Annu. Rev. Astron. Astrophys. 1994. 32: 227-275
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3. STELLAR MODELS AND OBSERVATIONS

3.1 Recent Progress in the Input Physics

Only in the Milky Way and a few galaxies of the Local Group are individual observations of massive stars feasible. Before interpreting the integrated spectra of more distant galaxies, we must first proceed to careful tests of the current models by observing stars in nearby galaxies and checking whether these models are realistic.

Over recent years many grids of stellar models of massive stars at different metallicities Z have been produced with various physical assumptions (Brunish & Truran 1982a, b; Chin & Stothers 1990; Maeder 1990; Arnett 1991; Baraffe & El Eid 1991; Schaller et al 1992; Schaerer et al 1993a, b; Charbonnel et al 1993: Woosley et al 1993; Alongi et al 1993; Bressan et al 1993; de Loore & Vanbeveren 1994; Meynet et at 1994). The general input physics for stellar models has been extensively discussed (Iben 1974, Iben & Renzini 1983, Chiosi & Maeder 1986. Maeder & Meynet 1989, Chiosi et al 1992a, Schaller et al 1992). We shall limit our review to those points most critical for massive stars. Evaporation by stellar winds is a dominant feature of massive star evolution and all model predictions are influenced. Stellar wind models have been developed by several groups (Abbott 1982, Pauldrach et al 1986, Owocki et al 1988, Kudritzki et al 1987, Schaerer & Schmutz 1994). However, the observed mass loss rates, M, and the wind momentum in O-type stars are generally still larger than predicted (Lamers & Leitherer 1993). Thus, taking theoretical Mdot is probably not yet a comfortable choice and it seems preferable to base the models on the empirical values. Those compiled by de Jager et al (1988) have commonly been adopted. These mainly depend on luminosity and to a smaller extent on Teff; the role of rotation is still uncertain (Nieuwenhuijzen & de Jager 1988, 1990; Howarth & Prinja 1989). These average Mdot might be too low (Schaerer & Maeder 1992). Considerable uncertainties remain in the adopted Mdot, particularly for the red supergiants which lose mass at very high values (de Jager et al 1988, Stencel et al 1989, Jura & Kleinmann 1990). Presently there is no complete theory for the winds of red supergiants. For these, evidence of dust ejection is provided by IRAS observations, which show that some of them possess extended circumstellar shells (Stencel et al 1989) potentially leading to OH/IR sources (Cohen 1992). Evidence for strong winds in the previous red supergiant phase of SN 1987A has also been presented by Fransson et al (1989), and for the more recent SN 1993J by Hoflich et al (1993).

Different treatments of convection and mixing in stellar interiors have been advocated, giving a major uncertainty in massive star models. We identify the following different assumptions regarding convection and mixing in massive star models:

All of these models are claimed by their authors to fit the observations and the debate has been lively in recent years (Chiosi & Maeder 1986; Maeder & Meynet 1989; Brocato et al 1989; Lattanzio et al 1991; Stothers 1991a, b; Stothers & Chin 1990, 1991, 1992a, b). There is at present no definite theoretical or observational proof in favor of any model. However, a few useful indications on the limits and possibilities of the various models must be mentioned.

Although claims have been made in favor of substantial overshooting from convective cores with respect to what is predicted by Schwarzschild's criterion, it now seems clear that the overshooting distance is limited to about (0.2-0.4 ) Hp (Maeder & Meynet 1989, Stothers & Chin 1991, Napiwotzki et al 1993, Meynet et al 1993). The main effect is to increase the main sequence width and lifetimes, while the helium burning lifetimes are reduced (due to higher L); the blue loops are also shorter. In contrast to overshooting, which extends convective zones, the Ledoux criterion tends to prevent convective mixing in zones with variable mean molecular weights (Kippenhahn & Weigert 1990). Some recent comparisons with observations seem to favor Ledoux's rather than Schwarzschild's criterion for convection (Stothers & Chin 1992a, b); however, the result may depend on the adopted Mdot. Other recent theoretical work (Grossman et al 1993) shows that the Ledoux criterion has no bearing at all in stratified stellar layers. Thus, both at the theoretical and observational levels, the convective criteria remain uncertain.

