5 Discussion

The adapted SEI model reproduces the HM-effect in HD 77581/Vela X-1, and naturally explains the lack of any (clear) HM-effect observed in the dense wind system HD 153919/4U1700-37. Absorption components that are seen both in the strong wind lines and additional weaker lines in the UV spectrum of HD 77581/Vela X-1 and that could not be reproduced by the adapted SEI model can be explained by a photo-ionization wake. The orbital modulation of the resonance lines in the UV spectra of the five HMXBs studied here shows a clear trend of the size of the Strömgren zone to increase with higher X-ray luminosity.

5.1 Terminal velocities and turbulence

The terminal velocity $v_\infty$ may be estimated from a comparison of an appropriate model line profile with the observed line profile. It is best derived from moderately saturated profiles, in which case the maximum absorption in the $\phi =0.5$ line profile occurs at the terminal velocity.

The terminal velocity for HD 77581/Vela X-1 is thus estimated to be $v_\infty=600$ kms^-1, much slower than previously reported (1105 kms^-1 Prinja et al. 1990). For HD 153919/4U1700-37 we estimate $v_\infty=1700$ kms^-1 (Prinja et al. 1990: 1820 kms^-1). These values should be correct to within 100 kms^-1. The low resolution IUE spectra show that the wind in HDE 226868/Cyg X-1 is faster (marginally resolved: $1000<v_\infty<1500$ kms^-1) than in Sk-Ph/LMC X-4 and Sk 160/SMC X-1 (unresolved:  kms^-1, see also Hammerschlag-Hensberge et al. 1984 for Sk 160/SMC X-1). HST/STIS spectra of the N  V, Si  IV and C  IV lines in Sk-Ph/LMC X-4 around $\phi =0$ suggest a terminal velocity of ${\sim}500$ kms^-1, even though wind velocities up to ${\sim}1200$ kms^-1 occur (Kaper et al. in preparation).

Single (galactic) O-type and early-B-type stars are known to have a fairly constant ratio of terminal over escape velocity $v_\infty/v_{\rm esc}\sim2.5$, where

$$v_{\rm esc} = \sqrt{ \frac{2 G M (1-\Gamma)}{R} }$$ (20)

with

$$\Gamma = 2.658 \times 10^{-5} \frac{L}{L_\odot} \left( \frac{M}{M_\odot} \right)^{-1}$$ (21)

for a typical Population I star (e.g. Groenewegen et al. 1989). This is also true at temperatures between $\sim$1.1 and $2.1\times10^4$ K but then $v_\infty/v_{\rm esc}\sim1.3$ (Lamers et al. 1995). The terminal velocities for the primary stars of the five HMXBs studied here are, with the possible exception of HDE 226868, low compared to single stars of the same effective temperature (Table 4 and Fig. 15). Especially the two magellanic stars have very low terminal velocities, which may at least partly be due to their subsolar metallicity and hence lower radiation-force multiplier (Garmany & Conti 1985; Prinja 1987; Kudritzki et al. 1987) - which is also found for dust-driven winds of Asymptotic Giant Branch stars (e.g. van Loon 2000).

The Strömgren zone could inhibit acceleration of the stellar wind material flowing through it, not only resulting in an ionization wake but possibly also leading to a slower terminal velocity of the wind. This scenario only works if the Strömgren zone is sufficiently large and the orbital period sufficiently short that all wind material leaving the star passes through the Strömgren zone before having reached the terminal velocity. The magellanic systems indeed must have very extended Strömgren zones leaving only a small region at the opposite side of the primary unaffected by the X-ray source (i.e. a shadow wind), and their orbital periods are rather short. However, the Strömgren zones in HD 77581/Vela X-1 and HD 153919/4U1700-37 are closed surfaces, and certainly do not occupy more than half of the circumstellar space. Within half their orbital periods of 4.5 and 1.7 days, respectively, their stellar winds ought to already have accelerated to velocities approaching the terminal velocity as if it were from a single star.

