Lecture course in Asteroseismology

Part IPart I

Margarida S. Cunha

Christensen-Dalsgaard

Stellar pulsatorsStellar pulsators

Christensen-Dalsgaard

Classification

Intrinsically unstableClassical

Origin Intrinsically stable

Solar-like

Acoustic wavesp modes

Nature Internal Gravity waves

g modes

Asteroseismology

Observables: some kind of periodic variation

intensity; Velocity

Properties of the oscillations

frequencies, angular form

Properties = f (interior)

Hydrodynamics

Continuity

Motion++−∇=

=+

Fgpt

t

D

vD

0v div D

D

ρρ

ρρ

rrr

r

Energy

Energy

Eqs ofstate

Γ−

−Γ=

=+=

t

p

t

p

tp

t

E

t

q

D

D

D

D

)1(

1

D

)1(D

D

D

D

D

1

3

ρρρ

ρ

Hydrodynamics

++−∇=

=+

Fgpt

t

D

vD

0v div D

D

ρρ

ρρ

rrr

r Continuity

Motion

Γ−

−Γ=

=+=

t

p

t

p

tp

t

E

t

q

D

D

D

D

)1(

1

D

)1(D

D

D

D

D

1

3

ρρρ

ρEnergy

Energy

Eqs ofstate

Hydrodynamics

++−∇=

=+

Fgpt

t

D

vD

0v div D

D

ρρ

ρρ

rrr

r Continuity

Motion

Small perturbations about a background state which is:static and in thermal equilibrium

Γ−

−Γ=

=+=

t

p

t

p

tp

t

E

t

q

D

D

D

D

)1(

1

D

)1(D

D

D

D

D

1

3

ρρρ

ρEnergy

Energy

Eqs ofstate

Hydrodynamics

′+′+′−∇=∂∂

=+∂

∂

ggpt

t

ρρρ

ρδρ

v

0v div

rrr

r

Small perturbations about a state:

• static

∂

∂Γ−

∂∂

−Γ=

∂∂

∂

t

p

t

p

t

q

t

δρρ

δρ

δ 1

3 )1(

1

• static • thermal equilibrium

Hydrodynamics

′+′+′−∇=∂∂

=+∂

∂

ggpt

t

ρρρ

ρδρ

v

0v div

rrr

r

Small perturbations about a state:

• static

∂

∂Γ−

∂∂

−Γ=

∂∂

∂

t

p

t

p

t

q

t

δρρ

δρ

δ 1

3 )1(

1

• static • thermal equilibrium

fff ∇⋅+′= rr

δδ

Hydrodynamics

Most common approximations

′+′+′−∇=∂∂

=+∂

∂

ggpt

t

ρρρ

ρδρ

v

0v div

rrr

r

∂

∂Γ−

∂∂

−Γ=

∂∂

∂

t

p

t

p

t

q

t

δρρ

δρ

δ 1

3 )1(

1

Hydrodynamics

Most common approximations

′+′+′−∇=∂∂

=+∂

∂

ggpt

t

ρρρ

ρδρ

v

0v div

rrr

r

Adiabatic approximation

∂

∂Γ−

∂∂

−Γ=

∂∂

∂

t

p

t

p

t

q

t

δρρ

δρ

δ 1

3 )1(

1

Hydrodynamics

Most common approximations

′+′+′−∇=∂∂

=+∂

∂

ggpt

t

ρρρ

ρδρ

v

0v div

rrr

r

Adiabatic approximation

∂

∂Γ−

∂∂

−Γ=

∂∂

∂

t

p

t

p

t

q

t

δρρ

δρ

δ 1

3 )1(

1

Why is the adiabatic approximation adequate?

