Good Vibrations and Stellar Pulsations

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Good VibrationsandStellar Pulsations
l 2, m 0
l 5, m 3

• Brad Carroll
• Weber State University
• April 11, 2007

www.univie.ac.at/tops/intro.html
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In August of 1596, David Fabricius (Lutheran
pastorand amateur astronomer) observed o Ceti,
a 2nd magnitude star in the constellation Cetus.
As it declined in brightness, the star vanished
by October. Later it reappeared, and was renamed
Mira (the Wonderful) By 1660 its 11-month
period had been established. The light
variations were believed to be caused by
blotches on the surface of a rotating star.
web.njit.edu/dgary/728/Lecture12.html
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In 1784, John Goodricke of York discovered that
d Cephei is variable P 5 days, 8 hours
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d Cephei is the prototype of the Classical
Cepheids
From Wycombe Astronomical Society,
wycombeastro.org.uk/news.shtm
magnitude varies from 3.4 to 4.3,so luminosity
changes by factor of100(Dm/5) 100(0.9/5) 2.3
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Edward Charles Pickerings Computersat Harvard
Observatory
From left to right Ida Woods, Evelyn Leland,
Florence Cushman, Grace Brooks,Mary Van,
Henrietta Leavitt, Mollie O'Reilly, Mabel Gill,
Alta Carpenter, Annie Jump Cannon, Dorothy
Black, Arville Walker, Frank Hinkely,
andProfessor Edward King (1918).
www.astrogea.org/surveys/dones_harvard.htm
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Henrietta Swan Leavitt
(1868 1921)
Found 2400 Classical Cepheids In 1912,
discovered the Period-Luminosity Relation
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Small Magellanic Cloud
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Cepheids in the SMC
From Shapley, Galaxies, Harvard University Press,
Cambridge, MA, 1961.
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Calibration The Distance to a Cepheid
The nearest Cepheid is Polaris (over 90 pc), too
far for trigonometric parallax. d (pc) 1/p (in
arcsec)
In 1913, Ejnar Hertzsprung of Denmark used least
squares mean parallax to determine the average
magnitude M -2.3 for a Cepheid with P 6.6
days. d (pc) 4.16/slope (in arcsec/yr)(4.16
AU/yr is the Suns motion)
www.cnrt.scsu.edu/dms/cosmology/DistanceABCs/dist
ance.htm
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Period Luminosity Relation
MltVgt -2.81 log10 Pd 1.43
d (pc) 10(m-M5)/5
Sandage and Tammann, The Astrophysical Journal,
151, 531, 1968.
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How to Find the Distance to aPulsating Star

• Find the stars apparent magnitude m (just by
  looking)
• Measure the stars period (bright-dim-bright)
• Use the Period-Luminosity relation to find the
  stars absolute magnitude M
• Calculate the stars distance (in parsecs) using

d (pc) 10(m-M5)/5
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What is the Milky Way?
antwrp.gsfc.nasa.gov/apod/ap060801.html
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Thomas Wright, An Original Theory on New
Hypothesis of the Universe, 1750.
Kapteyn, The Astrophysical Journal, 55, 302, 1922.
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In 1913, Hertzsprung calculated that the distance
to the Small Magellanic Cloud was 33,000 light
years. This was the greatest distance ever
determined for an astronomical object. In 1917,
Harlow Shapley used Hertzsprungs calibration of
the period-luminosity relation to determine the
distance to the globular clusters (some of which
contain Cepheids).
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The Milky Way has about 120 globular
clusters,each containing perhaps 500,000
stars. One-third of all known globular
clusters covers only 2 of the sky, in the
constellation Sagittarius. Shapley found the
globular clusters had a spherical distribution.
homepage.mac.com/kvmagruder/bcp/aster/constellatio
ns/Sgr.htm
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The Sun was removed from the center of the
universe, and placed at an inconspicuous spot
near the edge.
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The Great Debate Are the spiral Nebulae
(such as M31 Andromeda) comparable in size with
the Milky Way, or are they much smaller and near?
Harlow Shapley (left)vsHeber D. Curtis (right)
In 1925, Edwin Hubble discovered a Classical
Cepheid in M31. Hubble used Hertzsprungs
calibration of the period-luminosity relation
to calculate that M31 was over 300,000 pc
distant. At this distance, M31 would be 10 kpc
in diameter. The spiral nebulae are galaxies
like our own.
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Classical Cepheids are the standard candles of
the universe
www.pas.rochester.edu/afrank/A105/LectureXV/Lectu
reXV.html
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Embarrassments!

