A Practical Introduction to Stellar Nonradial Oscillations
Rich TownsendUniversity of Delaware
ESO Chile ̶ November 2006
Objectives
• What?
• Where?
• Why?
• How?
Overview
• Historical Perspective– 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 evolutionL / M3
• Constant T instabilityL / 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
P3 = 49.1236 d
8<
:
P1 = 0:25002dP2 = 0:25130dP3 = 49:1236d
(1)P1 = 0:25002dP2 = 0:25130dP3 = 49:1236d
(1)
Analogy: Hydrogen Spectrum
Nonradial Oscillations
Global Standing Waves
RadialAngular
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 kh
2 = 0
Acoustic cutoff frequency : ac = /2 g/a
Buoyancy frequency : N = (-1)1/2 g/a
|k|
kh
kz
Limit: No Stratification (g!0)
= a |k|
4 - [ac2 + a2 |k|2] 2 + N2 a2 kh
2 = 0
Acoustic waves
Limit: Vertical Propagation (kh!0)
= (a2 |k|2 + ac2)1/2 > ac
4 - [ac2 + a2 |k|2] 2 + N2 a2 kh
2 = 0
Modified acoustic waves
Limit: Incompressible (a!1)
4 - [ac2 + a2 |k|2] 2 + N2 a2 kh
2 = 0
= N kh/|k| = N sin < N
|k|
kh
kz
Gravity waves
Gravity Waves in a Liquid
Vertical Wavenumber
4 - [ac2 + a2 |k|2] 2 + N2 a2 kh
2 = 0
kz2 = (2 - ac
2)/a2 + (N2 - 2) kh
2/2
kz2 > 0 ! Propagating (wave)
kz2 < 0 ! Evanescent (exponential)
|k|
kh
kz
Isothermal Diagnostic Diagram
Acoustic wavesAcoustic waves
Gravity wavesGravity waves
WKBJ Diagnostic Diagram
Acoustic wavesAcoustic waves
Gravity wavesGravity waves
Sectoral
Spherical Harmonics
kh2 = ℓ(ℓ+1)/r2
Radial
Zonal
Tesseral
Propagation Diagram ̶ Polytrope
ℓ=2 modes
Lℓ2
N2
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
n [s a-1 dr]-1
Dispersion : ¼ a |k|
Trapping : R ¼ n / |k|
¼ n a/R
g-mode Frequencies
[ℓ(ℓ+1)]1/2/n [s N/r dr]
Dispersion : ¼ N kh / |k| = [ℓ(ℓ+1)]1/2 / |k|R
Trapping : R ¼ n / |k|
¼ [ℓ(ℓ+1)]1/2/n N
Frequency Spectra
Polytrope 5 M¯
p-mode Surface Variations
g-mode Surface Variations
p modes vs. g modes
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