Theory of stellar oscillations

Theory of stellar oscillations I. Solar-like oscillators J. Philidet, LIRA / Observatoire de Paris Ecole Evry Schatzmann 2025 •Asteroseismology = study of oscillations of stars and what they teach us about what happens underneath •Periodic motions of the gas, observed at the surface through Doppler-shifted absorption lines , or temperature (luminosity) variations •Fourier transform of light curve shows forest of narrow peaks è resonant modes of oscillation •All stars have the potential to oscillate, but we do not detect oscillations in all of them Introduction What is asteroseismology? [Garcia & Ballot 19] [Chaplin & Miglio 13] 1 •Many kinds of classical pulsators •Lie along specific regions in HR diagram (instability strips) Introduction Two classes of oscillating stars [Aerts+10] log𝑇 eff log 𝐿/𝐿! Classical pulsators Oscillations are thermally unstable Large oscillation amplitudes 2 •Excitation requires convective envelope à focus on cold, lowand intermediate-mass stars on main-sequence or after ( subgiants, red giants ) •Core of PLATO stellar science Introduction Two classes of oscillating stars [Garcia and Ballot 19] Solar-like oscillators Oscillations excited by vigorous convective gas motions Small oscillation amplitudes •Many kinds of classical pulsators •Lie along specific regions in HR diagram (instability strips) Classical pulsators Oscillations are thermally unstable Large oscillation amplitudes 3 Stellar oscillations: a local view Equations of hydrodynamics 𝜕𝜌 𝜕𝑡+∇·(𝜌u)=0 𝜌!𝜕 𝜕𝑡+u·∇"u+∇𝑝+𝜌∇Φ=−∇#𝜌uconv ⊗uconv$ 𝜌𝑇!𝜕 𝜕𝑡+u·∇"𝑠−𝜌𝜖𝑁+∇·Frad =(u·∇)#𝜌uconv ⊗uconv$−𝜕 𝜕𝑡%1 2𝜌u2 conv&−∇·%𝜌!ℎ+1 2u2 conv"uconv& ∇2Φ=4𝜋𝐺𝜌 +equation of state 𝑠(𝜌,𝑇),𝑝(𝜌,𝑇) +equations for 𝜖𝑁,Frad Hypotheses •Scale of convective motions much smaller than scale of oscillations •Turbulent pressure and convective flux can be neglected 𝜕𝜌 𝜕𝑡+∇·(𝜌u)=0 𝜌!𝜕 𝜕𝑡+u·∇"u=−∇𝑝−𝜌∇Φ+∇:𝝉 𝜌𝑇!𝜕 𝜕𝑡+u·∇"𝑠=𝜌(𝜖𝑁+𝜖𝑉)−∇·Frad ∇2Φ=4𝜋𝐺𝜌 •Turbulent convection needs to be filtered out è spatial average •Allows to discard small scale viscous effect… •… but introduces non-linear contributions from convection everywhere •In general, all effects of convection (turbulent pressure, convective flux, kinetic flux, …) are neglected + equation of state s(ρ, T), p(ρ, T) + equations for εN, Frad 4 •All variables (density, velocity, pressure, gravitational potential) linearised around spherically symmetric state •Full non-adiabatic equations read Stellar oscillations: a local view Adiabatic wave equation (1) 𝛿𝜌 𝜌0 +∇·𝝃=0 𝜕2𝜉𝑟 𝜕𝑡2+1 𝜌0 𝜕𝑝! 𝜕𝑟+!𝛿𝜌 𝜌0 −𝜉𝑟 d ln 𝜌0 d𝑟"𝑔0+𝜕Φ! 𝜕𝑟=0 𝜕2𝝃ℎ 𝜕𝑡2+1 𝜌0 ∇ℎ𝑝!+∇ℎΦ!=0 𝛿𝜌0 𝜌0 −1 Γ1 𝛿𝑝 𝑝0 =−∇ad 𝜌0𝑇0 𝑝0 𝛿𝑠nad ∇2Φ!=4𝜋𝐺𝜌! Hypotheses •Scale of convective motions much smaller than scale of oscillations •Turbulent pressure and convective flux can be neglected •Small wave perturbation •Spherical symmetry 𝑋(r,𝑡)=𝑋0(r,𝑡)+𝑋!(r,𝑡) =𝑋0(r0,𝑡)+𝛿𝑋(r0,𝑡) Eulerian perturbation Lagrangian perturbation 𝝃=r−r0⇒𝛿𝑋=𝑋#+𝝃·∇𝑋0 Wave displacement Contains all non-adiabatic terms 𝜌0𝑇 0 𝜕𝛿𝑠nad 𝜕𝑡 =(𝜌𝜖𝑁)!−∇·Frad +all terms related to convection Usually discarded in stellar oscillation calculations 5 •No time dependence of equilibrium è look for harmonic solutions •Oscillations equation expressed in terms of radial displacement, pressure, and gravitational potential only •Separating radial and angular variables •All wave variables satisfy Stellar oscillations: a local view Adiabatic wave equation (2) 1 𝜌0 𝜕𝑝! 𝜕𝑟+𝑔0𝜌0 Γ1𝑝0 𝑝!−!𝜔2−𝑁2"𝜉𝑟+𝜕Φ! 𝜕𝑟=0 1 𝑟2 𝜕𝑟2𝜉𝑟 𝜕𝑟+ 1 Γ1 d ln 𝑝0 d𝑟𝜉𝑟+ 1 Γ1𝑝0 𝑝!+ 1 𝜌0𝜔2∇2 ℎ𝑝!+ 1 𝜔2∇2 ℎΦ!=0 1 𝑟2 𝜕 𝜕𝑟#𝑟2𝜕Φ! 𝜕𝑟$+∇2 ℎΦ!−4𝜋𝐺#𝑝! 𝑝0Γ1 +𝑁2 𝑔0 𝜉𝑟$ Hypotheses •Scale of convective motions much smaller than scale of oscillations •Turbulent pressure and convective flux can be neglected •Small wave perturbation •Spherical symmetry •Adiabatic treatment ∝exp𝑗𝜔𝑡 Brünt-Väisälä frequency 𝑁2=𝑔0!1 Γ1 d ln 𝑝0 d𝑟 − d ln 𝜌0 d𝑟" !𝑟2∇2 ℎ+Λ"𝑋ang =0 𝑋!