## Effects of Rotation in the MS Phase

A rotating star is distorted and radiates anisotropically: hotter and brighter at the poles and cooler and fainter at the equator (Sect. 4.2.2). There is also a shift of luminosity which amounts up to A logL = +0.5 [371] for a fast rotating star seen pole-on and to -0.15 for a star seen equator-on. The average Teff (Fig. 27.7 left) is defined by the total L and the total stellar surface. This Teff corresponds to the value at P2(cos = 0, i.e., to a colatitude $ « 54°. The difference of the average Teff is well visible for stars on the zero-age sequence, later it is masked by other effects.

Fig. 27.7 Left: evolutionary tracks of a 20 [email protected] star with Z = 0.02 with different initial velocities. Right: MS tracks in the plot of the effective gravity (including centrifugal force) vs. Teff. The dashed and continuous lines are for zero rotation and for rotating models with initial v = 300 km s-1. Observations by Herrero et al. [245, 246] are shown. From G. Meynet and the author [409]

As evolution proceeds, the increase of the central condensation produces some differential rotation, which leads to hydrodynamical effects, such as shear and horizontal turbulence (meridional circulation is also present in solid body rotating stars). The resulting mixing brings some He out of the convective core into the radiative envelope, slightly increasing its mean molecular weight. The higher resulting P and T in the interior favor a larger convective core. For a 20 M0 mass with an average v = 200 km s^1 during the MS phase, the relative core increase at the end of the MS phase is ~ 20%. This increase makes a milder ^ gradient than what overshooting would do. The larger core leads to a higher luminosity during subsequent evolution.

The He enhancement in the envelope lowers its opacity, which increases L and makes the star hotter, favoring a blueward track. Figure 27.7 (right) shows the tracks resulting from these various effects. Figure 27.8 (top) shows a global view of the tracks with and without rotation for stars with M > 9 M0. Rotation enlarges the MS width for M < 40 M0 and reduces it for larger masses. The enlargement results from the bigger He cores. For the highest masses this effect is dominated by the He enrichments. An extreme situation may occur for stars with M > 60 M0 with high initial rotation: the convective core instead of shrinking expands during the MS phase and evolves to the blue in a quasi-homogeneous evolution to WR stars (Fig. 25.15).

Rotation by increasing the core makes the stars overluminous for their masses, which introduces some scatter in the M-L relation. Figure 27.7 (right) shows the MS evolutionary tracks in the log gef vs. log Tef diagram. If a value of the mass of a rotating star is assigned from the geff and Teff of non-rotating models (broken lines), a too high value of mass is obtained. This may be part of the origin of the mass discrepancy often found between the spectroscopic determinations and the values from evolutionary tracks.

### 27.3.5 Lifetimes and Age Estimates

The MS lifetimes are generally increased by rotation, from 15% at 120 M0 to 25% at 9 M0 [411]. This is due to the larger core and to the shear diffusion which continuously brings new H from the envelope into the core. An isochrone with rotation is almost identical to an isochrone without rotation with log age smaller by 0.1 dex [Fig. 27.8 (bottom)]. This means that ignoring the effect of rotation in the estimate of cluster ages leads to ages too small by about 25%. The spread in rotation and orientation angles introduces some scatter together with the effect of unsolved binaries.

The He-burning lifetimes depend on mass loss and mixing. For a 20 M0 model at Z = 0.02, the ratio tHe/tH of the He- to the H-burning lifetimes passes from 0.11 to 0.08 for an initial rotation velocity passing from 0 to 300 km s-1. For a 12 M0, these values are 0.124 and 0.074 [411]. The effects are similar at lower Z. The

Log Td

Fig. 27.8 Top: tracks at Z = 0.02 for non-rotating stars (dotted lines) and for stars with initial velocities of 300 km (continuous lines), corresponding to a MS average v = 160-235 km . Bottom: isochrones of log(age) = 6.9 to 7.3 for zero rotation (dashed lines) and for initial v = 200 km (continuous lines). The dashed line of log(age) = 6.9 is close to the continuous line with log(age)=7.0. From G. Meynet and the author [409]

Fig. 27.8 Top: tracks at Z = 0.02 for non-rotating stars (dotted lines) and for stars with initial velocities of 300 km (continuous lines), corresponding to a MS average v = 160-235 km . Bottom: isochrones of log(age) = 6.9 to 7.3 for zero rotation (dashed lines) and for initial v = 200 km (continuous lines). The dashed line of log(age) = 6.9 is close to the continuous line with log(age)=7.0. From G. Meynet and the author [409]

reasons for the shorter tne/tH ratio are the higher luminosity and the longer MS lifetime in rotating models. However, extreme mass loss in the most massive stars (M > 85 M0) leads to a lower luminosity in the advanced stages and increases their lifetimes.