Semiconvection occurs in zones that are convectively unstable according to Schwarzschild's criterion, but not according to Ledoux's. Semiconvection may thus produce some mixing in zones with a gradient of the mean molecular weight. Various treatments of the problem have been made (Chiosi & Maeder 1986, Langer at al 1989, Arnett 1991, Chiosi et al 1992a, Langer 1992, Alongi et al 1993). In a semiconvective zone, the nonadiabatic effects (radiative losses) produce a progressive increase of the amplitudes of oscillations at the Brunt-Vaisala frequency around a stability level (Kippenhahn & Weigert 1990). The growth of amplitudes is generally rapid compared to the evolutionary timescale, so that a situation equivalent to Schwarzschild's criterion is established. However, this might not be true in massive stars, as shown by Langer et al (1985), who discussed the timescales involved in semiconvective mixing. They propose a diffusion treatment that is equivalent to Schwarzschild's criterion when the mixing timescale is short compared to the evolutionary timescale, and to the Ledoux criterion in the opposite case. Models with such diffusion are of special relevance to the discussion about the blue progenitor of SN 1987A (Langer et al 1989; Langer 1991a, c) as well as about the evolutionary status of blue supergiants (cf Section 3.5).

The effects of rotationally induced mixing may be important for massive stars. The radiative viscosity is so large that dissipative processes may have a timescale comparable to the evolutionary timescalcs of massive stars (Maeder 1987b). Mixing could produce chemically homogeneous or nearly homogeneous evolution on the main sequence and thus lead directly as a result of nuclear burning to the formation of He stars, which would be observed as W-R stars. Models of massive stars losing mass and angular momentum have been calculated by Sreenivasan & Wilson (1985). As Langer (1991b) emphasized, models with semiconvective and rotational mixing may solve several problems: the existence of the WN+WC stars (Langer 1991b; cf Section 4.2), the origin of nitrogen enhancement in OB supergiants, the alleged mass discrepancy for OB main sequence stars (cf Section 3.3), and the nature of the blue progenitor of SN 1987A. Curiously enough, the claims in favor of semiconvection mean less mixing in the convective zone with varying mean molecular weight, while the claims in favor of rotational mixing mean more mixing from inner material into the outer radiative zone (Langer 1993). The situation is still uncertain, but we think it likely that mass and metallicity are not sufficient to describe massive star evolution and that rotational velocity will be an unavoidable additional parameter, as well as (for some) membership in close binary systems.

3.2 Metallicity Effects in Massive Stars

Metallicity, like other effects such as nonconstant star formation rates and peculiar initial mass functions (Section 5), is a key factor influencing massive star populations in galaxies. Metallicity effects can enter evolution through at least four possible doors:

  1. Nuclear production. Metallicity Z may influence the nuclear rates; a good example occurs for the CNO cycle. A very slight contraction or expansion to a new equilibrium state may compensate for a change in nuclear rates (Schwarzschild 1958). In massive stars, a lower Z also produces a more active H-burning shell in the post-main sequence evolution and this favors a blue location in a part or the whole of the He-burning phase (Brunish & Truran 1982a, b; Schaller et al 1992). This was one of the initial explanations proposed for the blue precursor of SN 1987A (Truran & Weiss 1987).

  2. Opacity effects. In the interiors of massive stars, electron scattering, which is independent of Z, is the main opacity source. Thus, in contrast to the case of low and intermediate mass stars, metallicity has no great direct effect on the inner structure of massive stars.

  3. Stellar winds. In the very external layers, Z may strongly influence the opacity and thus the atmospheres and winds. Wind models for O stars by Abbott (1982) suggested a Z-dependence of the mass loss rates Mdot of the form Mdot propto Zalpha, with alpha = 1.0. Other models gave a value of alpha between 0.5 and 0.7 (Kudritzki et al 1987, 1991; Leitherer & Langer 1991; Kudritzki 1994). It is likely that this is the main effect by which Z may influence massive star evolution (Maeder 1991a). For yellow and red supergiants, there are no models (Lafon & Berruyer 1991) nor observations (Jura & Kleinmann 1990) giving reliable Mdot vs Z information; thus a major uncertainty in post-MS evolution remains.

  4. Helium content. A ratio DeltaY / DeltaZ greater than 3 between the relative enrichments in helium and heavy elements has been established from low-Z H II regions (Peimbert 1986, Pagel et al 1992). Thus, changes in Z imply large changes in Y, which have a direct effect on the models.

