Transcript 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
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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).
Modeling Stellar Pulsation
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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.
Linear Models
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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.
Non-Linear Models
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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).
Non-Linear Models
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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.
Pulsation Modeling
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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.
Non-Radial Oscillations
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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.
Helioseismology
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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.
One Zone Models
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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
One Zone Models
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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.