Asteroseismic inversions in the context of Plato

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ASTEROSEISMIC INVERSIONS

IN THE CONTEXT OF PLATO

Gaël Buldgen Pr. Marc-Antoine Dupret

Dr. Daniel R. Reese University of Liège December 2014

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SEISMIC FORWARD MODELLING

Definition:

Usingseismic (and non-seismic) constraints to determine theoptimal

set of parameters describing the theoretical model of a real star.

Typically: Mass, age, Y, Z, αMLT, αov, ... witha given physics (opacities, convection treatment, extra-mixing, ...)

Mass Age Y Best Model Limitations? Physically representative? Bias on determinations of fundamental parameters

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USING INVERSIONS

Improving the use of seismic information:

1 _{By using seismic indicators (glitches, fine analysis of frequency}

combinations,...).

2 By relating seismic information to structural corrections using a

less model-dependent approach.

Basu et al. 1996

Not subject to the limitations of forward modelling

Sufficient number of frequencies required

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INVERSIONS OF INTEGRATED QUANTITIES

Using theSOLA method (Pijpers & Thompson 1994), we determine

integrated quantities.

Illustration for the mean density (Reese et al. 2012): ¯ ρ = Z R 0 4π ρ R3r 2_dr ₍₁₎ δνi νi = Z R 0 K_ρ,Γi 1 δρ ρ dr + Z R 0 K_Γi 1,ρ δΓ1 Γ₁ dr (2) δ ¯ρinv ¯ ρ = X i ci δνi νi ≈ Z R 0 4π ρ R3_ρ_¯r 2δρ ρ dr (3)

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GENERAL APPROACH (1)

But we are not limited in the choice of theintegrated quantity:

A = Z R 0 f (s1, s2, r )dr δA A = Z R 0 T_Aδs1 s₁ dr + Z R 0 T_A,crossδs2 s₂ dr

With s₁,s₂ beingρ, c2, Γ₁, u = P_ρ, Y , ... defining custom-made targets functionsT_A and T_A,cross.

Combining approaches:

Observational constraints

Forward

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GENERAL APPROACH (2): NEW INDICATORS 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 Position r/R Tρ T_ρ= 4πx2_ρ_˜ 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 Position r/R Tτ T_τ₌1 ˜ c 0 0.2 0.4 0.6 0.8 1 −20 −10 0 10 20 30 40 Position r/R Tt T_t₌1 x d˜c dx 0 0.2 0.4 0.6 0.8 1 −0.5 0 0.5 1 1.5 2 Position r/R Ttu T_t u= f (x) !d˜u dx "2 ¯ ρ=R0R4πr2ρdr ∝ (∆ν)2 τ= RR 0 1cdr ∝ (∆ν)−1 tu=R0Rf (r) !_du dr "2 dr t =R₀R1 r dc drdr ∝νδν∆ν

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PROOF OF CONCEPT

Test strategy (see Buldgen et al. 2014, arXiv:1411.2416)

Build target includingcomplex physics

Seismic modelling usingsimple physics

Carry outinversions for indicators

3455 3460 3465 3470 3475 −2.465 −2.46 −2.455 −2.45 x 10−3 Acoustic Radius τ (s) In d ic a to r t (s − 1)

1.05 M⊙, 1.5 Gyr, αMLT= 1.7 (< ∆ν > < δν > with Mass, Age and αMLT= 1.522)

T argetαconv SOLAαconv Referenceα_conv 3020 3025 3030 3035 3040 3045 3050 −1.91 −1.9 −1.89 −1.88 x 10−3

0.9 M⊙, 6.0 Gyr, non-adiabatic frequencies (< ∆ν > < δν > with Mass, Age)

Acoustic Radius τ (s) In d ic a to r t (s − 1) T argetNad1 SOLANad1 ReferenceNad1 3280 3300 3320 3340 3360 3380 3400 −2.3 −2.25 −2.2 −2.15 x 10−3

Acoustic Radius τ (s) In d ic a to r t (s −

1) T arget_SOLANad2

Nad2 ReferenceNad2 3405 3410 3415 3420 3425 3430 3435 −1.91 −1.9 −1.89 −1.88 x 10−3

1.0 M⊙, 4.0 Gyr, with turbulent pressure (< ∆ν > < δν > with Mass, Age)

Acoustic Radius τ (s) In d ic a to r t (s − 1) T argetturb SOLAturb Referenceturb ∆t= −0.5% ∆τ= 0.5% ∆t= 0.32% ∆τ= 0.001% ∆t= 1.8% ∆τ= 0.03% ∆t= 4.3% ∆τ= 3.2% ∆t= 0.2% ∆τ= 0.5% ∆t= 0.08% ∆τ= 0.03%

