11institutetext: Zentrum für Astronomie der Universität Heidelberg,
Astronomisches Rechen-Institut, Mönchhofstr. 12-14, 69120 Heidelberg
11email<EMAIL_ADDRESS>22institutetext: Institut für Physik
und Astronomie, Universität Potsdam, Karl-Liebknecht-Str. 24/25, D-14476
Potsdam, Germany 33institutetext: Anton Pannekoek Institute for Astronomy,
University of Amsterdam, 1090 GE Amsterdam, The Netherlands 44institutetext:
Instituto Universitario de Físca Aplicada a las Ciencias y las Tecnologías,
Universidad de Alicante, E-03690 Alicante, Spain 55institutetext: Sternberg
Astronomical Institute, M.V. Lomonosov Moscow University, Universitetskij pr.
13, 119234 Moscow, Russia 66institutetext: Department of Physics, North
Carolina State University, Raleigh, NC 27695-8202, USA 77institutetext:
Department of Astronomy, University of Geneva, Chemin d’Ecogia 16, CH-1290
Versoix, Switzerland 88institutetext: Space Science Institute, 4750 Walnut
Street, Suite 205, Boulder, CO 80301, USA

# Phase-resolved spectroscopic analysis of the eclipsing black hole X-ray
binary M33 X-7: System properties, accretion, and evolution††thanks: Based on
observations made with the NASA/ESA Hubble Space Telescope (HST), obtained
from the data archive at the Space Telescope Science Institute. STScI is
operated by the Association of Universities for Research in Astronomy, Inc.
under NASA contract NAS 5-26555. These observations are associated with the GO
program 15636. Based on observations obtained with XMM-Newton, an ESA science
mission with instruments and contributions directly funded by ESA Member
States and NASA, these observations are associated with id numbers 0831590201
& 0831590401.

V. Ramachandran 1122 L. M. Oskinova 22 W.-R. Hamann 22 A. A. C. Sander 11 H.
Todt 22 D. Pauli 22 T. Shenar 33 J. M. Torrejón 44 K. A. Postnov 55 J. M.
Blondin 66 E. Bozzo 77 R. Hainich 22 D. Massa 88

M33 X-7 is the only known eclipsing black hole high mass X-ray binary. The
system is reported to contain a very massive O supergiant donor and a massive
black hole in a short orbit. The high X-ray luminosity and its location in the
metal-poor galaxy M33 make it a unique laboratory for studying the winds of
metal-poor donor stars with black hole companions and it helps us to
understand the potential progenitors of black hole mergers. Using phase-
resolved simultaneous HST\- and XMM-Newton-observations, we traced the
interaction of the stellar wind with the black hole. We observed a strong
Hatchett-McCray effect in M33 X-7 for the full range of wind velocities. Our
comprehensive spectroscopic investigation of the donor star (X-ray+UV+optical)
yields new stellar and wind parameters for the system that differ
significantly from previous estimates. In particular, the masses of the
components are considerably reduced to $\approx 38\,M_{\odot}$ for the O-star
donor and $\approx 11.4\,M_{\odot}$ for the black hole. The O giant is
overfilling its Roche lobe and shows surface He enrichment. The donor shows a
densely clumped wind with a mass-loss rate that matches theoretical
predictions. An extended ionization zone is even present during the eclipse
due to scattered X-ray photons. The X-ray ionization zone extends close to the
photosphere of the donor during inferior conjunction. We investigated the
wind-driving contributions from different ions and the changes in the
ionization structure due to X-ray illumination. Toward the black hole, the
wind is strongly quenched due to strong X-ray illumination. For this system,
the standard wind-fed accretion scenario alone cannot explain the observed
X-ray luminosity, pointing toward an additional mass overflow, which is in
line with our acceleration calculations. The X-ray photoionization creates an
He ii emission region around the system emitting $\sim
10^{47}\mathrm{ph\,s^{-1}}$. We computed binary evolutionary tracks for the
system using MESA. Currently, the system is transitioning toward an unstable
mass transfer phase, possibly resulting in a common envelope of the black hole
and the O-star donor. Since the mass ratio is q$\gtrsim$3.3 and the period is
short, the system is unlikely to survive the common envelope, but will rather
merge.

##  1 Introduction

The majority of massive stars are commonly found in binary or higher-order
multiple systems (e.g., Sana et al., 2012, 2014). The evolution of these
massive stars is largely governed by the loss of mass via stellar winds, and
by the exchange of mass among binary components (e.g., Langer, 2012; Postnov &
Yungelson, 2006). When the initially more massive (i.e. primary) component in
a binary system collapses into either a neutron star (NS) or a black hole (BH)
at the end of its evolution, some systems remain bound after the supernova
explosion. The accretion of mass transferred from the former secondary star
onto its compact companion can power strong X-ray emission (Shklovsky, 1967).
Such systems are called high-mass X-ray binaries (HMXBs). The majority of
HMXBs contain NSs, while systems containing BHs are quite rare. In this paper,
we focus on M33 X-7, an HMXB with a BH companion.

Black hole X-ray binaries (BH XRBs) are essential natural laboratories for
studying stellar-mass BHs. These systems provide an indispensable
observational test for BHs that are formed by core collapse, therefore
providing observational constraints for binary evolution and compact object
formation. The majority of the detected BH XRBs have low-mass donor stars and
are X-ray transients (Remillard & McClintock, 2006; Özel et al., 2010; Corral-
Santana et al., 2016). BH XRBs with massive donor stars are persistently
strong X-ray sources that are powered by massive stellar winds and/or Roche
lobe overflow. Cyg X-1 was the first discovered BH HMXB system (BH + O9I
companion) and it remains the only confirmed one in the Milky Way (e.g.,
Paczynski, 1974; Gies & Bolton, 1986; Miller-Jones et al., 2021a). Other BH
HMXB candidates in the Galaxy are SS433 (Cherepashchuk et al., 2020), Cyg X-3
(Zdziarski et al., 2013), and MWC656 (Casares et al., 2014). In the extra-
galactic population of BH HMXBs, dynamical evidence has been presented for LMC
X-1 (BH + O7 III companion; Orosz et al., 2009), LMC X-3 (BH + B3–5, V star
Orosz et al., 2014), and M33 X-7 (BH with an O7-8 III companion; Orosz et al.,
2007). There is indirect evidence for the presence of a BH with a Wolf-Rayet
companion in NGC300 X-1 (Binder et al., 2021; Crowther et al., 2010) and IC10
X-1 (Laycock et al., 2015; Silverman & Filippenko, 2008) systems.

If the donor star in a BH HMXB is massive enough, its core collapse can result
in the formation of a relativistic binary comprising two BHs. In the case of a
tight system, the binary components may merge within a Hubble time. Such a
merger may be observed as a gravitational wave (GW) event (e.g., Abbott et
al., 2016a, b, 2017). Thus, HMXBs with BHs represent a key transitional stage
toward a binary BH merger. It is crucial to understand the physics of these
systems and their dependence on metallicity in order to establish whether the
BHs in present-day massive binaries and those detected by GWs during a merger
are part of the same stellar progenitor pool.

In HMXBs, accretion onto the compact object is typically powered by the
stellar wind of the donor star. For Bondi-Hoyle-Lyttleton accretion (Bondi &
Hoyle, 1944; Davidson & Ostriker, 1973; Edgar, 2004), the X-ray luminosity
crucially depends on the wind parameters of the donor star. In particular,
Bondi-Hoyle mass accretion is highly dependent on the velocity at which the
donor wind is captured by the gravitational field of the compact object
($\varv^{-3}$, from Bondi & Hoyle, 1944). The X-ray luminosity is directly
dependent on the accretion rate and accretion mode and can vary rapidly due to
the complexity of the stellar wind and the possible clumping of material.
Hence it is of the utmost importance to understand the wind properties of the
massive donor. Sophisticated spectral analyses of the donor stars using state-
of-the-art model atmospheres to determine the stellar and wind parameters of
the donor stars in HMXBs are therefore a pivotal instrument for a better
understanding of these systems.

The winds of OB stars are driven by the scattering of UV photons in metal
lines (Castor et al., 1975a). At lower metallicity, the winds of the OB stars
are much weaker (Bouret et al., 2015; Ramachandran et al., 2019) and hence
remove less mass and angular momentum during stellar evolution. Thus, a metal-
poor star is expected to undergo its core collapse at a higher mass and at
faster rotation than a star with initially the same mass, but higher
metallicity. As of yet, virtually nothing is known about the winds of metal-
poor donor stars in HMXBs with BH companions. Our current understanding of the
wind of massive stars in BH binaries is mostly based on the Cygnus X-1 system
in our Milky Way. Yet, this prototypical system has not been analyzed with the
current generation of expanding stellar atmosphere codes. Spectroscopic
analyses of the BH HMXBs were limited to the Sobolev with Exact Integration
(SEI) method (e.g., Cyg X-1 in Gies et al., 2008; Vrtilek et al., 2008) or
using plane-parallel models (e.g., for M33 X-7 by Orosz et al. 2007 and for
Cyg X-1 by Caballero-Nieves et al. 2009) which are optimized for hot stars
with no significant wind. More advanced analyses of donors done in the recent
past are limited to NS HMXB systems. For example, a sample of HMXBs with NS
companions was recently studied by Hainich et al. (2020) with detailed
expanding stellar atmosphere models for radiation-driven winds. In this study,
we carry out a detailed atmospheric analysis of multiwavelength spectra of the
metal-poor donor star in M33 X-7, taking into account line-driven winds, wind
clumping, complex effects of millions of spectral lines, and X-ray
photoionization.

Table 1: M33 X-7 system parameters from the literature used in this work.

Parameter | Value | Ref.
---|---|---
$d$ [kpc] | 840 $\pm$ 40 | 1,2,3
$m_{\mathrm{v}}$ [mag] | 18.9 | 1
$L_{\mathrm{x}}$ (0.5-5 kev) [erg s-1] |  $\sim$(0.02-2) $\times 10^{38}$ | 1,4,5
$T_{\mathrm{0}}$ [HJD] | $2,453,967.157\pm 0.05$ | 1
$P_{\mathrm{orb}}$ [days] | 3.45301 | 1,4
systemic velocity [km s-1] | $152\pm 5$ | 1
eccentricity $e$ | $0.0185\pm 0.0077$ | 1
$f(M)$ [$M_{\odot}$] | $0.46\pm 0.08$ | 1

(1) Orosz et al. (2007); (2) Galleti et al. (2004); (3) Gieren et al. (2013);
(4) Pietsch et al. (2006); (5) Pietsch et al. (2004).

### The M33 X-7 system

M33 X-7 is reported to contain one of the most massive known stellar mass BHs
with a very massive O7 III star in a tight binary orbit (Orosz et al., 2007).
The system is located in the metal-poor galaxy of Messier 33 (M33). M33 is
known to have a steep metallicity gradient (Cioni, 2009; Magrini et al.,
2007). The M33 X-7 system is located in an OB association (HS 13)
approximately $2$ kpc from the center of M 33. Based on H ii region abundance
measurements by Magrini et al. (2007), the metallicity at this galactic radius
is roughly half solar.

Importantly, M33 X-7 is the only known eclipsing BH HMXB (Pietsch et al.,
2004, 2006). Pietsch et al. (2006) derived a period of 3.45 days and an X-ray
luminosity of $5-11\times 10^{37}$ erg s-1 out of eclipse. The study conducted
by Orosz et al. (2007) provided a dynamical estimate of the mass of the
system, reporting a $\approx\\!70M_{\odot}$ O supergiant donor, and an
$\approx\\!16M_{\odot}$ BH. The mass of the donor is very high for an O7 III
spectral type (Markova et al., 2018; Ramachandran et al., 2018) as well as
compared to O stars in other BH HMXBs. Such high masses for the donor and
compact companion are exceptional among HMXBs, making M33X-7 a unique
laboratory for studying the potential progenitors of massive BH mergers
detected by LIGO.

Despite a rather tight orbit, it was reported that the O-type donor does not
fill its Roche lobe radius ($R_{\ast}\approx 0.77R_{\mathrm{rl}}$, Orosz et
al., 2007), and therefore the X-ray luminosity is expected to be powered by
accretion of the donor’s stellar wind alone. Orosz et al. (2007) estimated the
donor mass-loss rate from the X-ray luminosity by adopting an ad hoc accretion
efficiency. The estimated mass-loss rate turned out to be five times higher
than theoretically expected. Furthermore, an ad hoc wind velocity law (and
consequently the radial density structure of the wind) was adopted for the
donor star. On the basis of these assumptions, the eclipse width was measured
from the light curve with a suggested accuracy of 1 degree in terms of orbital
phase angle. As a result, it was claimed that the component masses are
estimated with 10% accuracy.

