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 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 (ℓ = 0 and 2 & 1 and 3) separated in frequency by (see slide 21 & 18): δν ≈ (1+ 2ℓ 3)d ; ℓ=0 (1) for even (odd) degree modes) The different groups are separated by approximately constant Δ ν 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 of4 this maximum is ν max Solar Oscillation Spectrum
5 c) Waves and interference
6 Resonant Frequencies
−1 ds # ds& Phase lag: φ = 2πν Resonant frequencies: ν n ∝% ∫ ( ∫ c $ cs ' s For spherical systems where the sound speed only depends on r : " %−1 Wn (r) ν n = $ ∫ dr' # cs (r) &
Weight function, Wn(r), expresses that different resonant sound waves link to different regions in the star.
cs T GM Scaling: ν n ∝ ∝ ∝ 3 ∝ ρ R R R 7 ! 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
Blue coming towards us. Red moving away. Colors invert after half an oscillation: = 3, m = 3, 2,1, 0 For spherical systems, three indices are needed: n : Number of radial nodes
: Number of surface nodal lines
m : Number of surface nodal lines passing through the rotation axis 9 d) Wave equation & frequency
∂2z ∂2z ρ = k ∂t 2 ∂x2 2 2 ∂ z 2 ∂ z − c = 0 ∂t 2 s ∂x2 written in terms of displacements: 2 ∂ ξ 2 2 − c ∇ ξ = 0 ∂t 2 s
10 d-1) Rectangular Membrane
2 2 2 ∂ ξ 2 ∂ ξ 2 ∂ ξ − c − c = 0 ∂t 2 s ∂x2 s ∂y2 say, edges are free; e.g., ∂ξ / ∂x = 0 or ∂ξ / ∂y = 0 # 2 2 2 2 % $−ω + cs (kx + ky )&ξ = 0 # 2 2 % 2 2 2 ' n +1* 'l +1* ωn,l = π cs -) , +) , . $-( X + ( Y + &.
n and l are the number of nodelines along x and y
11 d-2) Circular membrane
In cylindrical coordinates the wave equation is:
2 2 2 ∂ ξ 2 #∂ ξ 1 ∂ξ 1 ∂ ξ & − cs % + + ( = 0 ∂t 2 $∂r2 r ∂r r2 ∂φ 2 '
Resonance implies periodicity in f.
ξ = f1(t) f2 (r) f3(φ) # 2 2 2 & 2 2 ∂ f2 1 ∂f2 m ∂ f2 −ω f2 − cs % + + ( = 0 $ ∂r2 r ∂r r2 ∂φ 2 '
Substitute f2 (r) = gm (r) / r
12 Substitute f2 (r) = gm (r) / r, 2 2 ( 2 + d g !ω $ 4m −1 m +*# & − -g = 0 dr2 c 4r2 m )*" s % ,-
Now introduce the (phase-like) parameter, x ≡ ωr / cs 2 ( 2 + d gm 4m −1 +*1− -gm = 0 dx2 ) 4x2 ,
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 c x ≈ 2n + m − 1 π / 2 or ν ≈ 1 2n + m − 1 s 0 ( 2) n,m 4( 2) R
ν n,m ≈ ν n+1,m−2 13 d-3) Uniform sphere
For uniform spheres, the derivation is very similar
2 ) 2 2 2 , ∂ ξ 2 ∂ ξ 2 ∂ξ 1 # ∂ ξ cosθ ∂ξ 1 ∂ ξ & 2 − cs + 2 + + 2 % 2 + + 2 2 (. = 0 ∂t *∂r r ∂r r $∂θ sinθ ∂θ sin θ ∂φ '-
ξ = f (r)Yl,m (θ,ϕ) ∂2 f 2 ∂f )#ω 2 & l(l +1), 2 + ++% 2 (− 2 . f = 0 ∂r r ∂r *$ cs ' r -
Bessel functions and in asymptotic limit:
1 cs ν n,l ≈ (2n + l) 4 R
(near) degenerate: ν n,l ≈ ν n+1,l−2 and independent of m 14 d-4) Stellar oscillations
• Small perturbations: linear perturbation analysis
• Take the equations describing the structure
• Write each variable as: q = qo +δq • Drop terms involving terms of two or more δq
• Equilibrium values, qo, are solutions of eqns. e.g., drop terms involving no δq • Result: set of linear equations in δq
• Solve with resonance conditions
• Here, we will not detail this derivation but just discuss the results
15 ∂2 X + K(r)X = 0 ∂r2 where X is related to the displacement vector, δr 2 1/2 X ≡ cs ρ ∇δr This 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: ω > ω+ (r)
