Pmodes in rapidly rotating stars  looking for regular patterns in synthetic asymptotic spectra
ABSTRACT According to a recent raybased asymptotic theory, the highfrequency pmode spectrum of rapidly rotating stars is a superposition of frequency subsets associated with dynamically independent regions of the raydynamics phase space. At high rotation rates corresponding to typical $\delta$ 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' diskaveraging 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 equatoron configurations, (iii) in the absence of differential rotation, the $2 \Omega$ rotational splitting between $m=1$ and $m=1$ modes is an easy feature to detect. Comment: 4 pages, 4 figures, proceedings of the HELASIV International Conference, accepted for publication in Astronomische Nachrichten
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ABSTRACT: The effects of rapid stellar rotation on acoustic oscillation modes are poorly understood. We study the dynamics of acoustic rays in rotating polytropic stars and show using quantum chaos concepts that the eigenfrequency spectrum is a superposition of regular frequency patterns and an irregular frequency subset respectively associated with nearintegrable and chaotic phase space regions. This opens fresh perspectives for rapidly rotating star seismology and also provides a potentially observable manifestation of wave chaos in a largescale natural system.Physical Review E 08/2008; 78(1 Pt 2):016215. · 2.31 Impact Factor
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arXiv:1009.5939v1 [astroph.SR] 29 Sep 2010
Astron. Nachr. / AN 000, No.00, 1–4 (0000) / DOI please set DOI!
Pmodes 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 ToulouseTarbes (LATT); F31400 Toulouse, France
2CNRS; Laboratoire d’Astrophysique de ToulouseTarbes (LATT); F31400 Toulouse, France
3Universit´ e de Toulouse; UPS; Laboratoire de Physique Th´ eorique (IRSAMC); F31062 Toulouse, France
4CNRS; LPT (IRSAMC); F31062 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 raybased asymptotic theory, the highfrequency pmode spectrum of rapidly rotating stars is a
superposition of frequency subsets associated with dynamically independent regions of the raydynamics 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’ diskaveraging 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 equatoron 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 WILEYVCH 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 welldefined sets of integers. This mode structure al
lowed us to extract information on the stellar interior from
the spectrum of oscillation modes (ChristensenDalsgaard,
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: email: francois.lignieres@ast.obsmip.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 lowfrequency pmodes 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 pmode 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).
c ? 0000 WILEYVCH Verlag GmbH&Co.KGaA, Weinheim
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2 F. Ligni` eres, B. Georgeot & J. Ballot: Pmodes 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 150daylong 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 diskaveragingfac
tor computed for highfrequency pmodes in a polytropic
stellarmodel(seeformula(37)inLigni` eres & Georgeot(2009)).
Above a certain rotation rate, the pmodesspectrum 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 + ∆mm − 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 parameterfree 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
diskaveragingfactor of the chaotic modes tends to increase
towards equatoron configurations while the diskaveraging
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
between1 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
poleon 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
pmodes asymptotic regime, the Coriolis force has indeed
a negligible effect on the frequency because its characteris
tic timescale is much larger then the oscillation timescale.
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 xaxis 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 highfrequency pmodes
the inertial frame. In addition, the number of m = 1 and
m = −1 modes in the spectrum is relatively high because
their diskaveragingannulationeffect 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 pmode spectra of rapidly
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4 F. Ligni` eres, B. Georgeot & J. Ballot: Pmodes 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 pmode 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
nonlinear 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 nonadiabatic, limbdarkening 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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Garc´ ıa Hern´ andez, A., et al.: 2009, A&A 506, 79
Ligni` eres, F., Rieutord, M., & Reese, D.: 2006, A&A 455, 607
Ligni` eres, F., & Georgeot, B.: 2008, Pys. Rev. E 78, 016215
Ligni` eres, F., & Georgeot, B.: 2009, A&A 500, 1173
Reese, D., Ligni` eres, F., & Rieutord, M.: 2006, A&A 455, 621
Reese, D., Ligni` eres, F., & Rieutord, M.: 2008, A&A 481, 449
Reese, D. R., MacGregor, K. B., Jackson, S., Skumanich, A., &
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