Asteroseismic helium abundance determination

1. Asteroseismic helium abundance determination Günter Houdek Leiden 8 June 2004

2. Outline • Motivation • Variational principle • Application to an inhomogeneous vibrating string • Application to solar-type stars • Further improvements

3. Power (spectral density) of low-degree p modes (SOHO/GOLF)

5. Solar model

6. Second frequency differences Sun (1 year MDI/SOI data)

7. βHydrae

8. ξ Hya (1.04) Procyon (2.49) β Hyi (1.48) α Cen A (0.93)

9. The variational principle

10. The variational principle (I) Necessary condition for minimum (Euler-Lagrange equation): With additional constraint (e.g., isoperimetric problems): Seek continuous function y(x), satisfying BCs y(a)=ya, y(b)=yb such that

11. The variational principle (II) Consider: Solution of which is: Sturm-Liouville equations as Euler-Lagrange equations as an isoperimetric problem: Property: trial function y(x) which is good to “first order” yields approximate EV  which is good to “second order”.

12. The variational principle (III) Find eigenfunctions (EF) and eigenvalues (EV) : Suppose is nearly equal to with EF and EV : Let ( small) and expand in power of : First order: Perturbation theory applied to eigenvalue problem:

13. The vibrating (inhomogeneous) string

14. Wave equation

15. Point weight The homogeneous string

16. Point weight The inhomogeneous string

17. Point weight The inhomogeneous string

18. Assumed perturbation of

19. Second frequency differences of an inhomogeneous string

20. Application to solar-type star

21. Linearized, adiabatic, nonradial conservation equations: Eliminating with (1) leads to: Cowling approximation: operator is hermitian for at boundary: Variational principle in (nonrotating) stars (I)

22. with Small perturbations in equilibrium configuration by c2 and/or : Variational principle in (nonrotating) stars (II)

23. Fitting function (I) we approximate: in the asymptotic limit (JWKB): (e.g., Gough 1987) oscillatory cpt of : and

24. Fitting function (II) with from the base of the convection zone: with smooth term: Fitting function: from the HeII ionization zone:

25. Solar model

26. HeII ionization zone lower boundary of the convection zone Sun

27. with The effect of the remaining terms in

28. Adiabatically stratified models

29. - exact (numerical) frequencies

30. - exact (numerical) frequencies

34. - exact (numerical) frequencies

35. - exact (numerical) frequencies

36. - exact (numerical) frequencies

37. - exact (numerical) frequencies

38. Testing the formulation on toy models

39. Complete evolutionary (solar) models (Eva Novotny) Model  X Z Age (y) Y T (s) ---------------------------------------------------------------------------------------------------------- 1 1.3688297 0.6989868 0.0201400 4.1514980E+09 0.280873 3592.0524 2 1.3850422 0.7012563 0.0201400 4.3699937E+09 0.278604 3591.5498 0 1.4021397 0.7036170 0.0201400 4.5999964E+09 0.276243 3591.3059 3 1.4202404 0.7060764 0.0201400 4.8421028E+09 0.273784 3590.9296 4 1.4394131 0.7086334 0.0201400 5.0969477E+09 0.271227 3590.3508 5 1.3990221 0.7069156 0.0195005 4.5999964E+09 0.273584 3591.4168 6 1.4004459 0.7052699 0.0198177 4.5999964E+09 0.274912 3591.4939 0 1.4021397 0.7036170 0.0201400 4.5999964E+09 0.276243 3591.3059 7 1.4042702 0.7019591 0.0204676 4.5999964E+09 0.277573 3591.1314 8 1.4063854 0.7002957 0.0208005 4.5999964E+09 0.278904 3591.0566 All models are calibrated to have R=R¯ and L=L¯.

40. Z 8 7 age 2 0 1 3 4 6 5 Complete evolutionary (solar) models (Eva Novotny)

42. Z 8 7 age 2 0 1 3 4 6 5 1 0 5

45. 2nd frequency difference (2nd frequency derivative): Amplitude A = Y (needs to be calibrated): Amplitude of the oscillatory component of the HeII ionization as a measure for Y

50. Further improvements of the fitting formula