arXiv subjects

E. Caffau

Publications and source records attributed to E. Caffau.

At least 73 records · Page 4Linked to original sources

arXiv2018

Chemical analysis of very metal-poor turn-off stars from SDSS-DR12

The most metal-poor stars are the relics of the early chemical evolution of the Galaxy. Their chemical composition is an important tool to constrain the nucleosynthesis in the first generation of stars. The aim is to observe a sample of extremely metal-poor star (EMP stars) candidates selected from the Sloan Digital Sky Survey Data Release 12 (SDSS DR12) and determine their chemical composition. We obtain medium resolution spectra of a sample of six stars using the X-Shooter spectrograph at the Very Large Telescope (VLT) and we used ATLAS models to compute the abundances. Five stars of the sample have a metallicity [Fe/H] between -2.5 dex and -3.2 dex. We confirm the recent discovery of SDSS J002314.00+030758.0. As a star with an extremely low [Fe/H] ratio. Assuming the alpha-enhancement [Ca/Fe]=+0.4 dex, we obtain [Fe/H]=-6.1 dex. We could also determine its magnesium abundance and found that this star exhibits a very high ratio [Mg/Fe] >= +3.60~dex assuming [Fe/H] = -6.13 dex. We determined the carbon abundance and found A(C)=6.4 dex. From this carbon abundance, this stars belongs to the lower band of the A(C)-[Fe/H] diagram.

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arXiv2018

3D non-LTE corrections for Li abundance and 6Li/7Li isotopic ratio in solar-type stars I. Application to HD 207129 and HD 95456

Convective motions in solar-type stellar atmospheres induce Doppler shifts that affect the strengths and shapes of spectral absorption lines and create slightly asymmetric line profiles. 1D LTE studies of elemental abundances are not able to reproduce this phenomenon, which becomes particularly important when modeling the impact of isotopic fine structure, like the subtle depression created by the 6Li isotope on the red wing of the Li I resonance doublet line. The purpose of this work is to provide corrections for the lithium abundance, A(Li), and the 6Li/7Li ratio that can easily be applied to correct 1D LTE lithium abundances in G and F dwarf stars of approximately solar mass and metallicity for 3D and non-LTE (NLTE) effects. The corrections for A(Li) and 6Li/7Li are computed using grids of 3D NLTE and 1D LTE synthetic lithium line profiles, generated from 3D hydro-dynamical CO5BOLD and 1D hydrostatic model atmospheres, respectively. The 3D NLTE corrections are then approximated by analytical expressions as a function of the stellar parameters. These are applied to adjust the 1D LTE isotopic lithium abundances in two solar-type stars, HD 207129 and HD 95456. The derived 3D NLTE corrections range between -0.01 and +0.11 dex for A(Li), and between -4.9 and -0.4% for the 6Li/7Li ratio, depending on the adopted stellar parameters. In the case of the 6Li/7Li ratio, our corrections are always negative, showing that 1D LTE analysis can significantly overestimate (up to 4.9 percentage points) the presence of 6Li in the atmospheres of solar-like dwarf stars. These results emphasize the importance of reliable 3D model atmospheres combined with NLTE line formation for deriving precise isotopic lithium abundances. Although 3D NLTE spectral synthesis implies an extensive computational effort, the results can be made accessible with parametric tools like the ones presented in this work.

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arXiv2018

Abundance of zinc in the red giants of Galactic globular cluster 47 Tucanae

We investigate possible relations between the abundances of zinc and the light elements sodium, magnesium, and potassium in the atmospheres ofred giant branch (RGB) stars of the Galactic globular cluster 47~Tuc and study connections between the chemical composition and dynamical properties of the cluster RGB stars. The abundance of zinc was determined in 27 RGB stars of 47~Tuc using 1D local thermal equilibrium (LTE) synthetic line profile fitting to the high-resolution 2dF spectra obtained with the Anglo-Australian Telescope (AAT). Synthetic spectra used in the fitting procedure were computed with the SYNTHE code and 1D ATLAS9 stellar model atmospheres. The average 1D~LTE zinc-to-iron abundance ratio and its RMS variations due to star-to-star abundance spread determined in the sample of 27 RGB stars is $\langle{\rm[Zn/Fe]}\rangle^{\rm 1D~LTE}=0.11\pm0.09$. We did not detect any statistically significant relations between the abundances of zinc and those of light elements. Neither did we find any significant correlation or anticorrelation between the zinc abundance in individual stars and their projected distance from the cluster center. Finally, no statistically significant relation between the absolute radial velocities of individual stars and the abundance of zinc in their atmospheres was detected. The determined average [Zn/Fe]$^{\rm~1DLTE}$ ratio agrees well with those determined in this cluster in earlier studies and nearly coincides with that of Galactic field stars at this metallicity. All these results suggest that nucleosynthesis of zinc and light elements proceeded in separate, unrelated pathways in 47~Tuc.

