1. Variable Stars: Pulsation, Evolution and applications to Cosmology Shashi M. Kanbur, June 2007.

2. Lecture IV: Modeling Stellar Pulsation A pulsating star is not in hydrostatic equilbrium. For example A pulsating star is not in hydrostatic equilbrium. For example ρd 2 r/dt 2 = -GM r ρ/r 2 – dP/dr. ρd 2 r/dt 2 = -GM r ρ/r 2 – dP/dr. Mass continuity equation still holds. Mass continuity equation still holds. Energy equation: Energy equation: dE/dt + PdV/dt + dL/dm = 0, where dE/dt + PdV/dt + dL/dm = 0, where L(r) = -4πr 2 4σ/3κ. dT 4 /dm L(r) = -4πr 2 4σ/3κ. dT 4 /dm ρ(r) = 1/V(r), P = P(ρ,T), E=E(ρ,T), κ=κ(ρ,T). ρ(r) = 1/V(r), P = P(ρ,T), E=E(ρ,T), κ=κ(ρ,T).

3. Modeling Stellar Pulsation Boundary Cnditions: L 0 =L cons., dr/dt) 0 = 0. Boundary Cnditions: L 0 =L cons., dr/dt) 0 = 0. P surface = 0. T surface = f(T eff ) ie. a grey solution to the equationof radiative transfer. P surface = 0. T surface = f(T eff ) ie. a grey solution to the equationof radiative transfer. 1D radiative codes. Now there are “numerical recipes” to model time dependent turbulent convection. 1D radiative codes. Now there are “numerical recipes” to model time dependent turbulent convection.

4. Linear Models Assume displacement from equilbrium, δr, are small. Write variables as Assume displacement from equilbrium, δr, are small. Write variables as P = P 0 + δP, r = r 0 + δr, ρ 0 + δρ etc. P = P 0 + δP, r = r 0 + δr, ρ 0 + δρ etc. Expand pulsation equations and drop second order terms. This is linear stellar pulsation. Expand pulsation equations and drop second order terms. This is linear stellar pulsation. Assume δr = |δr|e iωt, solve resulting eigenvalue problem. Leads to linear periods and growth rates ie. Whether a given perturbation is stable or will continue to grow. Assume δr = |δr|e iωt, solve resulting eigenvalue problem. Leads to linear periods and growth rates ie. Whether a given perturbation is stable or will continue to grow. Can investigate boundaries of “instability strip” with such a technique. Can investigate boundaries of “instability strip” with such a technique.

5. Non-Linear Models Write differential equations as difference equations over a computational grid covering the star. Write differential equations as difference equations over a computational grid covering the star. Zones 1,……,N, with interfaces 0,1,….N+1. Zones 1,……,N, with interfaces 0,1,….N+1. Extensive variables r, velocity, v r, luminosity, L r, defined at zone interfaces. Extensive variables r, velocity, v r, luminosity, L r, defined at zone interfaces. Intensive variables defined at zone centers, T, ρ, P, κ etc. Intensive variables defined at zone centers, T, ρ, P, κ etc. Sometimes may need to extrapolate intensive/extensive variables to zone interface/centers. Sometimes may need to extrapolate intensive/extensive variables to zone interface/centers. Time mesh: t n+1 = t n + Δt n+1/2,t n+1/2 – t n-1/2 = Δt n, Δt n = ½(Δt n-1/2 + Δt n+1/2 ). Time mesh: t n+1 = t n + Δt n+1/2,t n+1/2 – t n-1/2 = Δt n, Δt n = ½(Δt n-1/2 + Δt n+1/2 ).

6. Non-Linear Models Momentum equation: Momentum equation: v n+1/2 (I) = v n-1/2 (I) – Δt n (GM(I)/r n (I) 2 + 4π(r n (I)) 2 /ΔM(I)[P n (I) – P n (I-1) + Q n-1/2 (I) – Q n+1/2 (I-1)]) v n+1/2 (I) = v n-1/2 (I) – Δt n (GM(I)/r n (I) 2 + 4π(r n (I)) 2 /ΔM(I)[P n (I) – P n (I-1) + Q n-1/2 (I) – Q n+1/2 (I-1)]) Leads to a matrix equation Ax=d to be solved for the increments to the physical variables at each time step. Leads to a matrix equation Ax=d to be solved for the increments to the physical variables at each time step. Q: Artifical vsicosity. Q: Artifical vsicosity. Field in its own right. Field in its own right.

7. Pulsation Modeling Linear model to find set of L,M, X,Z,T eff. Linear model to find set of L,M, X,Z,T eff. Also get eigenvector showing ampltide of rafial displacement. Also get eigenvector showing ampltide of rafial displacement. Non-linear model with an initial “kick” scaled by linear eigenvector for that model Non-linear model with an initial “kick” scaled by linear eigenvector for that model Continue pulsation until amplitude increase levels of: several hundred cycles, maybe 1-2 hours on a modern fast PC. Continue pulsation until amplitude increase levels of: several hundred cycles, maybe 1-2 hours on a modern fast PC. Need opacity tables, equation of state (usually Saha). Need opacity tables, equation of state (usually Saha). Result is a nonlinear full amplitude variation of L with T. Result is a nonlinear full amplitude variation of L with T. Stellar atmosphere converts this to magnitude and color. Stellar atmosphere converts this to magnitude and color. Compare with observations via Fourier analysis. Compare with observations via Fourier analysis. This is for radial oscillations. This is for radial oscillations. No time dependent code to model non-radial oscillations exists. No time dependent code to model non-radial oscillations exists.