Hydrodynamics

Adiabatic approximationMost common approximations

Typical values in the deep interior: 107 years (~10-15 s-1)Typical values in solar convection zone: 103 years (~10-11 s-1)

Hydrodynamics

Adiabatic approximationMost common approximations

Outer layers of a 2.2 Msun star

Typical values in the deep interior: 107 years (~10-15 s-1)Typical values in solar convection zone: 103 years (~10-11 s-1)

Hydrodynamics

Most common approximations

′+′+′−∇=∂∂

=+∂

∂

ggpt

t

ρρρ

ρδρ

v

0v div

rrr

r

Adiabatic approximation

∂

∂Γ−

∂∂

−Γ=

∂∂

∂

t

p

t

p

t

q

t

δρρ

δρ

δ 1

3 )1(

1

Hydrodynamics

Most common approximation

′+′+′−∇=∂∂

=+∂

∂

ggpt

t

ρρρ

ρδρ

v

0v div

rrr

r

Cowling approximation

Adiabatic approximation

∂

∂Γ−

∂∂

−Γ=

∂∂

∂

t

p

t

p

t

q

t

δρρ

δρ

δ 1

3 )1(

1

Hydrodynamics

Most common approximation

′+′+′−∇=∂∂

=+∂

∂

ggpt

t

ρρρ

ρδρ

v

0v div

rrr

r

Cowling approximation

( )dV

rr

trG

g

V

∫ ′−

′′=Φ′

Φ′∇=′

rr

r

r

,ρ

Adiabatic approximation

∂

∂Γ−

∂∂

−Γ=

∂∂

∂

t

p

t

p

t

q

t

δρρ

δρ

δ 1

3 )1(

1

Note: Cowling approximation not used for radial modes

Simple waves

gpt

rr

ρδ

ρ ′+′−∇=∂

∂2

2 r

Simple waves

gpt

rr

ρδ

ρ ′+′−∇=∂

∂2

2 r

Acoustic wavesNeglect last term on rhs

( )[ ]tii ω−⋅∝ rkexpperturrr

high ω

222

222

k

k

r

r

c

c

=

′=′

ω

ρρω

( )[ ]tii ω−⋅∝ rkexpperturrr

Simple waves

gpt

rr

ρδ

ρ ′+′−∇=∂

∂2

2 r

Acoustic waves Internal gravity waves

Motion cannot be purely radialNeglect last term on rhs

( )[ ]tii ω−⋅∝ rkexpperturrr ( )[ ]tii ω−⋅∝ rkexppertur

rr

low ω

222

222

k

k

r

r

c

c

=

′=′

ω

ρρω

hr

r

r

h

r

g

kk1

k

k1

22

2

2

+=

=

′=

+

ω

ξ

ρξρω

( )[ ]tii ω−⋅∝ rkexpperturrr ( )[ ]tii ω−⋅∝ rkexppertur

rr

Summary

Acoustic waves Internal gravity waves

Simple waves

• maintained by gradient of pressure fluctuation;

• maintained by gravity acting on density fluctuation;

• Radial or non-radial;

• Propagate in convectively stable or non-stable

regions

on density fluctuation;

• Always non-radial;

• Propagate in convectively stable regions only

p modes g modes

Summary

Simple waves

Waves in a star

])er(~

Re[),r( tiftf ωδδ −=rr

Separate solutions in time. Any perturbation δ f admits solutions :

])er(~

Re[),r( tiftf ωδδ −=rr

Waves in a star

Separate solutions in time. Any perturbation δf admits solutions :

δf (r)~Substituting in perturbed equations to get Eqs forf (r)

)~

(~2 ff δδω L=

Nontrivial solutions satisfying the Eqs + boundary

conditions exist only for particular values of ω

])er(~

Re[),r( tiftf ωδδ −=rr

Waves in a star

Separate solutions in time. Any perturbation δf admits solutions :

Separate solutions in angular space. Consider (r,θ,φ) . If the

equilibrium state has spherical symmetry, any scalar perturbation δfequilibrium state has spherical symmetry, any scalar perturbation δfadmits solutions:

]e),()(Re[),r( tim

lYrftf ωφθδδ −=(r

])er(~

Re[),r( tiftf ωδδ −=rr

Waves in a star

Separate solutions in time. Any perturbation δf admits solutions :

Separate solutions in angular space. Consider (r,θ,φ) . If the

equilibrium state has spherical symmetry, any scalar perturbation δfequilibrium state has spherical symmetry, any scalar perturbation δfadmits solutions:

]e),()(Re[),r( tim

lYrftf ωφθδδ −=(r

and the displacement δ r admits solutions :

∂∂

+∂

∂+= − ti

m

l

m

lh

m

lr

YYrYrt ω

φθ φθθξξδ e a

sin

1a)(a)(Re),r(r r

rr

Degree l : related to the horizontal wavenumber

Spherical Harmonics Ylm

R

L

R

llh =

+=

)1(k

Azimuthal order m : number of nodes along the equator

=> orientation on the sphere

l=0 l=1

m=0

l=1

m=1

Note: |m| ≤ l

]e),()(Re[),r( tim

lYrftf ωφθδδ −=(r

Waves in a star

Assume perturbation has the form

Substitute in perturbed equations to get Eqs for δf (r)

centre surfaceradius

ω=ω(n,l,m)

Spherical symmetry

Waves in a star

Spherical symmetry

Waves in a star

)~

(~2 ff δδω L=

Spherical symmetry

Waves in a star

)~

(~2 ff δδω L=

Eigenfrequencies ω of modes with the same n and l are degenerate.

∑=m

m

lmYaf~

δOne eigenmode any combination

ω=ω(n,l,m)

Waves in a star

Radial dependence

Under the adiabatic and Cowling approximations, the perturbed Eqs can be combined to give:

0d 2

2

=Ψ+Ψ

K )()( where rpur δ=Ψ0d

d 2

2=Ψ+

ΨrK

r)()( where rpur δ=Ψ

Waves in a star

Radial dependence

Under the adiabatic and Cowling approximations, the perturbed Eqs can be combined to give:

0d 2

2

=Ψ+Ψ

K )()( where rpur δ=Ψ0d

d 2

2=Ψ+

ΨrK

r)()( where rpur δ=Ψ

Kr2>0 => oscillatory behaviour => propagation

Kr2<0 => exponential behaviour => evanescent

Waves in a star

−−

−=

2

2

2

2

2

222

1ω

ωω N

r

L

cK c

r

Kh2

Waves in a star

p modes

−−

−=

2

2

2

2

2

222

1ω

ωω N

r

L

cK c

r

Kh2K2

Waves in a star

p modes

−−

−=

2

2

2

2

2

222

1ω

ωω N

r

L

cK c

r

Kh2K2 2

22 Kr

c=ω2

222 Kr

cc =− ωω

Waves in a star

p modes

−−

−=

2

2

2

2

2

222

1ω

ωω N

r

L

cK c

r

Kh2K2 2

22 Kr

c=ω

Near the surface Kr2 < 0 if ω2 < ωc

2 : ~independent of L

Deep in the interior Kr2 < 0 if ω2 < c2 Kh

2 : dependent of

2222 Kr

cc =− ωω

Waves in a star

p modes

Different colours

Different l

Near the surface Kr2 < 0 if ω2 < ωc

2 : ~independent of L

Deep in the interior Kr2 < 0 if ω2 < c2 Kh

2 : dependent of

Different l

Waves in a star

g modes

−−

−=

2

2

2

2

2

222

1ω

ωω N

r

L

cK c

r

Kh2

Waves in a star

g modes

−−

−=

2

2

2

2

2

222

1ω

ωω N

r

L

cK c

r

2N 2

Kh2

hr KK1

22

+=

Nω

hr

KK1

22

+=ω

Kr2 < 0 if ω2 < N2

g modes can propagate only in convectively stable regions.

Waves in a star

Propagation diagram

Lecture course in Asteroseismology

Part IIPart II

Margarida S. Cunha

Asteroseismology

Observables: some kind of periodic variation

intensity; Velocity

Properties of the oscillations

frequencies, angular form

Properties = f (interior)

Asteroseismology

Observables: some kind of periodic variation

intensity; Velocity

Properties of the oscillations

frequencies, angular form

Properties = f (interior)

counts

ω=ω(n,l,m)

Different colours

Different l

ω=ω(n,l,m)

Asymptotic analysis

How is the oscillation spectrum supposed to look like?

