1. Precision and accuracy in stellar oscillations modeling Marc-Antoine Dupret, R. Scuflaire, M. Godart, R.-M. Ouazzani, … 11 June 2014ESTER workshop, Toulouse1

2. 11 June 2014ESTER workshop, Toulouse2 Precision and accuracy in stellar oscillations modeling Precision: Precise solution of given differential equations Accuracy: Set of differential equations accurately modeling stellar oscillations

3. 11 June 2014ESTER workshop, Toulouse Precision in stellar oscillations modeling Numerical analyst point of view: Increasing the number of mesh points: “With 5000 mesh points, oscillation computations are precise …” Not enough in evolved stars 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 … 3

4. 11 June 2014ESTER workshop, Toulouse Precision in stellar oscillations modeling: choosing the good variables Lagrangian or Eulerian perturbations ? General rule: Compare the orders of magnitude and choose the smallest 1.Gravitational potential  The Cowling approximation is not so bad Always use the Eulerian perturbation of  2.Pressure P In dense cores, |P’| << |  P| Use the Eulerian perturbation of P 4

5. 11 June 2014ESTER workshop, Toulouse Lagrangian or Eulerian perturbations ? Pressure In a g-mode cavity where The Eulerian perturbation of P must be used 5 Precision in stellar oscillations modeling: choosing the good variables

6. 11 June 2014ESTER workshop, Toulouse Lagrangian or Eulerian perturbations ? Lagrangian Eulerian if and only if hydrostatic equilibrium of the structure model 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) 6 Precision in stellar oscillations modeling: choosing the good variables

7. 11 June 2014ESTER workshop, Toulouse Non-adiabatic oscillations in near-surface layers 7 Precision in stellar oscillations modeling: choosing the good variables must be used as variable in non-adiabatic oscillation codes or Lagrangian or Eulerian perturbations ? Lagrangian perturbation of state equation and opacities are simpler better to use them in the superficial non-adiabatic layers

8. 11 June 2014ESTER workshop, Toulouse8 Precision in stellar oscillations modeling The first integral of Takata, a good test of precision Dipolar modes Equation of momentum conservation for the center of mass of each sphere M r : 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, …)

9. 11 June 2014ESTER workshop, Toulouse9 Precision in stellar oscillations modeling The first integral of Takata, a good test of precision Proof: Integration on an arbitrary volume: ||

10. 11 June 2014ESTER workshop, Toulouse10 Precision in stellar oscillations modeling The first integral of Takata, a good test of precision First integral (general case): Dipolar mode, sphere:

11. 11 June 2014ESTER workshop, Toulouse11 Precision in stellar oscillations modeling Using asymptotic JWKB solutions Full non-adiabatic case: see Dziembowski (1977) Continuous match to the numerical solution Does not increase precision, but decreases the number of mesh points Useful in the core of high density contrast stars Adiabatic-Cowling approximation, g-mode cavity with : Numerous nodes in high density contrast stars Quasi-adiabatic approximation: Power lost by the mode through radiative damping:

12. 11 June 2014ESTER workshop, Toulouse12 Accuracy in stellar oscillations modeling Usual approximations in oscillation equations: Adiabaticity, slow rotation, no magnetic field, no tidal effects 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), … Magnetic field: Lorentz force + perturbed induction equation Direct effect on frequencies, mode geometry and driving Perturbative approach: see e.g. Hasan et al. 1992, 2005; Cunha & Gough 2000 Non-perturbative approach: see e.g. Bigot & Dziembowski (2003), Saio (2005) Tidal influence of a companion:

13. 11 June 2014ESTER workshop, Toulouse13 Accuracy in stellar oscillations modeling Rotation: Coriolis + centrifugal deformation Major effect on frequencies, mode geometry and driving Perturbative approach: see e.g. Dziembowski & Goode (1992), 2 nd order Soufi et al. 1998, 3 rd order Non-perturbative approach: Traditional approximation (spherical symetry, rigid rotation, horizontal Coriolis) Separabilityvery efficient computations Not so bad for g-modes of moderate rotators (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), See their talks ! Ouazzani et al. (2012) 2D structure models + oscillations with finite differences: Clement 1998, Deupree 1995, …

14. 11 June 2014ESTER workshop, Toulouse14 Accuracy in stellar oscillations modeling Non-adiabatic-energetic aspects in oscillations modeling: Predictions of mode excitation + normalized amplitudes and phases Heat engine pulsators: Range of unstable modes and instability strips Constrains opacities, time-dependent convection Stochastic excitation: Mode life-times line-widths in power spectrum Constrains time-dependent convection Improve accuracy of theoretical frequencies through a good oscillations modeling in the superficial layersPhysical treatment of surface effects Important for high-order p-modes (e.g. solar-like oscillations)

15. 11 June 2014ESTER workshop, Toulouse15 Accuracy in stellar oscillations modeling 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 1977 Balmforth 1992Houdek et al. (1999-…) Unno 1967 Gabriel 1996 Grigahcène, Dupret et al. (2005-…) Beyond the mixing-length theory: Xiong et al. (1997-2010) All these theories introduce free parameters !

16. 11 June 2014ESTER workshop, Toulouse16 Accuracy in stellar oscillations modeling Non-adiabatic-energetic aspects in oscillations modeling: Time-dependent convection All current theories introduce free parameters or are contradicted by observations … What should be done ? What hydrodynamical simulations are telling us ? (Gastine & Dintrans 2011, Mundprecht et al. 2012) Going beyond the MLT, yes but …