Using Pulsations to Test Stellar Evolution Models

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Using Pulsations to Test Stellar Evolution Models

• Joyce A. Guzik
• Thermonuclear Applications Group, X-2
• Los Alamos National Laboratory
• April 15, 2002

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Outline

• Pulsating star locations on the HR Diagram
• Pulsation modes and driving
• Period-mean density relation
• nonradial oscillations, p-modes and g-modes
• kappa/gamma effect driving
• Solar oscillations (helioseismology)
• Attempts at asteroseismology (example delta
  Scuti star FG Vir)

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From Christensen- Dalsgaard 2000
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Period-Mean Density Relation
dynamical frequency 1 / free fall time over
the distance of a stellar radius fundamental
model frequency
For homologous stars (same density
stratification), oscillation frequencies scale
approximately as sqrt(mean density)
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Solutions of stellar pulsation equations (1)

• Assume spherical symmetry, linear perturbations,
  sinusoidal time dependence
• Allow both radial and horizontal motions
• Solve coupled equations of continuity, Poisson,
  and energy
• sometimes make adiabatic approx. and Cowling
  approx. (neglect gravitational potential
  perturbations) to simplify solutions
• Solutions exist for discrete values of frequency
  (eigenvalues of equations)

See, e.g., Cox 1974 Hansen and Kawaler 1986
Christensen-Dalsgaard and Dziembowski 2000
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Solutions of stellar pulsation equations (2)
Pressure perturbation, e.g., can be written in
form with separated variables in spherical
coordinates as
From equation of motion, solution for
displacement vector has the form
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Solutions of stellar pulsation equations (3)
are radial and horizontal displacements, and are
functions of
and
Lamb Frequency
Brunt-Vaisala or buoyancy frequency
These two frequencies determine the propagation
regions for pressure (acoustic) and gravity
waves within a star
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From J.P. Cox 1974
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p-and g-mode propagation regions for ZAMS solar
model from Hansen and Kawaler 1994
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Spherical Harmonics
From Toomre 1984
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Solar radial eigenfuction amplitudes
From Toomre 1984
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Pulsation driving by kappa/gamma effect (1)

• Radial and nonradial oscillations of most types
  of variable stars are driven by the kappa/gamma
  (kappa as in opacity, gamma as in adiabatic
  indices) effect, due to the effects of ionization
  of H, He, C, O, or Fe in stellar envelopes.
• H, He ionizes 10,000 K
• He ionizes 50,000 K
• k-enhancement due to Fe ionization 200,000 K

See, e.g., Bohm-Vitense 1992 Hansen and Kawaler
1986 Mihalas and Binney 1981
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Pulsation driving by kappa/gamma effect (2)

• Linearizing the diffusion equation, and assuming
  a power-law opacity

For a non-ionizing region, G3 5/3, s3.5, n1,
and
When region is compressed and heated, more
radiation leaks out.
For an ionization region, G3 --gt 1, s lt 0, and
(for C)
Less radiation leaks out when a region
is compressed and heated, and radiation is dammed
up.
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Kappa Effect Driving
Stellar Interior
H, He
He
Stellar Surface
k
Fe

T, r
Layer compressed
Layer expands
T, k
T, k
Radiation blocked, pressure increases
Radiation escapes, pressure decreases
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Pulsation driving by kappa/gamma effect (3)

• Particular oscillation modes can be stable or
  unstable depending on whether
• region of star contributing significantly to
  mechanical response is in a driving region
• driving in one region is not overcome by
  radiative damping in another region (integrated
  work/zone)
• The kappa/gamma effect is weakened or disabled if
  convection can instead transport enough of the
  luminosity in the ionization region.
• (Mechanisms such as convective driving and
  convective blocking have also been proposed to
  explain pulsations of white dwarf and g Dor stars)

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Example Work plot for g Doradus model g-mode
Guzik et al. ApJL 542, L57 (2000)
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Helioseismology