(𝑟,𝜃,𝜙,𝑡)=𝑋𝑟(𝑟)𝑋ang(𝜃,𝜙)exp𝑗𝜔𝑡 Spherical harmonics 6 Horizontal wave vector •Regularity at the poles imposes with ℓ integer •Periodic longitude imposes m integer as well •In integrated light, only see the first few angular degrees Stellar oscillations: a local view Spherical harmonics Hypotheses •Scale of convective motions much smaller than scale of oscillations •Turbulent pressure and convective flux can be neglected •Small wave perturbation •Spherical symmetry •Adiabatic treatment Λ=ℓ(ℓ+1) 𝑋ang(𝜃,𝜙)=𝑌𝑚 ℓ(𝜃,𝜙)=𝑃𝑚 ℓ(cos 𝜃)exp𝑗𝑚𝜙 𝑚=0 𝑚=1 𝑚=2 𝑚=3 𝑚=4 𝑚=5 ℓ=0 ℓ=1 ℓ=2 ℓ=3 ℓ=4 ℓ=5 ∇2 ℎ𝑌𝑚 ℓ=−ℓ(ℓ+1) 𝑟2𝑌𝑚 ℓ 𝑘2 ℎ↔ℓ(ℓ+1) 𝑟2 [Garcia & Ballot 19] 7 Qualitative local treatment (small wavelength) Lamb frequency •Cowling approximation : 4th order à 2nd order •Canonical form Stellar oscillations: a local view Local JWKB analysis Hypotheses •Scale of convective motions much smaller than scale of oscillations •Turbulent pressure and convective flux can be neglected •Small wave perturbation •Spherical symmetry •Adiabatic treatment •Cowling approximation 1 𝜌0 d𝑝! d𝑟+𝑔0 𝜌0𝑐2 𝑠 𝑝!+)𝑁2−𝜔2"𝜉𝑟=0 1 𝑟2 d𝑟2𝜉𝑟 d𝑟−𝑔0 𝑐2 𝑠 𝜉𝑟+#1− 𝑆2 ℓ 𝜔2$𝑝! 𝜌0𝑐2 𝑠 =0 (Φ!=0) 𝑆2 ℓ= ℓ(ℓ+1)𝑐2 𝑠 𝑟2 =𝑘2 ℎ𝑐2 𝑠 d! 𝜉 d𝑟 = 𝑟2ℎ(𝑟) 𝑐2 𝑠"𝑆2 ℓ 𝜔2−1#! 𝑝 d! 𝑝 d𝑟 = 1 𝑟2ℎ(𝑟)$𝜔2−𝑁2%! 𝜉 ! 𝜉(𝑟),! 𝑝(𝑟)∝exp𝑗𝑘𝑟𝑟 𝑘2 𝑟= !𝜔2 −𝑆2 ℓ"#𝜔2 −𝑁2$ 𝑐2 𝑠𝜔2 8 Going global: asymptotic theory Wave resonance condition (3) •Solutions must match between the two turning points •If two turning points of the same type •If two turning points of different types cos !∫𝑟 𝑟𝑙 𝑘𝑟(𝑟!)d𝑟!−𝜋 4#=cos !∫𝑟𝑢 𝑟 𝑘𝑟(𝑟!)d𝑟!−𝜋 4# Resonant condition 1 ∫𝑟𝑢 𝑟𝑙 𝑘𝑟(𝑟!)d𝑟!=𝑛𝜋 cos !∫𝑟 𝑟𝑙 𝑘𝑟(𝑟!)d𝑟!−𝜋 4#=cos !∫𝑟𝑢 𝑟 𝑘𝑟(𝑟!)d𝑟!+𝜋 4# Resonant condition 2 ∫𝑟𝑢 𝑟𝑙 𝑘𝑟(𝑟!)d𝑟!="𝑛+1 2#𝜋 Matching done here 15 Large separation p-mode frequencies, asymptotic, 1st order [Tassoul 80] Asymptotic p-modes Large separation •Stratification close to the surface cannot be neglected in kr2 • Acoustic cutoff frequency defines maximum frequency contained below the atmosphere è only important at surface, its effect can be incorporated in extra phase α •Implicit relation can be inverted for low degree modes: using ω >> Sl everywhere except close to the turning point Acoustic cutoff frequency 𝜔2 𝑐≡ 𝑐2 𝑠 4𝐻2 𝜌!1−2 d𝐻𝜌 d𝑟" 𝜔𝑛,ℓ=2𝜋!𝑛+ℓ 2+ 1 4+𝛼"Δ𝜈 ∫𝑟𝑢(𝜔) 𝑟𝑙(𝜔) 1 𝑐2 𝑠"𝑆2 ℓ#𝑁2 𝜔2−1$+𝜔2−𝜔2 𝑐%d𝑟"=#𝑛+1 2$𝜋 𝜔)𝑅∗ 𝑟𝑙(𝜔) 1 𝑐𝑠"1− 𝑆2 ℓ 𝜔2#1/2 d𝑟#=(𝑛+𝛼(𝜔))𝜋 Δ𝜈𝑛,ℓ≡𝜈𝑛+1,ℓ−𝜈𝑛,ℓ∼!2∫𝑅∗ 0 d𝑟 𝑐𝑠#−1 è marker of sound travel time between core and surface [Mosser 15] 𝜔𝑐[2𝜋×mHz] 16 p-mode frequencies, asymptotic, 2nd order Asymptotic p-modes Small separation •Accounting for core variations of sound speed [Tassoul 80] •Lifts the degeneracy between ωnl and ωn+1,l-2 𝜔𝑛,ℓ=2𝜋!𝑛+ℓ 2+1 4+𝛼"Δ𝜈−(𝐴ℓ(ℓ+1)−𝛿2)2𝜋Δ𝜈2 𝜔𝑛,ℓ with 𝐴=1 4𝜋2Δ𝜈)𝑐(surf) 𝑠 𝑅∗−∫𝑅∗ 0 d𝑐𝑠 d𝑟 d𝑟 𝑟# Small separation 𝛿𝜈𝑛,ℓ≡𝜈𝑛,ℓ−𝜈𝑛−1,ℓ+2∼ −(4ℓ+6)Δ𝜈 4𝜋2𝜈𝑛,ℓ∫𝑅∗ 0 d𝑐𝑠 d𝑟 d𝑟 𝑟 è marker of sound speed profile in the core è marker of evolution [Chaplin 10] 𝛿𝑛,0 Δ𝑛,0/2 Δ𝑛,0 17 Weighted small separations Mean large separation •Even if individual frequencies not measured, global seismic indices can be used to characterise stars •Radial and dipolar modes more easily seen than quadrupolar modes è need seismic indices independent of l = 2 modes Asymptotic p-modes Asteroseismic diagram !Δ𝜈ℓ"≡!𝜈𝑛+1,ℓ−𝜈𝑛,ℓ"𝑛 𝑑01(𝑛)= 1 2!𝜈𝑛,1−2𝜈𝑛,0+𝜈𝑛−1,1" 𝑑(2) 01 (𝑛)= 1 8!𝜈𝑛−1,0−4𝜈𝑛−1,1+6𝜈𝑛,0−4𝜈𝑛,1+𝜈𝑛+1,0" Mean small separation 𝛿𝜈02 ≡"𝜈𝑛,0−𝜈𝑛−1,2$𝑛 Useful to probe core conditions •Allows to draw asteroseismic (or C-D) diagram above [Christensen-Dalsgaard 88] è more useful for main-sequence stars [White+11] 18 Asymptotic p-modes Echelle diagram •Real echelle diagram deviates from asymptotic echelle diagram •deviation from asymptotic regime (mitigated by numerical calculations of frequencies) •irregularities in the structure (glitches, see later) •surface effects (structural and dynamical effect of convection cannot be neglected at the surface) [Garcia & Ballot 19] [Chaplin & Miglio 13] 19 •Surface layers of the star crudely included in asymptotic formulation, badly modelled in numerical calculations of frequencies