3.3 Main Sequence Evolution

3.3.1 HR DIAGRAM, LIFETIMES, MASSES     Let us examine a few of the main properties of the models of massive stars at various metallicities. At low metallicity Z, the zero-age main sequence (ZAMS) is shifted to the blue due to the lower opacity in the external layers. Between the sequences at Z 0.001 and 0.04. the shift in log Teff amounts to +0.06 dex at 20 Msun (Schaller et al 1992, Schaerer et al 1993b) and a lowering in luminosity by 0.10 dex. The reason is that at low Z the hydrogen content is higher, thus the electron scattering opacity is larger. The width of the MS band is predicted to change considerably according to metallicity. The main feature is a prominent "paunch," which is displaced to lower luminosities for lower mass loss rates and metallicities. Two physical effects are responsible for this paunch (Maeder 1980). First, the large mass fraction of the He core, resulting from the removal of the outer layers, favors the redward extension of the tracks, Second, when the surface hydrogen content becomes lower than Xs = 0.3 or 0.4 as a result of mass loss (Chiosi & Maeder 1986), the lowering of the surface opacity moves the star back to the blue. Thus, the paunch appears in the range of masses where mass loss is sufficient to increase the core mass fraction, but not high enough to lower Xs below the critical limit. An increase of overshooting or opacity may enhance the paunch. Models with enhanced opacities may have a MS band covering all the HR diagram (Stothers & Chin 1977, Nasi & Forieri 1990).

The lifetimes in the various nuclear phases change with Z. For the H-burning phase, the lifetimes t(H) are typically longer by 35% for a 20 Msun model at Z = 0.001 compared to Z = 0.040. The reason rests on the lower luminosity and the larger reservoir of hydrogen. The lifetimes t(He) in the He-burning phase are generally longer in models with higher Z, due to the higher mass loss rates which lead to a drastic decrease of the luminosities in this phase. For models of 15 to 120 Msun, the t(He) / t(H) is typically 9 to 10%; these ratios are between 11 and 19% at Z = 0.04 and they may amount to 50% if the mass loss rates are increased by a factor of 2 (Meynet et al 1994). These large factors show how our ignorance of the exact mass loss rates at various Z may affect massive star models.

There is an apparent lack of O stars close to the theoretical zero-age sequence (Garmany et al 1982). This is also quite clear in recent gravity and Teff determinations by Herrero et al (1992). We notice that for massive stars, the accretion timescale of the protostellar cloud is longer than the Kelvin-Helmholtz timescale (Yorke 1986). The consequence is that no massive pre-MS star should be visible (Palla et al 1993) - a fact that could contribute to obscured stars close to the ZAMS, Wood & Churchwell (1989) and Chiosi et al (1992a) suggest that 10% to 20% of the O stars are still embedded in their parent molecular clouds. An alternative explanation is that there is no true ZAMS corresponding to a chemically homogeneous stage for O stars, because nuclear reactions ignite early during the contraction phase (Appenzeller 1980) and may thus make stars inhomogeneous before the end of the contraction phase.

Another potential problem is the so-called mass discrepancy for O stars. Spectroscopic masses derived from gravity and terminal velocity determinations were claimed to be smaller than predicted by stellar models (Bohannan et al 1990, Groenewegen et al 1989, Herrero et al 1992, Kudritzki et al 1992). In other words, spectroscopy suggests that O stars are overluminous for their masses and the discrepancy amounts up to about 50%. Langer (1992) interprets the overluminosity of O stars as a sign of rotational or tidal mixing enlarging the helium core. Apart from the fact that the force multiplier may not be correctly predicted by non-LTE wind models, the reality of the mass discrepancy has been questioned recently by Lamers & Leitherer (1993). They show that large discrepancies exist between theoretical and observed mass loss rates, as is true for the terminal velocities; they also argue that the discrepancies cannot be solved by adopting smaller masses for O stars. According to Schaerer & Schmutz (1994), the use of plane-parallel models for O stars may lead to significant errors for spectroscopic gravities, masses, and helium abundances. It is thus possible that the mass discrepancy is due to the inadequate modeling of stellar atmospheres. This view seems confirmed by the most recent work of Pauldrach et al (1994) and Kudritzki (1994), who do not find the mass discrepancy once additional wind opacity due to iron transitions is taken into account.

