Transcript Folie 1

Helioseismology
I.
Basic Principles of Stellar Oscillations
II.
Global Helioseismology
III. Local Helioseismology
The birth of helioseismology
A Dopplergram
Take a postive
image in a blueshifted band pass
Take a negative
image in a redshifted band pass
Add the two. No velocity is gray, postive, negative
velocities appear white/dark
A modern Dopplergram from space (SOHO)opplergram
I. Basic Principles of Stellar Oscillations
1. 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:
∞ ∞
l
n=0 l=0
m=–l
m(q,f)e –imf
V
(r)
Y
∑
∑
∑
v(r,q,f,t) =
n
l
where Ylm(q,f) is the associated Legendre function
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
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
Zonal mode
Sectoral mode
Low degree
modes
High degree
modes
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
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:
Ll2 = (khcs)2 =
k = 2p/l
Travel time t = (1/k)/cs
Frequency = 1/t
l(l +1)cs2
r2
Characteristic Frequencies
Brunt-Väisälä Frequency
The frequency at which a bubble of gas may oscillate vertically with
gravity the restoring force:
N2
=g
(
1

dlnr
dlnP
–
dr
dr
)
 is the ratio of specific heats = Cv/Cp
g is the gravity
Where does this come from? Remember the convection criterion?
r2
Convection: The Brunt-Väisälä Frequency
P2
r2 *
P2*
DT
Dr
Consider a parcel of gas that is perturbed
upwards. Before the perturbation r1* = r1
and P1* = P1
For adiabatic expansion:
Dr
P = r
 = CP/CV = 5/3
Cp, Cv = specific heats at constant V, P
r1
P1
r1 *
P1 *
After the perturbation:
P2 = P2*
r2 * = r1
P2* 1/
P1*
( )
Convection: The Brunt-Väisälä Frequency
Stability Criterion:
Stable: r2* > r2 The parcel is denser than its surroundings
and gravity will move it back down.
Unstable: r2* < r2 the parcel is less dense than its
surroundings and the buoyancy force will cause it to rise
higher.
Convection: The Brunt-Väisälä Frequency
Stability Criterion:
r2 *
r1 =
P2
P1
dP
1/
( )
=
{
P1 + dr Dr
P1
r2 * = r1 +
1/
}
1

(
1/
{
r1 dP
P dr
) Dr
dr
r2 = r1 +
dr Dr
1

(
r1 dP
P dr
)
>
}
1 dP
= 1 + P dr Dr
dr
dr
This is the Schwarzschild criterion for stability
r2
Convection: The Brunt-Väisälä Frequency
P2
r2 *
P2*
DT
Dr
r1
P1
r 2 = r1 +
dr
dr Dr
(
r1 dP
P dr
) Dr
Difference in density between inside the
parcel and outside the parcel:
Dr
r1 *
P1 *
r2 * = r1 +
1

1
Dr = 
=
=
1

r
(
r
dP
P dr
(
r dlnP
dr
(
1 dlnP
 dr
) Dr
–
dr
dr Dr
)
dlnr
r
Dr –
dr Dr
)
dlnr
r
Dr –
dr Dr
A
=
(
1 dlnP
 dr
)
–
dlnr
dr
Dr = rADr
Buoyancy force:
FB = –DrVg
V = volume
FB = –grAVDr
w2 = k/m
For a harmonic oscillator: FB = –kx
In our case x = Dr, k = grAV, m ~ rV
N2 = grAV/Vr
= gA = g
(
1 dlnP
 dr
–
dlnr
dr
)
The Brunt-Väisälä Frequency is the just the harmonic oscillator frequency of a
parcel of gas due to buoyancy
Characteristic Frequencies
The frequency of the oscillations indicate the type of the restoring force.
If s is the frequency of the oscillation:
s2 > Ll2, N2: For high frequency oscillations the restoring force is mainly
pressure and oscillations show the characteristic of acoustic (p) modes.
s2 < Ll2, N2: For low frequency oscillations the restoring force is mainly
due to buoyancy and the oscillations show the characteristic of gravity
waves.
Ll2 < s2 < N2, Ll2 > s2 > N2 : In these regions of the star evanescent waves
exist, i.e. the wave exponentially decreases with distance from the
propagation region.
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
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 = √p/r
c = 2ps/kh
=
(
∂lnp
∂lnr
)
S
Red: high degree modes have shorter wavelengths
and do not propagate deeper into the sun.
Decreasing density
causes the wave to
reflect at the surface
Increasing density
causes the wave to
refract in the interior.
Blue: low degree modes have longer wavelengths and
propagate deeper into the sun.
By observing modes with a range of frequencies one
can sample the sound speed with depth:
Assymptotic Relationship for P-mode oscillations (n>> l)
p-modes:
nnl ≈ Dn0 (n + l/2 + e) + dn
Tassoul (1980)
e is a constant that depends on the stellar structure
R
Dn0 = [2∫0 dr/c]–1 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)
Dn0
l=0
l=1
l=2
n
p-modes are equally spaced in frequency
The Small Frequency Spacing
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.
A(l + 1)l – h
nn,l = Dn0 (n + l/2 + d) –
(n + l/2 + e)
A,h,e depend on the structure of the star
dn,l = Dn0
(l + 1)
2p2nn,l
∫
0
R
dc dr
dr r
The small separation is sensitive to sound speed gradients
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.
Assymptotic Relationship for G-mode oscillations (n>> l)
g-modes:
Pn,l ≈
n+½l+
[l(l+ 1)]½
P0
–1
dr
P0 = 2p2[∫ N r ]
rc
0
P0
l=0
l=1
l=2
n
g-modes are equally spaced in Period
The basis of Helioseismology
P-modes enable you to probe the sound speed with
depth. The sound speed is related to the pressure and
density, thus you probe the pressure and density with
depth.
c = √p/r
G-modes enable you to probe the Brunt-Vaisaila frequency with
depth. This frequency depends on the gravity, and gradient of the
pressure and density
N2
=g
(
1

dlnr
dlnP
–
dr
dr
)
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 temperature. 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.
Excitation of Modes
 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.
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
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
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
GONG: Global Oscillation Network Group
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•
•
•
•
•
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
SOHO: Solar and Heliospheric Observatory is a
ESA/NASA project to observe the sun. It is located at
the L1 point
L1 is where
gravity of Earth
and Sun balance.
Satellites can
have stable orbits
with minimum
energy use
Three helioseismology experiments: MDI
(Michelson-Doppler Imager), GOLF (Global
Oscillations Low Frequency) and VIRGO
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.
The low-frequency range of the p modes from above spectrum,
showing low-n order modes.
Rise to low frequency
due to stochastic noise of
convection
Two useful methods of plotting the bewildering number of
pulsation modes on the Sun are via „Ridge“ Diagrams and
„Echelle“ diagrams
Ridge Diagrams are more common in Helioseismology,
while Echelle diagrams are more common for
Asteroseismology (explained in next lecture).
Ridge diagrams plot the amplitudes of solar modes as a function of
frequency and degree number.
f-mode is the
fundamental
mode, n=0
Christensen-Dalsgaard Notes
Ridge Diagrams for the Sun
Color symbolizes power
Results from Helioseismology
There are two ways of deriving the internal structure of the sun
Direct Modelling
Inversion Techniques
• Computationally easy
• Model independent
• Results depend on model
• Computationally difficult
Thick line: inversion
Thin line model
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.
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.
Possibly due to increased
turbulence
Note change of
scale from
previous graph
Deviations of the observed sound speed from the model. The
differences are mostly less than 0.2%
Simple convection (mixing length theory) does not
adequately model observed frequencies
Mixing length theory:
L
A hot blob moves a
certain distance
upwards and
deposits all of its
excess energy into
the surrounding
region. It is a flux
source.
Rotation:
With no rotation all m modes from a given l are degenerate. Rotation
lifts this degeneracy and the m. For an l=1, 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.
Rotation profiles of the Sun‘s interior at 3 latitudes.
Note that differential rotation only occurs in the
convective core.
The internal rotation of the Sun in nice color picture
Rotation
period in
days
The sun shows differential rotation throughout the convection zone,
and almost solid body rotation in the radiative zone
The radial displacement of
g- and p-modes. P-modes
have the highest
amplitudes at the solar
surface, g-modes in the
core of the Sun.
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 r=0.2 R‫סּ‬
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.
Detecting solar g-modes with ASTROD
ASTROD is a space mission that
will use interferometric ranging to
test relativity. It can detect the
gravitational effects of solar gmodes.
The Solar Neutrino Problem and Helioseismology‘s
Role in its solution
Or
How to observer the core of the Sun?
The Solar Core Neutrinos
The PP chain produces neutrinos. Neutrinos have no mass and interact
very weakly with matter. They are the only means of probing the core.
37Cl
+ n → 37Ar + e–
The 8B neutrinos from the PPIII chain have sufficient energy for
this. Unfortunately, this is the least likely (0.3%) branch of the PP
chain.
The predicted neutrino flux comes from solar models. The solar neutrino
flux of the 8B neutrinos in Solar Neutrino Units (1 SNU = 10–36
captures/target atom/sec)
Bahcall 1968: 7.5 ± 3.3 SNU
Bahcall & Pinsonneault 1992: 8.0 ± 1.0 SNU
Turck-Chieze & Lopez 1993: 6.4 ± 1.0 SNU
So just what is the solar neutrino flux?
The Homestake Experiment
In 1965 the Homestake Mining Company completed a 30 x 30 x 32 foot (1
foot = 30.5 cm) cavity at a depth of 1478 meters (4200 meters equivalent
water depth) to house the experiment of Ray Davis, Jr. 400.000 liters of the
cleaning fluid tetrachloroethylene (C2Cl4) were placed in a tank. After
exposing to solar neutrinos for a certain amount of time the tank was emptied
and the number of 37Ar atoms (half life = 35 days) was counted. Davis
demonstrated that 95% of the Ar atoms can be recovered.
Result:
In 1968: Upper bound of 3 SNU
After 25 years flux of neutrinos = 2.55 ± 0.17 SNU
The Solar Neutrino Problem: The sun does not produce enough
neutrinos due to the predicted fusion reactions. This problem has
persisted for over 30 years.
One solution:
The Cl experiment only detects neutrinos from the 8B branch
of the PPIII chain which is only a small fraction of the total PP
chain. If one could detect neutrinos from the PPI chain there
may be no inconsitency.
Remember where the neutrinos come from:
1H
+ 1H → 2D + e+ + n PPI
7Be
8B
+ e– → 7Li + n
PPII (31%)
→ 8Be +  + e+ + n PPIII (0.3%)
Other Neutrino Experiments
Kamiokande
4.5 kiloton cylindrical imaging water Cerenkov detector.
It was located 1000 m underground in the Mozumi
Mine. The detector was a tank containing 3000 tons of
pure water and 1000 photomultiplier tubes. It detects
neutrinos produced by the Cerenkov light of recoiling
electrons.
Result: Only 50% of predicted neutrinos dectected. It
confirmed the Cl result at Homestake
The Kamiokande was remarkable in that it measured solar
neutrinos in real time and could separate solar neutrinos from
isotopic events. There was also directional information so one
knew the neutrinos came from the sun.
Super-Kamiokande
A 50.000 ton ring imaging
water Cerenkov detector at
2700 m depth. Bigger and
Better.
On 12 Nov 2001 several
thousand photomultiplier
tubes imploded in a chain
reaction. Detector restarted
in 2003
Another Cherenkov
detector is the Sudbury
Neutrino Observatory
(SNO) in Canada
Other Neutrino Experiments
Sage and Gallex
Uses gallium to detect neutrinos:
71Ga
+ n → 71Ge + e
71Ge
→ 71Ga (11.43 day half life)
Gallex: In Italy, uses 30 tons of GaCl3:
Sage: In Caucusus, uses metallic Ga
These experiments can detect neutrinos from the initial PPI chain!
Sensitivity of Experiments to P-P Neutrinos
Possible Solutions to the Solar Neutrino Problem
1. Experimental Error
Highly unlikely. The solar neutrino problem has persisted for over 30
years and had been confirmed by 3 experiments.
2. Low Z-model
Lowering the abundance of heavy elements in the core reduces the
opacity and leads to a smaller temperature gradient and thus lower
temperature in the core. The higher surface abundance is explained by
accretion of dust as the sun formed.
Problems: Initial helium abundance has to be adjusted and this is much
lower than the primordial abundance of helium.
Not supported by helioseisimolgy
Possible Solutions to the Solar Neutrino Problem
3. Rapidly Rotating Core
If the core is rapidly rotating centrifugal force can provide some support
against gravity and thus one has a lower central temperature. This
requires a rotation rate 500 times faster than the surface layers!
Problem: Produces oblateness which adds a quadrupole moment to
the gravitational field. The outer layers will thus be deformed. This is
not observed. Not supported by helioseismology.
4. Internal Magnetic Field
Magnetic pressure provides some support against gravity, thus a
lower central temperature.
Problems: Field would not survive due to ohmic dissipation
Not supported by helioseismology
Possible Solutions to the Solar Neutrino Problem
5. Internal Mixing or Convection
Mixing would replace He in the core with H and thus lower the
molecular weight: A lower temperature could provide the pressure
support against gravity.
Problem: Not supported by helioseismology.
6. Weakly Interacting Massive Particles (WIMPS)
The long mean free path would make the core more isothermal.
Problems: Not supported by helioseismology
Possible Solutions to the Solar Neutrino Problem
7. Particle Physics
Neutrinos transform from one type to another. Neutrinos come in 3
flavors: electron, muon, and tauon neutrinos. The detectors only
detect electron neutrinos which is what the sun produces. But if the
neutrino were to change flavor on the way to the earth.
The Sudbury Neutrino Observatory (SNO) in Canada can
distinguish between the types of neutrinos. Researchers are 99%
certain that the sun is producing neutrinos in the right amount, but
that a fraction changes flavors. Oscillating neutrinos require a mass
which can account for 20% of the dark matter in the universe.
Note: Once again the study of the Sun points to new and
unknown physics.
The Nobel Prize in Physics 2002
Raymond Davis: "for
pioneering contributions to
astrophysics, in particular
for the detection of cosmic
neutrinos"
The solar neutrino problem is one that has stood the test of time. Ray
Davis‘s experiment was a brilliant experiment that ultimately led to new
physics. He was awarded the 2002 Nobel Prize for his work.
The Solar Abundance Problem
Abundance analyses: Martin Asplund and collaborators (MPA
Garching) have determined an abundance of „metals“ in the sun that
is lower than previous values (z = 0.0178 versus z = 0.0229).
This results in a radius of the convection zone of RCZ = 0.724 R‫ סּ‬and
a helium abundance of Y = 0.0248
Problem: Not supported by Helioseismology
Helioseismology gives RCZ = 0.713 R‫( סּ‬5s difference) and Y = 0.2485 (11s
difference) in strong conflict with the abundance results.
Local Helioseismology
Up until now we have been discussing „Global Heliosesismology“ a
recent development is the field of „Local Helioseismology“ which is
only possible because the sun is resolved.
Acoustic power at 3 mHz
Acoustic power at 6 mHz
Location of active regions
Sunspots and magnetic
regions strongly absorb pmode oscillations (Braun et al.
1992, ApJ, 392, 739). There is
also a „halo“ of enhanced
power at 5-6 mHz
Halo at 6 mHz
From the amplitude of the p-modes in the ridge diagram one can
regress these acoustic amplitudes at the surface into the solar
interior. One can map out the time delays across the surface
Ginzon et al. 2002, Space Sci. Rev. 144,249
Left: The reconstructed „moat velocities“ (outflow from sunspots) at a
depth of 1 Mm (million meters) below the solar surface reconstructed
from the time delay of p-mode obsrvations from the ridge diagrams.
Right: The moat velocities constructed from Moving Magnetic Features
Velocity flow
sunspot
Helioseismic Holography
P-modes emanating from the far side of the Sun act
as a wave front that can be used to „image“ the far
side
Helioseismic Holography
Helioseismic Holography is a new technique proposed by Braun &
Lindsey (2000). It takes advantage of the fact that magnetic active
regions absorb p-mode oscillations which can be propagated back in
time to reconstruct magnetic images on the far side of the sun.
Useful links:
http://www.cora.nwra.com/~dbraun/farside/
http://soi.stanford.edu/data/farside/
Summary: Or what to remember from this Lecture
1. Two main modes: p-modes (pressure is the
restoring force), g-modes (gravity is the restoring
force)
2. g-modes cannot propagate through the convection
zone, p-modes cannot propagate through the
radiative zone
3. High order p-modes are equally-spaced in
frequency, g-modes equally-spaced in period
4. Low degree modes probe deeper into the sun/star
5. Many modes means you can study the sound speed
(p-modes) or BV versus depth → internal structure