It has been known since the first classification scheme proposed by Cowling (1941) and later developed by Scuflaire (1974) and Osaki (1975), that the classification of modes for relatively evolved stars is ambiguous. This problem in the classification arises from the fact that the complete fourth-order system of stellar nonradial adiabatic oscillations has been obtained by a simple extension of the classification scheme proposed by Cowling (1941) for a second-order system, obtained by neglecting the Eulerian perturbation of the gravitational potential in the full problem. Furthermore, any classification scheme depends on the wave function choosen to determine the eigenstates of the oscillatory system and on its number of zeros.
In this work, we have addressed the problem determining a unique classification scheme for the equation of motion, independent of the eigenfunction used to characterize the oscillatory motion and its number of zeros.
The principle consists in determining two self-adjoint forms of the equation of motion, build by using the propagation diagram which is associated with the different types of stellar perturbations, that overcome from the restoring force due to the competition between the pressure and the gravity. All the properties of the eigenmodes depend only on the topological properties of the propagation diagram. For each of the self-adjoint forms of the equation of motion, a phase representation of the eigenmode is made which consists in describing the oscillatory motion of adiabatic oscillations by a system of two first-order nonlinear differential equations for the phase and the amplitude. The equations are coupled only in the amplitude equation, leaving a single phase equation to determine the eigenfrequencies. A convenient phase representation is determined for each type of propagation, corresponding to each case, whether is the pressure or the gravity that dominates in the restoring force. This competition between the pressure and gravity, can generate 3 types of waves for which the local properties can be defined on a propagation diagram together with the respective linear phase diagram, for which the linear phase properties are distinct from each other. Furthermore, the linear phase diagram proposed, constitutes a powerful method to determine an unique radial order number n. Similarly to the classical methods, the sign of order number n determines the dominant nature of the wave.
A possible atempt to classified the eigenvalues of the linear adiabatic nonradial oscillation system (of the fourth-order problem of nonradial adiabatic oscillations), is to determine the radial order n of the eigenmode by using the phase equations Eqs. (36) and (37) with the respective boundary conditions given by Eqs. (44) and (47) or (45) and (48). In that way, this technique can be used to constrain the linear phase diagram, which can determine the unique radial order number n. However, it is expected that this technique will not work in all the cases, once the second-order system have been obtained by an approximative method. A more extensive study must be made to identify those cases that this porcedure can be applied.
Finally, the classical scheme proposed by Scuflaire (1974) and Osaki (1975) corresponds to a particular case of this general scheme, for which the order number n is equal to the algebraic counting of the number of modes in the different propagative regions, with a positive sign for modes in the acoustic propagative region and a negative sign for modes in the gravity propagative region.
Acknowledgements
Ilídio Lopes thanks Douglas Gough for stimulating discussions on the oscillatory properties of acoustic waves and stellar modelling. He also acknowledges support of a grant from the Particle Physics and Astronomy Research Council (UK). Ilídio Lopes would like to thank the referee J. Provost for the careful reading of the paper as well the valuable comments that have allow me to improve the original manuscript.
Copyright ESO 2001