!"

Lecture 7.2: Asteroseismology

1

a) Introduction

!  Asteroseismology is the determination of stellar structure using oscillations as seismic waves

!  For the Sun, many, many, many normal modes have been identified providing deep insight

!  CoRoT and Kepler missions have provided multiyear, high S/N, photometric data for a large number of solar type and giant stars

2

Solar Oscillations

3

b) Asteroseismic diagnostic signatures

4

1) Solar-like oscillations show a range of high order acoustic modes excited to low amplitudes by turbulence in the convective envelope beneath the photosphere where convective timescales are comparable to the driving period. 2) Spectra are characterized by groups of modes of alternating even and odd degree separated in frequency by (see slide 21 & 18): The different groups are separated by approximately constant 3) The individual modes can show substructure (m) related to rotation. 4) The spectra show a maximum in their power distribution related to the cut-off frequency of the atmosphere. The frequency of this maximum is

(ℓ = 0 and 2 & 1 and 3)

δν ≈ (1+ 2ℓ 3)d ; ℓ=0 (1) for even (odd) degree modes)

Δν

νmax

Solar Oscillation Spectrum

5

c) Waves and interference

6

φ =2πν ds

cs

∫Phase lag: Resonant frequencies: νn ∝dscs∫#

$%

&

'(

−1

For spherical systems where the sound speed only depends on r :

νn =Wn (r)cs (r)

dr∫"

#$

%

&'

−1

Weight function, Wn(r), expresses that different resonant sound waves link to different regions in the star.

Resonant Frequencies

Scaling: νn ∝csR

∝TR

∝GMR3

∝ ρ7

!  Stellar oscillations: pressure or gravity (buoyance) is restoring force

!  Radial modes are pressure modes

!  Transverse modes are predominantly buoyance modes

!  Can also classify modes as pressure (p) or gravity (g) modes

!  Labeled with number of radial nodes: e.g., p3

8

Spherical Harmonics

n : :m :

Number of radial nodes Number of surface nodal lines Number of surface nodal lines passing through the rotation axis

Blue coming towards us. Red moving away. Colors invert after half an oscillation: For spherical systems, three indices are needed:

= 3, m = 3, 2,1, 0

9

ρ∂2z∂t2 = k

∂2z∂x2

∂2z∂t2 − cs

2 ∂2z∂x2 = 0

written in terms of displacements:∂2ξ∂t2 − cs

2∇2ξ = 0

d) Wave equation & frequency

10

d-1) Rectangular Membrane

∂2ξ∂t2 − cs

2 ∂2ξ∂x2 − cs

2 ∂2ξ∂y2 = 0

say, edges are free; e.g., ∂ξ /∂x = 0 or ∂ξ /∂y = 0

−ω 2 + cs2 kx

2 + ky2( )#

$%&ξ = 0

ωn,l2 = π 2cs

2 n+1X

'

()

*

+,

2

+l +1Y

'

()

*

+,

2#

$--

%

&..

n and l are the number of nodelines along x and y

11

d-2) Circular membrane

In cylindrical coordinates the wave equation is:

∂2ξ∂t2

− cs2 ∂2ξ∂r2

+1r∂ξ∂r

+1r2∂2ξ∂φ 2

#

$%

&

'(= 0

Resonance implies periodicity in f.

ξ = f1(t) f2 (r) f3(φ)

−ω 2 f2 − cs2 ∂2 f2∂r2 +

1r∂f2∂r

+m2

r2∂2 f2∂φ 2

#

$%

&

'(= 0

Substitute f2 (r) = gm (r) / r12

Substitute f2 (r) = gm (r) / r,

d 2gmdr2 +

ωcs

!

"#

$

%&

2

−4m2 −1

4r2

(

)**

+

,--gm = 0

Now introduce the (phase-like) parameter, x ≡ωr / csd 2gmdx2 + 1− 4m2 −1

4x2

(

)*

+

,-gm = 0

The solution is gm (x) = xJm (x) with Jm the Bessel fct

For the nth resonant mode , the nth zero point of the Bessel fct coincides with the rim: Asymptotically (large x), this yields

x0 ≈ 2n+m− 12( )π / 2 or νn,m ≈ 1

4 2n+m− 12( ) csR

νn,m ≈ νn+1,m−213

d-3) Uniform sphere

∂2ξ∂t2

− cs2 ∂2ξ∂r2

+2r∂ξ∂r

+1r2

∂2ξ∂θ 2

+cosθsinθ

∂ξ∂θ

+1

sin2θ∂2ξ∂φ 2

#

$%

&

'(

)

*+

,

-.= 0

ξ = f (r)Yl,m (θ,ϕ )

∂2 f∂r2

+2r∂f∂r+

ω 2

cs2

#

$%

&

'(−

l(l +1)r2

)

*+

,

-. f = 0

Bessel functions and in asymptotic limit:

νn,l ≈142n+ l( ) cs

R

(near) degenerate: νn,l ≈ νn+1,l−2 and independent of m

For uniform spheres, the derivation is very similar

14

15

•  Small perturbations: linear perturbation analysis

•  Take the equations describing the structure

•  Write each variable as:

•  Drop terms involving terms of two or more

•  Equilibrium values, qo, are solutions of eqns. e.g., drop terms involving no

•  Result: set of linear equations in

•  Solve with resonance conditions

•  Here, we will not detail this derivation but just discuss the results

q = qo +δq

δq

δqδq

d-4) Stellar oscillations

∂2X∂r2 +K(r)X = 0

where X is related to the displacement vector, δrX ≡ cs

2ρ1/2∇δrThis is similar to the wave equation for a uniform sphere. The fct K(r) is quite complex. Important here is that oscillations can exist for positive K(r) if either ω>ω+(r) or ω <ω−(r) where ω± depend on the sound speed, gravity, and the pressure/density gradients Pressure waves (p-modes) for: Gravity waves (g-modes) for: If composition changes, modes can have mixed character

ω >ω+(r)ω <ω−(r)

16

e) Probing stellar interiors

17

Theoretical Mode Spectrum

18

f) Asymptotic Frequencies for Pressure Waves

νn,l ≈ n+ l2+ε

"

#$

%

&'Δν − Al l +1( )−β( )Δν

2

νn,l

Δν= 2 drcs0

R

∫+

,-

.

