model concerning the di¬erences between roAp and noAp stars (that is,

non-oscillating Ap stars). In the last section we will present some conclu-

sions.

4.2 Magnetic ¬eld versus convection

In order to try to understand how strongly the magnetic ¬eld in¬‚uences

convection, one may compare the force exerted by buoyancy on a ¬‚uid ele-

ment with the magnetic restoring force experienced as that element moves.

The buoyancy force per unit mass driving the convection is roughly minus

the square of the buoyancy frequency times the displacement of the element:

’N 2 δr. The magnetic restoring force per unit mass exerted on the displaced

element is estimated by (|B|2 k2 /µρ)δr, where B is the magnetic ¬eld, k is

the characteristic wavenumber of the magnetic ¬eld-line distortion, µ is the

magnetic permeability and ρ is the density. Thus for the magnetic ¬eld to

inhibit the motion one expects

’N 2 ,

v2 k2 (4.1)

where v is the Alfv´n speed. According to equation (4.1), the comparison

e

of forces depends on the length scale, k’1 , and thus, on the ¬eld geometry.

In particular, for a magnetic ¬eld with a dipolar structure one expects that

convection may be suppressed in the polar regions, but not necessarily near

the equator of these stars. Having this in mind BCDGV constructed a model

Understanding roAp stars 53

for roAp stars which incorporated two distinct regions: polar and equatorial.

In each region, the magnetic stresses were neglected entirely in the balance

of forces, and were acknowledged only through their presumed e¬ect on the

convective energy and momentum ¬‚uxes. In practice, for each stellar model

two envelope models were generated, one with the characteristics of the

equatorial region and the other with the characteristics of the polar regions.

These two models were matched in the interior to assure that they di¬ered

only in their surface layers. This procedure is oversimpli¬ed, but without

substantial further sophistication it is not easy to proceed.

If the polar regions of stellar models like these occupy the domain ’‘ <

θ < ‘ and π ’ ‘ < θ < π + ‘ in the spherical polar coordinate system

(r, θ, φ), with θ = 0 along the magnetic axis, then the frequency νnlm of a

mode of the composite model, that is, the model composed by polar and

equatorial regions with the characteristics described above, is (e.g. Dolez et

al. 1988, Cunha 1999)

2π 1 2π ˜

µ

p eq

(Ylm )2 dµdφ + νnl (Ylm )2 dµdφ,

νnlm ≈ νnl (4.2)

0 ˜ 0 0

µ

eq p

where νnl and νnl are the oscillation frequencies in envelope models with

the equatorial structure and the polar structure, respectively, µ = cos θ and

µ = cos ‘.

˜

Equation (4.2) allows us to calculate the real and imaginary part of the

frequencies for a given mode in the composite model, from the frequencies

of the same mode in spherically symmetric models with the characteristics

of the magnetic poles and equator, respectively.

The abnormal chemical abundances of some elements, and their inho-

mogeneous distribution over the stellar surface, is another characteristic of

roAp stars that should be kept in mind when studying the indirect e¬ects

of the magnetic ¬eld on the oscillations. It is well known that the magnetic

¬eld in¬‚uences the surface and depth distribution of chemical elements in

roAp stars, by interacting with the ions. Also, if we are to believe that the

magnetic ¬elds suppress convection around the magnetic poles of roAp stars,

then the pro¬les of chemical abundances will also be in¬‚uenced by that sup-

pression, and further di¬erences will be introduced between the polar and

equatorial regions of the star. Having this in mind, BCDGV have used

di¬erent chemical pro¬les when computing the oscillations in the envelope

models representing the polar and the equatorial regions of the stars.

Cunha

54

Fig. 4.1. Growth rates in a model with M = 1.87M , log Tef f = 3.91 and

log L/L = 1.164. Filled circles show the results in the polar region and open

circles show the results in the equatorial region. For details on the chemical pro¬les

see panel (a) of Figure 5 of BCDGV.

Fig. 4.2. The envelope polar model used in Fig. 4.1 (¬lled circles) is compared

with similar models but with di¬erent helium pro¬les. The open triangles show

an envelope polar model with a strong accumulation of helium near the ¬rst he-

lium ionization zone, while the open squares show an envelope polar model with

homogeneous chemical composition.

4.3 Mode excitation and eigenfrequencies

4.3.1 Excitation

Several ideas concerning the excitation mechanism of roAp stars were pro-

posed over the years (Dolez & Gough 1982, Dolez et al. 1988, Shibahashi

1983, Dziembowski 1984, Dziembowski & Goode 1985, Gautschy et al. 1998,

Understanding roAp stars 55

Matthews 1988). However, with the exception of one of them, these ideas

either have not been pursued or have failed to show that models appropriate

to roAp stars should be unstable to high frequency oscillations. Gautschy et

al. (1998) did ¬nd high frequency unstable modes in some roAp star mod-

els, but their models assumed the presence of a chromosphere, for which

observational evidence has not yet been found (e.g. Shore et al. 1987).

Using models like those described above, BCDGV have found high fre-

quency unstable modes for temperatures and luminosities typical of roAp

stars. Because the problem of the excitation mechanism in roAp stars can-

not be looked at without considering, simultaneously, the problem of the

acoustic cuto¬ frequency, in an attempt to bracket the physically plausible

range of atmospheric conditions, the authors computed pulsational stability

with two di¬erent outer mechanical boundary conditions. The ¬rst bound-

ary condition considered was perfectly re¬‚ective, while the second boundary

condition was that appropriate to a plane-parallel isothermal atmosphere

whose temperature matches continuously with that of the underlying en-

velope, thus representing a star with no chromosphere. In both cases high

frequency unstable modes were found.

p

In Fig. 4.1 we show the growth rates (imaginary part of the frequency) ·n0

eq

and ·n0 of the oscillations of the polar and equatorial models that constitute

a composite model with M = 1.87M , log Te¬ = 3.91 and log L/L = 1.164.

All high-order modes are stable in the equatorial model. The polar model,

on the other hand, shows unstable subcritical modes of high frequency. A

comparison between the growth rates of modes of the same polar model, with

the growth rates of similar polar models but with di¬erent chemical pro¬les

is shown in Fig. 4.2 . It is clear form these ¬gures that according to the

model of BCDGV the possibility of exciting high frequency oscillations in

roAp stars depends on the extent of the region in which envelope convection

is suppressed and that the frequencies of the modes excited depend on some

of the model details, e.g. the chemical composition pro¬le.

4.3.2 E¬ect on the power spectrum

All of those characteristics of roAp stars that contribute to deviate their

structure from spherical symmetry, also contribute to deviate the frequencies

of their oscillations away from the values derived from asymptotic theory in

spherically symmetric models (Tassoul 1980). The magnetic ¬eld is one of

the e¬ects that should be kept in mind when such deviations are considered,

even if only indirect e¬ects of the latter, like those assumed in the model of

BCDGV, are under study.

Cunha

56

As illustrated by equation (4.2), in a model like that of BCDGV, in which

di¬erent angular regions of the star are characterized di¬erently, but within

each of them the structure is independent of latitude and longitude, the fre-

quency of a given mode in the composite model is a weighted average of the

frequencies that the same mode would have in spherically symmetric mod-

els corresponding to each of the regions considered. Because the average

is weighted by the spherical harmonic corresponding to the mode consid-