Basics of helio- and asteroseismology

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1 Basics of helio- and asteoseismology Anna Ogozałek, Supeviso: Pofesso Hiomoto Shibahashi Jagiellonian Univesity, Univesity of Tokyo Summe Reseach Intenship Pogam Octobe 20, 20 ABSTRACT This is a epot fom my poject at Univesity of Tokyo Reseach Intenship Pogam 20. Duing my stay at Todai I was leaning about basic physics, mathematical and computational tools that ae cuently being used in helio- and asteoseismology. This science is the only tool allowing us to lean about stella inteios. Fo bette undestanding of the poblem I was developing and using simple model polytope model) to constuct solutions of equations, calculate fequencies and physical pofiles of the sta. They expand and shink, what leads to gowth and op of luminosity. It was ealized that peiod of these oscillations is appoximately given by the dynamical time scale of the sta, which is popotional to sta s density. Such a simple consideation aleady gives us valuable infomation. It was soon undestood that many othe stas pulsate in fa moe complicated way than Cepheids. In many cases we deal with moe than one excited mode, so many diffeent fequencies will be obseved. The moe fequencies we see, the moe infomation about stella inteio we ae able to gain. Cuently we obseve aound 0 6 modes of Sun s pulsations. Unfotunately we ae still not able to gathe much data fom futhe stas, but still scientists managed to obseve many types of stas pulsating, what can be seen in Figue 2). Figue. Stella inteio and waves popagating inside a sta. Waves penetate egions of stat which we have no othe method of investigating. Souce: [3]. Intoduction Helio- and asteoseismology ae sciences that investigate pulsations of Sun and othe stas. These oscillations ae studied in ode to find what mechanisms ae esponsible fo exciting them in diffeent kinds of stas, but moe impotantly they ae studied to lean about stella inteios. As is shown on Figue ) the waves ae tapped in diffeent inne pats of sta poviding us with impotant infomation. Since huned yeas ago astonomes obseve stas whose luminosities vay peiodically. But only ecently it has been established that these changes ae due to some inne mechanisms. Fist studied pulsating stas wee Cepheids. anna.ogozalek@uj.edu.pl 2 Constucting simple model 2. Basic equations Stas ae gaseous objects, which means that we can apply theoy of fluid dynamical systems in ode to descibe them. Basic equations ae adapted fom hyodynamics. All the consevation laws ae also valid. Let us stat with mass consevation, which can be expessed as: ϱ + ϱv) = 0, ) t whee ϱ is density and v is mass element s vecto velocity. Equation is tue when no additional souces of mass ae pesent. Next we adapt simple equation of motion in the fom of: v t + v )v = p + Φ, 2) ϱ p denotes pessue and Φ is the gavitational potential. Φ has 20

2 A. Ogozałek: Helio- and asteoseismology 2 Figue 2. The Hetzspung Russell diagam shows goups of stas with simila popeties. Left-hand side gaph shows classical pulsating stas that have been known to pulsate since centuies. Right-hand side figue pesents cuent state of known oscillating stats. to satisfy Poiosson equation G is the gavitational constant): 2 Φ = 4πGϱ. 3) One can easily notice that by assuming t = 0 and v = 0 hyostatic equilibium is obtained. All popeties of gas used above density, pessue o othe themodynamic quantities) and instantaneous velocity ae functions of position vecto and time t. This appoach equies fixed coodinate system. It is called Eule s appoach and it coesponds to what stationay obseve can see. Anothe desciption of the situation, Lagangian appoach, follows gas element descibed unambiguously by its position in equilibium state 0 ). Hyodynamical desciption of a sta is difficult to solve, even numeically. Consideing stella oscillations some appoximations can be adapted. Pulsations amplitudes ae small compaed to typical stella time-scales, so we can teat them as a petubation aound equilibium state. Accoding to what was aleady said about diffeent appoaches, we can denote fo any physical quantity f its Euleian and Lagangian petubation accodingly f 0 coesponds to the equilibium state): Petubations ae elated by: f E, t) = f 0) + f, t) 4) f L 0, t) = f 0 0 ) + δf 0, t). 5) δf 0, t) = f, t) + ξ )f 0), 6) whee ξ is called displacement vecto, satisfying: = 0 + ξ. 7) Set of equations we obtained so fa is still not sufficient to solve the poblem of descibing stella pulsations. Let us conside case, in which all heat tansfe mechanisms can be neglected, so motion occus adiabatically. Pessue and Figue 3. Relation of dispesion. Two asymptotes ae depicted as dashed lines. Fo wave to popagate its fequency has to be lage than both N and k h c, o smalle than both of them at the same time. Shaded aeas coespond to kz 2 > 0. Top aea is whee pessue waves popagate p-modes), bottom one is whee gavitational waves popagate g-modes). density ae then connected by adiabatic elation: δp p 0 = γ δϱ ϱ 0. 8) γ is constant fo given gas, γ = 5 fo ideal gas. 3 Applying petubation and adiabatic elation to equations ), 2) and 3) as well as using theoetical model specific foms of ϱ, p and othe themodynamic quantities) will make it possible to descibe pulsations fo given model). 2.2 Relation of dispesion Let us now conside vey simple model of plane isothemal atmosphee with constant gavity foce acting along z axis with acceleation g). Then pessue and density will change as: p 0, ϱ 0 exp z H ) 9) whee H is the constant scale height. Hyodynamics equations with adiabatic appoximation lead to petubations of the fom of: v, p, ϱ exp z ) expk x), 0) 2H whee k is a wave vecto and σ is a fequency. In this situation we can sepaate wave into hoizontal coodinates and pat that popagates along the z axis, k = k h, k z). Using equations pesented in pevious section we can deive the dispesion elation between fequency and wavenumbes: σ 2 σ 2 ac)σ 2 k 2 h + k 2 z)c 2 σ 2 + k 2 hc 2 N 2 = 0. ) is the sound speed, σac = c Hee c 2 = γ p ϱ cut-off fequency and N 2 = g d ln ϱ 2H d ln p is the acoustic ) is the Buntγ Väisälä fequency. Fo given σ and k h above elation detemines kz, 2 what allows us to descibe vetical popagation of wave. If kz 2 > 0

3 A. Ogozałek: Helio- and asteoseismology N.5 0 Figue 4. Pue sound waves and thei popagation egion. N.5 0 k h c Figue 6. Thee layes with diffeent physical popeties: I - pink solid lines, II - blue dotted lines, III - puple dashed lines. Shaded aea epesents waves tapped in egion II. Thee ae still zones whee no waves can popagate. 2.3 Popagation egions N Figue 5. Pue gavitational waves and thei popagation egion. wave will popagate. On the othe hand, if k 2 z < 0 we have evanescent wave, so it will not popagate. Situation is pesented on Figue 3). As we can see, we have two egions whee k 2 z is positive, maked with lettes P and G. Let us fist conside simplified case, depicted on Figue 4) whee thee is no gavity. Then σ 2 = c 2 k 2 = c 2 k 2 z + k 2 h), so we have sound acoustic) waves, whee estoing foce is pessue. They ae called p-modes. When we have vey solid mateial, meaning c 2 no acoustic waves, as on Figue 5)), we will have only gavitational buoyancy) waves, g-modes, which occu unde gavity and buoyancy is the estoing foce. As we can see fom Figue 5) the condition fo g-modes to be pesent is positive Bunt- Väisälä fequency, N 2 > 0, what is consistent with adiabatic condition fo pesence of buoyancy oscillations. In conclusion, the aeas whee gavitoacoustic waves can popagate ae the ones whee given fequency is eithe lage than both N and ck h o smalle than both of them as shown on Figue 3)). N Waves ae usually tapped in some egion in the sta. To undestand this situation let us get back to the simple model used in pevious section. Now we conside thee egions I, II and III with diffeent physical popeties, so diffeent values of N and c, what is shown on Figue 6). We assume, that c III > c I > c II and N I > N III > N II. As can be seen on Figue 6) each egion of gaph vay in its popagation popeties. Some aeas ae still zones whee thee is no popagation. Some waves can popagate in only one of two of consideed layes. It is clea now that some waves ae tapped inside specific aeas inside the sta. Moving fom plane paallel laye to sphee, we know that the multiple of the hoizontal wavelength λ h at given adius has to be equal to 2π fo wave to be a standing wave. In othe wods: λ h l = 2π, l = 0,, 2, ) Knowing that k h = 2π λ h, we can wite: c 2 kh 2 = l2 c 2. 3) 2 As mentioned befoe, both c and N depend on stella stuctue. They will diffe fo adapted model. Stating hee we will be doing ou calculations using polytope model, which assumes the pessue-density elation to be: p) = Kϱ + n ), 4) whee n is called polytopic index. We will use n = 3 which, to some extent, descibes sola-type stas. Density, pessue and tempeatue pofile calculated fo polytope ae shown on Figue 7). Using discussed model we can now plot ck h and N against adius and study popagation zones fo diffeent fequencies, as shown on Figue 8). We have two diections: along the adius and hoizontal, so we will have two integes l, m. Petubations we conside will be popotional to spheical hamonics: Yl m Θ, ϕ) = Pl m cos Θ) expimϕ), 5) whee Pl m cos Θ) is the associated Legene polynomial.

4 A. Ogozałek: Helio- and asteoseismology 2.4 Radial pulsations Let us estict discussion to only adially symmetic motions fo now, which means that we put l = 0. We divide sphee into shells and use the Lagangian petubation. Mass m is chosen as an independent coodinate mass inside the sphee of a adius 0 in the equilibium state). Applying petubation to adius and density leads to: 0.8 fc D δm, t) m, t) = 0 m) + δm, t) = 0 m) + 0 m) R Figue 7. Density blue), pessue puple) and tempeatue pink) functions inside sta, fo polytope with index n = 3 in units of thei maximum value. δ%m, t) %m, t) = %0 m) + δ%m, t) = %0 m) + %0 m), 6). 7) In ode fo petubation to be small following must be tue: δm,t), δ%m,t). 0 m) %0 m) Now we eplace adius and mass in mass consevation equation and equation of motion with thei petubed values. Only tems with the fist o lowe ode of δm,t) and δ%m,t) 0 m) %0 m) will be taken into account. Using adiabatic elation and stating ζ = δ0 we obtain set of petubed diffeential equations: dζ = 3ζ + δp γ P 4ζ + σ2 3 δp ζ+ GM P 8) d δp d ln P P = ) whee M is the total mass of a sta. To solve these diffeential equations we need bounday conditions. Since ou poblem has adial symmety, the cente of the sta cannot move. Thus: δ = 0 fo = 0. 20) R Figue 8. Popagation egions fo polytope model. N is depicted as pink line, while puple line coesponds to l = and blue one to l = 2 sound speed. Second bounday condition is also vey natual. We define suface as a place, whee no foce is applied. Hence: δp = 0 Figue 9. Spheical hamonics inside sta fo exemplay values of l, m. Shaded aeas show the outwad motion, white aeas inwad. = R, 2) R being the total sta adius. Poblem equies some kind of scale. We adapted following condition: ζ= Looking at the stat at any adius) l will be the total numbe of nodal lines sufaces which ae not petubed) and m will denote numbe of lines going though the pole. That means thee will be aeas moving in diffeent diections between nodal lines while pulsating, as is shown on Figue 9). fo fo = R. 22) We see that we deal with second ode diffeential opeato which is of Stum-Liouville type, whee σ 2 is the eigenvalue: Lζ) = σ 2 ζ, what gives us additional infomation: opeato is Hemitian and its eigenvalues and eigenfunctions ae eal minimum value of σ 2 exists its coesponding to quantum mechanical gound state) eigenfunctions ae othonomal base constucted of eigenfunctions is complete. We can see the similaity with quantum mechanical wave functions in infinite dwell. Using numeical methods eigenvalues and coesponding eigenfunctions wee calculated fo polytope model, what is shown on Figues 0) and ). Quantum mechanical gound state coesponds to fundamental tone depicted on Figue 0) with its ovetones. Exemplay esults ae shown in Table ).

5 A. Ogozałek: Helio- and asteoseismology Figue 0. Radial pulsations: eigenfunctions fo the ζ eigenfunctions. Fist mode puple) has no nodal points, fist ovetone pink) has one node and so on R Figue. Thid pink) and foth puple) ovetones as solutions to adial pulsations poblem. 2.5 Nonadial pulsations To make moe geneal case we have to obtain equations fo nonadial oscillations fo any given l). The displacement vecto ξ will be oscillating, so ξ expiσt). Taking into account that by definition v = dξ, dt we can wite: v = iσξ. Mass consevation and equation of motion take fom of: R ϱ + ϱξ) = 0. 23) σ 2 ξ + ϱ p + Φ + ϱ Φ = 0 24) ϱ The second one is a vecto equation and it can be divided into adial and hoizontal components: σ 2 ξ + ϱ 0 p + Φ + ϱ ϱ 0 Φ = 0 25) σ 2 ξ h + h p ϱ + Φ ) = 0. 26) By combining equations 23) and 26) we can get id of ξ h : ) δϱ ϱ ξ ) + p σ 2 2 h ϱ + Φ = 0, 27) whee: 2 h = 2 sin 2 Θ sin Θ Θ sin Θ ) ) ) Θ ϕ 2 Knowing that dφ = g and using adiabatic elation we obtain set of two diffeential equations: 2 ϱ p + ϱgp γp 2 ξ ) + γ ) + N 2 σ 2 )ξ Φ = 0 29) ) d ln p ξ + p γ p + p σ 2 2 h p + Φ = 0. 30) We will assume that Φ 0, which is a eally good appoximation. Solution can be sepaated into adial and agula pat. As was mentioned befoe, the angle-dependent pat of the solution will be expessed as a spheical hamonics Y Θ, ϕ), which is an eigenfunction of diffeential opeato 2 h: 2 hy Θ, ϕ) = Λ 2 Y Θ, ϕ). 3) Taking that into account and putting Λ 2 = ll + ) equation 30) can be ewitten as: 2 2 ξ ) + γ d ln p ξ + p ll + ) p + = 0. 32) γ p σ 2 ϱ Equations 29) with Φ = 0) and 32) ae Hemitian and tend to a Stum-Liouville type with σ 0 o σ. Fo given l a set of eigenvalues σ nl can be calculated, whee diffeent l will coespond to diffeent modes. This has been done fo polytope model and esults will be discussed late in next section. Fist few obtained values ae pesented in Table 2). Table. Exemplay esults of solving Stum-Liouville poblem fo adial oscillations fo polytope model. n σ 2 n σ Asymptotic theoy As was aleady mentioned couple of times befoe, thee is an analogy in mathematical desciption of stella pulsations and quantum mechanics. Waves ae tapped within some egions of sta, just like paticle in the potential well. Let s say that well is eally steep in points a and b, so eigenfunction is tapped between these points. That means evey mode, which is eally a standing wave, has to have intege o half-intege numbe of wavelengths between a and b. We want to obtain numbe n of diffeent wavelengths in [a, b].

6 A. Ogozałek: Helio- and asteoseismology 6 Table 2. Exemplay esults of solving Stum-Liouville poblem fo nonadial oscillations fo polytopic model. Note that not all values of n ae always pesent. l n σ 2 l n σ Figue 3. Waves with diffeent spheical numbe l ae eflected at diffeent adii inside a sta. The smalle l is, the deepe wave can penetate stella inteio. Souce: [4]..5 0 k h c Figue 2. Asymptotic behaviou of fequencies. Fo small n we have behavious as the pink line. Fo lage values of n we have blue dashed lines, pue p-modes. Note that k is a function of, so we have to pefom integation: b a k = n + )π. 33) 2 This is tue fo both waves in stas and quantum mechanics, whee it is called Sommefeld s quantization condition. Remembeing that k = σ in the case of acoustic wave with c l = 0, we can wite: σ = πn + b ) 2 ) 34) c Integal in this equation fo big values of n is constant. The asymptotic elation fo fequency is obtained: σ = n + )σ0. 35) 2 What we see is that the fequencies ae equally spaced. If we ecall the dispesion elation gaph, we notice that high fequencies will coespond to p-modes, what is depicted on Figue 2). We know that: k 2 c 2 = σ 2 and k 2 = kh 2 + k, 2 so we can wite in the case of non-adial l 0) p-modes: nπ = R i σ 2 a ll + ) c2 2 ) 36) whee i is the inne tuning point. Note that the inne tuning point depends on the spheical numbe l. This is tue fo waves which ae excited close to suface and go into the inne pats of sta, but because of the gadient of physical popeties they ae eflected as fo example light wave in the deset, when thee is lage tempeatue gadient, is eflected ceating optical illusions). Situation is shown on Figue 3). 2.7 Fowad/invese poblem Fom pevious sections we see how, given a specific physical model of a sta s inteio, can we obtain fequencies of waves popagating inside a sta. Having constucted model usually means developing sound speed pofile, as it is function containing all the vital infomation of stella inteio as it is a atio of pessue and density pofile). Then obtained fequencies ae compaed with obsevational data and depending on how well they match the model is being adjusted to descibe the physical popeties in a bette way. This appoach is called fowad poblem. Fowad poblem: c 2 ) model σ 2 n,l model Thee is one moe method of studying stella pulsations. We can ewite equation 36) as: n + )π R ) 2 ll + )c2 = σ 2 σ 2 c. 37) i Left-hand side of above equation is a function of n σ, while ight-hand side is a function of l σ : n+ 2 σ = f σ ). The elation l between these two is called Duvall s law and is an empiical law. We can plot obsevational data σ 2 n,l) and fit a function as is shown on Figue 4), whee calculated peviously fequencies wee teated as obsevational data. We can see that

7 A. Ogozałek: Helio- and asteoseismology n Figue 4. Duvall s law: empiical elation between fequencies, thei ode and spheical numbe with fitted function. function is fitted petty well. We will call the fitted function F. Equation 37) can be tansfomed into Abel-type integal equation of the fom: R = exp[ 2 π whee x = σ ll+) c)/ cr)/r x 2 ) 2 df c) 2 dx dx] 38). Fom above fomula we can obtain sound speed pofile c). Fo diffeent values of we do the integal and obtain coesponding adius. Of couse we should fist estimate what values of c should be used. Moeove we have to emembe that we should only use function F Duvall s law) in the inteval that we have obsevational data in, because in othe egions it is not valid. This way fom set of fequencies fom obsevations we can ecove sound speed pofile. This appoach is called the invese poblem and is somehow opposite to the one peviously discussed. c Figue 5. Measuements of matte s velocity on Sun s suface. We can see that some pats ae moving outwad while othes ae moving inwad. Souce: [5]. Invese poblem: σ 2 n,l obs c 2 ) obs 3 Pactice Main souce of obsevational data fo Sun s oscillations is measuing the velocity of matte on its suface using the Dopple effect. Exemplay map of Sun s suface is shown on Figue 5). Cuently aound 0 6 sola modes of pulsation have been detected. The only eason fo such a detailed esults is the Sun s close poximity. As fo othe stas, simila method is applied but with fa wose esults. Fo futhe objects the changes in amount of light in diffeent wavelengths is analysed in seach of peiodicities. Sola oscillations that ae obseved have peiods aound 5 minutes, which ae shote than hou fundamental adial mode fo the Sun the same mode that is obseved in Cepheids). They ae plotted on Figue 6). These modes ae acoustic modes caused by pessue fluctuations. Gavitational modes have not yet been obseved, though they ae eally desied. As can be seen in Figue ), gavitational Figue 6. Fequency of sola oscillations and thei spheical numbe. The ed aeas coespond to bigge amplitudes, blue aeas to smalle ones. Souce: [6].

8 A. Ogozałek: Helio- and asteoseismology 8 modes contay to acoustic modes give us infomation about the vey cente of the Sun. Though thee ae still many questions in asteoseismology that have not yet been answeed, thee ae some eally valuable esults. Geneally fo stas which pulsate we ae able to detemine thei age, mass and evolution status. This leads to significant impovements in theoies concening stella evolution. Bette esults wee obtained fo Sun, as we have significantly much moe data. Detailed Sun s sound speed pofile have been obtained along with its tempeatue pofile. Moeove otation pofile has been established what gives impotant insight into dynamo theoy mechanism that geneates magnetic field) and explained why sunspots move faste than Sun suface, which is because they ae actually deepe layes of Sun s inteio moving with geate velocity. In addition infomation about Sun s equation of state, opacity, depth of convection zone, unde-suface stuctues below sunspots and many moe has been acquied. I hope that I was able to explain why asteoseismology is eally beautiful science and also eally valuable one. Fo foeseeable futue it is ou only tool to look inside stas, what is eally impotant in developing all sta-elated theoies in many astophysical fields. ACKNOWLEDGEMENTS I am eally gateful fo the oppotunity to do intenship at Univesity of Tokyo, one of the best univesities in the wold. I would like to deeply thank my supeviso, Pofesso Shibahashi fo all the time and effot put into teaching me. I would also like to thank Ms. Soeda and OIP office fo oganizing such a wondeful pogam and all my UTRIP fiends fo making this one of the best summes in my life. Refeences [] Wasabuo Unno, Yoji Osaki, Hioyasu Ando, Hideyuki Saio, Hiomoto Shibahashi Nonadial Oscillations of Stas, Univesity of Tokyo Pess, 989 [2] Cal J. Hansen, Steven D Kawale, Viginia Timble Stella Inteios - Physical Pinciples, Stuctue, and Evolution, Van Nohand, 995 [3] [4] [5] [6]