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\title{\bf Oscillations of rapidly rotating stars}
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\author{D. Reese, F. Ligni{\`e}res and M. Rieutord\\
\vspace{1cm}\\
\normalsize Laboratoire d'Astrophysique,
Observatoire Midi-Pyr{\'e}n{\'e}es \\
\normalsize Universit{\'e} Paul Sabatier Toulouse 3 \\
\normalsize 14 avenue {\'E}. Belin, 31400 Toulouse, France}
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\begin{document}
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{\noindent\small{\bf Abstract:}
The effects of rapid rotation on stellar pulsation is examined using an
accurate 2D numerical method.  We compare the results of these 
non-perturbative
calculations with those of perturbative methods and find that frequency
differences exceed $0.08 \, \mathrm{\mu Hz}$ on half of the modes when the
rotation rate is $15 \percent$ of the keplerian (break-up) limit.  The
differences between the two results is mainly attributed to the approximate
treatment of the centrifugal force in perturbative methods.  We also explore
different levels of approximation for the Coriolis force.}

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\section{Introduction}

The pulsation of rotating stars is of astrophysical interest because of the
many unresolved questions on the structure of rotating stars, and the 
potential
information that can be deduced through asteroseismology.  With the launch 
of
space missions such as Corot, we can expect to obtain accurate measurements 
of
pulsation frequencies some of which will be of rapidly rotating stars.  It 
is
therefore important to understand and accurately quantify the effects of
rotation on stellar pulsations.

There are two main effects that lead to a modification of stellar 
oscillations.
The first is the centrifugal force which affects pulsation modes in a direct
way through the effective gravity and in an indirect way through the
deformation of the star.  The second is the Coriolis force, which only has 
an
effect through the momentum equation. In both cases, couplings appear 
between
the different spherical harmonics, which adds to the mathematical difficulty 
of
the problem.

\section{Formalism}

We calculate adiabatic acoustic oscillations of a uniformly rotating 
polytropic
model of a star, in which the polytropic index $N$ is $3$.  This can be
decomposed into two steps:  first we calculate the equilibrium model, second 
we
solve the linearised oscillation equations. The numerical method we employ 
for
both steps consists of a direct 2D numerical approach.  It is entirely 
spectral
as it uses spherical harmonics in the horizontal direction and Chebyshev
polynomials in the radial direction. We use the surface-fitting coordinates
of Bonazzola et al. (1998).

In order to project the oscillation equations onto the spherical harmonic 
base
we start by expressing the different unknowns as a sum of spherical 
harmonics
multiplied by radial functions that need to be determined.  These 
expressions
are introduced into the oscillation equations, which are then projected onto
the vectorial spherical harmonics (\textit{e.g.} Rieutord 1987). In this 
way,
we obtain a highly coupled system of ordinary differential equations in 
terms
of the radial coordinate $\zeta$, the solution of which gives the unknown
radial functions.  This system is discretised onto a Gauss-Lobatto grid
associated with the Chebyshev polynomials and the resulting algebraic
eigenvalue problem is solved numerically.

We have done different tests to verify the accuracy of the calculated
frequencies.  In the non-rotating case we obtain an agreement of $\Delta
\omega/\omega \sim 10^{-7}$ with the frequencies of Christensen-Dalsgaard 
and
Mullan (1994).  Also, for particular modes calculated without the Coriolis
force, we obtain an agreement of $\Delta \omega /\omega \sim 10^{-6}$ with
Ligni{\`e}res et al. (2005) for rotation rates up to  $0.59 \, 
\Omega_K^{eq}$,
where $\Omega_K^{eq} = (GM/R_{eq}^3)^{1/2}$ ($R_{eq}$ being the equatorial
radius).  If we deduce perturbative coefficients from our frequencies, we
obtain an agreement of 2 percent and usually better  with the coefficients 
of
Saio (1981).  Finally, we also compute a variational test and obtain an
accuracy of $10^{-7}$ or better.

\section{Results and physical aspects}

When examining the pulsation frequencies, it is interesting to compare them 
to
those obtained through perturbative methods, in order to find out up to what
point perturbative methods are valid.  In what follows, we will consider 
modes
with a harmonic degree $\l$ ranging from 0 to 3, an azimuthal order $m$ 
going
from $-\l$ to $\l$, and the radial order $n$ ranging from $1$ to $6$ 
(strictly
speaking an oscillation mode is composed of a whole set of harmonic degrees;
here $\l$ refers to  the harmonic degree of the non-rotating mode).  We 
follow
their frequencies from the non-rotating case to a rotation rate of $0.38 \,
\Omega_K^{eq}$.

