September 28 October 5, Monday. 8. Solar oscillations

Transcription

1 Septembe 8 Octobe 5, Monday 8. Sola oscillations

2 Outline Obsevations Theoy of p-, g-, and -modes. Excitation mechanisms.

3 Obsevations of Sola Oscillations Sola oscillations wee discoveed in 1960 by Leighton, Noyes and Simon by obseving Dopple shift in spectoheliogams. MDI single Dopplegam minus an aveage sola velocity image obseved ove 45 minutes eveals the suface motions associated with sound waves taveling though the Sun s inteio. The small scale light and dak egions epesent the up and down motions of the hot gas nea the Sun s suface. The patten falls off towads the limb because the acoustic waves ae pimaily adial.

4 The otation speed of the sola suface is km/s.

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7 A sample of sola oscillations obseved as a function of time and position on the disk Velocity vs. time (hoizontal axis) and position on the sola disk (vetical axis). The distance between the cuves fo adjacent points is equal to a velocity of 400 m/s.

8 Randomly excited sola oscillations

9 Seismic esponse to sola flaes (sunquakes) High-enegy flae paticles heat the sola chomosphee geneating a shock popagating downwad and hitting the suface.

10 Enhanced images of the flae ipples on the Sun s suface Compae with wate ipples

11 Time-distance diagam of the flae seismic esponse calculated by aveaging the wave font ove 360 degees The popagation speed of the seismic wave: V=δ(distance)/δ(time) inceases with time fom 10 km/s to 100 km/s because the sound speed inceases with depth.

12 Popagation of acoustic waves on The wave font on the suface acceleates because it is fomed by acoustic waves popagating though the sola inteio whee the sound speed is highe. the Sun The ay paths ae pependicula to the wave fonts.

13 The idea of helioseismology Measue tavel times τ o esonant fequencies ω of acoustic waves and infe the intenal popeties, e.g. c s () - sound speed

14 l=0, m=16 Nomal Mode of Sola Oscillations

15 Basic popeties of oscillations Behave like spheical hamonics: P lm (cos θ) cos(m φ - ω t) k h = π / λ h = [l(l+1)] 1/ / Fom esonant wave pattens in the inteio nomal modes

16 l=0, m=16 Nomal Mode of Sola Oscillations

17 Powe Spectum Velocity of oscillations v( x, y, t) can be epesented in tems of its Fouie components: i( k x+ k y+ ωt) x y a( k, k, ω) = v( x, y, t) e dxdydt, x y whee k x and k y ae components of the wave vecto, ω is the fequency. The powe spectum is: P k k a a a * ( x, y, ω) =, whee is complex conjugate. If thee is no pefeence in the diection of the wave popagation then P depends on two vaiables, the hoizontal wavenumbe In the spheical coodinates, θ, φ : m whee Y ( θ φ ) l k = k + k, and fequency. h x y m iωt a( l, m, ω) = v( θ, φ, t) Y ( θ, φ) e dθdφdt, l, is a spheical hamonic of the angula degee l and angula ode m. Degee l gives the total numbe of node cicles on the sphee; ode m is the numbe nodal cicles though the poles. The powe spectum in this case is: P( l, m, ω) = a a. Fo a spheically symmetical sta, P depends only on l and ω. The powe spectum is degeneate with espect of angula ode m. In this case the analog of the hoizontal wavenumbe is: k h l( l + 1) =, whee R is the sola adius. R

18 Oscillation powe spectum Spheical hamonic tansfom oscillation signal is epesented in tems of spheical hamonics of angula degee l. acoustic (p) modes p 4 p 3 p p 1 f-mode convection modes

19 Powe spectum of sola oscillations obtained fom the MDI data. Black points ae mode fequencies detemined fom the powe spectum. The lowest idge is the suface gavity wave (f-mode). The uppe idges ae acoustic (p) modes.

