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Stellar acoustic radii and ages from seismic inversion techniques

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Stellar acoustic radii and ages from seismic

inversion techniques

G. Buldgen

1,4

, D. R. Reese

2

, M. A. Dupret

1

and R. Samadi

3

1

Institut d´Astrophysique et G´eophysique, University of Li`ege, Li`ege, Belgium

2

School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom

3

LESIA, Observatoire de Paris,Universit´e Denis Diderot, Meudon, France

4

PhD Student, FRIA grant holder

Abstract

Context: Determining stellar characteristics such as the radius, mass or age is crucial for the study of stellar evolution, exoplanetary systems or the characterisation of stellar populations in the Galaxy. Asteroseismology is currently the most promising tool to accurately determine these characteristics. However, a key question is how to reduce the model dependence of asteroseismic methods.

Method: We extend the SOLA inversion technique to new global characteristics in addition to the mean density (see Reese et al. 2012). We apply our methodology to the acoustic radius and an age indicator based on the sound speed derivative. The results from SOLA inversions are compared with estimates based on the small and large frequency separations for several test cases, including differing mixing-lengths, and the presence or absence of non-adiabatic effects or turbulent pressure.

Results: We show that SOLA inversions yield accurate results in all test cases, unlike the other techniques which are more sensitive to surface effects. We observe that the acoustic radius and mean density inversions are more robust than the age indicator inversions, which are limited to relatively young stars with radiative cores.

Theoretical approach

The inversion procedure relies on the variational principle and the frequency-structure relation for adiabatic stellar oscillations. The fundamental equations used to carry out the inversions are the following:

δνn,ℓ νn,ℓ = Z 1 0 Ksn,ℓ1,s2δs1 s1 dx + Z 1 0 Ksn,ℓ2,s1δs2 s2 dx + G(ν) Qn,ℓ with: δs s = sobs − sref sref

where s1, s2 are structural variables such as Γ1, c2, ρ0, ... Ksn,li,sj the structural kernels, G an ad-hoc surface correc-tion and Qn,l a normalisation factor.

The relative perturbation of a global characteristic A (e.g. ¯ρ, τ, ...) can be related to structural variables: δAobs A = Z 1 0 T (x)δs1 s1 dx + Z 1 0 Tcross(x)δs2 s2 dx

T and Tcross being the target functions. In the SOLA method, we minimise the following cost function to find the optimal frequency combination to reproduce δAobs

A : JA = Z 1 0 Kavg(x) − T (x)2 dx + β Z 1 0

[Kcross(x) − Tcross]2 dx + tan(θ) X i

(ciσi)2 + λ X i

[ci − f ]

Application to the acoustic radius, mean density and age indicator

We apply inversion techniques to the acoustic radius, τ , and the age indicator, t, which are defined as follows:

τ = Z R 0 dr c t = Z R 0 1 r dc drdr

SOLA inversions are compared with other techniques based on aster-oseismic indices. Indeed, τ and t are related to the large and small frequency separations as follows:

τ ≃ 1

2∆ν t ≃

−4π2ν ˜δν (4ℓ + 6)∆ν

where ℓ is the degree of the mode and ˜δν the small frequency sepa-ration. We illustrate various kernels from SOLA inversions and from

estimates based on asteroseismic indices in the following figures:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −5 0 5 Position r/R KA v g SOLA < ∆ν > Target 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −10 −8 −6 −4 −2 0 2 Position r/R KC r o s s

Fig. 1: Kernels for τ inversions using ρ0, Γ1 as s1, s2.

Reliability: The quality of the inversion depends on how well the

averaging and cross-term kernels fit their respective targets.

Results for a grid of models

Method: we use 93 main sequence and pre-main sequence models (Marques et al. 2008) to determine the values of the τ and t indi-cators in the target (M = 0.9 M, age = 1.492 Gyr, R = 0.821 R which includes surface effects). We used 33 frequencies with ℓ = 0 − 2 and n = 15 − 25. The results are presented in figures 1 and 2. The vertical lines give the position of the best model from the grid in terms of ˜δν and ∆ν. Each point gives the result for a given reference model of the grid.

