Magnetic perturbations to the acoustic modes of roAp stars

Abstract

We study the effect of a large‐scale surface magnetic field on the non‐radial acoustic modes of roAp stars. Special attention is given to the use of a variational principle which is used for determining the shifts in the frequencies with relative ease, enabling us to avoid having to calculate the perturbed eigenfunctions. With knowledge of the frequency shifts we then estimate the eigenfunctions in a simpler, albeit approximate way. The results indicate frequency shifts of the order of few μHz, which depend on the order, degree and azimuthal order of the mode. The loss of energy through Alfvén waves is also estimated from the imaginary parts of the frequency shifts. The results indicate that the loss is particularly high near specific frequencies. This might indicate the presence of a selection effect, which could make some modes more likely to be excited than others. However, our results do not explain why the modes observed appear always to be aligned with the axis of the magnetic field. Finally, the estimated perturbed eigenfunctions contain strong components of spherical harmonics that differ from those of the original unperturbed modes.

Effects of Dipole Magnetic Fields on Axisymmetric p-Mode Pulsations

“…The eigenfrequency behaviour seen in Fig. 2 shows that our results for £ = 1 agree qualitatively with those obtained by Cunha & Gough (2000) using a variational principle. …”

“…To start with, we considered only axisymmetric (m = 0) modes under the adiabatic and the Cowling approximations. In contrast to previous studies of pulsationmagnetic field interaction (Dziembowski & Goode, 1996;Bigot et al, 2000;Cunha & Gough, 2000), we retained the latitudinal derivatives of the perturbed quantities.The magnetic perturbation variables oscillate rapidly in space when the magnetic pressure is small compared to the gas pressure. These rapid oscillations are caused by inwardly propagating magnetic slow waves which should be dissipated before they reach the stellar center.…”

“…To start with, we considered only axisymmetric (m = 0) modes under the adiabatic and the Cowling approximations. In contrast to previous studies of pulsationmagnetic field interaction (Dziembowski & Goode, 1996;Bigot et al, 2000;Cunha & Gough, 2000), we retained the latitudinal derivatives of the perturbed quantities.…”

Pulsation of magnetic stars

“…Three different methods have been developed to calculate oscillation frequencies taking into account the generation of the slow wave and its damping effect; the first one is by Dziembowski & Goode (1996), which is extended by Bigot et al (2000); the next one is based on a variational principle developed by Cunha & Gough (2000); and the third one (Saio & Gautschy 2004a) is somewhat similar to the first one, expanding the eigenfunction by a sum of terms associated with spherical harmonics. These three methods are summarized by Saio (2008).…”

“…The left panel compares the results of Cunha & Gough (2000) for ℓ = 1 and 3 axisymmetric modes of the polytropic model having (M, R) = (2M ⊙ , 2R ⊙ ) with the results of Bigot et al (2000) for a 1.8 M ⊙ ZAMS model. The results of Cunha & Gough (2000) show that the frequency shift by the magnetic field increases with frequency, but at a certain frequency it jumps down and then starts increasing again. This property, which is governed by a parameter, νB 0.25 p (for a polytrope of index 3), is confirmed by Saio & Gautschy (2004a).…”