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Inferring Solar Internal Structure and Rotation from p-mode Frequencies and Splittings II Michael Thompson University of Sheffield, UK Affiliate scientist, High Altitude Observatory Singular values for 1-D rotation problem with 834 p-modes and 100-point radial mesh 2-D rotation inversion The 1-D rotation example developed in the last lecture is straightforwardy generalized to the case of Ω = Ω (r, θ). e.g. Schou et al. (1994) ApJ 433, 389 a-coefficients Commonly the results of the analysis of the observations are not individual nlm frequencies. Rather, the frequencies in each nl multiplet are fitted as a polynomial in m: where the P’s are even or odd polynomials in m of degree j. The odd coefficients aj can be used as the data for rotation inversions. RLS OLA 2-D Rotational Averaging Kernels Close-Up RLS OLA (1 s.d. uncertainties on inversion are indicated in nHz, for a typical MDI dataset) Inferred rotation inside the Sun from MDI data (Schou et al. 1998, ApJ 505, 390) Radial cuts through inferred rotation profile of the solar interior (at latitudes indicated) tachocline Location and width of tachocline Tachocline shape in depth typically fitted by a smooth step like this, with centre location rc and width w. Finite resolution artificially broadens the tachocline: this must be compensated for. At equator, rc/R = 0.692 +/- 0.002; w = 0.03 – 0.04. Rotation in the deeper interior from LOWL + BiSON data (Chaplin et al. 1999, MNRAS 308, 405) Possibly slightly slow core rotation - but inversions consistent with uniform rotation in deep interior. Differences between inversions of data taken at successive times reveal the “torsional oscillation” 0.99 R 0.95 R 0.90 R 0.84 R GONG RLS MDI RLS MDI OLA Here we difference rotation inversions relative to solar minimum at successive 1-year epochs. The evolution of the rotation rate in the whole convection zone can be seen. The rotation rate appears to show quasi-periodic oscillations near the base of the convection zone at mid-latitudes Black: GONG, red: MDI. Filled: RLS, open: OLA Linearized inversion for solar structure The dependence of the frequencies on solar structure is inherently nonlinear. But we can use linear inversion techniques if we assume that the Sun’s structure and frequencies are small perturbations to those of a known reference model. You have seen in JCD’s lectures that the frequencies can be written in terms of a variational principle such that This can describe differences between the Sun and the reference model. δF depends on differences in quantities p, ρ and Γ1. Primary seismic variables The model quantities that appear in the equations governing the adiabatic oscillations are p, ρ, Γ1 (and combinations such as c). These are therefore the primary seismic variables. Except for additional physics (e.g. rotation) they are the ONLY quantities that can be inferred from the frequencies unless we introduce additional assumptions. For full details see Gough & Thompson (1991), in Solar Interior and Atmosphere, eds Cox, Livingston & Matthews, p. 519-561 (Univ. of Arizona Press) Invoking hydrostatic equilibrium There appear to be three independent unknown functions: δp/p, δρ/ρ, and δΓ1/Γ1. But the oscillations are presumed to take place about an equilibrium background in hydrostatic equilibrium: Perturbing this gives Likewise, using the mass equation, δm can be written in terms of δρ . Hence δp/p can finally be expressed in terms of δρ/ρ, and the number of unknown functions reduced from 3 to 2. Kernels for sound speed and density Kernel for c2 Kernel for ρ Mass conservation - an additional constraint Density perturbations cannot be chosen arbitrarily, as the mass of the Sun is known, i.e. Hence Transforming between variable pairs A key to computing kernels for other variable pairs is how to use hydrostatic equilibrium to transform between δρ/ρ and δu/u, where u=p/δ . with ψ=0 at r=0 and r=R. Then integration from 0 to R logarithmic derivatives Introducing additional assumptions E.g. assume the equation of state Γ1= Γ1(p,ρ,Y) known, where Y is the helium abundance. So one can express δΓ in terms of other perturbations and hence (after some work) derive kernels for e.g. u and Y. Kernels for density and helium abundance Kernel for ρ Kernel for Y Formulation of structure inversion For each observed mean-multiplet frequency we have a datum (or constraint) of the form e.g. • Knlc,(r), Knl,c(r) are known functions • Gsurf(nl) is a term from near-surface errors in the mode nl are errors in the observations. Also have in this case the mass-conservation constraint, which can be written in the same form as the data constraints. Problem: use these constraints to make inferences about e.g. δc2 and δρ between Sun and model. RLS inversion for structure Perhaps the most obvious approach. Adjust the unknown functions δc2/c2, δρ/ρ and Gsurf to get the best fit to the data. As for rotation, need to regularize, so include terms in the minimization to penalize solutions δc2/c2, δρ/ρ that have e.g. large second derivative. Typically choose Gsurf(ω) to be a low-order polynomial. OLA inversion for structure Try to choose inversion coefficients ci(r0) so that e.g. is localized near r=r0. If successful, then is a localized estimate of the relative difference in sound-speed squared between Sun and model near r=r0. OLA inversion for structure Choose inversion coefficients ci to minimize where averaging kernel cross-talk kernel subject to the constraints unimodular constraint surface constraints Results of OLA Sound speed inversion of solar data Fractional differences between Sun and a Density model, in sense (Sun minus model) from BiSON + LOWL data (Basu et al. 1997, MNRAS 291, 243) Depth of convection zone From an inversion for sound speed, can calculate W, which in the convection zone takes the approximately constant value -(Γ1-1) (except in regions of partial ionization). inversion model Seismically determined location of base of convection zone is rcz/R = 0.713 +/- 0.004 Helium abundance From inversions using u and Y, Richard et al. (1999) determined helium abundance in the solar convection zone to be 0.248 +/- 0.002 Can also (try to) use W the HeII bump in W at r=0.98R either by fitting or from its signature as a sharp feature Overshoot at base of c.z. Conclusion: extent of any overshoot is small – less than 0.1 HP (0.007R) Introduction to the hands-on session • Use an IDL-based workbench for 1-D rotation inversion • Try using both RLS and SOLA methods • Experiment with trade-off parameters (λ2 for RLS, θ and δ for SOLA) • See how solution and error bars, averaging kernels and inversion coefficients behave as you vary the trade-off parameters • Four datasets to try: - Artificial rotation profile - Artificial rotation profile with 10nHz s.d. noise - Uniform rotation profile - Uniform rotation profile with 10nHz s.d. noise • There is also a fifth dataset for a mystery rotation profile, with 2nHz s.d. noise You are asked to hand in written assessments of what you can deduce (qualitatively and quantitatively) about the mystery rotation profile. There will be a PRIZE (on Friday) for the best overall deduction. Further reading: Kosovichev, A. G. (1999). Inversion methods in helioseismology and solar tomography J. Comp. Applied Math. 109, 1-39. Thompson, M. J. et al. (2003). The internal rotation of the Sun Ann. Rev. Astron. Astrophys. 41, 599-643.
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