4 A line-blanketed model for WR 111

WR stars usually show a whole "forest'' of iron lines in the UV shortward of $\approx$ $1800~{\rm\AA}$ (see e.g. Herald et al. 2001). One of the best studied objects among these stars is the WC 5 star WR 111. It has already been analyzed in detail by Hillier & Miller (1999) utilizing their line-blanketed models. Therefore it shall also serve here as the prototype for a first application of our code, examining the effects of line-blanketing.

In the present section we present a model for WR 111 with solar iron abundance and compare it to a similar model with zero iron abundance. The model calculations are described in Sect. 4.1, and the results are discussed with regard to the emergent flux distribution (Sect. 4.2), the ionization stratification (Sect. 4.3), and the wind dynamics (Sect. 4.4).

   
4.1 Comparison to observations

Figure 3: Spectral fit for WR 111. The observation (thin line) is shown together with the synthetic spectrum (thick line). The model parameters are compiled in Table 3. Prominent spectral lines are identified. The observed flux has been divided by the reddened model continuum for normalization. A correction for interstellar Ly $\alpha$ absorption is applied to the model spectrum.

Figure 4: Comparison of the observed spectrum of WR 111 (dashed black line) and the model flux. A distance modulus of 11.0 mag and interstellar reddening with $E_{B-V} = 0.325~{\rm mag}$ are applied to the calculated spectrum (grey) and the pure continuum calculation (solid black line). Below $\approx$ $1800~{\rm\AA}$ a pseudo continuum is formed by a large number of iron group lines and affects the determination of the EB-V parameter.

Figure 5: Energy distribution in the flux maximum. The model flux in the CMF (grey) is compared to the model continuum (black), and the de-reddened observation of WR 111 (black, dashed). Below $\approx$ $400~{\rm \AA}$ the emergent flux is strongly reduced by a large number of absorption lines. Longward of $\approx$ $1000~{\rm \AA}$ the emergent flux is enhanced by a mixture of emission and absorption lines. This "iron forest'' forms a pseudo continuum which considerably affects the emergent flux distribution in the UV.

 

 

Table 3: Model parameters for WR 111: Luminosity $L_\star$, stellar radius $R_\star$ and corresponding stellar temperature $T_\star$, mass loss rate $\dot{M}$, terminal wind velocity $v_\infty$, clumping factor D, transformed radius $R_{\rm t}$, Doppler broadening velocity $v_{\rm D}$, and surface mass fractions of carbon, oxygen, silicon and iron $X_{\rm C}$, $X_{\rm O}$, $X_{\rm Si}$, and $X_{\rm Fe}$. Interstellar parameters: distance modulus M - m, interstellar color excess E_B-V, and Ly $\alpha$ hydrogen column density $n_{\rm H}$.

$L_\star$	$10^{5.45} L_\odot$
$R_\star$	$2.455~R_\odot$
$T_\star$	$85~{\rm kK}$
$\dot{M}$	$10^{-4.90}~M_\odot~\mbox{yr}^{-1}$
$v_\infty$	$2200~\mbox{km}~\mbox{s}^{-1}$
D	10
$R_{\rm t}$	$4.13~R_\odot$
$v_{\rm D}$	$250~\mbox{km}~\mbox{s}^{-1}$
$X_{\rm C}$	0.45
$X_{\rm O}$	0.04
$X_{\rm Si}$	$0.8\times 10^{-3}$
$X_{\rm Fe}$	$1.6\times 10^{-3}$
M - m	$11.0~{\rm mag}$
EB-V	$0.325~{\rm mag}$
$n_{\rm H}$	$1\times 10^{21}~{\rm cm}^{-2}$

Observational data for WR 111 are retrieved from the following sources. The UV data are from the IUE satellite, retrieved from the INES database (http://ines.vilspa.esa.es). Optical spectra come from the atlas of Torres & Massey (1987). In addition, optical narrow-band colors (b, v) from Lundström & Stenholm (1984) are considered. Infrared continuum fluxes are given by Eenens & Williams (1992), obtained from broad-band photometry and corrected for the contribution of emission lines. The line-blanketed model spectrum is compared to the observations in Fig. 3 (lines), Fig. 4 (flux distribution in the UV and optical) and Fig. 6 (IR and optical photometry).

