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arXiv:1009.5939v1 [astro-ph.SR] 29 Sep 2010

Astron. Nachr. / AN 000, No.00, 1–4 (0000) / DOI please set DOI!

P-modes in rapidly rotating stars –

looking for regular patterns in synthetic asymptotic spectra

F. Ligni` eres1,2,⋆, B. Georgeot3,4, and J. Ballot1,2

1Universit´ e de Toulouse; UPS; Laboratoire d’Astrophysique de Toulouse-Tarbes (LATT); F-31400 Toulouse, France

2CNRS; Laboratoire d’Astrophysique de Toulouse-Tarbes (LATT); F-31400 Toulouse, France

3Universit´ e de Toulouse; UPS; Laboratoire de Physique Th´ eorique (IRSAMC); F-31062 Toulouse, France

4CNRS; LPT (IRSAMC); F-31062 Toulouse, France

The dates of receipt and acceptance should be inserted later

Key words

Hydrodynamics - Waves - Chaos - Stars: oscillations - Stars: rotation

According to a recent ray-based asymptotic theory, the high-frequency p-mode spectrum of rapidly rotating stars is a

superposition of frequency subsets associated with dynamically independent regions of the ray-dynamics phase space.

At high rotation rates corresponding to typical δ Scuti stars, two frequency subsets are expected to be visible : a regular

frequency subset described by a Tassoul like formula and an irregular frequency subset with specific statistical properties.

In this paper, we investigate whether the regular patterns can be detected in the resulting spectrum. We compute the

autocorrelation function of synthetic spectra where the frequencies follow the asymptotic theory, the relative amplitudes

are simply given by the modes’ disk-averaging factors, and the frequency resolution is that of a CoRoT long run. Our first

results are that (i) the detection of regular patterns strongly depends on the ratio of regular over irregular modes, (ii) low

inclination angle configurations are more favorable than near equator-on configurations (iii) in the absence of differential

rotation, the 2Ω rotational splitting between m = 1 and m = −1 modes is an easy feature to detect.

c ? 0000 WILEY-VCH Verlag GmbH&Co.KGaA, Weinheim

1 Introduction

The launching of the space missions COROT and KEPLER

is bringing a new wealth of observational data on the as-

teroseismology of many different types of stars. For slowly

rotating stars, e.g. the sun, the approximate spherical sym-

metry of the system enables us to classify modes accord-

ing to well-defined sets of integers. This mode structure al-

lowed us to extract information on the stellar interior from

the spectrum of oscillation modes (Christensen-Dalsgaard,

2002).

However, for rapidly rotating stars, the deformation of

the star breaks the spherical symmetry. It has recently been

shown that perturbation theory fails to yield accurate pre-

dictions of oscillation spectra even for stars with moderate

rotation rates (Ligni` eres et al., 2006; Reese et al., 2006). In

Ligni` eres & Georgeot (2008, 2009), an asymptotic theory

of spectra of such stars was built, based on acoustic ray dy-

namics.Itwas shownthatthis dynamicsis Hamiltonian,and

undergoes a gradual transition from integrability to chaos

when the rotation increases. At moderately rapid rotation,

the phase space is divided into integrableand chaotic zones,

where the dynamics is qualitatively different. It was shown

inLigni` eres & Georgeot(2008,2009)thatthisstructuremod-

ifies the structure of the oscillation spectra. The acoustic

mode frequencies cannot be in general associated to well-

definedsets ofintegers.Instead,thespectrumis split intoin-

⋆Corresponding author: e-mail: francois.lignieres@ast.obs-mip.fr

dependentsubspectra,correspondingtodifferentphasespace

zones for the ray dynamics. The subspectra corresponding

tointegrablezonesgiverisetoregularsequencesoffrequen-

cies, whereas a superimposed subspectrum is associated to

chaoticdynamicsanddisplaysregularityonlyin a statistical

sense.