Gravity waves (g-modes) for: ω < ω− (r) If composition changes, modes can have mixed character
16 e) Probing stellar interiors
17 Theoretical Mode Spectrum
18 f) Asymptotic Frequencies for Pressure Waves
" l % Δν 2 ν n,l ≈ $n + +ε'Δν −(Al(l +1) − β) # 2 & ν n,l −1 + R dr. 1 +c (R) R 1 dc . = 2 and A s s dr Δν - ∫ 0 = 2 - − ∫ 0 , 0 cs / 4π Δν , R 0 r dr / 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 ≈ ν n+1,l −ν n,l Δν 2 (4l + 6) A ≈ ν n,l −ν n−1,l+2 ≡ δν n,l 19 ν n,l g) Asymptotic periods for gravity waves
n +η Pn,l ≈ P0 l(l +1)
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 h) Summary
The resonant modes of stellar vibration ( n , ℓ ) are formed by constructive interference of propagating sound waves. In the limit of large n / ℓ , the rays pass almost through the center of the star (see slide 22). The phase delay along the ray, ω τ , plus the retardation at the surface, α ˆ π , and the caustic surface, ( π / 2 ) , must resonate with the interference pattern on the stellar surface, ( ℓ + 1 / 2 ) π , e.g., an integral multiple of 2π . 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: R −1 τ = 2 ∫ cs dr 0 R 1 dc Δχ ∝ s dr ∫ 21 0 r dr 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). 22 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 Δ ν / 2 . For rapidly rotating stars, rotational splitting23 can also be measured. The asteroseismic diagram ( Δ ν versus δν ) for the α 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 age24 are based upon simulations Applications
Echelle diagram formed by stacking overtones above each other. Curvature relates to regions of abrupt change
(ionization zones, base of the25 convective envelope) Applications
Frequency of maximum set by surface structure: c ν ∝ s ∝ g / T 1/2 max H eff
main sequence
Msun
Sub giant Kepler
Sub giant
base RGB Oscillationspectrum1 of measuredstars by
27 Oscillation spectrum of 1 Msun stars measured by Kepler Red Clump RGB RGB RGB 28 Uncertainties in log(g) from uncertainties in the estimated maximum frequency and in the analysis models. A 1% change in ν max results in a 0.17% change in log g. With R (Gaia), masses can be determined. 29 j) Solar Oscillation Spectrum
Error bars are30 1000 s ! 31 k) Scaling Relations
ν versus Δν max ν is location of max maximum power: theoretical studies imply ν is 70% of max acoustic cut off frequency, ν ac
" 2 % (ν −ν max ) P(ν) = Pν exp$− ' max # 2σ 2 &
32 3 1/2 1/2 2 1/2 Δν ∝(M / R ) & ν max ∝ν ac ∝ g / Teff ∝ M / R Teff 3 3/2 # & 4 # & M ν max # Δνo & Teff = % ( % ( % ( Mo $ν max,o ' $ Δν ' $Teff ,o ' 1/2 # & 2 # & R ν max # Δνo & Teff = % (% ( % ( Ro $ν max,o '$ Δν ' $Teff ,o '
Teff would have to come from spectroscopic studies. This also yields luminosities
33 3 1/2 1/2 2 1/2 Δν ∝(M / R ) & ν max ∝ν ac ∝ g / Teff ∝ M / R Teff 2 4 L ∝ R Teff 3/2 2 # &6 M # L & # Δν & Teff ,o = % ( % ( % ( Mo $ Lo ' $ Δνo ' $ Teff ' 7/2 # &# &# & M ν max L Teff ,o = % (% (% ( Mo $ν max,o '$ Lo '$ Teff '
Teff would have to come from spectroscopic studies & luminosities from accurate distances.
34 l) Validation
Astero-sizes versus interferometry
35 Astero-Mass Validation
36