Lecture 21 Helioseismology

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1 ASTR 7500: Sola & Stella Magnetism Hale CGEP Sola & Space Physics Pofs. Bad Hindman & Jui Toome Lectue 1 Tues 9 Ap 013 zeus.coloado.edu/ast7500-toome Lectue 1 Helioseismology Obsevations of Sola Oscillations Instuments Dopplegams Powe specta Sola Wave Cavities Restoing foces Acoustic waves Gavity Waves Wave Excitation and Resonances Ganulation makes waves Resonant oscillations Eigenfunctions 1

2 Tiumphs of Helioseismology The sola neutino poblem The sun s intenal otation ate 3 Acoustic Tomogaphy Active Region NOAA Aug 9 Sep (GONG) sin(latitude) AR Mm Caington Longitude Acoustic imaging of the magnetic field on the faside of the sun Fine scale measuements of subsuface convection

3 Obsevations of Sola Oscillations 5 GONG Global Oscillation Netwok Goup 104 x 104 pixel camea Magnetogams Dopplegams 3

4 MDI Michelson Dopple Image 104 x 104 pixel camea Magnetogams Dopplegams L 1 SOHO Sola and Heliospheic Obsevatoy Sping HMI Helioseismic and Magnetic Image 4096 x 4096 pixel camea Vecto Magnetogams Dopplegams 11 Feb 010 4

5 Dopplegams (MDI) Dopplegam of the Sun Dopplegam sequence with Rotation emoved 9 Dopplegam Movies Supeposition of 10 million esonant acoustic oscillations 10 5

appeas in the fom of a time seies of ectangula images (x,y).

6 New HMI & AIA Images Fom August 4 th 01 AIA 171 Å (Low Coona ~ 6x10 5 K) HMI Dopplegams (Photosphee ~ 5000 K) 11 Obsevations Spheical Coodinates The aw data appeas in the fom of a time seies of ectangula images (x,y). Each image is conveted into spheical coodinates. v (,,) x y t v (,,) jqt los los Dopplegam sequence with otation emoved 1 6

7 m = 0 Acoustic Powe Specta spheical hamonic tansfoms p v d j sin q d q Y ( q, j) v ( q, j) lm p = ò ò 0 0 m l * los Fouie tansfoms in time v i t lm( ) dt e - pn n = ò v lm( t ) Powe specta P ( n) = v ( n) lm lm GONG (Global Oscillations Netwok Goup) 3 months of Dopplegams (line-of-sight velocity) 13 Wave Powe P ( n) = v ( n) lm lm The hamonic degee is popotional to the hoizontal wavenumbe Fequency (mhz) p 4 p 3 p Powe is concentated in discete idges Each idge coesponds to a wave with a diffeent numbe of adial nodes p 1 f 3 decades of powe Wave Numbe l = kr 7

8 Low Hamonic Degee If we look caefully, we can see that each idge is actually compised of individual modes 15 Sola Wave Cavities 16 8

9 Momentum Equation Conside the inviscid fom of the Navie-Stokes equation, in the pesence of gavity in a otating efeence fame. æ ö v + v = - P + ( gn + W R) -W v ç è t ø Pessue petubations dive sound waves The Coiolis foce dives Rossby waves Buoyancy petubations dive intenal gavity waves g g R º N + W We won t spend much attention on such issues Newtonian gavity and the centifugal foce can be combined into an effective gavity 17 How ae sound waves tapped? Acoustic waves ae tapped within a spheical shell Suface Reflection Waves eflect off the stella photosphee because of the apid decease in mass density. Hotte Deep Refaction Inceasing sound speed with depth efacts waves back towads the suface 18 9

10 How ae gavity waves tapped? In the convection zone N» 0 In the coe and adiative zones N > 0 Gavity waves popagate if w < N Tapped Gavity Waves 19 Excitation and Resonances 0 10

11 Wave Diving Sola convection in the fom of ganulation dives waves with a white spectum Hinode G-band image 1 Swedish Sola Telescope (La Palma) Sping

m = 0 Acoustic Powe Specta If the focing is ove a boad spectum, why is the obseved enegy confined to special fequencies? 3 Resonances Since we have tapped waves, we can have esonant oscillations.

