Propagation and Evanescence. Wave Cavities and Propagation Diagrams. Wave Cavities. Propagation Bands in the Sun. Lecture 23 Helioseismic Inversions

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1 AST 75: Sola & Stella Magetism Hale CGEP Sola & Sae Physis Letue 3 Helioseismi Ivesios Pofs. Bad Hidma & Jui Toome Letue 3 Tues 6 A 3 zeus.oloado.edu/ast75-toome Wave Cavities Wave avities ad oagatio diagams esoaes ad Eigefutios Ifomatio Cotet of a Mode s Fequey Eigeoblem ayleigh quotiet Sesitivity Keels Helioseismi Ivesios LS Ivesios egulaizatio Wave Cavities ad Poagatio Diagams Poagatio ad Evaesee d y + k () y = The Helmholtz equatio has two diffeet tyes of behavio: k () z > k () z < Poagatio Osillatoy solutios Evaesee Exoetially deayig o gowig y Evaeset k () z < Osillatoy Evaeset k () z > k () z < 3 4 Wave Cavities The waves ae lagely ofied to the egio whee they oagate. This aea is alled a wave avity. Wave Cavity Poagatio Bads i the Su y k () z < k () z > k () z < t The boudaies of the wave avity ae whee k z (z) =. t k ( t) = 5 6

2 - ad g-mode Poagatio Bads Thee ae two fequey bads ove whih waves a oagate. These bads vay as a futio of adius, w - w () æ N () ö () = + k h() ç - () çè w Note that the loal disesio elatio is quaati i w. Hee thee ae two oots (all them w + ad w -). We must be able to wite the loal disesio elatio as follows () = é w - w () é w w () ê + úê - - ú w () Poagatio ous whe k >. w > w- & w+ w < w- & w+ 7 High-Fequey Aousti Waves Low-Fequey Iteal Gavity Waves Citial Fequeies The itial fequeies w + ad w - may be obtaied by solvig the quaati equatio that esults fom the disesio elatio whe k =. w - w () æ N () ö () = + k h() ç - = () çè w ø w k æ h w k ö h w = + + kn h ç - çè If N is vey small omaed to w +» w + k h + N k h ( h ) w N N 4 k 4 4 -» + w L = k h 8 Poagatio if w > w + w L Aousti Waves w < N Lamb Fequey w L º k h Iteal Gavity Waves Si Hoae Lamb ( ) Lamb was a Bitish alied mathematiia who authoed seveal ifluetial books o lassial hysis (still i it) Hyodyamis (879) Dyamial Theoy of Soud (9) w + ad w - withi the Su w L l = l = Lamb Fequey l = 3 ll ( + ) wl º k h = w + I am a old ma ow, ad whe I die ad go to heave thee ae two mattes o whih I hoe fo elightemet. Oe is quatum eletodyamis, ad the othe is the tubulet motio of fluids. Ad about the fome I am athe otimisti. - Si Hoae Lamb, 93 N w - w 9 Zoes of Poagatio The boudaies of the wave avities ae seified by w = w l = Evaeset Waves Poagatig Aousti Waves Poagatio Bads I Evolved Stas Poagatig Gavity Waves

3 ed Giat a UMa wl m l = 4.5 M Fequey N Mixed Modes Aousti egime But He oe w N w < N, wl w > N, wl H buig shell Dee ovetive eveloe Mixed modes Gavity Wave egime w + adius 3 m = mea moleula weight This is a shemati examle of a mode that has both -mode ad g-mode haateistis. It has osillatios i both avities. 4 Limitig Disesio elatios High-Fequey Limit ( modes) w > k h Fo l > N w = k L Thus fo aousti waves we a igoe the buoyay fequey. h 5 Set the buoyay fequey to zeo i the loal disesio elatio w - w () æ N ö Total waveumbe = + k h - ç çè w k = k h + k z w - w () ()» -k h() w» k () () + w() () 6 Low-Fequey Limit (g modes) Fo l > w < N N w = k L Fo iteal gavity waves we a igoe tems ivolvig the ivese soud seed. h esoaes Set the soud seed to ifiity i the loal disesio elatio w - w () æ N ö = + k h - ç çè w æ N () ö () kh() kh()» - ç w» N () çè w ø k () 7 8 3