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arXiv2018

Gaia confirms that SDSS J102915+172927 is a dwarf star

The Gaia Data Release 2 provides a parallax of 0.734+/-0.073 mas for SDSS J102915+172927, currently the most metal-poor known object. This parallax implies that it is dwarf star, ruling out the scenario that it is a subgiant. The subgiant scenario had as a corollary that the star had been formed in a medium highly enriched in C, thus making line cooling efficient during the collapse, that was also highly enriched in Fe by Type Ia SNe. This scenario can also now be ruled out for this star, reinforcing the need of dust cooling and fragmentation to explain its formation.

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arXiv2018

Abundances of Mg and K in the atmospheres of turn-off stars in Galactic globular cluster 47 Tucanae

We determined abundances of Mg and K in the atmospheres of 53 (Mg) and 75 (K) turn-off (TO) stars of the Galactic globular cluster 47 Tuc. The obtained abundances, together with those of Li, O, and Na, were used to search for possible relations between the abundances of K and other light elements, Li, O, Na, and Mg, as well as the connections between the chemical composition of TO stars and their kinematical properties. Abundances of Mg and K were determined using archival VLT FLAMES/GIRAFFE spectra, in combination with the 1D NLTE spectral synthesis methodology. Spectral line profiles were computed with the MULTI code, using ATLAS9 stellar model atmospheres. We also utilized 3D hydrodynamical CO5BOLD and 1D hydrostatic LHD model atmospheres for computing 3D-1D LTE abundance corrections for the spectral lines of Mg and K, in order to assess the influence of convection on their formation in the atmospheres of TO stars. The determined average abundance-to-iron ratios and their RMS variations due to star-to-star abundance spreads were $\langle{\rm[Mg/Fe]}\rangle^{\rm 1D NLTE}=0.47\pm0.12$, and $\langle{\rm [K/Fe]}\rangle^{\rm 1D NLTE}=0.39\pm0.09$. Although the data suggest the possible existence of a weak correlation in the [K/Fe]-[Na/Fe] plane, its statistical significance is low. No statistically significant relations between the abundance of K and other light elements were detected. Also, we did not find any significant correlations or anti-correlations between the [Mg/Fe] and [K/Fe] ratios and projected distance from the cluster center. Similarly, no relations between the absolute radial velocities of individual stars and abundances of Mg and K in their atmospheres were detected. The 3D-1D abundance corrections were found to be small ($\leq 0.1$\,dex) for the lines of Mg and K used in this study, thus indicating that the influence of convection on their formation is small.

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arXiv2018

Investigation of a sample of carbon-enhanced metal-poor stars observed with FORS and GMOS

Carbon-enhanced metal-poor (CEMP) stars represent a sizeable fraction of all known metal-poor stars in the Galaxy. Their formation and composition remains a significant topic of investigation within the stellar astrophysics community. We analysed a sample of low-resolution spectra of 30 dwarf stars, obtained using the the visual and near UV FOcal Reducer and low dispersion Spectrograph for the Very Large Telescope (FORS/VLT) of the European Southern Observatory (ESO) and the Gemini Multi-Object Spectrographs (GMOS) at the GEMINI telescope, to derive their metallicity and carbon abundance. We derived C and Ca from all spectra, and Fe and Ba from the majority of the stars. We have extended the population statistics of CEMP stars and have confirmed that in general, stars with a high C abundance belonging to the high C band show a high Ba-content (CEMP-s or -r/s), while stars with a normal C-abundance or that are C-rich, but belong to the low C band, are normal in Ba (CEMP-no).

