3 Modelling the UV resonance lines in HMXBs

3.1 Adapting the SEI code

The high-mass X-ray binary is described as a spheroidal early-type star with a spherically symmetric radially outflowing wind, and an orbiting X-ray point source. The used coordinate system is based on the impact parameter p and the line-of-sight parameter z, in units of the stellar radius and with the center of the early-type star at (0,0):

$$\vec{x} = (p,z) \hspace{5mm} {\rm\&} \hspace{5mm} x = \vert \vec{x} \vert \, ,$$ (1)

and with the spectrograph at $(0,-\infty)$. The velocity v(x) of the radiation driven stellar wind is normalised to $v(\infty)=1$ and parameterised as (Lamers et al. 1987):

$$v(x) = v_0 + ( 1 - v_0 ) ( 1 - \frac{1}{x} )^{\gamma}$$ (2)

where v₀ is the velocity at the base of the wind (x=1 and $\gamma > 0$):

$$v_0 = v (1) \simeq 0.01 v(\infty) \, ,$$ (3)

which is of the order of the sound speed in the stellar wind.

Figure 3: Sketch of the used coordinate system (p,z) indicating the location of the early-type star (large gray circle) and the X-ray point source (tiny gray circle, not to scale) at a distance of 1.5 stellar radii, along with surfaces of constant q (using a velocity law with $\gamma =1$ and v0=0.01).

The X-ray point source at position ${\vec{x}}_{\rm X}$ affects the normal abundance q'(x) of a certain ion in the wind. The degree at which the wind is disturbed is set by the parameter $q(\vec{x})$ (Hatchett & McCray 1977):

$$q(\vec{x}) = \frac{ v(x) }{ v(x_{\rm X}) } \frac{ x^2 }{ ( \vec{x} - \vec{x}_{\rm X} )^2 } \cdot$$ (4)

Approximately, at all points on a surface of constant q, the X-rays remove the same fraction ${\eta}(q)$ of the ion under consideration (Tarter et al. 1969). This leaves only a $1-{\eta}(q)$ contribution of this ion to the source function $S(\vec{x})$ and optical depth ${\tau}(\vec {x})$. In our computations we adopt a region enclosed by a surface set by the value of q, with

$${\eta}( q(\vec{x}) ) = \left\{ \begin{array}{ll} \eta & \mbo... ... > q$\space } \\ 0 & \mbox{ else }\cdot \end{array} \right.$$ (5)

A negative value for $\eta$ enhances the abundance of the ion within the Strömgren zone. The X-ray point source is located inside a closed surface of constant q if

$$q < q_{\rm critical}$$ (6)

with

$$% q_{\rm critical} =\lim_{\begin{array}{c} \\ [-6mm] {\scrip... ...le p = 0} \end{array}} q({x}) = \frac{1}{v(x_{\rm X})} \cdot$$ (7)

Because of the singularity in $x_{\rm X}$ when calculating q(x), we excluded a small spherical region around the X-ray point source from the grid with a radius of one millionth of a stellar radius, which is of the order of the radius of a neutron star. As an example surfaces of constant q are shown (Fig. 3), for a velocity law with

$$\gamma = 1 \hspace{5mm} {\rm\&} \hspace{5mm} v_0 = 0.01$$ (8)

and putting the X-ray point source at

$$x_{\rm X} = 1.5 \, ,$$ (9)

which corresponds to

$$v(x_{\rm X}) = 0.34 \hspace{5mm} {\rm\&} \hspace{5mm} q_{\rm critical} = 2.941 \, .$$ (10)

The escape probability method (Castor 1970) makes a local approximation for the source function $S_{\nu}(x)$ at comoving frequency $\nu$, namely the Sobolev approximation

$$S_{\nu}(x) = \frac{ {\beta}_{c}(x) I_{\nu}^{\ast} + \epsilon B_{\nu}(x) } { \beta + \epsilon }\cdot$$ (11)

Here, $\beta$ is the escape probability of a line photon; ${\beta}_{\rm c}$ the penetration probability of a continuum photon; $\epsilon$ the ratio of collisional over radiative de-excitations; $B_{\nu}$ the Planck function at frequency $\nu$; and $I_{\nu}^{\ast}$ the stellar photospheric intensity at frequency $\nu$. Due to the relatively low density of in the stellar wind, the contribution of collisional (de)excitations to the source function of the UV resonance lines can be neglected:

$$\epsilon = 0.$$ (12)

The SEI method (Lamers et al. 1987) incorporates an exact integration of the radiative transfer equation, yielding the observed flux F_v at the observed velocity v:

$$\frac{ {\rm d}I_{\nu} }{ {\rm d}{\tau}_{\nu} } = I_{\nu} - S_{\nu} \longmapsto F_v.$$ (13)