12 m = 0 Acoustic Powe Specta If the focing is ove a boad spectum, why is the obseved enegy confined to special fequencies? 3 Resonances Since we have tapped waves, we can have esonant oscillations. Constuctive self intefeence amplifies the enegy fo the esonant fequencies. The esonant oscillations satisfy an eigenpoblem. Wave equation Bounday conditions = 0 = R Fist few modes of a guita sting Fist few modes of the sun 4 1

13 Fom of the Eigenfunctions Fo spheical symmetic stas the eigenfunction is sepaable -i t w (, fq,,) t = W () Y(, fq) e w Radial Eigenfunction Angula Eigenfunction The adial and angula eigenfunctions satisfy sepaate wave equations ll ( + 1) hy( fq, ) + Y( fq, ) = 0 ll ( + 1) { W} ( w, ) + W( ) = 0 Hoizontal Wavenumbe Fouth Ode w is the eigenvalue Sepaation Constant k ll+ ( 1) ll ( + 1) h = 5 Spheical Hamonics The ODE fo the angula function Y is well-known and its solutions ae functions called spheical hamonics Y = Y lm ( fq, ) m m im = (-1) C P (cos q) e f lm l Associated Legende Polynomials Fouie Modes Angula Bounday Conditions Thee ae two sepaation constants (l and m) and both have intege values. l must be an intege fo the solution to be finite at the poles m must be an intege fo the solution to be continuous at f = 0 and p Futhemoe m l, l = 1 has a tiplet m = {-1, 0, 1} l = has a quintuplet m = {-,-1, 0, 1, } 6 13

Quantum Numbes l detemines the hoizontal scale k ll ( + 1) h = hy h = -k Y m detemines the numbe of nodes in longitude l m detemines the numbe of nodes in latitude Acknowledgements: BISON goup m / l

14 Quantum Numbes l detemines the hoizontal scale k ll ( + 1) h = hy h = -k Y m detemines the numbe of nodes in longitude l m detemines the numbe of nodes in latitude Acknowledgements: BISON goup m / l detemines how close to the pole 7 Radial Ode { } W ( w, ) + k W ( ) = 0 n h n Radial Ode n The adial diffeential equation pemits a sequence of solutions with diffeent numbes of nodes. n = 0 R f mode (fundamental) The eigenfunctions ae labeled by the numbe of nodes n. Each solution has a diffeent coesponding eigenfequency w lmn. n = 1 R p 1 mode n = p mode R 8 14

A p mode and a g mode with the same values fo the quantum numbes l and m have identical angula planfoms.

15 A Vetical Velocity Eigenfunction p mode l = 0 m = 17 n = 1 m lmn ln l w (, jq, ) = W () Y ( jq, ) 9 p- and g-mode Eigenfunctions The esonant acoustic waves ae called p modes (p fo pessue). The esonant intenal gavity waves ae called g modes (g fo gavity). A p mode and a g mode with the same values fo the quantum numbes l and m have identical angula planfoms. Howeve, they have vey diffeent adial eigenfunctions. fo a spheically symmetic sta (non-otating) the eigenfunctions and the eigenfequencies ae independent of m

16 Powe Spectum f Fundamental mode (suface gavity wave) p 0 p n acoustic mode (Pessue wave) p 4 p 3 p p 1 g n gavity mode (intenal Gavity wave) f No g modes? Why ae only the acoustic modes visible? 31 Tunneling Tapped Gavity Waves g modes must tunnel though the convection zone befoe they can be obseved at the suface. Thus, they have vey low amplitude and have emained unobseved (in the sun). 3 16

Internal Thermodynamics. Results from Global Helioseismology. Closer Look. Overlapping Kernels. Lecture 24 Helioseismic Discoveries

Internal Thermodynamics. Results from Global Helioseismology. Closer Look. Overlapping Kernels. Lecture 24 Helioseismic Discoveries ASTR 7500: Sola & Stella Magnetism Hale CGEP Sola & Space Physics Lectue 4 Helioseismic Discoveies Results fom Global Helioseismology Sound speed and density invesions Rotation ate invesion Pofs. Bad Hindman