M33 X-7 has been subject to evolutionary modeling in past studies (Abubekerov
et al., 2009; Valsecchi et al., 2010; de Mink et al., 2010; Bogomazov, 2014).
A large theoretical effort is ongoing to explain the large masses of the
M33X-7 components, its tight orbit, and the large BH spin. Measuring the
stellar wind parameters of M33 X-7 will significantly constrain evolutionary
models and allow unique tests of the evolution of BH binaries at low
metallicity.

For this study, we have secured quasi-simultaneous multiwavelength
spectroscopy of M33 X-7 at three key binary orbital phases (see Fig. 1). Using
the combined analysis of the multiwavelength data (UV, optical, and X-rays) we
measured the properties of the donor star wind and the mass accretion rate
onto the BH, and we study the effect of X-rays on the donor star wind. These
wind parameter measurements are required to understand how much wind is
quenched by the X-rays, so as to test stellar wind theory, refine the
measurements of the BH mass, and determine the accretion efficiency onto the
BH. Subsequently, our new realistic wind parameters serve as input for
sophisticated massive binary evolution models. The basic parameters of the
system which are used in this study from the abovementioned literature are
listed in Table. 1. Throughout the paper, we assume a distance of 840 kpc to
M33 (Galleti et al., 2004; Gieren et al., 2013).

This paper is organized as follows: We discuss the new and archival
multiwavelength spectroscopic observations of M33 X-7 in Sect. 2. Based on the
phase-resolved UV spectroscopy, we report orbital variations in the wind line
profiles in Sect. 3. In Sect. 4, we describe the detailed analysis of the
spectra. In Sect. 5 we present the updated stellar parameters of the system.
We address the detailed modeling of the donor star wind in Sect. 6 and
estimate the accretion rate in Sect. 7. Section 8 discusses the evolution and
formation of the system using binary evolution models. Sect. 9 contains our
final conclusions about this BH HMXB system.

##  2 Observations

Table 2: Log of spectroscopic observations used in this work |  | HST | XMM
---|---|---|---
Label | Date | JD | Orbital phase | exposure | JD | Orbital phase | exposure
|  |  | $\phi$ | [ks] |  | $\phi$ | [ks]
Eclipse | 2020-09-19 | 2459111.86 | 0.92 - 0.98 | 4 orbits |  |  |
Quadrature | 2019-07-13 | 2458678.08 | 0.3 - 0.36 | 4 orbits | 2458678.21 | 0.33-0.39 | 20 ks
Inferior Conjunction | 2019-07-17 | 2458682.25 | 0.5 - 0.56 | 4 orbits | 2458682.16 | 0.47-0.53 | 15 ks

We carried out simultaneous HST and XMM-Newton observations of M33 X-7. These
includes observations at two different phases of the orbit: quadrature
($\phi\approx 0.33$) and inferior conjunction ($\phi\approx 0.53$). These two
observations were taken four days apart in July 2019. In addition we acquired
HST spectra during X-ray eclipse ($\phi\approx 0.95$) of the system in
September 2020. From the latter observations we want to obtain stellar and
wind parameters of the O-star largely undisturbed by the BH. The details of
the spectroscopic observations are summarized in Table 2. A schematic
illustration of orbit with the observed phase ranges is shown in Fig. 1.

The planning of the joint HST and XMM-Newton campaign relied on the X-ray
light-curve and ephemeris which were based on observations with the Chandra
X-ray telescope in 2005 (Pietsch et al., 2006). According to Pietsch et al.
(2006), the mid-eclipse time is occurring at HJD$(2453639.119\pm 0.005)\pm
N(3.453014\pm 0.000020)$. However, the BH binaries are dynamic systems,
therefore the ephemeris might not be stable and the BH might change states.

To check the validity of ephemeris we obtained monitoring observations of M33
X-7 with the Neil Gehrels Swift Observatory. Swift visited M33 X-7 seven times
between 2019-07-02 and 2019-07-08. During five visits, the count rates were
consistent with that of the Chandra’s ACIS-I detections in 2005. On the other
hand, on HJD 2458666.828 and HJD 2458670.147 the system was not detected in
X-rays which we interpret as X-ray eclipses. Using the ephemeris in Pietsch et
al. (2006), these correspond to $N=1456.035$ and $N=1456.996$ cycles
respectively, that is to say they are in agreement within uncertainties (the
eclipse duration is $\approx 0.2$ in phase). Thus we verified that the orbital
parameters of the system are still valid and hence were applied for scheduling
our observations at three key orbital phases.

Figure 1: Sketch of the binary orbit in M33 X-7. The observer is located along
the horizontal axis on the right. The thick lines span the range of orbital
phases covered by the HST observations. The orbital separation is taken as
$\approx 1.75$ stellar radius as obtained in this work (see Table 4).

###  2.1 UV spectroscopy using HST/COS

We secured a total of 12 HST orbits in cycle 26 under the GO program 15636
(PI: Oskinova). Each of the three selected phases (see Fig. 1) was observed
during four orbits. We selected the COS FUV G140L grating with a central
wavelength setting of 1280 to cover the entire 912–2150 Å wavelength range
(Green et al., 2012). This configuration includes two spectral segments A and
B with a gap in the range of 1192–1265 Å, omitting the Ly$\alpha$. We
disregard wavelengths $\lambda<1100$ Å and $>1800$ Å due to low signal-to-
noise ratio. The resulting UV spectra have a spectral resolving power of
1500-2000. The observations were carried out in TIME-TAG mode.

Figure 2: HST COS aperture around M33 X-7. The background image is an RGB
composite of HST images in F336W, F475W, and F814W bands. A nearby cluster
falls within the FOV of the COS spectrograph.

####  _Crowding in the COS aperture_ :

Since M33 X-7 resides in an OB association (Ma et al., 2001; Pietsch et al.,
2006), separating the contamination of the COS UV spectrum by nearby sources
is challenging. The COS circular aperture is 1.25 arcsec in radius, but the
aberrated beam entering the aperture allows objects up to 2 arcsec from the
center to contribute to the recorded target spectrum. The most noticeable
contribution comes from a bright neighboring cluster of stars, which is
separated by only $0.9^{\prime\prime}$ to the southwest (see Fig. 2). The
position angle of the instrument during observation further reduced the
distance along the cross-dispersion axis to $0.8^{\prime\prime}$.

We extracted photometry of M33 X-7 and the cluster from high-resolution HST
images in F275W, F336W, F475W, and F814W filters. The fluxes were measured
using $0.2^{\prime\prime}$ aperture photometry and applied to the appropriate
zero point. At 814 nm, the brightness of the nearby cluster is just 0.2 mag
lower than M33,X-7, however at 275 nm, the cluster becomes 0.6 mag fainter
than our target. In the FUV, only GALEX photometry of the region is available,
with a spatial resolution of $\approx 6^{\prime\prime}$. Even though M33 X-7
dominates in the FUV, we expect a non-negligible contribution from the nearby
cluster. This contamination could be larger in the optical and thus affect the
parameters obtained by Orosz et al. (2007).

In order to separate contributions from nearby sources in the COS aperture, we
optimally extracted the one-dimensional target spectrum from the two-
dimensional flat-fielded detector images. For this purpose we used
observations carried out at $\phi=0.33$ and 0.53. Observation at these two
phases were carried out with a position angle of $\approx
116\hbox{${}^{\circ}$}$, implying a separation of $0.8^{\prime\prime}$ between
our target and the nearby cluster along the cross-dispersion axis. At
$\phi=0.95$ the position angle is $\approx 96\hbox{${}^{\circ}$}$, so the
corresponding separation is only $\lesssim 0.6^{\prime\prime}$. In the two-
dimensional spectra we could see a double peak profile on the cross-dispersion
axis indicating contributions from both M33 X-7 and neighboring cluster. Hence
we simultaneously fit for both our target and the cluster by multiplying them
with the (wavelength-dependent) cross-dispersion instrumental profile and
subsequently subtract from the combined observed spectra. This has been done
along the cross-dispersion axis for all wavelengths. Finally, we extracted two
spectral components such that the residual is minimum. We found that the
disentangled spectra of M33 X-7 have the same spectral features as the
pipeline extracted spectra. Based on all exposures taken at $\phi=0.33$ and
0.53, we found that on average M33 X-7 accounts only for $\approx 75\%$ of
total flux, whereas the nearby cluster contributes the remaining 25%.

###  2.2 X-ray spectroscopy using XMM-Newton

We carried out dedicated X-ray observations of M33 X-7 with the XMM-Newton
telescope with a goal of obtaining X-ray spectroscopy as close in time as
possible to the HST observations in quadrature and inferior conjunction. The
log ofthe XMM-Newton observations is given in Table 2.

The XMM-Newton data were analyzed using the Science Analysis System (SAS)
111www.cosmos.esa.int/web/xmm-newton/what-is-sas. The standard analysis steps
were performed to extract the spectra measured by the European Photon Imaging
Camera (EPIC). EPIC consists of a PN (Strüder et al., 2001) and two MOS
(Turner et al., 2001) CCD detectors in the focal plane, which operate in the
0.2–12 keV energy range. During our observations, all cameras were operating
in the standard, full-frame mode with a thin filter. To analyze the EPIC
spectra we used the standard spectral fitting software XSPEC (Arnaud, 1996).

Figure 3: XMM-Newton EPIC PN, MOS1, and MOS2 spectra of M33 X-7 (black, red, and green curves, respectively) obtained at phase $\phi\approx 0.47$. The error bars correspond to 3$\sigma$; the best-fit combined thermal plasma (apec) and diskpbb model is shown by solid lines. The model parameters are shown in Table 3. The lower panel shows residuals between the data and the best-fit model. Figure 4: Same as in Fig. 3, but for phase $\phi\approx 0.33$. Table 3: Parameters of the best-fit spectral models to the observed XMM-Newton EPIC spectra of M33 X-7. Two models are fitted. The first one is the plasma apec model for the metallicity $0.5\,Z_{\odot}$ combined with a diskpbb model and corrected for ISM absorption tbabs. The second one is the Bremsstrahlung (brems) model corrected for ISM absorption tbabs. Fluxes and luminosities are in the 0.2-12.0 keV band.  Model parameter | Best-fit values for $\phi\approx 0.47$ | Best-fit values for $\phi\approx 0.33$
---|---|---
tbabs(apec+diskpbb)
$N_{\mathrm{H}}$ [$10^{21}$ cm-2] | $2.3\pm 0.3$ | $2.3\pm 0.3$
$kT$ [keV] | $0.3\pm 0.1$ | $0.3\pm 0.1$
$EM$ [$10^{59}$ cm-3] | $5.4\pm 3.8$ | $2.9\pm 2.8$
$T_{\rm in}$ [keV] | $1.17\pm 0.07$ | $1.25\pm 0.08$
Parameter $p$: $T(r)\propto r^{-p}$ | $0.54\pm 0.03$ | $0.53\pm 0.02$
norm | $0.014\pm 0.006$ | $0.007\pm 0.003$
reduced $\chi^{2}$ | with $201$ d.o.f. $1.0$ | with $217$ d.o.f. $1.0$
Model $F_{\rm X}$ [erg cm-2 s-1] | $1.1\times 10^{-12}$ | $9.7\times 10^{-13}$
Luminosity $L_{\rm X}$ [erg s-1] | $1.7\times 10^{38}$ | $1.5\times 10^{37}$
tbabs*brems
$N_{\mathrm{H}}$ [$10^{21}$ cm-2] | $2.1\pm 0.1$ | $1.9\pm 0.1$
$kT$ [keV] | $2.34\pm 0.08$ | $2.47\pm 0.09$
$EM$ [$10^{60}$ cm-3] | $4.5\pm 0.1$ | $3.6\pm 0.1$
Reduced $\chi^{2}$ | with $204$ d.o.f. $1.1$ | with $220$ d.o.f. $1.05$
Model $F_{\rm X}$ [erg cm-2 s-1] | $1.1\times 10^{-12}$ | $9.9\times 10^{-13}$
Luminosity $L_{\rm X}$ [erg s-1] | $1.7\times 10^{38}$ | $1.5\times 10^{38}$

The extracted spectra and the best fit spectral models are shown in Figs. 3
and 4, while the best fit model parameters are listed in Table 3. It is
interesting to note that unlike HMXBs with neutron star companions accreting
from the stellar wind of OB-type supergiants (Giménez-García et al., 2015;
Sanjurjo-Ferrín et al., 2021), no Fe K$\alpha$ line is seen in the XMM-Newton
EPIC spectra of M33 X-7. After testing various models often used for
describing the spectra of accreting BHs, the simplest model which well
describes the spectra is a combination of thermal plasma apec and a multiple
blackbody disk model diskpbb. The latter model is characterized by the inner
disk temperature $T_{\rm in}$ and the scaling of the local disk temperature
depends on the radius, $T\propto r^{-p}$, with $p$ being a free parameter
(Kubota et al., 2005).

