Transcript Lecture 2 - SUNY Oswego

Variable Stars: Pulsation,
Evolution and applications to
cosmology
Shashi M. Kanbur
SUNY Oswgo, June 2007
Lecture II: Stellar Pulsation
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Most stars are intrinsically stable.
If the Sun were to contract a bit, the core gets hotter,
nuclear reactions go at a faster rate creating more energy
and hence more pressure halting the contraction.
If the Sun were to expand a bit, the core gets cooler,
nuclear reactions go at a slower rate creating less energy
and hence less pressure halting the expansion.
Most stars are stable against departures from hydrostatic
equilbrium because of this.
In Cepheids and RR Lyraes, upon contraction, the extra
energy flowing out is “held up” in the outer layers for a
short while and released when the star is expanding again.
It thus amplifies the contraction or departure from
hydrostatic equilbrium, at least in the outer layers.
Swing analogy.
Held up due to opacity behaviour in outer layers.
Static Stellar Structure
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m(r) = mass inside a radius r.
L(r) = luminosity at radius r.
P(r) = total pressure at radius r (gas plus
radiation)
T(r) = temperature at radius r.
ρ(r) = density at radius r.
Spherically symmetric.
Equation of State, energy generation and
opacity as a function of T,ρ.
Static Stellar Structure
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dm/dr = 4πr2ρ(r): mass continuity
dP/dr = -Gm(r)ρ(r)/r2: hydrostatic equilbrium
dL(r)/dr = 4πr2ε(r) : thermal equilbrium
ε=ε(ρ,T): energy generation rate
κ=κ(ρ,T): opacity
P = P(ρ,T): Equation of State.
At r=0, L(r=0)=0, m(r=0)=0 : Boundary
Ar r=R, L(r=R)=L, m(r=R)=M: Conditions
Energy Transport
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dL(r)/dm relates to energy transport.
In such stars usually by radiation or
convection.
Convection is very complicated: 3D
problem – more later
Radiative Transport: diffusion
approximation
L(r) = -(1/3κ)64πacr4T4dlnT/dm
Why do Cepheids and RR Lyraes
pulsate?
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Pulsation is not due to variations in the rate of energy
generation in the core.
More to do with the variation of the rate at which this
radiation can escape.
Early astronomers thought they were binary stars.
Harlow Shapley suggested there was an internal
“breathing” mechanism.
Radial pulsations proposed by Arthur Ritter in 1879 but
overlooked until Eddington provided a mathematical
formulation.
Assume pulsations are due to sound waves resonating in
the stellar interior. Then, the pulsation period, Π, is
Π = 2R/vs, where R is the radius and vs, the sound speed is
Vs = √γP/ρ, and γ is the ratio of specif heats for the stellar
material.
Why do Cepheids and RR Lyraes
pulsate?
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To work this out, use hydrostatic equilbrium,
dP/dr = - Gm(r)ρ/r2 = -4Gπrρ2/3
Integrate this between the center and surface
and assume P(surface)=0, yields
P(r) = 2πGρ2(R2-r2)
Substituing back into the original expression for
vs and integrating between 0 and R yields
approx:
Π~√3π/2Gγρ
This is the period mean density theorem, where
we take ρ to be the mean density of the star.
More accurate treatments of this derivation exist.
Why do Cepheids and RR Lyraes
pulsate?
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This is why Cepheids have longer
periods than RR Lyraes.
Cepheids are much more tenous and
have a smaller mean density and
hence a longer period. RR Lyraes are
compact with a high mean density.
Pendulum with a short string has a
shorter period than another
pendulum with a longer string.
Why do Cepheids and RR Lyraes
pulsate?
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Suppose a layer in the outer part of the star contracts for
some reason due to some momentary loss of hydrostatic
equilbrium.
This layer heats up and becomes more opaque to radiation.
Radiation diffuses more slowly through the layer because of
its increased opacity and heat builds up beneath it.
Pressure rises below the layer and eventually starts to push
the layer out.
The layer expands, cools and becomes more transparent to
radiation.
Energy can now escape from below the layer and the
pressure beneath the layer drops.
The layer falls inwards and the cycle repeats.
Need a mechanism by which opacity increases as
temperature increases.
Kramer’s opacity: κ~ρT-3.5 decreases upon compression.
Why do Cepheids and RR Lyraes
pulsate?
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In hydrogen and helium partial ionization
zones, temperature does not increase
much upon compression since energy of
compression goes into fully ionizing
hydrogen and helium.
Likewise during the expansion phase,
temperature does not decrease
significantly because the ions release
energy when they combine with electrons.
This is the κ mechanism.
Why do Cepheids and RR Lyraes
pulsate?
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Increased temperature gradient
between partial ionization zones and
surrounding layers causes more heat
to flow into them increasing
ionization.
This is the γ mechanism and
reinforces the κ mechanism.
Partial Ionization Zones
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Important ones are
Hydrogen ionization zone,
H <―> H+ + e-,
At a temperature of around T ~ 10000K.
Helium ionization zones,
He <-> He+ + e-, close to the H ionization
zone and
He+<-> He++ + e- at a temperature of
around 4×104K.
Partial Ionization Zones
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The pulsation properties of a star are determined
primarily by where the partial ionization zones
are located with respect to the mass distribution
of the star.
This is determined by the temperature of the star
and hence the location of the star on the HR
diagram.
Too close to the surface and there is not enough
mass to drive the pulsations effectively.
Too deep in the star and convection becomes
efficient thus transporting energy out of the
partial ionization zone on compression.
This is why there is an instability strip.
The Instability Strip
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This strip corresponds to the range of
temperatures for which the partial ionization
zones are located in the right place to sustain
stellar pulsations.
The blue and red edges of the instability strip.
Red edge defined by a theory of time dependent
turbulent convection (deepest layer in the star at
which partial ionization zones can sustain
pulsation.
Blue edge: defined by the highest layer at which
partial ionization zones can sustain pulsation.
But not all stars in instability strip pulsate.
Pulsation Periods
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Pulsation is a transient phenomenon.
As a star evolves of the main sequence, it crosses
the instability strip and starts to pulsate.
As observational methods improve, smaller and
smaller amplitude pulsators will be discovered.
Long Period Variables: 100-700 days
Classical Cepheids:
1-100 days
W Virginis:
2-45 days
RR Lyraes:
1.5-24 hours
δ Scutis:
1-3 hours
β Cepheids:
3-7 hours
ZZ Cetis:
100-1000 seconds
Luminosity and the partial
Ionization zones
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HeII partial ionization zone is the main driving
zone,
Bu H ionization zone important as well for driving
and
Produces an observable phase lag between
maximum brightness and minimum radius.
Max brightness at minimum radius is to be
expected on a purely adiabatic approach.
Max. brightness when mass between H ionization
zone and star’s surface is a minimum.
This occurs slightly after maximum compression
or minimum radius.
Non-Radial Oscillations
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Can also model small departures
from hydrostatic equilbrium as