The optical depth $\tau$ is parameterised as a function of velocity v:

$${\tau}(v) = \left\{ \begin{array}{ll} T \frac{ f(v) }{ \int_... ... \leq v_1$\space } \\ 0 & \mbox{ else } \end{array} \right.$$ (14)

with

$$f(v) = \left( \frac{ v }{ v_1 } \right)^{{\alpha}_1} \left(... ...1 } \right)^{\frac{ 1 }{ \gamma }} \right)^{{\alpha}_2} \, .$$ (15)

The parameters $\alpha _1$ and $\alpha_{2}$ are defined as:

$${\alpha}_1 = \frac{ 1 }{ \gamma } - t -2 \hspace{5mm} {\rm\&} \hspace{5mm} {\alpha}_2 = s \, ,$$ (16)

where t and s parameterize the ionization fraction $\kappa _{\rm i}$ as a function of radial distance and velocity:

$$\kappa_{\rm i}(x) \propto x^{-s} v(x)^{-t} \, .$$ (17)

If m is the dominant stage of ionization and i is the ionization stage of the ion under consideration, then t=m-i. If the abundance of the dominant ion in the wind is constant, one gets

$${\alpha}_1 = \frac{ 1 }{ \gamma } -2 \hspace{5mm} {\rm\&} \hspace{5mm} {\alpha}_2 = 0 \, .$$ (18)

If v₁=1, the ion is present up to the regions where the wind has reached its terminal velocity.

The intrinsic line profile resembles a Gaussian, with isotropic broadening ${\sigma }_v$ due to the thermal and turbulent motions - hereafter referred to as the "turbulence'' (Lamers et al. 1987):

$${\Phi}( {\Delta}v ) = \frac{ 1 }{ \sqrt{ \pi } {\sigma}_v } \exp{ - \left( \frac{ {\Delta}v }{ {\sigma}_v } \right)^2 } \, .$$ (19)

This introduces multiple resonance zones in the wind, which results in broader and stronger absorption effectively moving the emission peak towards longer wavelengths (in agreement with P-Cygni line obserservations).

In the case of doublets (most of the important UV resonance lines are doublets), depending on the doublet separation the source function of the red component includes a contribution from the blue component. Except for this coupling, the principle of the SEI code is the same as for singlets (Lamers et al. 1987). However, since we now include a Strömgren zone we can no longer use a one-dimensional radial grid to calculate the source function of the red component (as is the case for the SEI method). The contribution of the source function of the blue component to the source function of the red component depends on whether the coupled blue component points are situated inside the Strömgren zone. Hence the radial profile of the source function depends on the position of the Strömgren zone. This is solved by replacing the one-dimensional grid when calculating the source function of the red component by a two-dimensional axi-symmetric grid.

The SEI method allows the inclusion of a photospheric component in the input line spectrum, but we did not use this option in the model profiles shown below.

Figure 4: Changing the size of the X-ray ionization zone: SEI model profiles for a HMXB with a wind velocity law with $\gamma =1$, v0=0.01 and ${\sigma }_v=0.2$, $x_{\rm X}=1.5$ ( $v_{\rm X}=0.34$), doublet separation of 1.5, integrated optical depth of the blue component of T=100 (twice that of the red component), dominant ionization stage in the uniformly ionized undisturbed wind, and no additional source of emission. The effect of different sizes q of the Strömgren zone is illustrated (closed surfaces for q>2.941). The top panel shows the profile at X-ray eclipse (drawn line, $\phi =0$) and at $\phi =0.5$; the bottom panel presents the difference profile. Note that (much) less absorption is observed at $\phi =0.5$ when the X-ray source is in the line of sight.

Figure 5: The effect of turbulence: same as Fig. 4, but for q=3.2 and different levels of wind turbulence ${\sigma }_v$.

Figure 6: From weak to strong: same as Fig. 4, but for q=3.2 and illustrating the effect of increased integrated optical depth T.

Figure 7: Same as Fig. 4, but for q=3.2 and illustrating the effect of different parameterisations of the optical depth (via $\alpha _1$) and the velocity law (via $\gamma$).

Figure 8: Same as Fig. 4, but for q=3.2 and illustrating the effect of different distances $x_{\rm X}$ of the X-ray source.

The accuracy of the calculations depends on the grid $I_{\rm g}$, which samples the intrinsic line profile. The number of points used in the grid is $I_{\rm g,max}$. In the original code a choice of $I_{\rm g,max}=11$ yields satisfactory results, but if a Strömgren zone is included this value should be some two orders of magnitude larger to suppress the numerical noise. Stronger turbulence demands larger $I_{\rm g,max}$.

3.2 Application of the modified SEI model

To illustrate how the important model parameters affect the shape of the spectral line profile, model spectra are shown for a hypothetical resonance doublet of a HMXB system. The canonical model consists of a HMXB with $x_{\rm X}=1.5$, wind velocity law with $\gamma =1$, v₀=0.01 and ${\sigma }_v=0.2$, doublet separation of 1.5, integrated optical depth of the blue component of T=100 (twice that of the red component), dominant ionization stage in the uniformly ionized undisturbed wind (t,s=0), no additional source of emission ( $\epsilon=0$), no photospheric absorption profile, and a Strömgren zone of size q=3.2.