A viable alternative is that the $\Gamma$ factors for single stars are not applicable to stars in interacting binaries. HMXBs primaries may have already lost a significant fraction of their initial mantle mass, causing them to be undermassive for their luminosity (Conti 1978; Kaper 2001). This seems to be confirmed when comparing the primary star masses (Table 1) with Table 3 in Howarth & Prinja (1989). Hence we may have under-estimated $\Gamma$ and over-estimated $v_{\rm esc}$.

 

 

Table 4: Ratios of terminal velocity $v_\infty$ and escape velocity  $v_{\rm esc}$, and turbulent velocities ${\sigma }_v$ of the primary stars in the HMXBs, together with their effective temperatures $T_{\rm eff}$ and $\Gamma$ values estimated from Lamers et al. (1995).

Primary	$T_{\rm eff}$ (K)	$\Gamma$	$v_\infty/v_{\rm esc}$	${\sigma }_v$
HDE 226868	30500	0.34	1.8-2.8
Sk-Ph	36000	0.16	0.8
Sk 160	26000	0.18	1.1
HD 77581	23400	0.18	1.3	0.45
HD 153919	37200	0.34	2.0	0.15

Figure 15: Ratio of terminal over escape velocity for the primary stars in the HMXBs studied here (solid dots), compared to single stars (circles) from Lamers et al. (1995). HMXB members have low terminal velocities for their effective temperatures.

A turbulence description has been employed to account for material at velocities deviating from the bulk flow that obeys a velocity law according to standard radiation-driven wind theory. The $\sigma$ is the typical deviation in units of $v_\infty$. To approximately describe both the undisturbed wind and X-ray ionized components in the line profiles of HD 77581/Vela X-1 a very large value for $\sigma$ needs to be invoked ( $\sigma_v\sim0.45$), whereas for HD 153919/4U1700-37 a much smaller value suffices ( $\sigma_v\sim0.15$). In absolute terms the turbulence in these two HMXBs is very similar, though: $v_{\rm turb}=270$ and 255 kms^-1, respectively. These values are typical for winds from O-type stars for which $v_{\rm turb}$ has been found to be largely independent of $T_{\rm eff}$ (Groenewegen et al. 1989). As mentioned before, we find indications for the deviations from the monotonic velocity law to resemble a shocked wind structure rather than uniform turbulence.

5.2 Ionization fractions in the stellar wind

 

 

Table 5: Ionization fractions $\kappa _{\rm i}$ for several ions in the stellar winds of HD 77581/Vela X-1 and HD 153919/4U1700-37 as derived from SEI model parameters, adopting mass-loss rates $\dot{M}$ from H$\alpha$ line profile modelling (Schröder et al. in preparation) and using solar element abundances A. Ionization fractions from work on single stars (Lamers et al. 1999) are given for comparison.

Line	$\lambda_0$ (Å)	$f_{\rm line}$	T	A	$\dot{M}_{{\rm H}\alpha}$ ($M_\odot$ yr-1)	$\kappa _{\rm i}$	$\kappa_{\rm i, Lamers}$
HD 77581/Vela X-1:
N  V	1238.821	0.152	3	$1.1\times10^{-4}$	$1\times10^{-6}$	$5\times10^{-4}$	$3\times10^{-5}$
Si  IV	1393.755	0.528	300	$3.5\times10^{-5}$	$1\times10^{-6}$	$4\times10^{-2}$	$1\times10^{-2}$
C  IV	1548.20	0.194	200	$3.6\times10^{-4}$	$1\times10^{-6}$	$6\times10^{-3}$	$1\times10^{-2}$
Al  III	1854.716	0.560	20	$3.0\times10^{-6}$	$1\times10^{-6}$	0.3	<1
HD 153919/4U1700-37:
N  V	1238.821	0.152	30	$1.1\times10^{-4}$	$1\times10^{-5}$	$2\times10^{-3}$	$3\times10^{-3}$
Si  IV	1393.755	0.528	$2\times10^4$	$3.5\times10^{-5}$	$1\times10^{-5}$	$\sim$1	$1\times10^{-3}$
C  IV	1548.20	0.194	5000	$3.6\times10^{-4}$	$1\times10^{-5}$	$6\times10^{-2}$	$3\times10^{-3}$

