Wolf-Rayet Stars W.-R. Hamann, A. Sander, H. Todt, eds. Potsdam: Univ.-Verlag, 2015 URL: http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-84268

The stellar Eddington limit N. Langer, D. Sanyal, L. Grassitelli, D. Sz´ecsi Universit¨atBonn, Germany

It is often assumed that when stars reach their Eddington limit, strong outflows are initiated, and that this happens only for extreme stellar masses. We discuss here that in models of up to 500 M , the Eddington limit is never reached at the stellar surface. Instead, we argue that the Eddington limit is reached inside the stellar envelope in hydrogen-rich stars above 30 M and in Wolf-Rayet stars above 7 M , with drastic effects for their struture and stability.∼ ∼ 1 Introduction

Massive stars are powerful engines and strongly af- fect the evolution of star forming galaxies through- out cosmic time (Langer 2012, Sz´ecsiet al. 2015). In particular the most massive ones produce copious amounts of ionising photons (Doran et al., 2013), emit powerful stellar winds (Kudritzki & Puls, 2000, Smith 2014) and are thought to produce the most energetic and spectacular stellar explosions, as pair- instability supernovae (Kozyreva et al., 2014), su- perluminous supernovae (Gal-Yam et al. 2009), and long-duration gamma-ray bursts (Larsson et al., 2007, Raskin et al., 2008).

2 The Eddington limit in main sequence stars Fig. 1: The stellar Eddington limit (cf., Eqs. 1 and 2) in the mass-luminosity plane, assuming three different The most massive stars are close to the so called dominant opacity sources. The first one, referring to Eddington limit. The Eddington luminosity is tra- the electron scattering opacity κe (full-drawn straight ditionally considered as the maximum luminosity line), is valid for a Solar hydrogen and helium mass which a star may have to avoid that the radiative fraction and assuming complete ionisation, such that 2 −1 acceleration at its surface exceeds gravity, when con- κe 0.34 cm g . The second one (dashed straight ' sidering only electron scattering as opacity source line) refers to a Rosseland mean opacity dominated by 2 −1 (Eddington 1926): iron, assumed here as κFe 1 cm g . The third one ' (dotted straight line) refers to the Rosseland mean opac- Le = 4πcGM/κe. (1) ity in the hydrogen recombination zone and is assumed 2 −1 here as κH 100 cm g (see text). The curved blue ' Massive main sequence stars, which we under- line gives the location of zero-age main sequence stellar stand here as those which undergo core hydrogen models, with dots marking the crossing of the H- and burning, have a much higher luminosity than the Fe-Eddington limit. The brown line marks the terminal- Sun, as they are known to obey a simple mass- age main sequence (TAMS) location of the models of luminosity relation, L M α, with α > 1. How- Brott et al. (2011) and K¨ohleret al. (2015) for LMC ∼ composition rotating initially with 100 km/s, and the ever, whereas this relation is very steep for low mass ∼ stars (α 5), it is shown in Kippenhahn & Weigert red line marks those models at which a surface helium (1990) that' α 1 for M , and K¨ohleret al. enrichment is starting to occur. (2015) find α → 1.1 for M→ ∞= 500 M . In fact, Kato (1986) showed' that zero-age main sequence models computed only with the electron scattering Figure 1 illustrates the situation in the mass- opacity do reach the Eddington limit at a mass of luminosity plane. The zero age main sequence about 105 M . Whereas it is debated in the lit- (ZAMS) bends such that the electron-scattering Ed- erature∼ whether real main sequence stars do reach dington limit is only met far outside the figure. I.e., the Eddington-limit (Langer 1998, Crowther et al. real massive stars never encounter this limit. How- 2010, Maeder et al. 2012, Langer & Kudritzki 2014, ever, the true opacity can be substantially larger Bestenlehner et al., 2014), we argue below that this than the electron scattering opacity. We thus fol- is indeed the case. low Sanyal et al. (2015; SGLB15) and define a local

241 N. Langer, D. Sanyal, L. Grassitelli & D. Sz´ecsi

Eddington luminosity anywhere inside the star as Eddington luminosity as defined by Eq. (2), the en- velope inflates, as shown for an example in Fig. 2. As LEdd(r) = 4πcGM(r)/κ(r), (2) the iron opacity decreases for lower densities, infla- tion leads to an increase of the Eddington luminos- where r is the radial coordinate and κ(r) now repre- ity and generally stops for the hot models when the sents the local Rosseland mean opacity. We consider Eddington limit is not violated any more. Figure 3 LEdd locally inside the star, since SGLB15 showed shows that, for LMC metallicity, inflation occurs for that, again, the condition L = LEdd according to log L/L > 5.5, and exceeds a factor of 2 for models Eq. (2) is not met by any stellar model of K¨ohler et which are located∼ to the right of the hot edge of the al. (2015) at the stellar surface. This implies that, S Doradus instability strip (see also Gr¨afener et al. based on these models, we do not expect stars of up 2012). to 500 M to drive a super-Eddington wind.

