A Practical Introduction to Stellar Nonradial Oscillations

1. Rich Townsend University of Delaware A Practical Introduction to Stellar Nonradial Oscillations ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

2. Objectives • What? • Where? • Why? • How?

3. Overview • Historical Perspective • Radial pulsators • Nonradial pulsators • Waves in stars • Global oscillations • Surface variations • Rotation effects • Driving mechanisms

4. Cephei John Goodricke (1784)

6. Cepheids in the HR Diagram

7. Henrietta Leavitt (1868-1921) SMC Stars: Mv = -2.76 log(P) - 1.4

8. Period-Luminosity Relation

9. Origin of the P-L Relation • Constant L evolution L / M3 • Constant T instability L / R2 • Dynamical timescale / R3/2 M-1/2 • Combine: / L0.6 • Compare: / L0.9

10. Extragalactic Distance Scale

11. Paul Ledoux (1914-1988) •  mechanism • Secular instability • Semiconvection • Nonradial pulsation

12. Canis Majoris Struve (1950): P1 = 0.25002 d P2 = 0.25130 d P3 = 49.1236 d

13. Analogy: Hydrogen Spectrum

14. Nonradial Oscillations

15. Global Standing Waves Angular Radial

16. NRO’s in the HR Diagram

17. Types of Wave Acoustic (pressure) Gravity (buoyancy)

18. Linearized Hydrodynamics ’/t + r¢(v’) = 0 v’/t = -rp’ - g’ p’/ t + v’¢rp = a2(’/ t + v’¢r)

19. Wave Equation Eliminate ’ and p’: 2v’/t2 = a2r(r¢v’) + (a2r¢v’)rln 1 + (1 - 1)(r¢v’)g + r(g¢v’) 1 = (ln p/ln )s = a2/p

20. Waves in Isothermal Atmosphere 2v’/t2 = a2r(r¢v’) + ( - 1)(r¢v’)g + r(g¢v’) Trial solutions: v’ / exp[i(k¢r - t) + z/2H] E = ½  |v’|2 = ½ 0 exp[-z/H] v0’2 exp[z/H] = ½ 0 v0’2

21. |k| kz kh Dispersion Relation 4 - [ac2 + a2 |k|2] 2 + N2 a2 kh2 = 0 Acoustic cutoff frequency : ac = /2 g/a Buoyancy frequency : N = (-1)1/2 g/a

22. Limit: No Stratification (g!0) 4 - [ac2 + a2 |k|2] 2+ N2 a2 kh2 = 0  = a |k| Acoustic waves

23. Limit: Vertical Propagation (kh!0) 4 - [ac2 + a2 |k|2] 2+ N2 a2 kh2 = 0  = (a2 |k|2 + ac2)1/2 > ac Modified acoustic waves

24.  |k| kz kh Limit: Incompressible (a!1) 4 - [ac2 + a2 |k|2] 2 + N2 a2 kh2 = 0  = N kh/|k| = N sin  < N Gravity waves

25. Gravity Waves in a Liquid

26. |k| kz kh Vertical Wavenumber 4 - [ac2 + a2 |k|2] 2 + N2 a2 kh2 = 0 kz2 = (2 - ac2)/a2 + (N2 - 2) kh2/2 kz2 > 0 ! Propagating (wave) kz2 < 0 ! Evanescent (exponential)

27. Isothermal Diagnostic Diagram Acoustic waves Gravity waves

28. WKBJ Diagnostic Diagram Acoustic waves Gravity waves

29. Sectoral Zonal Spherical Harmonics Tesseral Radial kh2 = ℓ(ℓ+1)/r2

30. Lℓ2 N2 Propagation Diagram ̶ Polytrope ℓ=2 modes

31. p modes f mode g modes Wave Trapping ̶ Modes ℓ=2 modes

32. p modes f mode g modes Propagation Diagram ̶ 5 M¯

33. Mode Frequencies rb - ra = n /2 = n / kr Limit of large n : kr¼ |k| ra - rb¼ R ! R ¼ n / |k|

34. p-mode Frequencies Trapping : R ¼ n / |k| Dispersion : ¼ a |k| ¼ n a/R  = n [s a-1 dr]-1

35. g-mode Frequencies Trapping : R ¼ n / |k| Dispersion : ¼ N kh / |k| = [ℓ(ℓ+1)]1/2 / |k|R ¼ [ℓ(ℓ+1)]1/2/n N  = [ℓ(ℓ+1)]1/2/n [s N/r dr]

36. Frequency Spectra Polytrope 5 M¯

37. p-mode Surface Variations

38. g-mode Surface Variations

39. p modes vs. g modes