8. Non-Radial Oscillations Expand perturbatin δr in terms of spherical harmonics, specified by 3 numerbs, n, l, m. Expand perturbatin δr in terms of spherical harmonics, specified by 3 numerbs, n, l, m. δr = R(r)Y(θ,φ): n is for the radial part, l, m the angular part. δr = R(r)Y(θ,φ): n is for the radial part, l, m the angular part. l=m=0, pulsation purely radial. l=m=0, pulsation purely radial. l=0,1,2,,,n-1 and m=-l+1,-l+2,….l-1 l=0,1,2,,,n-1 and m=-l+1,-l+2,….l-1 With l,m non-zero need to worry about Poisson’s equation as well. With l,m non-zero need to worry about Poisson’s equation as well. n: number of nodes radially outward from Sun’s center. m: number of nodes found around the equator. l: number of nodes found around the azimuth (great circle through the poles) n: number of nodes radially outward from Sun’s center. m: number of nodes found around the equator. l: number of nodes found around the azimuth (great circle through the poles) Hard mathematical/numerical problem. Hard mathematical/numerical problem. P-modes: pressure is the restoring force, G modes: gravity is the restoring force. P-modes: pressure is the restoring force, G modes: gravity is the restoring force.

9. Helioseismology Sun is a non-radial oscillator. Sun is a non-radial oscillator. Modes with periods between 3 an d8 minutes – five minute oscillations are p modes: l going from 0 to 1000. Modes with periods between 3 an d8 minutes – five minute oscillations are p modes: l going from 0 to 1000. Modes with longer periods – about 160 minutes could be g modes: l ~1-4. Modes with longer periods – about 160 minutes could be g modes: l ~1-4. Comparison of observed and theoretical frequencies can be used to calibrate solar models: helioseismology. Comparison of observed and theoretical frequencies can be used to calibrate solar models: helioseismology. Can reveal the depth of the solar convection zone, plus rotation and composition of the outer layers of the Sun. Can reveal the depth of the solar convection zone, plus rotation and composition of the outer layers of the Sun.

12. One Zone Models Central point mass of mass M. At a radius R is a thin spherical shell, mass m. There is a pressure P in this shell which provides support against gravity. Central point mass of mass M. At a radius R is a thin spherical shell, mass m. There is a pressure P in this shell which provides support against gravity. Newton’s second law: Newton’s second law: md 2 R/dt 2 = -GMm/R 2 + 4πR 2 P md 2 R/dt 2 = -GMm/R 2 + 4πR 2 P In equilbrium, GMm/R 0 2 = 4πR 0 2 P 0 In equilbrium, GMm/R 0 2 = 4πR 0 2 P 0 Linearize: R = R 0 +δR, P = P 0 +δP Linearize: R = R 0 +δR, P = P 0 +δP Insert into momentum equation, linearize, keep only first powers of δs and use d 2 R 0 /dt 2 = 0 to give Insert into momentum equation, linearize, keep only first powers of δs and use d 2 R 0 /dt 2 = 0 to give

13. One Zone Models md 2 (δR)/dt 2 = 2GMm(δR)/R 0 3 + 8πR 0 P 0 (δR) + 4πR 0 2 δP md 2 (δR)/dt 2 = 2GMm(δR)/R 0 3 + 8πR 0 P 0 (δR) + 4πR 0 2 δP Adiabatic oscillations:PV γ = const. Adiabatic oscillations:PV γ = const. Linearized version: δP/P 0 = -3γδR/R 0 Linearized version: δP/P 0 = -3γδR/R 0 Hydrostatic equilbrium means 8πR 0 P 0 = 2GMm/R 0 3. The the linearized equation for δR is Hydrostatic equilbrium means 8πR 0 P 0 = 2GMm/R 0 3. The the linearized equation for δR is d 2 (δR)/dt 2 = -(3γ – 4)GM(δR)/R 0 3 d 2 (δR)/dt 2 = -(3γ – 4)GM(δR)/R 0 3 Simple Harmonic Motion, δR = Asin(ωt) with Simple Harmonic Motion, δR = Asin(ωt) with ω 2 =(3γ-4)GM/R 0 3 ω 2 =(3γ-4)GM/R 0 3 Since, the pulsation period, Π = 2π/ω, we have Since, the pulsation period, Π = 2π/ω, we have Π = 2π/(√[4πGρ 0 (3γ-4)]), the period mean density theorem. Π = 2π/(√[4πGρ 0 (3γ-4)]), the period mean density theorem.