Inorderto connectsuch results to observedspectra,sev-

eral questions have to be explored. The first one is the va-

lidity of the asymptotic theory to the finite range of spectra

which can be observed. A first answer was given in

Ligni` eres & Georgeot(2008,2009),whereitwasshownthat

the predicted structure can be found in numerically

computed relatively low-frequency p-modes of polytropic

stellar models. An other important question is to separate

these different subspectra in observational data, where the

visibility of the modes plays an important role. A first step

in this direction is to construct synthetic spectra based on

the asymptotic theory where visibility varies from mode to

mode and try to extract from them important features of the

system, as if theywere observedspectra. This is the strategy

we follow here. We also note that a first evidence of regular

patterns in a p-mode spectrum of a rapidly rotating star has

been recently obtained by analyzing the spectra of a δ Scuti

CoRoT target (Garc´ ıa Hern´ andez et al., 2009).

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2 F. Ligni` eres, B. Georgeot & J. Ballot: P-modes in rapidly rotating stars

2 Construction of synthetic spectra in the

asymptotic regime

We havebuilt syntheticspectrawith the followingformulas:

S(ν,i) =

?

j

L(ν,νj,Aj(i),Γ)

(1)

L(ν,νj,Aj(i),Γ) = Aj(i)exp?−(ν − νj)2/Γ2?

(2)

where νjare the frequencies of the modes and Aj(i) their

amplitudes, which are functions of the inclination angle i.

We have represented modes with Gaussian functions of

width Γ to take into accountthe finite resolutionof thespec-

trum. We have considered in this work a resolution Γ corre-

sponding to 150-day-long runs. Our main assumptions are

that the frequencies follow the asymptotic behaviour de-

scribed in Ligni` eres & Georgeot(2009) and that the relative

amplitudes only depend on the relative mode visibilitites,

the visibility being approximatedby the disk-averagingfac-

tor computed for high-frequency p-modes in a polytropic

stellarmodel(seeformula(37)inLigni` eres & Georgeot(2009)).

Above a certain rotation rate, the p-modesspectrum is dom-

inatedby two families

modes and the chaotic modes, as the visibilities of the other

modes become negligible. The island modes are associated

with island chainstructuresofthe raydynamicsphasespace

and their spectrum follows a simple formula (Reese et al.,

2008).

of modes,the island

ω˜ n˜ℓm= ∆˜ n˜ n + ∆˜ℓ˜ℓ + ∆m|m| − mΩ + ωref

(3)

where˜ℓ ≤ 3and∆˜ n,∆˜ℓ,∆mdependonthestar’sstructure.

The quantum numbers ˜ n and˜ℓ are defined from the spa-

tial distribution of the modes in a meridional plane. They

correspond respectively to the number of nodes along and

across the stable periodic trajectory associated with the is-

land chain. By contrast, chaotic modes are associated with

chaotic regions of phase space, and their frequency spec-

trum is said to be irregular because it is not described by

a smooth function of 3 integers. The frequency spectrum

has nevertheless specific statistical properties. Indeed, if we

considerchaoticmodesofthe samesymmetryclass, thedis-

tributionoftheconsecutivefrequencyspacingsσi= ωi+1−

ωi(scaled by the mean frequency spacing < ωi+1− ωi>)

is close to the parameter-free Wigner distribution P(σ) =

πσ/2exp(−πσ2/4) in accordance with the prediction of

Random Matrix Theory. A symmetry class correspondsto a

given azimuthal number m and a given symmetry (+ or −)

with respecttothe equator.Foreachsymmetryclass, aspec-

trum is thus determined from a realization of the Wigner

distribution and the chaotic spectrum is the superpositionof

all these spectra. Finally in order to construct the total spec-

trum, we need to know the ratio between chaotic and island

modes. This ratio tends to increase with rotationand,for the

rotation rate considered in the following Ω = 0.6ΩKwhere

ΩK=?GM/R3, it is close to 3.7 (Ligni` eres & Georgeot,

2009).

3 Autocorrelation of the synthetic spectra

The autocorrelation of the synthetic spectra has been inves-

tigated, for Ω = 0.6ΩK, by varying three parameters: the

frequency range of the spectrum, the inclination angle of

the star, and an amplitude threshold that only retains the

the highest amplitude peaks of the spectrum. The parameter

domain that we explored is the following : The frequency

range spans nδlarge separations δn= 2∆˜ n, where nδhas

been varied from 1 to 6. The amplitude threshold is chosen

to keep Nδ frequency peaks per large separation interval;

Nδhas been varied between 10 and 100. Once nδand Nδ

are fixed, the total number of frequencies of the spectrum is

nδ× Nδ. The inclination angle has been varied from 0 to

90◦.

