The roAp stars are a special subgroup of Ap stars which show rapid light variations. Since their discovery by Kurtz (1982), their number has grown to about 30. They are cold Ap stars () having anomalous surface element abundances of Sr, Cr and Eu. With a mass of about 2 , they lie in the HR diagram at the intersection between the -Scuti instability strip and the main sequence. One important particularity of these stars is the presence of a strong magnetic field of about kilogauss amplitude, discovered first by Babcock (1947). It has roughly a dipolar geometry, see Borra et al. (1982).
From the asteroseismological point of view, these roAp stars constitute a specific class of variable stars: they oscillate with high frequencies, say with large radial orders from 10 up to about 30. This represents a range of periods from 6 to 14 minutes, similar to the 5-minute oscillations of the Sun, but with much higher amplitude light variations (about 1 mmag). Recent discoveries by Martinez (1999) and Martinez et al. (1999) show possible long periods of 30 and 29 min, in two roAp stars, HD 75425 and HD 13038 respectively. However, as in most cases, roAp stars have rapid light variations, we assume, in this paper, high frequencies for oscillations. Another particularity is the geometry of these oscillations, which are aligned with the magnetic axis. They appear essentially as dipole modes, characterized by a spherical harmonic whose degree is and an azimuthal degree , say with the same geometry as the magnetic field. Reviews on roAp stars are available in Kurtz (1990) and Matthews (1991). Recent theoretical works about roAp stars are discussed in Cunha (1998).
Any seismological study of roAp stars must take into account the presence of this strong magnetic field. A lot of studies have been done to explain the seismological observations of these objects. However, they have been limited to the linear and adiabatic approximations. Even in this case, the difficulties due to the magnetic field remain numerous. Roberts & Soward (1983) developed analytical solutions of the problem but for polytropic star models and for a weak magnetic field ( kG), whose angular dependence has been neglected. They introduced the concept of the magnetic boundary layer and the possibility of Alfvénic wave production as a source of damping for p-modes. Campbell & Papaloizou (1986) considered numerically the general problem, taking into account the angular dependence of the Lorentz force, still in the case of polytropic models. However, they solved the problem for each colatitude. The global problem was first treated by Dziembowski & Goode (1996), who considered non-radial axisymmetric stellar oscillations, with strong magnetic fields still in the case of the magnetic boundary layer concept. They have shown that the frequencies are shifted by the magnetic field in a range of 10-20 , for the real part of the frequency, and for the imaginary part due to Alfvénic waves. They also showed that the modes of oscillations are not pure single harmonics but a mixing of them due to the angular dependence of the Lorentz force inside the magnetic layer. These results are consistent with the observations of HR 3831, whose frequency spectrum has been described by Kurtz (1992) as a sum of spherical harmonics of low degrees. Recently Cunha & Gough (1998) presented an alternative approach based on degenerate perturbation theory.
The aim of the present paper is to generalize the work of Dziembowski & Goode (1996) to non-axisymmetric oscillations, i.e. with the possibility of azimuthal dependence of modes and for magnetic fields up to 1.5 kG. We still use the same assumptions: we neglected non-adiabatic processes, like excitation, radiative damping, coupling with convection, and the effect of rotation.
© European Southern Observatory (ESO) 2000
Online publication: March 28, 2000