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RESULTS WITH MULTIPLE INDICATORS (1) 3455 3460 3465 3470 3475 −2.465 −2.46 −2.455 −2.45 x 10−3 Acoustic Radius τ (s) In d ic a to r t (s − 1)

1.05 M⊙, 1.5 Gyr, αMLT= 1.7 (< ∆ν > < δν > with Mass, Age and αMLT= 1.522)

T argetα_conv SOLAαconv Referenceαconv 3020 3025 3030 3035 3040 3045 3050 −1.91 −1.9 −1.89 −1.88 x 10−3

1.0 M⊙, 4.0 Gyr, with turbulent pressure (< ∆ν > < δν > with Mass, Age)

Acoustic Radius τ (s) In d ic a to r t (s − 1) T argetturb SOLAturb Referenceturb ∆t= −0.5% ∆τ= 0.5% ∆t= 0.32% ∆τ= 0.001% ∆t= 1.8% ∆τ= 0.03% ∆t= 4.3% ∆τ= 3.2% ∆t= 0.2% ∆τ= 0.5% ∆t= 0.08% ∆τ= 0.03%

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RESULTS WITH MULTIPLE INDICATORS (2) 1.68 1.682 1.684 1.686 1.688 1.69 1.692 1.694 1.696 1.698 1.7 4.1 4.2 4.3 4.4

0.9 M⊙, 3.08 Gyr, with diffusion (∆ν(ν), δν(ν) with αMLT, Y0, M ass, Age)

Mean Density ¯ρ(g/cm3₎ In d ic a to r tu ( × 1 0 65g 2) T arget7 SOLA7.2 Reference7.2 0.84 0.845 0.85 0.855 0.86 0.865 0.87 0.875 0.88 0.885 0.89 10.5 11 11.5 Mean Density ¯ρ(g/cm3₎ In d ic a to r tu ( × 1 0 65g 2)

1.0 M⊙, 4.02 Gyr, with diffusion (r02(ν), r01(ν), < ∆ν > with Z0, Y0, M ass, Age)

T arget8 SOLA8.2 Reference8.2 ∆tu= 4.6% ∆ρ¯= 0.35% ∆tu= −6.5% ∆¯ρ= −2.8% ∆tu= 0.68% ∆ρ¯= 0.06% ∆tu= 1.9% ∆¯ρ= −0.08%

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CONCLUSIONS AND UPCOMING WORK

Objectives:

Offer new constraints on stellar models and overcome the current accuracy limitations in ages, mass and radii determinations.

What we have:

1 _{accurate (usually < 1%) determinations of τ , t, t}_u _{and ¯}ρ

2 method tested for various sets of modes(50 freq, ` = 0, 1, 2,(3)) 3 method tested for various constraints on forward modelling

Prospects:

1 new indicators(helium, surface temperature gradient,...)

2 test on new generation of models

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APPENDICES

General expression for any A:

δA A = Z R 0 TA δs1 s₁ dr + Z R 0 TA,cross δs2 s₂ dr (4) Target functions: T_τ = 1 2cτ − Gm(r ) r2 ρ Z x 0 1 2cτ Pds T_{τ cross}= −1 2cτ − 4πGr2 " Z R r ( ρ s2 Z s 0 1 2cτ Pdt) # ds T_ρ_¯= 4π ρ R3_ρ_¯r 2 T_t = 1 r dc dr RR 0 1r dc drdr Ttu = −2u0 t_u d dr f (r )du dr (5)

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APPENDICES

Obtaining new kernels: From Masters et al. 1979:

Z R 0 K_sn,l₁_,s₂δs1 s1 dr + Z R 0 K_sn,l₂_,s₁δs2 s2 dr = Z R 0 K_sn,l₃_,s₄δs3 s3 dr + Z R 0 K_sn,l₄_,s₃δs4 s4 dr

Relate (s1, s2) to (s3, s4) (e.g. (ρ, Γ1) to (u0, Γ1) in the simplest cases) ⇒ new kernelsby solving a differential equation.

Ex: (u0, Γ1), (u0, Y ), (c2, Γ1), (c2, Y ), (Γ1, Y ), (P0, Γ1), (P0, Y ),

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APPENDICES

Table : Set of modes used in the accuracy tests

Set 1 Set 2 Set 3 Set 4

` = 0 n = 5 − 24 n = 9 − 28 n = 5 − 27 n = 9 − 34 ` = 1 n = 5 − 24 n = 9 − 28 n = 5 − 27 n = 9 − 34 ` = 2 n = 5 − 24 n = 9 − 28 n = 5 − 27 n = 9 − 34