Wefindthattheinclinationangleplaysanimportantrole

in the search for regular patterns because it affects the rel-

ative visibility of the chaotic and island modes. Indeed, the

disk-averagingfactor of the chaotic modes tends to increase

towards equator-on configurations while the disk-averaging

factor of island modes tends to decrease.

Let us first consider a high inclination configurationi =

63◦. In figures 1 and 2, the autocorrelation of the whole

spectrum (top panel), the chaotic spectrum (middle) and the

island spectrum (bottom)are displayed for Nδ= 57 and for

two different frequency ranges nδ= 3 (Fig. 1) and nδ= 6

(Fig. 2). The autocorrelation is defined as in statistics: it is

computed after substracting the mean and it is normalized

by the variance. The autocorrelation is therefore comprised

between-1 and 1.The verticalscale is differentfor the auto-

correlation of the island spectrum because, as expected, the

Tassoul like formula leads to strong peaks associated with

the ∆˜ nregular spacing and smaller peaks corresponding to

linear combinaisons of ∆˜ n,∆˜ℓ,∆m.

No structure is seen in the chaotic mode spectrum ex-

cept fora peaks at 2Ω (correspondingto the2Ω/δn= 0.836

peak of Fig. 1). This lack of structure can be understood

from the fact that the chaotic spectrum is a superposition

of the subspectra corresponding to the different symmetry

classes m±. These subspectra are statistically independent

and, as a consequence, the consecutive difference statistics

ofthechaoticspectrumisgenerallyclosetoaPoissonstatis-

tics thus explaining the featureless autocorrelation. The

pole-on configuration is an exception to this rule as only

axisymmetric modes m = 0 are visible in this case and the

superposition of only two independent subspectra does not

lead to Poisson statistics. The 2Ω peaks is due to a rota-

tional splitting between m = 1 and m = −1 modes. In the

p-modes asymptotic regime, the Coriolis force has indeed

a negligible effect on the frequency because its characteris-

tic time-scale is much larger then the oscillation time-scale.

As shown in Reese et al. (2006), this asymptotic property is

actually already correct at relatively low frequencies. Since

the centrifugal force does not distinguish between −m and

m modes, their frequencies can be considered as degener-

ate in the rotating frame thus leading to a 2mΩ splitting in

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Astron. Nachr. / AN (0000)3

Fig.1

spectrumresultingfromthe superpositionofa chaoticmode

spectrum and an island mode spectrum. The top panel dis-

plays the autocorrelation function of the whole spectrum,

while the middle and bottom panels shows the autocorre-

lation of the chaotic and island mode spectra, respectively.

The x-axis represents the frequency lag in units of the large

separation δn. The inclination angle is equal to i = 63◦, the

frequency range corresponds to 3 large separations and the

171 highest amplitude frequencieshave been retained in the

spectrum. The only significative feature of the autocorrela-

tion function is the 2Ω peak also seen in the autocorrelation

of the chaotic spectrum.

Autocorrelation of a high-frequency p-modes

the inertial frame. In addition, the number of m = 1 and

m = −1 modes in the spectrum is relatively high because

their disk-averagingannulationeffect is not strong (and less

important than for higher m modes), thus leading to a sig-

nificative peak at 2Ω in the autocorrelation function.

As shown in the top panel of Fig. 1, signatures of the

island mode regular patterns are not apparent in the auto-

correlation of the total spectrum. This is due to the fact that

the ratio of chaotic modes over island modes is too high in

this case. Increasing the frequency range to 6 large separa-

tions (Fig. 2) enables us to detect a peak at the large sepa-

ration δn(the peak seen at a scaled frequency lag of unity

on Fig. 1 or Fig. 2). This is expected as a larger frequency

range leads to the build up of the peaks associated with the

periodic features of the frequencyspectrum. For the present

high inclinationangle configuration,limiting the analysis to

the highest amplitude peaks does not help because in aver-

age the visibility of the island modes is not higher than the

visibility of the chaotic modes.

