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: • x(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 P ⇢ subtract off the equilibrium equations, and = so = 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 @ = 1Pr + ⌘ [(31 4) P ] ⇢ r @(r/ro) @t2 ⇢r4 @r @r r⇢ @r = 3 ro ✓ ◆ ⇢ ⇢o ro @ro • linearized equation of motion • assume exponential (complex) time dependence 2 d r/ro r P @Po @(P/Po) r(t, ro) r(ro) it it ⇢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(⌘)= Pr4 ⌘ [(3 4) P ] = 2⌘ LAWE becomes ⇢r4 @r 1 @r r⇢ @r 1 • ✓ ◆ ⇢ 1 @P ⌘ (3 4) = 2⌘ ⇢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 ⇧ = = G⇢¯ ( 4 ) 1 3 • look familiar?! q 13 14 the LAWE: standing wave solutions the LAWE: asymptotic solution d2w(r) 2 + (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 kr = 2 (r) • clever change of variables renders LAWE as: cs(r) d2w(r) 2⇢ for a standing wave, we need an integral + (r) w(r)=0 • dr2 P number of half-wavelengths between inner  1 • recognizing the sound speed when we see it: and outer reflection points: b d2w(r) 2 + (r) w(r)=0 kr dr =(n + 1)⇡ dr2 c2 a  s Z

15 16 the LAWE: asymptotic solution Nonradial oscillations

b 2 2 preserve angular derivatives in LAWE kr dr =(n + 1)⇡ where k = (r) • r c2(r) Za s • similar operator structure for radial part (as 2 before), now along with angular part if then • c2 1 2 s b d r P 0 1P dr = + 0 + A r =(n + 1)⇡ =(n + 1) dt2 r ⇢ ⇢ r · c o ✓ ◆ "Za s # where the quantity A is: d ln ⇢ 1 d ln P 1 T i.e. high-frequency (high overtone, n) radial A = = [ ad] • dr 1 dr P ⇢ 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 r = r er + r✓ e✓ + r sin ✓ e • produces (after some work): i=70 2 1 @(r r) 1 2 2 P 0 r = r L + 0 r · r2 @r 2 ⇢ ✓ ◆ 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 (✓, )=l(l + 1)Yl (✓, )

19 20 using Spherical Harmonics the two characteristic frequencies now we have so: • • 2 2 2 d⌘r l(l + 1) 1 @(r r) l(l + 1) P 0 r = g 2r ⌘r + l(l + 1) 1 r⌘t r = + 0 dr S2 S2 r · r2 @r 2r2 ⇢  l  l ✓ ◆ 2 2 d⌘t N 2 r expanding into components: r = 1 r⌘r + N 1 r⌘t frequency-2 dr 2 g •   2 2 2 d⌘r gr 2 l(l + 1) 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 2 g l r2 s  2  another frequency • the Brunt-Väisälä (buoyancy) frequency N: d ln ⇢ 1 d ln P N 2 = Ag = g dr 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 ( N )( Sl ) cs

23 page 24 asymptotic analysis Propagation diagram, ZAMS solar model 2 1 2 2 2 2 kr = 2 2 ( N )( Sl ) 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 = ( N )( S ) r 2c2 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 : nl =(n + l/2)o ; o = 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 =(n + l/2) ; = 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. 33 p. 34 Frohlich et al. 1997: SOHO/VIRGO How to observe all these low 10 5 1 hr min min amplitude modes?

20 parts per million!

have your friends buy a $600,000,000 photometer!

p. 35 p. 36 16 Cyg A and B (Metcalfe et al. 2012) asteroseismic radius determination (i.e. Chaplin et al. 2011)

• νmax scales with acoustic cutoff frequency ~ gTe-1/2

2 0.5 M R T max e max, M R Te,

• Δν measures mean density: 2 3 M R M R p. 37 p. 38 Kepler 93b Kepler 93b (Ballard et al. 2014) (Ballard et al. 2014)

p. 39 40 Kepler 93b g-modes: ~ equally spaced in period 1 b (Ballard et al. 2014) ⇧o 2 N ⇧nl = n ; ⇧o =2⇡ dr l(l + 1) r "Za # n-2 pn-1 n n+1 n+2 n+3 n+4 n+5 n+6

...... l=1 Period

... n-2 n-1 n n+1 n+2 n+3 n+4 n+5 n+6 ...