4 y esoae Coditio Fequey Sequee Diveges at = Fiite at = Fiite as Diveges as g g g3 w 7 w 5 w 3 w The adial esoae oditio is that the solutios is fiite at the ete of the sta ad that the eegy is fiite. This last equiemet equies that the solutio deays (istead of gows) with height above the hotoshee. g 4 g 5 ( = 5) 9 The sequee of modes is obseved i the su. The g modes would be vey low fequey ad have yet to be obseved. modes Eigefutios l =, = 3 = 33 mhz l =, = -5 = mhz g modes 4 3 l =, = 7 = 3375 mhz l =, = - = 3 mhz f t l = 6, = = 334 mhz t g modes? l = 4, = -9 = mhz Ifomatio Cotet of a Mode Fequey Eigevalue Equatio emembe last time I showed that the su s osillatios obeyed a Helmholtz equatio i adius, ad give bouday oditios fomed a eigevalue-eigefutio oblem. d y + k () () y = w - w() æ N () ö () = + k h() () ç - çè w If we take the high-fequey limit ( modes), the we a exess this diffeetial equatio i a stadad eigeoblem fom d y é w - w() + k h() - y() = ê () ú 3 4 4

5 d y éw - w() + k h() - y() = ê () ú Multily though by the squae of the soud seed ad eaage () d () h () () í - éw + k ýy () + w y () = ê ú î Defiig the diffeetial oeato, d º í -( w + kh ) ý î we a wite this ODE i the fom of a stadad eigevalue equatio. y + w y = o Note: is a diffeetial oeato that deeds o the soud seed ad the aousti utoff fequey (o desity). y l + wlyl = 5 Stum-Liouville System y + wy = The oeato is a Stum-Liouville oeato. Theefoe, we immediately kow that its eigefutio ae othogoal with a weight futio. The othogoality itegal is ove the etie iteio of the sta, fom its ete = to its sufae =. Itegate ove the etie sta ò W () y() y() = d W () = Mg () Allows the omalized eigefutios to etai thei hysial uits. Do l idex fo otatioal simliity. M = Stella Mass = Stella adius g() = Sufae gavity 6 What use ae the Eigefequeies? Joh William Stutt (84-99) aka Lod ayleigh The eigefequeies ae a weighted satial aveage of the iteio oeties of the sta. To see this we will omute the ayleigh Quotiet. The ayleigh Quotiet is a useful quatity that allows oe to ove all sots of mathematial ieties about the eigevalues of a diffeetial equatio. 7 Lod ayleigh had his figes i eveythig hysial ad mathematial. His ame aeas ove ad ove agai i a vaiety of otexts. Bitish Physiist Aoustis (Soud loalizatio i huma heaig, 9) Otis (ayleigh Citeio, ayleigh-jeas law 9, ayleigh satteig why is the sky blue?) Feomagetism (ayleigh Law 887) Chemisty (disoveed the elemet ago with amsay, Nobel Pize 94) Hyodyamis (ayleigh-béad ovetio 96, otatig Couette flows 88 & 96, ayleigh-taylo istability 883) 8 Comutig the ayleigh Quotiet To omute the ayleigh Quotiet, we multile the ODE by a eigefutio ad the weight futio ad the itegate ove the domai (if you kow what this meas, we take the ie odut i Hilbet sae of the ODE with a eigefutio). The solve fo the eigevalue. y + wy = Fo oveiee, o the hoizotal quatum umbe l Wy é y + wy ò ê ú = Itegate ove the etie sta eaage tems wòwy =-òwyy wòwy =-òwyy The itegal o the left-had side is ou othogoality itegal ò Wyy = d =-òw w y y d =- Wy ( w kh ò í - + ) ýy î Theefoe, =- é ò ê ( - h )- w Wy y k y wwy ú d º í î - + ( w kh ) ý 9 3 5