/0

−1

and A = 14π 2Δν

cs (R)R

−1rdcsdr

dr0

R

∫+

,-

.

/0

Dn = inverse of the phase lag (acoustic diameter). The A-term depends on the sound speed gradient in the central region which depends critically on the stellar evolutionary state.

We can measure the large separation and small separation in the spectra and relate those to these stellar characteristics Δν ≈ νn+1,l −νn,l

4l + 6( )Δν2

νn,l

A ≈ νn,l −νn−1,l+2 ≡ δνn,l19

g) Asymptotic periods for gravity waves

Pn,l ≈n+ηl(l +1)

P0

P0 is related to the period with which a gas bubble oscillates vertically around its equilibrium position

Gravity modes are generally only important for White Dwarfs

20

21

h) Summary

The resonant modes of stellar vibration are formed by constructive interference of propagating sound waves. In the limit of large , the rays pass almost through the center of the star (see slide 22). The phase delay along the ray, , plus the retardation at the surface, , and the caustic surface, , must resonate with the interference pattern on the stellar surface, , e.g., an integral multiple of . Of course, rays are not straight but bent due to refraction caused by the gradient in the sound speed in an inhomogeneous star. Key parameters are thus:

(n,ℓ)

n / ℓ

ωτα̂π (π / 2)

ℓ+1/ 2( )π

τ = 2 cs−1 dr

0

R

∫

Δχ ∝1rdcsdr

dr0

R

∫

2π

22

The large frequency separation measures thus the mean density while the small frequency separation is sensitive to the compositional gradient in the star. The latter is driven by thermonuclear transmutation in the core.

Left: Rays on a circular drum. Right: Rays in a sphere with a sound speed gradient (plotted on the acoustic radius scale, t).

i) Applications

Oscillation spectrum of 16 Cyg observed by Kepler. The diagnostic structure of the seismic spectrum can be readily recognized (see slide 4). Peaks corresponding to different n modes are labeled with their angular degree parameter, l. Modes of even/odd degree are separated by . For rapidly rotating stars, rotational splitting can also be measured. 23

Δν /2

The asteroseismic diagram ( ) for the Cen binary. Evolutionary tracks for fixed mass (solid lines). Dashed lines show constant core-hydrogen content, (related to age). Uncertainties for the measurements of mass, radius and stellar age are based upon simulations 24

Δν versusδν α

Echelle diagram formed by stacking overtones above each other. Curvature relates to regions of abrupt change (ionization zones, base of the convective envelope)

Applications

25

Applications

Frequency of maximum set by surface structure:

νmax ∝csH∝ g /Teff

1/2

26

main sequence

main sequence

Sub giant

base RGB

Sub giant

Osc

illat

ion

spec

trum

of 1

Msu

n

star

s m

easu

red

by K

eple

r

27

Osc

illat

ion

spec

trum

of 1

Msu

n

star

s m

easu

red

by K

eple

r RGB

RGB

RGB

Red Clump

28

Uncertainties in log(g) from uncertainties in the estimated maximum frequency and in the analysis models. A 1% change in

results in a 0.17% change in log g. With R (Gaia), masses can be determined. 29 νmax

j) Solar Oscillation Spectrum

Error bars are 1000 s ! 30

31

k) Scaling Relations

is location of maximum power: theoretical studies imply is 70% of acoustic cut off frequency,

P(ν ) = Pνmax exp −(ν −νmax )

2

2σ 2

"

#$

%

&'

νmax versusΔν

32

νac

νmax

νmax

Δν ∝ M / R3( )1/2

& νmax ∝νac ∝ g /Teff1/2 ∝M / R2Teff

1/2

MMo

=νmax

νmax,o

#

$%%

&

'((

3Δνo

Δν

#

$%

&

'(

4 TeffTeff ,o

#

$%%

&

'((

3/2

RRo

=νmax

νmax,o

#

$%%

&

'((Δνo

Δν

#

$%

&

'(

2 TeffTeff ,o

#

$%%

&

'((

1/2

Teff would have to come from spectroscopic studies. This also yields luminosities

33

Δν ∝ M / R3( )1/2

& νmax ∝νac ∝ g /Teff1/2 ∝M / R2Teff

1/2

L∝R2Teff4

MMo

=LLo

#

$%

&

'(

3/2ΔνΔνo

#

$%

&

'(

2Teff ,oTeff

#

$%%

&

'((

6

MMo

=νmax

νmax,o

#

$%%

&

'((LLo

#

$%

&

'(Teff ,oTeff

#

$%%

&

'((

7/2

Teff would have to come from spectroscopic studies & luminosities from accurate distances.

34

35

l) Validation

Astero-sizes versus interferometry

Astero-Mass Validation

36