![Page 1: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/1.jpg)
A Practical Introduction to Stellar Nonradial Oscillations
Rich TownsendUniversity of Delaware
ESO Chile ̶ November 2006
![Page 2: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/2.jpg)
Objectives
• What?
• Where?
• Why?
• How?
![Page 3: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/3.jpg)
Overview
• Historical Perspective– Radial pulsators– Nonradial pulsators
• Waves in stars• Global oscillations• Surface variations• Rotation effects• Driving mechanisms
![Page 4: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/4.jpg)
Cephei
John Goodricke (1784)
![Page 5: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/5.jpg)
![Page 6: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/6.jpg)
Cepheids in the HR Diagram
![Page 7: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/7.jpg)
Henrietta Leavitt (1868-1921)
SMC Stars:
Mv = -2.76 log(P) - 1.4
![Page 8: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/8.jpg)
Period-Luminosity Relation
![Page 9: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/9.jpg)
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
![Page 10: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/10.jpg)
Extragalactic Distance Scale
![Page 11: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/11.jpg)
Paul Ledoux (1914-1988)
• mechanism• Secular instability• Semiconvection• Nonradial pulsation
![Page 12: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/12.jpg)
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)
![Page 13: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/13.jpg)
Analogy: Hydrogen Spectrum
![Page 14: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/14.jpg)
Nonradial Oscillations
![Page 15: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/15.jpg)
Global Standing Waves
RadialAngular
![Page 16: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/16.jpg)
NRO’s in the HR Diagram
![Page 17: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/17.jpg)
Types of Wave
Acoustic (pressure) Gravity (buoyancy)
![Page 18: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/18.jpg)
Linearized Hydrodynamics
’/t + r¢(v’) = 0
v’/t = -rp’ - g’
p’/ t + v’¢rp = a2(’/ t + v’¢r)
![Page 19: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/19.jpg)
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
![Page 20: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/20.jpg)
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
![Page 21: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/21.jpg)
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
![Page 22: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/22.jpg)
Limit: No Stratification (g!0)
= a |k|
4 - [ac2 + a2 |k|2] 2 + N2 a2 kh
2 = 0
Acoustic waves
![Page 23: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/23.jpg)
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
![Page 24: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/24.jpg)
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
![Page 25: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/25.jpg)
Gravity Waves in a Liquid
![Page 26: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/26.jpg)
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
![Page 27: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/27.jpg)
Isothermal Diagnostic Diagram
Acoustic wavesAcoustic waves
Gravity wavesGravity waves
![Page 28: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/28.jpg)
WKBJ Diagnostic Diagram
Acoustic wavesAcoustic waves
Gravity wavesGravity waves
![Page 29: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/29.jpg)
Sectoral
Spherical Harmonics
kh2 = ℓ(ℓ+1)/r2
Radial
Zonal
Tesseral
![Page 30: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/30.jpg)
Propagation Diagram ̶ Polytrope
ℓ=2 modes
Lℓ2
N2
![Page 31: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/31.jpg)
Wave Trapping ̶ Modes
p modes
f mode
g modes
ℓ=2 modes
![Page 32: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/32.jpg)
Propagation Diagram ̶ 5 M¯
p modes
f mode
g modes
![Page 33: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/33.jpg)
Mode Frequencies
rb - ra = n /2 = n/ kr
Limit of large n : kr ¼ |k| ra - rb ¼ R
! R ¼ n/ |k|
![Page 34: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/34.jpg)
p-mode Frequencies
n [s a-1 dr]-1
Dispersion : ¼ a |k|
Trapping : R ¼ n / |k|
¼ n a/R
![Page 35: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/35.jpg)
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
![Page 36: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/36.jpg)
Frequency Spectra
Polytrope 5 M¯
![Page 37: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/37.jpg)
p-mode Surface Variations
![Page 38: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/38.jpg)
g-mode Surface Variations
![Page 39: A Practical Introduction to Stellar Nonradial Oscillations Rich Townsend University of Delaware ESO Chile ̶ November 2006 TexPoint fonts used in EMF. Read](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649f155503460f94c2aeec/html5/thumbnails/39.jpg)
p modes vs. g modes