è effect of surface convection cannot be neglected • Affects high frequency modes (because higher upper turning point) •Essentially independent of angular degree if rescaled with mode inertia •If not corrected, leads to biased estimation of stellar global properties Asymptotic p-modes Surface effects (1) [Schmidt & Basu 15] Scaling factor 𝑄𝑛ℓ =ℐ 𝑛ℓ /ℐ ℓ=0,ℐ≡ ∫𝒱 |𝝃|2 20 •Separation ratios insensitive on surface layers [Roxburgh & Vorontsov 03, Oti-Floranes+05] , but lose information about surface layers •Empirical corrections to remove surface effects [Kjeldsen+07, ChristensenDalsgaard 12, Ball & Gizon 14, Sonoi+15] Asymptotic p-modes Surface effects (2) [Oti-Floranes+05] Separation ratios 𝑟01 =𝑑(2) 01 /!Δ𝜈" Empirical surface corrections 𝜈obs −𝑟𝜈model =𝑎!𝜈obs 𝜈0"𝑏 𝜈obs −𝑟𝜈model = 1 ℐ)𝑎−1"𝜈obs 𝜈0#−1 +𝑎3"𝜈obs 𝜈0#3$ [Kjeldsen+08] [Ball & Gizon 14] 21 Convection modifies the equilibrium structure of the surface layers è structural surface term •turbulent pressure adds to gas pressure (pturb / pgas ~ 15% for the Sun) è radius elevated by ~ ½ density scale height •temperature gradient modified by convection è radius elevated by another ~ ½ density scale height • mitigated by patched models Convection modifies the propagation of sound waves in surface layers è modal surface term •Needs time-dependent treatment of convection •Mainly through MLT [Balmforth 92b, Grigahcène+05, Houdek+17, Sonoi+17] è limitations (many free parameters, hard to disentangle from other sources of frequency perturbations) •Alternative approaches [Xiong+00, Belkacem+19, Zhou+19, Schou & Birch 19, Kupka+22, Ahlborn+22, Philidet+21b, Philidet+22] è remains an open problem Asymptotic p-modes Surface effects (3) [Rosenthal 99] Obs. – patched model 22 •In real oscillation spectra - only first few angular degrees (up to 3 or 4) visible è integrated light of higher order modes cancel out - only a few radial orders are visible (not many more than 10) è due to excitation process •Solar-like modes linearly stable, not self-excited •Oscillations continually excited by vigorous, turbulent flows close to the surface è stochastic excitation , need convective envelope •Oscillations damped over time by effect of convection and non-adiabaticity Amplitude of solar-like p-modes Driving/damping balance Stochastically excited damped harmonic oscillator d2𝐴 d𝑡2+2𝜂 d𝐴 d𝑡+𝜔2 0𝐴=𝑆(𝑡) !!!" 𝐴(𝜔)!!! 2 = !!!" 𝑆(𝜔)!!! 2 /4𝜔2 0 #𝜔2−𝜔2 0$+𝜂2 linewidth ∝𝜂 Mode physical amplitude Mode energy = balance between excitation and damping 𝐴2 mode ∝𝒫 𝜂 23 •Source term comes from turbulent fluctuations of Reynolds stress in momentum equation •Excitation rate related to statistics of turbulent velocity, i.e. turbulent spectrum Amplitude of solar-like p-modes Stochastic excitation Mode physical amplitude Mode energy = balance between excitation and damping 𝐴2 mode ∝𝒫 𝜂 S(r,𝑡)=−∇!𝜌uconv ⊗uconv " 𝒫∝∫𝑀∗ 0 d𝑚"""" d𝜉𝑟 d𝑟"""" 2∫+∞ 0 d𝑘𝐸(𝑘)2 𝑘2∫+∞ −∞ d𝜔𝑘𝜒𝑘(𝜔0+𝜔𝑘)𝜒𝑘(𝜔𝑘) Dependence on eigenfunctions Dependence on spatial turbulent spectrum Dependence on temporal turbulent spectrum [Samadi+15] 24 • Asymptotic description : same strategy as with one cavity, but star subdivided in 4 regions, each containing 1 turning point •Limit q = 0: either… - p-mode phase Θp = (n+1/2)π è pure p-mode - g-mode phase Θg = nπ è pure g-mode Asymptotic low-frequency modes Resonant condition for mixed modes oscillating evanescent Resonant condition for mixed modes tan )∫𝑟𝑢,𝑝 𝑟𝑙,𝑝 𝑘𝑟d𝑟#tan−1)∫𝑟𝑢,𝑔 𝑟𝑙,𝑔 𝑘𝑟d𝑟#=𝑞 Coupling factor 𝑞=1 4exp )−2∫𝑟𝑙,𝑝 𝑟𝑢,𝑔 𝑘𝑟d𝑟# Asymptotic: needs many radial nodes in each cavity Only valid for weak coupling [Takata 2016] (𝑞!1) 31 •Asymptotic p-mode and g-mode phases as 1-cavity •g-mode period spacing perturbed by coupling with p-modes Asymptotic low-frequency modes Stretched period spacing Pure p-mode phase Pure g-mode phase Θ𝑔=𝜋 ΔΠ1!1 𝜈 − 1 𝜈𝑔" Θ𝑝=𝜋 Δ𝜈!𝜈−𝜈𝑝" ν g arbitrary pure g-mode frequency ν p arbitrary pure p-mode frequency Stretching function Π𝑛+1,1−Π𝑛,1 ΔΠ1 =!""""