Theoretical Interpretation of Power Spectra of Stellar Oscillations H.Ando National Astronomical Observatory of Japan 9th, December, 2004

1. Observed Power Spectra of Stellar Oscillations Procyon(F5 subg) Kambe (2000) Sun, αCen A, βHyi(G2 subg), η Boo(G0 subg), ξ Hya(G7 g) Bedding and Kjeldsen (2003) PASA, 20, 203-212

Spectrum of Procyon on 25,26,27,28, and 29, Dec., 2000

Summed spectrum of Procyon on 25, 28, and 29, Dec., 2000

Power Spectra

Charactristics of Spectra ・shape of envelope (peak freq.) ・Asymptotic Formula Large separation Small Separation

Newly recognized points ・Not a simple distribution of amplitudes ・Deviation from equal spacing pattern

2. Theoretical background of Oscillations ・Radial Pulsation (l=0) acoustic modes(n=0,1,2,....) fn(r) ・non-radial oscillation(l≠0) acoustic modes (p-mode) (n,l,m) gravity modes (g-mode)(n,l,m) fn(r)Ylm(θ,φ)

Propagation diagram in Stars ex. Procyon M=1.42 M⦿ (Prieto et. al 2002) #1(ZAMS) #51 (Procyon; L=7L⦿, Te=6530) #211(giant; L=7.7L⦿, Te=4490) Observed parameters of Procyon L= 7.04L⦿ , Te= 6530

Propagation diagram for #1 36 36 0 2236.61 47.29 5.90E-07 35 35 0 2116.27 46.00 8.05E-07 34 34 0 1999.02 44.71 9.92E-07 33 33 0 1884.98 43.42 1.15E-06 32 32 0 1774.07 42.12 1.29E-06 31 31 0 1666.27 40.82 1.40E-06 30 30 0 1561.78 39.52 1.48E-06 29 29 0 1460.84 38.22 1.53E-06 28 28 0 1363.62 36.93 1.55E-06 27 27 0 1270.12 35.64 1.54E-06 26 26 0 1180.19 34.35 1.52E-06 25 25 0 1093.67 33.07 1.49E-06 24 24 0 1010.51 31.79 1.45E-06 23 23 0 930.78 30.51 1.42E-06 22 22 0 854.58 29.23 1.40E-06 21 21 0 781.89 27.96 1.40E-06 20 20 0 712.57 26.69 1.42E-06 19 19 0 646.44 25.43 1.46E-06 18 18 0 583.36 24.15 1.49E-06 17 17 0 523.35 22.88 1.51E-06 16 16 0 466.58 21.60 1.52E-06 15 15 0 413.22 20.33 1.50E-06 14 14 0 363.33 19.06 1.47E-06 13 13 0 316.95 17.80 1.41E-06 12 12 0 274.08 16.56 1.33E-06 11 11 0 234.74 15.32 1.25E-06 10 10 0 198.94 14.10 1.19E-06 9 9 0 166.66 12.91 1.23E-06 8 8 0 137.71 11.73 1.42E-06 7 7 0 111.79 10.57 1.91E-06 6 6 0 88.70 9.42 2.97E-06 5 5 0 68.21 8.26 5.43E-06 4 4 0 50.49 7.11 1.14E-05 3 3 0 35.37 5.95 2.91E-05 2 2 0 22.92 4.79 1.02E-04 1 1 0 12.80 3.58 7.99E-04 -1 0 1 4.23 2.06 6.11E-01 -2 0 2 1.97 1.41 7.14E-01 P G P G