In the observation at phase $\phi\approx 0.47$, i.e. close to the inferior
conjunction when the observer has the direct view on the accreting BH, the
emission measure of the collisionally ionized plasma (described by the apec
model) appears to be higher than at $\phi\approx 0.33$, i.e. close to
quadrature. We interpret this apec plasma component as originating in the
accretion disk wind. Furthermore, close to the inferior conjunction, the
parameter norm which scales with the accretion disk inclination as
$\cos\theta$ (with $\theta=0$ being face-on) is twice as high as quadrature.
This might imply that we see a larger fraction of the disk at inferior
conjunction. Interestingly, the X-ray absorption, as characterized by the
parameter $N_{\rm H}$, does not change between the two orbital phases. This
implies an insignificant contribution to the X-ray absorption from the general
wind of the O-star beyond the BH orbit which could be due to the wind
quenching by strong X-ray irradiation (see Sect. 6.1)

###  2.3 Archival optical spectra

We made use of the optical spectra of M33 X-7 published by Orosz et al. (2007)
obtained from August 18 to November 16, 2006, with the 8.2-m Gemini North
Telescope at wavelengths $\lambda$ = 4000-5000 Å. The final spectra is an
average of spectra taken over 22 nights, and used these data for our spectral
analysis. Although the optical data is not phase-resolved like UV or X-ray, it
is sufficient for our spectral analysis since we do not expect any notable
variation in the absorption lines at the blue optical range.

##  3 Stellar atmosphere modeling

We carefully analyzed the UV spectra obtained at three different phases
(eclipse, quadrature, and inferior conjunction). As a first approximation, we
used the stellar parameters obtained by Orosz et al. (2007), where they used
plane-parallel models from the TLUSTY grid to analyze the spectra. This
approximation is only suitable for stars without winds where the spectrum is
purely formed below the sonic point in a quasi-hydrostatic photosphere.
However, since the donor in M33 X-7 is an O giant or supergiant with a dense
wind, the spectrum is altered in the layers beyond. This can cause an impact
on the derived stellar parameters such as $\log\,g$. So-called unified model
atmospheres are necessary to consistently describe the outermost layers of the
star and their winds (e.g., Gabler et al., 1989; Sander et al., 2017).
Finally, X-rays need to be accounted for the spectral analysis of the HMXB
donor since they can have a noticeable effect on the ionization structure of
the wind. In turn this affects the derived parameters of the donor and its
wind.

Figure 5: Spectral energy distribution of M33 X-7. The model SED (red solid
line) is adjusted to fit the photometry in UV and optical bands (blue boxes).
Figure 6: Iron forest in the FUV range observed in the M33 X-7 spectra (black
line). Over plotted in blue, red, and green are synthetic spectra obtained for
models with the same parameters but with different iron abundances, ranging to
1, 1/2, and 1/5 Fe⊙ respectively.

To analyze the UV and optical spectra, we use the Potsdam Wolf–Rayet (PoWR)
model222http://www.astro.physik.uni-potsdam.de/PoWR/, which is a state-of-the-
art non-local thermodynamic equilibrium (NLTE) code. The PoWR code assumes a
spherically symmetric outflow and accounts for iron-line blanketing, wind
inhomogeneities, a consistent stratification in the quasi-hydrostatic part,
and irradiation by X-rays, all of which are necessary to measure stellar and
wind parameters. To achieve a consistent solution, the equations of
statistical equilibrium and radiative transfer are iteratively solved to yield
the population numbers, while accounting for energy conservation. The
radiative transfer is solved in the comoving frame, which avoids
simplifications such as the Sobolev approximation. Once an atmosphere model is
converged, the synthetic spectrum is calculated via a formal integration along
emerging rays. Details of the PoWR code are described in Gräfener et al.
(2002), Hamann & Gräfener (2003), Todt et al. (2015), and Sander et al.
(2015). However, it should be noted that the standard version of the PoWR code
does not account for deviations from spherical geometry.

A PoWR model is specified by the stellar temperature $T_{\ast}$, the
bolometric luminosity $L$, the surface gravity $g_{\ast}$, the mass-loss rate
$\dot{M}$, the wind terminal velocity $\varv_{\infty}$, the velocity law, and
the chemical abundances. The stellar radius $R_{\ast}$ (inner boundary) is
defined at a Rosseland continuum optical depth of $\tau_{\mathrm{Ross}}=20$
from $T_{\ast}$ and $L$. Following the Stefan-Boltzmann law
$L=4\pi\sigma_{\mathrm{SB}}\,R_{\ast}^{2}\,T_{\ast}^{4}$ the stellar
temperature $T_{\ast}$ is the effective temperature corresponding to the
stellar radius $R_{\ast}$. The outer boundary in our models is set to
$R_{\mathrm{max}}=100R_{\ast}$.

In the main iteration, thermal broadening and turbulence are approximately
accounted for by assuming Gaussian line profiles with a Doppler width of 40 km
s-1. In the formal integral for the calculation of the emergent spectrum, the
Doppler velocity is split into the depth-dependent thermal velocity and a
microturbulence velocity $\xi(r)$. A microturbulent velocity of 10 km s-1 was
adopted in the photosphere, growing proportional with the wind velocity.

The velocity field in PoWR models consists of two parts. In the inner part of
the stellar atmosphere, the velocity field is calculated consistently such
that the quasi-hydrostatic density stratification is fulfilled (Sander et al.,
2015). In the supersonic region, the wind velocity field $\varv(r)$ is
prescribed assuming a so-called $\beta$-law
$\varv(r)=\varv_{\infty}\left(1-R_{\ast}/r\right)^{\beta}$ (Castor et al.,
1975b), where $\varv_{\infty}$ is the terminal wind velocity.

In PoWR models, wind inhomogeneities are accounted for in the microclumping
approach that assumes optically thin clumps (Hillier, 1991; Hamann &
Koesterke, 1998) with a void interclump medium and described by a so-called
density contrast $D(r)$. The matter density in the clumps is enhanced by a
factor $D=1/f_{\mathrm{V}}$, where $f_{\mathrm{V}}$ is the fraction of volume
filled by clumps. In the current study, we account for a depth-dependent
clumping factor.

The models account for complex atomic data of H, He, C, N, O, Mg, Si, P, S,
and the iron-group elements. Iron-group elements with millions of lines are
included in the PoWR code through the superlevel approach (Gräfener et al.,
2002).

The PoWR code can account for ionization due to X-rays. The X-ray emission is
modeled as described by Baum et al. (1992), assuming that the only
contribution to the X-ray flux is coming from free–free transitions. Since the
current generation of PoWR models is limited to spherical symmetry, the X-rays
are assumed to arise from an optically-thin spherical shell around the star.
The X-ray emission is specified by three free parameters, which are the
fiducial temperature of the X-ray emitting plasma $T_{\rm X}$, the onset
radius of the X-ray emission $R_{0}$ $(R_{0}>R_{\ast})$, and a filling factor
$X_{\rm fill}$, describing the ratio of shocked to nonshocked plasma. We set
the onset radius to the orbital distance between the donor star and the BH. To
obtain the fiducial temperature which describes the spectral shape in the
X-ray regime, we fit the X-ray spectra with the brems model. The resulting
temperatures are shown in Table 3. As the next step, the X-ray filling factor
is adjusted such that the wavelength integrated X-ray flux from the
observations are reproduced by the model. This is a similar procedure as
adopted in Hainich et al. (2020).

##  4 Stellar parameters

Table 4: Stellar and wind parameters derived for M33 X-7 in this work. The
estimated or assumed mass fractions of elements are also listed. By adopting
the derived parameters of the donor and orbital parameters from literature
(see Table 1), we calculated further system parameters. See text for more
details.

Parameter | value
---|---
Spectral type | O9 II
$T_{\ast}$ (kK) | $31^{+2}_{-2}$
$T_{2/3}$\textdagger\textdaggerfootnotemark: $\textdagger$ (kK) | $31^{+2}_{-2}$
$\log g_{\ast}$ (cm s-2) | $3.4^{+0.2}_{-0.2}$
$\log L$ ($L_{\odot}$) | $5.54^{+0.1}_{-0.1}$
$\beta$ | $0.8^{+0.2}_{-0.2}$
$R_{\ast}$ ($R_{\odot}$) | $20.5^{+2}_{-2}$
$D$ | $40^{+10}_{-10}$
$\log\dot{M}$ ($M_{\odot}\mathrm{yr}^{-1}$) | $-6.2^{+0.1}_{-0.2}$
$\varv_{\infty}$ (km s-1) | $1500^{+200}_{-100}$
$\varv\sin i$ (km s-1) | $250^{+30}_{-30}$
$\varv_{\mathrm{mac}}$ (km s-1) | $150^{+30}_{-30}$
$\xi$ (km s-1) | $10^{+5}_{-5}$
$X_{\rm H}$ (mass fr.) | $0.60^{+0.1}_{-0.1}$
$X_{\rm He}$ (mass fr.) | $0.395^{+0.1}_{-0.1}$
$X_{\rm C}/10^{-5}$ (mass fr.) | $8^{+2}_{-2}$
$X_{\rm N}/10^{-4}$ (mass fr.) | $5^{+2}_{-2}$
$X_{\rm O}/10^{-3}$ (mass fr.) | $2^{+1}_{-2}$
$X_{\rm Si}/10^{-4}$ (mass fr.) | $2^{+1}_{-2.0}$
$X_{\rm Mg}/10^{-4}$ (mass fr.) |  $2$**$*$corresponds to typical values for OB stars in the LMC based on Trundle et al. (2007)
$X_{\rm Al}/10^{-6}$ (mass fr.) |  $7.6$**$*$corresponds to typical values for OB stars in the LMC based on Trundle et al. (2007)
$X_{\rm P}/10^{-6}$ (mass fr.) |  $2$**$*$corresponds to typical values for OB stars in the LMC based on Trundle et al. (2007)
$X_{\rm S}/10^{-5}$ (mass fr.) |  $4.4$**$*$corresponds to typical values for OB stars in the LMC based on Trundle et al. (2007)
$X_{\rm Ne}/10^{-4}$ (mass fr.) |  $1.8$**$*$corresponds to typical values for OB stars in the LMC based on Trundle et al. (2007)
$X_{\rm Fe}/10^{-4}$ (mass fr.) | $7^{+1}_{-2.0}$
$E_{B-V}$ (mag) | $0.22^{+0.03}_{-0.04}$
$R_{V}$ | $2.4$
$M_{\mathrm{spec}}$ ($M_{\odot}$) | $38^{+22}_{-10}$
$M_{\mathrm{BH}}$ ($M_{\odot}$) | $11.4^{+3.3}_{-1.7}$
$d_{\mathrm{BH}}$ ($R_{\ast}$) | $1.75^{+0.2}_{-0.1}$
inclination $i$ [∘] | $65$
$\varv_{\mathrm{orb}}$ at BH (km s-1) | $510^{+100}_{-40}$
$\varv_{\mathrm{wind}}$ at BH (km s-1) | $760^{+100}_{-100}$
$R_{\mathrm{rl}}\,(R_{\ast})$ | $0.84^{+0.2}_{-0.1}$
$\log\,Q_{\mathrm{H}}$ (s-1) | 49
333† $T_{2/3}$ is the effective temperature which refers to the radius where
the Rosseland mean optical depth in the continuum is 2/3

We primarily used UV spectra taken at the time of eclipse to measure accurate
stellar parameters of the donor star since in this phase we see the part of
the star which is largely undisturbed by the BH. We also re-analyzed the
optical spectra from Orosz et al. (2007) (see Appendix Figs. 24, 26 and 25).
We started the model calculations with the parameters provided by Orosz et al.
(2007) and adjusted them to get the best fit. The best-fitting model is
selected after several iterations of visual inspection and systematic
variation of the parameters. We simultaneously reproduce the normalized line
spectrum and the spectral energy distribution (SED) of M33 X-7 using our best-
fit PoWR model, as shown in Figs. 5, 15, and 24. Details on the wind
parameters are described in Sect. 6.

