Variable Stars: Pulsation, Variable Stars: Pulsation, Evolution and applications Evolution and applications

to Cosmologyto Cosmology

Shashi M. Kanbur,Shashi M. Kanbur,

June 2007.June 2007.

Lecture IV: Modeling Stellar Lecture IV: Modeling Stellar PulsationPulsation

A pulsating star is not in hydrostatic A pulsating star is not in hydrostatic equilbrium. For exampleequilbrium. For example

ρρdd22r/dtr/dt22 = -GM = -GMrrρρ/r/r22 – dP/dr. – dP/dr. Mass continuity equation still holds.Mass continuity equation still holds. Energy equation:Energy equation: dE/dt + PdV/dt + dL/dm = 0, wheredE/dt + PdV/dt + dL/dm = 0, where L(r) = -4L(r) = -4ππrr2244σσ/3/3κκ . dT . dT44/dm/dm ρρ(r) = 1/V(r), P = P((r) = 1/V(r), P = P(ρρ,T), E=E(,T), E=E(ρρ,T), ,T),

κκ==κκ((ρρ,T).,T).

Modeling Stellar PulsationModeling Stellar Pulsation

Boundary Cnditions: LBoundary Cnditions: L00=L=Lcons.cons., dr/dt), dr/dt)00 = 0.= 0.

PPsurfacesurface = 0. T = 0. Tsurfacesurface = f(T = f(Teffeff) ie. a grey ) ie. a grey solution to the equationof radiative solution to the equationof radiative transfer.transfer.

1D radiative codes. Now there are 1D radiative codes. Now there are “numerical recipes” to model time “numerical recipes” to model time dependent turbulent convection.dependent turbulent convection.

Linear ModelsLinear Models Assume displacement from equilbrium, Assume displacement from equilbrium, δδr, are r, are

small. Write variables assmall. Write variables as P = PP = P00 + + δδP, r = rP, r = r00 + + δδr, r, ρρ00 + + δρδρ etc. etc. Expand pulsation equations and drop second Expand pulsation equations and drop second

order terms. This is linear stellar pulsation.order terms. This is linear stellar pulsation. Assume Assume δδr = |r = |δδr|er|eiiωωtt, solve resulting eigenvalue , solve resulting eigenvalue

problem. Leads to linear periods and growth rates problem. Leads to linear periods and growth rates ie. Whether a given perturbation is stable or will ie. Whether a given perturbation is stable or will continue to grow.continue to grow.

Can investigate boundaries of “instability strip” Can investigate boundaries of “instability strip” with such a technique.with such a technique.

Non-Linear ModelsNon-Linear Models Write differential equations as difference Write differential equations as difference

equations over a computational grid covering the equations over a computational grid covering the star.star.

Zones 1,……,N, with interfaces 0,1,….N+1.Zones 1,……,N, with interfaces 0,1,….N+1. Extensive variables r, velocity, vExtensive variables r, velocity, vrr, luminosity, L, luminosity, Lrr, ,

defined at zone interfaces.defined at zone interfaces. Intensive variables defined at zone centers, T, Intensive variables defined at zone centers, T, ρρ, ,

P, P, κκ etc. etc. Sometimes may need to extrapolate Sometimes may need to extrapolate

intensive/extensive variables to zone intensive/extensive variables to zone interface/centers.interface/centers.

Time mesh: tTime mesh: tn+1n+1 = t = tnn + + ΔΔttn+1/2n+1/2,t,tn+1/2n+1/2 – t – tn-1/2n-1/2 = = ΔΔttnn, , ΔΔttnn = ½(= ½(ΔΔttn-1/2n-1/2 + + ΔΔttn+1/2n+1/2).).

Non-Linear ModelsNon-Linear Models

Momentum equation:Momentum equation: vvn+1/2n+1/2(I) = v(I) = vn-1/2n-1/2(I) – (I) – ΔΔttnn(GM(I)/r(GM(I)/rnn(I)(I)22 + +

44ππ(r(rnn(I))(I))22//ΔΔM(I)[PM(I)[Pnn(I) – P(I) – Pnn(I-1) + Q(I-1) + Qn-1/2n-1/2(I) (I) – Q– Qn+1/2n+1/2(I-1)])(I-1)])

Leads to a matrix equation Ax=d to Leads to a matrix equation Ax=d to be solved for the increments to the be solved for the increments to the physical variables at each time step.physical variables at each time step.

Q: Artifical vsicosity.Q: Artifical vsicosity. Field in its own right.Field in its own right.

