11institutetext: Dept of Physics, P.O. Box 84, FI-00014 University of
Helsinki, Finland
11email<EMAIL_ADDRESS>22institutetext: Lab. for High-Energy
Astrophysics, NASA/GSFC, Greenbelt, Maryland, USA
22email<EMAIL_ADDRESS>33institutetext: Finnish Centre for
Astronomy with ESO (FINCA), University of Turku, Väisäläntie 20, 21500
Piikkiö, Finland
33email<EMAIL_ADDRESS>44institutetext: Aalto University Metsähovi
Radio Observatory, PO Box 13000, Fi-00076 Aalto, Finland
44email<EMAIL_ADDRESS>55institutetext: Sky & Telescope, One Alewife
Center, Cambridge, MA 02140, USA
55email<EMAIL_ADDRESS>
# Wind suppression by X-rays in Cygnus X-3

O. Vilhu 11 T.R. Kallman 22 K.I.I. Koljonen 3344 D.C. Hannikainen 55

(Received 22.2.2021; accepted 25.3.2021 )

###### Abstract

Context. The radiatively driven wind of the primary star in wind-fed X-ray
binaries can be suppressed by the X-ray irradiation of the compact secondary
star. This causes feedback between the wind and the X-ray luminosity of the
compact star.

Aims.  We aim to estimate how the wind velocity on the face-on side of the
donor star depends on the spectral state of the high-mass X-ray binary Cygnus
X-3.

Methods.  We modeled the supersonic part of the wind by computing the line
force (force multiplier) with the Castor, Abbott & Klein formalism and XSTAR
physics and by solving the mass conservation and momentum balance equations.
We computed the line force locally in the wind considering the radiation
fields from both the donor and the compact star in each spectral state. We
solved the wind equations at different orbital angles from the line joining
the stars and took the effect of wind clumping into account. Wind-induced
accretion luminosities were estimated using the Bondi-Hoyle-Lyttleton
formalism and computed wind velocities at the compact star. We compared them
to those obtained from observations.

Results.  We found that the ionization potentials of the ions contributing the
most to the line force fall in the extreme-UV region (100-230 Å). If the flux
in this region is high, the line force is weak, and consequently, the wind
velocity is low. We found a correlation between the luminosities estimated
from the observations for each spectral state of Cyg X-3 and the computed
accretion luminosities assuming moderate wind clumping and a low mass of the
compact star. For high wind clumping, this correlation disappears. We compared
the XSTAR method used here with the comoving frame method and found that they
agree reasonably well with each other.

Conclusions.  We show that soft X-rays in the extreme-UV region from the
compact star penetrate the wind from the donor star and diminish the line
force and consequently the wind velocity on the face-on side. This increases
the computed accretion luminosities qualitatively in a similar manner as
observed in the spectral evolution of Cyg X-3 for a moderate clumping volume
filling factor and a compact star mass of a few (2 - 3) solar masses.

###### Key Words.:

Stars: individual: Cyg X-3 - X-rays: individuals: Cyg X-3 – Accretion,
accretion disks - Radiation mechanisms: general - Stars: winds, outflows -
Stars: binaries: close

††offprints: O. Vilhu

##  1 Introduction

Cygnus X-3 (4U 2030+40) was one of the first X-ray binaries discovered
(Giacconi et al. 1967) and is also one of the brightest. It is listed in the
Catalogue of High-Mass X-ray Binaries in the Galaxy (Liu et al. 2006), where
most entries consist of an OB star whose wind feeds a companion neutron star
or a black hole, releasing X-rays in the process. The donor star of Cyg X-3
(optical counterpart V1521 Cyg) is a massive Wolf-Rayet (WR) star of either of
type WN 5-7 (van Keerkwijk et al. 1992 and 1996) or a weak-lined WN 4-6
(Koljonen & Maccarone 2017). The relatively small size of this helium star
allows for a tight orbit with a period of 4.8 hours that is more typical of
low-mass X-ray binaries. In this paper we study the effect of soft X-rays (EUV
radiation) on various wind models of the WR star and estimate the
corresponding accretion rates and luminosities during different spectral
states of Cyg X-3.

Sander et al. (2017) presented three different approaches for handling the
radiative transfer of line-driven winds of hot stars. The first is an
analytical description based on the concept of the CAK theory (Castor, Abbott
& Klein 1975). This method has also opened up the whole area of time-dependent
and multidimensional calculations (see, e.g., Sundqvist et al. 2010). The CAK
theory assumes that line broadening is dominated by the bulk motion of the
wind. In a rapidly expanding wind, line saturation is avoided, which leads to
a strong line force. The second approach is to calculate the radiative force
with the help of Monte Carlo (MC) methods. Computationally, the MC method is
much more time-consuming than the CAK method. The third approach is to
calculate the radiative transfer in the comoving frame (CMF; see Krticka &
Kubat 2017; Krticka et al. 2018; Sander at al. 2018). This method has also
been applied to WR stars (Sander et al. 2017; Hamann et al. 2006; Hainich et
al. 2014).

X-rays (mainly soft X-rays in the extreme-UV, EUV, region) from the compact
star affect the wind through photoionization. An increase in highly ionized
ions may decrease the line-driven mechanism and reduce the wind velocity. As
noted by Karino (2014), pioneering work on these ionized wind dynamics was
done by Stevens and Kallman (1990), who used the XSTAR code (Kallman 2018) to
approximate the wind with a photoionized plasma; the line force was computed
with the CAK method. In the present paper, we revisit and further develop this
method and apply it to Cyg X-3. We adopt the first approach (CAK) classified
by Sander et al. (2018). The results we present agree with those of Krticka et
al. (2018) for HMXRBs: X-rays reduce the wind velocity, and wind clumping
weakens the effect of X-rays. Szostek and Zdziarski (2008) found that the wind
of Cyg X-3 has to be clumpy in order to explain the observed X-ray absorption.
We therefore include clumping with different filling factors here.

Our goal is to study the penetration of X-rays from the compact star of Cyg
X-3 of the wind, and to determine how it affects the wind properties, in
particular, the wind velocity between the two stars. Based on this, we
estimate the accretion luminosities of the five main spectral states of Cyg
X-3 as classified by Hjalmarsdotter et al. (2009). MacGregor and Vitello
(1982) applied a somewhat similar approach to X-ray binary systems with wind
accretion and found that X-ray irradiation diminishes the wind velocity. A
relevant recent paper (and references therein) is on Cyg X-1 by Meyer-
Hofmeister et al. (2020). They found that the outflow from the OB-star facing
the X-ray source is noticeably slower than from the opposite hemisphere in the
X-ray shadow. Furthermore, they give evidence for two stable states: the soft
high-luminosity state, and the hard low-luminosity state.