3.3.2 ABUNDANCES ON THE MS     The surface abundances in He and CNO elements offer a powerful test of stellar evolution. Evidence of CN processing is provided by He and N enhancements together with C depletion, while O depletion only occurs for advanced stages of processing. The abundances may cover a range from solar values (C/N = 4, O/N 10) to CNO equilibrium values in the extreme case which is reached in WN stars (C/N = 0.02, O/N = 0.1; Maeder 1983, 1987a). Models with mass loss but no extra-mixing predict He and N enrichment in MS stars only for initial masses larger than about 50 Msun depending on the mass loss rates. Models with rotational mixing may lead to a precocious appearance of the products of the CNO cycle (cf Maeder 1985, Langer 1992). The observations of 25 OB stars by Herrero et al (1992) show that most MS stars have normal He and N abundances. The same is true for MS B-type stars (Gies & Lambert 1992). For example, even the most massive object, Melnick 42 (O3f), appears to show normal abundance ratios (Pauldrach et al 1994; but see Heap et al 1991). However, there are also exceptions for O and B stars. For example, the O4f star zeta Pup presents evidence of an atmosphere with CNO burned material (Bohannan et al 1986, Pauldrach et al 1994). Fast rotators are also an exception and they generally show He and N enhancements (Herrero et al 1992). Another ease is the group of ON stars, i.e. O stars with N-enrichments (Walborn 1976, 1988; Howarth & Prinja 1989); this group contains at least 50% short-period binaries (Bolton & Rogers 1978).

An analysis of the association Per OB1 (Maeder 1987b) suggests that there is a bifurcation in stellar evolution: While most stars follow the tracks of inhomogeneous evolution, a fraction of about 15%, mainly composed of fast rotators and binaries, may evolve homogeneously and become ON blue stragglers.

3.4 The Eddington Limit and LBV Stars

The value of the mass of the most massive stars in galaxies has been a much debated subject. Recent photometric and spectroscopic studies suggest stellar masses up to about 100 Msun (Section 2.2.2; Table 1; Divan & Burnichon-Prevot 1988, Kudritzki 1988, Heydari-Malayeri & Hutsemekers 1991, Massey & Johnson 1993). Recently Pauldrach et al (1994) have suggested that the most massive star known is Melnick 42 in the LMC, which may have a mass of up to 150 Msun.

There is an upper luminosity limit to the distribution of stars in the HR diagram. It runs from log L / Lsun = 6.8 at Teff = 40,000 K to log L / Lsun = 5.8 at 15,000 K and it stays constant at lower Teff (Humphreys & Davidson 1979: Humphreys 1989, 1992). The theoretical location of the Eddington limit has been examined by Lamers & Fitzpatrick (1988) on the basis of model atmospheres including metal line opacities. The limit was shown to agree with the observed limit in the Milky Way and in the LMC. Subsequent investigations indicate that the Eddington limit rises again at low Teff since the opacities decrease considerably there (Lamers & Noordhoek 1993). Thus, the lowest part of the limit was called the "Eddington trough"; its location in the HR diagram is, of course, higher for stars of lower Z since they have lower opacities in the external layers.

The Eddington limit or "trough" may prevent the redward evolution of very massive stars in the HR diagram (Maeder 1983, Lamers & Noordhoek 1993). The region inside the trough will be empty except for unstable stars during their outbursts. The upper luminosity limit is determined by stars that can just pass under the Eddington trough. Thus, because the location of the trough depends on Z, the upper luminosity of red supergiants may not be an ideal standard candle, contrary to expectations (Humphreys 1983b).

The Luminous Blue Variables (LBVs, Conti 1984), also called hypergiants, S Dor, or Hubble-Sandage Variables, are optically the brightest blue supergiants. They show irregular and violent outbursts, with average mass loss rates up to about 10-3 Msun yr-1 (Davidson 1989, Lamers 1989). The group continues toward lower Teff as the so-called cool hypergiants (Humphreys 1992), which are the most luminous F, G, K, and M stars. These also show evidence of variability, of high mass loss, and extensive circumstellar dust. The He and CNO abundances in LBVs (Davidson et al 1986) are in agreement with products of the CNO cycle at equilibrium, which confirms that LBVs are post-MS supergiants (Maeder 1983). About 30 LBVs have been identified by various authors in nearby galaxies (see list by Humphreys 1989) including the LMC, M31, M33, NGC 2403, M81, and M101. Among hypergiants, the OH/IR supergiants, revealed by radio and IR observations, are the most extreme M supergiants, likely having optically thick dust shells. About two dozen cool hypergiants are currently known in the Milky Way (Humphreys 1991).

