A Practical Introduction to Stellar Nonradial Oscillations

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A Practical Introduction to Stellar Nonradial
Oscillations

• Rich Townsend
• University of Delaware

ESO Chile ? November 2006
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Objectives

• What?
• Where?
• Why?
• How?

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Overview

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

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? Cephei
John Goodricke (1784)
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Cepheids in the HR Diagram
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Henrietta Leavitt (1868-1921)
SMC Stars Mv -2.76 log(P) - 1.4
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Period-Luminosity Relation
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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

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Extragalactic Distance Scale
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Paul Ledoux (1914-1988)

• ? mechanism
• Secular instability
• Semiconvection
• Nonradial pulsation

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? Canis Majoris

• Struve (1950)
• P1 0.25002 d
• P2 0.25130 d
• P3 49.1236 d

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Analogy Hydrogen Spectrum
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Nonradial Oscillations
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Global Standing Waves
Angular
Radial
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NROs in the HR Diagram
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Types of Wave
Acoustic (pressure)
Gravity (buoyancy)
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Linearized Hydrodynamics
??/?t r(?v) 0
??v/?t -rp - g?
? p/? t vrp a2(??/? t vr?)
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Wave Equation

• Eliminate ? and p

?2v/?t2 a2r(rv) (a2rv)rln ?1
(?1 - 1)(rv)g r(gv)
?1 (?ln p/?ln ?)s a2?/p
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Waves in Isothermal Atmosphere
?2v/?t2 a2r(rv) (? - 1)(rv)g r(gv)
Trial solutions v / expi(kr - ?t) z/2H
E ½ ? v2 ½ ?0 exp-z/H v02 expz/H
½ ?0 v02
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Dispersion Relation
?4 - ?ac2 a2 k2 ?2 N2 a2 kh2 0
Acoustic cutoff frequency ?ac ?/2 g/a
Buoyancy frequency N (?-1)1/2 g/a
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Limit No Stratification (g!0)
?4 - ?ac2 a2 k2 ?2 N2 a2 kh2 0
? a k
Acoustic waves
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Limit Vertical Propagation (kh!0)
?4 - ?ac2 a2 k2 ?2 N2 a2 kh2 0
? (a2 k2 ?ac2)1/2 gt ?ac
Modified acoustic waves
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Limit Incompressible (a!1)
?4 - ?ac2 a2 k2 ?2 N2 a2 kh2 0
? N kh/k N sin ? lt N
Gravity waves
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Gravity Waves in a Liquid
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Vertical Wavenumber
?4 - ?ac2 a2 k2 ?2 N2 a2 kh2 0
kz2 (?2 - ?ac2)/a2 (N2 - ?2) kh2/?2
kz2 gt 0 ! Propagating (wave) kz2 lt 0 ! Evanescent
(exponential)
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Isothermal Diagnostic Diagram
Acoustic waves
Gravity waves
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WKBJ Diagnostic Diagram
Acoustic waves
Gravity waves
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Spherical Harmonics
Tesseral
Radial
kh2 l(l1)/r2
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Propagation Diagram ? Polytrope
l2 modes
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Wave Trapping ? Modes
l2 modes
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Propagation Diagram ? 5 M
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Mode Frequencies
rb - ra n ?/2 n? / kr
Limit of large n kr ¼ k ra - rb
¼ R
! R ¼ n? / k
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p-mode Frequencies
Trapping R ¼ n? / k
Dispersion ? ¼ a k
? ¼ n? a/R
? n? s a-1 dr-1
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g-mode Frequencies
Trapping R ¼ n? / k
Dispersion ? ¼ N kh / k l(l1)1/2 / kR
? ¼ l(l1)1/2/n? N
? l(l1)1/2/n? s N/r dr
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Frequency Spectra
Polytrope
5 M
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p-mode Surface Variations
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g-mode Surface Variations
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p modes vs. g modes