Set 5 Set 6 Set 7 Cyg 16A

` = 0 n = 11 − 24 n = 11 − 26 n = 9 − 23 n = 12 − 27 ` = 1 n = 11 − 24 n = 11 − 26 n = 9 − 23 n = 11 − 27 ` = 2 n = 11 − 24 n = 11 − 26 n = 9 − 23 n = 11 − 24 ` = 3 n = 9 − 20 n = 12 − 22 n = 15 − 24 n = 15 − 21 Cyg 16B Set 33 ` = 0 n = 13 − 26 15 − 25 ` = 1 n = 13 − 26 15 − 25 ` = 2 n = 12 − 25 15 − 25 ` = 3 n = 17 − 24 −

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APPENDICES

Target properties: Total number: 57

Mass range: 0.9 − 1.1 M

Age range: 1.1 − 8.1 Gyr

Z0 range: 0.01 − 0.02 X₀ range: 0.73 − 0.68

No convective cores!

Including various mismatches: Y0, Z0, diffusion, heavy elements

mixing.

Observational constraints:

∆ν(ν), δν(ν), r₀₂(ν), r₀₁(ν), < ∆ν >, < δν >. Free parameters:

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APPENDICES 2000 2500 3000 3500 4000 10 15 20 25 Observed frequencyµHz D0 2 µ H z ˜ δνT ar1 n2 ˜ δνT ar1 n3 ˜ δνM od1 n2 ˜ δνM od1 n3 2000 2500 3000 3500 4000 0.02 0.025 0.03 0.035 0.04 Observed frequencyµHz r01 r01T ar1 rM od1 01 2000 2500 3000 3500 4000 146 148 150 152 154 Observed frequencyµHz ∆ ν µ H z ∆νT ar1 n0 ∆νT ar1 n1 ∆νT ar1 n2 ∆νT ar1 n3 ∆νM od2 n0 ∆νM od2 n1 ∆νM od2 n2 ∆νM od2 n3 2000 2500 3000 3500 4000 10 15 20 25 Observed frequencyµHz ˜ δνn ,l µ H z ˜ δνT ar1 n2 ˜ δνT ar1 n3 ˜ δνM od2 n2 ˜ δνM od2 n3 2000 2500 3000 3500 4000 148 150 152 154 Observed frequencyµHz ∆ ν µ H z ∆νT ar1 n0 ∆νT ar1 n1 ∆νT ar1 n2 ∆νT ar1 n3 ∆νM od3 n0 ∆νM od3 n1 ∆νM od3 n2 ∆νM od3 n3 2000 2500 3000 3500 4000 10 15 20 25 Observed frequencyµHz ˜ δνn ,l µ H z ˜ δνT ar1 n2 ˜ δνT ar1 n3 ˜ δνM od3 n2 ˜ δνM od3 n3

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APPENDICES 1400 1600 1800 2000 2200 2400 2600 2800 0.055 0.06 0.065 0.07 0.075 0.08 0.085 Observed frequency µHz r02 rT ar2 02 rMod4 02 rMod5 02 1400 1600 1800 2000 2200 2400 2600 2800 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Observed frequency µHz r01 rT ar2 01 rMod5 01

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APPENDICES 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 Position r/R R K A v g 0 0.2 0.4 0.6 0.8 1 −40 −20 0 20 40 Position r/R KC r o s s 0 0.2 0.4 0.6 0.8 1 −5 0 5 Position r/R KA v g 0 0.2 0.4 0.6 0.8 1 −15 −10 −5 0 5 10 Position r/R KC r o s s 0 0.2 0.4 0.6 0.8 1 −2 0 2 4 6 Position r/R KA v g 0 0.2 0.4 0.6 0.8 1 −20 0 20 Position r/R KC r o s s

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APPENDICES 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 Position r/R R K A v g 0 0.2 0.4 0.6 0.8 1 −40 −20 0 20 40 Position r/R KC r o s s 0 0.2 0.4 0.6 0.8 1 −5 0 5 Position r/R KA v g 0 0.2 0.4 0.6 0.8 1 −15 −10 −5 0 5 10 Position r/R KC r o s s 0 0.2 0.4 0.6 0.8 1 −2 0 2 4 6 Position r/R KA v g 0 0.2 0.4 0.6 0.8 1 −20 0 20 Position r/R KC r o s s 0 0.2 0.4 0.6 0.8 1 −20 0 20 40 60 80 Position x=r/R K A v g(x ) TAvg KAvg u0,Y(x) 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 Position x=r/R K C ro s s(x ) TC ross KC ross u0,Y(x)

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APPENDICES 0 1 2 3 4 5 6 x 109 5 5.5 6 6.5 7 7.5x 10 65 Age (Gyr) tu (g 2) 0 1 2 3 4 5 6 x 109 −3 −2.5 −2 −1.5 −1x 10 −3 Age (Gyr) t (s − 1)