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Helioseismology

• Basic Principles of Stellar Oscillations
• Helioseismology
• Asteroseismology

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The birth of helioseismology
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A Dopplergram
Add the two. No velocity is gray, postive,
negative velocities appear white/dark
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I. Basic Principles of Stellar Oscillations

• Notation for Normal Modes of Resonance
• Oscillations within a spherical object can be
  represented as a superposition
• of many normal modes , each which varies
  sinusoidally in time. The
• velocity due to pulsations can thus be expressed
  as

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l
imf
?
?
?
v(r,q,f,t)
Vn(r) Ylm(q,f)e
ml
n0
l0
Vn(r) is the radial part of the velocity
displacement n,l, and m are the quantum numbers
of the stellar oscillations where n is the
radial quantum number and is the number of radial
nodes (order) l is the angular quantum number
(degree) m is the azimuthal quantum number
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The number of nodes intersecting the pole m The
number of nodes parallel to the equator l
m The angular degree l measures the horizontal
component of the wave number
kh l(l 1)½ /r at radius r
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Sectoral mode
Zonal mode
Low degree modes
High degree modes
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Types of Oscillations
To get stable oscillations you need a restoring
force. In stars oscillations are classified by 3
major modes depending on the nature of the
restoring force
p-modes pressure is the restoring force
(example Cepheid variable stars)
g-modes gravity is the restoring force (example
ocean waves). As we shall see this is also
related to the buoyancy force.
f-modes fundamental modes. g- or p-modes that do
not have a radial node
In the sun p-modes have periods of minutes,
g-modes periods of hours
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Characteristic Frequencies
Stellar oscillations are characterized by two
frequencies, depending on whether pressure or
gravity is the restoring force.
Lamb Frequency The Lamb frequency is the
reciprocal time scale defined by the horizontal
wavelength divided by the local sound speed
k 2p/l
Travel time t (1/k)/c Frequency 1/t
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Characteristic Frequencies
Brunt-Väisälä Frequency The frequency at which a
bubble of gas may oscillate vertically with
gravity the restoring force
)
1
(
dlnP

N2 g
g
dr

• is the ratio of specific heats Cv/Cp
• g is the gravity

Where does this come from? Remember the
convection criterion?
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Dr rADr
FB grAVDr
In our case x Dr, k grA, m rV
dlnr
(
)
dlnP
1
N2 grAV/Vr
gA
g

g
dr
dr
The Brunt-Väisälä Frequency is the just the
harmonic oscillator frequency of a parcel of gas
due to buoyancy
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Characteristic Frequencies
The frequency of the oscillations indicate the
type of the restoring force.
If s is the frequency of the oscillation
s2 gt Ll2, N2 For high frequency oscillations the
restoring force is mainly pressure and
oscillations show the characteristic of acoustic
(p) modes.
s2 lt Ll2, N2 For low frequency oscillations the
restoring force is mainly due to buoyancy and the
oscillations show the characteristic of gravity
waves.
Ll2 lt s2 lt N2, Ll2 gt s2 gt N2 In these regions
of the star evanescent waves exist, i.e. the wave
exponentially decreases with distance from the
propagation region.
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Propagation Diagrams
p-modes
p-modes
Decaying waves
g-modes
Decaying waves
g-modes
g modes cannot propagate through the convection
zone. Why?
Buoyancy force is a destabilizing force.
Propagation diagrams can immediately tell you
where the p- and g-modes propagate
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Probing the Interior of the Sun p-modes
The period is determined by the travel time of
acoustic waves in a cavity defined by two turning
points one just below the photosphere where the
where the density decreases rapidly (reflection),
and a lower turning point caused by the gradual
increase of the sound speed, c, with depth
(refraction).
At the lower reflection point the wave is
traveling horizontally and the reflection occurs
at a depth d where
c 2ps/kh
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Horizontal wavelength
Decreasing density causes the wave to reflect at
the surface
Increasing density causes the wave to refract in
the interior.
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By observing modes with a range of frequencies
one can sample the sound depth with speed
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Assymptotic Relationship for P-mode oscillations
(ngtgt l)
p-modes
nnl Dn0 (n l/2 e) dn

• is a constant that depends on the stellar
  structure
• Dn0 2?0R dr/c1 where c is the speed of
  sound (i.e. this is the return sound travel time
  from the surface to the core)
• dn small spacing (related to gradient in sound
  speed)

p-modes are equally spaced in frequency
n
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Probing the Interior of the Sun g-modes
For g-modes wave propagation is generally only
possible in regions of the Sun below the
convection zone. A particular g-mode is trapped
in regions where its frequency s is less than the
local buoyancy frequency N. The upper and lower
reflection points of any given cavity correspond
to where N has approached s.
G-modes thus sample the Brunt-Väisälä frequency,
N, as a function of depth
The g-modes all share the reflection point near
the base of the convection zone and their
amplitudes decay throughout that zone
(evanescent). Since the decay rate increases with
l only low degree modes are likely to be detected
in the atmosphere.
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Assymptotic Relationship for G-mode oscillations
(ngtgt l)
g-modes
n ½ l g
P0
Pn,l
l(l 1)½
rc
1
dr


P0 2p2
?
N
r
0
P0
g-modes are equally spaced in Period
l0
l1
l2
n
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Excitation of Modes
Normally, when a star undergoes oscillations
dissipative forces would cause the oscillations
to quickly damp out. You thus need a driving
force or excitation mechanism to sustain the
oscillations. Two possible mechanisms
e Mechanism The energy generation depends
sensitively on the temparature. If a star
contracts the temperature rises and the energy
generation increases.This mechanism is only
important in the core, and is not an important
mechanism in the Sun.
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Excitation of Modes
k Mechanism If in a region of the star the
opacity changes, then the star can block energy
(photons) which can be subsequently released in a
later phase of the pulsation. Helium and and
Hydrogen ionization zones of the star are
normally where this works. Consider the Helium
ionization zone in the interior of a star. During
a contraction phase of the pulsations the density
increases causing He II to recombine. Neutral
helium has a higher opacity and blocks photons
and thus stores energy. When the star expands the
density decreases and neutral helium is ionized
by the emerging radiation. The opacity then
decreases. This mechanism is reponsible for the
5 minute oscillations in the Sun.
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II. Helioseismology
The solar 5 min oscillations were first thought
to be just convection motion. Later it was
established that these were acoustic
modes trapped below the photosphere. The sun is
expected to have millions of these modes. The
amplitude of detected modes can be as small as
0.2 m/s
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Currently there are several thousands of modes
detected with l up to 400. These are largely the
result of global networks which remove the 1-day
alias. These p-mode amplitude have a Gaussian
distribution centered on a frequency of 3 mHz
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To find all possible pulsaton modes you need
continuous coverage. There are three ways to do
this.
Ground-based networks Telescopes that are well
spaced in longitude. South pole in
Summer Spaced-based instruments
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• GONG Global Oscillation Network Group
• Big Bear Solar Observatory California,
  USA
• Learmonth Solar Observatory Western
  Australia
• Udaipur Solar Observatory India
• Observatory del Teide Canary Islands
• Cerro Tololo Interamerican Observatory
  Chile
• Mauno Loa Hawaii, USA

• BiSON Birmingham Solar Oscillation Network
• Carnarvon, Western Australia
• Izaqa, Tenerife
• Las Campanas, Chile
• Mount Wilson, California
• Narrabri, New South Wales, Australia
• Sutherland, South Africa

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L1 is where gravity of Earth and Sun
balance. Satellites can have stable orbits with
minimum energy use
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The p-mode Fourier spectrum from GOLF, using a
690-day time series of calibrated velocity
signal, which exhibits an excellent signal to
noise ratio.
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The low-frequency range of the p modes from
above spectrum, showing low-n order modes.
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Rise to low frequency due to stochastic noise of
convection
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The Small Frequency Spacing
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The Small Frequency Spacing
Normally modes of different n and l that differ
by say 1 in n and 2 in l are degenerate in
frequency. In reality since different l modes
penetrate to different depths this degeneracy is
lifted.
nn,l
Dn0 (n l/2 d)
A,h,e depend on the structure of the star
The small separation is sensitive to sound speed
gradients
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Echelle Diagrams
nn,l n0 kDn n1
n0 a reference k integer n1 0 ? Dn
ln1
ln
nn,l n0 kDn
An echelle diagram basically cuts the frequency
axis into chunks of Dn and stacks them on each
other
n1
0
Dn
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Results from Helioseismology
There are two ways of deriving the internal
structure of the sun

• Direct Modelling
• Computationally easy
• Results depend on model

• Inversion Techniques
• Model independent
• Computationally difficult

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Sound Speed
P-modes give information about the sound speed as
a function of depth. The sound speeds in the
mid-region of the radiative zone were found to be
off by 1. This suggested that the opacity below
the convection zone was underestimated. This has
since been confirmed by new opacities.
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Deviations of the sound speed from the solar
model red is positive variations (hotter) and
blue is negative variations (cooler regions).
From SOHI MDI data.
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Possibly due to increased turbulence
Deviations of the observed sound speed from the
model. The differences are mostly less than 0.2
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Simple convection (mixing length theory) does not
adequately model observed frequencies
Mixing length theory
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Chemical Composition Low abundance model
solution to the neutrino problem can be rejected
Z0.02
Z0.018
One explanation to the solar neutrino problem is
that the metal abundance (Z) in the interior is
lower. But lower Z moves the models away from the
observed frequencies
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Depth of the convection Zone
Early helioseismic results showed that the depth
of the convection zone was 50 greater than
current models The mixing length parameter a
2-3 (l aHp, Hp is the scale height)
In convection zone W 1 g 1 5/3 0.67
Negative values
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Rotation With no rotation all m modes from a
given l are degenerate. Rotation lifts this
degeneracy and the m. For an l1, m 1,0,1.
Thus rotational splitting will be a triplet.
Analogy Zeeman splitting of energy levels of
atom.
l 1 stellar oscillator with modes split into
triplets by rotation.
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The internal rotation of the Sun
Rotation period in days
The sun shows differential rotation throughout
the convection zone, and almost solid body
rotation in the radiative zone
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G-modes propagate only in the radiative core and
are evanescent in the convection zone. The
amplitude of these waves exponentially decay
while passing through the convection zone.
Consequently, their amplitudes at the observable
surface is expected to be small.
G-modes are important in that they can probe the
interior of the sun all the way down to the core
(r 0). P-modes can only get to about r0.2 R?
There have been many claims for detecting solar
g-modes, but none have been verified. Theoretical
work suggest that the amplitudes of these modes
at the surface should be 0.01 5 mm/s. It is
easy to see why these have not been detected. The
search for these, however, continues.
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Asteroseismology
When observing the sun we have the advantage of
having lots of photons and a resolved disk. When
searching for 5-min like observations in solar
type stars we are looking in integrated light.
Because of cancellation effects only the lowest
degree modes can be detected.
High degree larger numbers of / regions,
cancellation is more
Low degree Fewer numbers of / regions,
cancellation is less

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Asteroseismology
Scaling to solar values Kjeldsen Bedding,
Astronomy and Astrophysics, 293, 87, 1995
M, R in solar units
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3.05 mHz
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Peak power at P 7 min so either the Mass is
smaller than the sun, or the radius is larger ,
or both
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Pollux (from M. Zechmeister)
3 hrs
R 8.8 R? (from interferometry)
dL/L 0.0001 (predicted) dL/L 0.0001 (MOST)
M 1.4 M? (from scaling relations)
L 33 L? (from brightness and distance)
V 5 m/s (from scaling relations)
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Przybylskis Star (HD 101065)
Velocities taken with HARPS HD 101065 is an
pulsating Ap star (12 min) with a large inclined
dipole field.
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Przybylskis Star (HD 101065)
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Effective temperature 6538 K Luminosity 5.88
L? Mass 1.5 M? Magnetic field 8600 Gauss
From scaling law R 1.78 R?
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CoRoT
A space telescope (27 cm) designed to obtain
precise light curves of stars for
asteroseismology and extrasolar planet searches
(transits) for 150 days continuously in one field.

• Launched 27 Dec 2006
• French mission with partners from Germany,
  Austria, Belgium, Spain, and ESA
• German CoIs (Tautenburg, DLR, Köln)
• First results just released

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The CCDs of CoRoT
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Die COROT-Mission Focal Plane
Defocusd images on asteroseismology CCDs
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From the CoRoT press release
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Science from press release What can we say?
Dn0

• Equally spaced in frequency, these are p-modes

• Large spacing Dn0 2 42.5 85 mHz

• Assuming 1 M? and using expression for
  spacing, R 1.36 R?. This is an evolved star.
  Consistent with 10.4 min period.

• Maximum power at nmax 1.65 mHz 0.00165 c/s

• Order of maximum power nmax 18

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And lets not forget CoRoT transits
CoRoT Transit light curve
Ground Based transit light curve (different star)