6 Sesitivity Keels =- é ò ê ( - h )- w Wy y k y wwy ú This a be ewitte i a moe illustative fom by defiig keel futios. = é ò ê + w w () () () () ú ( yy hy ) () º-W -k () º Wy Keel fo the soud seed Keel fo the ut-off fequey The fequeies ae a weighted aveage of the oeties of the stella iteio. The weights deed o the eigefutios. Theefoe, the fequeies ae a aveage of the stella oeties oly withi the aousti avity fo that mode. 3 Petubative Tehiques The ayleigh Quotiet is aely used dietly i atie. Whe tyig to use measued fequeies to dedue the iteal oeties of a sta, the eigefutios (ad theefoe the keels) ae t kow util you have aleady solved the oblem. Beause of this diffiulty, etubatio theoy is usually emloyed. Assume that a stella stutue afiioado has ovided us with a efeee stella model, whih does a good ob of editig the eigefequeies (it s a good model but ot efet). efeee model ovides both atmoshei ofiles ad modes () = () The tilde idiates the y + w y = w() = w () efeee model Wyy ò = d 3 Petubatios Sesitivity Keels Assume that the tue stella ofiles of soud seed ad utoff fequey ae small etubatios fom the efeee model (emembe that the efeee model is a good model). () = () + d () = + w () w () dw () We a the safely assume that the tue eigefutios ad eigefequeies a be teated etubatively as well. = + w w dw y () = y () + dy () = + d Note: The eigefequeies ad eigefutios fo the efeee model ae kow. The tue eigefequeies (ad hee the etubatio to the fequeies) a be measued. What is t kow is the atmoshei etubatios d ad dw. 33 Usig stadad etubatio theoy, whee oe assumes that diffeees betwee the efeee model ad the eal sta ae small, oe a deive the followig itegal equatio dw é d dw = ò () + () w ê ú ( yy hy ) () º- W -k () º w y W Sesitivity Keel fo the fatioal etubatio i the soud seed Sesitivity Keel fo the fatioal etubatio i the utoff fequey See the ed of this letue fo a full deivatio 34 Mode Keels ò d æ () () ö ç dw = + çè ø =, = () Positive Helioseismi Ivesios () Positive & Negative These ae keels fo soud seed (squaed) ad desity

7 Couled Itegal Equatios Coside a ase whee we wish to igoe the effets of ioizatio. (I m doig this solely fo simliity of agumet.) I suh a situatio, the desity etubatio is liealy elated to the soud seed etubatio. Fequey obsevatios a the be haateized as follows d dwl = ò l() s l Obsevatioal Data Sesitivity Keels (efeee Model) Ukow! Obsevatioal Eo Ou goal is to ivet this set of ouled itegal equatios to obtai the iteal stutue. egulaized Least Squaes (LS) Ivesio We ty to desibe the iteal soud seed etubatio with a aameteizatio. d = f ( ; a) a = { a } The a ae the model s fee aametes ad the aameteizatio ould be a exasio i a set of basis futios (suh as sies, Chebyshev Polyomials, et.). The aametes ould also be the value of the soud seed etubatio d at a edetemied gid of adii. The etubed soud seed at adii i betwee gid oits might be defied though iteolatio (slie, biliea, et.). = å a T () f d a = = f f () = SPLINE(, a ; ) Goodess of Fit I a LS ivesio the aametes a ae hose to miimize the goodess of fit (i a least squae sese). Sum ove Measued Modes é ( a ) = å dw ; l -ò l f l s ê l ( a ) Paametes (Ukow) ú egulaizatio The solutio obtaied is ot esued i ay way of beig smooth. Oe ould (ad will) get a solutio whee f ossesses may exteme wiggles with adius. The way to etify this oblem is to ilude a tem i the goodess of fit that ealizes wiggly futios. This is alled egulaizatio. Oe a hoose ay ealty (o egulaizatio) tem that aomlishes this, but i atie oe ofte takes é æ f ö ( a) = dw ( ; ) l l f l å - ò a + ò l s ê ú ç è l Estimate of the measuemet eos Data (Obseved) Sesitivity Keel (efeee Model) Fittig futio 39 Tade-off Paamete The tade-off aamete l a be feely hose. The highe its value, the smoothe the solutio. Howeve, the highe the value, the highe the eo i the solutios as well (hee tade-off). 4 Bous Featue Petubatio Theoy 4 4 7

8 Petubatio Theoy What follows is a demostatio of how etubatio theoy a be used to alulate sesitivity keels fo a give stella efeee model. efeee Model The ayleigh Quotiet is aely used dietly i atie. Whe tyig to use measued fequeies to dedue the iteal oeties of a sta, the eigefutios (ad theefoe the keels) ae t kow util you have aleady solved the oblem. Beause of this diffiulty, etubatio theoy is usually emloyed. Assume that a stella stutue afiioado has ovided us with a efeee stella model, whih does a good ob of editig the eigefequeies (it s a good model but ot efet). 43 efeee model ovides both atmoshei ofiles ad modes () = () The tilde idiates the w() = w () efeee model y + w y = The eigefutios of the efeee model fom a omlete othogoal set. Wyy = d ò 44 Petubatios Assume that the tue stella ofiles of soud seed ad utoff fequey ae small etubatios fom the efeee model (emembe that the efeee model is a good model). () = () + d () = + w () w () dw () We a the safely assume that the tue eigefutios ad eigefequeies a be teated etubatively as well. = + w w dw y () = y () + dy () = + d Note: The eigefequeies ad eigefutios fo the efeee model ae kow. The tue eigefequeies (ad hee the etubatio to the fequeies) a be measued. What is t kow is the atmoshei etubatios d ad dw. 45 Petub the Oeato emembe the oeato deeds o the soud seed ad aousti utoff fequey. = í - + î d h ý ( w k ) æ ö d = d ç - -dw çè d ç k h d = í -( w + kh ) ý î d = íd - dw + k d î d h ý The etubed oeato deeds liealy o the etubed themodyami vaiables (ukows) 46 Petubatio Equatio Petub the wave equatio ad kee oly the tems that ae liea i the etubatio ( w ) y + = efeee Model Eigeoblem ( w ) + = Lieaized ODE fo etubatios ( ) ( ) + w dy + d + dw y = Kows y Ukows emembe, that the etubed oeato otais the atmoshei etubatios that we would like to dedue. æ d ö d = d k ç - h -dw çè Exasio i efeee Eigefutios The eigefutios of the efeee model fom a omlete othogoal set. Theefoe, we a eeset ay easoable futio as a liea ombiatio of eigefutios. Exad the etubed eigefutios i the eigefutios of the efeee model dy () = å A y () Iset this exasio ito the ODE fo the etubatios ( ) ( ) + w dy + d + dw y = ( + ) å A = -( + ) w y d dw y

9 ( + ) å A = -( + ) w y d dw y Ie Podut i Hilbet Sae Tasfe the oeato o the left iside the summatio ( + ) = -( + ) å A w y d dw y Use the eigevalue equatio fo the efeee model y w y =- ( - ) = -( + ) A w w y å d dw y Multily by W y ad itegate ove the etie sta å A ( w - w ) òwy y = - òwy ( d + dw ) y (- + ) = -( + ) A w w y å d dw y ( - ) = -( + ) A w w y å d dw y 49 Use the othoomality of the eigefutios of the efeee model to elimiate the LHS ad ewite the tem with etubed fequey. W =-ò dw y d y Wyy = d ò 5 Petubed Fequey W =-ò dw y d y I the alae of quatum mehais, the etubed fequey is ootioal to the diagoal matix elemets of the etubed oeato dw y d y d =- =- emembe ou evious deivatio of the etubed oeato d =- - h - ò ê ç çè dw Wy é d æ k ö y dw y ú æ d ö d = d k ç - h -dw çè 5 We a ow defie sesitivity keels fo the fatioal etubatios i the atmoshei ofiles. dw Sesitivity Keels d =- - h - ò ê ç çè dw Wy é d æ k ö y dw y ú é d dw = ò () + () w ê ú ( h ) () º- W y y -k y () º w y W Sesitivity Keel fo the fatioal etubatio i the soud seed Sesitivity Keel fo the fatioal etubatio i the utoff fequey 5 9