1. Wave chaos and regular frequency patterns in rapidly rotating stars F. Lignières Laboratoire d’Astrophysique de Toulouse et Tarbes - France in collaboration with B. Georgeot (IRSAMC), D. Reese (postdoc at Sheffield Univ.), M. Rieutord (LATT)

2. Motivations Helioseismology revolutionized our knowledge of the sun’s 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). HR diagram of pulsating stars

4. Rapidly rotating stars are really not spherical ! Altair: 1.14 < Re/Rp < 1.21  Scuti,  Cep: 1 < Re/Rp < 1.17 Existing perturbative models limited to small flatness ( Saio 1981, Soufi et al. 1998)  Need for a method able to handle significant centrifugal distortion The shape of Achernar Rp Re Re/Rp ~ 1.5 !! Domiciano et al. 2003 Spherical case  (r,  ) = f(r) Y l m (  ) a 1D boundary value problem Non-spherical  (r,  ) = f(r,  ) e im  a 2D boundary value problem

5. 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 Outline

6. The method  The coordinate system  The spatial discretization in the radial and latitudinal direction  A « large » matrix eigenvalue problem : ( N r. N . N f ) x ( N r. N . N f ) An oscillation code for rapidly rotating stars L(f)=0, L is a linear operator + boundary conditions A linear boundary value problem

7. A surface-fitting coordinate system (Bonazzola et al. 1998)

8. The method  The coordinate system  The spatial discretization  Matrix eigenvalue problem QZ or Arnoldi-Chebyshev algorithm  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) The tests An oscillation code for rapidly rotating stars

9.  Polytropic model of star (N=3)  Adiabatic perturbations  Uniform rotation The present simplifying assumptions are:

10. Accuracy 0,6  Hz Frequency range l=0,1,2,3 n=1, …,10 Validity of the perturbative methods 1 st order 2 nd order (Reese et al. 2006) Accuracy 0,08  Hz

11. Regular spacings in the frequency spectrum Degree of the spherical harmonic at  = 0 Frequency (mHz) complete calculation empirical formula  n, l ~  n n +  l +  m m +  (Lignières et al 2006, Reese et al, submitted 2007 )  K = 0.59

12. 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: integrable  modes are obtained from constructively interfering acoustic rays (Gough 1993) and the Tassoul’s asymptotic formula is recovered non-integrable  quantum (or wave) chaos looks for the fingerprints of classical chaos on the wave phenomena (frequency statistics). is the wave vector

13. Schrödinger equation Wave function and energy level Classical limit Linearized equations Acoustic modes and frequencies Ray dynamics e (-i E t/h) e (-i  t) h  0    Wave chaos in stars ? The (asymptotic) dynamical system is:  integrable  semi-classical quantization (e.g. Bohr’s atomic model)  chaotic  quantum or wave chaos looks for the fingerprints of classical chaos on the wave phenomena Quantum mechanics Harmoni c solution WKB approximation Acoustics &

14. Acoustic ray dynamics at  = 0

15. Poincaré section at r=0.92 r s k    k kk 

16. Acoustic ray dynamics at  K = 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 ?

17. Relating the modes with the phase space structures The Husimi distribution H(k   ) provides a phase space representation of the modes by projecting them onto localized wave packets

18. Enables to unambiguously define « island » modes, « chaotic » modes, whispering gallery modes, … What are the properties of the frequency subsets associated with each mode family ? Relating the modes with the phase space structures

19. The island p-modes frequency subset  K = 0.84 = 0, n =11= 1, n =11= 2, n =11 n nodes nodes  

20. 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  n the numerical within 2.2 percent  l also depends on along the periodic orbit

21. The chaotic p-modes frequency sub-set X i =  i+1 -  i  i+1 -  i > Statistics of consecutive frequency spacings Statistical frequency repulsion Compatible with the Wigner distribution from Random Matrix Theory, a generic distribution for chaotic systems  

22. 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  n,  l and  m  Chaotic mode frequencies are highly sensitive to small changes in the stellar model  The chaotic modes are non-radial p-modes probing the star’s center !

23.  Regular frequency patterns of island modes  Statisical frequency repulsion of chaotic modes  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: Conclusion 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)

24.  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 ? The next steps towards realistic models Don’t be afraid of rapidly rotating stars !