∆r/r 7.2 core 5 + 2 7 4.5

0 6.8 4

6.6 3.5

-2 Mini = 300M ) ]) ⊙ 6.4 3 ⊙ -3 Mcurrent = 285M ⊙ 6.2 2.5 -4 log (L/L 6 2

log ( ρ [g cm -6 non-inflated core 5.8 1.5

S Dor instability 5.6 1 strip -8 5.4 0.5

inflated envelope 5.2 0 -10 60 50 40 30 20 10 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Teff [kK] r/rcore

Fig. 3: Hertzsprung-Russell diagram showing core- Fig. 2: Density profile of a non-rotating 285 M star hydrogen burning inflated models from the LMC grid with Teff = 46600 K and log(L/L ) = 6.8, showing an of K¨ohleret al. (2015). The color indicates the degree inflated envelope and a density inversion. The X-axis of inflation as defined through ∆r/rcore, where r is the has been normalised by the core radius rcore of 25.3 R . stellar radius, rcore would be the stellar radius if inflation was absent, and ∆r = r rcore (cf., Fig. 2). Dots cor- − responding to models with ∆r/rcore > 5 are colored in However, when assuming that even in hot stars, yellow. Below the solid black line, no inflated models are the Rosseland mean opacity may become as large 2 1 found, and above the dotted black all models were found as 1 cm g− due to the iron opacity peak (cf., fig. 9 to be inflated. The hot part of the S Dor instability strip in SGLB15), the Eddington luminosity according to (Smith et al., 2004) is also marked. Cf., SGLB15. Eq. (2) is about a factor of three smaller than that according to Eq. (1). Correspondingly, the ZAMS line crosses it at about M = 100 M (upper blue dot in Fig. 1). When looking at the luminosity in- Since the inflated envelopes are convectively un- crease during core hydrogen burning by including stable (Langer 1997), and the convectively trans- the terminal age main sequence (TAMS) in Fig. 1, ported fraction of the stellar luminosity does not we see that even core hydrogen burning stars with contribute to the inflation, the extent of inflation M = 30 M may violate the local Eddington limit in depends on the assumed convection theory. The their envelopes. The iron opacity peak is metallicity shown results are obtained with the standard Mixing and density dependent, such that the corresponding Length Theory (cf. SGLB15). Eddington limit is only approximate. However, we Observational evidence for inflation in Galactic show below that the expectation derived from Fig. 1 main sequence stars, as predicted by the extremely is actually met by detailed stellar models. extended main sequence band for 5.5 < log L/L <

The largest opacities are found in the hydro- 6.5 of the K¨ohleret al. (2015) models,∼ is found in∼ gen recombination zones of cool stars. Thus, for fig. 5 of Clark et al. (2014) and fig. 1 of Castro et illustration, we plot a third Eddington limit in al. (2015). These figures show an abundance of Fig. 1, which is again approximate, by assuming B supergiants which is unexpected if they would have 2 1 κH =100 cm g− , which then we expect to be vi- to be explained by core helium burning stars. The olated by cool giants with M > 5 M . inflated envelopes are also likely to be pulsationally SGLB15 demonstrated that∼ when the local radia- unstable (cf. Grassitelli et al., this volume), which tive luminosity in the stellar envelope exceeds the may lead to observable consequences.

242 The stellar Eddington limit

Inflated stars may be found well below the increase of the Eddington-luminosity and to an oc- quoted Eddington luminosities for various reasons. currence of the kink in the He-ZAMS at higher lu- E.g., stars which lost a significant amount of their minosity. An increase of the opacity has been sug- hydrogen-rich envelope to a wind or to a close binary gested, on the other hand, by Gr¨afeneret al. (2012), companion will, to first order, keep their luminosity, as this might lead to a better agreement of the sur- but since their mass is reduced their Eddington lu- face temperatures of Wolf-Rayet stars with the ob- minosity will be reduced. Similarly, rotation may re- servations. We conclude that the main parameters duce the Eddington luminosity (Langer 1997, 1998), of the Wolf-Rayet stars are strongly affected by their even though its effect is latitude dependent and van- proximity to the Eddington limit. ishes at the poles. SGLB15 suggested that latitude dependent inflation may give rise to the B[e] phe- nomenon in supergiants (Zickgraf et al. 1985). References