# 1+1 𝑞 𝜈2ΔΠ1 Δ𝜈 cos2𝜋 ΔΠ1$1 𝜈−1 𝜈𝑔% cos2𝜋 Δ𝜈&𝜈−𝜈𝑝'())))* −1 ≡𝜁(𝜈) •Stretching function corresponds to amount of mode energy in g-mode cavity •Pure g-modes are close to ζ = 1, pure p-modes close to local minima •Coupling factor q determines narrowness of bumps pure g-modes pure p-modes 32 •In classical echelle diagram, dipolar mixed modes do not form a vertical ridge •Need to stretch the periods to recover regular pattern •Mixed mode density as a proxy for evolution Asymptotic low-frequency modes Stretched echelle diagram Stretched periods 𝜏(𝜈)=∫𝜈1 𝜁(𝑢) d𝑢 𝑢2 Mode density 𝒩≡ Δ𝜈 ΔΠ1𝜈2 max 𝒩modes [Mosser+15] 33 •Asymptotic mixed modes characterised by three main quantities - period spacing Π 1 - coupling factor q - g-mode extra phase (or gravity offset) ε g •Period spacing measures core mass (size and density) •Allows to discriminate between hydrogenburning shell phase (RGB) and helium-burning core phase (red clump) è asteroseismic HR diagram Asymptotic low-frequency modes Mixed modes asteroseismic HR diagram evolution Π1[𝑠] Δ𝜈[𝜇Hz] [Mosser+14] 34 •Coupling factor sensitive on region between hydrogen burning shell and base of convective envelope à radial extent à scale height of Brünt-Väisälä and Lamb frequencies •Expected coupling peaks at subgiant-RGB transition •In general, dense core = strong coupling Asymptotic low-frequency modes Coupling factor evolution [Mosser+17] 35 •Gravity offset εg related to phase change incurred by internal gravity waves at the turning points of the g-mode cavity •Asymptotic, weak coupling formulation does not work [Takata 2006, Takata et al. 2016, Pinçon et al. 2019] •Signature of Brünt-Väisälä profile at base of convective envelope •Sharp transition of gravity offset when starting RGB ascension Asymptotic low-frequency modes Gravity offset evolution [Pinçon+19] 36 •The perturbed equation of motion can be expressed solely as a function of displacement ξ •Introduce Hilbert space of vector functions in whole star, and arming it with inner product Linear perturbation theory: basics Adiabatic limit and operator symmetry −𝜔2𝝃+ 1 𝜌0 ∇𝑝"+∇Φ"+𝜌" 𝜌0 ∇Φ0=0 Linear operator for stellar oscillations 𝜔2𝝃=ℒ[𝝃] Depends on equilibrium profiles ρ 0(r), p0(r), Γ 1(r) Inner product ⟨𝝃1|𝝃2⟩≡∫𝒱 𝝃∗ 1·𝝃2𝜌0d3r            𝜌′=−∇·(𝜌0𝝃) 𝑝′=−Γ1𝑝0∇·𝝃−𝝃·∇𝑝0 Φ′=𝐺∫𝒱 ∇·(𝜌0𝝃) |r′−r|d3r′ Linear operator is symmetric (with proper BC) !𝝃1"""ℒ[𝝃2]#=!ℒ[𝝃1]"""𝝃2# 37 Eigenfrequency action •Squared eigenfrequencies are real è mode-like solutions (ω2 > 0) have zero imaginary part è modes cannot grow nor decay exponentially, no self-excitation, no damping è both described in non-adiabatic setting •Eigenfunctions are orthogonal è •Eigenfrequencies follow a variational principle : •eigenfunctions are local extrema of the action Σ defined by •First-order perturbation of the eigenfunction leads to second-order perturbation of the action Linear perturbation theory: basics Variational principle !𝝃𝑛,ℓ,𝑚" "𝝃𝑛!,ℓ!,𝑚!#=𝛿𝑛𝑛!𝛿ℓℓ!𝛿𝑚𝑚! Σ[𝝃]≡!𝝃"""ℒ[𝝃]# !𝝃"""𝝃# Variational principle Σ[𝝃0+𝛿𝝃]=𝜔2 0+𝒪!|𝛿𝝃|2" Can be used to refine numerical computation of eigenfrequencies 38 •Direct consequence of variational principle: how does perturbing the linear operator perturb the eigenfrequencies? Linear perturbation theory: basics Eigenfrequency perturbations Linear frequency perturbations Σ[𝝃0+𝛿𝝃]= 𝜔2 0!𝝃0"""𝝃0#+2𝜔2 0Re $!𝝃0"""𝛿𝝃#%+!𝝃0"""𝛿ℒ[𝝃0]#+𝒪$|𝛿𝝃|2% !𝝃0"""𝝃0#+2Re $!𝝃0"""𝛿𝝃#%+𝒪$|𝛿𝝃|2%=𝜔2 0+!𝝃0"""𝛿ℒ[𝝃0]# !𝝃0"""𝝃0#+𝒪$|𝛿𝝃|2% →→→+𝛿→,𝜔2 0→𝜔2 0+𝛿𝜔2 ,𝝃0→𝝃0+𝛿𝝃0 𝛿𝜔 = !𝝃0"""𝛿ℒ[𝝃0]# 2𝜔0!