workshop: « doradus stars in the corot fields » nice, 26-28 may, 2008 1 driving mechanism and...

Workshop: « Doradus stars in the COROT fields »Nice, 26-28 May, 2008

1

Driving mechanism and energetic aspects of pulsations

in Doradus stars

A. Miglio J. Montalban

Liège, Belgium

M.-A. DupretLESIA, Paris Observatory, France

A. Grigahcène

Algiers, Algeria


2

Driving mechanism and energetic aspects in Doradus stars

ExcitationAmplitudeand phases

Mode identification


3

Convection and partial ionization zones Doradus starsInternal physics:

•1 convective core•1 convective envelope

He an H partial ionization zonesare inside the convective envelope

Fc/F

rad = (Γ3-1) / 1


4

Driving mechanism Doradus stars

Main driving occurs in the transition region where the thermal relaxation time is of the same order as the pulsation periods

For a solar calibrated mixing-length, the transition region for the g-modes is near the convective envelope bottom.

Quasi-adiabaticregion

Non-adiabaticregion

g50

Log(th)

S 0

Coupling between

• the dynamical equations and

• the thermal equations


5

Driving mechanism Doradus

Flux blocking at the base of the convective envelope

Motor thermodynamical cycle

t i tml e e )( Y)( )( ,rr t , , r,r

M

0 d

M

0 d

d d

2

1

22r

m r

mm

LTT


6


M = 1.6 M0 , Teff = 7000 K , = 2 , Mode =1, g50

Flux blocking at the base of the convective envelope

Motor thermodynamical cycle

Work integral Luminosityvariation


7

Driving mechanism DoradusM = 1.6 M0

Teff = 7000 K = 2Mode =1, g50

Role of time-dependent

convection

c : Life-time of convective elements: Angular frequency

Log ( c )

Fc / F


8

Convection – pulsation interaction

3-D hydrodynamic simulations

All motions are convective ones

In particular the p-modes(present in the solution)

Nordlund & SteinSamadi, Belkacem (Meudon)

Analytical approach

Separation between convection and pulsationin the Fourier space of turbulence

Convective motions:short wave-lengths

Oscillations:long wave-lengths

1. Static solution without oscillations

2. Stability study of this solution

Perturbation Oscillations

Gough´s theory Gabriel´s theory

MLT

Gabriel´s theory


9

Hydrodynamic equations

Mean equations Convective fluctuations equations

Equations of linear non-radial

non-adiabatic oscillationsCorrelation terms

PerturbationPerturbation

Convection – pulsation interaction: Gabriel´s theory

• Convective flux• Reynolds stress• Turbulent kinetic energy dissipation

Perturbation of


10

Radiative luminosity Convective luminosity

Turbulent pressure

Turbulent kineticenergy dissipation

Convection – pulsation interaction: Work integral


11


WFRr: Radial radiative

flux term

M = 1.6 M0

Teff = 7000 K = 2Mode =1, g50


12



flux term

WFcr: Radial convective

flux term

M = 1.6 M0

Teff = 7000 K = 2Mode =1, g50


13


Teff = 7000 K = 2Mode =1, g50


flux term


flux term

Wpt: Turbulent pressure

W2: Turbulent kinetic

energy dissipation


14


Teff = 7000 K = 2Mode =1, g50


flux term


flux term

WFh: Transversal convective

and radiative flux


15


Teff = 7000 K = 2Mode =1, g50


flux term


flux term

Wtot: Total work


16

Instability strips Doradus

= 1


17

Instability strips Doradus

= 1


18

Unstable modes Doradus

Dor g-modes

Sct p-g modes


19



20


Period range decreases with


21



22


Key point: Location of the convective

envelope bottom

Instability region verysensitive to the effective

temperature and thedescription of convection

(, …)


23

Hybrid Sct –

Dor

Comparison : Sct red edge (=0, p1) Dor instability strip (=1)

HD 209295 Handler et al.

(2002)Tidally excited ?

