Precision and accuracy in stellar oscillations modeling

1. Precision and accuracy instellar oscillations modeling Marc-Antoine Dupret, R. Scuflaire, M. Godart, R.-M. Ouazzani, … ESTER workshop, Toulouse

2. Precision: Precise solution of givendifferentialequations Precision and accuracy in stellar oscillations modeling Accuracy: Set of differential equations accurately modeling stellar oscillations ESTER workshop, Toulouse

3. Numerical analyst point of view: • Increasing the number of mesh points: • “With 5000 mesh points, oscillation computations are precise …” • Not enough in evolved stars Precision in stellar oscillations modeling • Increasing the precision of the numerical scheme: • High order of precision of finite differences. • But don’t forget numerical stability (Reese 2013, A&A 555, 12, • GYRE: Townsend & Teitler 2013, MNRAS 435, 3406) • Spectral approach with orthogonal polynomials (TOP, ESTER, …) • But sharp variations in stellar interiors Multi-domain • (convective boundaries, opacities, …), approach • huge core-surface contrast This is not always enough … ESTER workshop, Toulouse

4. Lagrangian or Eulerian perturbations ? General rule: Compare the orders of magnitude and choose the smallest Precision in stellar oscillations modeling:choosing the good variables Gravitational potential F The Cowling approximation is not so bad Always use the Eulerian perturbation of F Pressure P In dense cores, |P’| << |d P| Use the Eulerian perturbation of P ESTER workshop, Toulouse

5. Lagrangian or Eulerian perturbations ? Pressure In a g-mode cavity where Precision in stellar oscillations modeling:choosing the good variables The Eulerian perturbation of P must be used ESTER workshop, Toulouse

6. Lagrangian or Eulerian perturbations ? LagrangianEulerianif and only if hydrostatic equilibrium of the structure model Precision in stellar oscillations modeling:choosing the good variables In high density contrast stars, 10.000-50.000 points required Interpolating the structure models ? No: hydrostatic equilibrium too imprecise … Non-radial oscillations in high-density contrast stars (blue and red supergiants): - Eulerian pressure perturbation in the g-mode cavity - Models in hydrostatic equilibrium with enough mesh points (avoid interpolations) ESTER workshop, Toulouse

7. Lagrangian or Eulerian perturbations ? Non-adiabatic oscillations Precision in stellar oscillations modeling:choosing the good variables in near-surface layers must be used as variable in non-adiabatic oscillation codes or Lagrangian perturbation of state equation and opacities are simpler better to use them in the superficial non-adiabatic layers ESTER workshop, Toulouse

8. The first integral of Takata, a good test of precision Dipolar modes Equation of momentum conservation for the center of mass of each sphere Mr: Precision in stellar oscillations modeling Takata 2005, PASJ 57, 375 • Reduce by two orders the differential system • Can be used as an a posteriori precision test in each layer • Valid in the full non-adiabatic case Could be generalized to fast rotating stars Good test of precision of non-perturbative oscillation codes for fast rotating stars (ACOR, TOP, …) ESTER workshop, Toulouse

9. The first integral of Takata, a good test of precision Proof: Precision in stellar oscillations modeling Integration on an arbitrary volume: || ESTER workshop, Toulouse

10. The first integral of Takata, a good test of precision First integral (general case): Precision in stellar oscillations modeling Dipolar mode, sphere: ESTER workshop, Toulouse

11. Using asymptotic JWKB solutions Adiabatic-Cowling approximation, g-mode cavitywith : Numerousnodes in high density contrast stars Precision in stellar oscillations modeling Full non-adiabatic case: seeDziembowski (1977) Continuousmatch to the numericalsolution Does not increaseprecision, but decreases the number of mesh points Usefulin the core of high densitycontraststars Quasi-adiabatic approximation: Power lost by the mode through radiative damping: ESTER workshop, Toulouse

12. Usual approximations in oscillation equations: Adiabaticity, slow rotation, no magnetic field, no tidal effects Magneticfield: Lorentz force + perturbed induction equation Direct effecton frequencies, mode geometry and driving Perturbative approach: see e.g. Hasan et al. 1992, 2005; Cunha & Gough 2000 Non-perturbativeapproach: seee.g. Bigot & Dziembowski (2003), Saio (2005) Accuracy in stellar oscillations modeling Tidal influence of a companion: Acts as a forcing term in oscillation equations, boosting some modes through resonances and complicating spin-orbit synchronisation:Savonije et al. 1995, … Affects frequencies: Saio (1981), … ESTER workshop, Toulouse

13. Rotation:Coriolis + centrifugal deformation • Major effect on frequencies, mode geometry and driving • Perturbativeapproach: see e.g. Dziembowski & Goode (1992), 2nd order • Soufi et al. 1998, 3rd order Accuracy in stellar oscillations modeling • Non-perturbativeapproach: • Traditional approximation (sphericalsymetry, rigid rotation, horizontal Coriolis) • Separabilityvery efficient computations • Not sobad for g-modes of moderaterotators(Ballot et al. 2011) • Perturbative structure models + full spectral expansion: Lee & Baraffe (1995), … • Full 2D structure models + full spectral expansions: • Major works of the Toulouse team (Dintrans, Lignières, Reese, Ballot 2000-2014), • Seetheirtalks ! Ouazzani et al. (2012) • 2D structure models + oscillations withfinitedifferences: Clement 1998, • Deupree 1995, … ESTER workshop, Toulouse

14. Non-adiabatic-energetic aspects in oscillations modeling: • Predictionsof mode excitation + normalized amplitudes and phases • Heat engine pulsators: Range of unstable modes and instability strips • Constrainsopacities, time-dependent convection • Stochastic excitation:Mode life-times line-widths in power spectrum • Constrains time-dependent convection Accuracy in stellar oscillations modeling Improveaccuracy of theoreticalfrequenciesthrougha good oscillations modeling in the superficiallayersPhysicaltreatment of surface effects Important for high-order p-modes (e.g. solar-like oscillations) ESTER workshop, Toulouse

15. Non-adiabatic-energetic aspects in oscillations modeling: • Main challenges: • Non-adiabaticity + rotation See talk of Daniel Reese • Non-adiabaticity + magnetism: Saio (2005) • Oscillations in the atmosphere: Dupret et al. (2002) • Time-dependent convection • Non-linear radial oscillations: e.g. Stellingwerf 1982, Kuhfuß 1986 • Linear oscillations: • Gough 1977Balmforth 1992Houdek et al. (1999-…) • Unno 1967 Gabriel 1996 Grigahcène, Dupretet al. (2005-…) • Beyond the mixing-lengththeory: Xiong et al. (1997-2010) • All thesetheoriesintroduce free parameters ! Accuracy in stellar oscillations modeling ESTER workshop, Toulouse

16. Non-adiabatic-energetic aspects in oscillations modeling: • Time-dependent convection • All currenttheoriesintroduce free parameters or are contradicted by observations … • Whatshouldbedone ? • Whathydrodynamical simulations are telling us ? (Gastine & Dintrans 2011, • Mundprecht et al. 2012) • Goingbeyond the MLT, yes but … Accuracy in stellar oscillations modeling ESTER workshop, Toulouse