Asteroseismology

Steve KawalerIowa State University

observed pulsations• operate on the dynamical time scale

• accessible on convenient time scale

• probe global and local structure

• periods change on ‘evolutionary’ time scale (thermal or nuclear) - depend on global properties

• amplitudes change on ~ ‘local’ thermal time scale

2

dynamical stability• “stable” configuration represents a stable

mean configuration

• on short time scale, oscillations occur, but the mean value is fixed on longer time scales

• simple example: a pendulum (single mode)• most likely position - extrema• mean position is at zero displacement• with no damping would oscillate forever

• more complex example: a vibrating string• multiple modes with different frequencies• enumerated by number of nodes

3

a more complex example: a star

• multiple oscillation modes

• radial modes - enumerated by number of nodes between center and surface

• non-radial modes - nodes also across surface of constant radius

• modes frequencies determined by solution of the appropriate wave equation

4

stability, damping, and driving

• zero energy change: constant amplitude oscillation

• energy loss via pulsation: oscillation amplitude drops with time

• if net energy input: amplitude increases with time (if properly phased)

5

Okay, start your engines...•PG 1159: light curve

• what kind of star might this be?

• what kind of star can this not possibly be?

• what about the amplitude over the run?

•PG 1336 light curve

• huh? what time scale(s) are involved

• what kind of star (or stars)?

• tell us everything you can about this!

6

Multimode pulsation

• Oscillations at “normal mode” frequencies

• mode = specific eigensolution of equations of motion within the confines of a stellar structure

• normal mode frequencies parallel structural properties

• simple example: radial fundamental is one mode, 1st overtone (a node within) is another mode

7

towards the wave equation I • continuity equation:

• equation of motion (HSE when RHS=0):

• perturb r, P, and ρ:

• and assume δx << x so we can linearize

8

@2r

@t2= �GMr

r2� 4⇡r2 @P

@Mr

@Mr

@r= 4⇡r2⇢

x(t, Mr) = xo(Mr)1 +

�x(t, Mr)xo(Mr)

�

• replace x with x+δx in the two equations, subtract off the equilibrium equations, and keep only 1st-order terms to find:

• linearized continuity equation

• linearized equation of motion

9

�⇢

⇢o= �3

�r

ro� ro

@(�r/ro)@ro

⇢orod2�r/ro

dt2= �

✓4�r

ro+

�P

Po

◆@Po

@ro� Po

@(�P/Po)@ro

towards the wave equation II • assume adiabatic relationship between P and ρ

• combine continuity and equation of motion:

• assume exponential (complex) time dependence

10

towards the wave equation III

�r(t, ro)ro

=�r(ro)

roei�t = ⌘(ro)ei�t

�1 =✓

@ lnP

@ ln ⇢

◆

ad

so�P

Po= �1

�⇢

⇢o

@2⌘

@t2=

1⇢r4

@

@r

✓�1Pr4 @⌘

@r

◆+ ⌘

1r⇢

⇢@

@r[(3�1 � 4) P ]

�

• substitute to yield the Linear Adiabatic Wave Equation (LAWE):

• This is a wave equation: L(η) = σ2η in the displacement η.

• the eigenvalue σ2 corresponds to the oscillation frequency

11

towards the wave equation IV

L(⌘) = � 1⇢r4

@

@r

✓�1Pr4 @⌘

@r

◆� ⌘

1r⇢

⇢@

@r[(3�1 � 4) P ]

�= �2⌘

the LAWE: a simple case• assume Γ1 and η both constant throughout

the star (homologous motion)

• LAWE becomes

• now, assume a constant density, and use HSE to replace the pressure derivative to find

• look familiar?!

12

�⌘1⇢r

(3�1 � 4)@P

@r= �2⌘

⇧ =2⇡

�=

p⇡q

G⇢̄ (�1 � 43 )

the LAWE: standing wave solutions

• boundary conditions:

• center: zero displacement (η = 0)

• surface: perfect wave reflection [d(δP/P)/dr = 0]

• asymptotic analysis:

• clever change of variables renders LAWE as:

• recognizing the sound speed when we see it:

13

d2w(r)dr2

+

�2⇢

�1P� �(r)

�w(r) = 0

d2w(r)dr2

+�2

c2s

� �(r)�

w(r) = 0

the LAWE: asymptotic solution

• represent the eigenfunction as: where kr is the (local) radial wavenumber and varies slowly with radius so, locally:

• for a standing wave, we need an integral number of half-wavelengths between inner and outer reflection points:

14

d2w(r)dr2

+�2

c2s

� �(r)�

w(r) = 0

w(r) / eikrr

k2r =

�2

c2s(r)

� �(r)

Z b

akr dr = (n + 1)⇡

the LAWE: asymptotic solution

• if then

• i.e. high-frequency (high overtone, n) radial modes are equally spaced in frequency, with

σo2 ≈ G<ρ>

15

Z b

akr dr = (n + 1)⇡

�2

c2s

� �

� = (n + 1)⇡

"Z b

a

dr

cs

#�1

= (n + 1) �o

k2r =

�2

c2s(r)

� �(r)where

Nonradial oscillations• preserve angular derivatives in LAWE

• similar operator structure for radial part (as before), now along with angular part

where the quantity A is:

A < 0 when radiative A > 0 when convective

(the ‘Schwarzschild A’)

16

A =d ln ⇢

dr� 1

�1

d lnP

dr=

1�P

�T

�⇢[r�rad]

d2�rdt2

= �r✓

P 0

⇢+ 0

◆+ A

�1P

⇢r · �r

Decompose into Spherical Harmonics

• position perturbation decomposition

• produces (after some work):

where the operator L2 (the Legendrian) is:

• which has eigenstates Ylm such that:

17

�r = �r er + r�✓ e✓ + r sin ✓�� e�

r · �r =1r2

@(r2�r)@r

� 1�2

r2L2

✓P 0

⇢+ 0

◆

L2 = � 1sin ✓

@

@✓

✓sin ✓

@

@✓

◆� 1

sin2 ✓

@2

@�2

L2Y ml (✓, �) = l(l + 1)Y m

l (✓, �)

l=1, m=0 l=1, m=1

i=70

l=3, m=0 l=3, m=1 l=3, m=3

Spherical Harmonics.... courtesy asteroseismology.org (Travis Metcafe)

• now we have

• expanding into components:

using Spherical Harmonics19

r · �r =1r2

@(r2�r)@r

� l(l + 1)�2r2

✓P 0

⇢+ 0

◆

r2 d⌘t

dr=

1 +

Ag

�2

�r⌘r +

(�Ag)

r

g� 1

�r⌘t

r2 d⌘r

dr=

gr

c2s

� 2�

r⌘r + r2 l(l + 1)r2

1� �2r2

c2s

1l(l + 1)

�r⌘t

frequency-2

another frequency2

• so:

• where we’ve defined 2 “structural” frequencies

• the acoustic (Lamb) frequency Sl :

• the Brunt-Väisälä (buoyancy) frequency N:

the two characteristic frequencies20

r2 d⌘r

dr=

gl(l + 1)

S2l

� 2r

�⌘r + l(l + 1)

1� �2

S2l

�r⌘t

r2 d⌘t

dr=

1� N2

�2

�r⌘r +

N2 r

g� 1

�r⌘t

S2l =

l(l + 1)r2

c2s

N2 = �Ag = �g

d ln ⇢

dr� 1

�1

d lnP

dr

�

page

Propagation diagram, ZAMS solar model21

Sl2, l=1

N2

CZbase

NRP dispersion relation• identify the horizontal wave number(s)

• allows the wave equation(s) to reduce to a local dispersion relation, as with the radial case, to provide relationship between kr and σ:

22

k2t =

l(l + 1)r2

=S2

l

c2s

k2r =

1�2c2

s

(�2 �N2)(�2 � S2l )

asymptotic analysis

• kr2 > 0 (kr real) when

• σ2 > N2, Sl2 - or - σ2 < N2, Sl2

• kr real means oscillatory eigenfunctions

• kr2 < 0 (kr imaginary) when

• Sl2 > σ2 > N2 or Sl2 < σ2 < N2

• kr real means evanescent (exponentially decreasing or increasing) eigenfunctions

23

k2r =

1�2c2

s

(�2 �N2)(�2 � S2l )

page

Propagation diagram, ZAMS solar model24

Sl2, l=1

N2

CZbase

l=1 n=1l=1 n=2l=1 n=3l=1 n=4

page 25

Sl2, l=1

N2

CZbase

l=1 n=1l=1 n=2l=1 n=3l=1 n=4

Propagation diagram, ZAMS solar model

evanescentzonesin gray

page 26

Sl2, l=1

N2

CZbase

l=1 n=1l=1 n=2l=1 n=3l=1 n=4

evanescentzonesin gray

Propagation diagram, ZAMS solar model

“low” σoscillatory

zonesin green

“high” σoscillatory

zonesin pink

the NRP LAWE: asymptotic solutions

• again, integrate dispersion relation over propagation regions:

• two classes of solutions:

• σ2 > N2, Sl2:

• “p-modes”; pressure as the restoring force

• σ2 < N2, Sl2:

• “g-modes”: buoyancy as the restoring force

27

k2r =

1�2c2

s

(�2 �N2)(�2 � S2l )

Z b

akr dr = (n + 1)⇡ where

�nl = (n + l/2)�o ; �o =Z b

a

dr

cs

⇧nl = n⇧op

l(l + 1); ⇧o = 2⇡2

"Z b

a

N

rdr

#�1

page 28

Sl2, l=1

N2

CZbase

l=1 n=1l=1 n=2l=1 n=3l=1 n=4

evanescentzonesin gray

Propagation diagram, ZAMS solar model

“low” σoscillatory

zonesin green

“high” σoscillatory

zonesin pink

l=1 n=1l=1 n=2l=1 n=3l=1 n=4

page 29 page

Pulsation PeriodsPeriod of ‘radial fundamental’ ~ tff

g-modes p-modes

Periods Π > tff Π < tffrestoring force buoyancy pressure

asymptotic behavior

Π ∝ Πo x n σ ∝ σo x n

examples white dwarfs Cepheids,the Sun

p-modes: ~ equally spaced in frequency31

n n+1 n+2 n+3 n+4 n+5 n+6n-1n-2

......frequency

�nl = (n + l/2)�o ; �o =Z b

a

dr

cs

l l l l l l l l l

n-2l+1

n-1l+1

nl+1

n+1l+1

n+2l+1

n+3l+1

n+4l+1

n+5l+1

• if modes of different l present, observed spacing ~ σo / 2

p. 32

20 parts per million!

Solar Oscillations: Full-disk photometry Frohlich et al. 1997: SOHO/VIRGO

1 hr 10 min

5 min

p.Solar Oscillations: Full-disk photometry Frohlich et al. 1997: SOHO/VIRGO

33

20 parts per million!

1 hr 10 min

5 min

p. 34

How to observe all these low amplitude modes?

have your friends buy a $600,000,000 photometer!

p.

16 Cyg A and B (Metcalfe et al. 2012)35 p.

asteroseismic radius determination (i.e. Chaplin et al. 2011)

36

• νmax scales with acoustic cutoff frequency ~ gTe-1/2

• Δν measures mean density:�

��

���

�2

��

M

M�

� �R

R�

��3

��max

�max,�

��

�M

M�

� �R

R�

��2 �Te�

Te�,�

��0.5

p.

Kepler 93b (Ballard et al. 2014)

37 p.

Kepler 93b (Ballard et al. 2014)

38

p. 39

Kepler 93b (Ballard et al. 2014)

g-modes: ~ equally spaced in period40

n n+1 n+2 n+3 n+4 n+5 n+6n-1n-2

......Period

n n+1 n+2 n+3 n+4 n+5 n+6n-1n-2

.........

...Period

l=1

l=2

⇧nl = n⇧op

l(l + 1); ⇧o = 2⇡2

"Z b

a

N

rdr

#�1

• spacing depends on l

PG 1159-035: a g-mode pulsator

P P-390

390 0

424 34 ?

451 61 3x20.3

495 105 5x21.0

516 126 6x21.0

539 149 7x21.3

645 255 12x21.3

832 442 21x21.0

an example: hot white dwarf PG 1159-035 (Corsico et al. [WET] 2008)

in white dwarfs: Πodepends on total stellar mass

PG 1159 as an example(Winget et al. 1991)

Rotational splitting of nonradial oscillations (uniform, slow rotation)

equal frequency spacing: triplets (l=1), quintuplets (l=2) etc.

page

Pulsating stars in the HR diagram

from J. Christensen-Dalsgaard

“Solar-like” oscillations• globally stable (damped) but constantly excited

• damping time τ generally ~ days

• continuously re-excited by turbulence

• frequencies locked to normal modes of star

• excited mode periods ~ minutes

• broad mode selection, low amplitude

• (integrated) velocity amplitude < meters / second

• photometric amplitude ~ parts per million

Consequences of stochasitic excitation

• lots of modes present . . . but

• coherence time of (only) days lowers peak amplitudes• reduces detectability

• phase instability broadens FT peaks• Lorentzian envelope

• reduces frequency accuracy

• confuses mode identification and rotational splitting effects

from J. C.-D.

