic binaries (e.g., Vogel & Nussbaumer, 1994;
Skopal, 2006; Skopal et al., 2017), classical novae (e.g., Friedjung, 1966;
Bath & Shaviv, 1976; Kato & Hachisu, 1994; Skopal, 2019), and in SSSs (e.g.,
Pakull et al., 1985, 1993; Charles et al., 2010; Oliveira et al., 2010).
Simultaneously, the hot and luminous central WD and the boundary layer of
accretion disk generate a large flux of hydrogen ionizing photons ($\sim
10^{48}$ s-1 for $T_{\rm BB}\sim 300\,000$ K and $L_{\rm SSS}\sim
10^{38}$$\rm\,erg\,s^{-1}$), which ionizes the out-flowing material giving
rise to nebular emission. This reprocessed radiation dominates the Rayleigh-
Jeans tail of a hot source radiating at $T_{\rm BB}\gtrsim 10^{5}$ K, usually
from the near-UV to longer wavelengths (see Skopal, 2005; Nielsen & Gilfanov,
2015). This nebular component of radiation is aimed on explaining the strong
UV–NIR excess observed for bright SSSs.

  2. 2.

Below we summarize indications of the nebular emission observed in the
spectrum of our targets.

     * •

Mass-loss from the accreting nuclear-burning WD is indicated by P-Cyg profiles
of H lines, their often strong flux and broad emission wings, and/or satellite
components to main H$\beta$ and He II $\lambda$4686 emission cores as for RX
J0513.9-6951 (see references above and those in Sects. 1.1 to 1.4). Therefore,
the H$\alpha$ line flux as a tracer of mass loss from hot stars has been
usually used to estimate its rate (e.g., Klein & Castor, 1978; Leitherer,
1988; Lamers & Cassinelli, 1999; Skopal et al., 2002; Skopal, 2006). The out-
flowing material ionized by the central hot source then gives rise to nebular
emission.

     * •

During the optical high state of RX J0513.9-6951, the resonance O VI
$\lambda$1032, $\lambda$1038 doublet shows broad wings due to electron
scattering, with the superposed central P Cyg-type of the profile (see
Hutchings et al., 2005). By comparison with accreting nuclear-burning WDs in
symbiotic stars, such the wings are created in the layer of electrons,
throughout which the line photons are transferred, with the electron optical
depth $\tau_{\rm e}\sim 0.05-0.6$, depending on the system and its activity
(see Sekeráš & Skopal, 2012, and Fig. 7 here for illustration). As there is no
resolved ionization nebula surrounding RX J0513.9-6951, the layer of free
electrons has to be spread in the vicinity of the ionizing source, which
indicates the presence of a compact (unresolved) circumstellar nebula in RX
J0513.9-6951, independently to our SED model. For a comparison, in the case of
the X-ray symbiotic binary AG Dra the layer of scattering electrons extends to
$\approx$10 – 100 $R_{\odot}$, which corresponds to the mean electron
concentrations of $10^{10}-10^{12}$ cm-3 to satisfy the measured $\tau_{\rm
e}$ (see Skopal et al., 2009a).

     * •

The presence of emission lines directly indicates the presence of the
corresponding nebular continuum, because both continuum and lines originate in
the same ionized volume. Example here is the case of RX J0058.6-7135 (LIN
333), for which Aller et al. (1987) revealed the presence of a strong nebular
emission radiating at a high $T_{\rm e}\sim 30000$ K. In particular, the
measured (extinction free) flux in the H$\beta$ line, $F_{{\rm
H}\beta}=1.45\times 10^{-13}$$\rm\,erg\,cm^{-2}\,s^{-1}$ corresponds to the
emission measure,

$\textsl{EM}=4\pi d^{2}\frac{F_{{\rm H}\beta}}{h\nu_{\beta}\,\alpha({\rm
H}\beta)}=1.4\times 10^{60}\,(d/60\,{\rm kpc})^{2}\,{\rm cm}^{-3},$ (A1)

which agrees well with our value determined from the nebular continuum (see
Table 3)333Note that values of parameters derived from the emission lines are
usually smaller than those derived from the continuum, because nebulae are
usually more opaque in the lines than in the continuum.. In Eq. (A1), we used
an effective recombination coefficient for the H$\beta$ transition and $T_{\rm
e}=30000$ K, $\alpha({\rm H}\beta)=1.088\times 10^{-14}$ cm3 s-1 (Hummer &
Storey, 1987).

