2 observed pulsations • operate on the dynamical time scale Asteroseismology • accessible on convenient time scale • probe global and local structure Steve Kawaler • periods change on ‘evolutionary’ time scale Iowa State University (thermal or nuclear) - depend on global properties • amplitudes change on ~ ‘local’ thermal time scale 3 4 dynamical stability a more complex example: a star • “stable” configuration represents a stable mean configuration • multiple oscillation modes • on short time scale, oscillations occur, but the • radial modes - enumerated by number of mean value is fixed on longer time scales nodes between center and surface • simple example: a pendulum (single mode) • most likely position - extrema • non-radial modes - nodes also across • mean position is at zero displacement surface of constant radius with no damping would oscillate forever • • modes frequencies determined by solution of • more complex example: a vibrating string the appropriate wave equation • multiple modes with different frequencies • enumerated by number of nodes 5 6 stability, damping, and driving Okay, start your engines... • PG 1159: light curve • zero energy change: what kind of star might this be? constant amplitude oscillation • • what kind of star can this not possibly be? • energy loss via pulsation: • what about the amplitude over the run? oscillation amplitude drops with time • PG 1336 light curve • if net energy input: • huh? what time scale(s) are involved amplitude increases with time • what kind of star (or stars)? (if properly phased) • tell us everything you can about this! 7 8 Multimode pulsation towards the wave equation I Oscillations at “normal mode” frequencies • continuity equation: • @M r =4⇡r2⇢ • mode = specific eigensolution of equations @r of motion within the confines of a stellar • equation of motion (HSE when RHS=0): structure 2 @ r GMr 2 @P 2 = 2 4⇡r • normal mode frequencies parallel structural @t − r − @Mr properties perturb r, P, and ρ: • δx(t, M ) x(t, M )=x (M ) 1+ r simple example: radial fundamental is one r o r x (M ) • o r mode, 1st overtone (a node within) is • and assume δx << x so we can linearize another mode 9 10 towards the wave equation II towards the wave equation III replace x with x+δx in the two equations, • assume adiabatic relationship between P and ρ • @ ln P δP ⇢ subtract off the equilibrium equations, and Γ = so = Γ 1 @ ln ⇢ P 1 ⇢ keep only 1st-order terms to find: ✓ ◆ad o o • combine continuity and equation of motion: linearized continuity equation 2 • @ ⌘ 1 @ 4 @⌘ 1 @ = Γ1Pr + ⌘ [(3Γ1 4) P ] ⇢ δr @(δr/ro) @t2 ⇢r4 @r @r r⇢ @r − = 3 ro ✓ ◆ ⇢ ⇢o − ro − @ro • linearized equation of motion • assume exponential (complex) time dependence 2 d δr/ro δr δP @Po @(δP/Po) δr(t, ro) δr(ro) iσt iσt ⇢oro = 4 + Po = e = ⌘(ro)e dt2 − r P @r − @r ro ro ✓ o o ◆ o o towards the wave equation IV 11 the LAWE: a simple case 12 substitute to yield the Linear Adiabatic Wave • • assume Γ1 and η both constant throughout Equation (LAWE): the star (homologous motion) 1 @ @⌘ 1 @ L(⌘)= Γ Pr4 ⌘ [(3Γ 4) P ] = σ2⌘ LAWE becomes −⇢r4 @r 1 @r − r⇢ @r 1 − • ✓ ◆ ⇢ 1 @P ⌘ (3Γ 4) = σ2⌘ − ⇢r 1 − @r • This is a wave equation: L(η) = σ2η in the • now, assume a constant density, and use HSE displacement η. to replace the pressure derivative to find • the eigenvalue σ2 corresponds to the 2⇡ p⇡ oscillation frequency ⇧ = = σ G⇢¯ (Γ 4 ) 1 − 3 • look familiar?! q 13 14 the LAWE: standing wave solutions the LAWE: asymptotic solution d2w(r) σ2 + φ(r) w(r)=0 boundary conditions: dr2 c2 − • s η represent the eigenfunction as: w(r) eikr r • center: zero displacement ( = 0) • / where kr is the (local) radial wavenumber and surface: perfect wave reflection [d(δP/P)/dr = 0] • varies slowly with radius so, locally: 2 • asymptotic analysis: 2 σ kr = 2 φ(r) • clever change of variables renders LAWE as: cs(r) − d2w(r) σ2⇢ for a standing wave, we need an integral + φ(r) w(r)=0 • dr2 Γ P − number of half-wavelengths between inner 1 • recognizing the sound speed when we see it: and outer reflection points: b d2w(r) σ2 + φ(r) w(r)=0 kr dr =(n + 1)⇡ dr2 c2 − a s Z 15 