Frequency dependence of the Frequency dependence of the solar-cycle frequency variationsolar-cycle frequency variation

M. Cristina Rabello-SoaresM. Cristina Rabello-Soares

Stanford UniversityStanford University

IntroductionIntroduction

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

• ascending phase (1999-2002)

• descending phase (2002-2007)

---- Both (1999-2007)

δ = a1 * [Δs/max(Δs)] + a2 * [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<0c1>0, c2<0• c1<0, c2>0c1<0, c2>0

HysteresisHysteresis

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

Certain pairs of solar activity indices present Certain pairs of solar activity indices present hysteresis.hysteresis.

δδ(0,2) [μHz] δδ(1,3) [μHz]

Jiménez-Reyes et al. (1998)A&A 329, 1119

Fig. 4

-0.2 0.0 0.2 0.2 0.4

l=0,2l=1,3

descendingde

scen

ding

• asc.• desc.

anti-correlated

1999

Area = abs[∫ δ(desc.)]

– abs[∫ δ(asc.)]

• high l• medium l

• ascending phase (1999-2002)

• descending phase (2002-2007)

δδ(n,l) behavior(n,l) behavior

δδ at solar maximum (2 at solar maximum (2ndnd degree fitting full cycle) 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 Libbrecht and Woodard (1990, Nature, 345, 779) were the first to suggest that the observed frequency shift is the first to suggest that the observed frequency shift is linearly proportional to the inverse mode inertia.linearly proportional to the inverse mode inertia.

Recently, Rabello-Soares, Korzennik and Schou (2008, Recently, Rabello-Soares, Korzennik and Schou (2008, Solar Physics, 251, 197) extended the analysis to high-Solar Physics, 251, 197) extended the analysis to high-degree modes and observed that scaling the frequency degree modes and observed that scaling the frequency shift with the mode inertia normalized by the inertia of a shift with the mode inertia normalized by the inertia of a radial mode of the same frequency follows a simple radial mode of the same frequency follows a simple power law, with a different exponent for the f and p power law, with a different exponent for the f and p modes.modes.

• using only positive δ

ConclusionsConclusionsI. I. δδ(solar index)(solar index)

Quadractic term (a2) is important.Quadractic term (a2) is important.

Linear relationship: a2 = b0 – b1 * a1Linear relationship: a2 = b0 – b1 * a1c1>0, c2<0c1>0, c2<0c1<0, c2>0c1<0, c2>0

Hysteresis with radio flux (FHysteresis with radio flux (F1010): ): | | δδ(declining)| > | (declining)| > | δδ(ascending)|(ascending)|

Some modes are anti-correlated with the solar cycle Some modes are anti-correlated with the solar cycle (specially n=2, but also n=1 and 3).(specially n=2, but also n=1 and 3).

ConclusionsConclusionsII. II. δδ(())

Qnl * Qnl * δδ αα ννγγ with a different coefficient with a different coefficient γγ for frequencies smaller and larger than: for frequencies smaller and larger than: – 2.5 mHz (p modes).2.5 mHz (p modes).– ~2 mHz (f modes).~2 mHz (f modes).

ENDEND

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

Comparison results solar physicsComparison results solar physics

New results (this work)New results (this work)– **** ONLY p modes and only delnu > 0. ******** ONLY p modes and only delnu > 0. ****

>2500: gamma = 3.7729806 +- 0.00069806367>2500: gamma = 3.7729806 +- 0.00069806367

<2500: gamma = 6.2307610 +- 0.0010213932 <2500: gamma = 6.2307610 +- 0.0010213932

Sol Physics: nu>2200: 3.56841 Sol Physics: nu>2200: 3.56841 chaplin: chaplin:

1.6-2.5mHz: (alpha=0) and gamma=7.59+-0.18 1.6-2.5mHz: (alpha=0) and gamma=7.59+-0.18 2.5-3.9mHz: (alpha=1.91) and gamma=3.58+-0.03 2.5-3.9mHz: (alpha=1.91) and gamma=3.58+-0.03

Area =

∫ δ(rising) –

∫ δ(declining)

• medium l• high l