HD 8801Henry et al. (2005)Am star

HD 49434


24

Dor g-modes

Sct p-g modes

Unstable modesHybrid Sct – Dor


25

Unstable modes

Stableregions

Hybrid Sct – Dor


26

HD 49434

HD 49434

Hybrid Sct – Dor


27

Hybrid Sct – Dor Unstable modes (HD 49434)

2 4 20 400.5


28

Hybrid Sct – Dor Work integral : = 1

p2


29

Hybrid Sct – Dor Work integral

p2

p1

Work integral : = 1


30


p2

p1

g1

Work integral : = 1


31


p2

p1

g1

g2

Work integral : = 1


32


p2

p1

g1

g2

g3

Work integral : = 1


33


p2

p1

g1

g2

g3

g4

Work integral : = 1


34


p2

p1

g1

g2

g3

g4

g6

Work integral : = 1


35


g6

Work integral : = 1


36


g6

g8

Work integral : = 1


37


g6

g8

g10

Work integral : = 1


38


g6

g8

g10

g12

Work integral : = 1


39


g6

g8

g10

g12

g15

Work integral : = 1


40


g6

g8

g10

g12

g15

g21

Work integral : = 1


41


g21

Work integral : = 1


42


g21

Work integral : = 1


43


g21

g30

Work integral : = 1


44


g21

g30

g35

Work integral : = 1


45


g21

g30

g35

g40

Work integral : = 1


46

Hybrid Sct – Dor Work integralWork integral

= 1 , g6

= 1 , g25


47

Hybrid Sct – Dor Radiative damping mechanism

Growth-rate


48

Hybrid Sct – Dor Radiative damping mechanism

Growth-rate

< 0


49

Hybrid Sct – Dor Work integralPropagation diagrams


50


g-mode cavity Evanescent

g-mode p-mode

g-mode cavity


51

Hybrid Sct – Dor Work integralAsymptotic behaviour

In propagation regions : 2 < N2 , L2


52

Hybrid Sct – Dor Work integralAsymptotic behaviour

In evanescent regions : L2 < 2 < N2


53

Hybrid Sct – Dor Work integralEigenfunctions behaviour

= 1 , g25

Small amplitudeIn the g-mode cavity

Large amplitudeat the bottom of theconvective envelope


54


= 1 , g25

Radiative dampingnegligible

Significantdriving


55


= 1 , g6

Large amplitudein the g-mode cavity

Smaller amplitudeat the bottom of theconvective envelope

EVANESCENT


56

Hybrid Sct – Dor

= 1 , g6

Eigenfunctions behaviour

Significantradiative damping

Drivingnegligible


57

Hybrid Sct – Dor Work integralWork integral

= 1 , g6

= 1 , g25


58

Hybrid Sct – Dor Unstable modes (HD 49434)

2 4 20 400.5


59


= 1

= 2


60


=2, g25

= 1

= 2


61

Spectro-photometric amplitudes and phasesand mode identification

Doradus

Phase-lags

Line-profile variations

Moment method

Amplitude ratios

Very sensitive to the non-adiabatictreatment of convection

Mode identification

Spectroscopy Photometry


62

• Distortion of the stellar surface described by the lagrangian displacement of matter

• Temperature :

• Flux :

• Limb darkening :

lnln

lnln

lnln

e

e

eff

eff

eff

T

gg

gT

TT

TT

TT

e

e

eff

eff

eff ln

ln

ln

ln

g

g

g

F

T

T

T

F

F

F

lnln

lnln

lnln

e

e

eff

eff

eff

h

gg

gh

TT

Th

hh

• Thermal equilibrium in the local atmosphere

Hypotheses

Photometric amplitudes and phasesand mode identification


63

Non - adiabaticcomputations

Influence of the local effective temperature

variations

Influence of the localeffective gravity

variations

Equilibriumatmosphere models

Stellar surfacevariation

)( cos ln

ln

ln

ln

) ( cos ln

ln

ln

ln )( cos )2)(1(

)(cos 10ln

2.5

e

e

eff

eff

eff

eff

tg

g

g

b

g

F

tT

T

T

b

T

Ft

bi Pm

T

m

Filters Integration on the pass-band

Linear computations Amplitude ratios and phase differences

Dependence with the degree Identification of

Monochromatic magnitude variation


64

Multi-colourphotometricObservations

Amplitude ratios& Phase lags

Range of frequencies

Mode identificationRange of frequencies

Non-adiabatic asteroseismology

Improving the fit

Stellar parameters

Scuti

Doradus

Cephei

SPB

Convection

Chemical composition

Atmosphere models - Limb darkening

Time-dependence

Mixing Length ()

