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7

The oscillations of rapidly rotating stars

MICHEL RIEUTORD

Observatoire Midi-Pyr´n´es and Institut Universitaire de France

ee

14 av. E. Belin, F-31400 Toulouse, France

We review the e¬ects of rotation on the oscillation spectrum of rapidly ro-

tating stars. We particularly stress the novelties introduced by rotation: for

instance, the disappearance of modes in the low frequency band due to the

ill-posed natured of the underlying mathematical problem. This is mainly an

e¬ect of the Coriolis acceleration. The centrifugal e¬ect changes the shape

of the star in the ¬rst place. The possible consequences of this deformation

on the oscillation spectrum are brie¬‚y analyzed. We also describe other pos-

sibly important e¬ects of the centrifugal acceleration which come about on

the time scale of star evolution.

7.1 A short introduction to rapidly rotating stars

All stars are a¬ected by rotation but some of them, the rapid rotators, are

more a¬ected than the others! Astronomers usually qualify as rapid rotators

all the stars with v sin i ≥ 50 km s’1 , i.e. those with an equatorial velocity

larger than 50 km s’1 . Such a value should be compared to the Keplerian

limiting velocity which is

Vkep ∼ 440 km s’1 (M/M )0.1

for stars on the main sequence (we used the mass-radius relation given by

Hansen and Kawaler 1994). Thus, for these stars the limiting velocity is

weakly mass-dependent and rapid rotators appear as stars whose centrifugal

acceleration exceeds 10% of the surface gravity; since this ratio measures the

impact of rotation on the star structure, rapid rotators are those stars whose

shape is signi¬cantly distorted by rotation.

Although important as we shall see later, the centrifugal deformation of

the star is not the ¬rst e¬ect to a¬ect the spectrum of oscillations of a star:

99

Rieutord

100

the spectrum is indeed ¬rst perturbed by the new time-scale imposed by the

period of rotation which interfere with other natural periods of the star. In

Figure 7.1, we sketch out the frequency bands of modes typical of a stellar

situation. Rotation comes into play through the appearance of the Corio-

lis acceleration, which ensures for any motion that angular momentum is

conserved. Note that it not only modi¬es already existing modes (p- and

g-modes) but also brings in new ones called inertial modes†. Since the Cori-

olis acceleration introduces the time-scale (2„¦)’1 (where „¦ is the angular

frequency of rotation), one should expect that the spectrum is strongly per-

turbed around and below this frequency. Hence, from the point of view

of the oscillations, rapid rotators would be those stars whose pulsation fre-

quencies are close to 2„¦. However, because of the rather high values of the

Brunt-V¨is¨l¨ frequency N found in stellar radiative zones, stars whose

a aa

pulsation frequencies are strongly perturbed by rotation need to be rapid

rotators in the classical sense.

Gravity modes Acoustic modes

c

N

2„¦

R

Inertial modes

Fig. 7.1. The typical placing of the di¬erent spectral bands in the stellar context.

Here N and c/R should be understood as the cut-o¬ frequencies of the gravity and

acoustic modes respectively, c being a mean value of the sound speed and R the

radius of the star.

The foregoing discussion shows that the changes in the spectrum of oscil-

lations of a star brought about by rotation come from two sides: the Coriolis

and centrifugal accelerations. As we shall see both are important; the former

is certainly better understood and I shall explain the recent progress on this

subject, while the latter has mostly indirect consequences, more involved,

but likely inescapable for a correct interpretation of rapid rotator pulsations.

Before developing the di¬erent aspects of these two non-Galilean e¬ects,

I would like to discuss brie¬‚y the method to deal with them, perturbative

or non-perturbative.

† Inertial modes are the vast family of modes whose restoring force is the Coriolis force. They

should not be confused with r-modes originally introduced by Papaloizou and Pringle (1978).

r-modes are indeed a small subset of inertial modes which have a purely toroidal velocity ¬eld;

in the ¬eld of geophysics they are called Rossby waves, hence the term r-modes which generated

much confusion recently in the ¬eld of neutron stars (Rieutord 2001).

The oscillations of rapidly rotating stars 101

7.2 Perturbative versus non-perturbative methods

The determination of the eigenspectrum of a star is not an easy task. Al-

though the set of equation is linear, the resolution of an eigenvalue problem

is intrinsically a nonlinear operation requiring iterative algorithms to com-

pute some subset of the spectrum. As this is a costly operation it is natural

to avoid it or use its results in an optimal way. This is the idea of pertur-

bative methods: using the spectrum and eigenmodes of a non-rotating star,

one derives the modi¬cations introduced by rotation to eigenfrequencies and

eigenmodes.

In practice, such an approach is useful for slowly rotating stars like the

sun. For rapidly rotating stars it is still useful but should be handled with

care. Indeed, the main drawback of perturbative methods is their inability

to “invent” new modes. They only make an adiabatic transformation of the

spectrum of a non-rotating star. Hence they miss modes speci¬c to rotation

like inertial modes; they also miss ™singular™ modes or modes which do not

exist in a perfectly spherical star. For all these modes one should resort to

non-perturbative methods which do not assume a separation of variables as

done in non-rotating stars.

7.3 The part played by the Coriolis acceleration

The Coriolis acceleration is the term by which the operator governing os-

cillation modes loses its spherical symmetry. Operator 2„¦— is indeed not

invariant under the rotations, except those around the rotation vector „¦.

Hence, when using spherical coordinates, separability of the radial r and

polar angle θ variables is lost. This loss shows up in the occurrence of a

coupling between the equations governing the lth spherical harmonic of the

¬elds; for instance, if we write the radial velocity u = l ul (r)Ylm , its lth

component ul (r) is coupled to the (l + 1)th and (l ’ 1)th component of the

toroidal velocity components themselves coupled to ul+2 , ul and ul’2 ; hence

an in¬nite chain of equations appears. These series converge rapidly when

the coupling coe¬cient is small, but at frequencies close to or less than 2„¦

this coe¬cient is of order unity.