2013 Lecture 21 - Helioseismologyzeus.colorado.edu/astr7500-toomre/Lectures/SolStelMag...–...
Transcript of 2013 Lecture 21 - Helioseismologyzeus.colorado.edu/astr7500-toomre/Lectures/SolStelMag...–...
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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
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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).