Full spectrum

Non-adiabaticcomputations

Equilibrium models,Stellar evolution code


65



Time-dependent convection

Frozen convection

Phase-lag


66

Frozen convection Time-dependent convection

-mechanism Not allowed by time-dependent convection




67

Doradus

3 frequencies: f1=1.32098 c/d, f2=1.36354 c/d, f3=1.47447 c/d

Balona et al. 1994 Strömgren photometry

Balona et al. 1996Simultaneous photometryand spectroscopy

Spectroscopic mode identification: (1, m1) = (3, 3),(2, m2) = (1, 1),(3, m3) = (1, 1)



68

Doradus


Phase-lag ( Vmagnitude – displacement ):

Observations: 1 = - 65° ± 5° 3 = - 29° ± 8°

Theory: Time-dependent convection Frozen convection( = 2) = - 30° = - 165°

Spectro-photometric phase differences


69

Photometric mode identification Doradus

Fro

zen

= 2 - MLT = 1 - MLT

Tim

e-de

pend

ent

Best models: Time-dependent convection = 1 modes


70

Influence of the effective temperaturevariations

Influence of the effective gravity variations

)( cos ln

ln

ln

ln

) ( cos ln

ln

ln

ln )( cos )2)(1(

)(cos 10ln

2.5

e

e

eff

eff

eff

eff

tg

g

g

b

g

F

tT

T

T

b

T

Ft

bi Pm

T

m

Importance of ultraviolet observations(bracketing the Balmer discontinuity)

Gravity derivatives vary quickly in u-v

Changes the weight of Teff and ge terms

Helps for the mode identification and

gives constraints on | Teff/Teff|

Stromgren, Genevasystems are perfects


71

Conclusions

• Driving mechanism:

Main driving due to convective blocking at the base of the convective envelope

Time-Dependent convection does not inhibits the mechanism

Driving mechanism and energetic aspects in Doradus stars

The size of the convective envelope is the key point) predictions very sensitive to the treatment of convection


72

Conclusions

• Mode identification, multi-color amplitudes and phases:Time-Dependent Convection required, gives constrain on :

• Convective envelope• Interaction convection – oscillations• Atmosphere models

Mode identification and non-adiabaticasteroseismology of Doradus stars

• Hybrid Scuti - Doradus• Driving at the bottom of the convective envelope• Radiative damping, deep in the g-mode cavity

Instability ranges depend on:• Location of convective envelope• Spherical degree • Local wave-length ) Brunt-Vaisala, Lamb frequencies


73

Doradus Stabilization mechanism

Radiative dampingin the g-modes cavity


74

Doradus Stabilization mechanism

Radiative dampingin the g-modes cavity


75

Comparison : Sct red edge (=0, p1) Dor instability strip (=1)

= 1.8


76

Stabilization mechanism Doradus


flux term

M = 1.6 M0

Teff = 7000 K = 2Mode =1, g50


77



78



79



80

Non-adiabatic stellar oscillations: utility

Excitation mechanisms Mode identification

Solar-like oscillations

Unstable modes: growth rates

Stable modes: damping rates Line-widths in the power spectrum

Observations

Stochastic excitation models Amplitudes

Stable modes: damping ratesStable modes: damping rates


81

Radiative luminosity Convective luminosity

Turbulent pressure

Turbulent kineticenergy dissipation

Convection – pulsation interaction: Work integral


82

Solar-type oscillations

Difficulties : 1. Treatment in the efficient part of convection

Very short wave-length oscillations of the eigenfunctions

Local treatment


83

Difficulties: 1. Treatment in the efficient part of convection

Origin of the problem:

dm

LdsTi C

dssd

i

LLL

C

CC

1

0 (...) 2

sidr

sdi

lC2

2

C

)/exp( )/exp( 2 lriclric ... s cc1

Wavelength much shorter than the mixing-length !



84


Non-local (Balmforth 1992) Local (Gabriel 2003)

Introduction of new free parameters

Solutions



85


Solutions

1

22

RRC FF

VsTVsT

Ts

1

C

CC

CC ssss

0 (...) (

2

sidr

sdi

lC2

2

C

Local (Gabriel 2003)



86

Difficulties: 2. Treatment of turbulent pressure perturbation

Increases the order of the system

Very stiff problem at the boundaries

Numerical instabilities



87

Models of stochastic excitation

• The Sun is a vibrationally stable oscillator.• Excitation of the mode is due to stochastic forcing coming from turbulent convective motions.

