11

Influence of the Convective Influence of the Convective Flux Perturbation on the Flux Perturbation on the

Stellar Oscillations:Stellar Oscillations:δ Scuti and γ Doradus cases δ Scuti and γ Doradus cases

A. Grigahcène, M-A. A. Grigahcène, M-A. Dupret, Dupret,

R. Garrido, M. Gabriel R. Garrido, M. Gabriel

and R. Scuflaireand R. Scuflaire

22

PlanPlan

I. IntroductionI. Introduction II. II. The TreatmentThe Treatment III. Instability StripIII. Instability Strip IV. IV. Photometric Amplitudes and PhasesPhotometric Amplitudes and Phases VV. Conclusion. Conclusion

33

I. IntroductionI. Introduction

44

I. IntroductionI. Introduction

Only the convection-pulsation Only the convection-pulsation interaction allows the retrievement of interaction allows the retrievement of the red edge of the the red edge of the δδ Scuti Instability Scuti Instability Strip.Strip.

The outer convection zone grows with The outer convection zone grows with the age.the age.

In this case we can’t neglect any more In this case we can’t neglect any more the Convective Flux and its Fluctuation the Convective Flux and its Fluctuation (Frozen convection is no longer valid).(Frozen convection is no longer valid).

I. Introduction

55

Convective zone vs. temperatureConvective zone vs. temperatureM=1.8 M0, α=1.5

Teff=8345.5 K

Teff=6119.5 K

I. Introduction

Figure 1

66

II. The TreatmentII. The Treatment

77

Hydrodynamic Equations

Mean Equations Convection Fluctuation Equations

Non-adiabatic Linear Pulsation Equations

Correlation Terms

The Treatment of the Convection-The Treatment of the Convection-Pulsation CouplingPulsation Coupling

Vuv

yyy

PerturbationPerturbation

Perturbation of ● Convective Flux

● Reynolds Stress Tensor

● Turbulent Kinetic Energy Dissipation

88

II.1. Hydrodynamic equationsII.1. Hydrodynamic equations

0vρtρ

Pp--vvρtvρ v-vvρUtρU PFRN

II. Theoretical BackgroundII. 1. Hydrodynamic Equations

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

99

II. 3. Convective FluctuationII. 3. Convective Fluctuation

22

2

2

2

1

0

FppVV

dt

d

ppVFFdt

sdT

pppdt

ud

uρdt

ρd

RG

RGNCR

TRGTRG

Vu

yyy

v

Mean EquationsMean Equations

Splitting the variablesSplitting the variables

II. Theoretical BackgroundII. 3. Convective Fluctuations

(M. Gabriel’s Formulation)(M. Gabriel’s Formulation)

TTpVV 1

VVF

22 2

1 Flux of the kinetic energy

of turbulence

Tensor of Reynolds2rT Vp Turbulence pressure

1010

II. 3.1. M. Gabriel’s TreatmentII. 3.1. M. Gabriel’s Treatment

ssV

dt

sd

dt

sd

T

T

uVV

ppdt

Vd

Vdt

d

C

C

1

3

8

0

1

Fluctuation EquationsFluctuation Equations

C

Convective efficiency.

Life time of the convective elements.

II. Theoretical BackgroundII. 3. Convective Fluctuations

sT

FF

s

T

FFVsTVsT

V

RRR

C

RR

CTRGTRG

22

3

8

CR 1

VRG

2

Dissipation rate of kinetic energy of turbulence into heat per unit volume.

R The inverse of the characteristic time 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

1111

0

1122

2

2

p

r

ll

dr

rrd

r

jrTjT

r

r

r

p

A

Ag

dr

pd

dr

dr

1212

Linear pulsation equations

II. Theoretical BackgroundII. 3. Convective Fluctuations

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 HTH

1212

Perturbation of the mean equations

1212

Equation of Energy conservation

II. Theoretical BackgroundII. 3. Convective Fluctuations

RG

HCR

CR

N

ppVFCH

r

ll

r

rL

r

r

drdTr

TL

r

ll

dm

LLd

dm

dLrr

rrsTi

2

3

22

1

/4

1

1

FCH Amplitude of the horizontal component of the convective flux

1313

II.4. Convective Flux FluctuationII.4. Convective Flux Fluctuation

VsVsTT

TFF CC

tirki eeXX

The unknown correlation terms can be obtained from the fluctuation equations.The solutions have the form:

Finally, the perturbed convective flux takes the following form:

Isotropic turbulence

Integration 222with, rkAkkkk

2/1A

and the problem is naturally separated in spherical harmonics.δFCr(r) and δFCh(r) are related to the perturbed mean quantities by first order differential equations.

,, mlhChr

mlCrC YrrFeYrFF

II. Theoretical BackgroundII. 4. Convective Flux Fluctuation

Convective Flux : VsTFC

Perturbation :

1414

II.5. ML PerturbationII.5. ML PerturbationThe main source of uncertainty in any ML theory of convection-pulsation interaction is in the way to perturb the mixing-length.In the results presented below, we used :

P

P

P

P

C

H

H

H

H

l

l

21

1

II. Theoretical BackgroundII. 5. ML Perturbation

C Life time of the convective elements

Angular pulsation frequency

Time-dependent treatment 1

Time-dependent treatment 2

PH Pressure scale

1when0

1when1

1

12

C

C

C

1515

II.6.II.6. Models Models Models M=1.4-2.2 M0, α=0.5, 1, 1.5, 2

II. Theoretical Background

1.4 M0

1.6 M0

1.8 M0

2 M0

2.2 M0

II. 6. Models

Obtained with the standard physics input by the Evolution Code of Liege. MAD.

