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Variable Stars: Pulsation, Evolution and applications to Cosmology

Variable Stars: Pulsation, Evolution and applications to Cosmology. Shashi M. Kanbur, June 2007. Lecture IV: Modeling Stellar Pulsation. A pulsating star is not in hydrostatic equilbrium. For example ρ d 2 r/dt 2 = -GM r ρ /r 2 – dP/dr. Mass continuity equation still holds.

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Variable Stars: Pulsation, Evolution and applications to Cosmology

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  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 • ρd2r/dt2 = -GMrρ/r2 – dP/dr. • Mass continuity equation still holds. • Energy equation: • dE/dt + PdV/dt + dL/dm = 0, where • L(r) = -4πr24σ/3κ . dT4/dm • ρ(r) = 1/V(r), P = P(ρ,T), E=E(ρ,T), κ=κ(ρ,T).

  3. Modeling Stellar Pulsation • Boundary Cnditions: L0=Lcons., dr/dt)0 = 0. • Psurface = 0. Tsurface = f(Teff) ie. a grey solution to the equationof radiative transfer. • 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 • P = P0 + δP, r = r0 + δr, ρ0 + δρ etc. • Expand pulsation equations and drop second order terms. This is linear stellar pulsation. • Assume δr = |δr|eiωt, 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.

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

  6. Non-Linear Models • Momentum equation: • vn+1/2(I) = vn-1/2(I) – Δtn(GM(I)/rn(I)2 + 4π(rn(I))2/ΔM(I)[Pn(I) – Pn(I-1) + Qn-1/2(I) – Qn+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. • Q: Artifical vsicosity. • Field in its own right.

  7. Pulsation Modeling • Linear model to find set of L,M, X,Z,Teff. • Also get eigenvector showing ampltide of rafial displacement. • 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. • Need opacity tables, equation of state (usually Saha). • Result is a nonlinear full amplitude variation of L with T. • Stellar atmosphere converts this to magnitude and color. • Compare with observations via Fourier analysis. • This is for radial oscillations. • 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. • δr = R(r)Y(θ,φ): n is for the radial part, l, m the angular part. • l=m=0, pulsation purely radial. • 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. • 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. • P-modes: pressure is the restoring force, G modes: gravity is the restoring force.

  9. Helioseismology • 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 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. • Can reveal the depth of the solar convection zone, plus rotation and composition of the outer layers of the Sun.

  10. 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. • Newton’s second law: • md2R/dt2 = -GMm/R2 + 4πR2P • In equilbrium, GMm/R02 = 4πR02P0 • Linearize: R = R0+δR, P = P0+δP • Insert into momentum equation, linearize, keep only first powers of δs and use d2R0/dt2 = 0 to give

  11. One Zone Models • md2(δR)/dt2 = 2GMm(δR)/R03 + 8πR0P0(δR) + 4πR02δP • Adiabatic oscillations:PVγ = const. • Linearized version: δP/P0 = -3γδR/R0 • Hydrostatic equilbrium means 8πR0P0 = 2GMm/R03. The the linearized equation for δR is • d2(δR)/dt2 = -(3γ – 4)GM(δR)/R03 • Simple Harmonic Motion, δR = Asin(ωt) with • ω2=(3γ-4)GM/R03 • Since, the pulsation period, Π = 2π/ω, we have • Π = 2π/(√[4πGρ0(3γ-4)]), the period mean density theorem.

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