For the spectral analysis the mass fractions of silicon and iron are set to solar values and a distance modulus of 11.0 mag is adopted, which corresponds to a distance of 1.58 kpc (see discussion in Hillier & Miller 1999). The Doppler broadening velocity is set to $v_{\rm D} = 250~\mbox{km}~\mbox{s}^{-1}$ which reproduces well the blue shifted absorptions of P-Cygni type line profiles. The density contrast D (clumping) is estimated by fitting the line wings of highly excited ions (He II and C IV). The value of D=10, which reduces the derived mass loss rates by a factor of $\sqrt{10}$, is commonly accepted for early-type WC stars (Hillier 1996; Hamann & Koesterke 1998; Hillier & Miller 1999).

The C III/C IV ionization structure is determined by means of spectral lines of C III (e.g. 1620, 1923, 2010, 2297, 4650, 5696 and 6750 Å) versus C IV (e.g. 2405, 2525, 2595, 2699 and 5812 Å). These lines are reproduced well, except for C III 2297 Å (which is mainly fed by dielectronic recombination) and the classification line C III 5696 Å. Simultaneously, the ratio $X_{\rm C}/X_{\rm He}$ is derived from the neighboring lines He II/C IV 5412/5470 Å, the He II lines at 1640, 2734 and 3204 Å, and the He II Pickering series.

For the determination of the oxygen mass fraction $X_{\rm O}$, ionization stages O III to O V (O III 3120, 3270, 3961 Å, the O III/O IV/C IV complex around 3700 Å, O III/O VI at 5270 Å, O III/O V at 5590 Å, O IV 3070 and 3410 Å, and O V at 3146 and 5114 Å) are used. Most of these lines fit very well, except O III 3120 and 3961 Å. One of these two lines, O III 3120 Å, is a Bowen emission line (Bowen 1935), i.e. it shares a common upper level with O III 303.6 Å, and is therefore pumped by interaction with the He II resonance line at 303.8 Å. As demonstrated by Schmutz (1997), this effect is sensitive to the value of the He II 303.8 profile function at the frequency of the O III 303.6 blend. The difficulties with O III 3120 Å may therefore arise from the simplified treatment of line broadening by pure Doppler profiles.

The O VI lines at 3811 and 5270 Å are not reproduced simultaneously with C III. A similar discrepancy is already known for the O VI resonance line in O-star atmospheres. For O-stars, it can be resolved by the inclusion of an additional X-ray emissivity from shock heated material (Pauldrach et al. 1994). Preliminary model calculations show evidence, that the inclusion of an X-ray emission according to Raymond & Smith (1977) has significant influence on the O VI lines in WC stars. If already marginally visible in the emergent spectrum, these lines can be strengthened by a factor of about two.

The derived model parameters are listed in Table 3. Compared to the work of Hillier & Miller we obtain similar values for the stellar temperature $T_\star$ (85 kK vs. 90 kK), the terminal wind velocity $v_\infty$ (2200 km s^-1 vs. 2300 km s^-1), the clumping factor D=10, and the carbon mass fraction $X_{\rm C}=0.45$, whereas we find a considerably lower oxygen mass fraction ( $X_{\rm O} = 0.04$ vs. 0.15), and a higher luminosity ( $L_\star = 10^{5.45} L_\odot$ vs. $10^{5.3} L_\odot$).

For a comparison of the derived mass loss rates ( $\dot{M} = 10^{-4.90}~M_\odot~\mbox{yr}^{-1}$ vs. $10^{-4.82}~M_\odot~\mbox{yr}^{-1}$) it is necessary to consider the different luminosities of both models. From Eq. (3) follows that the spectroscopic mass loss rates scale as $\dot{M}/v_\infty \propto L_\star^{3/4}$, i.e. our value of $10^{-4.90}~M_\odot~\mbox{yr}^{-1}$ would scale down to a considerably lower value of $10^{-4.99}~M_\odot~\mbox{yr}^{-1}$ for a luminosity of $10^{5.3} L_\odot$ and $v_\infty=2300~\mbox{km}~\mbox{s}^{-1}$.

   
4.2 The emergent flux distribution

Figure 6: Comparison of the model continuum to optical and infrared photometry (grey blocks). A distance modulus of 11.0 mag and interstellar reddening with $E_{B-V} = 0.325~{\rm mag}$ are applied to the calculated continuum. The red part of the optical spectrum is also included (black).