Stochastically excited modeheat-engine mode

1/(observing time)

1/lifetime

Observational Differences

1980 1990 2000 2010 2020

0.00

0.20

0.40

0.60

0.80

1.00T = 5d

1980 1990 2000 2010 2020

T = 10d

1980 1990 2000 2010 2020

T = 30d

coherent pulsators: frequency precision ~ 1/T

1980 1990 2000 2010 2020

0.00

0.20

0.40

0.60

0.80

1.00T = 5d

1980 1990 2000 2010 2020

T = 10d

1980 1990 2000 2010 2020

T = 30d

0.00

0.20

0.40

0.60

0.80

1.00T = 5d T = 10d T = 30d

0.00

0.20

0.40

0.60

0.80

1.00T = 5d T = 10d T = 30d

stochastic pulsators - freq. precision poorer than 1/T

Asymptotics of low-degree p modes

Large frequency separation:

Frequency separations:

Small frequency separations

Asteroseismic HR diagram

from J. Christensen-Dalsgaard

A Search for Habitable Planets

Borucki et al. 2009 -

Kepler’s Optical Phase Curve of the Exoplanet HAT-P-7b

HAT-P7 asteroseismology HAT-P7 asteroseismology

1.40 Mo

1.45 Mo

Asteroseismic HR diagram

from J. Christensen-DalsgaardSunα Cen A

HAT-P-7

HAT-P7 asteroseismology

spectroscopyPal et al. 2008

M=1.40±0.02

t = 1.6±0.4 Gyr

r = 1.94±0.05 Rsun

Xc=0.19

quick seismic fitISUEVO

Solar-like pulsators: KIC 6603624 Chaplin et al. 2010

Solar-like pulsators: KIC 3656476 Chaplin et al. 2010

Solar-like pulsators: KIC 11026764 Chaplin et al. 2010

Solar-like - FTs

Sunα Cen A

HAT-P-7

Asteroseismic HR diagram

KIC 11026764 is post-MS!

p.oscillations beyond the MS w/ Kepler(Chaplin & Miglio 2013)

64

p.Solar-like oscillations w/ Kepler(Chaplin & Miglio 2013)

65

MS

MS (16 Cyg A)

Subgiant

Subgiant

Base of RGB

500 1000 2000 3000 4000 uHz

p.Solar-like oscillations w/ Kepler(Chaplin & Miglio 2013)

66

MS

MS (16 Cyg A)

Subgiant

Subgiant

Base of RGB

500 1000 2000 3000 4000 uHz

p. 67

First-ascent RGB

First-ascent RGB

First-ascent RGB

Red Clump

20 40 60 80 100 uHz

oscillations beyond the MS w/ Kepler(Chaplin & Miglio 2013)

p.

scaling relations MS to RGB68

R/Rsun log g Δν [uHz] νmax

MS 1 4.44 135 3300

RGB base 5 3.04 12.1 140

RGB top 20 1.84 1.5 9 < 1/d1/wk

p. 69

multiplel=1 modesper order?!

Bedding et al. (2010) Ensemble asteroseismology of solar-type stars with Kepler

Chaplin et al. 2011 (Science, today!)

• determination of Δν and νmax for ~ 500 solar-type stars

• νmax scales with acoustic cutoff frequency ~ gTe-1/2

• Δν measures mean density:�

��

���

�2

��

M

M�

� �R

R�

��3

��max

�max,�

��

�M

M�

� �R

R�

��2 �Te�

Te�,�

��0.5

Chaplin et al., Science, 7 April 2011 Chaplin et al., Science, 7 April 2011

• Δν measures mean density:

• νmax scales with acoustic cutoff frequency ~ gTe-1/2

• then one can determine M and R:

• and, with Teff from multicolor photometry, one gets L

���

���

�2

��

M

M�

� �R

R�

��3

��max

�max,�

��

�M

M�

� �R

R�

��2 �Te�

Te�,�

��0.5

Chaplin et al., Science, 7 April 2011

R

R��

��max

�max,�

� ���

���

��2 �Te�

Te�,�

�0.5

M

M��

��max

�max,�

�3 ���

���

��4 �Te�

Te�,�

�1.5

Chaplin et al., Science, 7 April 2011