A. Grigahcène, M-A. Dupret, R. Garrido, M. Gabriel and R. Scuflaire
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Transcript of A. Grigahcène, M-A. Dupret, R. Garrido, M. Gabriel and R. Scuflaire
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neglect any more the Convective Flux and its Fluctuation the Convective Flux and its Fluctuation (Frozen convection is no longer valid).(Frozen convection is no longer valid).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.