The neffect

When the metallicity increases, ^ increases. This tends to make the star more luminous according to the mass-luminosity relation, provided that the opacity does not increase too much. At very high metallicity (Z > ~0.04), this is what happens for the Mq and 3M0 models. For the 2OM0-stellar-mass model, the ¡-effect is also present, as can be seen from Figure 36.1 (left-hand part). The luminosity already begins to increase as a function of Z at Z ~ 0.02 since the k-effect on L is negligible.

1.4 The snuc-effeet

The nuclear energy production snuc sustains the stellar luminosity. If snuc is arbitrarily increased, the central regions of the star expand, leading to decreases in the temperatures and densities, and an increase in the stellar radius. The temperature gradient therefore decreases, leading to decreases both of the luminosity and of the effective temperature.

The way snuc depends on the metallicity is closely related to the mode of nuclear burning. When the CNO cycles are the main mode of H-burning, snuc increases with Z. In contrast, when the main mode of burning is the pp chain, snuc is related to X. It is thus about independent of Z at Z < 0.02, and decreases with increasing Z at Z > 0.02. These properties determine the behavior of Tc and pc in MS stars as a function of Z.

Usually, the CNO cycles are the main mode of core H-burning for stars more massive than about (1.15-1.30)M0 (depending on metallicity), whereas the pp chain operates at lower masses. At Z = 0.1, however, all stars are found to burn their hydrogen through the CNO cycles, at least down to M = 0.8M0 (Mowlavi etal. 1998a).

When the CNO cycle is the main mode of burning, pc and Tc decrease with increasing Z (see above). This is clearly visible in the left-hand part of Fig. 36.2 for the 3M0 and 20M0 stars at Z < 0.04. At Z = 0.1, however, the higher surface luminosities of the models relative to those at Z = 0.04 result in concomitantly higher central temperatures.

When the pp chain is the main mode of burning, on the other hand, the location in the (log pc, log Tc) plane is not very sensitive to Z. This is well verified for the mass-M0 models at Z < 0.1.

The combined effects of k, and snuc on the stellar surface properties can be estimated more quantitatively with a semi-analytical approach using homology relations. This is developed in the appendix of Mowlavi et al. (1998b). We just recall below that a variation A¡i of the mean molecular mass of a star of mass M affects its position in the HR diagram according to i 7.31 11.021 10.021

Alog L = i7.4 Alog ^ "11.08 } Alog K0 " (0.08 } Alog ^ (U)

Alog Teff = \\ Alog x - j 0 35 J Alog K0 - j 0 10 J Alog €0, (1.2)

/

20 M

/ Jfm 0.001

-

X / W 0.004 -

* 0.008 _

0.10 0.02 _^

0.04

CO

1 2 log Pc

120 85

120 85

25 20 15 12

Figure 36.2. Left panel: the same as the left-hand part of Figure 36.1 but for the (log pc, log Tc) diagram. Right panel: masses of convective cores relative to the stellar masses as labeled (in Solar masses) next to the curves, as a function of metallicity. Figures taken from Mowlavi et al. (1998b).

where e 0 is the temperature- and density-independent coefficient in the relation for the nuclear energy production snuc = e0pxTv (Cox and Giuli 1969, p. 692), and k0 is the opacity coefficient in Kramers' law k = k0pT -3 5. The numbers on the first line in Equations (1.1) and (1.2) apply when the nuclear energy production results from the pp chain, whereas the second line applies when the CNO cycles provide the main source of nuclear energy. Variation of u is related to variation of Z by

Bl ln 10

The mass of the convective core is mainly determined by the nuclear energy production and by L. At Z < 0.04, it slightly decreases with increasing Z in low-and intermediate-mass stars, whereas it remains approximately constant in massive stars (see the right-hand part of Figure 36.2). At Z > 0.04, on the other hand, it increases with Z at all stellar masses.1 We recall furthermore that at Z = 0.1 all stars with M as low as 0.8MQ possess convective cores.

The MS lifetimes for several stellar masses as a function of metallicity are summarized in the left-hand part of Figure 36.3. They can be understood in terms of two factors: the initial H abundance, which determines the quantity of available fuel; and the luminosity of the star, which fixes the rate at which this fuel burns.

1 It is interesting to note that we could have expected smaller convective cores in metal-rich massive stars, for which the main source of opacity is electron scattering (since this opacity is positively correlated with the H content), but the effect of higher luminosity in those stars overcomes that k-effect, and the core mass increases with increasing Z.

^ 0.008. . - • .

" 0^02. - - -•

0.10 '^N.

