where LlV is the luminosity at the frequency of line i, kl a x1 /p the dimensionless line-strength, x the frequency-integrated line opacity and p the density. Because of the large number of lines present and needed to accelerate the wind (our database comprises roughly 4 million lines from 150 ions), a statistical approach is well suited, and the above sum can be approximated by an integral over the line-strength distribution, N(k). From early on, it turned out that this distribution closely follows a power law, dN(k)/dk ~ ka-2, where a is of the order of 0.6-0.7 under typical conditions (see Figure 31.1, left). Details can be found in Puls et al. (2000). With this distribution and integrating over all (optically thin and thick) lines, the line acceleration turns out to depend on the luminosity and the spatial velocity gradient! Namely,

¿andes = E gad ^ / / grad( V, k )d N (V, k) a Nf L (, with Neff the so-called effective (flux-weighted) number of lines.

3 4 log kL

metallicity Z (log)

metallicity Z (log)

Figure 31.1. Left panel: a logarithmic plot of the line-strength distribution function for an O-type wind of 40 000 K and the corresponding power-law fit. Right panel: predictions from line statistics of the dependence of mass-loss M (via Nf , cf. Equation (31.1)) on metallicity Z, for reff = 40 kK (crosses) and 10 kK (asterisks). The slopes are 0.56 and 1.35, respectively. Adapted from Puls et al. (2000).

Scaling laws

By inserting this expression into the hydrodynamical equation of motion, the latter can be solved, together with the equation of continuity, (almost) analytically (e.g. Kudritzki etal. 1989), and the resulting wind parameters obey the following scaling laws (with rotation neglected):

where r is the Eddington factor (Thompson scattering diminishes the effective gravity), a the power-law index of the line-strength distribution function and a' = a — S, with S ~ 0.1 the so-called ionisation parameter.

The wind-momentum-luminosity relation

Exploiting the fact that a' is close to 2/3, the so-called modified wind-momentum rate, Dmom (Kudritzki et al. 1995; Puls et al. 1996), becomes almost independent of mass:

log Dmom « — log L + constant(Z, spectral type). (2.5)

This wind-momentum-luminosity relation (WLR) constitutes one of the most important predictions of radiation-driven-wind theory, and can be applied at least in two ways. (i) From spectral analyses of large samples of massive stars, one can construct observed WLRs, and calibrate them as a function of spectral type and metallicity Z (Nef and a' depend on both parameters). The derived relations can then be used as an independent tool to measure extragalactic distances from the wind properties, effective temperatures and metallicities of distant stellar samples. (ii) Observed WLRs can be compared with theoretical predictions in order to test the validity of the theory itself.

Predictions from line statistics

Since a higher metallicity translates into higher opacities, an increase in abundance leads to a larger number of lines which can accelerate the wind. Denoting the global metallicity by Z (normalized with respect to the Solar value), it turns out that the effective number of lines scales via Neff a Z1-a and thus, via Equation (2.1),

For O-type winds then (a ~ 2/3), this means a scaling with Z0 6, whereas for A-supergiant winds (a ~ 0.4.. .0.5) a dependence of Z13 - 2 is predicted (see Figure 31.1, right). For (so-far-hypothetical) massive-star winds with z = 2, this implies mass-loss rates a factor of 1.5 .. . 2.8 higher than for winds of Solar composition.

Not only the global abundance but also the specific composition affects the wind. Owing to their different line-strength statistics, lines from Fe-group elements and light elements (CNO and similar) have different impacts on the wind properties. In particular, Fe-group elements dominate the acceleration of the lower wind, and thus determine M, whereas lines from light elements dominate the acceleration of the outer wind, and thus determine v^. For details, see Puls et al. (2000), but also similar results from hydrodynamical calculations by Pauldrach (1987), Vink et al. (1999, 2001) and Krticka (2006).

Predictions from hydrodynamical models

The most frequently quoted predictions for the wind properties of OB-type stars result from the hydrodynamical models provided by Vink et al. (2000), summarized by them in terms of a 'mass-loss recipe'. These predictions are in very good agreement with independent models by Kudritzki (2002) (v^ a Z012), Puls et al. (2003) and Krticka & Kubat (2004). Similar approaches have been used to predict the metallicity dependence. In particular, Vink etal. (2001) derived M a Z0 69 for O stars and M a Z0 64 for B-supergiants, and Krticka (2006) found M a Z0 67,

HD 14947

HD 14947

1530 1535 1540 1545 1550 1555 1560 1565 Wavelength (A)

6520 6540 6560 6580 6600 Wavelength (A)

Figure 31.2. Left panel: derivation of vm from UV P-Cygni lines (here, from CIV1548/50). From Kudritzki (1998). Right panel: M from Ha. Synthetic profiles (dashed) from models varied by ±30% in M. Figure adapted from Puls et al. (1996).

1530 1535 1540 1545 1550 1555 1560 1565 Wavelength (A)

6520 6540 6560 6580 6600 Wavelength (A)

Figure 31.2. Left panel: derivation of vm from UV P-Cygni lines (here, from CIV1548/50). From Kudritzki (1998). Right panel: M from Ha. Synthetic profiles (dashed) from models varied by ±30% in M. Figure adapted from Puls et al. (1996).

v^ a Z0 06 for O stars. Note that these results are in very good agreement with the results from line statistics alone (see above). With respect to the winds of Wolf-Rayet stars, which depend strongly on the Fe-content, we refer the reader to the work by Grafener & Hamann (2005) (see also Chapter 33 of this volume), Vink & de Koter (2005), Crowther & Hadifield (2006) and Chapter 29 of this volume.

The derivation of stellar/wind parameters from observations via quantitative spectroscopy requires the assumption of a physical model (incorporated into the atmosphere codes which synthesise the spectra). Most of the results presented in the following are based on a standard, one-dimensional description with a smooth wind (but see Section 4). The parameters have to be derived by using suitable diagnostics:

• photospheric parameters, Teff, log g and helium content from optical lines and NLTE atmospheres (including wind);

• wind parameters, vm (Figure 31.2 left), M and the velocity law from UV P-Cygni lines and/or optical/IR (emission) lines (Ha - Figure 31.2, right, He ii 4686, Bra), M also from the radio free-free excess;

• stellar radius, from distance, V-band magnitude and theoretical fluxes (reddening!); and

• metallicity, from spectra (UV and optical) and models.

During the last decade, many such spectroscopic investigations have been conducted. In Table 31.1 we have compiled important contributions regarding OBA stars and their winds (excluding Galactic Centre objects; see Chapter 13 of this

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