As shown in Fig. 4 the emergent flux of our model reproduces very well the observed energy distribution in the optical and UV spectral range. In the infrared (Fig. 6) the measured continuum fluxes of Eenens & Williams (1992) lie about 0.1 dex above the model flux. Because Hillier & Miller use a lower reddening parameter, they encounter an even higher discrepancy in the infrared ($\approx$ $0.2~{\rm dex}$), and resolve it through the application of a non-standard extinction law (Cardelli et al. 1988).

As recognized first by Koenigsberger & Auer (1985), the "iron forest'' of emission and absorption lines of Fe IV, Fe V and Fe VI forms a pseudo continuum below $\approx$ $1800~{\rm\AA}$ (see Fig. 4). The blanketed model spectra therefore show a steeper apparent continuum slope in the UV. Consequently, a higher reddening parameter - with respect to un-blanketed models - must be adopted to reproduce the observed flux distribution. As a result the derived stellar luminosity is increased. On the other hand the flux in the far UV is blocked by iron lines, and is redistributed to the UV (see Fig. 5), which leads to lower derived luminosities. In total, recent studies (Crowther et al. 2000; Dessart et al. 2000) show a trend to derive higher luminosities for WC stars when line-blanketed models are applied.

In the present work we derive values of $L_\star=10^{5.45}~L_\odot$ and E_B-V=0.325 mag for WR 111 using the standard extinction law of Seaton (1979). Hillier & Miller derive a lower luminosity of $L_\star=10^{5.3}~L_\odot$ with E_B-V = 0.30 mag, and Koesterke & Hamann (1995) deduce a luminosity of $L_\star=10^{5.0}~L_\odot$ with E_B-V = 0.25 mag based on un-blanketed models. Obviously, the derived luminosity depends on the treatment of iron group line-blanketing, as it affects the emergent flux distribution considerably. In addition, uncertainties arise from the interstellar extinction law. In the UV below $\approx$ $1500~{\rm\AA}$ the extinction law of Cardelli et al. (1988) applied by Hillier & Miller shows substantial differences compared with Seaton (1979).

The ionizing fluxes from our model compare very well to the blanketed model of Hillier & Miller (as listed in Dessart et al. 2000) after a scaling by 0.15 dex due to the difference in luminosity is performed. For the number of Lyman continuum photons we obtain a value of $10^{49.33}~{\rm s}^{-1}$ compared to $10^{49.35}~{\rm s}^{-1}$ from Hillier & Miller, and for the number of He I photons we get a slightly higher value of $10^{48.78}~{\rm s}^{-1}$ compared to $10^{48.65}~{\rm s}^{-1}$. The comparison to our un-blanketed model shows that line blanketing diminishes the number of He I photons by 0.26 dex (compared to $10^{49.04}~{\rm s}^{-1}$ for the un-blanketed model), whereas the number of Lyman continuum photons is nearly unaffected ( $10^{49.33}~{\rm s}^{-1}$ vs. $10^{49.36}~{\rm s}^{-1}$). The small discrepancy in the number of He I photons compared to Hillier & Miller might be explained by the different oxygen abundances.

   
4.3 The ionization structure

Figure 7: The populations of the ground states of the included ions are plotted over the atomic density as a depth index. The line-blanketed model for WR 111 (solid black) is compared to its un-blanketed counterpart (dashed grey). In the outer part of the envelope the recombination from C IV to C III is considerably enhanced.

In Fig. 7 the ionization stratification of the blanketed model is compared to its un-blanketed counterpart. The most striking effect of line-blanketing is the enhanced recombination from C IV to C III in the outer part of the wind, where the population of the C III ground state is increased from $\approx$10^-3 up to $\approx$1. The emergent C III line emissions are strengthened by a factor of $\approx$2 (with strong scatter) compared to the un-blanketed model. The other ions are also affected, but the corresponding changes are of minor importance for the spectral appearance of the models.

Because the line emission of WR stars is dominated by recombination processes, the emission line intensity is basically a measure for the wind density and therefore also for the mass loss rate. For this reason the enhanced recombination leads to a reduction of the derived mass loss rates.

In the outer part of the wind the electron temperature is only marginally changed (cf. Fig. 8). The back warming effect, which results from the increase of the Rosseland mean opacity, appears in the optically thick part of the envelope. In these layers the temperature is increased by about 20% without effect on the model spectrum. The changes in ionization result from very complex radiative interactions between the different ions. Therefore the various effects can not be clearly separated. Nevertheless, a closer inspection allows some interpretations.