Figure 36.3. Left panel: main-sequence (MS) lifetimes of models of initial masses as labeled on the curves, as a function of their metallicity. The lifetimes are normalized for each stellar mass to their values at Z = 0.02. Right panel: stellar masses at the end of the MS phase as a function of initial mass for various metallicities as labeled on the curves. Dotted lines correspond to models computed with moderate mass-loss rates, while thick lines correspond to models computed with twice those mass-loss rates. See Mowlavi et al. (1998b) for the references to the models used. Figures taken from Mowlavi et al. (1998b).

At Z < 0.02, tH is mainly determined by L, being shorter at higher luminosities. The left-hand part of Figure 36.3 confirms, as expected from L (see the left-hand part of Figure 36.1), that tH increases with Z for low- and intermediate-mass stars, and is about independent of Z for massive stars.

At Z > 0.02, on the other hand, the initial H abundance decreases sharply with increasing Z (the H-depletion law is dictated by the adopted AY/AZ law; for AY/AZ = 2.4 and Y0 = 0.24, X drops from 0.69 at Z = 0.02 to 0.42 at Z = 0.1). The MS lifetimes are then mainly dictated by the amount of fuel available. This, combined with the higher luminosities at Z = 0.1, leads to MS lifetimes that are about 60% shorter at Z = 0.1 than at Z = 0.02. This result is independent of the stellar mass for M < 40M0. Above this mass, however, the action of mass loss (see below) extends the MS lifetime, as can be seen from the 60MQ curve in the left-hand part of Figure 36.3.

1.5 The effect of M

When mass loss is driven by radiation, Kudritzki et al. (1987), see also Vink et al. (2001,) showed that M in O stars is proportional to Z 0-5-0-8. As a result, the effects of mass loss dominate in more metal-rich stars. The stellar masses remaining at the end of the MS phase for various mass-loss-rate prescriptions are shown in the

log Teff Age (106 years)

Figure 36.4. Left panel: evolution during the MS phase of the N/C number ratios at the surface of rotating stellar models as a function of the effective temperature. The differences in N/C ratios are given with respect to the initial values. Right panel: evolution of the surface equatorial velocity as a function of time for 60M© stars with uinit = 300 km s-1 for various initial metallicities.

log Teff Age (106 years)

Figure 36.4. Left panel: evolution during the MS phase of the N/C number ratios at the surface of rotating stellar models as a function of the effective temperature. The differences in N/C ratios are given with respect to the initial values. Right panel: evolution of the surface equatorial velocity as a function of time for 60M© stars with uinit = 300 km s-1 for various initial metallicities.

right-hand part of Figure 36.3. The most striking result at metallicities higher than twice Solar is the rapid evaporation of massive stars with M > 50M©, and the consequent formation of Wolf-Rayet (WR) stars during core H-burning.

1.6 The effects of rotation

The effects of shellular rotation on the evolution of massive-star models at high metallicity have been discussed in Meynet & Maeder (2005). As at lower metallicity, rotation induces mixing in the stellar interiors. However, as was discussed in Maeder & Meynet (2001), rotational mixing tends to be less efficient in metal-rich stars. This can be seen by looking at the 9M© stellar model in the left-hand part of Figure 36.4. This comes from the fact that the gradients of Q are much less steep in the higher-metallicity models, so they trigger less-efficient shear mixing. The Q gradients are shallower because less angular momentum is transported outwards by meridional currents, whose velocity scales as the inverse of the density in the outer layers; see the Gratton-Opik term in the expression for the meridional velocity in Maeder & Zahn (1998). On looking at the 40M© stellar model in the left-hand part of Figure 36.4, one sees that the higher-metallicity model presents the highest surface enrichments, in striking contrast with the behavior of the 9M© model. This comes from the fact that the changes occurring at the surface of the 40M© star are due not only to rotation but also to mass loss, which is more efficient at higher Z.

In the high-mass range at high metallicity, the stars have less chance to reach the critical velocity during the MS phase, as can be seen from the right-hand part of Figure 36.4. This is due to the fact that stellar winds remove angular momentum at the surface, thus preventing the outer layers from reaching the critical limit. Let us note that, for smaller initial masses, a high metallicity may favor the approach to the critical limit. Indeed, in that case, the stellar winds are too weak, even at high metallicity, to remove a lot of mass and therefore of angular momentum, while the meridional currents, which bring angular momentum from the inner regions of the star to the surface, are more rapid due to the lower densities achieved in the outer layers of metal-rich stars. Another point that should be kept in mind at this point is that, at very high metallicity, the stars may lose large amounts of mass without losing too much angular momentum. This is due to the fact that, in the high-velocity regime, the stellar winds are polar (Maeder & Meynet 2001). Let us stress, however, that this situation has little chance to be realized at high metallicity. Indeed, the timescales for mass loss are likely shorter than the timescale for the torque due to wind anisotropy to affect the surface velocity.

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