The observed spectra are best reproduced by a model with $T_{\ast}$ = 31 kK
and $\log g_{\ast}$= 3.4, which is in line with the spectral class of a late O
giant. To constrain the temperature, we used photospheric diagnostic lines
such as C iii and Si iii in the UV along with He i and He ii lines in the
optical spectra. Our estimated values of $T_{\ast}$ and $\log g_{\ast}$ are
lower by 4 kK and 0.35 dex, respectively, than previously suggested by Orosz
et al. (2007). How the variations in $T_{\ast}$ and $\log g_{\ast}$ affect
different line profiles is depicted in Figs. 24, 26 and 25). Hotter models
would result in stronger He ii lines in the optical and weaker C iv and Si iv
lines in the UV. We measured the surface gravity using the pressure-broadened
wings of the Balmer lines. Since the ionization balance also reacts on $\log
g_{\ast}$, we readjusted the temperature accordingly. Overall, our best-fit
model agrees well with the observed UV and optical lines. Subsequently, the
emergent spectrum of the PoWR model was convolved with a rotational velocity
of 250 km s-1 and macro-turbulent velocity of 150 km s-1 to reproduce the
shape of the observed line profiles. Taking into account the updated
inclination of the system, the star rotates with
$\varv_{\mathrm{rot}}/\varv_{\mathrm{critical}}\approx 0.55$.

The luminosity, color excess $E_{\mathrm{B-V}}$, and the extinction-law
parameter $R_{V}$ were determined by fitting the model SED to the photometry
and flux calibrated UV spectra (see Fig. 5). We have used UBV broad-band
magnitudes from Pietsch et al. (2006) and R magnitude from Massey et al.
(2006). The extinction $A_{V}=R_{V}\times E_{\mathrm{B-V}}$ derived by the
final SED fit is in agreement with that of Orosz et al. (2007). Subsequently,
the model flux is scaled with the adopted distance modulus of 24.62 mag (see
Orosz et al., 2007, for more details). The foreground Galactic extinction is
set to 0.052 mag (according to the NED extinction calculator, Schlegel et al.,
1998), and the intrinsic extinction in M33 is negligible according to the
detailed dust maps by Hippelein et al. (2003). The derived luminosity was
found to be lower than obtained by Orosz et al. (2007).

Figure 7: Mass plane diagram for components in the M33 X-7. Masses derived in
this work are marked in red whereas gray points represent previously
determined values from Orosz et al. (2007). The uncertainty in masses is shown
as error bars. The solid lines represent loci of constant orbital inclination
angle of $65\hbox{${}^{\circ}$}$ derived in this work. The dashed line
corresponds to the inclination angle of $75\hbox{${}^{\circ}$}$ derived by
Orosz et al. (2007).  Figure 8: Comparison of HST/COS spectra of M33 X-7 taken
during different orbital phases. Line colors indicate different orbital phases
as given in the legend. The upper panel shows the full spectra range and lower
panels focus on wind line profiles. The observed line variations are due to
H-M effect (see Sect. 5).

The Iron forest present in the FUV is most useful to constrain the metal
content of the stars. Previous studies assumed a low metallicity of
$Z=0.1\,Z_{\odot}$ for the system. However, we find that the iron forest in
the UV range is best reproduced with models assuming an LMC-like metallicity
of $Z=0.5\,Z_{\odot}$ (see Fig. 6). Considering the galactocentric distance of
$\approx$ 2 kpc from the nucleus of M33, this is in agreement with ISM
abundances reported in the M33 star-forming regions (Magrini et al., 2007;
Alexeeva & Zhao, 2022).

We adjusted the abundances of individual elements to reproduce the observed
strength of their respective absorption lines. The $\alpha$-element abundances
are also in agreement with LMC-like metallicity. We found evidence for
nitrogen enrichment and carbon deficiency in the observed spectra. We needed
to reduce the hydrogen mass fraction to 60% and enrich helium up to 40% to
account for the strength of the He i lines in the optical spectra. The He
enrichment could be a result of the fast rotation of the donor. Alternatively,
the outer layers might have been stripped by interaction with the BH
companion. The UV spectra also show indications for CNO processing in the
star. In order to account for the line strengths, we had to reduce the carbon
abundance and increase the nitrogen abundance by a factor of five from
baseline values. Individual element abundances and parameters derived from the
spectral fitting are tabulated in Table. 4. For elements Mg, P, S, Al, and Ne,
we adopted typical LMC abundance values derived from OB stars (Trundle et al.,
2007).

#### The masses of the components in M33 X-7:

The reduction in luminosity, temperature, and gravity resulted in a much lower
spectroscopic mass of $\approx 38\,M_{\odot}$ for the O star than previously
found in Orosz et al. (2007). The spectroscopic mass is calculated from
$\log\,g_{\ast}$ and $R_{\ast}$ ($g_{\ast}=G\,M_{\ast}\,R_{\ast}^{-2}$), and
the uncertainties in these parameters propagates into the final uncertainty of
the donor mass. The newly estimated mass falls in a similar range as that of O
star donors in other BH HMXBs such as Cyg X-1, and LMC X-1 (Miller-Jones et
al., 2021b; Orosz et al., 2009).

Considering the orbital parameters derived by Orosz et al. (2007), we
estimated the BH mass, the orbital radius and the semi-eclipse angle for
different values of the inclination angle. We found that an inclination angle
of 65.5∘ best matches the observed semi-eclipse angle of 26.5∘ (Pietsch et
al., 2006). If we take into account the effects of an extended wind from the
O-star, the donor radius slightly increases, therefore reduce the inclination
angle to 65∘. In comparison, the inclination angle derived in Orosz et al.
(2007) was 74.6∘. The newly derived BH mass is $\approx 11\,M_{\odot}$, which
is much lower than the previous estimate of $\approx 15.65\,M_{\odot}$. The
scaling between the estimated masses of the donor and the BH along with their
respective uncertainties are illustrated in Fig. 7.

The mass ratio of the system is now $\approx 3.3$. Based on the revised
masses, we recalculated the orbital separation to $35.3\,R_{\odot}$, whereas
the radius of the donor is $\approx 20.5\,R_{\odot}$. The distance to the BH
is always the same as the orbital separation, since the system has a nearly
circular orbit. According to our results the BH is located much closer to the
O giant than suggested by Orosz et al. (2007). Abundances and stellar
parameters of the O star indicate that it is close to the main-sequence turn-
off, overfilling its Roche-lobe radius.

##  5 Orbital modulations in the UV wind lines

In HMXBs, the X-ray source produces an extended zone of high ionization that
moves around the orbit. When this region is in front of the donor ($\phi$ =
0.5), the stellar wind gets over-ionized by X-ray irradiation. Therefore the
intensity of the UV resonance lines is reduced due to the depletion of the
absorbing ion species, while the opposite occurs at X-ray eclipse. These
orbital variations were first suggested by Hatchett & McCray (1977) and they
are expected to be observable in the ultraviolet resonance lines of ions such
as C iv, Si iv, or N v.

In Fig. 8 we present the UV spectra of M33 X-7 taken at three different
orbital phases. We identified prominent P-Cygni profiles in the resonance
lines of C iii at 1176 Å, Si iv $\lambda\lambda$1393-1403 and C iv
$\lambda\lambda$1548-1551 at X-ray eclipse. By comparing UV and optical
spectral features to that of normal O stars, we re-classify the donor to an O9
II spectral type. We notice a significant change in these P-Cygni profiles at
phases when the compact object is in the line of sight or in egress compared
to that during eclipse. Such phase-dependent profiles variation of UV
resonance lines across different orbital phases can be attributed to the X-ray
ionization of the wind.

The Hatchett-McCray effect (HM effect) is not observed for all HMXB systems
since the orbital configuration influences the appearance of the wind line
profiles. We report a strong HM effect in M33 X-7 (compared to Galactic
HMXBs), in both Si iv and C iv lines, indicating that the donor wind is
significantly disturbed by the accreting BH. During the eclipse, the donor
star wind profiles are much stronger and broader whereas the other two phases
display a large reduction in the extent and depth of the blue-shifted
absorption component. Although variations at Si iv and C iv are most prominent
at high velocities (-500 to -1500 km s-1), we found notable changes also at
low velocities (-100 to -500 km s-1). The variation between phase 0.33 and
phase 0.53 is mostly negligible, except at lower velocities, suggesting that
the wind ionization extends all the way toward the photosphere of the O-star
when the BH is in the line of sight, leaving only a “shadow wind” (see e.g.,
Blondin, 1994) unaffected.

The redward emission component of the P-Cygni lines is significantly reduced
or absent in the eclipse spectra compared to that found in typical O giants.
We find evidence for an increase in the emission component during phase 0.33
and phase 0.53. The He ii at 1640 Å emission is also found to be stronger when
the BH is in the line of sight. The reduction in the strength of the red
emission component in the UV resonance lines is possibly due to the loss of
ions by strong X-ray photoionization, because the red emission component forms
in the opposite part of the star facing the BH.

##  6 Modeling the donor wind

Winds of massive stars are characterized mainly by their mass-loss rate
$\dot{M}$, terminal wind velocity $\varv_{\infty}$, velocity law exponent
$\beta$, and clumping factor $D$. In this work, we used UV spectra taken at
$\phi=0.95$ to yield parameters of the O star wind and spectra taken at
$\phi=0.53$ and 0.33 to understand the wind quenching by X-rays.

The main diagnostic lines used to determine the terminal velocity are Si iv
$\lambda\lambda$1393-1403, and C iv $\lambda\lambda$1548-1551. The maximum
blue edge velocity in the unsaturated P-Cygni lines is
$\varv_{\mathrm{edge}}\approx 1900$ km s-1. We find a narrow absorption
feature (narrow absorption component, NAC) at a blueshifted velocity of
$\approx 1500$ km s-1. Based on this, we adopted the terminal velocity to be
1500$\pm 100$ km s-1, which is also in agreement with $\varv_{\infty}\approx
0.8\,\varv_{\mathrm{edge}}$ (Howarth & Prinja, 1989; Prinja et al., 1990). The
ratio of terminal velocity to the escape velocity of the donor is found to be
$\varv_{\infty}/\varv_{\mathrm{esc}}\approx 2$, lower than for typical O stars
(Lamers et al., 1995; Kudritzki & Puls, 2000).

Figure 9: Comparison of observed Si iv (left panel) and C iv (right panel)
line profiles (blue solid) at eclipse to the models calculated with different
wind parameters and X-rays. Various colors represent changes in the adopted
model parameters as given in the legend. Please see text for more details.
Figure 10: Same as Fig. 9, but at inferior conjunction

A microturbulent velocity of 10 km s-1 in the photosphere is derived based on
the optical lines, and is also in agreement with the UV spectra. The turbulent
velocity grows proportional to the terminal velocity in the wind regime. In
the outer part of the wind, it reaches typically up to a value of
$\xi(R_{\mathrm{max}})=0.1\varv_{\infty}$. However, for M33 X-7 we measure a
higher turbulence velocity, with a ratio
$\xi(R_{\mathrm{max}})/\varv_{\infty}\approx$ 0.3. The wind turbulent
velocities derived here are larger compared to the previous studies of OB
stars in the LMC and the Galaxy (e.g., 0.1 $\varv_{\infty}$ by Kudritzki &
Puls 2000, 0.14 $\varv_{\infty}$ by Herrero et al. 2001). A UV spectroscopy
study of early B supergiants in M33 by Urbaneja et al. (2002) also reported
high turbulent velocities in the range of 0.25-0.35 $\varv_{\infty}$.

The adopted microclumping is depth-dependent, assumming that clumping begins
at the sonic point, and increases outward. The maximum value of the clumping
factor is found to be $D=40$, which is required to reproduce the UV lines,
especially P v $\lambda\lambda$1118-1128 and N iv at 1718 Å. Our analysis
suggested that the clumping factor reaches the maximum value only beyond
3$R_{\ast}$. Compared to this distance, the location of the BH is much closer
(1.75$R_{\ast}$) to the photosphere of the donor star. Near the location of
the BH, the wind clumping is lower ($D=10$).

One noticeable feature in the UV spectra are the weak unsaturated P-Cygni
lines with negligible or no emission component. Whereas, OB supergiants with
similar spectral types typically show Si iv and C iv resonance lines with
saturated blue absorption and strong emission in the red part of the line. We
tried different approaches such as adjusting the mass-loss rate, the velocity
law, the outer boundary of the wind, and including X-rays, to avoid the red
emission part and to improve the fit quality to the observed UV lines. For
that, we computed models with $\log\dot{M}$ in the range of -6.0 to -9.0. To
model the supersonic part of the wind, we tested $\beta$ values in the range
0.6 to 3.0. A comparison of different possible solutions that can reproduce
the observed spectra and our final best-fit parameters are discussed below.