27.3.6 He-Burning: Blue and Red Supergiants at Different Z

The upper luminosity limit of the red supergiants depends on metallicity. The higher Z, the lower the cutoff of the L distribution of red supergiants: in NGC 6822 (Z = 0.005) the upper limit is around Mbol = -9.0 (M - 30 M0), while in M31 (Z = 0.036) it is around -7.5 (M - 15 M0) [382]. This is an effect of the Z dependence of the mass loss rates (in the MS as well as in the red-supergiant stages): for higher Z, M values are higher and the stars more quickly become WR stars. Below about 25 M0 at Z = 0.02, the He-burning phase is shared between the blue- and the red-supergiant stages. The stars are often close to a neutral state between a blue and a red location: minor structural changes can produce major differences of the radii. This is a limiting case of the Vogt-Russel theorem (Sect. 24.1.1).

An important question is the number ratio B/R of blue (Types B and A) to red (K andM) supergiants [311]. Figure 27.9 (left) shows the B/R ratio as a function of Z in the Milky Way and SMC. In the galactic interior, there is almost no red supergiants, this ratio is above 10; in the solar neighborhood it is about 3, and 0.6 in the SMC where there are lots of red supergiants (cf. also [384]). The red supergiants in Fig. 27.9 (left) are defined in a large spectral interval in order to encompass the changes of the mean types with Z: K5-K7 I in the SMC, M1 I in the LMC and M2 I in the Milky Way [383].

Figure 27.9 (right) shows that some bluewards shifts occur even at Z = 0.02 for certain rotational velocities. The blue shifts are influenced by mass loss, mixing and the intermediate convective zone (cf. Sect. 26.2.4). This is interesting in relation

Fig. 27.9 Left: the blue to red-supergiant ratios B/R at different Zin clusters of log(age) = 6.8-7.5. "B" includes B- and A-type supergiants, "R" the K and M supergiants. From P. Eggenberger et al. [173]. Right: evolutionary tracks for 20 Mq stars at Z = 0.02 with different initial velocities. From Hirschi et al. [251]

Fig. 27.9 Left: the blue to red-supergiant ratios B/R at different Zin clusters of log(age) = 6.8-7.5. "B" includes B- and A-type supergiants, "R" the K and M supergiants. From P. Eggenberger et al. [173]. Right: evolutionary tracks for 20 Mq stars at Z = 0.02 with different initial velocities. From Hirschi et al. [251]

Fig. 27.10 Left: evolution of the Teff as a fraction of the lifetime spent in the He-burning phase for different initial velocities. Right: distribution of He at the middle of the He-burning phase in a 20 M® model at Z = 0.004 with zero rotation (dashed-dotted line) and in a model with an initial velocity of 300 km s_1 (continuous line). From the author and G. Meynet [367]

with the B/R ratio at lower Z like in the Small Magellanic Cloud (SMC). There, the numerous red supergiants were difficult to explain for long, because models at low Z have little mass loss and they normally spend most of their He-burning phase in the blue [Fig. 27.10 (left)]. The B/R ratio decreases for higher rotation. A 20 M® rotating star spends about the half of its He-burning phase in the blue and the half in the red (this is the same at 15 M®). The physical reason rests on the change of the internal He profile [Fig. 27.10 (right)]. Without rotation, there is a large intermediate convective zone, which keeps the star on the blue side. With rotation, the mild mixing increases the amount of He near the H-burning shell, reducing its efficiency. Thus, there is no intermediate convective zone keeping the star to the blue. These results are consistent with those about semiconvection in Sect. 6.2.1.

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