1. 1 Influence of the Convective Flux Perturbation on the Stellar Oscillations: δ Scuti and γ Doradus cases A. Grigahcène, M-A. Dupret, R. Garrido, M. Gabriel and R. Scuflaire

2. 2 Plan I. Introduction I. Introduction II. The Treatment II. The Treatment III. Instability Strip III. Instability Strip IV. Photometric Amplitudes and Phases IV. Photometric Amplitudes and Phases V. Conclusion V. Conclusion

3. 3 I. Introduction

4. 4 Only the convection-pulsation interaction allows the retrievement of the red edge of the δ Scuti Instability Strip. Only the convection-pulsation interaction allows the retrievement of the red edge of the δ Scuti Instability Strip. The outer convection zone grows with the age. The outer convection zone grows with the age. In this case we can’t neglect any more the Convective Flux and its Fluctuation (Frozen convection is no longer valid). In this case we can’t neglect any more the Convective Flux and its Fluctuation (Frozen convection is no longer valid). I. Introduction

5. 5 Convective zone vs. temperature M=1.8 M 0, α=1.5 T eff =8345.5 K T eff =6119.5 K I. Introduction Figure 1

6. 6 II. The Treatment

7. 7 Hydrodynamic Equations Mean Equations Convection Fluctuation Equations Non-adiabatic Linear Pulsation Equations Correlation Terms The Treatment of the Convection- Pulsation Coupling Perturbation Perturbation of ● Convective Flux ● Reynolds Stress Tensor ● Turbulent Kinetic Energy Dissipation

8. 8 II.1. Hydrodynamic equations II. Theoretical Background II. 1. Hydrodynamic Equations P: Pressure tensor ; p : its diagonal component. Radiative Flux

9. 9 II. 3. Convective Fluctuation Mean Equations Splitting the variables II. Theoretical Background II. 3. Convective Fluctuations (M. Gabriel’s Formulation) Flux of the kinetic energy of turbulence Tensor of Reynolds Turbulence pressure

10. 10 II. 3.1. M. Gabriel’s Treatment Fluctuation Equations Convective efficiency. Life time of the convective elements. II. Theoretical Background II. 3. Convective Fluctuations Dissipation rate of kinetic energy of turbulence into heat per unit volume. 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

11. 11 Linear pulsation equations II. Theoretical Background II. 3. Convective Fluctuations Equation of mass conservation Radial component of the equation of momentum conservation Transversal component of the equation of momentum conservation Perturbation of the mean equations

12. 12 Equation of Energy conservation II. Theoretical Background II. 3. Convective Fluctuations Amplitude of the horizontal component of the convective flux

13. 13 II.4. Convective Flux Fluctuation 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 and the problem is naturally separated in spherical harmonics. δF Cr (r) and δF Ch (r) are related to the perturbed mean quantities by first order differential equations. II. Theoretical Background II. 4. Convective Flux Fluctuation Convective Flux : Perturbation :

14. 14 II.5. ML Perturbation The 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 : II. Theoretical Background II. 5. ML Perturbation Life time of the convective elements Angular pulsation frequency Time-dependent treatment 1 Time-dependent treatment 2 Pressure scale

15. 15 II.6. Models Models M=1.4-2.2 M 0, α=0.5, 1, 1.5, 2 II. Theoretical Background 1.4 M 0 1.6 M 0 1.8 M 0 2 M 0 2.2 M 0 II. 6. Models Obtained with the standard physics input by the Evolution Code of Liege. MAD. Figure 2

16. 16 III. Instability Strip

17. 17 Radial Modes – 1.8 M 0, α=1.5 Frozen Convection Time-dependent convection p8p8 p3p3 p4p4 p5p5 p6p6 p7p7 p1p1 p2p2 p1p1 p8p8 p7p7 p6p6 p5p5 p4p4 p3p3 p2p2 Figure 3 Figure 4 III. Instability Strip III. 1. δ Scuti stars

18. 18 Radial Modes – 1.8 M 0, α=1.5 Time-dependent convection p7p7 p8p8 p6p6 p5p5 p4p4 p3p3 p2p2 p1p1 Figure 5 III. Instability Strip III. 1. δ Scuti stars

19. 19 =2 modes – 1.8 M 0, α=1.5 p7p7 p2p2 p3p3 p4p4 p5p5 p6p6 g2g2 g1g1 f p1p1 g7g7 g8g8 g6g6 g4g4 g3g3 g5g5 Frozen Convection Time-dependent convection p7p7 p6p6 p4p4 p5p5 p3p3 g1g1 f p1p1 p2p2 g2g2 g3g3 g4g4 g5g5 g6g6 g7g7 g8g8 Figure 5Figure 6 III. Instability Strip III. 1. δ Scuti stars