𝝃0"""𝝃0# Can be used to related model uncertainty in ρ0(r), p0(r), Γ1(r) to difference between theoretical and observed frequencies See Gaël’s lecture on Wednesday First order term cancels out because of variational principle Independent of perturbed eigenfunction 39 •Without rotation, frequency does not depend on m (each mode is degenerated (2l+1)-fold •Naive picture: •wave has frequency ω0 in corotating frame, goes as •in observing frame, longitude is •so wave goes as •Helps with mode identification •But this does not account for non-inertial rotating frame è Coriolis force è Centrifugal force è what about differential rotation? Application to rotation Rotational splittings cos(𝜔0𝑡−𝑚𝜙corot) 𝜙=𝜙corot +Ω𝑡 cos((𝜔0+𝑚Ω)𝑡−𝑚𝜙) Degeneracy lifted because of rotation [Aerts et al. 2024] In naive picture, this is Ω 40 Angular variable separation without rotation Angular variable separation with uniform rotation Spherical harmonics Hough functions •If splitting too high, multiple crossing between multiplets è rotation rate cannot be inferred from splittings, even if linear perturbation valid •In sub-inertial regime (|ω| < 2Ω), Coriolis force cannot be treated perturbatively è gravito-inertial modes, pure inertial modes • Radial action of Coriolis force often neglected compared to buoyancy è Traditional Approximation of Rotation (TAR) [Lee&Saio 1987, Bildsten+96, Lee&Saio97, Townsend03a, Savonije05, Mathis09, Mathis&Prat19] Rotation beyond perturbative theory Traditional approximation of rotation [Gehan+18] 𝑋ang(𝜃,𝜙)=𝑌𝑚 ℓ(𝜃,𝜙)=𝑃𝑚 ℓ(cos 𝜃)exp𝑗𝑚𝜙,Λ=ℓ(ℓ+1) !d d cos 𝜃"sin2𝜃 1−𝑠2cos2𝜃 d d cos 𝜃#+1 1−𝑠2cos2𝜃"𝑚𝑠 cos 𝜃$1+𝑠2cos2𝜃% 1−𝑠2cos2𝜃 − 𝑚2 sin2𝜃#+Λ&𝑋ang =0 !d d cos 𝜃"sin2𝜃d d cos 𝜃#− 𝑚2 sin2𝜃 +Λ$𝑋ang =0 𝑋ang(𝜃,𝜙)=Θ 𝑠,Λ,𝑚(cos 𝜃)exp𝑗𝑚𝜙,Λ𝑠,ℓ,𝑚→→ →→ 𝑠→0ℓ(ℓ+1) (𝑠=2Ω/𝜔) •Angular part of eigenfunctionsmore confined around equator as s increases Radial wave equation identical to non-rotating case 47 •Splittings are so high that (ell,m)-sequences of modes cluster separately from one another •Rotation tilts the period spacing of retrograde (m<0) modes upwards, and of prograde (m>0) modes downwards Rotation beyond perturbative theory Effect of rotation on low-frequency modes Asymptotic period spacing in TAR ΔΠco =Π0 !Λ𝑠,ℓ,𝑚"1+ 1 2 d ln Λ d ln 𝑠#−1 Inertial frame periods VS corotating periods Πinertial =Πco 1−𝑚Πco/𝑃rot Slope related to rotation rate [Van Reeth+15] [Ouazzani+17] 48 •Extra class of oscillations: inertial modes •Without rotation, zero-frequency solutions characterised by purely horizontal motions (toroidal modes) •In rotating stars, frequency is not zero, and of the order of the rotation period (< 2Ω) è sub-inertial regime Rotation beyond perturbative theory Gravito-inertial modes (1) [Ouazzani+20] 49 •In γ-Doradus (fast rotators), inertial mode frequencies of the convective core coincide with g-mode frequencies of radiative envelope è coupling efficient if modes have similar geometry •Rotation distorts period spacing , and coupling with core inertial modes creates dips in g-mode period spacing [Saio et al. 2018, 2021 ; Ouazzani et al. 2021] •Dips occur at specific core spin parameters s = 2Ωcore/ ω è allows to probe rotation of the core •Dip shape gives info about chemical composition gradient at convective/ radiative interface [Tokuno and Takata 2022] , radial differential rotation or core magnetism [Aerts and Mathis 2023, Galoy et al. 2024, Barrault et al. 2025] Gravito-inertial modes (2) Rotation beyond perturbative theory [Ouazzani+20] [Saio+18] 50 •Accounting for Lorentz force in momentum equation + induction equation •Linear perturbation theory predicts frequency change under influence of magnetic field Application to magnetic fields Effect of magnetic fields on frequencies −𝜔2𝝃+ 1 𝜌0 ∇𝑝"+∇Φ"+𝜌" 𝜌0 ∇Φ0=0 −𝜔2𝝃+1 4𝜋𝜇0𝜌0 [(∇×B#)×B0+(∇×B0)×B#]+1 𝜌0 ∇𝑝#+∇Φ#+𝜌# 𝜌0 ∇Φ0=0 B#=∇×(𝝃×B0) Frequency perturbed by internal magnetic field 𝛿𝜔 =−1 8𝜋𝜇0𝜔0 ∫𝑅 0 𝝃∗·"(∇×(∇×(𝝃×B0))) ×B0+(∇×B0)×(∇×(𝝃×B0))#𝑟2d𝑟 ∫𝑅 0 |𝝃|2𝜌0𝑟2d𝑟 ∼− 1 8𝜋𝜇0𝜔0 ∫𝑅 0$$$∇×(𝝃×B0)$$$ 2 𝑟2d𝑟 ∫𝑅 0 |𝝃|2𝜌0𝑟2d𝑟 for potential field B0 51 •Signature of core magnetic fields on evolved stars mixed mode detectable [Loi2020,2021, Bugnet+21, Bugnet22, Li+22b, Mathis&Bugnet23, Deheuvels+23, Hatt+24, Li+24a] •Signature takes the form of multiplet asymetry •Unlike rotation, magnetic fields also impact zonal (m=0) modes •Magnetic fields always increase mode frequencies Application to magnetic fields Multiplet asymmetry caused by magnetic fields without rotation with rotation [Bugnet+21] 52 •Study of seismic signature of internal magnetic field usually confined to dipolar modes Application to magnetic fields What magnetic field do we see? Dipolar triplet shift !