IP

Vs 2

Non-adiabatic models

Damping rate:

Stochastic models

Acoustical noise generation rate: P

Velocity

Theory

ObservationsLine-widths Observed amplitudes

Solar-type oscillations: confrontation to observations


88

Theoretical damping rates ↔ line-widths observed by BiSON (Chaplin et al. 1997)



89




90




91




92

Scuti Stables and unstable modes

Time-dependent convection

p7

p6

p4

p5

p3

g1

f

p1

p2

g2g3g4 g6 g8

Frozen convection

p7

p6

p4

p5

p3

g1

f

p1

p2

g2g3g4 g6 g8

= 2 - 1.8 M0 - = 1.5


93

Scuti Instability strips

Radial modes


94

Scuti Instability strips

= 2 modes


95

Photometric amplitudes and phases

=0=1

=3

=2

=2

=3

=0=1=2

=3

phase(b-y)-phase(y) (deg) phase(b-y)-phase(y) (deg)

Amplitude ratios vs. phase difference - Stroemgren photometry

= 0.5 = 1

Scuti


96

Conclusions

Damping rates Line-widths

Convection-pulsation interaction in solar-like stars

Efficient part of convective envelope

Local treatment → Spatial oscillationsof the eigenfunctions→ Introduction of a freeparameter in the perturbationof the closure equations

Perturbation of turbulent pressure

→ Numerical instabilities

AmplitudesStochasticexcitation

We found a model fitting theobserved damping rates but …

Theoretical difficulties Confrontation to observations


97


98

Oscillations de type solaire

Taux de dámortissement confrontables aux observations (largeurs de raie)

Mécanisme déxcitation : Excitation stochastique

Oscillateur vibrationellement stableforcé stochastiquement par la convection

Contrainte sur les modèles non-adiabatiquesdíntéraction convection-oscillations


99


Difficulté des modèles non-adiabatiques:intéraction convection-oscillations

Grande enveloppe convectiveConvection efficace ( tc >> tp )

Oscillations spatiales non-physiquesdes fonctions propres


100





Solutions

Non-locales (Balmforth 1992) Locales (Gabriel 2003)

Introduction de paramètres libres supplémentaires


101

1

22

RRC FF

VsTVsT

Ts





Solutions

Locales (Gabriel 2003)

1

C

CC

CC ssss


102

Conclusions

Mécanismes d’excitation

Bandes d’instabilité

Amplitudes et phasesphotométriques

Identification des modes

Astérosismologienon-adiabatique

Astérosismologie“classique” (fréquences)


103


Conclusions


Bandes d’instabilitéAstérosismologienon-adiabatique


Mécanisme (Fe) Contraintes sur lamétallicité

Marchetrès bien

Idem Idem OK mais effetde la rotation ?

Cephei

Modes p

SPB

Modes g


104


Conclusions


Bandes d’instabilitéAstérosismologienon-adiabatique


Frontière bleue: mécanisme (HeII)Frontière rouge: convection

Contraintes sur lesmodèles de convection et d’intéractionconvection - pulsation

Bon accordavec solaire

IdemNettement mieuxavec l’intéraction convection-pulsation

OK mais restedifficile

Effet de larotation ?

Scuti

Modes p

Dor

Modes g


105

Conclusions

Type solaire

Modes p

Mécanismes d’excitationAmplitudes et phases

photométriques

Excitation stochastique,modélisation difficile de l’intéraction convection – oscillations

Amplitudes données parles modèles d’excitation stochastique.

Grands espoirs futurs :


106


Difficulté des modèles non-adiabatiques:intéraction convection-oscillations:

Solutions

Non-locales (Balmforth 1992) Locales (Gabriel 2003)


107

Scuti Taille de l´enveloppe convective pourdifférentes températures effectives

Teff=8345.5 K

Teff=6119.5 K

M=1.8 M0, α=1.5


108

Teff / Teff|

Longueur de mélange - différents - convection gelée

Sensibilité à la structure de l´enveloppe convective

Scuti Amplitudes et phases photométriques


109

Photométrie multi-couleur et astérosismologie non - adiabatique

Observations en photométrie

multi-couleur

Rapports d´amplitude& déphasages

Modèles d´équilibreCode d´évolution stellaire

Calculs non-adiabatiques

Identification de mode

Paramètres stellaires

Scuti

Doradus

Cephei

SPB

Convection

Metallicité (Z)

Modèles d´atmosphère - Limb darkening

Mixing length ()