20. 20 Time-dependent convection p7p7 p2p2 p3p3 p4p4 p5p5 p6p6 g1g1 f p1p1 g8g8 g7g7 g6g6 g5g5 g4g4 g3g3 g2g2 Figure 8 III. Instability Strip III. 1. δ Scuti stars =2 modes – 1.8 M 0, α=1.5

21. 21 III. Instability Strip III. 1. δ Scuti stars δ Scuti Instability Strips M=1.4-2.2 M 0, α=1.5, =0 Figure 7 p1p1 p7p7 p1p1 1.4 M 0 1.8 M 0 2 M 0 2.2 M 0 1.6 M 0

22. 22 Figure 8 III. Instability Strip III. 1. δ Scuti stars δ Scuti Instability Strips M=1.4-2.2 M 0, α=1, =0 p7p7 p1p1 1.4 M 0 2 M 0 1.8 M 0 1.6 M 0 2.2 M 0

23. 23 Figure 9 III. Instability Strip III. 1. δ Scuti stars δ Scuti Instability Strips M=1.4-2.2 M 0, α=0.5, =0 p7p7 p1p1 1.4 M 0 1.6 M 0 1.8 M 0 2 M 0 2.2 M 0

24. 24 Figure 10 III. Instability Strip III. 1. δ Scuti stars δ Scuti Instability Strips M=1.4-2.2 M 0, α=1.5, =2 p6p6 fBfB g7g7 fRfR 1.4 M 0 2 M 0 2.2 M 0 1.8 M 0 1.6 M 0

25. 25 γ Dor Instability modes M=1.5 M 0, α=1, =1 Figure 11 III. Instability Strip III. 2. γ Dor stars

26. 26 γ Dor Instability modes M=1.5 M 0, α=1.5, =1 Figure 12 III. Instability Strip III. 2. γ Dor stars

27. 27 γ Dor Instability modes M=1.6 M 0, α=1.5, =1 Figure 13 III. Instability Strip III. 2. γ Dor stars

28. 28 Comparison between γ Dor Instability Strips ( =1) for α=1, 1.5, 2 Figure 14 III. Instability Strip III. 2. γ Dor stars

29. 29 Comparison between δ Scuti Red Edge ( =0, P 1 ) and γ Dor Instability Strip ( =1) for α=1.8 Figure 15 III. Instability Strip III. 2. γ Dor and δ Scuti stars - comparison

30. 30 IV. Photometric Amplitudes and Phases

31. 31 Non-adiabatic Amplitudes and Phases M=1.8 M 0, Te=7151.4 K, α=0.5 IV. Photometric Amplitudes and Phases IV. 1. δ Scuti stars Figure 16 =0-3

32. 32 IV. Photometric Amplitudes and Phases IV. 1. δ Scuti stars Non-adiabatic Amplitudes and Phases M=1.8 M 0, Te=7128 K, α=1 Figure 17 =0-3

33. 33 IV. Photometric Amplitudes and Phases IV. 1. δ Scuti stars Non-adiabatic Amplitudes and Phases M=1.8 M 0, Te=7148.9 K, α=1.5 Figure 18 =0-3

34. 34 Figure 19Figure 20 IV. Photometric Amplitudes and Phases IV. 1. δ Scuti stars Strömgren Photometry Phase-Amplitude Diagram M=1.8 M 0, Te=7148.9 K α=0.5 α=1 =0-3 =0 =1 =3 =2 =3 =0 =1 =2 =3 Phase(b-y)-phase(y) (deg)

35. 35 IV. Photometric Amplitudes and Phases IV. 2. γ Dor stars Non-adiabatic Amplitudes and Phases M=1.5 M 0, Te=6981.5 K, α=1.8 Figure 21 =0-3

36. 36 IV. Photometric Amplitudes and Phases IV. 2. γ Dor stars Strömgren Photometry Ratio Model: M=1.5 M 0, Te=6981.5 K, α=1.8 Star: HD 164615, freq=1.23305 cycles/day Figure 22Figure 23 FST atmosphere Kurucz atmosphere Time-dependent Convection 1 Frozen convection

37. 37 V. Conclusion With our Convection-Pulsation interaction, we have been able to give the red edge of the δ Scuti Instability Strip. With our Convection-Pulsation interaction, we have been able to give the red edge of the δ Scuti Instability Strip. We can explain the γ Dor instability. We can explain the γ Dor instability. Our results are very sensitive to the α value: Our results are very sensitive to the α value: -Increasing α => Red edge shifts to hotter models. - α ~ 1.8 => better results for γ Dor and δ Scuti stars and is not in contradiction with observed phase lag and amplitudes. The same stars can show at the same time δ Scuti and γ Dor oscillation modes. The same stars can show at the same time δ Scuti and γ Dor oscillation modes.