1. A Practical Introduction to Stellar Nonradial Oscillations
Rich Townsend
University of Delaware
ESO Chile ̶ November 2006
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2. Objectives
What?
Where?
Why?
How?

3. Overview Historical Perspective Waves in stars Global oscillations
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 ( )
SMC Stars:
Mv = 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

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. 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
|k| kh kz

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

25. Gravity Waves in a Liquid

26. kz2 = (2 - ac2)/a2 + (N2 - 2) kh2/2
Vertical Wavenumber
4 - [ac2 + a2 |k|2] 2 + N2 a2 kh2 = 0
kz2 = (2 - ac2)/a2 + (N2 - 2) kh2/2
|k| kh kz
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. Spherical Harmonics
Sectoral
Radial
Tesseral
Zonal
kh2 = ℓ(ℓ+1)/r2

30. Propagation Diagram ̶ Polytrope
ℓ=2 modes

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

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

33. Mode Frequencies rb - ra = n /2 = n / kr Limit of large n : kr ¼ |k|
ra - rb ¼ 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. Dispersion :  ¼ N kh / |k| = [ℓ(ℓ+1)]1/2 / |k|R
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