• Solar Oscillations Discovered in 1960
• (Leighton, Noyes Simon)
• Explained as global resonant acoustic modes of
  the Sun in 1970 (Ulrich, Leibacher, Stein)
• Extensive application to test physics of solar
  models and derive internal structure of Sun in
  1980s (helioseismology)
• Attempt to apply technique to other stars in
  1990s (asteroseismology)

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Doppler velocity observations of solar disk
From Toomre 1984
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From Toomre 1984
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From Christensen- Dalsgaard 2000
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(No Transcript)
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Sound speed differences between model and seismic
results using LLNL OPAL and LANL LEDCOP opacities
Neuforge-Verheecke et al., ApJ 561, 450 (2001)
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From Christensen-Dalsgaard et al. 1991
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Solar Interior Rotation as inferred from
inversions at 0, 30 and 60 latitude
From Chaplin et al. 1999
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What did we learn from helioseismology?

• Convection zone helium abundance
• 0.248 0.001 (e.g. Basu 1998)
• Radius of convection zone base
• 0.7135 0.0005 Rs (e.g. Basu 1998)
• Sound speed versus depth
• Agrees with modern standard models to within 1
• (e.g. Basu et al. 1998)
• Evidence for diffusive settling of about 10 of
  initial helium from convection zone during Suns
  lifetime
• Rotation profile versus latitude and depth
• (e.g. Chaplin et al. 1999)

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Evidence for Diffusive Helium Settling

• Diffusion calculations predict that about 10 of
  Helium by mass should settle out of solar
  convection zone during the Suns lifetime
• Observational Evidence
• Convection zone He abundance (Y) determined from
  signature of He ionization 0.248 0.001 (Basu,
  1997), whereas initial Y needed to calibrate
  solar model to present solar luminosity is 0.274.
• Much improved agreement between calculated and
  helioseismically inferred sound speed below solar
  convection zone (Gabriel 1997) when He diffusion
  included.
• Direct helioseismic inversion for hydrogen
  abundance profile (Kosovichev 1995).

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Solar composition profiles near convection zone
base including diffusion
From Christensen-Dalsgaard et al. 2000
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Sound speed differences for solar models
with/without diffusion and extra mixing
No diffusion
From Gabriel 1997
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Inversions to determine solar H abundance gradient
From Kosovichev 1995
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Helioseismology Results

• Latest opacities and EOS give excellent results
  in comparisons between calculated and observed
  solar oscillation frequencies

Guzik and Swenson 1997
Agreement is to within 0.1, but frequency
observations are accurate to 0.01!
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Results for d Scuti stars

• Example attempted detailed match to FG Vir
• 29 observed frequencies 106 to 395 µHz
• Evolution models matching observational
  constraints (L, Teff, g, 140.5 µHz mode radial
  fundamental) nonunique
• No model found that matches all observed
  frequencies
• Frequency predictions are sensitive to
• composition and gradients
• extent of convective core and overshoot
• rotation rate and differential rotation
• More modes predicted than observed (problem worse
  for more evolved d Sct stars)

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FG Vir Evolution Models
Guzik et al. in Delta Scuti and related stars,
ASP Vol. 210, 2000
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FG Vir Model Properties
1.82 M? (Z0.02)
1.95 M? (Z0.03)

• R (R?) 2.26 2.31
• M/R3 0.1575 0.1580
• Teff (K) 7368 7412
• log g 3.99 4.00
• log L/L? 1.13 1.16
• Xcc 0.257 0.345
• Rcc/R? 0.155 0.181
• Mcc/M? 0.175 0.220
• Age (Gyr) 0.879 0.731

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1.82 Ms model frequencies vs. 1.95 Ms model
frequencies
Guzik et al. in Delta Scuti and related stars,
ASP Vol. 210, 2000
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Observed and calculated FG Vir
frequencies (observations from Breger et al. 1997)