In principle, the mass-loss rate may be estimated from the integrated optical depth T of a resonance line:

$$T = \frac{\pi {\rm e}^2}{m_{\rm e} c} f_{\rm line} \lambda_0 N_{\rm i} v_{\infty}^{-1}$$ (22)

with $f_{\rm line}$ the oscillator strength of the transition in terms of the harmonic oscillator $\pi {\rm e}^2/m_{\rm e} c$, $\lambda_0$ the rest wavelength of the transition, and $N_{\rm i}$ the column density of the ion:

$$N_{\rm i} = \int_{R_\star}^{\infty} n_{\rm i}(r) {\rm d}r$$ (23)

with $n_{\rm i}$ the number density of the ion. The continuity equation yields

$$\dot{M} = 4 \pi r^2 v(r) \left<m\right> \frac{n_{\rm i}(r)}{A_{\rm i}}$$ (24)

with $\rho$ the mass density, $\left<m\right>$ the mean mass of an ion in the wind, and $A_{\rm i}$ the ion abundance by number. The mass-loss rate $\dot{M}$ can now be estimated from observed quantities in the following way:

$$\dot{M} = \frac{4 m_{\rm e} c}{{\rm e}^2 f_{\rm line} \lambda_0} \left<m\right> R_\star v_{\infty}^2 \psi \frac{T}{A_{\rm i}}$$ (25)

with

$$\psi = \left\{ \begin{array}{lll} (1-\gamma) & \mbox{\hspace{... ...\hspace{1mm} and \hspace{0.8mm} $v_0>0$ }. \end{array} \right.$$ (26)

For our choice of $\gamma =1$ and v₀=0.01 the numerical factor $\psi=0.215$. Adopting a mean nucleus mass $\left<m\right>=2.2624\times10^{-24}$ g (Lamers et al. 1999) the product of mass-loss rate $\dot{M}$ and ion abundance $A_{\rm i}$ can be calculated for the resonance lines in HD 77581/Vela X-1 and HD 153919/4U1700-37.

In practice, however, the limited knowledge of the ionization balance in the winds of OB supergiants makes the derived mass-loss rates highly unreliable. Instead, the ionization fractions of the ions may be derived when other, more reliable estimates for the mass-loss rate are available. For resonance lines the excitation fraction of the ion that produces the line is unity, and the ion abundance $A_{\rm i}$ is the product of the ionization fraction $\kappa _{\rm i}$ and elemental abundance A (by number). Assuming solar abundances (Anders & Grevesse 1989), the ionization fraction may be derived using Eq. (25).

Lamers et al. (1999) found $\kappa _{\rm i}$ to depend on both radiation temperature and density, with the radiation temperature scaling approximately linearly with $T_{\rm eff}$. The empirical dependencies of $\kappa _{\rm i}$ on density were contrary to the expected ionization balance, however, and it was argued that the empirical relations may suffer from selection effects. Therefore, we only consider their empirical relations between $\kappa _{\rm i}$and $T_{\rm eff}$.

Schröder et al. (in preparation) model the H$\alpha$ profiles of OB supergiants in HMXBs. They derive mass-loss rates for HD 77581 ( $\dot{M}\sim1.0\times10^{-6}~M_\odot$ yr^-1) and HD 153919 ( $\dot{M}\sim1.0\times10^{-5}~M_\odot$ yr^-1), which are comparable to those derived for single stars of similar spectral type (Howarth & Prinja 1989). Combining their results with the results from the SEI models, ionization fractions $\kappa _{\rm i}$ are estimated for the N⁴⁺, Si³⁺ and C³⁺ (and Al²⁺) ions in the (undisturbed) stellar winds of HD 77581 and HD 153919 (Table 5). These are then compared with the ionization fractions estimated from Fig. 3 in Lamers et al. (1999), taking into account their lower and upper limits.

