A Vivace Introduction to Solar Oscillations and Helioseismology Matthew Kerr Department of Physics University of Washington Nuclear Astrophysics, 2007
Outline 1 Introduction and Motivation 2 Non-radial Solar Oscillations The Equations of Motion Linear Oscillations Characterization of Solutions 3 Linear Inversion Eigenvalue Formulation Solutions
A Musical Motivation The Problem: can you tell me what the inside of a piano looks like using only your ears? 88 notes >Solar Normal Modes Struck Notes >Excited Normal Modes Hammers >Non-adiabatic Mechanism Strings, sounding board, air, wood >Structure Resonance >Nonlinear Effects
A Musical Motivation The Problem: can you tell me what the inside of a piano looks like using only your ears? 88 notes >Solar Normal Modes Struck Notes >Excited Normal Modes Hammers >Non-adiabatic Mechanism Strings, sounding board, air, wood >Structure Resonance >Nonlinear Effects
A Little More Formal The forward problem: Create a solar model. Calculate frequency spectrum analytically or with simulations. Fit model to observations. The inverse problem: Create a solar model. Express observables as integral equations over physical quantities. Invert equations to determine functional form of physical quantities.
Fluid Equations Conservation Laws: ρ t + (ρ u) = 0 d dt u = f p ρ Φ + S Supplementary Equations: T d dt S = P N(ρ, T ) + P V F R 2 Φ = 4πGρ F R = ac 3ρ κ eff (ρ, T ) T 4 p = p(ρ, T )
Perturbations Eulerian: f ( r, t) = f 0 ( r) + f ( r, t) Lagrangian: δf ( r, t) = f (r, t) + ξ f 0 (r 0 ) Work to linear order in the primed quantities: t ( ρ0 + ρ ) + [(ρ 0 + ρ )(0 + v ) ] ρ t + (ρ 0 v ) = 0.
Oscillation Equations Inviscid Flow ρ 0 v t ρ t + (ρ 0 v ) = 0 + p + ρ 0 Φ + ρ Φ 0 = 0 ( ) S ρ 0 T 0 + v S 0 t = (ρp N ) F Form of the equation lends itself to separation of variables: [ ξ = ξ r (r), ξ h (r) ] θ, ξ h(r) Yl m (θ, φ)e iωt. sin θ φ
Simplified Oscillation Equations Further approximations: Adiabaticity (δs 0): valid in most of sun, when photon mean free path «dynamic length scale Cowling Approximation (Φ 0): valid for higher harmonics, when abundant over- and under-densities wash out perturbed potential The equations of motion: 1 d r 2 dr (r 2 ξ r ) g ξ c 2 r + ( 1 L2 l ω 2 ) p ρ c 2 = 0, 1 dp ρ dr + g p ρ c 2 p + (N2 ω 2 )ξ r = 0. The characteristic frequencies: L 2 l = l(l+1)c2 r ( 2 1 d ln p Γ1 dr N 2 = g ) d ln ρ dr
A Wave Equation at Last Change of variable: Ψ c 2 ρ ξ Ignore curvature effects: drop O(1/r), dφ/dr, so forth Acoustic cutoff frequency: ωc 2 ( ) c2 4H 1 2 dh 2 dr The payoff: d 2 Ψ dr 2 = 1 [ c 2 ω 2 ωc 2 L 2 l (1 N2 ω 2 )] Ψ.
Limiting Cases G-modes: ( ) ω < L l, N k(r) N L l c ω 1 ω2 2N 2 Buoyancy most significant restoring force > gravity waves Propagation zones close to core; unstable in convective zones P-modes: ( ) ω > L l, N k(r) ω c 1 L2 l 2ω 2 Pressure most significant restoring force > acoustic waves Propagation in mantle; reflect from low density scale height in outer layers
In Pictures I Christensen-Dalsgaard, 2002
In Pictures II Osaki, 1975
In Pictures III The Sun Christensen-Dalsgaard, 2002
Linear Operators Adiabatic, Cowling approximation > Linear wave equation for ξ Only a function of two perturbed quantities (e.g. ρ and p ) Write as L(ρ, p )ξ = ω 2 ξ ω 2 = ξ L(ξ) ξ ξ
Model Dependence Chandrasekhar showed ω 2 stationary under perturbations to ξ So, to good accuracy, δω 2 = ξ δl(ξ) ξ ξ Only dependence on unknown physical quantities is in δl! Method: postulate a model; calculate ξ and ω 2 ; form above quantity from differences in observed frequencies with model frequencies
Formal Inversion Express linear relation as O i = d rk ij ( r)q j ( r) = j j d rk ij ( r) m q jm ψ jm ( r) Define a collective index i over the physical quantities and the basic vectors, i.e., j, m i; let this index define a second inner product space O = i q i (K ψ) i q A
Quality of Solution O q A dim[ q]: pieces of information to solve for dim[ O]: pieces of information at hand Can determine reliability of solution by well-known matrix methods, e.g., singular value decomposition of A Can regularize result by, e.g., adding a matrix that washes out small scale (Tikhonov regularization)
Appendix For Further Reading I F. Pijpers. Methods in Helio- and Asteroseismology. Imperial College Press, 2006. Unno et al. Nonradial Oscillations of Stars. University of Tokyo Press, 1989. J. Christien-Dalsgaard. Helioseismology. Review of Modern Physics, 74, 2002.