Figure 11: Comparison of model SEDs (left upper panel) and emerging UV spectra
(lower panel) for different X-ray illuminations. The model SEDs are compared
with XMM-Newton and HST spectra taken at phase $\phi=0.53$. A zoom on C iv and
Si iv line profiles are displayed in the upper panels (middle and right).
Figure 12: Comparison of total wind acceleration normalized to the
gravitational acceleration for models including different amounts of X-rays.
The location of the sonic and the BH radius are marked with dashed and solid
gray lines.

As a first attempt, we tried to change the wind parameters in the model
without incorporating X-rays. A model with a low mass-loss rate
($\log\dot{M}\approx-7.2$) assuming a shallow wind velocity law ($\beta$=2) is
found to be in good agreement with the observations at phase 0.95 (orange line
in Fig. 9). The observed profile can also be reproduced by models with a
moderate mass-loss rate ($\approx$-6.6) and adopting a velocity law exponent
of $\beta$=0.8 if we assume the wind to be shut off by X-rays beyond 900 km
s-1. On the other hand, the theoretically predicted mass-loss rate from Vink
et al. (2000, 2001) for the derived donor parameters is much higher,
$\log\dot{M}\approx-6.2$. Assuming such a higher $\dot{M}$ and $\beta$=0.8 in
the model would result in much stronger P-Cygni profiles compared to the HST
spectra taken during the X-ray eclipse (see the black line in Fig. 9). Models
with a lower mass-loss rate would make the Si iv P-Cygni lines much weaker,
but not have much impact on the C iv profile.

Compared to eclipse, the observed spectra taken at $\phi=0.53$ and 0.33 do not
show any wind line features. Both Si iv and C iv appear to be mostly
photospheric (see Fig. 10). To mimic the observed spectral features at this
phase without accounting for the effect of X-ray photoionization, we need to
adopt a very low mass-loss rate of $\log\dot{M}\lesssim-8.6$.

Figure 13: Contributions to the radiative acceleration from the different ions
for models with no X-ray (top) and including $L_{\mathrm{x}}\sim 10^{36}$ erg
s-1 (bottom). Symbols and colors representing different elements and their
ionization state is shown in the figure. The plot is limited to the ions that
contribute at least $\sim 1$% to the radiative acceleration. Line acceleration
due to scattering by free electrons (magenta solid curve) and gas pressure
(orange dashed curve) and total radiative acceleration (red dashed curve) are
plotted for comparison. The radius at which the BH is located is marked by a
gray solid line and the location of the sonic point marked by gray dotted
line. Figure 14: Same as Fig. 13, but for model including $L_{\mathrm{x}}\sim
10^{38}$ erg s-1

###  6.1 Impact of X-rays on the UV spectra

X-ray irradiation from the BH accretion disk makes a significant impact on the
ionization stratification of the donor wind. While the photospheric absorption
lines are unaffected, the appearance of spectral lines that originate in the
wind, most notably in the UV range, changes significantly. Even adding a small
amount of X-rays ($L_{\mathrm{x}}\lesssim 10^{32}$ erg s-1) in the model to
account for wind intrinsic X-ray emission of the star, changes the ionization
structure of the donor (e.g., Oskinova et al., 2011). However, the line
profiles can get either stronger or weaker depending on the wind density near
the BH and the X-ray intensity. To understand the impact of X-ray
photoionization on the observed UV spectra, we have calculated models with
different X-ray luminosities ($10^{32-38}$ erg s-1). In our spherically
symmetric models, the X-rays are assumed to be generated from an optically-
thin spherical shell around the star, whereas the geometry of the HMXB is more
complex. In this system, the BH is behind the donor during the eclipse and in
the foreground during other phases. On top of this, the radius of the donor
star is large enough to fill its Roche-lobe (see also Sect. 7). Nevertheless,
this is one of the first attempts to study the detailed wind driving of a
massive star in a BH HMXB.

The effects of the X-ray field on the emergent spectra are illustrated in Fig.
11. The mass-loss rate is fixed to $\log\dot{M}\approx-6.2$ for all cases. The
synthetic spectra generated by models with no X-rays and moderate X-ray
irradiation ($L_{\mathrm{x}}\lesssim 10^{34}$ erg s-1) show strong saturated
Si iv and C iv P-Cygni profiles. Including X-rays makes a pronounced effect on
wind lines of highly ionized species like O vi and N v. These lines are absent
in models which do not account for X-rays, but the inclusion of X-rays causes
sufficient photo- and Auger ionization to populate the O vi and N v states.
The overall wind acceleration remains about the same for both cases (see black
and cyan lines in Fig. 12). A further increase in the X-ray field to
$L_{\mathrm{x}}\sim 10^{36}$ erg s-1 affects the emerging UV spectrum
drastically. The strength of the wind lines (such as Si iv, C iv, C iii, and P
v) reduces significantly whereas O vi becomes stronger. A model including a
strong X-ray field on the order of $L_{\mathrm{x}}\sim 10^{38}$ erg s-1
considerably reduces the strength of Si iv and C iv lines, making them appear
to be photospheric lines.

Since the important wind-driving ions are depopulated, the line acceleration
in the outer wind decreases correspondingly. As shown in Fig. 12, strong X-ray
illumination causes a strong decline in the wind accelerations from the sonic
radius onward and hence stalling of the wind. This is because photoionization
destroys the ions that are capable of absorbing photons in the UV. The
normalized wind acceleration drops to unity for models with
$L_{\mathrm{x}}\sim 10^{36}$ erg s-1. The wind driving drops further below
unity for models irradiated by $L_{\mathrm{x}}\sim 10^{38}$ erg s-1 (see red
line in Fig. 12). The total wind acceleration is a factor of seven lower at
the side of the donor facing BH and a factor of three lower in the shadow
region compared to that of an unperturbed O-star. Our investigations thus
suggest a breakdown of wind toward the BH in M33 X-7. For this model, the blue
O vi and N v absorption components only reach up to 1000 km s-1 instead of
1500 km s-1 measured at eclipse. The decrease of the wind terminal velocity
due to X-ray irradiation is supported by previous HMXB studies (see e.g.,
Watanabe et al., 2006; Sander et al., 2018; Krtička et al., 2018).

###  6.2 Wind driving

In order to have a detailed picture of wind driving in these situations, we
need to look at the contributions of the various elements in different
ionization stages. This is illustrated in Figs. 13 and 14 by showing the
radiative acceleration contribution from different ions and how this is
affected by X-rays. While these are not dynamically consistent models, the
changes due to the X-ray illumination are representative nonetheless. For a
model without including X-rays (Fig. 13 upper panel), the leading ions are Fe
iv, N iii, and C iii. O iii becomes dominant in the far outer part of the
wind. Near the sonic point, iron group opacities are the leading contributors
to the wind along with the electron scattering and gas pressure. Beyond
$R>1.1R_{\ast}$, CNO and other higher elements become important agents in wind
driving. Higher ions such as Si iv, C iv, S iv and N iv also make a
significant contribution to the radiative acceleration. The contribution by C,
N, O, and iron lines outweigh the free electron (Thomson) scattering
contribution in the outer wind.

In contrast, the model with $L_{\mathrm{x}}\sim 10^{36}$ erg s-1 (Fig. 13
lower panel) depopulates all the lower ions, especially beyond the radius at
which the BH is located. In the inner part, the picture remains the same as
for the model with no X-rays. Beyond $R>1.1R_{\ast}$, the acceleration by Fe
iv reduces rapidly and becomes unimportant. In this case O vi and N v are the
leading contributors in the outer wind as we see can from the emerging UV
spectra in Fig. 11. The total radiative acceleration by these ions is
comparable to that of free electrons. Si iv and C iv also make a noticeable
contribution to the wind driving, but their influence is a factor of three
lower compared to the model with no X-ray ionization. Increasing the X-ray
field to $L_{\mathrm{x}}\sim 10^{38}$ erg s-1 makes a remarkable further
difference as shown in Fig. 14. Strong X-ray photoionization results in the
super-ionization of lower species hence only a few highly ionized species are
contributing to the wind driving. O vi is still the leading ion in this case,
and other higher ions such as Fe vii, N iv and O vii are also important.
However, the contributions from these ions are lower than that by electron
scattering. In comparison to the other two scenarios, here we can see a
completely different picture in the inner part of the wind, where a plethora
of ions contributing to the wind driving including the iron group components,
are missing. It should be noted that apart from CNO and Fe, elements like P,
S, Si, Mg, and Ne also make important contributions in wind driving. Excluding
higher elements and higher ionization stages would significantly affect the
wind structure and the emerging spectra.

Figure 15: Spectral fit for M33 X-7 UV spectra at the X-ray eclipse.
Normalized HST spectra (blue solid line) is overplotted with the PoWR model
(red dashed line). The atmospheric parameters and abundances of this best-fit
model are given in Tables 4. Figure 16: Same as Fig 15, but spectra taken at
inferior conjunction ($\phi\approx 0.53$) where BH is in front of the donor
star.  Figure 17: Same as Fig 15, but spectra taken at Quadrature
($\phi\approx 0.33$).

As the calculation of hydrodynamically consistent atmosphere models is beyond
the scope of the present paper, one has to be careful with drawing deeper
conclusions about the absolute radiative acceleration in the deeper wind
layers. However, our studies give us very valuable insights regarding relative
changes of the wind acceleration between the different situations. Without
X-rays, the acceleration in the outer wind is approximately constant with
$1<\Gamma_{\text{rad}}\approx 3$, which is quite similar to the result for the
much cooler Vela X-1 donor obtained with a dynamically consistent model
(Sander et al., 2018). With the amount of X-rays measured during eclipse in
M33 X-7, the situation reduces to $\Gamma_{\text{rad}}\approx 1$. Taking this
result literally would imply that there is hardly any wind acceleration, but
as we do see wind signatures in the UV spectrum during eclipse we know that
$\Gamma_{\text{rad}}>1$ must be fulfilled at least in an inner region where
the wind would be launched and which we do not model dynamically consistent in
this work. The consistent model from Sander et al. (2018, their figure 6)
reflects such a situation where even for an illumination of
$L_{\text{X}}\approx 10^{37}$ there is still a meaningful radiative
acceleration close to the star before a complete breakdown occurs further out.
Moreover, the inner acceleration in our M33 X-7 models is not affected when
comparing the model without X-rays to those using the $L_{\text{X}}$ from the
eclipse. This changes drastically, when looking at the strong irradiation case
where the full $L_{\text{X}}\approx 10^{38}$ from the BH are irradiated. Now,
even the acceleration in the inner region decreases by about 40%. Close to the
BH, the acceleration further decreases by another 60% compared to the eclipse
situation. With $\Gamma_{\text{rad}}<1$ now, the acceleration of the wind
breaks down. For an isolated star, this would lead to a significant
deceleration of the wind, as also discussed by Krtička et al. (2018, 2022) for
a wider range of system parameters. On top, the presence of the BH and its
gravitational potential will need to be considered here, which we will discuss
in Sect. 7.

###  6.3 X-ray luminous HMXB

The typical wind-intrinsic X-ray emission of an O-giant donor is
$L_{\mathrm{x}}\sim 10^{32}$ (Nebot Gómez-Morán & Oskinova, 2018). However,
this is much lower than the observed X-ray luminosity at eclipse estimated by
Pietsch et al. (2004) using XMM-Newton observations. They derived
$L_{\mathrm{x}}=2-3\times 10^{36}$ erg s-1 at phase $\phi\approx 0$. Such a
high X-ray luminosity could arise from the photoionized region around the
accretion source if the inclination angle of the system is low or if the size
of the donor is too small to fully cover the photoionization zone. However,
M33 X-7 is an eclipsing system with a high inclination angle ($\approx
75\hbox{${}^{\circ}$}$) and the donor has a large radius over-filling its
Roche-lobe. Typically in Roche-lobe overflow systems, due to the high X-ray
luminosity, the stellar wind is interrupted by the photoionization. Thus, the
wind can only develop in the X-ray shadow behind the OB supergiant, a so-
called “shadow wind” region (e.g., Blondin, 1994; Kaper & van der Meer, 2005).
In M33 X-7, the strong X-ray luminosity near phase $\phi\approx 0$ suggests
the presence of a significant ionization zone in the line of sight of the
donor with the shadow wind region being very limited. Such an extended
Strömgren zone could be formed due to the scattering of X-rays by ions in the
stellar wind. As a result, the absorption and emission components of UV wind
lines are diminished. By incorporating such a high amount of X-rays we are
able to reproduce the observed UV spectra at the eclipse as shown in Fig. 15.

