Precision and accuracy in stellar oscillations modeling

ESTER workshop, Toulouse 1

Precision and accuracy instellar oscillations modeling

Marc-Antoine Dupret, R. Scuflaire, M. Godart, R.-M. Ouazzani, …

11 June 2014

ESTER workshop, Toulouse 211 June 2014

Precision and accuracy in stellar oscillations modeling

Precision: Precise solution of given differential equations

Accuracy: Set of differential equations accurately modeling stellar oscillations


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, …), approachhuge core-surface contrast This is not always enough …


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 FThe Cowling approximation is not so bad

Always use the Eulerian perturbation of F2. Pressure P

In dense cores, |P’| << |d P|Use the Eulerian perturbation of P



PressureIn a g-mode cavity where

The Eulerian perturbation of P must be used




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)



Non-adiabatic oscillations

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



The first integral of Takata, a good test of precisionDipolar modes Equation of momentum conservation for the center of mass of each sphere Mr:

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, …)



The first integral of Takata, a good test of precision Proof:

Integration on an arbitrary volume:

||



The first integral of Takata, a good test of precision First integral (general case):

Dipolar mode, sphere:



Using asymptotic JWKB solutions

Full non-adiabatic case: see Dziembowski (1977)Continuous match to the numerical solutionDoes 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 densitycontrast stars

Quasi-adiabatic approximation:

Power lost by the mode through radiative damping:


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:



Rotation: Coriolis + centrifugal deformation

Major effect on frequencies, mode geometry and driving

Perturbative approach: see e.g. Dziembowski & Goode (1992), 2nd order

Soufi et al. 1998, 3rd orderNon-perturbative approach:• Traditional approximation (spherical symetry, rigid rotation, horizontal Coriolis)

Separability very efficient computationsNot 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, …



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 layers Physical treatment of surface effects

Important for high-order p-modes (e.g. solar-like oscillations)



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 1992 Houdek 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 !



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 …

Precision and accuracy in stellar oscillations modeling

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