second e¬ect appears in radiative zones only: unlike in a non-rotating star,

the radiative zone of rotating stars cannot be in hydrostatic equilibrium:

an azimuthal ¬‚ow forced by the baroclinic situation (isobars and isotherms

do not match) appears. Such ¬‚ows, which may be turbulent, are not very

strong but are active during a large fraction of the lifetime of the star,

their in¬‚uence is therefore understandable only on a long time scale. It is

indeed on such time scales that the element distribution is modi¬ed; since

this distribution may control the excitation of the modes, via the kappa-

mechanism, we see here an indirect in¬‚uence of the centrifugal acceleration

on mode visibility. Another one may be the state of di¬erential rotation

Rieutord

106

which is known to perturb the spectrum of oscillations (Dziembowski and

Goode 1992).

Obviously, these e¬ects are still largely unknown and we shall thus restrict

our discussion to the ¬rst e¬ect which has the advantage of being universal.

It indeed a¬ects the whole star and not speci¬cally the radiative or the

convective zone.

The centrifugal deformation of a star is an e¬ect controlled by „¦2 and

therefore often thought to be unimportant; for instance, we know that acous-

tic modes are only slightly perturbed by the Coriolis acceleration and usually

¬rst order perturbations in „¦ are su¬cient. Logically, second order pertur-

bations should be irrelevant.

The relevance of centrifugal corrections comes from the fact that they

should be compared to the di¬erence between the frequencies of two neigh-

bouring modes. Indeed, it is clear that mode identi¬cation will be vitiated if

these corrections are larger than the frequency spacing and are not included.

To make things quantitative somehow, we need to compare the typical

wavelength of the mode to the actual centrifugal deformation of a star. We

may characterize this deformation by the polar ¬‚attening µ de¬ned by

µ = (Re ’ Rp )/Re

where Re and Rp are respectively the equatorial and polar radii. For an

n = 3/2 polytrope the maximum value of µ is 0.37; we shall take a typical

value of µ = 0.1. If we consider the case of a δ-Scuti star with a typical radius

of 2R , its equatorial distortion amounts to 0.2R , while the wavelength of

a mode pulsating with a period of 20min is 0.17R (assuming a mean sound

speed of 100 km s’1 ). It is therefore clear that the centrifugal deformation is

an inescapable feature for a correct identi¬cation of acoustic modes in rapid

rotators.

In order to better appreciate these e¬ects, Ligni`res et al. (2001) studied

e

the case of acoustic modes of a perfect gas contained in an axisymmetric

ellipsoidal vessel. One interesting property of such a system is that it shows

quite clearly the qualitative di¬erences which arise when one goes from the

sphere to the ellipsoid. Within a sphere acoustic modes have a maximum

pressure on the boundary (recall that their radial structure is described by

Bessel functions of semi-integer order). Such modes are called “whispering

gallery” modes. Their associated ray path never penetrates a central ball.

When one shifts to an ellipsoid, whispering gallery modes still exist but a

new family appears: these are the central modes which are associated with

ray paths crossing the equatorial plane between the foci of the meridional

The oscillations of rapidly rotating stars 107

ellipse. Such modes, which do not exist in a sphere, prove the qualitative

di¬erences introduced by a slight change of shape. The two types of modes

are shown in Figure 7.4.

Fig. 7.4. The kinetic energy viewed in a meridional section of an axisymmetric

ellipsoid for two types of acoustic modes: on left a whispering gallery mode at

ω = 25.3, on right a central mode at ω = 21.0. Both are axisymmetric; note the

absence of amplitude at the equator for the central mode. The polar ¬‚attening is

µ = 0.13 and no dissipation has been included. We used 30 spherical harmonics

and 50 radial nodes to compute them.

Now real stars are not true ellipsoids and the sound speed varies radially.

Thus acoustic modes live in a cavity which has no special symmetry except

the equatorial one. In these conditions it is very likely that the dynamics of

acoustic rays is chaotic which would endow the spectrum of acoustic modes

with properties of systems experiencing quantum chaos, a possibility which

was already pointed out in the masterful work of Gough (1993).

7.5 Conclusions

To conclude this contribution, I would like to stress our progress in the

understanding of the e¬ect of rotation in rotating stars but also stress the

problems which remain to be solved.

We have seen that the introduction of rotation modi¬es in many respects

the spectrum of oscillations of a star: new modes appear while others dis-

Rieutord

108

appear. These drastic changes are obviously not captured by perturbative

methods.

Rotation introduces through Coriolis acceleration its own modes, namely

inertial modes, which are low frequency modes; as such, they strongly in-

teract with gravity modes. This interaction is not visible for slowly rotating

stars since only very high order modes are perturbed. However, in rapid

rotators the cut-o¬ frequency of inertial modes, 2„¦, may be close to the

cut-o¬ frequency of gravity modes; in this case, the low frequency band of

the oscillation spectrum of the star is deeply modi¬ed: as I have shown, the

mathematical nature of the adiabatic problem is such that almost all the

low frequency band is ™polluted™ by singularities. Some di¬usion is therefore

required to compute modes in this band.

Besides the Coriolis acceleration, the centrifugal one does not play a less

important rˆle. Its ¬rst e¬ect, which is to change the shape of the “con-

o

tainer”, may perturb noticeably the acoustic spectrum especially in the high

frequency range. However, order-of-magnitude arguments show that even

acoustic modes of period similar to those observed in δ-Scuti stars are likely

perturbed by the centrifugal e¬ects. Looking back to the gravito-inertial

modes, I note that these modes are likely less perturbed by the change of

shape since observable ones have a large-scale structure.

Besides these results, it is quite clear that many questions are left open.

Especially the e¬ects of the centrifugal acceleration call for more work: the

hand-waving arguments need to be re¬ned with more realistic models such

as fast rotating polytropes for a ¬rst step. Then, a far more challenging

issue concerns the evolutionary e¬ects related to the transport and mixing

of elements so as to determine the excitation mechanism yielding observable

modes. The investigation of such e¬ects clearly calls for two-dimensional

stellar models which include transport processes and losses of angular mo-

mentum through stellar winds.

Finally, to end with a note of exotism, we should also mention the case of

neutron stars which are very rapidly rotating objects. Since the discovery

by Andersson (1998) that inertial r-modes are unstable when coupled to

gravitational radiation, the oscillation spectrum of neutron stars is actively

investigated and there too rotation is an unavoidable ingredient for all types

of modes.

Acknowledgements The results presented here have been obtained with the

enthusiastic collaboration of B. Dintrans, B. Georgeot, F. Ligni`res and

e

L. Valdettaro who are all gratefully thanked. Figure 4 owes much to the

help of F. Ligni`res.

e

The oscillations of rapidly rotating stars 109

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