Asymptotic analysis

Adiabatic oscillations in the Cowling approximation.

High n, low l, acoustic oscillations:

1

0

0

0

2 where

2

−

=∆

+∆

++≈

∫R

nl

c

dr

termsorderhigherl

n

ν

ναν

0 c

• ∆ν0 prop (M/R3)1/2

• α function of ν and is due to surface effects

• Note: ν=ω/2π

Asymptotic analysis

Adiabatic oscillations in the Cowling approximation.

High n, low l, acoustic oscillations:

∆ν0 prop (M/R3)1/2

...2

0 +∆

++≈ νανl

nnl

Asymptotic analysis

Large separations ∆νnl

termsorderhigherl

nnl 2

0 +∆

++≈ ναν

Asymptotic analysis

Large separations ∆νnl

termsorderhigherl

nnl 2

0 +∆

++≈ ναν

0,,1∆ ννν ∆≈−= + lnlnnlν α (M/R3)1/2

νννν

∆νnlSchematicPowerSpectrum

n-1,0 n-1,1 n,0 n,1 n+1,0

( )[ ]

−

∆=

+∆

−+−∆

++≈

∫R

nl

nl

r

dc

R

RcA

lAll

n

00

2

00

)(

4

1 where

...12

νπ

νν

δναν

Asymptotic analysis

Adiabatic oscillations in the Cowling approximation.

High n, low l, acoustic oscillations:

∆ rR004 νπ

( )[ ]

−

∆=

+∆

−+−∆

++≈

∫R

nl

nl

r

dc

R

RcA

lAll

n

00

2

00

)(

4

1 where

...12

νπ

νν

δναν

Asymptotic analysis

small separations δνnl

∆ rR004 νπ

( ) ∫∆

+−≈−= +−

R

ln

lnlnnlr

dcl

0,

2

02,1,

464

νπν

ννδν

νννν

∆νnlSchematicPowerSpectrum

n-1,2

δνnl

n-1,0 n-1,1 n,0 n,1

Asymptotic analysis

Sun as a star

Asymptotic analysis

Sun as a star

Harder for classical pulsators

Asymptotic analysis

( ) ( ) ∫∆

+−≈+−= ++−

R

ln

lnlnlnnlr

dcl

0,

2

01,1,1,

)1(

422

2

1

νπν

ννννδ

Alternative small separations δ(1)νnl

νννν

∆νnlSchematicPowerSpectrum

δ(1)νnl

n-1,0 n-1,1 n,0 n,1

Asymptotic analysis

MsunZAMS

Asymptotic analysis

JWKB breaks down

Msun8 Gyr

Signatures of sharp transitions

e.g: boundaries of convective regions;regions of different chemical composition;Ionization regions, etc

...2

0 +∆

++≈ νανl

ns

nlDeviations from smooth ν

Signatures of sharp transitions

20

nl

nl

s

nlnl νδνν ~+=

Modes of different frequency ‘feel’ the transition region differently.

e.g: boundaries of convective regions;regions of different chemical composition;Ionization regions, etc

Signatures of sharp transitions

Transition region

differently.

( )[ ]ϕτπννδ +dnlnl A 22cos~~

Signatures of sharp transitions

( )[ ]ϕτπννδ +dnlnl A 22cos~~

∫==R

ddc

drr )(ττ ∫==

r

dd

dc

r )(ττ

Signatures of sharp transitions

( )[ ]ϕτπννδ +dnlnl A 22cos~~

∫==R

ddc

drr )(ττ ∫==

r

dd

dc

r )(ττ

How to isolate the perturbation due to the sharp transition?