Fig.~1 shows a comparison between complete numerical frequencies and 
$2^{nd}$
order perturbative ones (we calculate the perturbative coefficients using a
least squares fit near $\Omega = 0$).  As can be seen from the figure,
perturbative calculations lead to errors exceeding $1 \percent$ when the
rotation rate is higher than $0.28 \, \Omega_K^{eq}$.  If we consider a star
with a mass of $1.9 M_{\odot}$ and a polar radius of $2.3 R_{\odot}$, a 
typical
$\delta$ Scuti star, and we if impose an error bar of $0.6 \, \mathrm{\mu 
Hz}$
(the accuracy of Corot's secondary program), we find that the mode
$(n=6,\l=2,m=0)$ is erroneous at a rotation rate of $0.12 \, \Omega_K^{eq}$
(which corresponds to an equatorial velocity $v_{eq} = 47 \,
\mathrm{km.s^{-1}}$) and half the modes are incorrect beyond $0.24 \,
\Omega_K^{eq}$ ($v_{eq}=95 \, \mathrm{km.s^{-1}}$).  If instead we impose an
error bar of $0.08 \, \mathrm{\mu Hz}$ (the accuracy of Corot's primary
program), the $(n=6,\l=2,m=0)$ mode is erroneous at $0.075 \, \Omega_K^{eq}$
($v_{eq}=30 \, \mathrm{km.s^{-1}}$) and half the modes are incorrect at 
$0.15
\, \Omega_K^{eq}$ ($v_{eq}=60 \, \mathrm{km.s^{-1}}$).

\begin{figure}[ht!]
\begin{tabular}{lr}
\includegraphics[width=8cm]{figures/comp1_4.eps} &
\includegraphics[width=8cm]{figures/diff1_1.eps} \\
\parbox[][3.5cm][t]{7.8cm}{\caption{Non-dimensional frequencies as a 
function
of the rotation rate.
\textit{Solid lines:} non-perturbative frequencies;
\textit{dotted lines:} $2^{nd}$ order perturbative frequencies;
\textit{dashed lines:} non-perturbative frequencies without the Coriolis 
force.}} &
\parbox[][3.5cm][t]{7.8cm}{\caption{Behaviour of the relative error from
$2^{nd}$ order perturbative methods as a function of the radial order $n$.
The error increases with $n$. In all the figures,
\mbox{$\Omega_K^{eq}=(GM/R_{eq}^3)^{1/2}$},
\mbox{$\Omega_K^{pol}=(GM/R_{pol}^3)^{1/2}$} and
\mbox{$\epsilon = 1 - R_{pol}/R_{eq}$,} where $R_{eq}$ and $R_{pol}$ are the
equatorial and polar radii, resp.}}
\end{tabular}
\end{figure}

It is then interesting to try to ascertain where these errors come from. 
Fig.~2
and~3 give some clues as what may cause these differences. In fig.~2,  we 
show
the relative error $(\omega_{pert.}-\omega)/\omega$ from perturbative 
methods
as a function the rotation rate.  As the radial order $n$ increases, so does
the relative error.  In fact this trend is quite general, as it applies for 
all
values of $\l$ and $m$.  In fig.~3, we show another comparison between
perturbative calculations and complete ones. This time, however, the 
Coriolis
force has been suppressed in both sets of calculations. Once more, the 
relative
error increases with the radial order, which shows that the centrifugal 
force
alone reproduces the same behaviour as in fig.~2.  By contrast, we know that
the effects of the Coriolis force decrease as the frequency increases.
Therefore the error resulting from the perturbative treatment of the 
Coriolis
force is not expected to increase with the radial order.  This suggests that
the centrifugal force plays the dominant role in the differences between
perturbative and complete calculations.  Fig.~1 confirms this conclusion: 
the
dashed lines represent calculations in which the Coriolis force is neglected
but the centrifugal force is treated in a non-perturbative way.  They give a
better approximation than the perturbative calculations.