##  2 Cyg X-3 system parameters

Using Chandra High Energy Transmission Grating data (HETG, ObsID 7268 and
6601), Vilhu et al. (2009) derived radial velocity curves of the FeXXVI
emission line at 1.78 Å (H-like), which is thought to originate in the
vicinity of the compact star. These data comprise ten phase bins and give a
radial velocity K amplitude of 454$\pm{109}$ km/s. When this is coupled with
that of the infrared HeI absorption line (thought to originate in the WR
component) of 109$\pm{13}$ km/s (Hanson et al. 2000), it yields a mass ratio
$M_{C}$/$M_{WR}$ = 0.24$\pm{0.06}$. Zdziarski et al. (2013) arrived at a
similar number using the same data, but selecting a slightly different subset
of the data. Based on the statistics of the masses and mass-loss rates of WR
stars, Zdziarski et al. (2013) fixed the WR companion mass at about 10
$M_{\odot}$. This implies a mass of 2.4 $M_{\odot}$ for the compact star. We
use these values in the present paper. Zdziarski et al. (2013) estimated the
mass-loss rate to be between (4.8–8.2)$\times$10-6 $M_{\odot}$/year. We
adopted the mean value 6.5$\times$10-6 $M_{\odot}$/year, which is close to
that derived from the orbital period increase
$\dot{M}$=$M_{tot}$($\dot{P}$/2$P$) (Ergma & Yungelson 1998).

Based on Gemini/GNIRS infrared spectroscopy, Koljonen and Maccarone (2017)
suggested that the WR component is of spectral class WN 4-6 and that its
bolometric luminosity lies between (5.5–11.5)$\times$ 1038 erg/s. We adopted
8.0$\times$1038 erg/s as a compromise value. For the photospheric blackbody
temperature we adopted 1.25$\times$105 K. The WR component radius at the wind
base and the binary separation are 0.95 and 3.31 solar radii, respectively.
The temperature and radius of the adopted model are close to those of the
Potsdam WNE model 15-21 (in http://www.astro.physik.uni-
potsdam.de/$\sim$wrh/PoWR/WNE/).

As the WR component is a hydrogen-deficient WN star, we adopted hydrogen,
helium, and heavy element mass fractions of X=0.085, Y=0.90, and Z=0.015,
respectively (Nugis & Lamers 2000). We used solar abundances for individual
heavy elements (Asplund et al. 2009). The effect on the force multiplier of
using equilibrium CNO abundances is negligible. We assigned a distance of 7.4
kpc based on McCollough et al. (2016). The system parameters are collected in
Table 1.

Table 1: Parameters of the Cyg X-3 binary components and stellar wind used in this paper. Masses of the compact star and WR star ($M$C and $M$WR) are given in $M_{\odot}$, the mass-loss rate $\dot{M}$ in 10-6$M_{\odot}$/year, the luminosity of the WR star $L$WR in 1038 erg/s, the blackbody temperature $T_{bb}$ at the WR surface in Kelvin degrees, the orbital period P in hours, and the distance d in kiloparsecs. $M_{C}$ | $M_{WR}$ | $\dot{M}$ | $L_{WR}$ | $T_{bb}$ | P | d
---|---|---|---|---|---|---
2.4 | 10.0 | 6.5 | 8.0 | 125000 | 4.8 | 7.4

##  3 EUV photon fluxes in the spectral states of Cyg X-3

The hybrid EQPAIR model originally developed by Coppi (1992) is suitable for
modeling the broadband X-ray spectrum of Cyg X-3 (Vilhu et al. 2003;
Hjalmarsdotter et al. 2009). At soft X-rays below 0.1 keV, which are not
observable due to high circumstellar and interstellar absorption, the model is
dominated by disk blackbody and Compton scattering from thermal electrons.

Using a large dataset of RXTE X-ray observations, Hjalmarsdotter et al. (2009,
see their Fig. 7, reproduced here as Fig. 1) classified the model into five
main states (hard, soft nonthermal, intermediate, ultrasoft, and very high).
Table A.1 gives the model fluxes at 100 eV (unabsorbed at Earth). From these
fluxes and spectral shapes of Fig. 1, the unabsorbed number of photons between
100–230 Å (54–124 eV, where 54 eV is the HeII ionization edge) was estimated.
These EUV photons are important because they decrease the size of the line
force. A similar effect is in the increase of ionization due to X-rays (see
Appendix B). The EUV flux of the hard state is uncertain due to the
absorption, as noted by Zdziarski et al. (2016).

The ionization potentials of NiV and FeV (four electrons stripped off), the
lines of which contribute to the line force, fall into the 100–230 Å region.
As an example, photons between 70–90 eV can ionize FeV (Fe4+; see, e.g., El
Hassan et al. 2009). If the radiator has significant flux in this region,
these ions are photoionized and the line force decreases. Other smaller
contributors to the line force are the lines of CIV, NV, and CrV ions, for
instance, whose ionization potentials also fall in this region. For the effect
of EUV photons on the line force as well as for a computation of the line
force, see Appendix B.

Figure 1: Unabsorbed model spectra of the five main accretion states of Cyg
X-3 reproduced from Hjalmarsdotter et al. (2009). The models corresponding to
the hard state (solid blue line), the intermediate state (long-dashed cyan
line), the very high state (short-dashed magenta line), the soft nonthermal
state (dot-dashed green line), and the ultrasoft state (dotted red line) are
shown (unabsorbed at Earth). Figure 2: Number of photons between 100–230 Å (in
units of that from a 125 000 K blackbody) as a function of the distance from
the WR star surface (the intermediate model of Table A.1 ($f_{vol}$ = 0.3,
$\Phi$ = 0) along the line joining the stars). The compact star is located at
2.4$R_{WR}$. The solid line shows the contribution from the WR star and the
dashed line that from the compact star (X-ray source). The dotted line shows
the coadded contribution.