The bolometric luminosities of LBVs are constant (Appenzeller & Wolf 1981) during an outburst. However, the matter ejection, particularly during the outbursts, modifies the photospheric radius and Teff, and as a consequence also the bolometric correction and visual luminosity. During their outbursts, LBVs essentially move back and forth horizontally along the HR diagram. Obscuration by gas and dust may also affect the emitted light (Davidson 1987). The circumstellar environment of these stars is peculiar and may affect the distance estimates (Viotti et al 1993). The evolutionary changes of P Cyg over the past two centuries have been recently discussed by Lamers & de Groot (1992), de Groot & Lamers (1992), and El Eid & Hartman (1993) and have been shown to correspond to recent theoretical estimates. At its minimum visual light, the star is hotter (T = 20,000-25,000 K) than at its maximum light (where T = 9000 K).

In the past, LBVs have been assigned to all possible evolutionary stages, but they are currently interpreted as a short stage in the evolution of massive stars with initial M > 40 Msun. A likely scenario is

O star -> Of / WN -> LBV -> Of / WN -> LBV ... -> WN -> WC.

After central H-exhaustion, the star undergoes redward evolution in the HR diagram and is likely to reach the Eddington limit or the "trough". Strong mass loss occurs with shell ejection (LBV). As a result, stability and bluer location in the HR diagram are restored (Of/WN). Internal evolution again brings the star to the red in a few centuries - a time that may depend on the stellar mass and amount of ejected mass (cf Maeder 1989, 1992b). The star again moves toward the Eddington limit, and the cycle of evolution between the Of/WN and LBV stages continues until, as a result of mass loss, the surface hydrogen content is low enough (Xs leq 0.3) so that the star definitely settles in the Wolf-Rayet stage. The overall duration of the LBV phase is fixed by the amount of mass DeltaM to be lost between the end of the MS phase and the entry in the W-R phase. For a typical DeltaM = 10 Msun and an average Mdot of 10-3 to 10-4 Msun yr-1, the typical duration would be approx 104 to 105 yr. This general scenario is consistent with several properties of LBVs: their location in the Hertzsprung-Russell diagram (Humphreys 1989, Massey & Johnson 1993), their Mdot rates (Lamers 1989), their high N/C and N/O abundance ratios (Davidson et al 1986), and the existence of transition objects, as discussed below.

Many observational studies have been made of these transition objects, which are often of spectral type Of/WN and present spectral variability. Examples are S Dor (Appenzeller & Wolf 1981, Wolf et al 1988), R 71 (Appenzeller & Wolf 1981, Wolf et al 1981), AG Car (Caputo & Viotti 1970, Viotti et al 1993), R 127 (Stahl et al 1983; Stahl 1986, 1987; Wolf 1989), R 84 (Schmutz et al 1991), and He 3-519 (Davidson et al 1993). It is also possible that after the LBV phase, some stars go to the stage of OH/IR object and then become W-R stars. This different, but not contradictory scenario, could happen to stars with masses low enough to enable them to go below the "trough". The special cases of Var A in M33 (Humphreys 1989) and IRC+104020 - an extreme galactic F-supergiant with a very large IR excess from circumstellar dust (Jones et al 1993) - might correspond to such a scenario.

The physical origin of the outbursts in LBVs and hypergiants is still a matter of controversy and several models have been considered (e.g. Stothers & Chin 1983; Doom et al 1986; Appenzeller 1989; de Jager 1992; Maeder 1989, 1992b). The most striking property of these models is the strong density inversion occurring in the outer layers, where a thin gaseous layer floats upon a radiatively supported zone. This zone results from the opacity peak which leads to supra-Eddington luminosities in some layers. The idea of a density inversion has a 40 year history (Underhill 1949, Mihalas 1969, Osmer 1972, Bisnovatyi-Kogan & Nadyozhin 1972, Stothers & Chin 1983). A review of the literature shows that essentially three different kinds of conclusions were drawn: 1. A Rayleigh-Taylor instability occurs as a result of the density inversion, which is therefore washed out by the instability. 2. The supra-Eddington luminosity drives an outward acceleration and mass loss without a density inversion. 3. Strong convection and turbulence develop and the inversion is maintained.