𝛿𝜔"ℓ=1= ℐ 𝜇0𝜔3∫𝑅 0 𝐾𝐵(𝑟)𝐵2 𝑟d𝑟 𝐾𝐵(𝑟)∝1 𝜌0"𝑁 𝑟#3 Average triplet shift sensitive on radial field strength at the Brünt-Väisälä frequency maximum è H-burning shell Dipolar triplet distorsion (è asymmetry) 𝑎≡1 3!(𝛿𝜔1,1−𝛿𝜔1,0)−(𝛿𝜔1,0−𝛿𝜔1,−1)"=∫𝑅 0 d𝑟∫𝜋 0 sin 𝜃d𝜃∫2𝜋 0 d𝜙𝐾𝐵(𝑟)𝐵2 𝑟(𝑟,𝜃,𝜙)𝑃2(cos 𝜃) ∫𝑅 0 d𝑟∫𝜋 0 sin 𝜃d𝜃∫2𝜋 0 d𝜙𝐾𝐵(𝑟)𝐵2 𝑟(𝑟,𝜃,𝜙) Triplet asymmetry sensitive on angular structure of magnetic field è probes B topology and inclination 53 •Distorsion of dipolar ridges particularly visible in echelle diagram •Splitting and triplet asymmetry both go down as frequency increase Application to magnetic fields Magnetic signature in echelle diagram [Deheuvels+23] [Hatt+24] m = 1 m = 0 m = -1 54 •Seismic signature of core magnetic fields in evolved stars seems to indicate a decrease along evolution •For the moment, too much degeneracy to measure topology (would need l=2 modes as well [Das+24] ) Application to magnetic fields Magnetic field strength VS evolution [Deheuvels+23] [Hatt+24] •Possible to extend to rapidly rotating stars in TAR [Lignières+24, Barrault+24] è magnetic field detection in γ-Doradus? 55 •Perturbation in ρ0(r), p0(r), Γ1(r) è perturbation in linear operator è perturbation in mode frequencies •Rapid (localised) variation of sound speed profile (treated as discontinuities) cause frequency variations called acoustic glitches [Gough 90] Application to glitches Acoustic glitches Glitches caused by ionisation zones Partial ionisation of abundant elements (H, HeI, HeII) causes sharp decrease of Γ1, i.e. of sound speed Glitches caused by convective/radiative interface Transition between two energy transport processes causes sharp change in thermal gradient, i.e. in sound speed gradient Base of convective envelopes [Monteiro+94, Roxburgh & Vorontsov 94, Ballot+04, Roxburgh 09, Mazumdar+12, Lebreton & Goupil 12, Verma+17, Pereira+17, Farnir+19, Deal+23, Deal+25] Top of convective cores [Miglio+08, Goupil+11, Silva Aguirre+13, Cunha+15, Appourcheaux+15, Van Reeth+15, Mosser+15, Deheuvels+16, Cunha+19, Mombarg+22, Hatta 23] Helium second-ionisation zones [Houdek & Gough 07, Miglio+10, Mazumdar+12, Verma+17, Farnir+19, McKeever+19, Dréau+21] 56 •Oscillations in pre-main sequence stars •Oscillations in compact objects •Oscillations in binaries (tidal effects, …) •Oscillations in fast rotators deformed by centrifugal force •Helioseismology •Classical pulsators è see you on Thursday •Seismic inferrence of stellar parameters •Seismic inversion (of structure or rotation) è see you on Wednesday Conclusion What we have not talked about… 63 Reviews papers… •… on solar-like oscillations in general •Chaplin and Miglio (2013) , Annual Review of Astronomy and Astrophysics , vol. 51, issue 1, pp. 353-392 •Garcia and Ballot (2019) , Living Reviews in Solar Physics , Volume 16, Issue 1, article id. 4 •Jackiewicz (2021) , Frontiers in Astronomy and Space Sciences , Volume 7, id.102 •Aerts (2021) , Reviews of Modern Physics , Volume 93, Issue 1 •… on oscillations in evolved red giants •Hekker and Christensen-Dalsgaard (2017) , The Astronomy and Astrophysics Review , Volume 25, Issue 1, article id.1 •Basu and Hekker (2020) , Frontiers in Astronomy and Space Sciences , Volume 7, id.44 •… on oscillations in fast rotators •Aerts & Tkachenko (2024) , Astronomy & Astrophysics , Volume 692, id.R1, 38 pp. Books •Unno et al. (1989) , Non-radial oscillations of stars , University of Tokyo Press •Aerts, Christensen-Dalsgaard and Kurtz (2010) , Asteroseismology , Springer Science & Business Media •Proceedings on Evry Schatzman School 2014 , Asteroseismology and next generation stellar models , EDP Sciences Conclusion To go deeper… 64 Theory of stellar oscillations