##  4 Computation of wind velocities

Wind models were computed for each of the five spectral states taking into
account the effect of X-rays from the compact star. The wind equations were
solved along the line joining the centers of the two stars ($\Phi$ = 0) and
for different orbital angles $\Phi$ (see Fig. 3). The geometry is the same as
in Krticka et al. (2012, see their Fig. 1). We also repeated their Vela X-1
computations with the present XSTAR method, and these are outlined in Appendix
C, along with comparisons with the CMF method for O stars from Krticka and
Kubat (2017).

Figure 3: Geometry of the system in the orbital plane in units of the WR star
radius. The compact star (CS) is located at distance 3.4 from the center of
the WR star. The orbital angle $\Phi$ (PHI) is marked in the direction of the
motion of the compact star.

Soft X-rays from the compact star penetrate the wind from the donor star and
change the radiation field and the number of photons in the critical EUV range
100–230 Å. In numerical computations, fluxes from inside (WR star) and outside
(compact star) were computed at each wind layer. Fig. 2 shows the relative
number of photons between 100–230 Å as a function of the distance from the WR
surface (intermediate state at $\Phi$ = 0). The local force multiplier (line
force) was then computed using the coadded 100–230 Å fluxes from the WR and
compact stars, respectively. Higher EUV flux corresponds to a weaker line
force.

We considered only the supersonic part of the wind flow (above
$\delta$($r/R_{WR}$) = 0.01 from the WR surface). As shown in Fig. 2, EUV
photons from the compact star do not penetrate the sonic point, only the outer
part of the supersonic wind is affected. At this boundary, the wind velocity
was set to 25 km/s, slightly above the sonic point for a 105 K He gas, which
is about 20 km/s. A better estimate of the sonic point properties can be found
in Grassitelli et al. (2018). For a He-star model of 10 $M_{\odot}$ and mass-
loss rate of 8$\times$10-6$M_{\odot}$/year (close to the Cyg X-3 value in
Table 1), they find a velocity of 19.62 km/s at the sonic point (see their
Table 1). Below this boundary, the wind particle column was assumed to be
1023cm-2 (the whole wind column is 20 times larger). The wind velocity scales
with this column loosely as $(column)^{0.1}$. For the model without X-rays,
the assumed column gives $V_{inf}$ $\approx$ 2200 km/s (NOX in Fig. 4).

The wind was divided into 1 000 shells with $\delta$($r/R_{WR}$) = 0.01. The
mass-loss rate was kept constant, and we assumed that it is determined in the
subsonic part of the wind that X-rays from the compact star do not penetrate.
The gravitational pull of the compact star was not included. Clumping was
included assuming that all the wind mass is in the clumps and that the space
between clumps is empty. In this case, the particle number density $N$ in the
ionization parameter $\xi$ (Eq. B.4) is replaced by $N$/$f_{vol}$ , where
$f_{vol}$ is the volume-filling factor. Clumping makes $\xi$ smaller,
increasing the line force and consequently the wind velocity (see Fig.B.4).

The two equations which determine the flow are mass conservation

$\dot{M}=M_{\mathrm{{dot}}}=4\pi r^{2}\rho v$ (1)

and momentum balance (neglecting gas pressure)

$vdv/dr=GM/r^{2}[\Gamma_{e}(1+g_{line})-1]\,{\rm},$ (2)

where the line force is

$g_{line}=F_{M}(r)F_{d}(r)$ (3)

and

$\Gamma_{e}=\sigma_{e}L/(4\pi GMc).$ (4)

Using the values in Table 1 for the WR star yields $\Gamma_{e}$ = 0.34. The
value of the finite disk correction factor $F_{d}(r)$ is about 0.7 at the WR
surface and increases to 1 at $x$ = $r/R_{WR}$ = 1.5. It remains constant
thereafter (Sundqvist et al 2014). The force multiplier $F_{M}$ depends on the
ionization parameter $\xi$, the absorption parameter $t$ (Eq. B.1), and the
spectral shape of the radiator, particularly in the EUV region around 100–230
Å. In practice, $F_{M}$ was interpolated from an extensive precomputed grid at
each wind layer using local values of these parameters (see Appendix B).

The wind equations were integrated numerically from $x$ = $r/R_{WR}$ = 1.01 up
to 10.0 , resulting in velocity stratification v(r) (see Fig. 4). Without
X-rays the resulting velocity stratification can be compared and fitted with a
standard $\beta$-velocity model to give the best model (see Appendix C.1).
However, with X-ray irradiation, no such standard model exists, and we used
our own formula in which the $\beta$-velocity model is multiplied by a
suppression factor $SF,$

$v_{model}=v_{\mathrm{{inf}}}(1-B/x)^{\beta}\times SF\,$ (5)
$SF=\cases{1}-slope\times(x-xkink),&\text{if}\ x\geq xkink\\\ 1,\text{if}\
x<xkink{}$ (6) $B=1.01\times[1-(v_{in}/v_{inf})^{1/\beta}]$ (7)
$x=r/R_{star},$ (8)

where the $slope$ and $xkink$ are fitting parameters. In the inner boundary,
$v_{in}$ = 25 km/s.

The fits were improved by integrating the wind equations several times
(without fitting), computing the EUV contamination from the previous model,
until the average difference between two subsequent models was smaller than 1%
. This typically required 10–20 runs. The results do not differ much from the
case where the X-rays were simply switched on in the NOX-model (no X-rays) and
the model integrated without fitting. The computation for $\Phi$ = 0 (along
the line joining the centers of the two stars) is straightforward. For larger
angles, we numerically integrated across the wind to obtain dilution factors
and absorption. Following Krticka et al. (2012), we assumed that the wind in
the direction of the orbital angle $\Phi$ can be modeled by a spherically
symmetric wind.

The resulting wind velocity in the NOX state is compatible with velocities of
WN stars derived from line widths of He lines (Hamann et al. 1995).
Computations with different orbital angles are relevant because the time in
which the wind element crosses the distance between the components (crossing
time) is about 0.35–0.45 hours for our models, corresponding to orbital angles
between 25–35 degrees (see Table A.1). The true wind velocity at the compact
star is likely a smear of computed velocities inside $\Phi$ = 0–35 degrees.