• Our galaxy seemed to be the largest.
• The globular clusters in M31 were
  underluminous by a factor of 4.

In 1952, Walter Baade discovered that there
are two types of Cepheids and two period
luminosity relations.
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Population I Cepheids (Classical Cepheids) are
relatively rich in heavy elements. Population II
Cepheids (W Virginis stars) are relatively poor
in heavy elements. Pop I Cepheids are four times
more luminous than Pop II Cepheids.outreach.atnf
.csiro.au/education/senior/astrophysics/variable_c
epheids.html
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So ..

• Hertzsprings Classical Cepheids (Pop I) were
  obscured by dust in the plane of the Galaxy,
  so luminosities of Classical Cepheids were
  calibrated too low by a factor of 4.
• Shapley mistook the Pop II Cepheids in globular
  clusters for Pop I Cepheids, so his Pop II
  Cepheids in the globular clusters were
  properly calibrated (luck!).
• Shapleys distances to the globular clusters
  were correct.
• Hubbles Pop I Cepheids in M31 were
  underluminous by a factor of 4, so M31(and
  all other galaxies measured using Classical
  Cepheids) was twice as far away as previously
  believed, and twice as large.
• The globular clusters around M31 are as bright
  as those surrounding our own galaxy.

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The Instability Strip on the HR Diagram
DT 600 1100 K
density period incr incr
lt hotter cooler gt
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• Luminous Blue Variables
• Wolf-Rayet stars
• Cephei stars
• Planetary Nebula Nuclei Variables
• Miras, Semi-Regular variables
• Slowly Pulsating B stars
• DO-type Variable white dwarfs
• DB-type Variable white dwarfs

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Some Pulsating Variables
Type Range of Periods Population Type Type of Oscillation
Long-Period Variables 100 700 days I, II R
Classical Cepheids 1 50 days I R
W Virginis stars 2 45 days II R
RR Lyrae stars 1.5 24 hours II R
d Scuti stars 1 8 hours I R, NR
b Cephei stars 3 7 hours I R, NR
DAV stars 100 1000 s I NR
R radial oscillations NR nonradial
oscillations
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RR Lyrae variables in the globular cluster
M3 (one nights observation)
cfa-www.harvard.edu/jhartman/M3_movies.html
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Light and radial velocity curves for d Cephei
receding
approaching
Schwarzschild, Harvard College Observatory
Circular, 431, 1938
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d Cephei radius
Schwarzschild, Harvard College Observatory
Circular, 431, 1938
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Schwarzschild, Harvard College Observatory
Circular, 431, 1938
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The star is brightest when its surface is moving
outward most rapidly, and not at minimum radius
a phase lag.
Schwarzschild, Harvard College Observatory
Circular, 431, 1938
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Consider the adiabatic, radial pulsation of a
gas- filled shell.
Linearize the equation of motion
by setting
to get
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For adiabatic motion,
Also,
Set
and plug into
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The result is
or
dynamical instability
P
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for the Sun,
P
Compare this with the time for sound to cross a
stars diameter
P
Estimate!
P
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The Period Mean Density Relation
density period incr incr
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Organ Pipes and Pulsating Stars
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Pulsating Stars are Heat Engines
The Otto cycle. 1. In the exhaust stroke, the
piston expels the burned air-gas mixture left
over from the preceding cycle. 2. In the intake
stroke, the piston sucks in fresh air-gas
mixture. 3. In the compression stroke, the piston
compresses the mixture, and heats it. 4. At the
beginning of the power stroke, the spark plug
fires, causing the air-gas mixture to burn
explosively and heat up much more. The heated
mixture expands, and does a large amount of
positive mechanical work on the piston.
www.lightandmatter.com/html_books/0sn/ch05/ch05.ht
ml
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• In 1918, Arthur Stanley Eddington proposed that
  pulsating stars are heat engines, transforming
  thermal energy into mechanical energy. He
  proposed two mechanisms
• Energy Mechanism Eddington suggested that when
  the star is compressed, more energy is generated
  by sources in the stellar core. Ineffective.
  The core pulsation amplitude is very small.
• Valve Mechanism Suppose that the cylinder of
  the engine leaks heat and that the leakage is
  made good by a steady supply of heat. The
  ordinary method of setting the engine going is to
  vary the supply of heat, increasing it during
  compression and diminishing it during expansion.
  That is the first alternative we considered.
  But it would come to the same thing if we varied
  the leak, stopping the leak during compression
  and increasing it during expansion. To apply
  this method we must make the star more heat-tight
  when compressed than when expanded in other
  words, the opacity must increase upon
  compression.

