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

Objectives What? Where? Why? How?

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

 Cephei John Goodricke (1784)

Cepheids in the HR Diagram

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

Period-Luminosity Relation

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

Extragalactic Distance Scale

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

 Canis Majoris Struve (1950): P1 = 0.25002 d P2 = 0.25130 d

Analogy: Hydrogen Spectrum

Nonradial Oscillations

Global Standing Waves Angular Radial

NRO’s in the HR Diagram

Types of Wave Acoustic (pressure) Gravity (buoyancy)

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

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

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

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

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

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

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

Gravity Waves in a Liquid

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)

Isothermal Diagnostic Diagram Acoustic waves Gravity waves

WKBJ Diagnostic Diagram Acoustic waves Gravity waves

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

Propagation Diagram ̶ Polytrope ℓ=2 modes

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

Propagation Diagram ̶ 5 M¯ p modes f mode g modes

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

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

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]

Frequency Spectra Polytrope 5 M¯

p-mode Surface Variations

g-mode Surface Variations

p modes vs. g modes