1. Fit a smooth function to the frequencies as function of n and remove that function from the frequencies => residual

2. Calculate the second differences:lnlnlnnl ,1,,12 2 −+ +−=∆ νννν

Signatures of sharp transitions

( )[ ]ϕτπννδ +dnlnl A 22cos~~

Direct fitting

Direct fitting

[ ] ref

nl

ref

surfeffi rRHFeLTy 020 ,,,,/,, νν ∆=

Set of non-seismic and seismic observables, yi errors σi

Direct fitting

[ ] ref

nl

ref

surfeffi rRHFeLTy 020 ,,,,/,, νν ∆=

Set of non-seismic and seismic observables, yi errors σi

Set of model parameters aj

ageYXMa MLj ,,,, 00 α=

Set of model parameters aj

Direct fitting

[ ] ref

nl

ref

surfeffi rRHFeLTy 020 ,,,,/,, νν ∆=

Set of non-seismic and seismic observables, yi errors σi

Set of model parameters aj

ageYXMa MLj ,,,, 00 α=

Set of model parameters aj

From models construct yimod and minimize

( )[ ]∑ −=i

iii yy2mod2 σχ

Inversion

Combine different seismic data (e.g. frequencies) to infer localised information.

Inversion

Combine different seismic data (e.g. frequencies) to infer localised information.

∫ Ω=V

ii rdrrkdrrr

)()(

We want localized information about Ω at position r0

Inversion

Combine different seismic data (e.g. frequencies) to infer localised information.

∫ Ω=V

ii rdrrkdrrr

)()(

[ ]∫∑∑ Ω=V i

ii

i

ii rdrrkadarrr

)( )(

We want localized information about Ω at position r0

),( 0 rrrr

K

Seismic inference

• Asymptotic properties to get information on mean density and about the core

• Deviation from asymptotic behaviour to characterize sharp transitions characterize sharp transitions

• Direct fitting to determine model parameters

• Inverse methods to derive localize information

Deviations from spherical symmetry

Deviations from spherical symmetry

Fgpt

rrr

++−∇= ρρD

vDe.g. rotation; magnetic field

ω=ω(n,l,m)

Deviations from spherical symmetry

Fgpt

rrr

++−∇= ρρD

vDe.g. rotation; magnetic field

ω=ω(n,l,m)

Deviations from spherical symmetry

Fgpt

rrr

++−∇= ρρD

vDe.g. rotation; magnetic field

ω=ω(n,l,m)

l=1

m=-1

l=1

m=1

Deviations from spherical symmetry

Fgpt

rrr

++−∇= ρρD

vDe.g. rotation; magnetic field

ω=ω(n,l,m)

l=1

m=-1

l=1

m=1)cos(

]e),(Re[),r(

tm

Ytf tim

l

ωφ

φθδ ω

−∝

∝ −r

Axisymmetric Star

)~

(~2 ff δδω A=

AAAA

Reference frame aligned with axis of symmetry

one eigenmode one single m state: Ylm

Axisymmetric Star

)~

(~2 ff δδω A=

AAAA

Axisymmetric Star

)~

(~2 ff δδω A=

AAAA

1 – perturbation invariant under reflection about the equator:- degeneracy partially lifted: ω=ω(n,l,|m|)

2 – perturbation not invariant under reflection about the equator:- degeneracy totally lifted: ω=ω(n,l,m)

Axisymmetric Star

Slowly rigid rotating star

‘Geometrical’ effect

Ω

Axisymmetric Star

Slowly rigid rotating star

‘Geometrical’ effect

Ω

mΩobs

m += 0ωω

Axisymmetric Star

Slowly rigid rotating star

Coriolis effect

In a reference frame rotating with the star, to first order in Ω/ω

rirrrrrr

δωδδω ×Ω−= 2)(2L rirr δωδδω ×Ω−= 2)(L

Axisymmetric Star

Slowly rigid rotating star

Coriolis effect

In a reference frame rotating with the star, to first order in Ω/ω

rirrrrrr

δωδδω ×Ω−= 2)(2L rirr δωδδω ×Ω−= 2)(L

ΩmCnl

rot

m −= 0ωω

Where Cnl is the Ledoux constant: typically << 1 and ->0 for high n

Axisymmetric Star

Slowly rotating star

Coriolis effect

In a reference frame of the observer, to first order in Ω/ω

ΩCm nl

obs

m )1(0 −+= ωω ΩCm nlm )1(0 −+= ωω

If Ω = Ω(r)

nlnl

obs

m ΩCm )1(0 −+= ωω

Axisymmetric Star

Slowly rotating star