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But this does not work for most stellar material!
Why?
The opacity is more sensitive to the temperature
than to the density, so the opacity usually
decreases with compression (heat leaks out). But
in a partial ionization zone, the energy of
compression ionizes the stellar material rather
than raising its temperature! In a partial
ionization zone, the opacity usually increases
with compression! Partial ionization zones are
the direct cause of stellar pulsation.
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• hydrogen ionization zone (H H and He
  He) T
  (1 1.5) x 104 K
• helium II ionization zone (He He)
  T 4 x 104 K

C
C
If the star is too hot, the ionization zones will
be too near the surface to drive the
oscillations. This accounts for the blue
edge of the instability strip. The red edge
is probably due to the onset of convection.
1 s t o v e r t o n e
f u n d a m e n t a l
n o p u l s a t I o n
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www.univie.ac.at/tops/dsn/texts/nonradialpuls.html
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Phase lag problem
A Cepheid is brightest when its surface is moving
outward most rapidly, and not at minimum radius
a phase lag.

• the emergent luminosity varies inversely with
  the mass lying above the hydrogen ionization
  zone
• the luminosity on the bottom of the hydrogen
  ionization zone is largest at minimum radius
• the hydrogen ionization zone is moving outwards
  (through mass) fastest at minimum radius
• the hydrogen ionization is farthest out ¼
  cycle later
• the luminosity peaks ¼ cycle after minimum
  radius

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Nonradial Oscillations
Pulsational corrections df to equilibrium model
scalar quantities f0 go as (the real part of)
l 0 radial m gt 0 retrograde m lt 0 prograde m
0 standing
click here
http//gong.nso.edu/gallery/images/harmonics
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Smith, The Astrophysical Journal, 240, 149, 1980
to Earth
In a rotating star, frequencies are rotationally
split ( Zeeman).
Si III
l 2, m 0, -1, -2
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Two Types of Nonradial Modes
www.astro.uwo.ca/jlandstr/planets/webfigs/earth/s
lide1.html
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Two Types of Frequencies
The acoustic frequency
The Brunt-V_at_is_at_l_at_ (buoyancy) frequency
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or
l 2
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p modes
a surface gravity wave
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g modes
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Seismology and Helioseismology
5-minute p15 mode with l 20 and m 16
www.geophysik.uni-muenchen.de /research/seismology
Courtesy NOAO
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GONG (Global Oscillation Network Group)
a six-station network of extremely sensitive and
stable velocity imagers located around the Earth
to obtain nearly continuous observations of the
Sun's "five-minute" oscillations
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SOHO (Solar and Heliospheric Observatory)

• Michelson Doppler Interferometer
• (MDI)
• measures vertical motion of
• photosphere at one million points
• can measure vertical velocity
• as small as 1 mm/s

