[1 rn

If a = 0, this ratio is equal to 1. It can also be expressed in terms of v/vcrit1, the ratio of the rotational velocity v to the critical velocity (4.34), since we have 1fip ~ — over a large range of values (see 4.38 and following remarks). For

2nGpm 9 vcrit,1

a star with a small Eddington factor r, it simplifies to

This equation shows that the effects of rotation on the M rates remain moderate in general. However, for stars close to the Eddington limit, rotation may drastically increase the mass loss rates, in particular for low values of a, i.e., for stars with log Teff < 4.30. In cases where r > 0.639, a moderate rotation may make the denominator of (14.45) to vanish, indicating large mass loss.

Table 14.2 shows some numerical results based on (14.44) for different initial stellar masses at the end of the Main Sequence (MS) phase in the Geneva models at Z =0.02 [513]. The values are given for the empirical force multipliers a [303], which span a large range of values as mentioned in Sect. 14.2.1. The ratio M(Q)/M(0) has a maximum value of about 1.60 for hot stars with a small r. The amplification of the mass loss rates is larger for stars with higher r and/or lower Teff. The indication "crit" means that the combination of the radiation pressure and rotation makes the surface layers unbound before the usual critical velocity ucrit1 (4.34) is reached. A very high mass loss is likely to result, determined by the evolution of M, R and rotation velocity. For r > 0.639, the critical velocity ucritj2 applies (Fig. 4.3). This is the regime of the Qr limit, which concerns objects like the LBV stars.

Table 14.2 r at the end of the MS for various initial masses and ratios M(Q)/M(0) of the M rates for a star at break-up rotation to that of a non-rotating star of the same mass and luminosity at log Teff > 4.35, at log Teff = 4.30, 4.00 and 3.90. The empirical force multipliers a by Lamers et al. ([303]) are used

Table 14.2 r at the end of the MS for various initial masses and ratios M(Q)/M(0) of the M rates for a star at break-up rotation to that of a non-rotating star of the same mass and luminosity at log Teff > 4.35, at log Teff = 4.30, 4.00 and 3.90. The empirical force multipliers a by Lamers et al. ([303]) are used

a = 0.15

120

0.903

crit

crit

crit

crit

85

0.691

crit

crit

crit

crit

60

0.527

4.00

101.8

1196

3731

40

0.356

2.26

14.4

58.5

112.1

25

0.214

1.86

7.43

21.3

34.5

20

0.156

1.77

6.21

16.1

25.0

15

0.097

1.69

5.33

12.8

19.1

12

0.063

1.66

4.95

11.4

16.7

9

0.034

1.63

4.67

10.4

15.0

The anisotropies are likely to play a great role in the evolution of rotating stars. They are beautifully confirmed by observations. Polar mass loss allows the fast rotating hot stars to lose lots of mass without losing too much angular momentum, thus these stars keep high rotation velocities. This effect has been advocated to explain how the precursors of gamma-ray bursts may have lost a lot of mass and at the same time kept a very high rotation [413].

Part IV

Acoustic and Gravity Waves. Helio- and

Asteroseismology

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