25 25 0 1224.07 34.99 3.05E-06 24 24 0 1131.91 33.64 3.21E-06 23 23 0 1043.65 32.31 3.38E-06 22 22 0 959.17 30.97 3.48E-06 21 21 0 879.00 29.65 3.59E-06 20 20 0 802.57 28.33 3.83E-06 19 19 0 746.14 27.32 1.42E-04 18 19 1 728.52 26.99 4.35E-06 17 18 1 659.75 25.69 4.33E-06 16 17 1 593.52 24.36 4.72E-06 15 16 1 530.53 23.03 5.13E-06 14 15 1 470.90 21.70 5.54E-06 13 14 1 414.70 20.36 5.86E-06 12 13 1 362.18 19.03 6.03E-06 11 12 1 319.80 17.88 9.31E-04 10 12 2 313.30 17.70 6.11E-06 9 11 2 268.43 16.38 5.81E-06 8 10 2 227.25 15.07 5.51E-06 7 9 2 189.91 13.78 5.02E-06 6 8 2 167.35 12.94 6.05E-03 5 8 3 156.58 12.51 4.58E-06 4 7 3 127.08 11.27 4.37E-06 3 6 3 105.14 10.25 1.12E-03 2 6 4 101.36 10.07 4.58E-06 1 5 4 79.28 8.90 5.75E-06 0 4 4 73.00 8.54 1.80E-03 -1 4 5 60.86 7.80 9.30E-06 -2 3 5 53.57 7.32 5.26E-04 -3 3 6 47.92 6.92 2.70E-05 -4 3 7 41.58 6.45 6.67E-05 -5 2 7 39.04 6.25 8.00E-05 -6 2 8 32.63 5.71 1.70E-04 -7 2 9 30.25 5.50 1.39E-04 -8 2 10 26.37 5.14 2.36E-04 -9 1 10 24.17 4.92 4.53E-04 P G P G

211 1 -23 22 45 1097.25 33.12 2.65E-05 -24 22 46 1060.79 32.57 6.50E-06 -25 21 46 1046.33 32.35 1.15E-05 -26 21 47 1009.25 31.77 1.01E-04 -27 21 48 975.16 31.23 1.78E-05 -28 21 49 962.36 31.02 1.82E-05 -29 20 49 930.81 30.51 2.01E-04 -30 20 50 897.73 29.96 8.23E-05 -31 20 51 882.12 29.70 1.95E-05 -32 19 51 860.46 29.33 2.05E-04 -33 19 52 830.45 28.82 3.23E-04 -34 19 53 806.29 28.40 3.59E-05 -35 19 54 795.75 28.21 6.40E-05 -36 18 54 770.81 27.76 5.44E-04 -37 18 55 745.08 27.30 4.06E-04 -38 18 56 726.23 26.95 4.20E-05 -39 17 56 716.05 26.76 1.23E-04 -40 17 57 694.02 26.34 8.00E-04 -41 17 58 671.87 25.92 6.44E-04 -42 17 59 653.85 25.57 7.07E-05 -43 17 60 645.80 25.41 9.67E-05 -44 16 60 627.81 25.06 8.81E-04 -45 16 61 608.60 24.67 1.22E-03 -46 16 62 590.72 24.30 3.20E-04 -47 16 63 580.86 24.10 6.02E-05 -48 15 63 570.18 23.88 4.37E-04 -49 15 64 553.83 23.53 1.64E-03 -50 15 65 537.77 23.19 1.58E-03 -51 15 66 522.89 22.87 3.66E-04 -52 15 67 514.61 22.68 7.27E-05 -53 14 67 505.59 22.49 4.93E-04 -54 14 68 491.91 22.18 1.96E-03 -55 14 69 478.35 21.87 2.31E-03 -56 14 70 465.50 21.58 9.38E-04 -57 14 71 455.57 21.34 1.08E-04 -58 14 72 450.26 21.22 1.92E-04 -59 13 72 439.52 20.96 1.54E-03 -60 13 73 428.07 20.69 3.16E-03 -61 13 74 417.01 20.42 2.85E-03 -62 13 75 406.54 20.16 1.09E-03 -63 13 76 398.22 19.96 1.27E-04 -64 13 77 393.84 19.85 1.89E-04 G P G P

Interaction bet ween Two potential wells ・ZAMS: Almost independent

・Advanced Evolution stage: Mixed character Avoided Crossing

Mixed Mode

3. Prediction of Power Spectra Basic assumptions ・Power ∝ (Input Energy)/(Kinetic Energy of mode), where KE is estimated with radial displacement, say(δr/r=1.0), at the surface. ・Input Energy Continuous spectrum by turbulent convection (Kolmogorov spectrum) We give Power ∝ f^n/(KE) n=2 : flat input energy (say, 1m/s at the surface) n=-2: Kolmogorov type input energy

Summary ・There are pulsation modes (l=0) beyond Cut-off frequency ・There are practical cut-off in lower end due to existence of g-modes’ territory ・Larger interaction of p-modes and g-modes in l=1 -modes with smaller amplitudes in mixed modes -frequencies of modes(l=1) shifted to modes with l=0 in lower frequency region

A possible suggestion Quantitative analysis of the Oscillation spectrum of ηBoo Guenther AJ, 612, 454-462, 2004