Figure 2

1616

III. Instability StripIII. Instability Strip

1717

Radial Modes – 1.8 M0, α=1.5Frozen Convection Time-dependent convection

p8

p3

p4

p5

p6

p7

p1

p2

p1

p8

p7

p6

p5

p4

p3

p2

Figure 3 Figure 4

III. Instability StripIII. 1. δ Scuti stars

1919

=2 modes – 1.8 M0, α=1.5

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

2121

III. Instability StripIII. 1. δ Scuti stars δ Scuti Instability Strips

M=1.4-2.2 M0, α=1.5, =0 P

P

CH

H

l

l

21

1

Figure 7

p1p7

p1

1.4 M0

1.8 M0

2 M0

2.2 M0

1.6 M0

2222Figure 8

III. Instability StripIII. 1. δ Scuti stars δ Scuti Instability Strips

M=1.4-2.2 M0, α=1, =0 P

P

CH

H

l

l

21

1

p7

p1

1.4 M0

2 M0

1.8 M0

1.6 M0

2.2 M0

2323Figure 9

III. Instability StripIII. 1. δ Scuti stars δ Scuti Instability Strips

M=1.4-2.2 M0, α=0.5, =0 P

P

CH

H

l

l

21

1

p7

p1

1.4 M0

1.6 M0

1.8 M0

2 M0

2.2 M0

2424Figure 10

III. Instability StripIII. 1. δ Scuti stars δ Scuti Instability Strips

M=1.4-2.2 M0, α=1.5, =2 P

P

CH

H

l

l

21

1

p6

fB

g7

fR

1.4 M0

2 M0

2.2 M0

1.8 M0

1.6 M0

2525

γ Dor Instability modes M=1.5 M0, α=1, =1

Figure 11

III. Instability StripIII. 2. γ Dor stars

P

P

CH

H

l

l

21

1

2626

γ Dor Instability modes M=1.5 M0, α=1.5, =1

Figure 12

III. Instability StripIII. 2. γ Dor stars

P

P

CH

H

l

l

21

1

2727

γ Dor Instability modes M=1.6 M0, α=1.5, =1

Figure 13

III. Instability StripIII. 2. γ Dor stars

P

P

CH

H

l

l

21

1

2828

Comparison between γ Dor Instability Strips (=1) for α=1, 1.5, 2

P

P

CH

H

l

l

21

1

Figure 14

III. Instability StripIII. 2. γ Dor stars

2929

Comparison between δ Scuti Red Edge (=0, P1)

and γ Dor Instability Strip (=1) for α=1.8 P

P

CH

H

l

l

21

1

Figure 15

III. Instability StripIII. 2. γ Dor and δ Scuti stars - comparison

3030

IV. Photometric Amplitudes and IV. Photometric Amplitudes and PhasesPhases

3131

Non-adiabatic Amplitudes and Phases M=1.8 M0, Te=7151.4 K, α=0.5

IV. Photometric Amplitudes and Phases

IV. 1. δ Scuti stars

Figure 16

=0-3

3232

IV. Photometric Amplitudes and Phases

IV. 1. δ Scuti stars

Non-adiabatic Amplitudes and Phases M=1.8 M0, Te=7128 K, α=1

Figure 17

=0-3

3333

IV. Photometric Amplitudes and Phases

IV. 1. δ Scuti stars

Non-adiabatic Amplitudes and Phases M=1.8 M0, Te=7148.9 K, α=1.5

Figure 18

=0-3

3434

Figure 19 Figure 20

IV. Photometric Amplitudes and Phases

IV. 1. δ Scuti stars

Strömgren Photometry Phase-Amplitude Diagram M=1.8 M0, Te=7148.9 K

α=0.5 α=1=0-3

=0=1

=3

=2

=2

=3

=0=1=2

=3

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

3535

IV. Photometric Amplitudes and Phases

IV. 2. γ Dor stars

Non-adiabatic Amplitudes and Phases M=1.5 M0, Te=6981.5 K, α=1.8

Figure 21

=0-3

3636

IV. Photometric Amplitudes and Phases

IV. 2. γ Dor stars Strömgren Photometry Ratio Model: M=1.5 M0, Te=6981.5 K, α=1.8

Star: HD 164615, freq=1.23305 cycles/day

Figure 22 Figure 23

FST atmosphere Kurucz atmosphere

Tim

e-de

pend

ent

Con

vect

ion

1F

roze

n co

nvec

tion

3737

V. ConclusionV. Conclusion With our Convection-Pulsation interaction, we have With our Convection-Pulsation interaction, we have

been able to give the red edge of the been able to give the red edge of the δδ Scuti Instability Scuti Instability Strip.Strip.

We can explain the We can explain the γγ Dor instability. Dor instability. Our results are very sensitive to the Our results are very sensitive to the αα value: value:

- Increasing Increasing α => Red edge shifts to hotter models.α => Red edge shifts to hotter models.- α ~ 1.8 => better results for γα ~ 1.8 => better results for γ Dor and Dor and δδ Scuti stars and is Scuti stars and is

not in contradiction with observed phase lag and not in contradiction with observed phase lag and amplitudes.amplitudes.

The same stars can show at the same time The same stars can show at the same time δδ Scuti and Scuti and γγ Dor oscillation modes. Dor oscillation modes.