     * •

Often measured flux ratio F(He II $\lambda$4686)/F(H$\beta$)$\gg$1 suggests
the presence of the nebular continuum from doubly ionized helium, we can
indicate as a discontinuity at $\sim 2050$ Å (see Figs. 1c, 2a and 8).

     * •

In spite that the He++ region around strong SSSs should be especially
prominent as compared to other astrophysical nebulae (Rappaport et al., 1994),
its presence within the resolved ionization nebula around CAL 83 had not been
detected for a long time (Remillard et al., 1995). The first evidence of an He
II$\lambda$4686 region around CAL 83 was presented by Gruyters et al. (2012)
as a faint (relative to its bright stellar component) asymmetrically
distributed zone confined to one side of CAL 83. In addition, the authors
found that the H II emission did not fit in with model predictions. This
unusual He II$\lambda$4686 nebular emission could be explained by the presence
of a compact (unresolved) circumstellar nebula around the burning WD of CAL
83, whose He++ zone is almost ionization-bounded, i.e., almost all photons
with $\lambda<228$ Å are absorbed within this zone. For a comparison, in the
case of the eclipsing symbiotic star AX Per, the He++ region extends up to
$\sim$50 $R_{\odot}$ from the WD within its dense wind escaping at a rate of
$\sim 2\times 10^{-6}$ $M_{\odot}\,{\rm yr}^{-1}$ and generating the total
emission measure of a few times $10^{59}$ cm-3 (from the eclipse profile, see
Skopal et al., 2011).

     * •

Similarly to the above point, the circumstellar nebulae in the brightest SSSs
as suggested by our SED models could attenuate also the flux of H-ionizing
photons to such an extent that it would not allow us to detect the
corresponding extended ISM nebulae in the optical. From this point of view,
the strange absence of ionization ISM nebulae surrounding the brightest SSSs
in LMC and SMC, up to CAL 83 (Remillard et al., 1995; Woods & Gilfanov, 2016;
Farias et al., 2020) could be explained by the presence of compact
(unresolved) circumstellar nebulae in these objects.

Figure 7:  Broad wings of the resonance O VI $\lambda$1032, $\lambda$1038
doublet observed in the FUSE spectra of LMC SSS RX J0513.9-6951 (top panel)
and of the X-ray symbiotic binary AG Dra with the model of the scattering of
line photons on free electrons (bottom panel, adopted from Sekeráš & Skopal,
2012). The similarity of these profiles suggests the presence of a dense
circumstellar nebula also in RX J0513.9-6951, showing mass-outflow (see text).
Note that the spectra of RX J0513.9-6951 and AG Dra are shifted in wavelengths
due to their space velocity by $\sim$+500 and $\sim$-150 km s-1, respectively
(Hutchings et al., 2005; Fekel et al., 2000).

##  Appendix B CAL 83: The SED of the central source and the surrounding
nebula

Figure 8 shows the UV–optical part of the CAL 83 SED compared with that of the
surrounding inner $7.5\times 7.5$ pc2 ionization nebula. The fluxes of the
latter (red crosses in the figure) were taken from Fig. 2 of Gruyters et al.
(2012). The spectrum of the outer nebula can be matched with the same type of
the nebular continuum (dotted green line) as the circumstellar nebula (solid
green line), but scaled with a factor of $\sim$17 times larger. This suggests
similar electron temperature ($\sim$30000 K), but $\sim$17 times larger
emission measure (i.e., $\sim$4.9$\times 10^{61}$ cm-3) in comparison with the
dense circumstellar nebula (see Table 3, Sect. 4.5).

