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

3. IR and radio continuum formation in a smooth or clumpy wind

In this section we describe the determination of formulae used to derive radio mass-loss rates, following the numerical technique, assumptions and nomenclature of Nugis (1990) adapted for clumpy winds by Nugis & Niedzielski (1995) to which the reader is referred for a detailed description. As usual, we adhere to spherical symmetry throughout.

3.1. Smooth wind solution

Continuum fluxes at IR ( 10µm) and radio wavelengths were found iteratively by direct numerical integration of Eqs. (42-44) in Nugis & Niedzielski (1995). At IR and radio wavelengths, the absorption coefficient is dominated by free-free transitions, with small bound-free contributions at IR wavelengths, and electron scattering unimportant (Nugis 1994, Nugis & Niedzielski 1995).

The transfer of radiation in the outer stellar winds of hot stars was considered by Panagia & Felli (1975) and Wright & Barlow (1975), assuming asymptotic values of temperature, wind velocity, a constant ionization state and smooth outflow. With these assumptions, the intensity at a radio wavelength from any line-of-sight with ( is the stellar radius, and is the impact parameter) is given by

where the optical depth is

where is the mass-loss rate, is the atomic unit mass, µ is the mean atomic weight, is the mean number of electrons per atom, and is given by Eq. (45) in the paper of Nugis & Niedzielski (1995). is given by

Here is the number of z -times ionized atoms of the element A, is the total number of atoms of all elements, is the ionization energy of level i of times ionized atoms of the element A and and are the Gaunt factors for free-free and bound-free transitions. Free-free Gaunt factors at long wavelengths () can be precisely found through the Spitzer formula:

At wavelengths with the free-free Gaunt factors can be found by interpolation from the Tables of Carson (1988).

To obtain the total emission, we must integrate the intensity, , along a line-of-sight over the entire wind and multiply by 4 to account for the isotropy of the emission. Thus we have:

where is the observed, de-reddened flux. Following Panagia & Felli (1975) we can divide this into two parts (the first part corresponds to the optically thick zone with small impact parameters ):

If to choose so that is much higher than unity, then . For numerical calculations it is enough to use so that (). The first integral in parenthesis is equal to and the second integral can be expanded into:

where

Therefore the observed flux becomes:

From this formula and the formula for (Eq.  2), we can write down the smooth wind mass-loss rate as follows

where is a numerical constant. As increases, the ratio quickly approaches an asymptotic value of 0.478. We obtain the mass-loss rate

using the usual units for mass-loss rate ( yr^-1), distance (kpc), terminal velocity (km s^-1), flux (Jy) and frequency (Hz). At IR and radio wavelengths 1, so the term involving tends towards unity. For an asymptotic smooth wind, the continuum fluxes depend on frequency as follows:

We can now calculate spectral indices for individual stars. Table 4 compares observations of WR 78 (WN7) with predicted indices, demonstrating the poor agreement achieved for a smooth wind. Even if we assume that helium is doubly ionized where the 12µm continuum flux is formed and singly ionized in the radio-emission zone, the predicted 12µm-6cm index of 0.68 still differs from the observed index of 0.77.

Table 4. Comparison between observed IR-mm-cm spectral indices for WR 78 (WN7) with predictions from the asymptotic smooth wind model, a clumped wind model (Model I) with clumps dominating the formation of IR and radio fluxes in the whole range and a clumped wind model (Model II) with clumps dominating the formation of IR and mm- fluxes, and an enhancement factor due to clumping approaching unity in the region where cm fluxes are formed. Two different solutions are presented for Model II with a normal ionization structure ("n") and with a higher ionization status in the far wind ("nhn"). Smooth wind and clumped wind Model I are in conflict with observations

At this stage we should highlight one limitation with our analysis. We assume that the entire IR-radio continuum flux is formed in the asymptotic regime, where the wind velocity, temperature and ionization structure have reached their final values. In WR stars, this regime is not reached at mid-IR wavelengths. Consequently this model underestimates the IR-radio spectral index by about 0.05-0.10 (early-type WR stars) or 0.02-0.06 (late-type WR stars). Therefore, longer wavelength indices ( mm) provide a more reliable indicator. Nevertheless, serious discrepancies remain for all WR stars under investigation assuming a smooth outflow.

3.2. Clumped wind solution - constant filling factor and density contrast

We now turn to the derivation of the observed continuum flux for a clumped wind. From the equation of mass continuity (see also Nugis & Niedzielski 1995):

The enhancement of continuum emission in the radio spectral region has been derived by Abbott et al. (1981) and Lamers & Waters (1984). They assumed that (the filling factor), and p (the density contrast) are constant with radial distance. In addition, we assume that clumps have the same and ionization state as interclump material, which should be reasonable provided matter is not predominantly neutral, and no substantial interactions take place in the wind.

The optical depth along a particular line-of-sight can be found from

which is identical to the smooth case (Eq.  2) except for an enhancement factor EF:

In the case of a constant EF we obtain an identical relationship for the mass-loss rate as Eq. (11) except for in the denominator. Equally, the continuum flux has an identical form to the smooth case (Eq.  12) except for a factor. Therefore, the IR-radio spectral index is identical to the smooth wind case, and so fails to explain the observed spectral indices. We therefore have to relax the assumption that clumping is constant throughout the wind.

3.3. Clumped wind solution - variable filling factor and density contrast

Antokhin et al. (1992), Nugis (1994) and Nugis & Niedzielski (1995) have studied conservative clumped wind models, in which clumps yield variable filling factors and density contrasts with radial distance. The matter density of the clumps is determined by their expansion rate which is close to the local sound speed (Eq. (46) of Nugis & Niedzielski 1995). Clumps are assumed to be uniform, spherical, formed at the stellar surface and move with an identical and velocity law ( of unity) as the interclump medium. EF is around unity near the stellar surface, increasing to a maximum at 5-10 , and subsequently returning to unity in the outer wind. The precise value of the enhancement factor depends on the inner matter contrast, filling factor and the mean size of clumps at the stellar surface.

Since we are interested in deriving mass-loss rates from radio fluxes in the present work we do not need to consider the precise inner wind structure, and instead develop equivalent relationships to the smooth case described earlier. We consider the wind from where EF reaches its maximum value up to the radius, , at which EF returns to unity, where the filling factor reaches its maximum value and p =1. In the asymptotic region the clumped volume behaves linearly with radius (). Incorporating the functional form of the enhancement factor into the optical depth relation along a particular line-of-sight (Eq.  14) we find

The observed flux then follows from

where is such that . Assuming clumps dominate the IR and radio continuum formation, the leading integral term in Eq. (16) becomes dominant so that:

In this case, the observed flux is

where is a numerical constant and

Finally, the mass-loss rate is:

where is a numerical constant.

Substantially higher IR-radio spectral indices result relative to a smooth wind case, as illustrated in Table 4 for WR 78 (clumped wind Model I). However, from the observed spectral indices, clumps do not appear to dominate the entire IR-radio continuum formation. Observations can only be reproduced if clumps dominate at IR wavelengths, in which case the spectral index in the IR is 0.9, decreasing to 0.6 in the radio range (clumped wind Model II). Although this reproduces the observed spectral indices for some WR stars, in other cases (e.g. WR 1, WR 110, WR 147) increases from the IR-mm to the mm-cm range, indicating an interaction between clumps and interclump medium. A likely explanation for this is the formation of a highly ionized zone, caused by shocks in the outer wind ( 30-100 ), the extent of which depends on the optical depth of the outer layers. We shall now elaborate on this in the following section.

© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998

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