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