1.5 1.6 1.7 1.8 1.9 x 10−4 2800 2805 2810 2815 2820 2825 2830 2835 2840

Average large separation (s−1 ) Ac o u st ic R a d iu s τ (s ) < ∆ν > SOLA Target

Fig. 2: Results for the acoustic radius for A′.

1 1.5 2 2.5 3 3.5 x 10−4 −4 −3.5 −3 −2.5 −2 −1.5 −1x 10 −3

Average small separation (s−1)

Ag e In d ic a to r t (s − 1 ) SOLA < ˜δν > Target

Fig. 3: Results for the age indicator and the acoustic radius.

Results: SOLA inversions yield better results when surface effects are present. Estimates based on seismic indices sometimes bene-fit from fortuitous error compensation, thereby leading to accurate results.

Prospects and conclusions

Conlusion: we set the basis of a new framework to determine stellar global characteristics using less model-dependent inversion techniques. We show that SOLA methods can handle the physical complexity of observed stars and overcome the limitations of forward modelling.

What’s next? Provide a set of global characteristics allowing us to determine more accurately the age of stars and their structural proper-ties.

How? The figure on the right shows evolutionary tracks of models with different masses in a t vs τ diagram. The sharp transition at higher masses result from the apparition of the convective core during the evo-lution of the star. This demonstrates the great sensitivity of global struc-tural characteristics and thus their diagnostic potential.

4000 4500 5000 5500 6000 6500 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5x 10 −3 τ(s) t( s − 1 ) 1.21M⊙ 1.22M⊙ 1.23M⊙ 1.24M⊙ 1.25M⊙ 1.26M⊙ 1.27M⊙ 1.28M⊙ 1.29M⊙

Fig. 4: Evolutionary tracks for massive stars in t vs τ plot.

Results using forward modelling

Analysis: the results for t show that we need a criterion for selecting a reference model. Indeed, the inversion equations rely on the variational principle, which assumes that the reference model is close to the star. Therefore, the reference model needs to be chosen carefully, so we choose

forward modelling. With the help of the OSM software, we fit h∆νi and ˜

δν(ν) using the same modes as in the previous tests.

Method: 3 targets are used, with masses ranging from 0.95 M to 1.05 M and ages ranging from 1.5 Gyr to 6 Gyr. They differ from the reference models by their mixing-length parameter (Modelαconv) and the

presence or absence of turbulent pressure (Modelturb) or non-adiabatic effects in their frequencies (Modelnad1,2). Results for the τ and t inver-sions are plotted in the figures below.

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 ) Modelαconv 3020 3025 3030 3035 3040 3045 3050 −1.9 −1.88 x 10−3 Acoustic Radius τ (s) In d ic a to r t (s − 1 ) ModelNad1 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 ) ModelNad2 3405 3410 3415 3420 3425 3430 3435 −1.91 −1.9 −1.89 −1.88 x 10−3 Acoustic Radius τ (s) In d ic a to r t (s − 1 ) Modelturb

Fig. 5: Results for τ and t for each of the test cases. The results from forward modelling are in black, the estimates based on seismic indices in red, the results from SOLA inversions in blue and the true values for the targets in green.

Results and conclusions

The combination forward modelling + SOLA inversions

can be used to accurately determine the indicators t, τ and ¯ρ (not presented here) in observed stars. SOLA inversions always improves the accuracy of the determination of the indicator.

The SOLA approach is able to overcome

lim-itations of forward modelling, is more

accu-rate than estimates based on asteroseismic

indices and allows us to choose the

struc-tural characteristics we wish to determine.

References

• Dupret, 2001, A&A, 366, 166

• Marques et al., 2008, Ap&SS, 316, 173 • Reese et al., 2012, A&A, 539, A63

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