Astron. Astrophys. 333, 956-969 (1998)

4. A high ionization zone in the outer winds of WR stars?

4.1. Formulation of empirical relationship between mass-loss and optical line strength

In general, the observed IR-cm spectral energy distribution of WR winds cannot be reproduced using either a smooth or clumpy wind, under the assumption of constant ionization in this region. We now use an empirical approach to determine the influence of such a high ionization zone on mass-loss rates derived from IR-radio observations.

We wish to determine whether this zone extends beyond the formation region of the cm-fluxes, which is necessary for mass-loss determinations. However, in many cases available data prevents a definitive determination of these radii. Consequently, due to the uncertain radio region ionization state, mass-loss rates suffer from a factor of three uncertainty. We have therefore developed an empirical relationship, based on the assumption that optical emission lines provide relative mass-loss rate estimates, as discussed below.

Let us first assume that and are the inner and outer radii over which such a high ionization zone extends. For clumped winds in the asymptotic regime, the optical depth (Eq.  16) can then be split into separate integrals:

while is given by Eq. (17), where the impact parameter obeys =3, is found from Eq. (2) and is obtained from Eq. (8). In general, we therefore have five unknown parameters, , , , and , with other variables, such as d, , given in Table 2. For cases where the spectral index changes smoothly over the whole IR-radio range we have only three parameters since = , and can be found from Eq. (16).

Following the nomenclature of Nugis & Niedzielski (1995), the energy emitted in the line in the clumped case is

where

Here, the terms with subscripts h and l denote the contributions from clumps and the interclump medium, respectively. Using the mass-loss rate formula (Eq.  13), we can express as follows

where

In this formulation, , , and ( is the number of atoms in the ionization stage above ).

We now assume that stars of the same spectral type have comparable values of () for high excitation lines. These lines are formed very close to the stellar surface, where the influence of the clumped structure on the line fluxes is minimal. Assuming that emergent and relative optical fluxes are fairly constant within each spectral type, we obtain our final expression for the mass-loss rate:

where d is in kpc, is in km s^-1, is in Jy, is in yr^-1 and is a numerical constant which will be determined in the following subsection.

4.2. Determination of ionization and clumping-corrected radio mass-loss rates

We have obtained clumping-corrected "radio" mass-loss rates by the following scheme. First we studied stars with many observed flux points covering the whole IR-mm-cm wavelength range, starting by assuming the normal (low) ionization state in the cm forming region. , and were adjusted to match observations, where we imposed the smooth wind solution as an upper limit to the mass-loss rate. In addition, the maximum value of the filling factor, (at ) cannot exceed 2/3 since this corresponds to the maximum filling factor of a spherical shell filled by equal size spherical clumps with diameters equal to the thickness of the shell. can also be constrained from the fact that large matter contrasts are not observed in WR winds; an upper limit of was found observationally by Brown et al. (1995) which we adhere to here.

In the present study we further introduce the parameter , which corresponds to the value of the impact parameter predicted at for the longest observed wavelength for a smooth wind (Eq.  9). Since the radio continuum formation zone lies 100 times further from the stellar surface than the IR formation zone we set as an upper limit for . (The maximum density contrast () is achieved where IR fluxes are formed, and the density contrast decreases to unity at times larger radius).

We investigated alternative solutions depending on the extent of the high ionization zone:

• "nh" in the case where the high ionization zone ends beyond the radio continuum formation zone ( and ),
• "nhn" where the high ionization zone ends between the IR and radio continuum formation zones (), )).

Figs. 3 - 4 demonstrate solutions to the radio ionization state for WR 147 (WN8+OB) and WR 134 (WN6b). Unfortunately, unique solutions were possible for only a few cases. For WR 11 (WC8+O 9) only interacting wind models agree with observation, but the IR/radio data alone are insufficient to distinguish between "nh" and "nhn" solutions (see Fig. 5). In general, IR-radio observations prevented us from distinguishing between the three possibilities. Of these, "n" and "nhn" produced almost identical mass-loss rates, with "nh" solutions yielding mass-loss rates a factor of three lower.

Fig. 3. The observed IR/radio continuum distribution of WR 147 compared to the predictions of smooth and clumped wind models. Smooth wind and constant EF clumped wind models (which produce equal predictions - solid line) and noninteracting type II clumped wind models (long dashes) disagree with the observed continuum distribution, indicated by dots, in contrast with the interacting type II clumped wind model (short dashes) with a "nhn" ionization structure

Fig. 4. The observed IR/radio continuum distribution of WR 134 compared with model predictions. Models are as for Fig. 3, with an interacting type II clumped wind model (short dashes, "nhn" ionization structure) required to reproduce observations

Fig. 5. The observed IR/radio continuum distribution of WR 11 compared with model predictions. Models are as for Fig. 3, with less definitive conclusions. Both interacting type II clumped wind models with "nh" (short dashes) and "nhn" (not shown for clarity) ionization structures are consistent with observations (see Sect. 4.2)

For us to obtain mass-loss rate estimates from radio observations we need to estimate the ionization state where the cm-radio continuuum is formed. We proceeded as follows. Primary standards were adopted for each subclass where a unique cm solution was obtained (namely WR 134, WR 147, WR 113 plus all WC9 stars). Stars with variable radio fluxes (e.g. WR 89) were interpreted as oscillating between a high and low cm-ionization regime. For these primary standards we then obtained coefficients that reproduced the derived mass-loss rates for individual optical lines (Eq.  25). Since many subclasses were without suitable standards (WN3-5 and WC4-7), it was necessary to make two further assumptions for the coefficients : (i) they are comparable within each WR spectral class and (ii) they change smoothly between spectral classes. Depending on the outer ionization balance, two alternative mass-loss rates were obtained from cm fluxes for each star, which could be compared to results from optical lines (Eq.  25). With the above assumptions, we were able to identify mean coefficients for each WR spectral class. These are presented in Table 5, based on stars with thermal radio emission, plus three additional WN3-4 stars, namely WR 127 (WN3b+O9.5 V) from Sect.  5, and WR 128 (WN4) and WR 3 (WN3b+O4) using (clumping-corrected) mass-loss rates derived by Nugis & Niedzielski (1995) from an optical-IR analysis. For individual stars without radio fluxes, the ionization state in the radio emission zone follows from , which allows mass-loss rate predictions using the empirical formula (Eq.  25). In the following section we present our clumping-corrected WR mass-loss rates, and compare these with estimates obtained from independent techniques.

Table 5. The constants to be used in Eq. (25) for the determination of mass-loss rate using observed optical line equivalent widths

© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998

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