As discussed in Sect. 2.2, we estimated the X-ray luminosity of the system at
phase $\phi=0.33$ and 0.5 to be around $L_{\mathrm{x}}=10^{38}$ erg s-1.
Hence, by including X-rays with $L_{\mathrm{x}}\sim 10^{38}$ erg s-1 and
adopting the same mass-loss rate ($\log\dot{M}\sim-6.2$), our model SED and
spectra are in agreement with the XMM-Newton and HST observations at phase
$\phi=0.33$ and 0.53 (see Figs. 11, 16, and 17). Since the X-ray and UV
spectra at phase 0.33 and phase 0.53 are similar and the UV wind lines in
these phases suffer large reductions in the absorption strength at all
velocities, we could not find any evidence for an accretion wake or focused
wind stream toward the BH in this system.

Figure 18: Velocity field for M33 X-7 donor adopting $\beta=0.8$. The location
of BH and the corresponding wind velocity and orbital velocity are marked.

HMXBs with high X-ray luminosities such as SMC X-1 and LMC X-4 (OB supergiant
+ neutron star) show similar orbital-dependent variations in the UV spectra.
They show a similarly high X-ray luminosity around deep eclipse ($\sim$ 1% of
peak). As a result, their corresponding UV spectra show strong P-Cygni
profiles of O vi and N v (Vrtilek et al., 2005; Sonneborn et al., 2007). These
systems are also reported to have a reduction in the strength of O vi and N v
due to two orders of magnitude higher X-ray luminosity in other phases. This
is exactly what is predicted by the models for M33 X-7 in Fig. 11.
Unfortunately, the UV spectra of M33 X-7 do not cover these O vi and N v
lines. In conclusion, our models are capable of explaining the UV wind line
variation of high luminosity HMXBs.

Moreover, we varied $\dot{M}$ in models including X-rays to check its impact
on the emergent spectral profiles. However, these results either over or
under-predict the strength of the UV wind lines compared to the observed
spectra. By fixing mass-loss rate and wind velocity, we have also tested
models with different $\beta$ values. The standard $\beta=0.8$ gives the best-
fit model whereas higher $\beta$ values could not reproduce the observed UV
line profiles. A low $\beta$ value implies a strong acceleration of the wind,
hence the wind velocity near the BH (at 1.75$R_{\ast}$) could reach up to 760
km s-1 (see Fig. 18). In comparison, the orbital velocity ($2\pi a/P$) at this
radius is only 510 km s-1. However, the actual wind velocity near the BH might
very well be comparable to the orbital velocity or even lower as we assume a
prespecified wind velocity field in this work. In this approach, the effect of
the X-ray illumination from the BH on the spectrum formation is taken into
account, but there is no immediate coupling of the velocity field to the
accompanying lower radiative acceleration derived in Sect. 6.2.

In conclusion, we prefer our final best-fit solutions to M33 X-7 using X-ray
irradiated models over other possible solutions (see Figs. 9 and 10) since
they simultaneously reproduce the X-ray, UV, and optical observations.
Moreover, we are able to account for phase-dependent variations in the spectra
with the same wind parameters. The wind parameters of our best possible
solution are listed in Table. 4.

###  6.4  He ii emission

One of the main sources of the He ii emission seen in high redshift galaxies
is thought to be HMXBs. Strong X-ray photoionization in HMXBs causes an excess
flux beyond the He ionizing edge. This is demonstrated in our model SEDs in
Fig. 19. The number of He ii ionizing photon flux provided by the O star donor
model alone (black lines in Fig. 19) is substantially lower than those
irradiated with X-rays. The He ii ionizing photon flux generated by M33 X-7 is
found to be approximately seven orders of magnitude higher than that from a
normal O star of the same spectral type.

Maggi et al. (2011) have reported nebular He ii 4686 Å emission in the region.
They inferred the luminosity of the line to be $\sim 2\times 10^{35}$ erg s-1.
Using a basic photoionization equation from Osterbrock & Ferland (2006), we
calculate the He ii photon flux to be $Q_{\mathrm{He\,\textsc{ii}}}=1.02\times
10^{48}\frac{L(\mathrm{He\,\textsc{ii}\,4686})}{10^{36}\mathrm{erg\,s^{-1}}}=2\times
10^{47}\mathrm{ph\,s^{-1}}$. This is in good agreement with our model
predictions for M33 X-7 considering an X-ray irradiation of
$L_{\mathrm{x}}=10^{38}$ erg s-1 out of eclipse.

Figure 19: Spectral energy distribution for models with different X-ray
luminosities. The corresponding number of He ii ionizing photons are displayed
in the legend.  Figure 20: Comparison of UV spectra of M33-X7 around He ii
1640 Å taken during different orbital phases. Line colors indicate different
orbital phases as given in Fig 8. The dashed black line represent our best fit
model including $L_{\mathrm{x}}=10^{38}$ erg s-1, corresponding to
$\phi=0.53$.

Interestingly, the UV spectra does not show an absorption line at He ii 1640 Å
despite this being predicted by the model (see Fig. 20). This absence of
absorption suggests that there is possibly an emission component filling the
line. This feature is also variable with orbital phases, with the maximum
“emission” occuring when BH is in our line of sight (green and blue lines in
Fig. 20). Such He ii emission features are not expected for a typical late O
giant star. Although we account for a maximum X-ray luminosity of
$\log(L_{\text{X}}/\text{erg}\,\text{s}^{-1})\approx 38$ in the model which
predicts He ii ionizing photons as high as 1047 s-1, we were unable to
reproduce these observed emission features. Broad He ii $\lambda$1640 emission
features were detected in another HMXB HD 153919 (alias: 4U1700-37) with a
neutron star companion (Kaper et al., 1990). These lines are expected to
originate from EUV emission lines close to 256Å, which is Raman scattered by
He ii ions in the stellar wind. Kaper et al. (1990) show that these emission
lines originate close to the compact object, causing the orbital modulation of
the Raman scattered lines. However, HD 153919 also shows a wind-instrinsic
P-Cygni profile in He ii $\lambda$1640 (Hainich et al., 2020), which never
appear in our M33 X-7 UV spectra, even at X-ray eclipse.

##  7 Accretion rate

Accretion of matter by the compact object is the most efficient way of
producing X-rays. In wind-fed supergiants, X-ray accretion is powered by the
stellar wind of the donor. Accretion of matter onto a star moving through a
medium was first described by Bondi & Hoyle (1944) and applied to HMXBs by
Davidson & Ostriker (1973). The mass accretion rate via the Bondi-Hoyle
formula can be written as follows:

$\dot{M}_{\mathrm{acc}}=\pi R_{\mathrm{acc}}^{2}\rho\varv_{\mathrm{rel}}$ (1)

Here $\rho$ is the wind density near the BH and $R_{\mathrm{acc}}$ is the
accretion radius.

In an HMXB, the compact object has a velocity relative to the stellar wind
flow:

$\varv_{\mathrm{rel}}^{2}=\varv_{\mathrm{wind}}^{2}+[\varv_{\mathrm{orbit}}-(R_{\ast}/d_{\mathrm{BH}})\,\varv_{\mathrm{rot}}]^{2}$
(2)

Here $\varv_{\mathrm{wind}}$ and $\varv_{\mathrm{orbit}}$ are the wind
velocity and orbital velocity near the BH which are 760 km s-1 and 513 km s-1
respectively. The eccentricity of the orbit is ignored in this case because it
is essentially zero for this system. Since the O star is rotating fast while
the wind is leaving its surface, there is an additional tangential component
to its velocity. Plugging in our derived values from the analysis, the
estimated velocity is $\varv_{\mathrm{rel}}\approx 845$ km s-1. The derived
relative velocity is comparable to $0.56\,\varv_{\infty}$. Stellar wind matter
is accreted onto the compact object if it approaches within an accretion
radius, which can be approximated to

$R_{\mathrm{acc}}=\dfrac{2GM_{\mathrm{BH}}}{\varv_{\mathrm{rel}}^{2}}$ (3)

Using the velocity calculated above, the accretion radius in M33 X-7 is $\sim
4\times 10^{6}$ km = $5.7R_{\odot}$. This is roughly $0.5R_{\ast}$ from the
photosphere of the donor star. The density of the wind at the distance of the
BH can be approximated to

$\displaystyle\rho=\dfrac{\dot{M}_{\mathrm{donor}}}{4\pi
d_{\mathrm{BH}}^{2}\varv_{\mathrm{wind}}}$ (4)

By substituting the derived wind parameters, we get a density of $\rho\approx
6.7\times 10^{-15}\mathrm{g\,cm^{-3}}$, and based on Eq. (1), we get
$\dot{M}_{\mathrm{acc}}=2.8\times 10^{17}\mathrm{g\,s^{-1}}$. The expected
X-ray luminosity can then be estimated using the relation $L_{X}=\eta
L_{\mathrm{acc}}$ with the accretion luminosity

$L_{\mathrm{acc}}=\dfrac{GM_{\mathrm{BH}}\dot{M}_{\mathrm{acc}}}{R_{\mathrm{BH}}}$
(5)

and $\eta$ being an accretion efficiency parameter.

In BHs, the accretion disk luminosity will be created while the material falls
in from the outermost radius to the innermost stable circular orbit. For a
nonrotating BH, the inner edge of the accretion disk should be at three times
the Schwarzschild radius $R_{\mathrm{sch}}=\dfrac{2GM_{\mathrm{BH}}}{c^{2}}$.
For a BH of $11M_{\odot}$, the accretion radius would be around 32 km. For a
rapidly spinning BH, the inner edge of the accretion disk can be considerably
closer to the event horizon than for a nonrotating BH. The accretion
efficiency would be in the range of $\sim 6-40\%$. Therefore, the accretion
luminosity can be in the range $1-8\times 10^{37}\mathrm{erg\,s^{-1}}$.
Previous studies have estimated the spin of M33 X-7 to be 0.84 (Liu et al.,
2008, 2010). However, the considerable reduction in the mass of the BH can
lower the spin parameter. Considering the effect of the BH mass on the spin
parameter (see Fig. 3 in Liu et al., 2008), the updated spin of M33 X-7 is
around $a_{\ast}\approx 0.6$. Therefore, the corresponding accretion
luminosity is only $\sim 4\times 10^{37}\mathrm{erg\,s^{-1}}$.

The estimated accretion luminosity corresponds approximately to the observed
X-ray luminosity for M33 X-7 at eclipse (Pietsch et al., 2006, 2004) but is a
factor of four lower than our estimated values for the other two phases out of
eclipse (Sect. 2.2). In comparison, the Eddington luminosity of the system is
$\sim 1.4\times 10^{39}\mathrm{erg\,s^{-1}}$. So the maximum observed
luminosity is only $\approx 12\%$ of the Eddington value, which excludes
super-Eddington-accretion.

Possible explanations for the observed high X-ray luminosity would be excess
mass transfer due to classical Roche-lobe overflow (RLOF) in the system or so-
called wind-RLOF, where the radius at which the wind is accelerated is
comparable to the Roche-lobe radius so that the wind material fills the
donor’s Roche-lobe and is transferred to the compact companion through the
inner Lagrangian point (Mohamed & Podsiadlowski, 2007). The different modes of
mass supply likely yield differences in the observed X-ray luminosity of X-ray
binaries. Typically, HMXBs which accrete only via stellar winds show a
constant X-ray radiation with luminosities in the range $10^{35-37}$ erg s-1,
while systems associated with RLOF have higher luminosities
($L_{\mathrm{x}}\gtrsim 10^{38}$ erg s-1).

In M33 X-7, the O supergiant companion is in a stage where it is already
filling the traditionally defined Roche lobe. However, the presence of a
radiative acceleration $\Gamma_{\text{rad}}$ can significantly alter the Roche
potential (e.g. Howarth, 1997; Dermine et al., 2009). To get a better idea
about this, we can use our insights gained from the radiative acceleration
study in Sect. 6.2. As discussed in Dermine et al. (2009), there is no longer
a Roche lobe for the donor star in regions with $\Gamma_{\text{rad}}>1$. We
know from UV spectroscopy that this must be the case for M33 X-7, at least in
some region not facing the BH. Hence, some of the material will directly
escape from the system and is not re-directed toward the BH. For
$\Gamma_{\text{rad}}>1$, there is no longer a clearly defined Roche lobe and
thus the destinction between wind capturing and RLOF actually vanishes.