...... l=2 Period • spacing depends on l PG 1159-035: a g-mode pulsator an example: hot white dwarf PG 1159-035 (Corsico et al. [WET] 2008)

P P-390 390 0 424 34 ? 451 61 3x20.3 495 105 5x21.0 516 126 6x21.0 539 149 7x21.3 645 255 12x21.3 832 442 21x21.0

in white dwarfs: Rotational splitting of nonradial oscillations Π odepends on total stellar mass (uniform, slow rotation)

PG 1159 as an example equal frequency spacing: triplets (l=1), quintuplets (l=2) etc. (Winget et al. 1991) page Pulsating stars in “Solar-like” oscillations the HR diagram • globally stable (damped) but constantly excited • damping time τ generally ~ days • continuously re-excited by turbulence • frequencies locked to normal modes of star • excited mode periods ~ minutes • broad mode selection, low amplitude • (integrated) velocity amplitude < meters / second • photometric amplitude from J. Christensen- ~ parts per million Dalsgaard

Consequences of stochasitic excitation Observational Differences

• lots of modes present . . . but • coherence time of (only) days lowers peak amplitudes reduces detectability • 1/(observing time) phase instability broadens FT peaks • 1/lifetime • Lorentzian envelope • reduces frequency accuracy • confuses mode identification and rotational splitting effects heat-engine mode Stochastically excited mode from J. C.-D. 1.00

0.80

0.60

0.40

0.20

0.00 Asymptotics oflow-degree pmodes

9019 0021 2020 2010 2000 1990 1980 coherent pulsators: frequency precision ~ 1/T

T = 5d Large frequency separation: frequency Large

9019 0021 2020 2010 2000 1990 1980

T = 10d

9019 0021 2020 2010 2000 1990 1980

T = 30d

Frequency separations: Frequency

0.00

0.20

0.40

0.60

0.80

1.00

0.00

0.20

0.40

0.60

0.80

1.00

0.00

0.20

0.40

0.60

0.80 1.00

stochastic pulsators - freq. precision poorer than 1/T

9019 0021 2020 2010 2000 1990 1980

T = 5d = T

T = 5d = T T = 5d = T

Small frequency separations

9019 0021 2020 2010 2000 1990 1980

T = 10d = T

T = 10d = T

T = 10d = T

9019 0021 2020 2010 2000 1990 1980

T = 30d = T

T = 30d = T T = 30d 30d = T Asteroseismic HR diagram Borucki et al. 2009 - A Search for Habitable Planets Kepler’s Optical Phase Curve of the Exoplanet HAT-P-7b

from J. Christensen-Dalsgaard

HAT-P7 asteroseismology HAT-P7 asteroseismology

1.45 Mo

1.40 Mo Asteroseismic HR diagram HAT-P7 asteroseismology

HAT-P-7 M=1.40±0.02 t = 1.6±0.4 Gyr

r = 1.94±0.05 Rsun

Xc=0.19

quick seismic fit spectroscopy ISUEVO Sun Pal et al. 2008 α Cen A from J. Christensen-Dalsgaard

Solar-like pulsators: KIC 6603624 Solar-like pulsators: KIC 3656476 Chaplin et al. 2010 Chaplin et al. 2010 Solar-like pulsators: KIC 11026764 Solar-like - FTs Chaplin et al. 2010

oscillations beyond the MS w/ Kepler p. 64 Asteroseismic HR diagram (Chaplin & Miglio 2013) HAT-P-7

KIC 11026764 is post-MS!

Sun α Cen A Solar-like oscillations w/ Kepler p. 65 Solar-like oscillations w/ Kepler p. 66 (Chaplin & Miglio 2013) (Chaplin & Miglio 2013)

MS MS

MS (16 Cyg A) MS (16 Cyg A)

Subgiant Subgiant

Subgiant Subgiant

Base of RGB Base of RGB

500 1000 2000 3000 4000 uHz 500 1000 2000 3000 4000 uHz

p. 67 p. 68 oscillations beyond the MS w/ Kepler scaling relations MS to RGB (Chaplin & Miglio 2013)

First-ascent RGB

First-ascent RGB

R/Rsun log g Δν [uHz] νmax

First-ascent RGB MS 1 4.44 135 3300 RGB base 5 3.04 12.1 140 Red Clump RGB top 20 1.84 1.5 9 < 1/d

20 40 60 80 100 uHz 1/wk p. 69 Bedding et al. (2010) Ensemble asteroseismology of solar-type stars with Kepler Chaplin et al. 2011 (Science, today!) • determination of Δν and νmax for ~ 500 solar-type stars

multiple • νmax scales with acoustic l=1 modes cutoff frequency ~ gTe-1/2 2 0.5 per order?! max M R Te max , M R Te,

• Δν measures mean density: 2 3 M R M R

Chaplin et al., Science, 7 April 2011 Chaplin et al., Science, 7 April 2011 Chaplin et al., Science, 7 April 2011 Chaplin et al., Science, 7 April 2011

2 3 Δν measures mean density: M R • M R

• νmax scales with acoustic cutoff frequency ~ gTe-1/2 2 0.5 max M R Te max, M R Te, • then one can determine M and R: 2 0.5 R T max e R max, Te, 3 4 1.5 M max Te M max, Te,

• and, with Teff from multicolor photometry, one gets L