Fig. 4 (left panel) shows the velocity stratifications for three spectral
states at $\Phi$ = 0 and moderate clumping ($f_{vol}$ = 0.3). In addition,
orbital angles of 30 and 90 degrees are included for the intermediate state.
The right panel of Fig. 4 shows velocity stratifications for the very high
state with several clumping filling factors. At large distances from the WR
star, the velocities approach constant values. This plot can be compared with
a similar one from Krticka et al. (2018, their Fig. 4). For small clumping
filling factors ($f_{vol}$ ¡ 0.1), the wind velocity and accretion luminosity
become rather insensitive to the EUV flux (see Table A1 and Fig. 5) because
clumping causes the ionization $\xi$ parameter to decrease and consequently
the EUV dependence to weaken (see Fig. B.4).

The wind velocity around the compact star during the very high state is
consistent with the P Cygni absorption of the SiXIV line at $-$900 $\pm$ 750
km/s, which is observed most clearly at about orbital phase 0.5 (compact star
in front, WR star behind; Vilhu et al. 2009). For more details on P Cygni
lines and photoionization in Cyg X-3, see also Kallman et al. (2019). In the
next section we estimate the accretion rates and luminosities from the
computed velocities at the compact star and compare them with those from the
EQPAIR models. The wind velocities were computed without the gravitational
pull of the compact star to show the effect of X-rays and to apply the Bondi-
Hoyle-Lyttleton model in the next section.

Figure 4: Left panel: Computed wind velocities as a function of the radial
distance from the center of the WR star (in units of stellar radius) for the
orbital angle $\Phi$ = 0 deg (solid lines, see Fig. 3). The clumping volume
filling factor $f$ = $f_{vol}$ = 0.3. VH = very high state, INT = intermediate
state, HARD = hard state, and NOX = the wind without X-rays from the compact
star (whose distance is marked as CS in the plot). Dashed lines are for the
intermediate state for angles $\Phi$ = 30 deg (lower) and 90 deg (upper).
Right panel: Wind radial velocities for the very high state with three
clumping volume filling factors $f$ = $f_{vol}$ = 1.0, 0.3, and 0.1 (dashed
lines). The solid line shows the wind without X-rays (NOX) and with $f$ = 0.3.
The gravitational pull of the compact star is not included.

##  5 Accretion luminosities during Cyg X-3 spectral states

Using wind velocities at the compact star and a simple analytic wind accretion
model described by Frank, King, and Raine (1992), we estimate the accretion
luminosity $L_{acc}$. This Bondi-Hoyle-Lyttleton model has been used widely
for instance by Watanabe et al. (2006), Koljonen and Maccarone (2017), and
Krticka et al. (2018). A review of the topics with original references can be
found in Martinez-Nunez et al. (2017). In the previous section, wind
velocities were computed without the gravitational pull of of the compact star
because in the Bondi-Hoyle-Lyttleton model, the wind velocity is taken at the
distance of the compact star and without its gravitational pull. This is taken
into account when deriving the formula itself, which should only be considered
as a sort of scaling law. In the model, $L_{acc}$ strongly depends on the
relative velocity $V_{rel}$,

$V_{rel}^{2}=V_{wind}^{2}+V_{orb}^{2},$ (9)

where the orbital velocity of the compact star is Vorb = 675 km/s using the
system parameters in Table 1. The accretion luminosity is

$L_{acc}=\dot{M}_{acc}\times\eta c^{2},$ (10)

where the accretion rate depends on the accretion radius as follows:

$\dot{M}_{acc}/\dot{M}_{wind}=\pi r_{acc}^{2}/(4\pi a^{2})$ (11)
$r_{acc}=2GM_{C}/V_{rel}^{2}.$ (12)

The mass-to-energy conversion factor $\eta$ = 0.1 was used, which is
approximately valid for neutron stars and black holes (see, e.g., Frank, King
and Raine 1992). Accretion luminosities were calculated for the five spectral
states using the computed wind velocities at the compact star (see Fig. 4 and
Table A.1) for three clumping filling factors $f_{vol}$ = 1.0 (no clumping),
0.3, and 0.1. These are shown in Fig. 5. The filled squares in Fig. 5 show the
EQPAIR model luminosities (Lbol of Table A.1).

The left panel of Fig. 5 shows the luminosities computed for the compact star
mass 2.4 $M_{\odot}$ , and the right panel shows the luminosities for a higher
compact star mass 3.0 $M_{\odot}$. Because the realistic wind velocity
probably lies between orbital angles 0 and 30 degrees (30 deg is close to the
crossing times in Table A.1), moderate clumping favors 2.4 $M_{\odot}$ and
higher clumping favors 3.0 $M_{\odot}$. Lower masses require less clumping. At
very high clumping, the correlation between different spectral states
disappears.

Fitting the mean bolometric luminosity value of spectral states
($L_{bol}$=2$\times$1038 erg/s) with similar means from our computations at
orbital angle $\Phi$ = 15 degrees, we derived the relation between the
clumping filling factor and compact star mass ($f_{vol}$, $M_{C}$):
(0.70,1.4), (0.28,2.0), (0.16,2.4), and (0.08,3.0). Szostek and Zdziarski
(2008) estimated values between 0.07 - 0.3 for the one-phase filling factor
(used in the present work). This points to a compact star mass between
(2-3)$M_{\odot}$.

Figure 5: Left panel: Computed accretion luminosities (in units of 1038 erg/s)
for the five spectral states of Cyg X-3 are shown for three clumping filling
factors $f$ = $f_{vol}$ = 1.0 (dotted lines), = 0.3 (dashed lines), and = 0.1
(solid lines). The orbital angle $\Phi$ = 0 deg for the upper curves and
$\Phi$ = 30 deg for the lower ones. The filled squares show the EQPAIR model
luminosities ($L_{bol}$ of Table A.1) The compact star mass is 2.4
$M_{\odot}$. In the abscissa 0 indicates NOX (no X-rays), 1 hard, 2 soft
nonthermal, 3 intermediate, 4 ultrasoft, and 5 very high state. Right panel:
Same as in the left panel, but the compact star mass is 3.0 $M_{\odot}$.

##  6 Discussion

We computed the force multiplier (line force) using the CAK method, and the
wind was approximated by the photoionized plasma code XSTAR (Stevens and
Kallman 1990; Kallman 2018). XSTAR is designed to calculate the ionization and
thermal balance in gases exposed to ionizing radiation. It contains a
relatively complete collection of relevant atomic processes and corresponding
atomic data. It does not by itself include processes associated with wind
flows, such as shock heating, adiabatic cooling, or time-dependent effects. It
also has a simple treatment of radiation transport that provides approximate
results in situations involving truly isotropic radiation fields or line
blanketing, for example.