Bestenlehner J M, Gr¨afenerG, Vink J S, et al., 2004, 3 The Eddington limit in A&A, 570, A38 Wolf-Rayet stars Brott I, de Mink S E, Cantiello M, et al., 2011, A&A, 530, A115 So far, the effect of envelope inflation has been dis- Crowther P A, Schnurr O., Hirschi R., et al. 2010, cussed most extensively for Wolf-Rayet stars (Ishii et MNRAS, 408, 731 al. 1999, Petrovic et al. 2006, Gr¨afeneret al. 2012). Doran E.I., Crowther P.A., de Koter, A., et al., 2013, Figure 4 demonstrates the situation in the HR dia- A&A, 558, A134 gram on the basis of zero-age helium star models. Eddington A.S., 1926, The Internal Constitution of Using standard Rosseland mean opacities, we find Stars inflation to occur for models above 7 M at so- Gal-Yam A, Mazzali P, Ofek E O, et al., 2009, Na- ∼ lar metallicity (Grassitelli et al., this volume), and ture, 462, 624 above 12 M for the SMC metallicity. Gr¨afenerG, Owocki, S P, Vink, J S 2012, A&A, 538, ∼ A40 Ishii M, Ueno M, Kato M, 1999, PASJ, 51, 417 7.5 Kato M, 1986, Ap&SS, 119, 57 500 7 κ ZLMC Kippenhahn R., Weigert A., 1990, Stellar Structure 100 300 80 ZMW 6.5 60 200 ZSMC and Evolution (Springer, Berlin) 35 100 6 K¨ohlerK., Langer N., de Koter A., et al., 2015, 35 25 ⊙ 25 A&A, 5.5 15 15 50

log L/L Kozyreva A, Blinnikov S, Langer N, Yoon S-C, 2014, κ 30 5 A&A, 566, A146 6 6 20 4.5 Kudritzki R.-P., Puls, J, 2000, ARA&A, 38, 613 4 4 He-ZAMS H-ZAMS 4 Langer N, 1997, in ASP Conf. Ser. Vol. 120, p. 83 10 3.5 Langer N, 1998, A&A, 329, 551 5.2 5.1 5 4.9 4.8 4.7 4.6 4.5 4.4 4.3 4.2 Langer N, 2012, ARA&A, 50, 107 log Teff/K Langer N., Kudritzki R P, 2014, A&A, 564, A52 Larsson J, Levan A J, Davies M B, Fruchter A S, 2007, MNRAS, 376, 1285 Fig. 4: Zero-age main sequence of the models of K¨ohler et al. (2015) for LMC metallicity (full drawn red line) Maeder A, Georgy C, Meynet G, Ekstr¨om,S. 2012, together with lines for zero-age main sequence helium A&A, 539, A110 star models for Galactic (blue, dashed line) and SMC Owocki S P, Gayley K G, Shaviv N J, 2004, ApJ, (blue, dash-dotted line) metallicity. Labelled dots rep- 616, 525 resent masses in units of the Solar mass. The labels Petrovic J, Pols O, Langer N. 2006, A&A, 450, 219 “κ ” and “κ ” indicate the vertical shift in the kink Raskin C, Scannapieco E, Rhoads J, Della Valle M, ↓ ↑ of the He-ZAMS lines if the opacity in the inflated stars 2008, ApJ, 689, 358 would decrease (the Eddington luminosity would move Sanyal D., Grassitelli L., Langer N., Bestenlehner up) or increase (the Eddington-limit would move down), J.M., 2015, A&A, 580, A20 respectively. Smith N, 2014, ARA&A, 52, 487 Smith N, Vink J S, de Koter A. 2004, ApJ, 615, 475 Sz´ecsiD., Langer N., Yoon, S.-C., et al., 2015, A&A, It has been suggested that the opacity in the in- 581, A15 flated layer may come down because of porosity ef- Zickgraf F-J, Wolf B., Stahl O., et al., 1985, A&A, fects (Owocki et al. 2004), which would lead to an 143, 421

243 N. Langer, D. Sanyal, L. Grassitelli & D. Sz´ecsi

Dany Vanbeveren: What is the effect of a com- Norbert Langer: The companion may suck off the panion on the inflation? inflated envelope (RLOF), but it will regrow, such that the mass transfer rate will be close to the criti- cal mass-loss rate above which inflation is supressed.

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