The ion fractions derived from the SEI modelling for HD 77581/Vela X-1 are in agreement with the predictions for single stars by Lamers et al. (1999), except for the N⁴⁺ abundance which is observed to be an order of magnitude higher than predicted. This may be due to either super-ionization or nitrogen over-abundance (or both). Auger ionization (Cassinelli & Olson 1979) is sometimes invoked to explain strong N  V resonance lines. Unless the density becomes very high, Auger ionization increases with the velocity in the wind, and may originate in the high-velocity extrema of a shocked wind. This might mimic a moderate increase of ionization fraction with distance, as required to reproduce the observed line profile and variability of the N  V line in HD 77581/Vela X-1. On the other hand, nitrogen over-abundance at the surface of HD 77581 may have resulted from (i) the transfer in the past of nitrogen-enriched material from the progenitor of Vela X-1 onto HD 77581, or (ii) strong mass loss exposing deeper layers mixed with the products of nuclear burning. Kaper et al. (1993) note that HD 77581 might be a BN star.

For HD 153919/4U1700-37 the opposite is found: the observed ionization fraction of N⁴⁺ agrees very well with the predictions, whereas the ionization fractions of Si³⁺ and C³⁺ are observed to be (much) higher than predicted. Perhaps these are the dominant ionization states for silicon and nitrogen in the stellar wind of HD 153919, rather than Si⁴⁺and C⁴⁺. Still, an ionization fraction $\kappa_{\rm i}\sim1$ for Si³⁺ is unrealistic for a multi-level atom, and hence the SEI model must have over-estimated the integrated optical depth of the Si  IV line in HD 153919/4U1700-37. If indeed the degree of ionization in the wind of HD 153919 is lower than that predicted for single stars, this would mean that - like in the wind of HD 77581 - N⁴⁺ is in fact overabundant in the wind of HD 153919.

5.3 Size of the Strömgren zone

The size of the Strömgren zone, indicated by a particular value of the parameter q, is related to the ionization parameter $\xi$ as

$$\xi = \frac{q L_{\rm X}}{n_{\rm X} a^2}$$ (27)

where a is the distance between the centres of the primary and the X-ray source. The ionization parameter can be interpreted as a measure for the number of X-ray photons per particle. The ionization balance in the stellar wind is mostly affected by soft X-ray photons. In common with previous models of the X-ray ionization of stellar winds in HMXBs, we assume the dominant source of opacity for soft X-rays is oxygen (Hatchett & McCray 1977). Masai (1984) has objected that this assumption may not be valid if He  IIis present, but incorporating the effects of a He  II/He  III ionization front would be beyond the scope of this paper. As a consequence of our assumption, the sharp boundaries between the X-ray ionized and the undisturbed stellar wind coincide for ions like C³⁺, Si³⁺ and N⁴⁺ (cf. Kallman & McCray 1982; McCray et al. 1984). Hence the edge of the Strömgren zone is given by a particular value for $\xi$ that corresponds to the boundary at which oxygen is being completely ionized by the X-ray source, and depends on the shape of the X-ray spectrum and the oxygen abundance (Hatchett et al. 1976).

Accretion of matter onto a star moving through a medium was first described by Bondi & Hoyle (1944). Their concept was applied to HMXBs by Davidson & Ostriker (1973). In an HMXB the compact object has a velocity $v_{\rm rel}$relative to the stellar wind flow:

$$v_{\rm rel}^2 = v_{\rm wind}^2 + ( \vert v_{\rm X}-v_\star\vert - \frac{R_\star}{a} v_{\rm rot} )^2$$ (28)

with stellar wind velocity $v_{\rm wind}$ near the X-ray source (which may be lower than implied by the undisturbed wind velocity law; see Sect. 5.1), orbital velocity $v_{\rm X}$ of the X-ray source, and orbital velocity $v_\star$, stellar radius $R_\star$ and rotation velocity $v_{\rm rot}$ of the primary. Kaper (1998) finds that generally $v_{\rm rel}\sim\frac{1}{2}v_{\rm wind}$. Stellar wind matter is accreted onto the compact object if it approaches within an accretion radius approximately given by