Pulsation ModelingPulsation Modeling Linear model to find set of L,M, X,Z,TLinear model to find set of L,M, X,Z,Teffeff.. Also get eigenvector showing ampltide of rafial Also get eigenvector showing ampltide of rafial

displacement.displacement. Non-linear model with an initial “kick” scaled by linear Non-linear model with an initial “kick” scaled by linear

eigenvector for that modeleigenvector for that model Continue pulsation until amplitude increase levels of: Continue pulsation until amplitude increase levels of:

several hundred cycles, maybe 1-2 hours on a modern fast several hundred cycles, maybe 1-2 hours on a modern fast PC.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 No time dependent code to model non-radial oscillations

exists.exists.

Non-Radial OscillationsNon-Radial Oscillations Expand perturbatin Expand perturbatin δδr in terms of spherical harmonics, r in terms of spherical harmonics,

specified by 3 numerbs, n, l, m.specified by 3 numerbs, n, l, m. δδr = R(r)Y(r = R(r)Y(θθ,,φφ): n is for the radial part, l, m the angular part.): 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-1l=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 With l,m non-zero need to worry about Poisson’s equation

as well.as well. n: number of nodes radially outward from Sun’s center. m: n: number of nodes radially outward from Sun’s center. m:

number of nodes found around the equator. l: number of number of nodes found around the equator. l: number of nodes found around the azimuth (great circle through the nodes found around the azimuth (great circle through the poles)poles)

Hard mathematical/numerical problem.Hard mathematical/numerical problem. P-modes: pressure is the restoring force, G modes: gravity P-modes: pressure is the restoring force, G modes: gravity

is the restoring force. is the restoring force.

HelioseismologyHelioseismology Sun is a non-radial oscillator.Sun is a non-radial oscillator. Modes with periods between 3 an d8 Modes with periods between 3 an d8

minutes – five minute oscillations are p minutes – five minute oscillations are p modes: l going from 0 to 1000.modes: l going from 0 to 1000.

Modes with longer periods – about 160 Modes with longer periods – about 160 minutes could be g modes: l ~1-4.minutes could be g modes: l ~1-4.

Comparison of observed and theoretical Comparison of observed and theoretical frequencies can be used to calibrate solar frequencies can be used to calibrate solar models: helioseismology.models: helioseismology.

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

One Zone ModelsOne Zone Models Central point mass of mass M. At a radius Central point mass of mass M. At a radius

R is a thin spherical shell, mass m. There is R is a thin spherical shell, mass m. There is a pressure P in this shell which provides a pressure P in this shell which provides support against gravity.support against gravity.

Newton’s second law:Newton’s second law: mdmd22R/dtR/dt22 = -GMm/R = -GMm/R22 + 4 + 4ππRR22PP In equilbrium, GMm/RIn equilbrium, GMm/R00

22 = 4 = 4ππRR0022PP00

Linearize: R = RLinearize: R = R00++δδR, P = PR, P = P00++δδPP Insert into momentum equation, linearize, Insert into momentum equation, linearize,

keep only first powers of keep only first powers of δδs and use s and use dd22RR00/dt/dt22 = 0 to give = 0 to give

One Zone ModelsOne Zone Models mdmd22((δδR)/dtR)/dt22 = 2GMm( = 2GMm(δδR)/RR)/R00

33 + 8 + 8ππRR00PP00((δδR) + R) + 44ππRR00

22δδPP Adiabatic oscillations:PVAdiabatic oscillations:PVγγ = const. = const. Linearized version: Linearized version: δδP/PP/P00 = -3 = -3γδγδR/RR/R00 Hydrostatic equilbrium means 8Hydrostatic equilbrium means 8ππRR00PP00 = =

2GMm/R2GMm/R0033. The the linearized equation for . The the linearized equation for δδR isR is

dd22((δδR)/dtR)/dt22 = -(3 = -(3γγ – 4)GM( – 4)GM(δδR)/RR)/R0033

Simple Harmonic Motion, Simple Harmonic Motion, δδR = Asin(R = Asin(ωωt) witht) with ωω22=(3=(3γγ-4)GM/R-4)GM/R00

33

Since, the pulsation period, Since, the pulsation period, ΠΠ = 2 = 2ππ//ωω, we have, we have ΠΠ = 2 = 2ππ/(√[4/(√[4ππGGρρ00(3(3γγ-4)]), the period mean density -4)]), the period mean density

theorem.theorem.