Determining the physical properties of WR stars, such as effective temperature Tef f,
stellar luminosity (L), chemical abundances, and wind parameters, requires detailed modelling of the emergent spectral energy distribution and absorption/emission features. Modelling the state of gasses constituting a Wolf-Rayet atmosphere, with the goal of cre- ating a synthetic spectrum, is a task littered with complexities. For cooler stars like the Sun, the assumption of local thermal equilibrium (LTE) can be applied to the gasses in their photospheres; in which case, the equilibrium relations of statistical mechanics and thermodynamics yield occupation numbers for bound and free energy levels based solely on the local temperature and density. For hot stars likeWRs, however, radiative tran- sitions dominate the state of the gas. Consequently, the simplifying assumption of LTE in invalid, and the equations of statistical equilibrium must be used to calculate energy level occupation numbers (as described byMihalas 1978). The solution of the equations of statistical equilibrium is an iterative process, as the radiation field is dependent on these populations, which in turn rely on the radiation field.
A further complication arises from the extended nature ofWRatmospheres compared to their O-star precursors. WRradii are characterised by R?, where the Rosseland optical
depth is high (τRoss= 10–100). Observed optical continuum radiation is emitted from
larger stellar radii (R2/3) where the Rosseland optical depth, τRoss=2/3. The separation
between R2
radius itself, prohibiting simplification of the radiative transfer equation by the commonly applied plane-parallel approximation (Gray, 2005).
The quantitative analysis of WR atmospheres requires extremely complex model atoms, which can be computationally demanding. This problem is alleviated by grouping together atomic levels with similar energies and properties in to a ‘super-level’ (Anderson,
1989). This allows the population of an individual atomic level to be calculated assuming it has the same departure coefficient (from LTE) as the super-level to which it belongs. The simplification brought about by this approach allows thousands of individual levels to be reduced to hundreds, allowing the inclusion of highly complex Fe atoms with millions of transitions, which have a profound effect on the radiation field throughout the atmosphere - known as ‘line-blanketing’.
Two codes well-suited to modelling WR atmospheres are in common usage: PoWR (Hamann & Schmutz, 1987; Gr¨afener et al., 2002) and CMFGEN (Hillier, 1987; Hillier & Miller, 1998). Both codes solve the radiative transfer equation in the co-moving frame, subject to statistical and radiative equilibrium, assuming an expanding spherically sym- metric atmosphere. Both have also been updated to account for line-blanketing by the super-level approach, and include an approximation to clumped winds. Wind clumping is parameterised by a volume-filling factor, whereby a fraction (f ) of the stellar wind is occupied by clumps, which are separated by voids. Alternatively, a clumping factor, Dcl= 1/f , is sometimes quoted. Values of f ∼ 0.05–0.25 generally provide adequate fits
to line profiles of stars with strong winds (Hillier & Miller, 1999; Dessart et al., 2000;
Crowther et al., 2002; Hillier et al., 2003).
Typical parameters and disagreement with theory
The physical parameters derived by spectral analyses of WR stars are summarised by
Crowther(2007), and in Table1.2I show typical values by subtype. I have touched upon many of the key parameters of WR stars throughout this section. In the remainder I will highlight some important trends and challenges.
The measured luminosities of H-free WN, and WC stars fall consistently below the predictions from single-star evolutionary models (see Figure 1.15). This is observed in large samples not only in the Galaxy (Hamann et al., 2006; Sander et al., 2012), where distance uncertainties are large, but also in the LMC (Hainich et al., 2014) which has a well-known distance. I discuss this problem further in Section 5.4.3, in relation to the
WRstars in Galactic cluster Westerlund 1.
Table 1.2: Typical parameters of Galactic WR stars by subtype. Data are taken from
Herald et al. (2001) and Hamann et al. (2006) for WN stars; and Sander et al. (2012), Tramper et al. (2015, in prep), and Williams et al.(2015) for WO and WC stars.