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arXiv2018

Using the CIFIST grid of CO5BOLD 3D model atmospheres to study the effects of stellar granulation on photometric colours. I. Grids of 3D corrections in the UBVRI, 2MASS, Hipparcos, Gaia, and SDSS systems

The atmospheres of cool stars are temporally and spatially inhomogeneous due to the effects of convection.The influence of this inhomogeneity, referred to as granulation, on colours has never been investigated over a large range of effective temperatures and gravities. We aim to study, in a quantitative way, the impact of granulation on colours. We use the CIFIST grid of CO5BOLD hydrodynamical models to compute emerging fluxes. These in turn are used to compute theoretical colours in the UBVRI,2MASS,Hipparcos,Gaia and SDSS systems. Every CO5BOLD model has a corresponding one dimensional (1D) plane-parallel LHD model computed for the same atmospheric parameters, which we used to define a "3D correction" that can be applied to colours computed from fluxes computed from any 1D model atmosphere code. The 3D corrections on colours are generally small, of the order of a few hundredths of a magnitude, yet they are far from negligible. We find that ignoring granulation effects can lead to underestimation of Teff by up to 200K and overestimation of gravity by up to 0.5dex, when using colours as diagnostics. We have identified a major shortcoming in how scattering is treated in the current version of the CIFIST grid, which could lead to offsets of the order 0.01mag, especially for colours involving blue and UV bands.We have investigated the Gaia and Hipparcos photometric systems and found that the (G-H_p),(BP-RP) diagram is immune to the effects of granulation. In addition, we point to the potential of the RVS photometry as a metallicity diagnostic. Our investigation shows that the effects of granulation should not be neglected if one wants to use colours as diagnostics of the stellar parameters of F,G,K stars. A limitation is that scattering is treated as true absorption in our current computations, thus our 3D corrections are likely an upper limit to the true effect. (Abridged)

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arXiv2018

Using the CIFIST grid of CO5BOLD 3D model atmospheres to study the effects of stellar granulation on photometric colours. II. The role of convection accross the H-R diagram

We studied the influence of convection on the spectral energy distributions, photometric magnitudes, and colour indices of different types of stars across the H-R diagram. The 3D hydrodynamical CO5BOLD, averaged <3D>, and 1D hydrostatic LHD model atmospheres were used to compute spectral energy distributions of stars on the main sequence (MS), main sequence turn-off (TO), subgiant branch (SGB), and red giant branch (RGB), in each case at two different effective temperatures and two metallicities, [M/H]=0.0 and -2.0. Using the obtained spectral energy distributions, we calculated photometric magnitudes and colour indices in the broad-band Johnson-Cousins $UBVRI$ and 2MASS $JHK_{\rm s}$, and the medium-band Strömgren $uvby$ photometric systems. The 3D-1D differences in photometric magnitudes and colour indices are small in both photometric systems and typically do not exceed $\pm0.03$ mag. Only in the case of the coolest giants located on the upper RGB are the differences in the $U$ and $u$ bands able reach $\approx-0.2$ mag at [M/H]=0.0 and $\approx-0.1$ mag at [M/H]=-2.0. Generally, the 3D-1D differences are largest in the blue-UV part of the spectrum and decrease towards longer wavelengths. They are also sensitive to the effective temperature and are significantly smaller in hotter stars. Metallicity also plays a role and leads to slightly larger 3D-1D differences at [M/H]=0.0. All these patterns are caused by a complex interplay between the radiation field, opacities, and horizontal temperature fluctuations that occur due to convective motions in stellar atmospheres. Although small, the 3D-1D differences in the magnitudes and colour indices are nevertheless comparable to or larger than typical photometric uncertainties and may therefore cause non-negligible systematic differences in the estimated effective temperatures.

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arXiv2018

TOPoS: IV. Chemical abundances from high-resolution observations of seven EMP stars