Figures 4 to 8 illustrate the effect of varying several of these parameters. The top panels present the UV resonance profiles at two orbital phases: at X-ray eclipse ($\phi =0$, drawn line) a large fraction of the Strömgren zone is behind the OB supergiant so that only a small part of the observable stellar wind is affected. When the X-ray source is located in the absorbing column in front of the OB supergiant ($\phi =0.5$), the blue-shifted absorption trough is reduced in strength. Note that also the emission peak of the P-Cygni profile is affected, though less pronounced because the wind volume contributing to the P-Cygni emission is much larger in comparison to the volume of the absorbing column. The bottom panels show the difference spectra ($\phi =0.5$ minus $\phi =0$).

The size q of the Strömgren zone strongly affects the spectral line variability (Fig. 4). Small Strömgren zones (large q) leave little trace in the spectral line profile, apart from some diminished absorption at $\phi =0.5$ at velocities between about $-v_{\rm X}$ and 0 (i.e. not centred at $-v_{\rm X}$). A larger Strömgren zone enhances the HM-effect especially near velocities ${<}\!\!-v_{\rm X}$, also reducing the absorption in the blue wing of the line profile, and reduces the emission at positive velocities at $\phi =0$ (notably in the red component of the doublet). Saturation is not maintained for Strömgren zones that cover a significant fraction of the hemisphere in front of the star. However, open Strömgren zones leave only a small area behind the primary unaffected (in the most extreme case this is a shadow wind), and hence cause diminished absorption and emission at all orbital phases, thereby reducing the contrast in line profile shape between transit and eclipse of the X-ray source. The appearance of the HM-effect is most prominent for $q\sim q_{\rm critical}$. The definition of q allows a Strömgren zone extending into the X-ray shadow behind the supergiant; obviously, this can only happen if scattering of X-rays by ions in the stellar wind is important.

The presence of turbulence in the wind causes the line profile to broaden, which results in less pronounced emission (Fig. 5). The HM-effect is also smoothed in velocity space. The amplitude of variability at positive velocities and between 0 and $-v_\infty$ decreases, whilst the amplitude of variability in the blue wing ( $v<-v_\infty$) increases. The blue wing variability becomes dominant in the case of strong turbulence. In the absence of turbulence it is difficult to hide the HM-effect unless the Strömgren zone is much smaller than the projected stellar disk.

Greater optical depth enhances the extent to which the absorption is maintained when the Strömgren zone moves through the line of sight (Fig. 6). Note that for heavily saturated lines with turbulence, as for unsaturated lines, it is not straightforward to determine the exact value of $v_\infty$, and careful modelling is required.

The ionization stage of the ion, relative to the dominant ionization stage in the (undisturbed) wind is of importance for the parameterisation and normalisation of the optical depth. If the ion corresponds to an ionization stage that is one level below the dominant ionization stage, then $\alpha_1=-2$ instead of -1 in the canonical example (see Eq. (16)). For the same integrated optical depth, this would yield unsaturated and rather triangular shaped absorption profiles with little emission. In the opposite case, if the ion corresponds to one level above the dominant ionization stage, then $\alpha_1=0$. This yields a heavily saturated line profile with stronger emission and somewhat enhanced HM-effect in the absorption part between $-v_{\rm X}$ and $-v_\infty$, compared to the canonical case. From the normalisation of the optical depth (Eqs. (14) and (15)) it follows that $\tau/T$at $v=v_\infty$ (=v₁) is 0.010, 0.22, or 1.0 for $\alpha_1=-2$, -1 and 0, respectively. Scaling the total optical depth T to 2200 and 22 for $\alpha_1=-2$ and 0, respectively, the line profiles and appearance of the HM-effect (Fig. 7) are similar to the canonical case. It may therefore be difficult to distinguish between different values of $\alpha _1$, and prior knowledge of the dominant ionization species is required.

The parameterisation of the optical depth involves the velocity law, and hence the parameter $\gamma$. Changing $\gamma$ from 1 to 0.5 requires rescaling to q=1.87, $\alpha_1=0$ and T=22 (Eqs. (4) and (15)). The resulting variability is not exactly a scaled version of the canonical case (Fig. 7), because qdepends on both v and x. Most notably, the blue wing HM-effect is stronger and saturation more difficult to maintain.

A different distance $x_{\rm X}$ of the X-ray source implies that different parts of the wind are probed. Changing to $x_{\rm X}=1.2$ and 2.0 requires rescaling to q=6.2 and 2.15, respectively. Increasing the distance of the X-ray source enhances the HM-effect in the blue wing at the expense of the HM-effect in the absorption part between $-v_{\rm X}$ and $-v_\infty$, and also makes it more difficult for a saturated line to remain saturated (Fig. 8). If the X-ray source is located close to the primary, then the HM-effect in the emission at positive velocities is less pronounced.