Hydrodynamique

FST


110

Full Spectrum of Turbulence Longueur de mélange

Teff / Teff|

Amplitudes et phases photométriques Scuti

Sensibilité à la structure de l´enveloppe convective


111


Full Spectrum of Turbulence Longueur de mélange

= 3 = 3

= 2

= 2 = 1

= 1 = 0

= 0

Q = 0.015 d Q = 0.033 d

Rapport d´amplitude vs. Différences de phase - photométrie Stroemgren


112

Doradus

• Types spectraux

F

• Masses

+/- 1.5 M0

• Périodes

0.3 à 3 jours

modes g


113

Doradus Amplitudes et phases photométriques

Comparaison : convection gelée convection dépendant du temps

M = 1.5 M0 - Teff = 7000 K - = 1.8


114

Type solaire

• Types spectraux

F et G

• Masses

1 M0 à 1.5 M0

• Périodes

Quelques minutes

Modes p élevés

Faibles amplitudes


115

• Excitation mechanisms

Photometric amplitudesand phases in different filters

Identification of the degree

Non - adiabaticasteroseismology

Non adiabatic oscillationsat the photosphere

Need of a non - adiabatic codefor the confrontation between theoryand observations

Utility of our non - adiabatic code

• Multi-colour photometry


116

Introduction

Stellar pulsations

• Pressure modes

Acoustic waves

• Gravity modes

Buoyancy force

Asteroseismology


117

stellar oscillations Non - radial non - adiabatic

Splitting in spherical harmonics

Non - radial

p - modes Acoustic waves

g - modes Buoyancy force

1)


118

stellar oscillations Non - radial non - adiabatic

S 0

non - adiabatic

Coupling between the dynamical thermaland equations

• Equation of momentum conservation

• Equation of mass conservation

• Poisson equation

dynamical

• Equations of transfer by radiation and convection

thermal

• Equation of energyconservation

1)


119

)( cos ln

ln

ln

ln

) ( cos ln

ln

ln

ln )( cos )2)(1(

)(cos 10ln

2.5

e

e

eff

eff

eff

eff

tg

g

g

b

g

F

tT

T

T

b

T

Ft

bi Pm

T

m

Stellar surfacedistortion

Influence of the local effective temperature

variations

Influence of the localeffective gravity

variations

Equilibriumatmosphere models

(Kurucz 1993)

