Wave chaos and regular frequency patterns in rapidly

rotating stars

F. LignièresLaboratoire d’Astrophysique de Toulouse

et Tarbes - Francein collaboration with B. Georgeot (IRSAMC), D. Reese (postdoc at Sheffield Univ.), M. Rieutord (LATT)

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

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) Yl

m()

a 1D boundary value problem Non-spherical (r,) = f(r,) eim

a 2D boundary value problem

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

The method The coordinate system

The spatial discretization in the radial and latitudinal direction

A « large » matrix eigenvalue problem : (Nr . N. Nf ) x (Nr . N. Nf )

An oscillation code for rapidly rotating stars

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

A linear boundary value problem

A surface-fitting coordinate system

(Bonazzola et al. 1998)

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

An oscillation code for rapidly rotating stars

Polytropic model of star (N=3)

Adiabatic perturbations

Uniform rotation

The present simplifying assumptions are:

Accuracy 0,6 Hz

Frequency range l=0,1,2,3 n=1, …,10

Validity of the perturbative methods

1st order 2nd order

(Reese et al. 2006)

Accuracy 0,08 HzAccuracy 0,08 Hz

Regular spacings in the frequency spectrum

Degree of the spherical harmonic at = 0

Fre

qu

en

cy

(mH

z)

complete calculation empirical formula

n,l ~ n n + l + m m +

(Lignières et al 2006, Reese et al, submitted 2007 )

K = 0.59

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

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

Harmonic

solution

WKB approximati

on

Acoustics

&

Acoustic ray dynamics at = 0

Acoustic ray dynamics at = 0

Poincaré section at r=0.92 rs

k

k

k

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 ?

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

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

The island p-modes frequency subset

K = 0.84

= 0, n =11

= 1, n =11

= 2, n =11

n nodes

nodes

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

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

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 !

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)

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 !