5. Diffusion and the mechanism
To determine the frequencies of modes of oscillation for a star requires only that we solve the adiabatic equations. Solving the full nonadiabatic equations of stellar oscillation allows us to calculate the growth rates of the modes, and hence to determine which of the modes are overstable; also, by considering the work integral we can investigate the contributions of the different parts of the star to the excitation and damping of the mode.
The nonadiabatic oscillation package used was generously provided to us by W. Dziembowski and follows the procedure first described by Dziembowski (1977). We are mainly concerned here with excitation via the mechanism on which abundance variations have a direct impact. We note, however, that the present calculations lack a good modeling of the effect of convection; this must be kept in mind in the analysis of the results.
The physics of the mechanism has been reviewed extensively (e.g. Cox 1980; Unno et al. 1979; Gautschy & Saio 1995). As a quick reminder, generating pulsations in a star requires that the energy gained by an oscillation mode over a complete cycle be larger than the energy lost. We are then looking for a positive net work over the entire star over one cycle. In the case of the mechanism, the energy is transferred from the outward radiation flux to the oscillation mode via the opacity. A mode becomes overstable by this mechanism if the opacity profile and its derivatives have the right features.
Following Unno et al. (1979), from the definition of the work (W) as the variation of the kinetic energy (E) over a cycle,
one can write
where denotes Lagrangian perturbations. Also, is the (angular) oscillation frequency, T is temperature, is the mass interior to the radius r, is the nuclear energy generation rate, and and are the radiative and convective fluxes. If one neglects the contribution from the nuclear () and convective terms (), and only keeps the perturbation of the radiative flux (), one can isolate the contribution of the mechanism to the driving of a given mode of oscillation. To obtain a simple estimate of this contribution to the work integral we make the quasi-adiabatic approximation (i.e. , evaluate the work integral by means of adiabatic eigenfunctions), and furthermore assume that the adiabatic thermodynamic derivatives , and are constant. Then the work done by the mechanism is proportional to
contribute to the excitation. Local increases in the logarithmic derivatives of are necessary and a decrease in in partial ionization zones of a dominant species (H or He) is helpful. It also follows that regions where the gradients of and are negative contribute to damping of the pulsation. The term usually dominates over the other term.
The numerical results reported in the text, including the growth parameters and work integrals, are computed using the full nonadiabatic procedure of the Dziembowski code.
In order that the excitation by the mechanism should not be cancelled by damping elsewhere, it is necessary that the driving region lie in the so-called transition zone between the quasi-adiabatic and nonadiabatic regimes; in that case, the oscillations are strongly nonadiabatic outside the driving region, and this part of the star therefore does not contribute to the damping, giving rise to net driving. This leads to an approximate relation between the period of a given mode of pulsation and the position of the transition region in a star (Cox 1980):
here is the mass outside the transition region, is the average over that part of the star, being the specific heat, and L is the luminosity.
The normalized growth rate is defined as
In this formulation, varies from , if there is driving in the entire star, to -1, if there is damping in the entire star. The value of zero defines neutral stability.
Diffusion affects the mechanism by decreasing driving from helium in favour of driving from metals. As a consequence of Eq. (6), the pulsation period of the unstable modes depends on the depth of the driving region. During a star's evolution the helium ionization zone gradually shifts deeper in the star, thereby increasing the period of the observed pulsation modes. Additionally, as the driving in the deeper iron-peak driving region increases while the driving due to helium decreases as a result of diffusion, one might expect the observed pulsation periods to shift to even longer periods. The effect of abundance variations on the opacity profiles is discussed below for selected models (see Fig. 3).
© European Southern Observatory (ESO) 2000
Online publication: August 17, 2000