HD 66811 (O4 If) HD 164270 (WC9) Figure 5 Comparisons between stellar radii at Rosseland optical depths of 20 ( = R∗ , orange) and 2/3 ( = R2/3 , red ) for HD 66811 (O4 If ), HD 96548 (WR40, WN8), and HD 164270 (WR103, WC9), shown to scale. The primary optical wind line-forming region, 1011 ≤ ne ≤ 1012 cm−3, is shown in dark blue, plus higher density wind material, ne ≥ 1012 cm−3, is indicated in light blue. The ﬁgure illustrates the highly extended winds of ownloaded from www.annualreviews.org ARY on 02/28/12. For personal use only. 20 R(Sun) HD 96548 (WN8) HD 66811 (O4 If) HD 164270 (WC9) Figure 5 Comparisons between stellar radii at Rosseland optical depths of 20 ( = R∗ , orange) and 2/3 ( = R2/3 , red ) for HD 66811 (O4 If ), HD 96548 (WR40, WN8), and HD 164270 (WR103, WC9), shown to scale. The primary optical wind line-forming region, 1011 ≤ ne ≤ 1012 cm−3, is shown in dark blue, plus higher density wind material, ne ≥ 1012 cm−3, is indicated in light blue. The ﬁgure illustrates the highly extended winds of WR stars with respect to Of supergiants (Repolust, Puls & Herrero 2004; Herald, Hillier & Schulte-Ladbeck 2001; Crowther, Morris & Smith 2006b). evolutionary models, namely Primary wind Review by Crowther (2007) Opaque core Rc Re R⇤
mass loss! • Restrictive boundary conditions • Purely radiative in envelope • Numerically integrates the equations of stellar structure Hydrostatic equilibrium, mass conservation, energy conservation & energy transport • Density perturbations in inflated region Changes the opacity in that region • Input mass, composition, inhomogeneity, boundary conditions on temperature and density Chosen so that star is nearly Eddington Assumptions Ingredients
-5 0 5 0 2 4 6 r/R log 10 (ρ/g cm-2 ) Fig. 1. Density vs. radius for our 23 M He model (cf. Table 1). 20 To achieve density scale h For our examp 3 × 103(1 − Γ) 4 × 10−4 (cf. E radiation press a large radial e A qualitati can be obtaine and the total p agram is parti limit, P is of t arithmic diagr (in accordance 1918) for whi uation is more similar relatio drops signiﬁca sure Pmin is rea NUMERICAL RESULTS: Density Profile
1. Density vs. radius for our 23 M He model (cf. Table 1). 0 5 10 15 20 0 2 4 6 r/R log 10 (P/dyn cm-2 ) P rad P gas P tot Fig. 2. Pressure vs. radius. Gas pressure Pgas , radiation pressure Prad , and total pressure P = P + P . arithmic di (in accorda 1918) for uation is m similar rel drops signi sure Pmin is Pgas drops peak in the More e follows a leads into a creases sig When κ re Eddington reduced. L ever, in our convection reduce κ, is has to drop (cf. Fig. 5) where Pgas tioned abo solution to
inﬂation near 5.0 5.5 6.0 6.5 0 2 4 6 r/R log 10 (L/L ) L tot L Edd Fig. 3. Total stellar luminosity L(r), compared to the Eddington lumi- assuming a an inter-clu would thus The same d f is commo winds (e.g. In the c no radiativ ing/porosity cally thin c or pancake Followi large envelo for Edding shown that the Fe-opac clumping, t densities, i.
lumi- nosity LEdd = 4πcGM/κ. 20 15 10 5 20 15 10 5 log 10 (P rad /dyn cm-2 ) log 10 (P/dyn cm-2 ) P rad P gas P tot Fig. 4. Stellar structure in the Prad –Pgas plane. Gas pressure Pgas , radi- ation pressure Prad , and total pressure Ptot = Pgas + Prad are plotted vs. Prad throughout the whole star. are a consequence of the strength, and shape of the Fe-opacity peak in the Prad –Pgas plane. 2.4. The inﬂuence of density inhomogeneities For our pure He models in Table 1, the described envelope ex- tension starts at luminosities of log(L/L ) ≈ 5.5, and fully de- velops for log(L/L ) ≈ 5.8 (cf. Fig. 6). As we will discuss in densities, i.e., towards lower (mean) values of Pg the envelope inﬂation is further enhanced (a more nation of this eﬀect is given in Sect. 3, Eqs. (19), In Fig. 6 we show He-ZAMS models that are clumping factors D = 4, and D = 16. As expected extension occurs much earlier in these models. I will show that these models cover the observed H sitions of the Galactic H-free WR stars, i.e., we c observed temperatures of these objects with mod factors. The adopted values for D are in notable a spectroscopically determined clumping factors fo WR stars, e.g. by Hamann & Koesterke (1998). 3. Analytical description of the envelope In the previous section we have shown that the pr ﬂated envelopes are chieﬂy determined by the fa lution has to follow a path with Γ = 1 in the P (cf. Fig. 5). In the following we elaborate on this analytical description of this process. As we con physics of the inﬂated envelopes alone, the resu do not depend on the internal structure of the sta generally applicable to stars with given (observe In Sect. 3.1, we start with the equations describin structure. In Sect. 3.2 we derive analytical expr radial extension of the envelope ∆R, and in Sect. a recipe to estimate ∆R based on observed/adop rameters. Furthermore, we derive an estimate fo the surrounding shell, ∆M, in Sect. 3.4. Finally, Sect. 3.5, to which extent the inﬂation eﬀect may model assumptions.
lumi- nosity LEdd = 4πcGM/κ. 20 15 10 5 20 15 10 5 log 10 (P rad /dyn cm-2 ) log 10 (P/dyn cm-2 ) P rad P gas P tot Fig. 4. Stellar structure in the Prad –Pgas plane. Gas pressure Pgas , radi- ation pressure Prad , and total pressure Ptot = Pgas + Prad are plotted vs. Prad throughout the whole star. are a consequence of the strength, and shape of the Fe-opacity peak in the Prad –Pgas plane. 2.4. The inﬂuence of density inhomogeneities For our pure He models in Table 1, the described envelope ex- tension starts at luminosities of log(L/L ) ≈ 5.5, and fully de- velops for log(L/L ) ≈ 5.8 (cf. Fig. 6). As we will discuss in densities, i.e., towards lower (mean) values of Pg the envelope inﬂation is further enhanced (a more nation of this eﬀect is given in Sect. 3, Eqs. (19), In Fig. 6 we show He-ZAMS models that are clumping factors D = 4, and D = 16. As expected extension occurs much earlier in these models. I will show that these models cover the observed H sitions of the Galactic H-free WR stars, i.e., we c observed temperatures of these objects with mod factors. The adopted values for D are in notable a spectroscopically determined clumping factors fo WR stars, e.g. by Hamann & Koesterke (1998). 