A difficulty with most models is that they look for a hydrostatic solution to the problem. However, the resolution likely lies in the context of hydrodynamical models. Although the second of the above conclusions seems preferable, it is still unclear whether or not the density inversion is maintained. Another noticeable peculiarity in the physics of LBVs is that the thermal timescale in the outer layers is shorter than the dynamical timescale. During an outburst, which is at the dynamical timescale, the ionization front is able to substantially migrate inward (Maeder 1992b), so that some layers of matter may participate in the ejection and produce the observed shells (Hutsemekers 1994).

3.5 Blue and Red Supergiants

Conti (1991b) recently reviewed the observations of hot massive stars in galaxies and a complete list of the observations of red supergiants in galaxies has been given by Humphreys (1991). Humphreys (1983b, 1991) has also reviewed the potential role of red supergiants as distance indicators. Amazingly, many problems and controversies remain about supergiants, for which evolution is even more uncertain than for W-R stars! The reason is that W-R stars are dominated by the overwhelming effect of mass loss, which washes out most effects related to uncertainties in convection and mixing. Supergiants are often close to a neutral state between a blue and a red location in the HR diagram (Tuchmann & Wheeler 1989, 1990); even minor changes in convection and mixing processes may greatly affect their evolution.

3.5.1 CHEMICAL ABUNDANCES     Walborn (1976, 1988) proposed that ordinary OB supergiants have an atmospheric composition enriched in helium and nitrogen and depleted in carbon, as a result of CNO processing. According to Walborn, it may just be the small group of the so-called OBC supergiants that have normal cosmic abundances (Howarth & Prinja 1989). Herrero et al (1992) showed that most OB supergiants and Of stars show helium enhancements. As for all rules, there are exceptions: A few B-supergiants do not show He and N excesses (Dufton & Lennon 1989). Herrero et al also show that fast rotators of all luminosities present evidence of CNO processing. Enhancements of nitrogen and helium abundances have also been found for post-MS B type stars by Gies & Lambert (1992), and by Voels et al (1989) in the 09.5 Ia star alpha Cam. As expected, the so-called OBN stars show evidence of He and N excesses with C depletion (Walborn 1988, Schonberner et al 1988).

Abundance determinations have also been made for B supergiants in the LMC and SMC, particularly interesting in relation to the progenitor of SN 1987A. These supergiants generally show He and N enhancements (Reitermann et al 1990, Kudritzki et al 1990, Lennon et al 1991). A recent high-dispersion study of LMC B-supergiants also confirms such enrichments (Fitzpatrick & Bohannan 1993). Among 62 stars of types B0.7 to B3, only 7 are OBC stars (Fitzpatrick 1991). These authors conclude, in agreement with the Walborn hypothesis, that the "typical" supergiants show contaminated surfaces, and only the rare nitrogen weak stars (OBC) have retained their original main sequence composition. The progenitor of SN 1987A, which was a B-supergiant, had N/C and N/O ratios larger than solar values by 37 and 12, respectively (Fransson et al 1989). From all these results, it is clear that most B-type supergiants in the Galaxy, the LMC, and the SMC generally show evidence of CNO processing on their surfaces.

The above observations place severe constraints on stellar models, which do not usually predict He and N enrichments in blue supergiants at solar composition. At solar Z, blue loops with the associated He and N enrichments (as a result of dredge-up in red supergiants) only occur for M leq 15 M. This is the case for the models with Schwarzschild's criterion and overshooting (Schaller et al 1992), and with the Ledoux criterion (Stothers & Chin 1992a, b; Brocato & Castellani 1993). Models with semiconvection (Arnett 1991) have the same difficulty: At solar composition, the evolution goes straight to the red supergiant phase and there are no enriched blue supergiants. At lower metallicity, the blue loops are generally more developed and thus blue supergiants are predicted with He and N enrichments. However, even in this case it seems necessary (Langer 1992) to advocate some, rotational mixing to account for the observed abundances.

The study of CNO abundances in three A-type supergiants (Venn 1993) reveals N-enrichments larger than predicted by the first convective dredge-up, if these stars have first gone to the red supergiant stage. This supports the idea of additional mixing. Analyses of four F-type supergiants by Luck & Lambert (1985) show material processed by the CN cycle at a level that may be higher than predicted. Analyses of some F and K supergiants in the SMC by Barbuy et al (1991) indicate solar N/Fe and C/Fe ratios, and thus no evidence of CNO processing. A further study by Barbuy et al (1992) of 14 Galactic F-supergiants shows an absence of CNO processed material in stars with low rotational velocities. For F supergiants with high rotation, the derivation of CNO abundances is unfortunately masked by the line broadening. Yellow supergiants also show sodium overabundance by a factor of 3 to 4 (Boyarchuk et al 1988: cf also Lambert 1992). Boyarchuk et al have suggested an increase of this overabundance with initial stellar masses. An interpretation put forward by Denissenkov & Ivanov (1987) and Denissenkov (1988, 1989) rests on proton capture by the isotope Ne22, supposed to be overabundant. However, it is not clear why Ne22 should be overabundant, whether it is present initially, or whether it results from N-burning in the helium core.