As noted above, there is a lack of doubly ionized helium in the surrounding
resolved nebula (Appendix A). Therefore, we also compared its fluxes with the
net hydrogen nebular continuum that corresponds to $EM\sim 8.0\times 10^{61}$
cm-3 and the same $T_{\rm e}$ (gray line in Fig. 8). Such the high $EM$
requires a high flux of the ionizing photons. Assuming that all H-ionizing
photons from CAL 83 balances the rate of recombinations in both nebulae, we
can estimate the corresponding $L_{\rm SSS}$ for the temperature of the
ionizing source $T_{\rm BB}$ according to expression,

$L_{\rm SSS}=\alpha_{\rm B}({\rm H},T_{\rm e})\,\textsl{EM}\frac{\sigma T_{\rm
BB}^{4}}{f(T_{\rm BB})},$ (B1)

(see Skopal et al., 2017), where the function $f(T_{\rm BB})$ determines the
flux of ionizing photons emitted by 1 cm2 area of the ionizing source, and
$\alpha_{\rm B}({\rm H},T_{\rm e})$ is the recombination coefficient to all
but the ground state of hydrogen (i.e., Case $B$). Then the total $EM\sim
8.29\times 10^{61}$ cm-3 (i.e., of both nebulae), $T_{\rm BB}=345000$ K,
$f(T_{\rm BB})=6.0\times 10^{27}$ cm-2 s-1, and $\alpha_{\rm B}({\rm
H},30000)=9.97\times 10^{-14}$ cm3 (Nussbaumer & Vogel, 1987) correspond to
$L_{\rm SSS}\sim 1.1\times 10^{39}$$\rm\,erg\,s^{-1}$. This value is close to
that given by the SED models, but in the order of magnitude higher than those
derived from the emission lines in previous works (see Sect. 1.3).
Consequently, the very high $EM$ ($8.0\times 10^{61}$ cm-3) of the $7.5\times
7.5$ pc2 nebula surrounding CAL 83 corresponds to its average particle density
$\sim$39 cm-3 and the mass of $\sim$1700 $M_{\odot}$, which are a factor of
$>$4, and a factor of $\sim$11, respectively, larger than those previously
derived from emission lines (e.g., Remillard et al., 1995; Gruyters et al.,
2012). Such the large difference of the surrounding nebula properties can be
given by a large difference in the opacity of the emitting medium in the line
and the continuum transitions, respectively.

Figure 8:  The UV–optical part of the CAL 83 SED from Fig. 1c including
representative flux-points from the resolved $7.5\times 7.5$ pc2 nebula
surrounding CAL 83 (red crosses), taken from Fig. 2 of Gruyters et al. (2012).
They can be matched by nebular continuum with the same $T_{\rm e}$ but
significantly larger $EM$ (dotted green and gray line for H+ \+ He++ and H+
nebula) than the unresolved circumstellar nebula (solid green line; see text).
Meaning of the other lines and symbols as in Fig. 1.

##  Appendix C LIN 333: Planetary nebula or a binary system?

Here we present two reasons suggesting that LIN 333 could be a binary system.

(i) First, we compare the nebular spectrum, both in the continuum and emission
lines, of LIN 333 with that observed in symbiotic binaries containing an
accreting nuclear-burning WD. Panels a and b of Fig. 9 show a comparison with
SY Mus in the 1200 – 2200 Å part of the spectrum. Both, LIN 333 and SY Mus
contain the same emission lines in the far-UV spectrum, except for the low-
excitation O I $\lambda\lambda$1302-1306 lines444The absence of O I and very
faint resonance O VI $\lambda\lambda$1032,1038 doublet probably reflect a low
abundance of oxygen in the LIN 333 nebula suggested by Leisy & Dennefeld
(1996)., which are characteristic for symbiotic stars. Strong resonance
emission lines, N V $\lambda\lambda$1238,1242, C IV $\lambda\lambda$1548,1550,
high-ionization line He II $\lambda$1640, and the intersystem lines of O IV]
$\lambda\lambda$1397-1405, N IV] $\lambda\lambda$1483-1486, O III]
$\lambda\lambda$1661,1666, N III] $\lambda\lambda$1746-1753, Si
III$\lambda$1892 and C III$\lambda\lambda$1907,1909 reflect the circumstellar
nebular conditions of symbiotic stars (Meier et al., 1994), and thus also of
SSS LIN 333.