16 the LAWE: asymptotic solution Nonradial oscillations b 2 2 σ preserve angular derivatives in LAWE kr dr =(n + 1)⇡ where k = φ(r) • r c2(r) − Za s • similar operator structure for radial part (as σ2 before), now along with angular part if φ then • c2 1 2 s b − d δr P 0 Γ1P dr = + 0 + A δr σ =(n + 1)⇡ =(n + 1) σ dt2 r ⇢ ⇢ r · c o ✓ ◆ "Za s # where the quantity A is: d ln ⇢ 1 d ln P 1 χT i.e. high-frequency (high overtone, n) radial A = = [ ad] • dr − Γ1 dr λP χ⇢ rr modes are equally spaced in frequency, with A < 0 when radiative A > 0 when convective σo2 ≈ G<ρ> (the ‘Schwarzschild A’) 17 Spherical Harmonics.... Decompose into Spherical Harmonics courtesy asteroseismology.org (Travis Metcafe) • position perturbation decomposition l=1, m=0 l=1, m=1 δr = δr er + r✓ e✓ + r sin ✓φ eφ • produces (after some work): i=70 2 1 @(r δr) 1 2 2 P 0 δr = r L + 0 r · r2 @r − σ2 ⇢ ✓ ◆ where the operator L2 (the Legendrian) is: l=3, m=0 l=3, m=1 l=3, m=3 1 @ @ 1 @2 L2 = sin ✓ −sin ✓ @✓ @✓ − sin2 ✓ @2 ✓ ◆ • which has eigenstates Ylm such that: 2 m m L Yl (✓, φ)=l(l + 1)Yl (✓, φ) 19 20 using Spherical Harmonics the two characteristic frequencies now we have so: • • 2 2 2 d⌘r l(l + 1) σ 1 @(r δr) l(l + 1) P 0 r = g 2r ⌘r + l(l + 1) 1 r⌘t δr = + 0 dr S2 − − S2 r · r2 @r − σ2r2 ⇢ l l ✓ ◆ 2 2 d⌘t N 2 r expanding into components: r = 1 r⌘r + N 1 r⌘t frequency-2 dr − σ2 g − • 2 2 2 d⌘r gr 2 l(l + 1) σ r 1 where we’ve defined 2 “structural” frequencies r = 2 r⌘r + r 1 r⌘t • dr c2 − r2 − c2 l(l + 1) s s • the acoustic (Lamb) frequency Sl : d⌘ Ag r 2 t 2 l(l + 1) 2 r = 1+ r⌘r + ( Ag) 1 r⌘t S = c dr σ2 − g − l r2 s 2 another frequency • the Brunt-Väisälä (buoyancy) frequency N: d ln ⇢ 1 d ln P N 2 = Ag = g − − dr − Γ dr 1 page 21 22 Propagation diagram, ZAMS solar model NRP dispersion relation • identify the horizontal wave number(s) Sl2, l=1 2 2 l(l + 1) Sl kt = 2 = 2 CZ r cs base • allows the wave equation(s) to reduce to a N2 local dispersion relation, as with the radial case, to provide relationship between kr and σ: 2 1 2 2 2 2 kr = 2 2 (σ N )(σ Sl ) σ cs − − 23 page 24 asymptotic analysis Propagation diagram, ZAMS solar model 2 1 2 2 2 2 kr = 2 2 (σ N )(σ Sl ) σ cs − − Sl2, l=1 2 • kr > 0 (kr real) when l=1 n=4 CZ l=1 n=3 • σ2 > N2, Sl2 - or - σ2 < N2, Sl2 base l=1 n=2 l=1 n=1 • kr real means oscillatory eigenfunctions N2 • kr2 < 0 (kr imaginary) when • Sl2 > σ2 > N2 or Sl2 < σ2 < N2 • kr real means evanescent (exponentially decreasing or increasing) eigenfunctions page 25 page 26 Propagation diagram, ZAMS solar model Propagation diagram, ZAMS solar model “high” σ oscillatory evanescent evanescent zones 2 2 zones Sl , l=1 zones Sl , l=1 in pink in gray in gray l=1 n=4 l=1 n=4 CZ l=1 n=3 CZ l=1 n=3 base l=1 n=2 base l=1 n=2 l=1 n=1 l=1 n=1 N2 N2 “low” σ oscillatory zones in green 27 page 28 the NRP LAWE: asymptotic solutions Propagation diagram, ZAMS solar model “high” σ • again, integrate dispersion relation over oscillatory evanescent zones propagation regions: zones 2 b Sl , l=1 in pink 2 1 2 2 2 2 in gray kr dr =(n + 1)⇡ where k = (σ N )(σ S ) r σ2c2 − − l l=1 n=4 Za s CZ l=1 n=3 • two classes of solutions: base l=1 n=2 b l=1 n=1 2 2 2 dr • σ > N , Sl : σnl =(n + l/2)σo ; σo = 2 l=1 n=1 a cs N “p-modes”; pressure as the restoring forceZ l=1 n=2 • l=1 n=3 l=1 n=4 1 b − 2 2 2 ⇧o 2 N “low” σ σ < N , Sl : ⇧nl = n ; ⇧o =2⇡ dr • l(l + 1) r "Za # oscillatory “g-modes”: buoyancy as the restoring force • p zones in green page 29 page Pulsation Periods Period of ‘radial fundamental’ ~ tff g-modes p-modes Π > t Π < t Periods ff ff restoring force buoyancy pressure asymptotic Π ∝ Π x n σ ∝ σ x n behavior o o Cepheids, white dwarfs examples the Sun 31 Solar Oscillations: Full-disk photometry p. 32 p-modes: ~ equally spaced in frequency Frohlich et al. 1997: SOHO/VIRGO 10 5 1 hr min min b dr σ =(n + l/2)σ ; σ = nl o o c Za s l l l l l l l l l n-2 n-1 n n+1 n+2 n+3 n+4 n+5 n+6 ... ... frequency l+1 l+1 l+1 l+1 l+1 l+1 l+1 l+1 n-2 n-1 n n+1 n+2 n+3 n+4 n+5 • if modes of different l present, observed 20 parts per million! spacing ~ σo / 2 Solar Oscillations: Full-disk photometry p.