As illustratedin figures3 and 4 fori = 30◦, low inclina-

tion configurations are more favorable in order to detect the

island mode regular patterns. The autocorrelation functions

are shown for a frequency range nδ = 3 and for two dif-

ferent amplitude threshold Nδ= 68 (Fig. 3) and Nδ= 18

(Fig. 4). In both cases, features associated with the island

mode regular patterns are detected in the total autocorre-

lation function. This is because, as compared to the high

Fig.2

of the spectrum now spans 6 large separations, the number

of frequencies retained per large separation remaining the

same. The autocorrelation spectrum now shows a peak cor-

responding to the large separation.

Same as figure 1 except that the frequency range

Fig.3

to i = 30◦, the frequency range of the spectrum spans 3

large separations and the 204 highest amplitude frequencies

have been retained. The main correlation peak of the island

spectrum can be retrieved in the autocorrelation of the total

spectrum.

Same as Fig. 1. The inclination angle is now equal

inclination case, the proportion of island modes in the total

spectrum has increased. The comparison of figures 3 and 4

also shows that, contrary to the high inclination case, the is-

land mode regular patterns are more easily detected if the

analysis is limited to the highest amplitude modes. Finally

we checked that, as expected,the signature of these patterns

is stronger if the frequency range is increased.

4 Discussion and conclusion

Although based on strong simplifying assumptions, the

presentstudy shouldprovidesome clues to conducta search

for regular patterns in observed p-mode spectra of rapidly

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4 F. Ligni` eres, B. Georgeot & J. Ballot: P-modes in rapidly rotating stars

Fig.4

plitude frequencies have been retained. This helps to detect

some of thecorrelationpeaks that werenot significantin the

previous case.

Same as Fig 3. except that only the 54 highest am-

rotating stars. For example, we find that limiting the search

forregularpatternstothehighestamplitudefrequencypeaks

is not necessarily helpful. This depends on the inclination

angle.Generally,lowinclinationconfigurationsaremorefa-

vorablebecause the island modes are relativelymorevisible

than the chaotic modes. A robust feature of the autocorrela-

tion functions computedin the present study is the 2Ω peak.

This featuredoesnotrelyontheasymptoticregimeassump-

tion, it ratherrequiresthatthe Coriolis forcehas a negligible

effect which is already true at relatively low p-mode fre-

quency. Possible effects of the differential rotation on this

peak should nevertheless be tested, for example using the

formalism described in Reese et al. (2009).

The two most important assumptions in the construc-

tion of the synthetic spectra concern the asymptotic regime

and the modeamplitudes.We assumedthat island modefre-

quencies strictly follow the asymptotic formula (3) while it

is in fact only approximate due to finite wavelength effects

(see the dicussion in Ligni` eres & Georgeot(2009)). The de-

partures from the asymptotic formula (3) have been deter-

mined for different stellar models and frequency ranges in

Ligni` eres et al. (2006), Reese et al. (2008) and Reese et al.

(2009). As a first step, a simple way to model these non-

asymptotic effects in the present study would be to decrease

the frequency resolution of the synthetic spectra. A second

step would be to compute numerically a complete spectrum

using the codeTOP (Reese et al., 2006). Suchcomputations

are nevertheless very time consuming. Thus, simple mod-

els of the spectra like the one presented in this study will

still be useful as they enable us to cover a wider range of

parameters. In what concerns the amplitudes of the modes,

we lack a consistent theory to describe their excitation and

non-linear saturation in rapidly rotating stars. The present

model could nevertheless be improvedthrough better deter-

minations of the mode visibilities, including more realistic

stellar models as well as non-adiabatic, limb-darkening and

gravity darkening effects. The mode inertia could also be

taken into account in modelling the amplitudes.

Acknowledgements. We thank D. Reese for fruitful discussions.

This work was supported by the Programme National de Physique

Stellaire of INSU/CNRS and the SIROCO project of the Agence

National de la Recherche.

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c ? 0000 WILEY-VCH Verlag GmbH&Co.KGaA, Weinheim

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