Also, the nebular continuum in the LIN 333 spectrum is unambiguously indicated
by our method as for other targets (green lines in Fig. 1). As there is no
resolved ionization nebula around LIN 333, the indicated nebular continuum
suggests the presence of a compact unresolved nebula in the system, the origin
of which is similar to that of dense symbiotic nebulae, i.e., resulting from
the interaction between the binary components, given by the ionized mass
outflows (e.g., Boyarchuk et al., 1966; Seaquist et al., 1984; Nussbaumer &
Vogel, 1987; Skopal, 2005). Here, it is of interest to note that the first
studies of symbiotic stars pointed to the similarity of their hot components
to the central star of planetary nebulae (e.g., Berman, 1932), and this
possibility was often considered until the first satellite observations that
unambiguously revealed the binary nature of symbiotic stars (e.g., Kenyon &
Webbink, 1984, and references therein).

Table 4 compares luminosities of the strongest emission lines in the UV
spectrum, $EM$ and $T_{\rm e}$ of the compact nebulae of LIN 333 and SY Mus.

Table 4:  Luminosities of the strongest permitted lines in the UV spectrum (in $\rm\,erg\,s^{-1}$), emission measure $EM$ ( cm-3), and electron temperature $T_{\rm e}$ (K) of LIN 333 and SY Mus. The distance dependent parameters are scaled with 1.0 kpc (Skopal, 2005) and 60 kpc for SY Mus and LIN 333, respectively. The corresponding spectra and models are shown in Fig. 9a and 9b.  Object | $L_{\rm N\,{\small V}\,\lambda 1240}$ | $L_{\rm C\,{\small IV}\,\lambda 1550}$ | $L_{\rm He\,{\small II}\,\lambda 1640}$ | $EM$ | $T_{\rm e}$
---|---|---|---|---|---
LIN 333 | 2.7$\times 10^{35}$ | 5.4$\times 10^{34}$ | 2.9$\times 10^{35}$ | 1.6$\times 10^{60}$ |  $\sim$37000
SY Mus | 2.1$\times 10^{34}$ | 2.2$\times 10^{34}$ | 1.1$\times 10^{34}$ | 3$\times 10^{59}$ |  $\sim$18500

Figure 9:  Left: Comparison of the far-UV spectrum emitted by the SSS LIN 333
(a) and by the symbiotic binary SY Mus (b), dereddened with $E_{\rm B-V}$ =
0.08 and 0.35. The similarity of the nebular spectrum of both systems suggests
their same origin – in a compact circumstellar nebula (see text). Right: The
UV–NIR SEDs of these objects. The case of LIN 333 allows the presence of a
yellow giant companion (orange dashed line) to the SSS, suggesting binary
nature of LIN 333 (see Appendix C). Panels a and c represent details of Fig.
1b, while the panels b and d were adapted according to Fig. 18 of Skopal
(2005).

(ii) According to the $UBV$ photometry and the HST spectroscopy of LIN 333
(Sect. 2.2), the measured fluxes in the visual band lie above our two-
components model (see Fig. 1b), which suggests contribution from the companion
in a binary system. To match the optical continuum of the HST spectrum555We
ignored the $U,B,V$ fluxes, because they are not corrected for emission lines,
and thus lie well above the true continuum.

with a minimum effect on the near-UV region, where the original two-component
model is satisfactory (see Fig. 1b), we selected a synthetic spectrum of a
yellow giant from a grid of models made by Hauschildt et al. (1999),
calculated for $T_{\rm eff}=5000$ K, $\log g=3.5$ and [M/H] = -1. Its scaling
in Fig. 9c corresponds to the luminosity of 278 $L_{\odot}$ and the radius of
22 $R_{\odot}$; parameters similar to a G8 III giant (e.g., van Belle et al.,
1999). If this were the case, LIN 333 would belong to the so-called yellow
symbiotic stars. However, new optical/NIR observations are highly desirable to
confirm or refute the binary nature of LIN 333.

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