One of the reasons that may explain the centrifugal force's dominant role in
the differences between the two approaches is the approximate treatment by
perturbative methods of the star's deformation.  Perturbative methods 
typically
only use the $\l=0$ and $\l=2$ harmonics to describe this deformation, which 
is
inaccurate, especially in the star's outer layers. Furthermore, differences
between two descriptions of the deformation will give rise to differences in
the frequencies which are roughly proportional to the frequencies.  Hence a
frequency twice as large as another will vary twice as much as the other due 
to
modifications of the star's deformation.  It is then helpful to bear in mind
that as the radial order increases, pulsation frequencies also increase, and
the associated modes become more concentrated in the star's outer layers.  
As a
result, the centrifugal force rapidly becomes the dominant factor in the
differences between the two approaches.

\begin{figure}[ht!]
\begin{tabular}{lr}
\includegraphics[width=8cm]{figures/diff_centrifugal1_1.eps} &
\includegraphics[width=8cm]{figures/comp1_3.eps} \\
\parbox[][3.5cm][t]{7.8cm}{\caption{The relative error between complete 
numerical calculations and
$2^{nd}$ order perturbative calculations, both of which exclude the Coriolis 
force.  As $n$
increases, the error has the same behaviour as the calculations
which include the Coriolis force (see fig.~2).}} &
\parbox[][3.5cm][t]{7.8cm}{\caption{The absolute error caused by either 
neglecting the Coriolis force (No
Cor.) or doing a $1^{st}$ order approximation of its effects (Pert.). The
$1^{st}$ order approximation is not substantially better, but does make the
errors symmetric in $m$, the same parity as $2^{nd}$ order corrections.}}
\end{tabular}
\end{figure}

We can then take a more detailed look at the effects of the Coriolis force.  
In
what follows, we will continue to treat the centrifugal force in a
non-perturbative way and will look at three different ways of treating the
Coriolis force.  The first way consists in neglecting the Coriolis force.  
The
second way corresponds to a first order perturbative approximation.  The 
third
way is a complete treatment. We denote these three levels of approximation 
by
$\omeganlm^0$, $\omeganlm^1$ and $\omeganlm^2$ respectively:
\begin{equation}
\omeganlm^2 =
\underbrace{\omeganlm^0 + \Dnlm \Omega}_{\displaystyle \omeganlm^1}
+ \mathcal{O}(\Omega^2)
\quad \mbox{where}
\quad
\Dnlm = \frac{i \displaystyle \int_V \rho_o \ez
\cdot \left( \unlm \times \unlm^* \right) dV}
{\displaystyle \int_V \rho_o \| \unlm \|^2 dV}
\end{equation}
The expression for $\Dnlm$ is very similar to the spherical expression (in 
both
cases, the linear fluid dynamics operator is self-adjoint), but the 
integration
domain is spheroidal, the equilibrium density is that of a rotating star and
the eigenfunction $\unlm$ takes the centrifugal force into account, but not 
the
Coriolis force.  However, in the spherical limit, we obtain the familiar
expression $\Dnlm = m\Cnl$ which can be found in Ledoux (1951).

Fig.~4 shows the absolute errors of $\omeganlm^0$ and $\omeganlm^1$ for an 
$\l
= 1$ triplet.  In the $m = 0$ case, $\omeganlm^0 = \omeganlm^1$ since the
corrections in that case start at second order in $\Omega$.  For $m = \pm 
1$,
it is interesting to note that the first order correction $\omeganlm^1$ 
brings
the absolute error of both $m = 1$ and $m = -1$ to roughly the same level 
(this
implies increasing the error on $m = -1$).  This shows that corrections
symmetric in $m$, which come from even orders, are necessary.  A second
order correction might then adequately account for the Coriolis force.

\section{Conclusion}

In this paper, we have been able to examine some of the differences between 
a
perturbative and a complete treatment of the effects of rotation on stellar
pulsations, and explore the relative effects of the centrifugal and Coriolis
forces.  Our results show that the errors of perturbative methods can mostly 
be
attributed to the approximate treatment of the centrifugal force.

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%\section*{Acknowledgements}
%This research was supported in part by contract ARC 90/94-140 ``Action de
%Recherche Concert\'ee de la Communaut\'e Fran\c{c}aise'' (Belgium).
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\beginrefer

\refer Bonazzola S., Gourgoulhon E., Marck J.-A., 1998, Phys. Rev. D 58, 
104020.

\refer Christensen-Dalsgaard J., Mullan D. J., 1994, MNRAS 270, 921.

\refer Ledoux P., 1951, ApJ 114, 373.

\refer Ligni{\`e}res F., Rieutord M., Reese D., in press. In:
Walker A.R., Bono G. (eds.)
Memorie della Societa' Astronomica Italiana, Vol. 76/4.


\refer Rieutord M., 1987, Geophys. Astrophys. Fluid Dynamics 39, 163.

\refer Saio H., 1981, ApJ 244, 299.

\endrefer
\end{document}


