Wave chaos and regular frequency patterns in rapidly rotating stars

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Wave chaos and regular frequency patterns in
rapidly rotating stars
F. Lignières
Laboratoire dAstrophysique de Toulouse et Tarbes
- France
in collaboration with B. Georgeot (IRSAMC), D.
Reese (postdoc at Sheffield Univ.), M. Rieutord
(LATT)
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Motivations
HR diagram of pulsating stars

• Helioseismology revolutionized our knowledge of
  the suns interior.
• Asteroseismology is due to revolutionize stellar
  evolution theory (Most, Corot, Kepler).
• But, the necessary mode identification is not an
  easy task (especially for early-type stars).

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Rapidly rotating stars are really not spherical !

• Existing perturbative models limited to small
  flatness (Saio 1981, Soufi et al. 1998)
• Need for a method able to handle significant
  centrifugal distortion

Altair 1.14 lt Re/Rp lt 1.21 d
Scuti, b Cep 1 lt Re/Rp lt 1.17
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Outline

• An accurate oscillation code for rapidly
  rotating stars
• Domain of validity of the perturbative methods
• The asymptotic organisation of the p-modes
  frequency spectrum at high rotation rates

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An oscillation code for rapidly rotating stars

• A linear boundary value problem

L(f)0, L is a linear operator boundary
conditions

• The method

• The coordinate system
• The spatial discretization in the radial and
  latitudinal direction
• A  large  matrix eigenvalue problem (Nr .
  Nq . Nf ) x (Nr . Nq . Nf )

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A surface-fitting coordinate system
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An oscillation code for rapidly rotating stars

• The method

• The coordinate system
• The spatial discretization
• Matrix eigenvalue problem QZ
  or Arnoldi-Chebyshev algorithm

• The tests

• The separable ellipsoïd case(Lignières et al.
  2001)
• Polytropic model of stars deformed by the
  centrifugal force (Lignières Rieutord 2004,
  Lignières et al. 2006, Reese et al. 2006)
• Effect of the Coriolis force Viriel test (Reese
  et al. 2006)

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An oscillation code for rapidly rotating stars

• The present simplifying assumptions are

• Polytropic model of star (N3)
• Adiabatic perturbations
• Uniform rotation

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Validity of the perturbative methods
(Reese et al. 2006)
Frequency range l0,1,2,3 n1, ,10
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Regular spacings in the frequency spectrum
(Lignières et al 2006, Reese et al, submitted
2007 )
W / WK 0.59
Frequency (mHz)
Degree of the spherical harmonic at W 0
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The asymptotic organisation of the p-modes
frequency spectrum
Travelling wave solutions in the small wavelength
(WKB) limit leads to the acoustic ray Hamiltonian
dynamics
is the wave vector

• integrable a modes are obtained from
  constructively interfering acoustic rays
  (Gough 1993) and the Tassouls asymptotic
  formula is recovered
• non-integrable a quantum (or wave) chaos looks
  for the fingerprints of classical chaos on the
  wave phenomena (frequency statistics).

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Wave chaos in stars ?
Quantum mechanics
Acoustics

• Schrödinger equation
• Wave function
• and energy level
• Classical limit

• Linearized equations
• Acoustic modes
• and frequencies
• Ray dynamics

Harmonic solution
e(-i E t/h)
e(-i w t)
WKB approximation
h g 0

• g

• The (asymptotic) dynamical
  system is
• integrable a semi-classical quantization (e.g.
  Bohrs atomic model)
• chaotic a quantum or wave chaos looks for
  the fingerprints of
  classical chaos on the wave phenomena

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Acoustic ray dynamics at W 0
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Acoustic ray dynamics at W 0
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Acoustic ray dynamics at W / WK 0.59
The phase space has a mixed structure (island
chains, central chaotic sea, region of surviving
KAM tori)
Does this phase space structure reflects in the
structure of the frequency spectrum ?
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Relating the modes with the phase space structures
The Husimi distribution H(kq,q)
provides a phase space representation of the
modes by projecting them onto localized wave
packets
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Relating the modes with the phase space structures
What are the properties of the frequency subsets
associated with each mode family ?
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The island p-modes frequency subset
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A simple model for the axisymmetric island p-modes

• Inspired from quantization of laser modes in
  cavities
• Gaussian wave beam propagating along the
  periodic orbit of the island (Permitin Smirnov
  1996)
• Quantization condition leads to the right
  formula with

and

• This model value approximates dn the numerical
  within 2.2 percent
• dl also depends on along the
  periodic orbit

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The chaotic p-modes frequency sub-set
Statistics of consecutive frequency spacings
w/w1
w i1 - w i
X i
lt w i1 - w i gt

• Statistical frequency repulsion
• Compatible with the Wigner distribution
• from Random Matrix Theory, a generic distribution
  for chaotic systems

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Possible implications for the asteroseismology of
rapidly rotating stars

• Regular patterns recognition would lead to
• identification of the island
  modes
• determination of the seismic
  observables dn, dl and dm
• Chaotic mode frequencies are highly sensitive to
  small changes in the stellar model
• The chaotic modes are non-radial p-modes probing
  the stars center !

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Conclusion

• Domain of validity of the perturbative methods as
  a function of the rotation rate
• Ray dynamics and quantum chaos tools reveal that
  the p-modes axisymmetric spectrum is the
  superposition of  independent  frequency
  subsets reflecting the phase space structure,
  involving

• Regular frequency patterns of island modes
• Statisical frequency repulsion of chaotic modes

Both results are unlikely to change in real
(non-polytropic) stars (except in the presence of
abundance discontinuities where, as in
non-rotating model, the WKB breaks down)

• A manifestation of quantum chaos phenomenology in
  a large scale natural system

(Lignières Georgeot submitted)
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The next steps towards realistic models

• Extension to non-axisymmetric modes
• Realistic stellar structure models for
  asteroseimic studies (D. Reese)
• Modes visibility and stability, synthetic
  frequency spectra, mode identifications
• Gravity modes in deformed stars (a postdoc
  position starting in 2008 should be advirtized
  soon).
• Are wave chaos features observable in observed
  spectra ?

Dont be afraid of rapidly rotating stars !