3. Analytical description of the envelope In the previous section we have shown that the pr ﬂated envelopes are chieﬂy determined by the fa lution has to follow a path with Γ = 1 in the P (cf. Fig. 5). In the following we elaborate on this analytical description of this process. As we con physics of the inﬂated envelopes alone, the resu do not depend on the internal structure of the sta generally applicable to stars with given (observe In Sect. 3.1, we start with the equations describin structure. In Sect. 3.2 we derive analytical expr radial extension of the envelope ∆R, and in Sect. a recipe to estimate ∆R based on observed/adop rameters. Furthermore, we derive an estimate fo the surrounding shell, ∆M, in Sect. 3.4. Finally, Sect. 3.5, to which extent the inﬂation eﬀect may model assumptions. A&A 538, A40 (2012) 6 R 4 R 2 R 7 6 5 4 3 8 7 6 5 4 3 log 10 (P rad /dyn cm-2 ) log 10 (P/dyn cm-2 ) 1.0 0.5 0.0 -0.5 -1.0 P gas P rad P tot Fig Th Ed cor & 23 fol and cat as e Minimum gas pressure & density corresponds to Fe opacity peak = Lrad 4 ⇡ c G M
D ¯ ⇢ (⇢, T) ! (D ⇢, T) 7 8 7 6 5 4 log 10 (P rad /dyn cm-2 ) P gas P rad P tot 10 M 15 M 20 M 30 M 40 M 40 M 60 M 85 M 120 M 5.0 5.5 6.0 6.5 5.2 5.0 4.8 4.6 4.4 4.2 log (T * /K) log (L /L ) T*/kK 15 20 25 30 35 40 50 60 70 80 100 120 140 160 ZAMS He-ZAMS X=0.0 X=0.0, D=4 X=0.0, D=16 X=0.2 X=0.4 X=0.7 fo w ∂ ∂ w ∂ o ∂ ∂ N i. o
G M P ¯ ⇢ The mass ∆M of the dense shell surrou (cf. Fig. 1) may be of interest, e.g. in rela get, and the timescales of variations in ∆ To compute ∆M we use the followin radial extension of the shell is small com radius, i.e., r = Re ; 2) the mass of the s to the stellar mass, i.e., m = M; 3) the p surface is small compared to the pressur of the inﬂated region Pe . Under these assumptions, the equati librium (Eq. (1)) can be integrated direct ∆M = Pe 4πR4 e GM · The mass of the shell thus simply follow pressure force Pe 4πR2 e at the rim of the lo to be in equilibrium with the gravitation ANALYTIC RESULTS: Size of Envelope ANALYTIC RESULTS: Mass of Surrounding Shell
6.5 5.2 5.0 4.8 4.6 4.4 4.2 log (T * /K) log (L /L ) T*/kK 15 20 25 30 35 40 50 60 70 80 100 120 140 WNL WNE-w WNE-s AG Car 1985-2003 ZAMS He-ZAMS Fig. 11. Herzsprung-Russel diagram of the Galactic WN stars, and the LBV AG Car. Red/blue symbols indicate observed HR diagram positions of WN stars from Hamann et al. (2006) (blue: with hydrogen (X > 0.05); red: hydrogen-free (X < 0.05)). Black symbols indicate the HR diagram positions of AG Car throughout its S Dor Cycle from 1985–2003, according to Groh et al. (2009b). Large symbols refer to stars with known distances from cluster/association membership. The symbol shapes indicate the spectral subtype (see inlet). Arrows indicate lower limits of T H-free with H X=0.7 X=0.4 X=0.2 D=1 D=4 D=16 T⇤ = L 4 ⇡ SB R2 ⇤ 1/4
6.5 5.2 5.0 4.8 4.6 4.4 4.2 log (T * /K) log (L /L ) T*/kK 15 20 25 30 35 40 50 60 70 80 100 120 140 WNL WNE-w WNE-s AG Car 1985-2003 ZAMS He-ZAMS Fig. 11. Herzsprung-Russel diagram of the Galactic WN stars, and the LBV AG Car. Red/blue symbols indicate observed HR diagram positions of WN stars from Hamann et al. (2006) (blue: with hydrogen (X > 0.05); red: hydrogen-free (X < 0.05)). Black symbols indicate the HR diagram positions of AG Car throughout its S Dor Cycle from 1985–2003, according to Groh et al. (2009b). Large symbols refer to stars with known distances from cluster/association membership. The symbol shapes indicate the spectral subtype (see inlet). Arrows indicate lower limits of T H-free with H X=0.7 X=0.4 X=0.2 D=1 D=4 D=16 T⇤ = L 4 ⇡ SB R2 ⇤ 1/4 Wc = Rc G M P ¯ ⇢
ut to have only e inﬂated zone he energy bud- mptions. 1) The the total stellar mall compared P at the stellar outer boundary drostatic equi- we obtain 6 4 2 0 8 6 4 log 10 (P rad /dyn cm-2 ) log 10 (P gas /dyn cm-2 ) 6 R 4 R 2 R T=50 kK ρ=5 10-12 g cm-3 T=50 kK ρ=5 10-8 g cm-3 T=80 kK T=80 kK T=160 kK T=160 kK T=50 kK ρ=5 10-10 g cm-3 Fig. 10. The eﬀect of ρ and T at the outer boundary, on the envelope solution. The solid blue line indicates our standard 23 M He-model in