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click
5 hours of MDI Medium-l data 96/09/01
Measurements of Frequencies of Solar Oscillations
from the MDI medium-l Program by E.J. Rhodes,
Jr., A.G. Kosovichev, P.H. Scherrer, J. Schou
J. Reiter
sohowww.nascom.nasa.gov/publications/CDROM1/papers
/rhodes/
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• 5-minute p modes have a very low amplitude, 10
  cm/s
• dL/L 10-6
• incoherent superposition of 10 million modes

p2
p1
f
sohowww.nascom.nasa.gov /publications/CDROM1/paper
s /rhodes/
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Theory (curves) vs. Data (circles)
Libbrecht, Space Science Reviews, 47, 275, 1988
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Some Results for the Sun

• base of convection zone at 0.714 Rsun, where T
  2.18 x 106 K
• mass fraction of helium at surface is Y
  0.2437
• helioseismologically measured sound speed and
  calculated sound speed for standard solar model
  agree to within 0.1

www.sns.ias.edu/jnb/Papers/Preprints/solarmodels.
html
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Rotational Frequency Splittingin Solar p-Mode
Power Spectra
l 20
Liebbrecht, The Astrophysical Journal, 336, 1092,
1989
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The Suns Internal Rotation

• angular velocity profile
• in the solar interior inferred
• from helioseismology

(b) angular velocity plotted as a function of
radius for several latitudes
Brandenberg, arXivastro-ph/0703711, 2007
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The contours of constant angular velocity do
not show a tendency of alignment with the axis of
rotation, as one would have expected, and as many
theoretical models still show. The angular
velocity in the radiative interior is nearly
constant, so there is no rapidly rotating core,
as has sometimes been speculated. There is a
narrow transition layer at the bottom of the
convection zone, where the latitudinal
differential rotation goes over into rigid
rotation (i.e. the tachocline). Below 30?
latitude the radial angular velocity gradient is
here positive, i.e. ?W/?r gt 0, in contrast that
what is demanded by conventional dynamo
theories. Near the top layers (outer 5) the
angular velocity gradient is negative and quite
sharp.
Brandenberg, arXivastro-ph/0703711, 2007
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Delta Scuti Stars
q2 Tauri

• A to early F stars
• Periods 30 min to 8 hrs
• radial, nonradial p (sometimes g)

Poretti et al, The Astrophysical Journal,
557,1021, 2002
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DAV White Dwarfs

• hydrogen atmospheres
• mass 0.6 Msun
• r 106 g cm-3
• Te 10,000 K 12,000 K
• periods 100 1000 s nonradial
  g-modestrapped in hydrogen surfacelayer
• hydrogen partial ionizationzone drives the DAV
  oscillations

dr/r
lt surface center gt
log(1-r/R)
Winget et al, The Astrophysical Journal Letters,
245, L33, 1981
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G191-16
G185-32
G191-16
G185-32
very complex!
McGraw et al, The Astrophysical Journal,
250,349,1981
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Don Winget predicted that the helium partial
ionization zone could drive oscillations in DB
(helium atmosphere) white dwarfs with Te 19,000
K
Te 26,000 K
rotationally split frequencies
Winget et al, The Astrophysical Journal Letters,
262, L11, 1982.
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White Dwarf Seismology

• Verify theories of white dwarf structure
• Determine white dwarf rotation rates
• Calibrate cooling rates Pg 1/T
  white dwarf cosmochronology!

white dwarfs are fossil stars
theoretical cooling rates observed white
dwarfs of different luminosities history of
star formation!
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This suggests an age of 11 Gyr or less for the
local disk
www.casca.ca/ecass/issues/2000-JS/fontaine.html
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Detailed Asteroseismology of Other Stars
The COROT (COnvection, ROtation, and planetary
Transits) satellite was launched on December 27,
2006. Equipped with a 27-cm diameter telescope
and a 4-CCD camera sensitive to tiny variations
of the light intensity from stars.
smsc.cnes.fr/COROT/
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The Basic Equations (if you really want To know!)