II. Classical pulsators J. Philidet, LIRA / Observatoire de Paris Ecole Evry Schatzmann 2025 •Classical pulsations (or stellar variability) à thermally unstable oscillations •Variable stars found in very specific regions of HR diagram (instability strips) à why? •Instability strips found all over HR diagram à very different classes of variables • Non-adiabatic effects are essential Introduction Oscillations in HR diagram [Aerts+10] log𝑇 eff log 𝐿/𝐿! 1 •Validity of adiabatic approximation depends on comparison between dynamical and thermal timescales •In most of the star τdyn << τth è oscillations are adiabatic (no time to transfer energy) •In very top layers, τdyn >> τth è energy transfer between oscillations and medium is instantaneous •Small region beneath surface where τdyn ~ τth è transition region Instability criteria Departure from adiabaticity Dynamical time scale 𝜏dyn ∼!𝐺𝑀 𝑅3"−1/2 Thermal time scale 𝜏th =𝑐𝑣𝑇Δ𝑚 𝐿 Time it takes sound to travel across stellar diameter Time it takes for shell to give off all its energy [Samadi+15] 2 •Brute-force approach would be to solve the full non-adiabatic oscillations equations •Usually too complicated, and not necessary to determine instability criteria Instability criteria Non-adiabatic oscillations equations 𝛿𝜌 𝜌0 +∇·𝝃=0 𝜕2𝜉𝑟 𝜕𝑡2+1 𝜌0 𝜕𝑝! 𝜕𝑟+!𝛿𝜌 𝜌0 −𝜉𝑟 d ln 𝜌0 d𝑟"𝑔0+𝜕Φ! 𝜕𝑟=0 𝜕2𝝃ℎ 𝜕𝑡2+1 𝜌0 ∇ℎ𝑝!+∇ℎΦ!=0 𝛿𝜌0 𝜌0 −1 Γ1 𝛿𝑝 𝑝0 =−∇ad 𝜌0𝑇0 𝑝0 𝛿𝑠nad 𝑇0 𝜕𝛿𝑠nad 𝜕𝑡=𝛿!𝜖𝑁−1 𝜌∇·Frad"+equations for nuclear reaction rate and energy fluxes ∇2Φ!=4𝜋𝐺𝜌! Extra part compared to adiabtic equations seen on Monday 3 •In general, departure from adiabaticity remains small è linear perturbation theory (again) •We are interested in mode growth rate γ à imaginary part of eigenfrequency •Since ω0 real, we simply have: Instability criteria Departure from adiabaticity as small perturbation Non-adiabaticity as small perturbation 𝛿𝜔2=−1 𝑗𝜔 ∫𝒱 𝝃∗·∇"(Γ3−1)#𝜖′ 𝑁−1 𝜌0 ∇·F′ rad −1 𝜌0 ∇·F′ conv$%d3r ∫𝒱 |𝝃|2𝜌0d3r −𝜔2𝝃+ 1 𝜌0 ∇𝑝"+∇Φ"+𝜌" 𝜌0 ∇Φ0=0 −𝜔2𝝃+ 1 𝜌0 ∇𝑝"+ 1 𝜌0 ∇𝑝" nad +∇Φ"+𝜌" 𝜌0 ∇Φ0=0 𝑝! nad =−1 𝑗𝜔(Γ3−1)!𝜖! 𝑁−1 𝜌0 ∇·F! rad −1 𝜌0 ∇·F! conv" Mode growth rate 𝛾= Im(𝛿𝜔2) 2𝜔0 4 •Mode growth rate takes the form of work integral Instability criteria Work integral (1) Mode growth rate as work integral 𝛾=1 2𝜔2 0 ∫𝒱 (Γ3−1)Re "𝛿𝜌∗ 𝜌0 𝛿#𝜖𝑁−1 𝜌0 ∇·Frad −1 𝜌0 ∇·Fconv$%d3r ∫𝒱 |𝝃|2𝜌0d3r Radial mode growth rate 𝛾=1 2𝜔2 0ℐ Re !∫𝑀 0 (Γ3−1)𝛿𝜌∗ 𝜌0#𝛿𝜖𝑁−d𝛿𝐿rad d𝑚−d𝛿𝐿conv d𝑚$d𝑚% Sources of entropy fluctuations include •fluctuation in nuclear reaction rate δε •fluctuation in radiative luminosity δ Lrad •fluctuation in convective luminosity δ Lconv 5 • Sign of γ defines whether the mode is stable (γ < 0) or unstable (γ > 0) • Sign of integrand defines whether a given region of the star is driving (negative integrand) or damping (positive integrand) • Excitation or damping occurs at transition region where τ dyn ~ τ th à in deep interior, τdyn << τth à entropy does not have time to change as the wave passes à γ ~ 0 à At the surface, τdyn >> τth à wave is too slow to break flux conservation (δε - d(δLrad+ δLconv)/dm = 0) à γ ~ 0 Instability criteria Work integral (2) Excitation mechanisms Usually, a star has only one of the three coloured terms causing an instability •ε -mechanism •κ -mechanism, γ -mechanism • Convective flux blocking, convective driving Radial mode growth rate 𝛾=1 2𝜔2 0ℐ Re !∫𝑀 0 (Γ3−1)𝛿𝜌∗ 𝜌0#𝛿𝜖𝑁−d𝛿𝐿rad d𝑚−d𝛿𝐿conv d𝑚$d𝑚% 6 •For better physical interpretation, growth rate can be recast as •Phase lag between δ⍴ and δp determines stability è if maximum pressure occurs after maximum density, stable è if maximum pressure occurs before maximum density, unstable • Same principle as a heat-engine è what is causing the engine? è where is the engine within the star? è under which conditions is the engine powerful enough? Instability criteria Heat-engine mechanism 𝛾= 1 2𝜔0ℐIm !