Distribution of kinetic energies of monotonically unstable modes 58

unsta-ble modes, can be derived from the corresponding eigenfunctions. It is shown in Figs.

5.5and5.6for three ZAMS models with metallicity Z = 0.03. As the work integral is not defined for non oscillatory modes, the distribution of the kinetic energy might provide

Chapter 5. Monotonically unstable modes in main sequence stars 59

FIGURE5.4: Real and imaginary parts of the eigenfrequencies (normal-ized by the global free fall time) of unstable modes associated with the monotonically unstable modes mu1and mu2as a function of harmonic degree (l) for three ZAMS models with metallicity Z = 0.03. Note the mode pairing of the monotonically unstable modes at a certain mass de-pendent value of the harmonic degree providing an unstable oscillatory

mode for highl.

0

FIGURE5.5: Normalized kinetic energy as a function of relative radius of the two monotonically unstable modes mu1 and mu2withl = 2for

two ZAMS models having the metallicity Z = 0.03.

0

some information on the excitation of the monotonically unstable modes and on their domain of existence. Figs. 5.5and5.6 show that a maximum of the kinetic energy is found at the bottom of the convection zone associated with the Fe-opacity bump. For the low mass model with 10 M, a second maximum appears between the Fe-opacity convection zone and He II convection zone. This is for the mode mu2, contrary to mode mu1, the position of the absolute maximum of the kinetic energy. Secondary, less pronounced maxima may appear close to the photosphere (see Figs.5.5and5.6).

5.5 Discussion and conclusions

The monotonically unstable modes identified in this study are present both for radial as well as non-radial perturbations. Hence their existence is independent of the ge-ometry of the perturbations. The present analysis is based on ZAMS models but we suspect that the monotonically unstable modes are also present in later stages of stel-lar evolution. Their growth rates are higher for non-radial perturbations compared to radial perturbations and their presence in models with 10M suggests that they can affectβ- Cepheid type stars close to the main sequence. The distribution of the kinetic

Chapter 5. Monotonically unstable modes in main sequence stars 61

0 0.2 0.4 0.6 0.8 1 1.2

0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

Lrad/LEdd

Relative Radius 10 M

50 M

100 M

FIGURE 5.7: Ratio of the radiative and the Eddington luminosity as a function of relative radius for three ZAMS models with metallicity Z =

0.03.

energy for these modes exhibits a local maximum at the bottom of the Fe-opacity con-vection zone. A second local maximum of the kinetic energy is found for 10 Mmodels close to the stellar surface. Our findings concerning the kinetic energy of the monoton-ically unstable modes for the massive ZAMS models are similar to those ofSaio(2011).

However, in the models studied bySaio(2011), the bottom of the Fe-opacity convection zone is situated deeper within the stellar envelope and the models studied by him have a more pronounced core envelope structure.

The dependence on the harmonic degree l of the monotonically unstable modes has been studied by following them up tol = 1000where an interesting phenomenon has been discovered. When increasingl, two monotonically unstable modes merge at a critical value of l(l ≈480 for 10 M,l ≈150 for 50 M andl ≈3 for 100 M, see Fig. 5.4) and form an unstable oscillatory mode with a frequency in the g-mode range (σr<1).

Saio(2011) suggested that the origin of his radial monotonically unstable modes is due to the radiative luminosity being close to the Eddington luminosity (Lrad/LEdd≈ 1) in a large fraction of the stellar envelope. The latter is also true for radial monoton-ically unstable modes in our main sequence models. However, Lrad/LEdd ≈1 is not a necessary condition for the occurrence of non-radial monotonically unstable modes:

Fig. 5.7 shows the ratio of the radiative to the Eddington luminosity (Lrad/LEdd) as a function of the normalized radius in the envelopes of ZAMS models with 10, 50 and 100 M. Although for the entire envelope of the 10 M model we have Lrad/LEdd <

0.3 nonradial monotonically unstable modes can be identified here. Thus the nonra-dial monotonically unstable modes cannot be related to the existence of the HD limit (Humphreys & Davidson,1979). In this respect we note thatSaio(2011) claims that the occurrence of the radial monotonically unstable modes studied by him coincides with the HD limit and leads to optically thick winds. We emphasize that this conclusion is premature, since the final result of an instability can only be determined by nonlinear calculations and never by a linear analysis.

Chapter 6

Summary and future work

6.1 Summary

Linear stability analyses have been performed for models of massive main sequence stars and models of the B-type supergiant 55 Cygni (HD 198478). The instabilities iden-tified have been followed into the nonlinear regime with the help of a time implicit fully conservative scheme. The final velocity amplitudes are in any case comparable with the escape velocity of the corresponding models. For selected models pulsation velocities exceed the escape velocity which is taken as direct evidence for mass loss.

The nonlinear simulations provide an estimate for the mass loss rate which appears to be consistent with observations.Saio(2011) reported on the presence of monotonically unstable modes in models of massive stars. The properties of these modes have been investigated systematically for main sequence models and their existence even in the nonradial spectrum has been established. The three main issues of this thesis may be summarized in more detail as follows: