Helioseismology

I. Basic Principles of Stellar Oscillations

II. Global Helioseismology

III. Local Helioseismology

The birth of helioseismology

A Dopplergram

Add the two. No velocity is gray, postive, negative velocities appear white/dark

Take a postive image in a blue-shifted band pass

Take a negative image in a red-shifted band pass

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

superpositionof many normal modes , each which varies sinusoidally in time. Thevelocity due to pulsations can thus be expressed as:

v(r,,,t) =∑ ∑ ∑n=0 l=0 m=–l

∞ ∞ l

Vn(r) Ylm(,)e –im

where Ylm(,) 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

Low degree modes

High degree modes

Zonal mode Sectoral mode

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 = l(l +1)cs

2

r2

k = 2/ Travel time t = (1/k)/cs

Frequency = 1/t

Brunt-Väisälä Frequency

The frequency at which a bubble of gas may oscillate vertically with gravity the restoring force:

Characteristic Frequencies

is the ratio of specific heats = Cv/Cp

g is the gravity

( )1 dlnPdr

dlndr

–N2 = g

Where does this come from? Remember the convection criterion?

Convection: The Brunt-Väisälä Frequency

T

r

1

*

P*

P

*

P*

2 P

Consider a parcel of gas that is perturbed upwards. Before the perturbation 1* = 1 and P1* = P1

For adiabatic expansion:

P = = CP/CV = 5/3Cp, Cv = specific heats at constant V, P

After the perturbation:

P2 = P2* 2* = 1(P2*

P1*)

1/

Stability Criterion:

Stable: 2* > 2 The parcel is denser than its surroundings and gravity will move it back down.

Unstable: 2* <2 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:

2*1

= ( P2

P1)

1/= {P1 +

dPdr r}

P1

1/

={ }1/

1 + r1P

dPdr

2* = 1 + (1

P )1 dPdr

r

2 = 1 +ddr r

(1

P )1 dPdr

ddr

>

This is the Schwarzschild criterion for stability

Convection: The Brunt-Väisälä Frequency

T

r

1

*

P*

P

*

P*

2 P

2* = 1 + (1

P )1 dPdr

r

2 = 1 +ddr r

Difference in density between inside the parcel and outside the parcel:

= (

P )1 dPdr

r –ddr r

= ( )1 dlnPdr

rdlndr r–

= ( )dlnP

drr dr r–1 dln

Convection: The Brunt-Väisälä Frequency

= Ar

= ( )dlnPdr dr–

1 dlnA

Buoyancy force:

FB = –Vg V = volume

FB = –gAVr

In our case x = r, k = gAV, m ~ V

( )dlnP dlnN2 = gAV/V = gA = g dr dr–

1

The Brunt-Väisälä Frequency is the just the harmonic oscillator frequency of a parcel of gas due to buoyancy

FB = –kx 2 = k/mFor a harmonic oscillator:

Characteristic Frequencies

The frequency of the oscillations indicate the type of the restoring force.

If is the frequency of the oscillation:

2 > Ll2, N2: For high frequency oscillations the restoring force is mainly

pressure and oscillations show the characteristic of acoustic (p) modes.

2 < Ll2, N2: For low frequency oscillations the restoring force is mainly

due to buoyancy and the oscillations show the characteristic of gravity waves.

Ll2 < 2 < N2, Ll

2 > 2 > N2 : In these regions of the star evanescent waves exist, i.e. the wave exponentially decreases with distance from the propagation region.

Propagation Diagrams

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

p-modes

g-modes

g-modes

p-modes

Decaying waves

Decaying waves

The period is determined by the travel time of acoustic wavesin a cavity defined by two turning points: one just below the photospherewhere the where the density decreases rapidly (reflection), and a lowerturning point caused by the gradual increase of the sound speed, c, with depth (refraction).

At the lower reflection point the wave is traveling horizontallyand the reflection occurs at a depth d where

Probing the Interior of the Sun: p-modes

c = 2/kh c = √p/

= ( )∂lnp∂ln S

Decreasing density causes the wave to reflect at the surface

Increasing density causes the wave to refract in the interior.

Red: high degree modes have shorter wavelengths and do not propagate deeper into the sun.

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:

nl ≈ (n + l/2 + ) +

is a constant that depends on the stellar structure

0 = [2∫0

R 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)

= small spacing (related to gradient in sound speed)

l=0 l=1 l=2

p-modes are equally spaced in frequency

Tassoul (1980)

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.

n,l = (n + l/2 + –

A,depend on the structure of the star

A(l + 1)l –

(n + l/2 + )

n,l = 0

(l + 1)

22n,l

∫0

R dcdr r

dr

The small separation is sensitive to sound speed gradients

For g-modes wave propagation is generally only possible in regionsof the Sun below the convection zone. A particular g-mode is trapped in regions where its frequency is less than the local buoyancyfrequency N. The upper and lower reflection points of any givencavity correspond to where N has approached .

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 convectionzone and their amplitudes decay throughout that zone (evanescent). Sincethe decay rate increases with l only low degree modes arelikely to be detected in the atmosphere.

Probing the Interior of the Sun: g-modes

Assymptotic Relationship for G-mode oscillations (n>> l)

g-modes:

l=0 l=1 l=2

P0

g-modes are equally spaced in Period

Pn,l ≈ n + ½ l +

[l(l+ 1)]½P0

P0 = 22 ∫[0

rc

Ndrr ]

–1

The basis of Helioseismology

c = √p/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.

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

( )1 dlnPdr

dlndr

–N2 = g

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:

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

• 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

Three helioseismology experiments: MDI (Michelson-Doppler Imager), GOLF (Global Oscillations Low Frequency) and VIRGO

L1 is where gravity of Earth and Sun balance.

Satellites can have stable orbits with minimum energy use

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).

Christensen-Dalsgaard Notes

Ridge diagrams plot the amplitudes of solar modes as a function of frequency and degree number.

f-mode is the fundamental mode, n=0

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

• Computationally easy

• Results depend on model

Inversion Techniques

• Model independent

• 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

Deviations of the observed sound speed from the model. The differences are mostly less than 0.2%

Note change of scale from previous graph

Mixing length theory:

Simple convection (mixing length theory) does not adequately model observed frequencies

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 thesun all the way down to the core (r = 0). P-modes can only get to about r=0.2 Rּס

There have been many claims for detecting solar g-modes, but nonehave 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 g-modes.

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 + → 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 Solar Neutrino Problem and Helioseismology‘s Role in its solution

Or

How to observer the core of the Sun?

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.

1H + 1H → 2D + e+ + PPI

Remember where the neutrinos come from:

7Be + e– → 7Li + PPII (31%)

8B → 8Be + e+ + PPIII

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:

Gallex: In Italy, uses 30 tons of GaCl3:

Sage: In Caucusus, uses metallic Ga

71Ga + → 71Ge + e

71Ge → 71Ga (11.43 day half life)

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 Rּס and a helium abundance of Y = 0.0248

Problem: Not supported by Helioseismology

Helioseismology gives RCZ = 0.713 Rּס (5 difference) and Y = 0.2485 (11 difference) in strong conflict with the abundance results.

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.

Local Helioseismology

Sunspots and magnetic regions strongly absorb p-mode oscillations (Braun et al. 1992, ApJ, 392, 739). There is also a „halo“ of enhanced power at 5-6 mHz

Acoustic power at 3 mHz

Acoustic power at 6 mHz

Location of active regions

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

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

Ginzon et al. 2002, Space Sci. Rev. 144,249

sunspotVelocity flow

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