Eddington and The Internal Constitution of the Stars

1. Eddington and The Internal Constitution of the Stars What appliance can pierce through the outer layers of a star and test the conditions within?” A S Eddington, The Internal Constitution of the Stars, p1, 1926.

2. The Goals of Asteroseismology with Eddington To test and develop our understanding of stellar evolution and to apply this new understanding to determinations of chemical evolution (large mass) and ages (small mass) the internal structure P,r,G1,W the size of convective cores – core overshooting signatures of mixing by rotation, … the depth of convective envelopes the entropy of convective zones and He abundance properties of convective turbulence (line shapes)

3. The Principles of Asteroseismology Stars oscillate in many modes, excited by turbulent convection, opacity, … Oscillation frequencies of a star are determined by its internal structure and dynamics: ni= f(r,G1,W) Measured frequencies give internal structure and dynamics: r(r) =F(ni), W(r,q) = G(ni) Compare model frequencies with observations Inversion: determine model that best fits frequencies

4. Known oscillating stars in the H-R diagram Eddington will Cover HR diagram Masses, ages, compositions Hipparcos Stars and theoretical evolutionary tracks Courtesy JC-D

5. Power spectrum of time series of flux F(t) gives frequenciesn(whole disc) Virgo on SoHO The Sun as a star Appourchaux et al 1966

6. Oscillations of a Spherical Star dY(r,t) =ynlm(r) Ylm(q,f)ei2pnt n = nnlm Ylm spherical harmonics n radial order  no of nodes l degree, m azimuthal order Only modes of low degree observable over integrated disc m degenerateW, B =0 W important see Goupil’s talk n=18, l=2, m=2

7. Equations of Spherical Oscillations dr=x(r)Ylmeiwt, dP=p/(r)Ylmeiwt, dF=f/(r)Ylmeiwt

8. N2 >> w2 N2 << w2 w2r2 << l(l +1) c2 w2r2 >> l(l +1) c2 n large n small n large n-1n,l -n-1n+1,lconst nn+1,l-nn,lconst Pressure and Gravity modes g-modes p-modes WD SPBs d Scuti/bCephs Solar Type/SubG

10. Can be much more complicated with mixed p- and g-modes Behave like g-modes in the interior and p modes in the outer layers eg b Hydri, HD57006

11. Model Fitting to Observed Frequencies Example Solar p-mode Frequencies BiSON data 1 Standard Solar ModelS 2 Inversions Here just 2 examples (IR&SV) Other techniques in use (eg OLA) 1 Chaplin et al 1998 2 Christensen-Dalsgaard et al 1996

14. Difference in frequencies between Solar modelS (JCD) and BiSON data ModelSxf is identical to modelS x<xf with modified surface layers for x>xf

15. Values of frequencies are strongly influenced by surface layers which are not well understood Better to use frequency differences and ratios Solar type p-modes Large separations, Small separations Asteroseismic Diagram (JC-D, 1988)

16. Large and Small Separations Dl =nn+1,l – nn,l dl,l+2 = nn,l – nn-1,l+2 Virgo on SoHO D0 l=0, n=21 d13 D1 d02

17. _ ∫ T= dr c(r) R 0 Large Separations Dl Measure of acoustic radius Strongly influenced by structure of the surface layers

18. Small Separations dl,l+2 Determined primarily by structure of the core c(r) Measure of evolution hence of age

19. Small separations from modes with l=0,1 The periodic signal in d01 is determined by the acoustic radius of the convective envelope

20. The C-D Asteroseismic Diagram Mass Xc Plots average small separation d02 vs average large separation D IF Theory correct Measure of mass and age age Alsod01 d13 AM & IR, 2003

21. The l=0,1 C-D Asteroseismic Diagram Mass Xc Plots average small separation d01 vs average large separation D Different measure of internal structure and age age AM & IR, 2003

22. Cluster Fitting

23. Classical Cluster fitting Principle: Fit theoretical HR diagrams to observations all stars assumed to have same age and initial composition Observable HR: Vi, (B-V)i { X, Z, D ?} : 2n+? observations Theoretical HR : Mi ,t, X, Z, D : n+4 parameters add mixing length parameter a (entropy of CZ) overshoot parameter b (core overshoot=bMcore ) 2n+? observations n+6 parameters Best fit gives t, Mi, X, Z, D, a, b Assumes theory is correct except for parameters a,b