Key simplifications employed by XSTAR are with regard to the radiative
transfer solution. Escape of line radiation is treated using an escape
probability formalism, and this affects the temperature through the net
radiative cooling. Transfer of the ionizing continuum is treated using a
single-stream integration of the equation of transfer, including opacities and
emissivities calculated from the local level populations. Thus, there is no
allowance for the inward propagation of diffusely emitted radiation.

This last approximation is likely to be most important for the results
presented here because strong winds can have large optical depths in the
ionizing continua of H and He II. It would be desirable to verify our current
results for the force multiplier using a transfer solution that does not have
this limitation. Limitations of the escape probability assumption for line
escape have been pointed out by Hubeny (2001), for example. However, for hot
star winds, the strong velocity gradient reduces most line optical depths to
approximately a few at most, and so the lines are effectively thin and the
radiative cooling is not strongly affected. However, the simplifications made
do not affect the EUV sensitivity of the force multiplier, which is the key
issue here.

The assumed masses for the components may represent just one possible solution
for Cyg X-3. A more massive possibility within errors is 3.0 + 10 $M_{\odot}$,
for example, yielding 50% higher computed accretion luminosities. A less
massive alternative 2.0 + 10 $M_{\odot}$ yields the opposite. A tolerable
match with the EQPAIR models can be provided by moderate clumping ($f_{vol}$ =
0.1 - 0.3) and $M_{C}$ = 2.4 $M_{\odot}$ or by $f_{vol}$ = 0.1 and $M_{C}$ =
3.0 (see Fig. 5). Smaller compact star masses require less clumping, which is
an unknown parameter in this study. However, typical volume filling factors
for WR-stars have been estimated to be 0.1 - 0.25 (e.g., Hamann and Koesterke
1998).

The region in which the photoionization of the main contributors to the line
force takes place was assumed to be 100 Å–230 Å. However, this region could be
shallower, for instance, 100–177 Å, because the ionization potentials of the
main contributors NiV and FeV are at 164 Å. In this case, the computed
accretion luminosities would increase (by a factor of 1.5). One possible
caveat is that the estimated EUV fluxes from the EQPAIR models do not
necessarily reflect reality. In a sense, the EUV flux is always model
dependent because we cannot obtain any observations in these wavelengths.

Accretion disks are frequently assumed in the context of wind accretion.
Accretion disk formation in radiatively driven wind accretion has recently
been studied by Karino et al (2019). A good illustration of the Cyg X-3 disk
can be found in Hjalmarsdotter et al. (2009). The origin of transitions
between the different states probably lies in some form of disk instability.
In the hard state, the radius of the inner accretion flow is large (and the
EUV flux low). In the soft states, the inner disk boundary shrinks, and
consequently, the EUV flux increases, the wind velocity decreases, and the
accretion luminosity $L_{acc}$ increases (see Fig. 7 in Hjalmarsdotter et al.
2009 and discussion therein).

The disk and wind may interact. This means that the changing EUV flux from the
compact star might account for the different luminosity and spectral states of
Cyg X-3. When the EUV flux is high, the X-ray accretion luminosity is also
high, and vice versa. Our computations together with the EQPAIR model
luminosities favor this picture if both the compact star mass and clumping are
suitable (see Fig. 5). We might speculate that during an eventual extremely
high state (flare) the computed wind at the compact star may stop, the X-ray
production ends, and the wind returns to the NOX state. During this state, the
accretion luminosity does not differ much from the hard-state luminosity (see
Fig. 5). The hard state may thus be the most natural and long-lasting state of
Cyg X-3.

Antokhin and Cherepashchuk (2019) extended the list of times of X-ray minima
and fit the $O-C$ diagram by a sum of quadratic and sinusoidal (with period
15.79 years) functions. They suggested the existence of either apsidal motion
or a third body. The mass loss, causing the period increase, is behind the
overall shape of the $O-C$ diagram. For the quadratic plus sinusoidal
ephemerides, Antokhin and Cherepashchuk find $\dot{P}$ =
5.629$\times$$10^{-10}$ , corresponding to a relative period change of
$\dot{P}$/$P$ = 1.029$\times$$10^{-6}$/year. $\dot{P}$ is related to the
coefficient $c$ of the quadratic term by $c$ = $P$$\dot{P}$/2. We estimate the
change of $\dot{P}$/$P$ in our models below.

For a nonconservative mass transfer by fast isotropic wind (Jeans mode, see
Ziolkowski and Zdziarski 2018, their eq. 15 with $\alpha$=1):

$\frac{\dot{P}}{P}=\frac{\dot{M}_{wind}}{M_{WR}}[3-3\beta\frac{M_{WR}}{M_{C}}-(1-\beta)\frac{M_{WR}}{M_{tot}}-3(1-\beta)\frac{M_{C}}{M_{tot}}],$
(13)

where $\beta$ is the fraction of the wind accreted by the compact star
$\dot{M}_{acc}$/$\dot{M}_{wind}$.

For an acceptable model (10$M_{\odot}$ \+ 2.4$M_{\odot}$, $f_{vol}$ = 0.16,
see Sect. 5), we have $\beta$ = 0.002 and 0.005 for the hard state and the
very high state, respectively. For the period change, these give $\dot{P}$/$P$
values 1.034$\times$$10^{-6}$/year (hard) and 1.012$\times$$10^{-6}$/year
(very high), which differ by 2%. Hence, the effect of spectral states on the
$O-C$ diagram is very small through the quadratic term coefficient. Moreover,
in the $ASM/RXTE$ data, Cyg X-3 spent 50% of the time in the low state (¡ 10
cps) and 22% in the high state (¿ 20 cps). Hence, the average of $\dot{P}$/$P$
is closer to the hard-state value as observed. Locally, like during and after
the jigsaw pattern detected in the $ASM/RXTE$ data between MJD = 50700 - 51600
(when Cyg X-3 was mostly in the low hard state), this effect is unobservable.
The jigsaw pattern is probably caused by the temporal irregular distribution
of $ASM$ data, as suggested by Antokhin and Cherepashchuk.

