3.5.1 75 M
According to the linear stability analysis the ZAMS model with 75 Mhas two unstable modes with periods of 3.25 hours (σr = 2.65;σi = −0.094) and 3.20 hours (σr = 2.69;
σi =−0.012). The evolution of the instabilities into the nonlinear regime is shown for this model in Fig. 3.3. Without any external perturbation the instability develops from numerical noise (of the order of 10−6cm/s in terms of the velocity), undergoes the lin-ear phase of exponential growth and finally saturates in the nonlinlin-ear regime. In the linear phase, periods and growth rates may be compared with the results of the entirely independent linear stability analysis. Differences between the periods and the growth rates amount to less than one per cent. In the nonlinear regime, the velocity amplitude reaches values up to 107 cm/s thus remaining well below the escape velocity of the model. In the regime of stationary finite amplitude pulsations, the temperature at the outermost grid point varies between 19000 K and 31000 K. We note that the position of the outermost grid point does not necessarily coincide with the photosphere. There-fore its temperature in general differs from the effective temperature. Together with temperature and density the variation of the bolometric magnitude exhibits a period of 0.14 day (≈3.36 hours) in the finite amplitude pulsation regime. Thus compared to the linear phase periods have changed in the nonlinear regime by at most 10 per cent.
As discussed in the previous section, the correct treatment of the energy balance is essential. Fig. 3.3illustrates that the dominant gravitational potential and internal energies exceed the kinetic energy and the time integrated acoustic flux by four orders of magnitude. This difference poses a challenging task for the numerical treatment. To prove that the scheme adopted satisfies the accuracy requirements and the simulations provide reliable results rather than numerical artifacts, the error in the energy balance is shown in Fig. 3.3(i). It amounts to a fraction of 10−8 of the gravitational potential
0.89
23.47 23.57 23.67 23.77 23.87
Temperature[K]
23.47 23.57 23.67 23.77 23.87
Density[10−10g/cm3]
23.47 23.57 23.67 23.77 23.87
∆Mbol
23.47 23.57 23.67 23.77 23.87
Ekinetic[1040erg]
23.47 23.57 23.67 23.77 23.87 E[1040erg]
FIGURE3.3: Evolution of instabilities and finite amplitude pulsations of a ZAMS model with 75 M: Radius (a), velocity (b), temperature (c) and density (d) at the outermost grid point, the variation of the bolometric magnitude (e), the time integral of the acoustic luminosity (f), the kinetic energy (g), the gravitational potential and internal energies (h) and the
error of the energy balance (i) as a function of time.
and the thermal energies (Fig. 3.3.h) and to a fraction of 10−4 of the kinetic energy and the time integrated acoustic flux. The time integrated acoustic luminosity is not a monotonic function (see Fig. 3.3.f) indicating that in each pulsation cycle the acoustic flux changes its sign. However, its mean over a pulsation cycle is positive implying that on average an outward acoustic flux is generated by the system. From Fig. 3.3.f we deduce that in the regime of stationary finite amplitude pulsations (after 18 days) the mean slope of the time integrated acoustic flux is constant. It corresponds to the mean acoustic luminosity of the pulsating star which is used to estimate the mass loss rate associated with a pulsationally driven wind in the way described in the previous section. Thus we obtain a mass loss rate of 0.59 ×10−7 M/yr (see alsoGrott et al., 2005).
3.5.2 90 M
For the 90 M model, the linear stability analysis provides one low order unstable radial acoustic mode with a period of 4.38 hours (σr = 2.19; σi = −0.13). The results of our nonlinear simulations for this model are presented in Fig. 3.4. From Fig. 3.4 we deduce that even quantitatively they are similar to those discussed in the previous sections for the 75 Mmodel. Thus the discussion given there also refers to the 90 M
Chapter 3. Massive main sequence stars 37
23.35 23.5 23.65 23.8 23.95
Temperature[K]
23.35 23.5 23.65 23.8 23.95
Density[10−10g/cm3]
23.35 23.5 23.65 23.8 23.95
∆Mbol
23.35 23.5 23.65 23.8 23.95
Ekinetic[1040erg]
23.35 23.5 23.65 23.8 23.95
E[1040erg]
Time (days) (h)
Epotential Einternal -0.0006
-0.0004
model except for the finite amplitude pulsation period and the mass loss rate which now amount to 0.19 day (4.56 hours) and 0.69×10−7M/yr, respectively.
3.5.3 150 M
Similar to the 75 Mmodel, this model exhibits two unstable low order acoustic modes with periods of 11.5 hours (σr= 1.33, σi =−0.288) and 15.9 hours (σr= 0.96, σi=
−0.17×10−5), respectively. The results obtained by following the instabilities into the nonlinear regime are shown in Fig. 3.5. Compared to the previously discussed models with 75 and 90 Mthey are qualitatively similar and the discussions presented there also refer to the 150 Mmodel except for the finite amplitude pulsation period and the mass loss rate which now amount to 0.99 day (23.76 hours) and 0.25×10−7M/yr, re-spectively. Contrary to the 75 and 90 Mmodels, the finite amplitude pulsation period of 23.76 hours is substantially longer than the initial pulsation period of 11.5 hours in the linear regime of the evolution of the instability (see table3.2). The difference is a nonlinear effect and caused by a considerable inflation of the envelope by the instabil-ity. This can be deduced from Fig.3.5(a) where the radius is found to increase from its initial hydrostatic value of1.65×1012cm to a mean value of2×1012 cm in the non-linear regime. An increased radius implies a lower mean density and, according to the period-density relation, a longer period.
1.5
(h) Epotential Einternal
-6
Thus, for very massive ZAMS stars the final finite amplitude pulsation periods may significantly differ from periods determined by a linear analysis. As a consequence, observed periods should not be compared to the latter but to periods determined by
Chapter 3. Massive main sequence stars 39 TABLE3.2: Pulsation periods and mass loss rates for the ZAMS models
selected.
Mass (M)
Linear Period (hours)
Nonlinear Period (hours)
Mass Loss Rate (M/yr)
75 3.25 3.36 0.59×10−7
90 4.38 4.56 0.69×10−7
150 11.50 23.76 0.25×10−7
nonlinear simulations. These findings are in agreement with previous investigations of models for 55 Cygni (Yadav & Glatzel,2016). These authors find substantial differ-ences between linear and nonlinear periods and emphasize that only nonlinear peri-ods should be compared with observations, in particular for strong instabilities such as strange mode instabilities.