Red supergiants of type G-K Ib exhibit some sodium overabundances, but less pronounced than in F supergiants (Lambert 1992). For red supergiants, the presence of CNO processed elements, as a result of dredge-up in the deep convective envelopes, is both expected and observed (Lambert et al 1984, Harris & Lambert 1984). Comparisons show a general agreement (Maeder 1987a), with possible indications that some extra mixing may be needed.

3.5.2 THE BLUE HERTZSPRUNG GAP     There are many more stars outside of the MS band than predicted (Meylan & Maeder 1982). The problem is particularly serious in the SMC and LMC. In the Milky Way, excesses of A-type supergiants have also been suggested (Stothers & Chin 1977, Chiosi et al 1978). The observed and theoretical numbers can be brought into agreement if the MS phase would also include the B- and A-type supergiant stages. This discrepancy is related to the problem of the so-called blue Hertzsprung gap (BHG), which is predicted by most stellar models to occur at the end of the MS and is not observed. Instead, the true star distribution appears continuous from the MS to the A-type supergiants (Nasi & Forieri 1990, Fitzpatrick & Garmany 1990, Chiosi et al 1992b).

Various explanations have been proposed for the lack of a BHG. Opacity effects may produce a "paunch" on the MS as discussed above (Section 3.3), but with present opacities (Iglesias et al 1992) and mass loss rates, the paunch occurs at luminosities too high to account for the observations (Schaller et al 1992). Extended atmospheres and the role of binaries (Tuchmann & Wheeler 1989, 1990) have also been advocated as explanations. Mixing reduces, but does not suppress, the gap (Langer 1991c). The temperature scale may also be a problem given the photometric nature of most of the observations of stars. A gap between Teff = 35,000 K and 20,000 K corresponds only to a difference of 0.04 in (B - V) color, which is quite small and may be blurred by other effects. The adjustment of individual isochrones on star clusters (Meynet et al 1993), together with a mapping of He and CNO abundances and the use of (U - B) colors, may eventually inform us as to the reality of the gap problem and the exact status of blue supergiants.

3.5.3 THE SUPERGIANT DISTRIBUTION AND THE SN 1957A PROGENITOR     A drop-off in the distribution of LMC supergiants in the HR diagram to the right of an oblique line between log Teff = 4.2 and 3.9 was noted by Fitzpatrick & Garmany (1990), and called a "ledge": A further study of 5050 LMC stars with new calibrations and reddening corrections (Gochermann 1994) shows that the "ledge" might be less significant. In data from the Galaxy it appears marginally (Blaha & Humphreys 1989). Two kinds of models are able to produce high numbers of blue supergiants and to produce a "ledge" or at least a marked decrease in the star distribution in the HR diagram: (a) models with low mass loss, (b) models with blue loops.

Models with low mass loss (Brunish & Truran 1982a, b: Schaller et al 1992) predict that most of the He core burning phase is spent in the blue supergiant phase directly after the MS. This may give a ledge; however such models do not provide red supergiants, in disagreement with observations in the LMC and SMC. The blue location of models with low mass loss is due to the large intermediate convective zone which homogenizes a part of the star (Stothers & Chin 1979, Maeder 1981). Mass loss, even if small, reduces this zone and favors the redward motion in the HR diagram, and thus the star becomes a red supergiant early during the He phase. However, the uncertainties about mass loss are critical. As an example, 25 Msun models at Z = 0.008 with typical Mdot (Schaerer et al 1993a) spend most of their He-burning phase in the blue with log Teff between 4.3 and 3.9. An enhancement of Mdot by a factor of 2 leads to a red location of the whole He-burning phase (Meynet et al 1994). Thus, as long as the Mdot rates are imprecise, it may be difficult to derive conclusions about semiconvection, diffusion, and rotational mixing from the distribution of supergiants.

Models with blue loops also enhance the number of blue supergiants, and are simultaneously able to account for some He and N enrichments in blue supergiants, but often not as much as required by the observations (Section 3.5.1). Models with Schwarzschild's criterion and overshooting at Z leq 0.008 have well-developed blue phases at all masses (Schaerer et al 1993a). This is also the case for models with the Ledoux criterion for Z < 0.004 (Brocato & Castellani 1993) and for models with semiconvection by Arnett (1991) at Z < 0.007, which well reproduce the numbers of blue and red supergiants in the LMC and lead to a blue location of the supernova progenitor at 20 Msun, as must be the case for SN 1987A (Arnett 1991, Langer 1991c). The physical connection leading to the blue progenitor is most interesting (Langer 1991a, c) and illustrates a general stellar property. The mild mixing reduces the He content in the He burning shell, which is therefore less efficient. At the end of the He core burning phase, when the CO core contracts, the He shell acts as a weak minor and produces only a moderate expansion in the intershell region between the He and H shells. Thus, the H shell is not extinct and it keeps very active. It acts as a strong mirror responding to the moderate intershell expansion by a strong contraction of the external envelope; therefore a blue final location results.

In conclusion, uncertainties in mass loss rates may allow many models to fit some features of the supergiant distributions. However, the situation is not fully satisfactory regarding the He and N enhancements, the BHG, and the "ledge." These features are usually not predicted at solar Z. At lower Z, most models predict blue loops and better fit the observations, but even there the agreement does not seem complete for either the BHG, or the He or N abundances. It is essential that the models reproduce the observations at all metallicities and this is not yet achieved.

3.5.4 THE RATIO OF BLUE TO RED SUPERGIANTS (B/R)     The B/R of supergiants was among the first stellar properties to be shown to vary through galaxies (van den Bergh 1968, Humphreys & Davidson 1979, Humphreys 1983a, Meylan & Maeder 1983, Humphreys & McElroy 1984, Brunish et al 1986). Studies have been made in the Milky Way, the LMC, the SMC, and also in M33 (Humphreys & Sandage 1980, Freedman 1985). The B/R depends on the range of luminosities considered, being found to be slightly larger at higher luminosities. The main trend is that B/R increases steeply with Z: for Mbol between -7.5 and -8.5. B/R is up to 40 or more in inner Galactic regions and only about 4 in the SMC (Humphreys & McElroy 1984). A difference in B/R by an order of magnitude between the Galaxy and the SMC was also found on the basis of well-selected clusters (Meylan & Maeder 1982). Further studies of the young SMC cluster NGC 330 confirm the high number of red supergiants (Carney et al 1985).

Many star models are able to account for the occurrence of blue supergiants, with predicted B/R more or less in agreement with the observations (Brunish et al 1986). As already mentioned, one reason for this is the flexibility offered by the uncertain mass loss rates. Indeed, B/R may change from infinity in the case of no mass loss to about 0 for high mass loss rates. Thus, we emphasize that the real difficulty is not to account for some average observed B/R in the LMC or the Galaxy, but to account also for its change with metallicity. The models with Schwarzschild's criterion and overshooting (Schaller et al 1992, Alongi et al 1993), the models with the Ledoux criterion (Brocato & Castellani 1993), and the models with semiconvection (Arnett 1991), even if they are able to fit some average B/R, all appear to predict higher B/R at lower Z, in contradiction to the observations. This is a major problem, which is not solved by any of the published models.

We also point out an interesting change is the Teff of red supergiants according to the metallicity of the parent galaxy. Red supergiants are hotter at lower Z, the difference amounting to about 800 K between models at Z = 0.001 and at Z = 0.040 (Schaller et al 1992). This is consistent with observations (Humphreys 1979, Elias et al 1985) which show that red supergiants in the Milky Way have spectral types between MO and M5, while in the SMC they are between types K3 and M2. This difference in the range of the spectral types of red supergiants is mainly the result of increased opacities at higher metallicities. We recall that a significant star formation rate is also a necessary condition for the presence of red supergiants in a galaxy. As an example, the paucity of red supergiants in M31 has been assigned to the low star formation rate rather than to the metallicity (Humphreys et al 1988). However, when B/R ratios are considered in galaxies, the effects of possible differences in the SFR and IMF are quite small and mainly result from effects of Z. These various tests show just how essential the studies of galaxies with different metallicities are for stellar evolution.

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