A. Grigahcène, M-A. Dupret, R. Garrido, M. Gabriel and R. Scuflaire

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

3. I. Introduction

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 • In this case we can’t neglect any more the Convective Flux and its Fluctuation (Frozen convection is no longer valid).

5. Teff=8345.5 K Teff=6119.5 K I. Introduction Convective zone vs. temperature M=1.8 M0, α=1.5 Figure 1

6. II. The Treatment

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

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

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

10. II. Theoretical Background II. 3.1. M. Gabriel’s Treatment II. 3. Convective Fluctuations Fluctuation Equations 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. Approximations of Gabriel’s Theory Life time of the convective elements. Convective efficiency. In the static case, assuming constant coefficients (Hp>>l !), we have solutions which are plane waves identical to the ML solutions.

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

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

13. II. Theoretical Background II.4. Convective Flux Fluctuation II. 4. Convective Flux Fluctuation Convective Flux : Perturbation : The unknown correlation terms can be obtained from the fluctuation equations. The solutions have the form: Integration Isotropic turbulence Finally, the perturbed convective flux takes the following form: 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.

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

15. 2.2 M0 2 M0 1.8 M0 1.6 M0 1.4 M0 II. Theoretical Background II.6. Models II. 6. Models Models M=1.4-2.2 M0, α=0.5, 1, 1.5, 2 Obtained with the standard physics input by the Evolution Code of Liege. MAD. Figure 2

16. III. Instability Strip

17. p8 p8 p7 p7 p6 p6 p5 p5 p4 p4 p3 p3 p2 p2 p1 p1 III. Instability Strip Radial Modes – 1.8 M0, α=1.5 III. 1. δ Scuti stars Time-dependent convection Frozen Convection Figure 3 Figure 4

18. p7 p7 p6 p6 p5 p5 p4 p4 p3 p3 p2 p2 p1 p1 f f g1 g1 g2 g2 g3 g3 g4 g4 g5 g6 g7 g8 g5 g6 g7 g8 III. Instability Strip l=2 modes – 1.8 M0, α=1.5 III. 1. δ Scuti stars Time-dependent convection Frozen Convection Figure 5 Figure 6

19. p7 p1 2.2 M0 2 M0 p1 1.8 M0 1.6 M0 1.4 M0 δScuti Instability Strips M=1.4-2.2 M0, α=1.5, l=0 III. Instability Strip III. 1. δ Scuti stars Figure 7

20. 2.2 M0 p7 2 M0 1.8 M0 p1 1.6 M0 1.4 M0 δScuti Instability Strips M=1.4-2.2 M0, α=1, l=0 III. Instability Strip III. 1. δ Scuti stars Figure 8

21. 2.2 M0 2 M0 p7 1.8 M0 1.6 M0 p1 1.4 M0 δScuti Instability Strips M=1.4-2.2 M0, α=0.5, l=0 III. Instability Strip III. 1. δ Scuti stars Figure 9

22. fB 2.2 M0 p6 fR 2 M0 g7 1.8 M0 1.6 M0 1.4 M0 δScuti Instability Strips M=1.4-2.2 M0, α=1.5, l=2 III. Instability Strip III. 1. δ Scuti stars Figure 10

23. III. Instability Strip γ Dor Instability modes M=1.5 M0, α=1, l=1 III. 2. γ Dor stars Figure 11

24. III. Instability Strip γ Dor Instability modes M=1.5 M0, α=1.5, l=1 III. 2. γ Dor stars Figure 12

25. III. Instability Strip γ Dor Instability modes M=1.6 M0, α=1.5, l=1 III. 2. γ Dor stars Figure 13

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

27. III. Instability Strip Comparison between δ Scuti Red Edge (l=0, P1) and γ Dor Instability Strip (l=1) for α=1.8 III. 2. γ Dor and δ Scuti stars - comparison Figure 15

28. IV. Photometric Amplitudes and Phases

29. IV. Photometric Amplitudes and Phases Non-adiabatic Amplitudes and Phases M=1.8 M0, Te=7151.4 K, α=0.5 IV. 1. δ Scuti stars l=0-3 Figure 16

30. IV. Photometric Amplitudes and Phases IV. 1. δ Scuti stars Non-adiabatic Amplitudes and Phases M=1.8 M0, Te=7128 K, α=1 l=0-3 Figure 17

31. 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 l=0-3 Figure 18

32. IV. Photometric Amplitudes and Phases l=3 l=3 l=3 l=2 l=2 l=2 l=1 l=1 l=0 l=0 Phase(b-y)-phase(y) (deg) Phase(b-y)-phase(y) (deg) IV. 1. δ Scuti stars Strömgren Photometry Phase-Amplitude Diagram M=1.8 M0, Te=7148.9 K α=1 α=0.5 l=0-3 Figure 19 Figure 20

33. 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 l=0-3 Figure 21

34. IV. Photometric Amplitudes and Phases IV. 2. γ Dor stars Strömgren Photometry Ratio Model: M=1.5 M0, Te=6981.5 K, α=1.8Star: HD 164615, freq=1.23305 cycles/day Kurucz atmosphere FST atmosphere Time-dependent Convection 1 Frozen convection Figure 22 Figure 23

35. V. Conclusion • 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. • 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.