For M33 X-7, the situation is likely even more complex. Even the concept of
wind-RLOF (Mohamed & Podsiadlowski, 2007) does not really apply here as it
would require $\Gamma_{\text{rad}}\lesssim 1$ to define a Roche lobe that
could be filled by a slow wind as we typically find them in late-type stars.
Instead, as our acceleration study shows, our donor star likely has a region
with $\Gamma_{\text{rad}}\ll 1$ facing the BH. In this case, there is a
definable modified Roche potential with generalized radius, which will be
smaller than the one for the unperturbed Roche lobe (see Dermine et al.,
2009). Since the derived donor star parameters for M33 X-7 are already
sufficient to yield a star that would fill the standard definition of a Roche
Lobe, the further reduction of this criterion for a region towards the BH
paves the way for an additional “easy” overflow-like mass transfer, that is
further boostered by the diminished radiative acceleration of the material due
to the X-rays from the BH. In contrast to the donor, a meaningful Roche lobe
can still be defined for the accretor, which would play a major role in
capturing a considerable ($>10\%$) fraction of the wind as indicated in recent
hydro-dynamical simulations by El Mellah et al. (2019).

Based on our atmosphere modeling, the maximum wind velocity outside eclipse
should be $\varv_{\infty}\approx 1000\,\mathrm{km}\,\mathrm{s}^{-1}$, which is
similar to other high-$L_{\mathrm{X}}$ HMXBs. Hence, we can assume that the
wind speed at the distance of the BH is lower than in our eclipse model (cf.
Fig. 18), i.e., close to the orbital speed. Subsequently, the accretion
luminosity in the Bondi-Hoyle scenario already increases to $\sim 1.4\times
10^{38}\mathrm{erg\,s^{-1}}$, in better agreement with the X-ray observations.
With the additional mass overflow in the highly ionized region, we have a
plausible “wind overflow” scenario for the observed $L_{\text{X}}$, even
taking into account the efficiency factor between $L_{\mathrm{acc}}$ and
$L_{\text{X}}$. While our scenario likely explains the higher accretion
luminosity compared to the classical Bondi-Hoyle wind-fed scenario, the
condition of $\Gamma_{\text{rad}}>1$ in other directions will avoid a standard
RLOF where all of the donor’s mass loss is redirected toward the BH.
Consequently, our derived $L_{\text{X}}$ is on the lower end compared to
systems that are considered to be powered by standard RLOF.

Recent studies of other BH HMXB systems, namely Cyg X-1 (Orosz et al., 2011;
Miller-Jones et al., 2021a) and LMC X-1 (Orosz et al., 2009; Hyde et al.,
2017), also suggest that the donor stars in these systems are close to filling
their Roche lobe. Yet, a detailed consideration of the radiative acceleration
has so far not been done. Depending the the impact of the X-ray illumination
of the donor atmosphere, these systems will either have a similar “wind
overflow” like M33 X-7 or a standard RLOF in case of strong wind inhibition.

Table 5: Comparison of system parameters from single and binary evolution models that best match the derived parameters from spectroscopic analysis of M33 X-7 Parameter | Single star model | Binary model 1 | Binary model 2 | Empirically derived
---|---|---|---|---
$T_{\ast}$ (kK) | $31.2^{+2}_{-1.5}$ | $31.4$ | $30.9$ | $31^{+2}_{-2}$
$\log g_{\ast}$ (cm/s-2) | $3.43^{+0.08}_{-0.17}$ | $3.44$ | 3.34 | $3.4^{+0.2}_{-0.2}$
$\log L$ ($L_{\odot}$) | $5.52^{+0.07}_{-0.06}$ | $5.54$ | 5.54 | $5.54^{+0.1}_{-0.1}$
$R_{\ast}$ ($R_{\odot}$) | $19.84^{+3}_{-3.5}$ | $20$ | 20.6 | $20.5^{+2}_{-2}$
$\log\dot{M}\mathrm{{}_{wind}}$ ($M_{\odot}\mathrm{yr}^{-1}$) | $-6.22^{+0.14}_{-0.12}$ | $-6.35$ | -6.1 | $-6.2^{+0.1}_{-0.2}$
$\varv_{\mathrm{rot}}$ (km s-1) | $250^{+13}_{-15}$ | $30$ | 278 | $280^{+30}_{-30}$
$X_{\rm He}$ (mass fr.) | $0.26^{+0.1}_{-0.1}$ | $0.28$ | 0.3 | $0.395^{+0.1}_{-0.1}$
$\log(C/H)+12$ | 7.58 | 7.64 | 7.56 | 7.05
$\log(N/H)+12$ | 7.84 | 7.63 | 7.67 | 7.85
$\log(O/H)+12$ | 8.27 | 8.3 | 8.32 | 8.45
$P$ (days) |  | 4.1 | 6.3 | 3.45
$M_{\mathrm{O}}$ ($M_{\odot}$) | $33.8^{+2.5}_{-2}$ | $39.4$ | 34.2 | $38^{+22}_{-10}$
$M_{\mathrm{BH}}$ ($M_{\odot}$) |  | $\lesssim 10.5$ | $\lesssim 12.2$ | $11.4^{+3.3}_{-1.7}$
$d_{\mathrm{BH}}$ ($R_{\ast}$) |  | $2$ | 2.5 | $1.75^{+0.2}_{-0.1}$
$R_{\mathrm{rl}}\,(R_{\ast})$ |  | $1$ | 1.1 | $0.84^{+0.2}_{-0.1}$
Age (Myr) | $4^{+0.27}_{-0.26}$ | 5.76 | 5.08 |

##  8 Formation and evolution of the HMXB

M33 X-7 has been identified as an evolutionary challenge, considering the
previously reported very high mass donor star along with a heavy BH in a tight
orbit. The updated stellar and binary parameters allow us to gain new insights
into the evolution of this system. In the first step, we explore the formation
channel of the system starting from a binary with two stars on the zero-age
main-sequence (ZAMS) to a BH - O star binary and then follow its path to later
evolution.

As a first approach, we compare the updated parameters of the O star donor to
that of single-star evolutionary tracks. For this purpose we used tracks
provided by Brott et al. (2011) at LMC metallicity along with the Bayesian
statistic tool “The BONN Stellar Astrophysics Interface”
(BONNSAI444https://www.astro.uni-bonn.de/stars/bonnsai/, Schneider et al.,
2014) . The results from the analysis are summarized in Table 5. Single-star
evolutionary tracks suggest lower masses for the O star donor given its
observed luminosity, i.e., the donor is underluminous compared to a normal
single star of the same mass. The observed wind mass-loss rate of the donor is
in good agreement with the prediction of Vink et al. (2000, 2001) which is
used in the stellar evolutionary models. The single star models could account
for the fast rotation of the star by assuming an initial velocity of 280 km
s-1. Abundances are also reproduced within the uncertainty limit except for
helium.

With our knowledge of the currently observed properties of the M33 X-7 system,
we tried to reconstruct its evolution from the moment the system was comprised
of two ZAMS stars through the intermediate phases such as mass transfer from
the primary to the secondary, and the formation of the BH, followed by the
currently observed HMXB phase and the final evolution of the system.

Figure 21: Best matching MESA evolutionary tracks reproducing the HRD position
of O-star donor in M33 X-7. The current position and errors of the O star are
taken from the spectroscopic analysis. Initial masses and periods are labeled
on the plot.

###  8.1 Modeling the binary system with MESA

We performed detailed binary evolution calculations to explore the possible
evolutionary history of M33 X-7. The tracks were computed using the stellar
evolution code MESA (“Modules for Experiments in Stellar Astrophysics”,
Version No. 10398) with a physics implementation as described in Paxton et al.
(2011, 2013, 2015, 2018). We adopted LMC metallicity (Z = 0.006) for the
calculation, which is in line with the derived metallicities based on the
spectroscopic analysis. To make our models comparable to the Brott (2011) LMC
models used for the single star evolution in BONNSAI, we modified MESA as
described in Marchant et al. (2016).

Stellar winds were implemented as in Brott et al. (2011), with mass-loss rates
for hot hydrogen-rich stars modeled as in Vink et al. (2001). For stars with
surface hydrogen of $X_{\mathrm{H}}<0.4$, we use the mass-loss rate of Hamann
et al. (1995) divided by a factor of 10 to account for clumping (Yoon et al.,
2010). We interpolate these two mass-loss rates for surface hydrogen fractions
between 0.7 and 0.4. For temperatures below the bi-stability jump, the mass-
loss rate was taken as the maximum between the Vink mass-loss rate and that of
Nieuwenhuijzen & de Jager (1990).

Convection was modeled using the mixing-length theory (Böhm-Vitense, 1958)
with a mixing-length of $\alpha_{ML}$ = 1.5. Semiconvective mixing is modeled
with an efficiency of ${\alpha_{\mathrm{SC}}=1}$.

Rotational mixing and angular momentum transport were modeled as diffusive
processes following Heger & Langer (2000), including the effects of Eddington-
Sweet circulations, the Goldreich-Schubert-Fricke instability, secular and
dynamical shear, and sensitivity to composition gradients. We also take into
account the transport of angular momentum due to the Spruit-Tayler dynamo
(Spruit, 2002).

Stars are assumed to be synchronized with the orbital period at the ZAMS. The
evolution of orbital angular momentum considers the effects of mass loss,
gravitational wave radiation, and spin-orbit coupling, as described in Paxton
et al. (2015).

Figure 22: Illustration of possible evolutionary scenario for M33 X-7 starting
from the ZAMS massive binary to HMXB stage followed by common envelope
merging. Figure 23: Similar to Fig. 23, but an alternative evolutionary track
assuming no tidal synchronization.

###  8.2 Evolutionary status of M33 X-7

To investigates possible scenarios of M33 X-7’s evolutionary history, we
considered progenitor masses in the range of $M_{1}=38-60\,M_{\odot}$ for the
primary, $M_{2}=18-40\,M_{\odot}$ for the secondary, and initial orbital
periods from 1.8 to 5 days. Guided by the mass derived from the spectroscopic
analysis of the donor, which is higher than what is expected from evolutionary
models of single stars, we searched for binary models where the secondary
accretes mass from the BH progenitor. Given the short orbital period, we
considered only binary systems that start their life in a tight orbit, hence
undergoing a phase of mass transfer during the main sequence phase of the
primary. In our model calculation, we allow both stars to evolve up to core
carbon depletion. After core carbon depletion, a star is assumed to directly
collapse into a BH without losing mass or the orbit receiving a kick due to a
possible supernovae explosion. In this approach, we attribute the final CO
mass to the compact companion and treat it as a point mass. Subsequently, we
model the evolution of the secondary together with the compact object in the
binary system.

While exploring the parameter space, we discovered that 1) models with high
initial masses and short orbital periods are predicted to undergo overflow
through the second Lagrangian point L2 during core-H burning and are expected
to merge; 2) models with lower mass ratios ($q\lesssim 0.5$) and short periods
($P<5$ days) had numerical issues during the fast Case A mass transfer phase;
and 3) models with mass ratios going toward unity ($q\gtrsim 0.85$) have a
reverse mass transfer from the secondary to the primary while both stars are
still core-H burning, leading to an unstable situation. These models terminate
before forming a BH + O star binary, and are therefore ruled out, limiting the
possible parameter space. For some models that successfully reach the HMXB
stage we found that they form a convective envelope during mass transfer on
the compact object, leading to unstable situations followed most likely by a
common envelope phase and a merger. We think that these models are unlikely to
represent the true nature of this system which is believed to be currently in
a stable mass transfer phase.

Based on our tests, the resulting models exhibit the following properties at
the HMXB phase: $M_{\mathrm{OB}}\approx 30-40\,M_{\odot}$,
$M_{\mathrm{BH}}\approx 8-13\,M_{\odot}$, and $P\approx 4-12$ days. Our best
possible evolutionary model which matches the current parameters of M33 X-7 is
shown in Fig. 21 (Binary model 1). The parameters derived from this model are
compared to the observations in Table 5. This model suggests the initial
masses of the stars to be $M_{1}=45\,,M_{2}=32\,M_{\odot}$ with a period of
2.7 days. Our best model satisfies the current observational properties
($T_{\ast}$, $\log g$, $\log L$, $R_{\ast}$) of the donor star. Moreover, the
evolutionary masses suggested by the model for the O-star donor and the BH are
in good agreement with the observations.

The initial masses of the M33 X-7 progenitors result from our tests are
comparable to typical main-sequence O-stars and are much lower than the
predictions from previous studies (Abubekerov et al., 2009; Valsecchi et al.,
2010; de Mink et al., 2010; Bogomazov, 2014). According to the evolutionary
model, the donor is overflowing its Roche lobe and transferring mass to the BH
at the currently observed stage as a HMXB. As a result, the total mass-loss
rate of the system is an order of magnitude higher in this model, despite the
wind mass-loss rate being in agreement with the empirical estimation. This
would qualitatively be in line with out findings in Sect. 7, where we conclude
an enhanced mass transfer due to the strong X-ray illumination and the
perturbance of the Roche potential due to the presence of a donor wind. Of
course such a detailed consideration of the wind is beyond the MESA treatment
where standard Roche-lobe overflow is assumed instead.