20 Low-Degee (Global) Modes When the Sun is obseved as a sta (integated whole-disk Dopple-shift measuements) the powe spectum consists only of low-degee p-modes of l = 0, 1, and 3. The distance between main peaks in the powe spectum is about 68 µ Hz. The coesponding time: 6 1 /(68 10 ) = 45 min is the tavel time fo acoustic waves popagate though the cente of the Sun to the fa side and come back. The low-degee mode povide infomation about physical conditions of the sola coe. This figue is a Fouie spectum of the longest continuous GOLF time seies (805 days). GOLF is an instument on SOHO that measues the oscillations in the line-of-sight velocity of the sola photosphee fom the whole Sun. These oscillations appea at pecise fequencies, visible as shap peaks in this spectum, mainly aound 3mHz, coesponding to peiods about 5min.

21 Excitation of Sola Oscillations Powe specta of A) l = 50, m = 3, n = 1 and B) l = 50, m = 0, n = 16. Sola oscillations ae andomly excited by tubulent convection. The andom excitation function appeas as multiplicative noise in the powe specta. This epesents the most seious poblem fo measuing mode fequencies. This figue shows examples of good and poo fits of an oscillation model to the powe specta.

22 Examples of spheical hamonics. Cyclic fequency ν = ω π is often used as fequency vaiable. Because only a hemisphee of the Sun is obseved in the powe spectum fo a given mode of taget l, m beside peaks coesponding to this mode peaks of othe modes appea (so-called mode leaks ). The spheical hamonics ae not othogonal on a hemisphee.

23 Rotational fequency splitting The modes with m 0 epesent azimuthally popagating waves. The modes with m > 0 popagate in the diection of sola otation and, thus, have highe fequencies in the inetial fame than the modes m < 0 which popagate in opposite diection. As a esult the modes with fixed n and l ae split in fequency: ν nlm = ν nlm ν nl 0. Thus, the intenal otation is infeed fom splitting of nomal mode fequencies with espect to the azimuthal ode, m. iωt m m imφ + iωt l l ξ θ φ θ e Y (, ) = CP ( ) e - displacement of the sola suface in sola modes ν = ω / π ν is cyclic fequency, measued in Hz - The oscillation peiod is 1/ν (in sec, min, etc). ω is the angula fequency, measued in ad/s

24 Illustation of the fequency shift due to the sola otation Typical powe specta of sola oscillation data fom the MDI instument on SOHO. Each hoizontal cuve shows thee lines of the powe spectum fo diffeent azimuthal ode m with adial ode n = 15 and angula degee l = 19, 0, and 1 (fom left to ight). The slope of the modal lines is due to the otational fequency shift: pogade modes with positive m have highe fequencies than etogade modes with negative m.

25 Basic Equations Basic assumptions: 1. lineaity: v/ c s << 1. adiabaticity: ds/ dt = 0 3. spheical symmety of the backgound 4. magnetic foces and Reynolds stesses ae negligible The basic equations ae consevations of mass, momentum, enegy and Newton s gavity law. 1. Consevation of mass (continuity equation): The ate of mass change in a fluid element of volume V is equal to the mass flux though the suface of this element (of aea A ): ρdv = ρvda = ( ρv) dv. t V A V ρv Then, ρ divegence + ( ρv) = 0, t o d ρ + ρ v = 0. dt A V

26 . Momentum equation (consevation of momentum of a fluid dv element): ρ = P + ρg, dt whee P is pessue, g is the gavity acceleation, which can be expessed in tems of gavitational potential Φ : g = Φ. dv v Also, = + ( v ) v. This is the 'mateial' deivative. dt t vx vx vx e. g. vx + vy + vz fo vx component x y z 3. Adiabaticity equation (consevation of enegy) fo a fluid element: d P dp 0 γ =, d ρ dt ρ o = c, dt dt whee c = γ P/ ρ is the adiabatic sound speed. 4. Poisson equation: 4π Gρ Φ =.

27 Plan to solve the sola oscillation equations 1. Lineaize - conside small-amplitude oscillations.. Neglect the petubations of the gavitational potential (Cowling appoximation). 3. Wite the lineaized equations in the spheical coodinates:, θ, φ. 4. Conside hamonic (peiodic) oscillations 5. Sepaate the adial and angula coodinates. 6. Show that the angula dependence can be epesented by spheical hamonics. 7. Deive equations fo the adial dependence, epesenting the eigenvalue poblem fo the nomal modes 8. Solve the eigenvalue poblem in the asymptotic (shot wavelength) JWKB appoximation. 9. Investigate popeties of p- and g-modes

28 1. Lineaization Conside small petubations of a stationay spheically symmetical sta in the hydostatic equilibium: v = 0, ρ = ρ ( ), P = P ( ) If ξ ( t) is a vecto of displacement of a fluid element then velocity of this element: dξ ξ v =. dt t Petubations of scala vaiables ρ, P,Φ ae two types: Euleian, at a fixed position : ρ(, t) = ρ0( ) + ρ (, t), and Lagangian petubation in the moving element: ρ( + ξ ) = ρ ( ) + δρ(, t). The Euleian and Lagangian petubations ae elated to each othe: d ρ0 d ρ0 δρ = ρ + ( ξ ρ0) = ρ + ( ξ e ) = ρ + ξ, d d whee e is a adial unit vecto. In ou case, the density gadient is adial. 0

29 Then, the lineaized equations ae: ρ + ( ρ ξ ) = 0, 0 the continuity (mass consevation) equation v ρ0 P = g0e ρ + ρ 0 Φ, t the momentum equation dp0 ( d ρ0 P + ξ ) = c0 ρ + ξ, the adibaticity (enegy) equation, o d d δ P = c 0 δρ fo the Lagangian petubations of pessue and density. Φ = 4π Gρ. the equation fo the gavitational potential. Cowling appoximation: Φ = 0.

30 3. Conside the lineaized equations in the spheical coodinates, θ, φ : ξ = ξ e + ξ θ eθ + ξ φ eφ ξ e + ξ, h whee ξ = ξ h θ e θ + ξφ e φ is the hoizontal component of displacement ξφ ξ div ξ = ( ξ ) (sin ) + θξθ + = sinθ θ sinθ φ 1 1 = +. ( ξ ) hξ h 4. Conside peiodic petubations with fequency ω : i t m m im i t ξ e ω Y ( θ φ) CP ( θ ) e φ +, = ω ν = ω / π, l l whee ν is the cyclic fequency (measued in Hz), and ω is the angula fequency (measue in ad/s).

31 Then, in the Cowling appoximation, we get (leaving out subscipt 0 fo unpetubed vaiables): 1 ρ ρ + ( ρξ ) 0 + hξ =, the continuity equation h P ω ρξ = + gρ, the adial component of the momentum equation 1 ω ρξ = h hp, the hoizontal component of the momentum equation 1 ρn ρ = P + ξ, the adiabatic equation c g 1 dp 1 d ρ whee N = g is the Bunt-Vaisala fequency. γ P d ρ d Bounday conditions: ξ ( = 0) = 0, - displacement at the Sun s cente is zeo, (o a egulaity condition fo l = 1). δ P( = R) = 0, - Lagangian pessue petubation at the sola suface is zeo. (this is equivalent to absence of extenal foces). Also, we assume that the solution is egula at the poles θ = 0, π.

32 5. Conside the sepaation of adial and angula vaiables in the fom: ρ (, θ, φ) = ρ ( ) f ( θ, φ), P (, θ, φ) = P ( ) f ( θ, φ), ξ(, θ, φ) = ξ( ) f ( θ, φ), ξ (, θ, φ ) = ξ h( ) f h ( θ, φ ). h Then, the continuity equation is: 1 ρ ρ + ( ρξ ) ( ) 0 f θ, φ + ξh h f =. The vaiables ae sepaated if = α f, h f whee α is a constant. This equation has non-zeo solutions egula at the poles, θ = 0, π only when α = l( l + 1), whee l is an intege. 6. The non-zeo solution of equation f h + l ( l + 1) f = 0 epesents the spheical hamonics: m m im f ( θ, φ) = Y ( θ, φ) = CP ( θ ) e φ, m whee P ( θ ) is the Legende function. l l l

33 7. Deive equations fo the adial dependence, epesenting the eigenvalue poblem fo the nomal modes Afte the sepaation of vaiables the continuity equation fo the adial dependence ρ ( ) is 1 l( l + 1) ρ + ρξ 0 ρξ h =. Compae with the oiginal equation: ρ + ( ρ ξ ) = 0, 0 and with this equation in the spheical coodinates: 1 ρ ρ + ρξ + ξ =, ( ) 0 h h ξ P Tansfom this equation in tems of vaiables: and - adial displacement and Euleian pessue petubation.

34 The hoizontal component of displacement ξ h can be detemined fom the hoizontal component of the momentum equation: ω ρξ h( ) = P ( ), 1 o ξh = P. ω ρ Substituting this into the continuity equation we get: dξ d ρ P ρn L ρ + ξh + ρξ + + ξ 0 P =, d d c g ω ρ 1 whee we define L = l( l + 1) (note the similaity to quantum mechanics). Using the hydostatic equation fo the backgound (unpetubed) state dp = gρ, d dξ g L c P finally get: + ξ ξ =, d c ω ρc o dξ g Sl P + ξ ξ =, d c ω ρc L c whee Sl = is the Lamb fequency, L =l(l+1), c ()=γp/ρ is the squaed sound speed, g()=gm()/ is the gavity acceleation at adius.

35 Similaly, the momentum equation is: dp g + P + N = d c ( ω ) ρξ 0, whee N is the Bunt-Vaisala fequency. The lowe bounday condition: 0 ξ =, dp δ = + ξ =, d P gρξ =. The uppe bounday condition: P P 0 o using the hydostatic equation: 0 Fom the hoizontal component of the momentum equation: P = ω ρξ, (o a egulaity condition). ξh g Then fom the uppe bounday condition: =, ξ ω that is the atio of the hoizontal and adial components of displacement is invese popotional to squaed fequency. Howeve, this elation does not hold in obsevations, pesumably, because of the extenal foce caused by the sola atmosphee. h

36 7. The deived equations with the bounday conditions constitute an eigenvalue poblem fo sola oscillation modes dξ g Sl P + ξ ξ =, d c ω ρc S l dp g + ( ) 0 P + N ω ρξ =. d c L c = is the Lamb fequency. dp dρ = g γ ρ 1 1 N P d d Popeties of oscillations depend on the signs of these coefficients in backets. is the Bunt-Vaisala fequency. The lowe bounday condition: ξ = 0, (o a egulaity condition). dp The uppe bounday condition: δ P = P + ξ = 0, d

37 Popagation diagam of sola oscillations l=1 p-modes S l = L c the Lamb fequency. g-modes 1 dp 1 d ρ N = g γ P d ρ d the Bunt-Vaisala fequency. Buoyancy (Bunt-Vaisala) fequency N, and Lamb fequency S l fo l = 1, 5, 0 and 100 vs. factional adius / R fo a standad sola model. The hoizontal lines indicate the tapping egions fo a g mode with fequency ν = 100µ Hz, and fo a p mode of degee l = 0 and ν = 000µ Hz.

38 8. JWKB (Jeffeys-Wentzel Kames Billouin) Solution (shot-wavelength asymptotic appoximation simila to quantum mechanics) We assume that only density ρ ( ) vaies significantly among the sola popeties in the oscillation equations, and seek fo an oscillatoy solution in the JWKB fom: ξ Aρ / e 1 ik P = B e, 1 ik =, ρ / whee the adial wavenumbe k is a slowly vaying function of. Then, whee H 1 ρ d log = d dξ 1/ 1 ik Aρ = ik + e, d H dp 1/ 1 ik = Bρ ik e, d H is the density scale height.

39 Fom the oscillation equations we get a linea system: 1 g 1 S 1 l ik + A A + B = 0, H c c ω 1 g ik A + A + ( N ω ) A = 0. H c The deteminant of this system is equal zeo when k ω ω c S c ω c l = + N ω c whee ω c = is the acoustic cut-off fequency H (use the elation: N = g/ H g / c ). The sola waves popagate in the egions whee If k > 0. k < 0, the waves exponentially decay ( evanescent ).

40 Popeties of Sola Oscillation Modes ω ω S Equation k c c ω epesents a dispesion elation of sola waves. It elates fequency ω with adial wavenumbe Conside two simple cases: 1: the high-fequency case. If ω >> N then ω ω S whee Then, Finally, o k c l = + N ω c l = c c ω = ω + k c + k c, c h k and angula ode l. L l( l + 1) kh = Sl / c is the hoizontal wave numbe. k = k + k is the squaed total wavenumbe. h c ω = ω c + k c, whee ωc = is the acoustic cut-off fequency. H This is the dispesion elation fo acoustic (p) modes; ω c is the acoustic cutoff fequency. Physically, the waves with fequencies below the acoustic cutoff fequency cannot popagate. Thei wavelength becomes shote than the density scale height. Fo the Sun ν c ωc / π 5 mhz. (c~10 km/s, H~150km).

41 k ω ω c S c ω c l = + N ω : conside the low-fequency case when ω << S l Sl then k ( ) = N ω (emembe S l = ckh = cl / ) c ω kh N Then, ω = N cos θ, whee k = k + kh k whee θ is the angle between wavevecto k and the hoizontal diection. This is a dispesion elation fo intenal gavity (g). modes. They popagate mostly hoizontally.

42 The fequencies of nomal modes ae detemined fo the Boh quantization ule (esonant condition): Nomal modes of sola oscillations 1 k d = π ( n + α), whee 1 and ae the adii of the tuning points whee k =0, n is a adial ode -intege numbe, and α is a phase shift which depends on popeties of the eflecting boundaies. k ω ω S = + N ω c c ω c l ω = c c() is the sound speed H S l c H = L c 1 ρ d log =, d is the acoustic cut-off fequency; it has vey shap incease at /R=1 L N = g/ H g / c = l( l + 1)

43 P-mode ay paths Inne tuning point ω ωc Sl k = c c The waves popagate whee 1 k >0. The waves ae evanescent whee k <0 The wave tuning points ae located whee k =0. Because ω c = c / H has a shap peak nea the suface the uppe tuning point ( ) is whee ω = ω c The lowe tuning point ( 1 ) is whee ω = Sl = ( L / ) c = khc whee the hoizontal phase speed is equal to the sound speed. ω = c / kh

44 g-mode ay paths g-modes popagate only in the adiative zone which is convectively stable N > 0

45 Calculation of nomal mode fequencies Estimate fequencies of nomal modes fo these cases. 1. p-modes: popagating egion: k > 0 L c tuning points k = 0 : ω = ωc +. Fo the lowe tuning point in the inteio: ωc << ω. Lc c( 1 ) ω Then, ω, o = is the equation fo the lowe tuning point. L 1 ω ( ) The uppe tuning point: c nea the suface, R. ω Then, the esonant condition fo p-modes is:. Since ω ( ) is a steep function of c R 1 ω L ( ) d = π n + α c Abel integal equation.

46 π ( n + L/ + α) Fo l << n, 1 0, and we get: ω. R d 0 c That is the spectum of low-degee p-modes is appoximately equidistant with fequency spacing: Low-degee p-modes 1 4 R d ν =. 0 c Maximum amplitude is aound 3,300 µhz, o 3.3 mhz. The coesponding oscillation peiod is 300 seconds o 5 minutes.

47 Asteoseismology Bedding & Kjeldsen (003)

48 Fequencies of g-modes: The tuning points ae detemined fom equation: N ( ) = ω. In the popagation egion, k > 0, fa fom the tuning points ( N >> ω ): LN k. ω Then, fom the esonant condition: L d N = π ( n + α). 1 ω L N d 1 we find: ω. π ( n + α )

49 Wavelength in Mm Spectum of nomal modes calculated fo a standad sola model. Note the avoided cossing effect fo f and g-modes.

50 Suface gavity waves (f-mode) These wave popagate at the suface bounday whee Lagangian pessue petubation δ P ~ 0. Conside the oscillation equations in tems of δ P by making use of the elation between Euleian and Lagangian vaiables: P = δ P + gρξ. + 1 = 0, d ω ω ρc dξ L g L c δ P ξ dδ P L g gρ f δ P ξ 0 + =, d ω ω L g whee f. g ω These equations have a peculia solution: δ P = 0, f = 0. Fo this solution: Lg R ω = = k g -dispesion elation fo f-mode. dξ L The eigenfunction equation: ξ = 0 d has a solution kh ( R) h ξ e exponentially decaying with depth.