∫𝑀 0 𝛿𝜌∗ 𝜌0 𝛿𝑝 𝜌0 d𝑚# corresponds to ‘PdV’ work exerted on the oscillation by the medium Unstable Stable Adiabatic 𝛿𝜌 𝛿𝑝unstable 𝛿𝑝adia 𝛿𝑝stable Im(𝛿𝜌∗𝛿𝑝) 7 Driving mechanisms Pulsators driven by convective effects Convective flux blocking γ-Doradus Convective driving ZZ Ceti (White-dwarf pulsators) [Aerts+10] log𝑇 eff log 𝐿/𝐿! Driving occurs at BSCZ Driving occurs in shallow convective envelope è what is causing the engine? ✅ è where is the engine within the star? ✅ è under which conditions is the engine powerful enough? ❓ 14 •Self-excitation can only be efficient at transition region where τdyn ~ τth à unstable oscillations can only occur if transition region coincides with driving region Instability strips Transition region (1) Relevant time scale ratio 𝜙≡𝜏th 𝜏dyn ∼ 𝑐𝑣𝑇Δ𝑚 Π𝐿 ɸ << 1 à strongly non-adiabatic ɸ >> 1 à adiabatic ɸ ~ 1 à transition Energy stored above local shell / energy radiated by shell in one mode period [Samadi+15] Temperature of transition region 15 •At what temperature within the star is the transition region located? •Unstable modes occur when… … TTR ~ ionisation temperature (δ-Scuti, RR Lyrae, Cepheids, SPB, β Cep, …) … TTR ~ temperature of BSCZ (γ-Doradus, white dwarf pulsators DAV), … Instability strips Transition region (2) 𝑃TR ∝ 𝑀 𝑅4Δ𝑚 Π∝𝑀−1/2𝑅3/2 𝑃∝𝑇𝑛+1∝𝑇4(if radiative) Hydrostatic equilibrium à Homology à Polytrop à Temperature of transition region 𝜙∝𝑇5𝑅5/2 𝑀1/2𝐿 𝑇 TR ∝𝑀1/10𝐿1/5𝑅−1/2 16 •At given mass and luminosity: •Position of transition region compared to ionisation depends on R à existence of a critical radius •Less evolved stars à smaller radius à transition region is too deep compared to ionisation region Instability strips Critical radius (1) 𝑇 TR ∝𝑅−1/2 [Aerts+10] log𝑇 eff log 𝐿/𝐿! [Gastine 09] star 17 •More evolved stars à higher radius à transition region is not deep enough to overlap with ionisation region Instability strips Critical radius (2) log𝑇 eff log 𝐿/𝐿! [Gastine 09] star 18 [Aerts+10] •There is a small range of stellar radii where transition and ionisation regions overlap à self-excitation becomes efficient, unstable modes Instability strips Critical radius (3) log𝑇 eff log 𝐿/𝐿! [Gastine 09] star 19 [Aerts+10] •Concerns γ-Dor, δ-Sct, roAp, SPB, and βCep stars •Not the same self-excitation mechanisms, but their instability strips ovelap (especially γ-Dor and δ-Sct) à hybrid pulsators •γ-Dor pulsate in high-order g-modes (0.5 – 3 days) •δ-Sct pulsate in p-modes (20 min – 8 hrs), but most also in g-modes •SPB pulsate in high-order gravito-inertial modes (0.5 – 4 days) •βCep pulsate in p-modes (2–8 hrs), some also in gravito-inertial modes Main sequence OBAF pulsators Overlapping instability strips log𝑇 eff log 𝐿/𝐿! 20 [Aerts+10] •In δ-Sct, often hundreds of unstable modes (p, g, mixed) + frequency combinations due to non-linear coupling à mode identification extremely challenging, but sometimes possible [Bedding+20] •Extra spurious frequencies due to modulation of mode amplitude in time [Bowman 16] •Some δ-Sct have stable solar-like oscillations in addition to unstable modes à Δ ν measurable as well [Antoci+11] Main sequence OBAF pulsators p-mode identification in δ-Sct [Mantegazza+12] [Antoci+11] 21 •Cold main-sequence self-excited pulsators (δ-Sct, γ-Dor) lie within classical instability strip •Blue (hot) edge of instability strip well predicted by fully non-adiabatic calculations, but red (cold) is not well predicted •Convective time scale smaller than thermal time scale near red edge à convection no longer frozen à need time-dependent convection treatment • Allowed to constrain convective properties Main sequence OBAF pulsators Insight from red edge of classical instability strip [Pamyatnykh+00] [Dupret+04] 22 •Rotation affects not only mode period, but also mode stability •Faster rotation à range of unstable prograde (resp. retrograde) modes shifted to shorter (resp. longer) periods • Extension of γ -Dor instability strip by rotation fills theoretical gap with δ-Scuti strip Main sequence OBAF pulsators Insight from γ-Doradus instability strip [Bouabid+13] 23