Sp. type T?(kK) log(L/L ) log M (M˙ yr−1)
v∞(kms−1) Mv WN 3 85 5.34 -5.3 2200 -3.1 4b 85 5.3 -4.9 1800 -4.0 5 60 5.2 -5.2 1500 -4.0 6b 70 5.2 -4.8 1800 -4.1 7 50 5.54 -4.8 1300 -5.4 8 45 5.38 -4.7 1000 -5.5 9 32 5.7 -4.8 700 -6.7 6ha 45 6.18 -5.0 2500 -6.8 9ha 35 5.86 -4.8 1300 -7.1 WC & WO WO 170 5.4 -5.0 5000 -2.8 5 83 5.3 -4.62 2800 -3.6 6 78 5.5 -4.61 2300 -3.6 7 71 5.3 -4.8 2000 -4.5 8 60 5.3 -4.8 1800 -4.0 9 40 5.0 -5.0 1000 -4.6
clumping factor (f ) to the fore in determining mass-loss rates. Typical f values of 0.05–0.25 cause a factor of 2–4 reduction in ˙M relative to homogeneous wind models ( ˙M ∝ f−1/2). Any resulting reduction in the mass-loss rates assumed by stellar models, as proposed bySmith(2014), would only worsen the aforementioned luminosity problem, by predicting more massive and hence luminousWR stars. Indeed, evolutionary models with artificially enhanced mass-loss rates have previously been proposed to account for low-luminosityWNE and WC stars (Meynet et al., 1994), which can now be ruled out. During the WR phases, models of stellar evolution adopt mass-loss rates dictated by empirical relations based on luminosity and chemical composition (Nugis & Lamers,
2000). Clumping-corrected mass-loss rates from spectroscopic analyses are required to thoroughly test the adequacy of these relations.
The temperatures ofWRstars are measured by modelling the intensity ratios of adja- cent ionisation stages of a particular element (typically Heii/Hei in WN, and Civ/Ciii/Cii in WC). However, due to the extension ofWRatmospheres, these spectral features orig- inate from layers well above the hydrostatic domain - typically where temperatures are defined in stellar models. WRtemperatures, T?, must therefore defined at (unobservable)
radii, where the optical depth is high (R?), for meaningful comparisons to models. This
relies on an assumption that the same velocity law holds throughout the wind. Differ- ences between R? and R2/3 can be extreme when the stellar wind is strong. For example,
in HD 50896 (WN4b), R?= 2.9 R and R2/3= 7.7 R , corresponding to T?= 85 kK and
T2/3= 52 kK (Morris et al., 2004). Differences in weak-lined WN stars are less extreme,
as R?∼ R2/3.
Furthermore, temperatures and mass-loss rates must be derived simultaneously using non-LTE atmosphere codes, due to the highly stratified winds of WR stars. This is because an increase in mass-loss rate would raise wind density, leading to more efficient recombination from high to low ionisation stages, thus affecting the ionisation-based temperature diagnostics. In Figure 1.21 I show the predicted wind structure of a WC8 star as a function of radius.
Values of R? derived by spectroscopic analysis (Table 1.2) are consistently an or-
der of magnitude higher than radii predicted by stellar structure models for H-freeWR
stars (e.g., Schaerer & Maeder 1992). Consequently, a comparison between the mea- sured T? and Tef f predicted by evolutionary models (at the hydrostatic surface) is not
straightforward. To explain this radius discrepancy, it has been proposed that someWR
stars close to the Eddington limit consist of a small hydrostatic core within an ‘inflated’ sub-photospheric envelope - the so-called “core-halo” model (Ishii et al., 1999; Petrovic
Figure 1.21: Stellar wind structure as a function of radius given by a extttCMFGEN model for a WC8 star (WR 135). (Top) Wind ionisation stratification showing carbon (solid) and helium (dotted). Note, R2/3 occurs at several R?. (Bottom) Electron density
(Ne, solid), wind velocity (v, dotted), and temperature (T , dot-dashed). Taken from
et al., 2006). Consequently, hydrostatic cores may be smaller and hotter than derived from atmosphere analyses. Such a scenario would also alleviate the radius problem in
WR stars. Gr¨afener et al. (2012) have recently shown that envelope inflation may not restricted to high-L H-richWR stars, but may occur in H-free WR stars, or in a lower mass regime if the clumping factor of the wind is high enough.