24. Fitting the cluster IC4651 The two isochrones with ages 1.6 and 2.0 109 y have no convective core oveshooting The best isochrone fit has an age t=2.3 109 y and is for models with convective core overshooting (b=0.6, IC) Dowler & Vandenberg 1996

25. Cluster fitting with frequencies More observables eg <D> and <d02> ; frequencies nnl Determine ai bi for each star ie a(M,X,Z,t) b(M,X,Z,t) Probing physics of convection not just parameter fitting!! • Group of 6 stars M=0.9, 1, 1.1, 1.5. 2, 2.2 Msun • A) V, B-V ± 0.1mag • <D>± 0.1mHz, <d02> ± 0.3mHz • nnl ± 0.4mHz 10 frequencies for each star Assume physics of stellar evolution correct except for ai bi Inconsistencies reveal other errors in models of stellar evolution

26. Errors in parameter estimation for group of 6 stars NA&IR 1998

27. a1 a2a3b4b5b6 Errors in estimation of convective parameters for group of 6 stars

28. b1 b2 b3 b4 Errors in estimation of parameters group of 4 massive stars 5,8,10,12 Ms NA & IR, 1998

29. Probing the interior structure The surface layers are not well understood: non adiabatic convection; atmospheric structure; 3-D not 1 D; non adiabatic oscillations Subtracting off the effect of the surface layers Ratio of small to large separations Modify surface layers of solar model leaving interior exactly the same P(r), r(r), Mr(r), G1(r) Note especially model C: Solar structure r<0.71Rs and adiabatic envelope with G1=5/3. 1.00001Ms, 1.06Rs

30. Effect of surface layers S = Solar ModelS A=ModelS, G1=5/3 B=ModelS, xf=0.90 C=ModelS, xf=0.71 adiabatic envelope G1=5/3 All models have same P(r),r(r),Mr(r) in interior r<rf

31. Diagnostics of interior structure The ratio of small to large separations for the 4 different “Solar models” These ratios are ‘independent’ of the structure of the outer layers of a star

32. Diagnostics of interior structure The ratio of (d01) small to large separations for the 4 different “Solar models” These ratios are ‘independent’ of the structure of the outer layers of a star

33. Inversion for the interior structure of a star using the ratio of separations Values of frequencies depend on structure of outer layers but the functional forms of the ratios Fk(n) = dk/D are determined solely by interior structure (Why? – see later) Inversion: find structure r(r) that best fits the Fk(n). Hydrostatic support gives P, Mr and take G1=5/3 Model the star by the values of Di=dLogr/dLogP at a set of mass points Mi - iterate to find best fit to the Fk(n) Example M=1.45Msun Xc= 0.35

34. Inversion for a 1.45 Msun star Frequency set n=600-1800 mHz, l=0,3 s=0.2mHz Inversion using ratio of small to large separationsG1=5/3 IR&SV, 2002

35. Ratio of small to large separations Fk(n) = dk/D are determined solely by the interior structure WHY?

36. Properties of Eigenmodes y(t) = r p/(rc)1/2 , t= ∫ dr/c acoustic radius Look like modified Spherical Bessel functions exact for isothermal non gravitating sphere Partial waves yi(w,t) – any w, interior out to some rf ye(w,t) – any w, exterior in to some rf Continuity of partial waves determines eigenfrequencies wnl Differential Response Inversion technique

37. Eigenmodes yl(w,t)= r p/(rc)1/2 t= ∫dr/c is the acoustic radius yl behaves like Spherical Bessel function Jl(wt): exact for isothermal non gravitating sphere Rayleigh (1896) yl(w,t) distorted Bessel function

38. Partial Waves Outward propagating wave at tf with any w determined only by interior structure t<tf Inward propagating wave determined by exterior structure t>tf Exact for l =0. For l0 apply Laplace boundary condition at xf : very good as r very small x>xf

39. Partial Waves Overlay of full eigenmode and partial wave with same frequency Zoom in at tf The partial wave is very good fit to full eigenmode

40. Eigenmode, standing wave, when inner and outer partial waves match: y and dy/dt continuous ℱ(w,t)=wy/(dy/dt) is scale independent Plot inner and outer solutions for different w