Frequency dependence of the solar-cycle frequency variation M. Cristina Rabello-Soares Stanford University

Introduction The correlation between solar acoustic mode frequencies and the magnetic activity cycle is well established and has been substantially studied during the last two solar cycles. However, its physical origin is still a matter of debate. Recently, Rabello-Soares, Korzennik and Schou (2008) extended the analysis to high-degree modes. As data from the end of solar cycle 23 is now available, this analysis is revisited.

Solar acoustic mode frequencies were obtained by applying spherical harmonic decomposition to MDI full-disk observations (medium-l: Larson & Schou; high-l: Rabello-Soares & Korzennik.

ftp://ftp.ngdc.noaa.gov MDI Dynamics program: 2-3 months continuous observation each year. Δs/max(Δs)

a scending phase ( ) descending phase ( ) ---- Both ( ) δ = a 1 * [Δs/max(Δs)] + a 2 * [Δs/max(Δs)] 2 Δs: solar cycle variation in relation to the 2008 period

asc. desc. anti-correlated

δ = a1 * (Δs/max(Δs)) + a2 * (Δs/max(Δs))^2 c1>0, c2 0, c2<0 c1 0 c1 0

Hysteresis The frequency shift as a function of solar activity follow different path for the ascending and descending phases (Jimenez-Reyes et al. 1998, Tripathy et al. 2001) Certain pairs of solar activity indices present hysteresis.

δ δ δ (0,2) [μHz] δ (1,3) [μHz] Jiménez-Reyes et al. (1998) A&A 329, 1119 Fig l=0,2 l=1,3 descending

asc. desc. anti-correlated 1999

Area = abs[∫ δ (desc.)] – abs[∫ δ (asc.)] high l medium l

ascending phase ( ) descending phase ( )

δ (n,l) behavior δ at solar maximum (2 nd degree fitting full cycle)

medium l high l Qnl is the mode inertia normalized by the inertia of a radial mode of the same frequency, calculated from model S (Christensen-Dalsgaard et al. 1996)

medium l high l

n > 5 Surface term (Fsurf = Qnl * δ / )

Libbrecht and Woodard (1990, Nature, 345, 779) were the first to suggest that the observed frequency shift is linearly proportional to the inverse mode inertia. Recently, Rabello-Soares, Korzennik and Schou (2008, Solar Physics, 251, 197) extended the analysis to high- degree modes and observed that scaling the frequency shift with the mode inertia normalized by the inertia of a radial mode of the same frequency follows a simple power law, with a different exponent for the f and p modes.

using only positive δ

Conclusions I. δ (solar index) Quadractic term (a2) is important. Linear relationship: a2 = b0 – b1 * a1 c1>0, c2 0, c2<0 c1 0 Hysteresis with radio flux (F 10 ): | δ (declining)| > | δ (ascending)| Some modes are anti-correlated with the solar cycle (specially n=2, but also n=1 and 3).

Conclusions II. δ ( ) Qnl * δ α ν γ with a different coefficient γ for frequencies smaller and larger than: –2.5 mHz (p modes). –~2 mHz (f modes).

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Abstract The correlation between solar acoustic mode frequencies and the magnetic activity cycle is well established and has been substantially studied during the last two solar cycles. However, its physical origin is still a matter of debate. Libbrecht and Woodard (1990, Nature, 345, 779) were the first to suggest that the observed frequency shift is linearly proportional to the inverse mode inertia. Recently, Rabello-Soares, Korzennik and Schou (2008, Solar Physics, 251, 197) extended the analysis to high-degree modes and observed that scaling the frequency shift with the mode inertia normalized by the inertia of a radial mode of the same frequency follows a simple power law, with a different exponent for the f and p modes. As data from the end of solar cycle 23 is now available, this analysis is revisited. Solar acoustic mode frequencies with degree up to 900 obtained by applying spherical harmonic decomposition to MDI full-disk observations are analysed. The dominant structural changes during the solar cycle, inasmuch as they affect the mode frequencies, is given by surface effects. After subtracting the surface effects, the frequency-shift residuals will be inverted to search for small variations of the sound speed with the solar cycle up to 0.99R.

Comparison results solar physics New results (this work) –**** ONLY p modes and only delnu > 0. **** >2500: gamma = <2500: gamma = Sol Physics: nu>2200: chaplin: mHz: (alpha=0) and gamma= mHz: (alpha=1.91) and gamma=

Area = ∫ δ (rising) – ∫ δ (declining) medium l high l