Monochromatic magnitude variation

Non - adiabaticcomputations

Filters Integration on the pass-band

Linear computations Amplitude ratios and phase differences

Dependence with the degree Identification of


120

4.2 Scuti Time dependent convection - MLTTheory of M. Gabriel

Red edge of the instability strip

Time-dependent convectionFrozen convectionRadial modes – 1.8 M0 , = 1.5

p7

p6

p5

p4

p3

p2

p1

p8

p7

p6

p5

p4

p3

p2

p1

p8


121

4.2 Scuti Red edge of the instability strip

Time-dependent convection = 2 modes – 1.8 M0 , = 1.5

Frozen convection

p7

p6

p5

p4

p3

p2

p1

fg1g2

g3 g4 g5 g6 g7 g8 g9 g10

p7

p6

p5

p4

p3

p2

p1

fg1g2

g3 g4 g5 g6 g7 g8 g9 g10


122

p7

p2

p3

p4

p5

p6

g2

g1

fp1

g7 g8g6

g4

g3

g5

Frozen Convection Time-dependent convection

p7

p6

p4

p5

p3

g1

fp1

p2

g2g3g4g5 g6 g7 g8

Figure 5 Figure 6

III. Instability StripIII. 1. δ Scuti stars


123

Scuti Bandes d´instabilité

p7

p1

1.4 M0

2 M0

1.8 M0

1.6 M0

2.2 M0

= 1 - = 0


124


p7

p1

1.4 M0

1.6 M0

1.8 M0

2 M0

2.2 M0

= 0.5 - = 0


125


p6

fB

g7

fR

1.4 M0

2 M0

2.2 M0

1.8 M0

1.6 M0

= 1.5 - = 2


126

Plan de l’exposé

1. Introduction

2. Oscillations stellaires non-adiabatiques : utilité

5. Conclusions

4. Applications • Cephei• Slowly Pulsating B• Scuti• Doradus• Type solaire

3. Modélisation du problème • Atmosphère• Convection


127

Z

Teff / Teff|

Multi-colour photometry

Cephei

16 Lacertae

High sensitivity of thenon - adiabatic resultsto the metallicity

L / L|

Z = 0.015Z = 0.02Z = 0.025

Z = 0.015Z = 0.02Z = 0.025

U

B

A

A

U

V

A

A

Johnsonphotometry


128

Modes excitation

Z > 0.016

Non-adiabatic constraints on the metallicity

4.1 Cephei : HD 129929

Unstable

Stable


129

Contraintes sismiques sur l´overshooting

ov = 0.1 ± 0.05

ov= 0.1ov= 0.2

ov= 0

Instable

Stable

Cephei : HD 129929


130

Excitation mechanism

M = 4 M0

Teff = 13 955 K

Z = 0.02

Mode = 1 g22

Work integral and luminosity variation from the centerto the surface of the star

Slowly Pulsating B stars


131

Identification photométrique des modes

HD 74560 = 1

= 2

= 3

Slowly Pulsating B stars


132


Transitionzone

mechanism of excitation

Partial ionization zone of Helium II =

Opacity and thermal relaxation time from the center to the surface

Scuti

PS

ad

log (th)


133

M = 1.8 M0

Teff = 7480 K

Z = 0.02 , = 1

Radial fundamental mode

Work integral and luminosity variation from the centerto the surface of the star

Scuti



134

Modes stables et instables Scuti= 0 - 1.8 M0 - = 1.5

p1

p8

p7

p6

p5

p4

p3

p2

Convection dépendant du tempsConvection gelée

p1

p8

p7

p6

p5

p4

p3

p2


135

Mécanisme d´excitation :

Doradus

?? Bloquage convectif ?? (Guzik et al. 2000)

Travail intégré

M = 1.6 M0

Teff = 7000 K = 2Mode =1 , g47


136

1.6 M0 - = 1 - = 1.5

Modes instables Doradus


137

0vρtρ

Pp--vvρt

vρ

v-vvρUt

ρU

PF RN


0vρt

ρ

PΦ-ρρt

)(ρ

vvv

vv

PF-ρερU

t

ρURN

P: Pressure tensor ; p : its diagonal component.

Radiative Flux

XXX

RG

RG

pP

ppp

PPP

1

RF

Hydrodynamic equations


138

Mean equations

RG

RGNCR

TRGTRG

ppVV

dtd

ppVFFdtsd

T

pppdtud

uρdtρd

2

2

2

21

0

TTpVV 1

RG ppV

2 Dissipation of turbulent kinetic energy into heat

Reynolds stress tensor

2rT Vp Turbulent pressure



139

ssV

dt

sd

dt

sd

T

T

uVV

ppdt

Vd

Vdt

d

C

C

1

3

8

0

1

C

Convective efficiency

Life time of the convective elements

s

T

FF

s

T

FFVsTVsT

V

RRR

C

RR

CTRGTRG

22

3

8

CR 1

R Characteristic frequency of radiative energy lost by turbulent eddies

In the static case, assuming constant coefficients (Hp>>l !), we have solutions which are plane waves identical to the ML solutions.

Approximations of Gabriel’s Theory

Convective fluctuations equations



140

Linear pulsation equationsPerturbation of the mean equations

0)1(

1 2

2 rr

lldr

rrdr

H

jrTj

T

rr

rp

AA

gdr

pd

dr

pd

drd

r

1211 turbg2

Equation of mass conservation

Radial component of the equation of momentum conservation

Transversal component of the equation of momentum conservation

r

r

r

rp

A

AViscHrp

rr HT

H

1212

A : Anisotropy parameterA=1/2 for isotropic turbulence

: Angular pulsation frequency



141


Equation of Energy conservation

RG

HCR

CRHN

ppVFCH

rll

rr

Lrr

drdTrT

Lr

ll

dmLd

dmLd

dmdL

rr

llsTi

2

3

1

/41

1

FCH : Amplitude of the horizontal

component of the convective flux


142

VsVsTT

TFF CC

Convective Flux : VsTFC

Perturbation :

Convective flux perturbation

efficient is convection where1

1

1 a

CCRC

CR7

i i

dssd

ll

cs

rr

drrd

rr

F

F

v

H

rC

rC

7654321

a a a a a a a



143

r

r

VV

P

P

2

turb

turb

dssd

ll

cs

rr

drrd

rr

VV

v

H

r

r

7654321 b b b b b b b

Turbulent pressure :2

rVP turb

Perturbation :


Turbulent pressure perturbation

surface near the a

bC

77

i

P

P

HH

ll

Perturbation of the mixing-length

Hp Pressure scale height


144

Convection – pulsation interaction: the Solar case


Local analysis tireXtrX ),( 0

...

