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ASTR 7500: Solar & Stellar MagnetismHale CGEP Solar & Space Physics

Profs. Brad Hindman & Juri ToomreLecture 21 Tues 9 Apr 2013

zeus.colorado.edu/astr7500-toomre

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Lecture 21Helioseismology• Observations of Solar Oscillations

– Instruments– Dopplergrams– Power spectra

• Solar Wave Cavities– Restoring forces– Acoustic waves– Gravity Waves

• Wave Excitation and Resonances– Granulation makes waves– Resonant oscillations– Eigenfunctions

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Triumphs of Helioseismology

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The sun’s internal rotation rateThe solar neutrino problem

Acoustic Tomography

Carrington Longitude

sin

(lat

itu

de)

Active Region NOAA-10808

Aug 29 Sep 9 2005 (GONG)

AR10808

Acoustic imaging of the magnetic field on the farside of the sun

Fine scale measurements of subsurface convection

378

Mm

3

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Observations of Solar Oscillations

GONGGlobal Oscillation Network Group

1024 x 1024 pixel cameraMagnetogramsDopplergrams

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Spring 2011 7

MDIMichelson Doppler Imager1024 x 1024 pixel camera

MagnetogramsDopplergrams

L1

SOHOSolar and Heliospheric Observatory

11 Feb 2010

HMIHelioseismic and Magnetic Imager

4096 x 4096 pixel cameraVector MagnetogramsDopplergrams

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Dopplergrams (MDI)

Dopplergram of the Sun Dopplergram sequence withRotation removed

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Dopplergram MoviesSuperposition of 10 million resonant acoustic oscillations

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New HMI & AIA Images

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From August 24th 2012

HMI Dopplergrams(Photosphere ~ 5000 K)

AIA 171 Å (Low Corona ~ 6x105 K)

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ObservationsSpherical Coordinates

Dopplergram sequence withrotation removed

The raw data appears in the form of a time series of rectangular images (x,y).Each image is converted into spherical coordinates.

los los( , , ) ( , , )v x y t v tj q

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spherical harmonic transforms

Fourier transforms in time

Power spectra

GONG (Global Oscillations Network Group)3 months of Dopplergrams

(line-of-sight velocity)

0m =

2*

los0 0

sin ( , ) ( , )mlm lv d d Y vp p

j q q q j q j= ò ò

Acoustic Power Spectra

2( ) ( )i tlm lmv dt e v tp nn -= ò

2( ) ( )lm lmP vn n=

Wave Power

Wave Number – l = kR

Freq

uenc

y (m

Hz)

The harmonic degree is proportional to the horizontal wavenumber

Power is concentrated in discrete ridges

Each ridge corresponds to a wave with a different number of radial nodes

2( ) ( )lm lmP vn n=

3 decades of power

p2

f

p1

p3p4

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Low Harmonic DegreeIf we look carefully, we can see that each ridge is actually comprised of individual modes

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Solar Wave Cavities

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Momentum Equation

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( ) 2N 2v v P g R vtr r ræ ö¶ ÷ç + ⋅ = - + + W - W´÷ç ÷ç ÷è ø¶

The Coriolis force drivesRossby waves

We won’t spend much attention on such issues

Buoyancy perturbations driveinternal gravity waves

Consider the inviscid form of the Navier-Stokes equation, in the presence of gravity in a rotating reference frame.

Pressure perturbations drivesound waves

2Ng g Rº + W

Newtonian gravity and the centrifugal force can be combined into an effective gravity

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Surface ReflectionWaves reflect off the stellar photosphere

because of the rapid decrease in mass density.

How are sound waves trapped? Acoustic waves are trapped within a spherical shell

Deep RefractionIncreasing sound speed

with depth refracts waves back towards the surface

Hotter

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How are gravity waves trapped?In the convection zone

2 0N »

In the core and radiative zones2 0N >

Trapped Gravity Waves

Gravity waves propagate if2 2Nw <

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Excitation and Resonances

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Wave DrivingSolar convection in the form of granulation drives waves with a white spectrum

Hinode G-band image

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Swedish Solar Telescope(La Palma)

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0m =Acoustic Power

SpectraIf the forcing is over a broad spectrum, why is the observed energy confined to special frequencies?

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ResonancesSince we have trapped waves, we can have resonant oscillations.

Constructive self interference amplifies the energy for the resonant frequencies.

The resonant oscillations satisfy an eigenproblem.•Wave equation•Boundary conditions

First few modes of a guitar string

r

r R=

First few modes of the sun

0r =

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Form of the EigenfunctionsFor spherical symmetric stars the eigenfunction is separable

Radial Eigenfunction

The radial and angular eigenfunctions satisfy separate wave equations

Angular Eigenfunction

Fourth Orderw is the eigenvalueSeparation Constant

( , , , ) ( ) ( , ) i tw r t W r Y e wf q f q -=

2h 2

( 1)( , ) ( , ) 0

l lY Y

rf q f q

+ + =

{ }2

( 1)( , ) ( ) 0

l lW r W r

rw

++ =

( 1)l l +

2h 2

( 1)l lk

r

+=Horizontal Wavenumber

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Spherical HarmonicsThe ODE for the angular function Y is well-known and its solutions are functions called spherical harmonics

Angular Boundary ConditionsThere are two separation constants (l and m) and both have integer values.l must be an integer for the solution to be finite at the polesm must be an integer for the solution to be continuous at f = 0 and 2p

Furthermore |m| £ l , l = 1 has a triplet m = {-1, 0, 1}l = 2 has a quintuplet m = {-2,-1, 0, 1, 2}

Associated Legendre Polynomials( , )mlY Y f q=

( 1) (cos )m m imlm lC P efq= -

Fourier Modes

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determines the horizontal scalel

Acknowledgements: BISON group

Quantum Numbers

2h 2

( 1)l lk

r

+=

2 2h hY k Y = -

m determines the number of nodes in longitude

l – m determines the number of nodes in latitude

m / l determines how close to the pole

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Radial Order

Radial Order nThe radial differential equation

permits a sequence of solutions with different numbers of nodes.

The eigenfunctions are labeled by the number of nodes n.

Each solution has a different corresponding eigenfrequency wlmn.

{ } 2h( , ) ( ) 0n nW r k W rw + =

0n =

rR

f mode(fundamental)

1n =

2n =

p1 mode

p2 mode

rR

rR

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29( , , ) ( ) ( , )mlmn ln lw r W r Yj q j q=

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l

m

n

===

p mode

A Vertical Velocity Eigenfunction

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p- and g-Mode EigenfunctionsThe resonant acoustic waves are called p modes (p for pressure).

The resonant internal gravity waves are called g modes (g for gravity).

A p mode and a g mode with the same values for the quantum numbers l and m have identical angular planforms. However, they have very different radial eigenfunctions.

for a spherically symmetric star(non-rotating) the eigenfunctionsand the eigenfrequencies are independent of m.

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f – Fundamental mode(surface gravity wave)

pn – acoustic mode(Pressure wave)

gn – gravity mode(internal Gravity wave)

p2

f

p1

p3p4

p20

Power Spectrum

No g modes?

Why are only the acoustic modes visible?

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Tunneling

Trapped Gravity Waves

g modes must tunnel through the convection zone before they can be observed at the surface.

Thus, they have very low amplitude and have remained unobserved (in the sun).