##  7 Conclusions

We constructed wind models in the supersonic part of the WR component of Cyg
X-3 using force multipliers (line force) computed with CAK formalism and XSTAR
physics (following the pioneering work of Stevens and Kallman 1990; for the
XSTAR-code, see Kallman 2018). The line force strongly depends on the EUV flux
of the radiator (100–230 Å), to which the X-rays from the compact star
contribute. The main contributors are lines of NiV and FeV ions (to a lesser
extent, e.g., CIV, NV, and CrV). Their (photo)ionization potentials fall
between 100–230 Å (54–124 eV). In this region the radiation field controls
these lines and consequently the size of the force multiplier. Higher EUV flux
from the compact star means a weaker line force, which in turn leads to a
lower wind velocity.

Because X-rays do not reach the subsonic part of the wind, we assumed that the
mass-loss rate is unaffected by X-rays. We included clumping by assuming that
all the wind mass is in the clumps. We modeled the wind at different orbital
angles (see Fig. 3). Computations with different orbital angles are relevant
because the time in which the wind element crosses the distance between the
components (crossing time) is about 0.35–0.45 hours in our models. This
corresponds to orbital angles between 25–35 degrees (see Table A.1).

Radiation from the compact component increases the EUV flux and diminishes the
wind velocity (Fig. 4). This enhances the accretion rate, and consequently,
the accretion luminosity (Fig. 5). The computed Bondi-Hoyle-Lyttleton
accretion luminosities $L_{acc}$ correlate moderately well with the
luminosities of the five main spectral states of Cyg X-3 defined by
Hjalmarsdotter et al. (2009), provided that the clumping is moderate
($f_{vol}$ = 0.1 - 0.3) and the mass of the compact star is about (2 -
3)$M_{\odot}$ (with the WR-star mass 10$M_{\odot}$, see Fig. 5). For higher
clumping ($f_{vol}$ ¡¡ 0.1) the EUV sensitivity disappears because clumping
decreases the ionization parameter $\xi$ and the line force dependence on EUV
flux (see Fig. 9).

We compare our XSTAR method with the CMF method (comoving frame) in Appendix
C. The results are qualitatively similar for both O stars (Krticka and Kubat,
2017) and X-ray suppression in Vela X-1 (Krticka et al. 2012).

###### Acknowledgements.

We thank Drs. Hjalmarsdotter and Krticka for permission to use their Figures
(Fig.1 and Fig.C.3) and Drs Andrzei Zdziarski and Linnea Hjalmarsdotter for
valuable comments. K.I.I.K was supported by the Academy of Finland projects
320045 and 320085.

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##  Appendix A Computed wind velocities

In Table we list A.1 the EUV/EQPAIR fluxes and bolometric luminosities of Cyg
X-3 spectral states (Hjalmarsdotter et al. 2009) and the computed wind
velocities at the compact star (this study).

Table 2: Luminosities and EUV-fluxes of the five basic states of Cyg X-3 from RXTE observations and EQPAIR models (Hjalmarsdotter et al. 2009, first two rows) and computed wind velocities at the compact star (this study, last rows). NOX is the state without X-rays. $L_{bol}$ (in units of 1038 erg/s) is the EQPAIR model luminosity. EFE (E$\times$FE; in units of KeV/cm2/s) is the unabsorbed flux at Earth (at 100 eV, 124 Å). Computed wind velocities at the compact star Vwind (km/s) are given for two orbital angles $\Phi$ = 0 and 30 degrees (the orbit = 360 deg, see Fig. 3) and for three clumping filling factors $f_{vol}$ = 1.0 (no clumping), 0.3, and 0.1. Time (in degrees, the orbital period = 360 deg) is the crossing time of a wind particle from the WR surface to the compact star distance. PARAMETER | NOX | HARD | SOFT NON-TH | INTERMEDIATE | ULTRASOFT | VERY HIGH |
---|---|---|---|---|---|---|---
$L_{bol}$ |  | 1.1 | 1.7 | 1.5 | 2.4 | 3.2 |
EFE |  | 0.03 | 0.07 | 0.11 | 0.14 | 0.19 |
$f_{vol}$ = 1.0 : |  |  |  |  |  |  |
Vwind $\Phi$=0 | 1779 | 1094 | 933 | 841 | 747 | 627 |
Vwind $\Phi$=30 | 1779 | 1340 | 1181 | 1113 | 983 | 885 |
Time $\Phi$=0 | 29 | 35 | 38 | 40 | 43 | 47 |
$f_{vol}$ = 0.3 : |  |  |  |  |  |  |
Vwind $\Phi$=0 | 1851 | 1382 | 1289 | 1253 | 1188 | 1121 |
Vwind $\Phi$=30 | 1851 | 1682 | 1590 | 1552 | 1483 | 1401 |
Time $\Phi$=0 | 28 | 30 | 31 | 31 | 32 | 33 |
$f_{vol}$ = 0.1 : |  |  |  |  |  |  |
Vwind $\Phi$=0 | 1967 | 1622 | 1557 | 1543 | 1490 | 1444 |
Vwind $\Phi$=30 | 1967 | 1913 | 1879 | 1873 | 1830 | 1776 |
Time $\Phi$=0 | 25 | 26 | 26 | 27 | 27 | 28 |

##  Appendix B  Methods for computing the line force (force multiplier).
Effect of EUV photons.

The force multiplier (line force) $F_{M}$ was computed following the method
described by CAK. An important difference is that CAK (and other compilations,
e.g., Gayley et al. 1995) assume that the ionization and excitation in the
wind is given by the Saha-Boltzmann distribution modified by an analytic
dilution factor applied to all elements. We computed the ionization
distribution in the wind by balancing the ionization and recombination due to
the stellar radiation field. We also employed the Boltzmann distribution to
determine the populations of excited levels. The radiation field was computed
using a single-stream integration outward from the stellar photosphere. In
this way, we calculated the line force appropriate to any spectral form of the
photospheric spectrum and at any layer in the wind. This permitted us to
integrate wind equations from the surface to infinity in a self-consistent
manner because the upward force is known.

The ionization balance and outward transfer of photospheric radiation were
calculated using the XSTAR photoionization code (Stevens & Kallman, 1990;
Kallman, 2018). The ion fractions at each spatial position in the wind were
used to calculate the force multiplier $F_{M}$ by summing the CAK line force
expression over an ensemble of lines. The list of lines was taken from Kurucz
(http://kurucz.harvard.edu/linelists.html) and permitted us to use local wind
parameters. The line list is more extensive than in the original CAK work. It
is assumed that the wind is spherically symmetric around the donor star (the
ionizing source).