Our suggested binary model 1 does not reproduce the fast rotation of the
secondary star (the current O-star donor). The current period of the system
suggested by the model is slightly higher than the observed period. This is
mainly because the initial orbit got widened as a result of the strong mass-
loss rate of the primary during the assumed Wolf-Rayet phase. When reducing
the initial period, we find no stable solution during the first mass-transfer
phase. However, adopting a reduced mass-loss rate in the WR stage compared to
the standard prescriptions as recently derived for lower metallicities by
theoretical calculations (Sander & Vink, 2020; Higgins et al., 2021) might be
able to resolve this conflict.

Considering the predicted abundances, we find that He is by far not as
enriched in our model as derived from our observations. There are major
effects that can lead to altered abundances, mixing and losing mass, so that
the inner enriched layers are exposed to the observer. As the fast rotation
rate of the OB star is not reproduced either, these two findings are likely
linked and the evolutionary abundances suffer from missing the additional
rotational mixing.

Figure 22 illustrates possible evolutionary sequences of M33 X-7 based on the
results from our best-match model. The progenitor comprises of a primary with
$45\,M_{\odot}$ (BH progenitor) and a secondary with $32\,M_{\odot}$ (O-star
donor progenitor). At this point, both stars are on the main sequence. During
the first $\approx 3$ Myr the system remains detached. After core-hydrogen
exhaustion, the primary leaves the main sequence and starts to expand quickly.
When its radius approaches the Roche lobe, a rapid phase of mass transfer
initiates to the less massive secondary which still resides on the main
sequence. The secondary accretes most of the transferred mass and acquires
large angular momentum so that it can spin up to critical velocity. During
this phase, the orbit shrinks to 2.5 days. Even though the secondary spun up
during the mass transfer, the rotation rate drops quickly to the earlier
values after this stage due to tidal locking. After the rapid interaction the
system experiences episodes of slow mass transfer until $\approx 4$ Myr. The
mass transfer ends when the primary’s hydrogen envelope is partially stripped,
with only $\approx 30\,M_{\odot}$ left. The primary will evolve to the core-He
burning phase, transitioning to a Wolf-Rayet star with a strong stellar wind.
At this point, the mass of the He core would be around $15\,M_{\odot}$. Due to
its strong mass loss, the orbital period would increase to 5.5 days. The Wolf-
Rayet phase will last for $0.6$ Myr. The remaining $He$ core of the primary
eventually collapses into a BH at 5.6 Myr.

At around $5.76$ Myr, the scenario described above forms a HMXB, resembling
the properties of M33 X-7. The total duration of this phase lasts for
$\lesssim 0.2$ Myr, implying that this stage has a reasonable probability to
be observed. At this stage, similar to our observations, the O-star donor has
already filled up its Roche lobe, and the BH is accreting this material
(though see Sect. 7).

According to our model, the mass-transfer will become unstable in a few 10,000
years and the system will merge. However, a more fine-tuned model (e.g., with
different mixing) might lead to a stable solution. Nonetheless, since the mass
ratio is large and the orbital period is short, it does not seem to be
unreasonable that the system will eventually merge555 However, the formation
of a supercritical accretion disk around the BH may stabilize the mass
transfer in a way similar to the case of SS433 with similar mass ratio, see
for e.g., Cherepashchuk et al. (2021). Under this assumption, our system will
not be a progenitor of a double-BH binary. Instead, the BH and the He core
will form a common envelope with an in-spiral of the BH eventually resulting
in a merger, leading to the formation of a more massive BH. This could be
accompanied by a $\gamma$-ray burst.

As a slightly different alternative we show a second binary model (Binary
model 2) that also reproduces the observed parameters quite well while having
a different fate. The results are shown in Table 5 and the corresponding
evolutionary tracks are plotted in Fig. 23. In this scenario, we have
discarded our assumption of tidal synchronization in the binary system.
However, it is most likely that the short period binaries would evolve into
tidal locking very soon, independent of the initial stellar spins. In binary
model 2, we assume an initial rotation velocity of 300 km s-1 for both stars.
Due to the change in our assumptions on rotation, now the secondary is only
accreting a small fraction of mass during the first mass transfer. Contrary to
the earlier model, the rotation rate of the secondary reduces only gradually
after the spin up. Hence, this model resembles the observed fast rotation of
the O-star donor at the current HMXB phase. Subsequently, the primary quickly
proceeds to the Wolf-Rayet phase at around 4.5 Myr with a He-core mass of
$\approx 20\,M_{\odot}$. At 5 Myr the primary collapses to a $12\,M_{\odot}$
BH. At this stage, the orbit has already widened due to the strong wind mass
loss. The HMXB phase in this model happens earlier than in the previous model.
However, the period of the system is almost a factor two higher than that of
M33 X-7. In addition, the mass of the donor is lower whereas the BH mass is
slightly higher compared to our empirical estimations. This translates to a
lower mass ratio for the system ($q\approx 2.8$). Because of this slight
decrease in the mass ratio and increase in the orbital period, the second
model predicts a stable mass transfer phase after the donor fills its Roche
lobe at 5.4 Myr, showing the complexity of stellar evolutionary modeling. At
this point, the orbital period shrinks down to two days. Subsequently, the
secondary starts to contract and evolves further as a Wolf-Rayet star.
Throughout these final phases, the primary BH accretes and gains around
$1\,M_{\odot}$ via stellar winds. At 6.3 Myr, the secondary will reach core
carbon burning exhaustion. The final system consists of a BH-BH binary with 13
and $9\,M_{\odot}$ in a four-day orbit. Based on their large separation, these
BHs will not merge in a Hubble time.

#### On the elusive detection of M33 X-7-like systems:

Although the initial masses of the M33 X-7 progenitors are comparable to that
of typical O-star binaries, such BH HMXB systems are rare. Using binary
evolutionary model calculations, we investigated the formation of OB + BH
binaries with a range of masses and periods. According to these models, the
duration of the OB + BH system is only $\approx 2-5$% of the total lifetime of
the binary before both stars become compact objects. Moreover, such a system
would be observed as X-ray binaries only when an accretion disk forms around
the BH. The accretion disk formation criterion for wind-fed systems mainly
depends on the wind and orbital velocity near the BH radius and on the mass
ratio (Illarionov & Sunyaev, 1975; Shapiro & Lightman, 1976). Thus, BH HMXBs
are mostly limited to short-period systems with donors evolved from the main
sequence. With the same observed parameters as of M33 X-7, a system would not
be detectable in X-rays if the orbital period is $>15$ days. Besides, if the
donor is a main-sequence star with a smaller radius and a higher terminal wind
velocity than in M33 X-7, the accretion disk will only form when the orbital
period is less than its current value.

##  9 Conclusions

We present a detailed spectroscopic analysis of the only known eclipsing BH
HMXB system, M33 X-7. For this study we obtained the first UV spectra of the
system accompanied by X-ray observations. The observations were carried out at
three key orbital phases, tracing the BH eclipse, egress, and inferior
conjunction using the HST. At the latter two phases, X-ray observations were
obtained simultaneously using XMM-Newton. These phase-resolved observations
shed light on the interaction of the stellar wind with the BH. The UV
resonance lines show the Hatchett-McCray effect with a large reduction in
absorption strength when the BH is in the foreground due to the strong X-ray
ionization. We performed a detailed analysis of X-ray, UV, and archival
optical spectra using stellar atmosphere models, arriving at the following
conclusions:

  * •

The donor is much cooler and dimmer than previously predicted. The mass-
accreted O star shows signs of He enrichment and fast rotation.

  * •

Our detailed analysis considering multi-wavelength data suggests a large
reduction in the mass of the system compared to previous results. The mass of
the donor star is reduced from $70\,M_{\odot}$ to $38\,M_{\odot}$.
Correspondingly, the BH mass changes from $15.6\,M_{\odot}$ to
$11.4\,M_{\odot}$ and spin of the BH decreases to $a_{\ast}\approx 0.6$.

  * •

Based on the newly derived parameters of the donor and mass of the BH, the
orbital separation has come down to $35\,R_{\odot}$. The comparison of the
radius of the star with a standard Roche-lobe approximation suggests an
overfilling factor of $f\approx 1.2$.

  * •

The strong X-ray radiation of M33 X-7 ionizes much of the wind and leaves a
large Strömgren zone. These extreme ionization conditions prevent the
formation of a normal radiatively driven wind from significant parts of the
donor. Our one-dimensional calculations confirm that the photoionization by
the X-ray radiation can significantly change the ionization structure and
diminish the wind accelerations. Our investigations on wind driving and the
impact X-ray irradiation in M33 X-7 can also be applied to other high-
luminosity HMXB systems in general.

  * •

We estimated the wind parameters of the donor based on the UV spectra taken at
X-ray eclipse. The derived mass-loss rate is in good agreement with the
theoretical prediction assuming a depth-dependent microclumping. By
incorporating X-rays corresponding to the observations at different orbital
phases, we were able to reproduce the observations at three different phases
including eclipse with the same mass-loss rate.

  * •

The observed X-ray luminosity derived from our analysis is $\sim 2\times
10^{38}\mathrm{erg\,s^{-1}}$, which is about 12% of the BH Eddington
luminosity. The accretion luminosity derived using the Bondi-Hoyle calculation
is found to be much lower than the observed X-ray luminosity. Fostered by the
wind inhibition due to the X-ray photoionization, we instead have a “wind
overflow” scenario where the BH not only captures a part of the wind, but it
is further fed by an additional mass overflow along the inner Lagrangian
point. In other directions, the presence of a significant stellar wind
significantly perturbs the Roche potential, allowing material to escape from
the system. As a consequence, the classical distinction between wind-fed and
Roche-lobe overflow systems becomes meaningless for our system. The observed
X-ray luminosity is exactly in line with such a “wind overflow” scenario.

  * •

The X-rays contribute to the formation of the He ii emission region around the
system emitting $\sim 10^{47}\mathrm{ph\,s^{-1}}$.

Taking the parameters of M33 X-7 from the observations into account, we
attempted to trace its evolution using binary evolution models. We modeled the
system from a ZAMS binary system until its current state. In addition, we
proposed two possible scenarios for its future evolution. The system is
transitioning toward an unstable mass transfer phase, resulting in a common
envelope with a deep spiraling of the BH to the envelope of the massive donor.
Since the mass ratio is q$\gtrsim$3.3 and the period is short, the common
envelope phase is likely to result in the merger of two companions resulting
in a heavier BH.

###### Acknowledgements.

The authors are grateful to the referee for the detailed and constrictive
report which helped to improve the paper, and also for providing the optical
spectroscopic data. We thank O. Kargaltsev for the useful discussions. We
thank the Swift telescope team for approving our ToO request. This work made
use of data supplied by the UK Swift Science Data Centre at the University of
Leicester. VR acknowledges support by the Deutsches Zentrum für Luft- und
Raumfahrt (DLR) under grant 50 OR 1912. VR and AACS acknowledge support by the
Deutsche Forschungsgemeinschaft (DFG - German Research Foundation) in the form
of an Emmy Noether Research Group (grant number SA4064/1-1, PI Sander). DP
acknowledges financial support by the Deutsches Zentrum für Luft und Raumfahrt
(DLR) grant FKZ 50 OR 2005. This publication has benefited from discussions in
a team meeting sponsored by the International Space Science Institute at Bern,
Switzerland. This research made use of the VizieR catalog access tool, CDS,
Strasbourg, France. The original description of the VizieR service was
published in A&AS 143, 23.

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##  Appendix A Appendix

Figure 24: Comparison of synthetic spectra in the blue optical range obtained
for models with different stellar temperatures and surface gravities. The mean
optical spectrum of M33 X-7 (Orosz et al. 2007) is shown as a black solid
line. The deep cores in some of the Balmer lines are likely due to an over-
subtraction of nebular emission (see supplementary information in Orosz et al.
2007). In contrast to the other He ii lines, He ii 4686Å appears as a weak
emission in the observation. This line might be filled up with nonstellar
emission, e.g. from the accretion disk.  Figure 25: Same as figure 24, but
zooming on to H$\delta$ comparing models with different surface gravities.

Figure 26: Same as figure 24, but zooming on to multiple He i and He ii lines