$$r_{\rm acc} = \frac{2 G M_{\rm X}}{v_{\rm rel}^2}$$ (29)

with $M_{\rm X}$ the mass of the compact object. The X-ray luminosity due to the release of gravitation energy of the matter being accreted onto the surface of the compact object with radius $R_{\rm X}$ is

$$L_{\rm X} = \pi \zeta r_{\rm acc}^2 v_{\rm rel} \rho_{\rm X} \frac{G M_{\rm X}}{R_{\rm X}}$$ (30)

where $\rho_{\rm X}$ is the mass density of the stellar wind near the compact object, and $\zeta$ is an efficiency parameter ( $\zeta \sim 0.1$ for accretion onto a neutron star). Hence we obtain

$$q = \frac{\xi R_{\rm X} a^2}{4 \pi \zeta \left<m\right>} \left(\frac{v_{\rm rel}}{G M_{\rm X}}\right)^3\cdot$$ (31)

The size of the Strömgren zone depends on: (1) the dimensions of the compact object via $R_{\rm X}$, $M_{\rm X}$ and $\xi$ (by the shape of the X-ray spectrum); (2) the orbit via a and $v_{\rm rel}$ (by the orbital velocity); (3) the wind flow via $v_{\rm rel}$; (4) and the abundances in the wind via $\left<m\right>$ and $\xi$ (by the oxygen abundance). The size of the Strömgren zone does not, however, explicitly depend on the mass-loss rate nor on the X-ray luminosity.

 

 

Table 6: Critical, expected and observed sizes of the Strömgren zones in the five HMXBs (the smaller q, the larger the Strömgren zone).

HMXB	$q_{\rm critical}$	$q_{\rm expected}$	$q_{\rm observed}$
HDE 226868/Cyg X-1	1.7	1.7-5.0
Sk-Ph/LMC X-4	2.7	<1.6-4.5	<2.7
Sk 160/SMC X-1	2.9	<6.3-17	${\ll}2.9$
HD 77581/Vela X-1	2.7	10-24	2.9
HD 153919/4U1700-37	2.0	177-215	>4.0

From the time sequence of the integrated line variability (Figs. 1 and 2) and the modelling of the line profile variability it is already clear that the Strömgren zone is largest for Sk-Ph/LMC X-4 and especially Sk 160/SMC X-1: their zones must extend far beyond the primary as seen from the X-ray source, and the stellar wind is undisturbed only in a small shadow region behind the primary. The Strömgren zone is smaller but still very extended for HDE 226868/Cyg X-1 where the variability due to the HM-effect is continuous over the orbit. HD 77581/Vela X-1 has an again smaller and now closed Strömgren zone, but it still occupies a considerable fraction of the wind volume. The (by far) smallest Strömgren zone is found for HD 153919/4U1700-37.

The critical and expected sizes of the Strömgren zones for the five HMXBs are listed in Table 6. The $q_{\rm critical}$ is calculated assuming a velocity law according to Eq. (2) with v₀=0.01 and $\gamma =1$. The $q_{\rm expected}$ is calculated assuming $\xi=10^3$ (Hatchett & McCray 1977) for a typical X-ray spectrum, $\zeta=0.1$, and $R_{\rm X}=10$ km for $M_{\rm X}=1.4~M_\odot$ and $R_{\rm X}{\propto}M_{\rm X}$. The range in values results from either assuming co-rotation of the primary with the orbit or no rotation of the primary (and the uncertainty about $v_\infty$ for Cyg X-1, LMC X-4 and SMC X-1). The observed sizes of the Strömgren zone are in reasonable agreement with the expected sizes that, in general, seem to have been somewhat under-estimated. This could easily be solved if for instance the accretion efficiency $\zeta$ were twice as large. In particular, $\zeta$ increases with larger mass and smaller radius of the compact object (Shakura & Sunyaev 1973), which would yield an accretion efficiency for the black-hole candidate Cyg X-1 significantly larger than 10%.