XSTAR calculates the ionization balance for all the elements with atomic
number Z $\leq$ 30 together with the radiative equilibrium temperature. It
calculates full nonlocal thermal equilibrium (NLTE) level populations for all
the ions of these elements. It includes a fairly complete treatment of the
level structure of each ion, that is, more than $\sim$50 levels per ion, and
up to several hundred levels for some ions. Many relevant processes affecting
level populations are included such as radiative decays, electron impact
excitation and ionization, photoionization, and Auger decays. The electron
kinetic temperature is calculated by imposing a balance between heating from
fast photoelectrons and Compton scattering with radiative cooling.

For a fixed radiator, the force multiplier $F_{M}(t,\xi,N)$ is a three-
dimensional function of the absorption parameter $t$, ionization parameter
$\xi$ (erg/s$\times$cm), and particle number density $N$ ($cm^{-3}$),

$t=\sigma_{\mathrm{e}}v_{\mathrm{th}}\rho(dv/dr)^{-1}$ (14)
$\xi=\frac{L_{\mathrm{ly}}}{Nr^{2}}\ .$ (15)

Here $L_{ly}$ is the radiation luminosity below the hydrogen ionization limit
912 Å, $\sigma_{e}$ is the electron scattering coefficient, $\rho$ and $N$ are
the gas and particle number density, respectively, $v$ is the outward wind
velocity, $r$ is the radial distance from the ionizing source, and $v_{th}$ is
the gas thermal velocity (typically $\sim$10 km/s). The dependence on $N$ is
weak (Vilhu and Kallman 2019). The force multiplier was computed assuming a
point source radiator. When applied to stars, the dilution factor $r^{-2}$ in
$\xi$ (Eq. B2) should be replaced by the finite disk factor (Mewe and
Schrijver 1978),

$W(r)=0.5[1-\sqrt{(}1-(R_{star}/r)^{2})]R_{star}^{-2}.$ (16)

In our paper the radiator consists of two sources: the WR-star and the (point
source) compact star. The ionization parameter was computed taking the
(absorbed) contributions $L_{lywr}$ and $L_{lyx}$ from both sources into
account,

$\xi=\frac{L_{\mathrm{lywr}}}{N}W(rwr)+\frac{L_{\mathrm{lyx}}}{N(rx)^{2}}\ ,$
(17)

where $rwr$ and $rx$ are the distances from the WR center and compact star,
respectively. $L_{lywr}$ and $L_{lyx}$ are component luminosities below the
Lyman edge at a specific point in the wind, absorbed along the path from the
WR-surface and the compact star (X-ray source), respectively (by
exp(-absorption$\times$column)). X-rays increase the ionization parameter
$\xi$ through the Lyman luminosities and diminish the line force. In the case
of Cyg X-3, the EUV-photons from the compact star are more important. Their
number effectively controls the line force.

The chemical abundances of the wind and the ionizing source spectrum can be
specified as input data. The method is in principle the same as in the
classical work of CAK. However, we explicitly included $\xi$ and $t$ and
computed it locally in the wind. In the case of clumping, with volume-filling
factor $f_{vol}$ and all mass in the clumps, the particle number density $N$
in Eq. B2 should be replaced by $N$/$f_{vol}$.

The force multiplier depends on $t$ and on the cross section for line
absorption,

$$$F_{M}(t,\xi)=\Sigma_{lines}{\frac{\Delta\nu_{D}F_{\nu}}{F}\frac{1}{t}(1-e^{\eta
t})}$$,$ (18)

where $\Delta\nu_{D}$ is the thermal Doppler width of the line, $F$ is the
total flux in the continuum, $F_{\nu}$ is the monochromatic flux at the line
energy, and

$$$\eta=\kappa_{line}/\sigma_{e}$$$ (19) $$$\kappa_{line}=\frac{\pi
e^{2}}{m_{e}c}g_{L}f\frac{N_{L}/g_{L}-N_{U}/g_{U}}{\rho\Delta\nu_{D}}$$,$ (20)

where $N_{L}$ and $N_{U}$ are the lower and upper level populations, $g_{i}$
are the statistical weights, and $f$ is the oscillator strength. Our code
(xstarfmult.f) uses the ion fractions calculated as described above to sum
over lines to calculate $F_{M}(t,\xi)$. Ground-state level populations were
taken directly from XSTAR; excited level populations were calculated assuming
a Boltzmann (LTE) distribution. This is necessitated by the use of the Kurucz
line list, which does not include collisional rates or other level-specific
quantities that would allow a full NLTE calculation of populations. Most
previous calculations of the CAK force multipliers have employed the LTE
assumption for excited levels as well (e.g., CAK; Abbott 1982; Gayley 1995).

$F_{M}$ grids were computed for the chemical composition of WNE stars for many
$\xi$\- and $t$-values and radiators around 125 000 K blackbody with different
amounts of flux cut below 230 Å. Cut blackbodies give the same size force
multipliers as realistic absorbed models (Vilhu and Kallman 2019, their Figs.
2 and 3). In the course of computations, at a fixed layer in the wind, the
real absorbed radiator spectrum was compared with the grid spectra and $F_{M}$
interpolated together with local $\xi$ and $t$ values. The EUV contamination
of the X-ray source was included.

The lines of NiV and FeV ions are the strongest contributors to the force
multiplier (in addition to, e.g., CIV, NV, and CrV). The lines are mainly in
the Lyman-continuum below 912 Å. When these ions ionize, the lines disappear,
and consequently, the force multiplier becomes smaller. This ionizing takes
place if the radiator has significant flux below 230 Å . Figures 6 and 7 show
this effect for a 135 000 K blackbody (slightly differing from 125 000 K of
Table 1) as a function of element atomic number and wavelength. Fig. 8 shows
the mean ionization weighted by the contribution to the force multiplier (4
means that four electrons are stripped off). The soft X-ray suppression
clearly vanishes when flux below 230 Å is absorbed, which increases the
contribution of the iron group elements. This property of force multiplier
also helps to explain the momentum problem of WR winds (Vilhu and Kallman
2019).