The ionization from C III to C IV is strongly influenced by shading effects. At the relevant depth the radiation field below $\approx$320 Å is effectively blocked by a large number of iron group lines. Therefore the ionization of C III (ionization edge at 258 Å) is strongly reduced. Test calculations show that the ionization stratification of carbon changes significantly if only iron line-blanketing is accounted for and the rest of the iron group elements is omitted. Probably this effect is responsible for the higher mass loss rate derived by Hillier & Miller. Because these authors do not account for the whole iron group they obtain a weaker recombination which is compensated by a higher wind density.

Figure 8: Temperature structure for the model including line-blanketing (solid line) and the un-blanketed model (dashed line). The temperature in the optically thick part of the blanketed model is increased by the back-warming effect of the additional opacities.

Obviously the inclusion of the complete iron group is of some importance, but the simplified treatment as one generic element is a questionable approximation. Especially, different ionization and excitation conditions may result when the detailed atomic models are accounted for separately. The splitting of the iron group into different model atoms will therefore be a subject of our future work.

The main ionization stages of iron in the outer envelope are Fe V and Fe VI with ionization edges at 165 Å and 128 Å. In the regions where He II or C IV with edges at 228 Å and 192 Å are the leading ions, the ionization of iron is only possible from highly excited energy levels. Actually this is the case because of the metastable nature of the low lying iron levels (cf. Sect. 3.1). From Fig. 1 it can be seen that for Fe V the levels up to $\approx$1/3 of the ionization energy are of same parity, and are therefore supporting the ionization of Fe V.

   
4.4 Wind dynamics

For a radiation driven wind, the average number of scatterings before a photon escapes from the atmosphere is indicated by the wind efficiency $\eta = \dot{M} v_\infty / (L_\star/c)$. In former model calculations for WC stars values of $\eta \approx 50$ are derived (Koesterke & Hamann 1995; Gräfener et al. 1998). Even though multiple photon scattering might be efficient in spectral regions with a very high line density, these high values led to the question if the mass loss of WR stars can be driven by radiation pressure alone.

For the current WR models this situation has changed, because lower mass loss rates are derived by accounting for clumping, and higher luminosities are predicted by line-blanketed models. Both effects lead to lower values for the wind efficiency $\eta$. In the present work we obtain a value of $\eta = 4.8$ for WR 111. This value even lies below the wind efficiencies calculated by Springmann & Puls (1998) in their Monte-Carlo simulations. Therefore, the mass loss of WR 111 seems to be easily explicable by radiative driving.

In our radiation transport calculations, the radiative acceleration

$$% a_{\rm rad}=\frac{1}{\rho}~\frac{4\pi}{c}\int_0^\infty \kappa_\nu H_\nu~{\rm d}\nu$$ (41)

can be integrated directly, because $\kappa_\nu$ and $H_\nu$ are provided over the complete relevant wavelength range. In Fig. 9 we compare the resultant radiative acceleration of the blanketed and the un-blanketed model to the wind acceleration

$$% a_{\rm wind}= v(r)~v'(r) + \frac{M_\star G}{r^2}$$ (42)

which is implied by the assumed velocity law v(r) from Eq. (2). For the stellar mass $M_\star$ we take the value of $14~M_\odot$ from the mass-luminosity relation from Langer (1989). In the outer part of the wind, the radiative acceleration of the blanketed model is strongly enhanced with respect to the un-blanketed one. The work performed on the wind per unit of time,

$$% W_{\rm rad} = \dot{M}\int_{R_\star}^\infty a_{\rm rad}~{\rm d}r ,$$ (43)

corresponds to a fraction of 0.42 of the mechanical wind luminosity

$$% L_{\rm wind} = \dot{M} \left( \frac{1}{2}~v_\infty^2 + \frac{M_\star G}{R_\star} \right)\cdot$$ (44)

Although the assumed velocity structure is not yet consistent to the radiative acceleration, we can conclude that at least a large fraction of the wind acceleration is achieved by radiation pressure. However, in the acceleration zone from the sonic point (at $\approx$12 km s^-1) up to $\approx$1000 km s^-1 there is still a large deficit of radiative force. One may hope that more complete opacities will provide the missing part.

Figure 9: Acceleration in units of the local gravity, as function of radius. The wind acceleration $a_{\rm wind}$ (Eq. (42), black) is compared to the radiative acceleration from the blanketed (grey) and the un-blanketed model (grey, dashed). At the top of the figure the wind velocity is indicated. The sonic point is reached at $\approx$12 km-1.