... /

... /

ggp

pgg

ppbs/c

s/cpp

Movement

Transfer

Characteristic polynomial 0 (( 40 PP3

New terms = 0

New root: = 1/(b) ∞ at the convective boundaries

The worse numerical case givesthe best fits with observations

Re(1/(b)) < 0 at the left boundaryRe(1/(b )) > 0 at the right boundary


145





Example:

z )1(

)1(2 22

2

xxix

dxdz

22 11

1exp)(

xx

ix

cxz


146





Example:

z )1(

)1(2 22

2

xxix

dxdz

22 11

1exp)(

xx

ix

cxz


147

Plan de l’exposé

1. Introduction

2. Convection-pulsation interaction

Plan of the presentation

1. Introduction

3. Confrontation to observations

2.1. The MLT theory of Gabriel

2.2. The case of solar-like oscillations

4. Conclusions

2. Convection-pulsation interaction

2.1. The MLT theory of Gabriel

2.2. The case of solar-like oscillations

3. Confrontation to observations

4. Conclusions


148

Non-adiabatic stellar oscillations

Quasi-adiabaticregion

Non-adiabaticregion

S 0

Coupling between

• the dynamical equations

• the thermal equations

Introduction


149

Theoretical damping rates ↔ line-widths observed by GOLF

Confrontation to observations


150

Problem:

This solution fitting very well the observationsis subject to numerical instabilities near the upper

boundary of the convective envelope.

They come from the turbulent pressure perturbation term.


Theoretical damping rates ↔ line-widths observed by GOLF


151

Theoretical work integrals for different (mode l=0, p22)



152


Convection dépendant du temps

M = 1.8 M0 - Teff = 7150 K - = 0.5

Teff / Teff|


153



M = 1.8 M0 - Teff = 7130 K - = 1

Teff / Teff|


154



M = 1.8 M0 - Teff = 7150 K - = 1.5

Teff / Teff|


155

Propagation cavities Doradus starsInternal physics:

High inertia of the g-modes near the

convective core top2

2 r 4 r

Main displacement isin the transversal direction

Small radial displacement

g50


156

rT

r

r

r

p

A

Ag

dr

pd

dr

pd

dr

dr

12

11 turbg2

r

r

r

rp

A

Arp

rr HTh

H

1212

Radial component of the equation of momentum conservation

Transversal component of the equation of momentum conservation

A : Anisotropy parameterA=1/2 for isotropic turbulence

: Angular pulsation frequency

Influence of turbulent Reynolds stress perturbation

TTpVV 1 ),(Y ),( Y m

lhhmlr rT


157

Work integral



158

Pturb 0 and d Pturb / dr discontinuous

at the bottom of the convective envelope

singularity of the equations

unphysical discontinuity of the eigenfunctions

r

rVV

V

VVVV

V

VVVVPr

rr

r

rhhr

r

rhhr

d

d C (...)

d

d

1 (...)

hh2

2turb3

3h



159

Non-local treatments:

PeVVVVb

hh log d d ; d 0

L0NL

Improves the things (continuity) but problem still present

r

rVV

V

VVVV

V

VVVVPr

rr

r

rhhr

r

rhhr

d

d C (...)

d

d

1 (...)

hh2

2turb3

3h



160

Influence of Reynolds stress: Work integral


161

Doradus

- d W / d log T

|L / L|

FR / F

W

For hot or small models:very thin convective envelope

the transition region is in the

radiative zone

Small -driving (Fe, ~ SPBs)compensated by

large radiative dampingbelow and above


162

9 Aurigae

3 frequencies: f1=0.795 c/d, f2=0.768 c/d, f3=0.343 c/d

Zerbi et al. 1994Simultaneous photometryand spectroscopy

Spectroscopic mode id. : (1, |m1|) = (3, 1), Aerts & Krisciunas (1996) (3, |m3|) = (3, 1)

Spectro-photometric amplitudes and phases


163

Aurigae


Phase-lag ( Vmagnitude – displacement ):

Observations: 1 = - 77° ± 12° 3 = - 41° ± 10°

Theory ( = 2): TDC = - 22° = - 39°FC = - 156° = - 140°

Spectro-photometric phase differences

workshop: « doradus stars in the corot fields » nice, 26-28 may, 2008 1 driving mechanism and...

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