Figure 6: Contributions to the force multiplier ($F_{M}$) as a function of the
element atomic number (26 for Fe). A 135 000 K blackbody (dashed line) and
below 230 Å totally cut blackbody (Nphot = 0 between 100–230 Å) with the same
temperature (solid line) were used as radiators, and the force multiplier was
calculated at log($\xi$) = 2.5 and log($t$) = $-$1.84. The y-axis has a
logarithmic scale.  Figure 7: Same as in Fig. 6, but contributions to the
force multiplier counted inside 50 Å wide bands are shown as a function of
wavelength.  Figure 8: Mean ionization (weighted by the contribution to the
force multiplier) as a function of atomic number shown for log($\xi$) = 2.5
and log($t$) = $-$1.84 (as in Figs. 6 and 7).  Figure 9: Force multiplier vs.
the number of photons in the radiator between 100–230 Å (in units of that in
125 000 K blackbody). Curves for four log($\xi$)-values (0, 1, 2, 3) with
log($t$) = -1.85 are plotted (solid lines). The dashed line is for the
(log($\xi$), log($t$))-pair (0,-1.36).  Figure 10: Spectral energy
distributions for the radiators of the intermediate spectral state at $\Phi$ =
0 (along the line joining the centers of components). Solid lines show the WR
spectra at the photosphere (125 000 K blackbody) and the absorbed spectra at
distance 2$R_{WR}$ from the WR surface. The dashed lines show the compact star
spectra (unabsorbed and absorbed at distance 0.1$R_{WR}$ from the WR surface).
The dotted lines show some partially cut blackbodies used to compute force
multiplier grids.

Fig. 9 shows the change in the force multiplier as a function of the number of
photons in the radiator between 100–230 Å (relative to that of a 125 000 K
blackbody). The plot shows curves for several (log($\xi$), log($t$)) pairs.
Decreasing $\xi$ or $t$ enhances the line force. Fig. 10 shows some spectral
energy distributions for the radiators of the intermediate spectral state at
$\Phi$ = 0 (along the line joining the centers of components) and some
examples of partially cut blackbodies with which force multiplier grids were
computed. At each wind layer, the coadded number of photons (from WR and
compact star) between 100–230 Å was computed and the force multiplier was
interpolated from a precomputed grid.

##  Appendix C Comparison of our method (XSTAR) and CMF methods

###  C.1 OB-star winds

Krticka and Kubat (2017) computed comoving frame winds (CMF) for a model grid
of O stars comprising main-sequence stars, giants, and supergiants (their
Table 1). Using the masses, radii, effective temperatures, and solar chemical
abundances (Asplund et al. 2009), we computed the wind models with the XSTAR
method. The data were reproduced from extensive computations in Vilhu and
Kallman (2019). The wind equations were integrated numerically, resulting in
velocity stratification v(r) that was compared with the widely used
$\beta$-velocity model,

$v_{model}=v_{\mathrm{{inf}}}(1-B/x)^{\beta}\,$ (21)
$B=1.-(v_{in}/v_{inf})^{1/\beta}$ (22) $x=r/R_{star}.$ (23)

The integrations were started above the subsonic region at $x$ = 1.01 and
ended at $x$ = 10, using 1000 grid points with $\delta x$ = 0.01. The subsonic
part of the wind enters only through the boundary condition $v_{in}$ = 10 km/s
(at x = 1.0). The resulting $v(x)$ was then compared with the model
$v_{model}(x)$ . The unknown free parameters mass-loss rate $M$dot and
$v_{inf}$ (velocity at infinity) were iterated using the IDL procedure
mpcurvefit.pro (written by Craig Markwardt) to obtain
($v(x)-v_{model}(x)$)/$v_{model}(x)$ to approach zero at each $x$. The
procedure mpcurvefit.pro performs a Levenberg-Marquardt least-squares fit, and
no weighting was used. Density clumping was not used by either Krticka and
Kubat (2017) or here. The formal one-sigma errors of each parameter were
computed from the covariance matrix.

$\beta$ was set to 0.6, giving overall the best-fitting results. Using
$\beta$= 1.0, $\dot{M}$ and $v_{inf}$ should be multiplied on average by 0.7
and 1.4, respectively (see details in Vilhu and Kallman, 2019). Fig. 11
compares our XSTAR models and the CMF models. Increasing $\beta$ from 0.6 to
0.8 would give better agreement at high mass-loss rates. However, the overall
agreement is moderately good.

###  C.2 X-ray contaminated wind of Vela X-1.

Krticka et al. (2012) applied the CMF method to the X-ray contaminated wind of
Vela X-1 pulsar. This is a binary system with an orbital period of 8.96 days
and a separation of 53.4 $R_{\odot}$. The masses of the neutron star and of
the companion O star are 1.88 and 23.5 solar masses, respectively. The O star
has an effective temperature of 27 000 K and a radius 30 $R_{\odot}$ , and the
wind mass-loss rate is 1.5$\times$ 10-6 $M_{\odot}$/year. The X-ray luminosity
of the neutron star 3.5$\times$ 1036 erg/s was assumed with a photon index
$\Gamma$ = 1 (Watanabe et al. 2006). Force multiplier grids with different EUV
contaminations were computed for the 27 000 K blackbody in a similar fashion
as explained for Cyg X-3 (Appendix B). In the Vela X-1 case, the number of EUV
photons relative to that of 27 000 K blackbody is larger than 1. The geometry
is illustrated in Fig. 1 of Krticka et al. (2012).

Our method was applied to Vela X-1 in a similar fashion as to Cyg X-3. Fig. 12
shows the computed velocity stratifications for different orbital angles
$\Phi$. The trend is similar to that in Krticka et al. (2012, their Fig. 3,
reproduced here as Fig. 13). Our study gives a weaker $\Phi$-dependence, but
the results are qualitatively similar. Quantitative differences are expected
because the approaches are very different.

Figure 11: Comparison of CMF models of Krticka and Kubat (2017) with XSTAR
models of our study. Mdot (solar masses/year) and $v_{inf}$ (km/s) are the
mass-loss rate and velocity at infinity, respectively. Figure 12: Radial wind
velocity stratifications of Vela X-1 for different orbital angles $\Phi$
marked (0–90 degrees). The label ’NO X’ shows the wind without X-rays from the
neutron star at radius 1.78. The radius is measured from the center of the O
star in units of 30 $R_{\odot}$ (= O-star radius)  Figure 13: Related Vela X-1
study reproduced from Krticka et al. (2012). Radial wind velocity
stratifications for different orbital angles $\Phi$. The position of the
neutron star is marked in the graph.