Axisymmetric p-mode pulsations of stars with dipole magnetic fields

The effect of a dipole magnetic field on adiabatic axisymmetric non-radial p-mode pulsations is studied numerically. The angular dependence of pulsation, which cannot be represented by a single spherical harmonic in the presence of a magnetic field, is expanded into a series of spherical harmonics with different degrees l. The presence of a magnetic field not only shifts the pulsation frequency, the pulsations are also damped due to the generation of magnetic slow waves. In agreement with the results of Cunha & Gough, who used a different approach from ours, we find that the effect of a magnetic field on the intermediate-to-high-order p-modes is not monotonic but cyclic with respect to the pulsation frequency and the magnetic field strength. The damping rate of a high-order p-mode becomes very small at about 3 kG and 8 kG; the corresponding field strengths are higher for lower overtones. The diminished magnetic damping is favourable for the corresponding modes, if they are excited by the classical kappa-mechanism, to survive even in the presence of a strong magnetic field. This picture could explain the mode selection as observed in the rapidly oscillating Ap stars. For a low-order p-mode, the damping rate increases as the strength of the magnetic field increases. We find that in the presence of a magnetic field of a few kG, magnetic damping seems to exceed the driving owing to the kappa-mechanism of oscillations representative of delta Scuti variability. This may explain why delta Scuti-type oscillations are unlikely to be seen in magnetic Ap stars. The amplitude of a mainly dipole (or quadrupole) mode is strongly confined to the magnetic axis in the outer layers. Furthermore, horizontal motion can be comparable to radial motion even for high-order p-modes. We discuss the influence of the magnetic distortion of the eigenfunction on the pulsation amplitude modulation with respect to the rotation phase.