Slide #1.

Variable Stars: Pulsation, Evolution and applications to Cosmology Shashi M. Kanbur, June 2007.
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Slide #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).
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Slide #3.

Modeling Stellar Pulsation    Boundary Cnditions: L0=Lcons., dr/dt)0 = 0. Psurface = 0. Tsurface = f(Tef) ie. a grey solution to the equationof radiative transfer. 1D radiative codes. Now there are “numerical recipes” to model time dependent turbulent convection.
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Slide #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.
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Slide #5.

Non-Linear Models       Write diferential equations as diference 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).
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Slide #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.
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Slide #7.

Pulsation Modeling           Linear model to find set of L,M, X,Z,Tef. 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.
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Slide #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.
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Slide #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.
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Slide #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. 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
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Slide #13.

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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