Extremely metal-poor stars provide us with indirect information on the first generations of massive stars. The TOPoS survey has been designed to increase the census of these stars and to provide a chemical inventory that is as detailed as possible. Seven of the most iron-poor stars have been observed with the UVES spectrograph at the ESO VLT Kueyen 8.2m telescope to refine their chemical composition. We analysed the spectra based on 1D LTE model atmospheres, but also used 3D hydrodynamical simulations of stellar atmospheres. We measured carbon in six of the seven stars: all are carbon-enhanced and belong to the low-carbon band, defined in the TOPoS II paper. We measured lithium (A(Li)=1.9) in the most iron-poor star (SDSS J1035+0641, [Fe/H] < -5.2 ). We were also able to measure Li in three stars at [Fe/H]~ -4.0, two of which lie on the Spite plateau. We confirm that SDSS J1349+1407 is extremely rich in Mg, but not in Ca. It is also very rich in Na. Several of our stars are characterised by low alpha-to-iron ratios. The lack of high-carbon band stars at low metallicity can be understood in terms of evolutionary timescales of binary systems. The detection of Li in SDSS J1035+0641 places a strong constraint on theories that aim at solving the cosmological lithium problem. The Li abundance of the two warmer stars at [Fe/H]~ -4.0 places them on the Spite plateau, while the third, cooler star, lies below. We argue that this suggests that the temperature at which Li depletion begins increases with decreasing [Fe/H]. SDSS J1349+1407 may belong to a class of Mg-rich EMP stars. We cannot assess if there is a scatter in alpha-to-iron ratios among the EMP stars or if there are several discrete populations. However, the existence of stars with low alpha-to-iron ratios is supported by our observations.

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arXiv2017

Galactic evolution of Copper in the light of NLTE computations

We have developed a model atom for Cu with which we perform statistical equilibrium computations that allow us to compute the line formation of Cu I lines in stellar atmospheres without assuming Local Thermodynamic Equilibrium (LTE). We validate this model atom by reproducing the observed line profiles of the Sun, Procyon and eleven metal-poor stars. Our sample of stars includes both dwarfs and giants. Over a wide range of stellar parameters we obtain excellent agreement among different Cu I lines. The eleven metal-poor stars have iron abundances in the range -4.2 <= [Fe/H] <= -1.4, the weighted mean of the [Cu/Fe] ratios is -0.22 dex, with a scatter of -0.15 dex. This is very different from the results from LTE analysis (the difference between NLTE and LTE abundances reaches 1 dex) and in spite of the small size of our sample it prompts for a revision of the Galactic evolution of Cu.

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arXiv2017

Carbon-enhanced metal-poor 3D model atmospheres

We present our latest 3D model atmospheres for carbon-enhanced metal-poor (CEMP) stars computed with the CO5BOLD code. The stellar parameters are representative of hot turn-off objects (Teff ~ 6250 K, log g=4.0, [Fe/H]=-3.0). The main purpose of these models is to investigate the role of 3D effects on synthetic spectra of the CH G-band (4140-4400 A), the CN BX-band (3870-3890 A), and several UV OH transitions (3122-3128 A). By comparison with the synthetic spectra from standard 1D model atmospheres (assuming local thermodynamic equilibrium, LTE), we derive 3D abundance corrections for carbon and oxygen of up to -0.5 and -0.7 dex, respectively.

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arXiv2017

Abundances of Na, Mg, and K in the atmospheres of red giant branch stars of Galactic globular cluster 47 Tucanae

We study the abundances of Na, Mg, and K in the atmospheres of 32 RGB stars in the Galactic globular cluster (GGC) 47 Tuc, with the goal to investigate the possible existence of Na-K and Mg-K correlations/anti-correlations, similar to those that were recently discovered in two other GGCs, NGC 2419 and 2808. The abundances of K, Na, and Mg were determined using high-resolution 2df spectra obtained with the AAT. The 1D NLTE abundance estimates were obtained using 1D hydrostatic ATLAS9 model atmospheres and spectral line profiles synthesized with the MULTI package. We also used 3D hydrodynamical CO5BOLD and 1D hydrostatic LHD model atmospheres to compute 3D-1D LTE abundance corrections, $Δ_{\rm 3D-1D~LTE}$, for the spectral lines of Na, Mg, and K used in our study. These abundance corrections were used to understand the role of convection in the formation of spectral lines, as well as to estimate the differences in the abundances obtained with the 3D hydrodynamical and 1D hydrostatic model atmospheres. The average element-to-iron abundance ratios and their RMS variations due to star-to-star abundance spreads determined in our sample of RGB stars were $\langle{\rm [Na/Fe]}\rangle^{\rm 1D~NLTE}=0.42\pm0.13$, $\langle{\rm [Mg/Fe]}\rangle^{\rm 1D~NLTE}=0.41\pm0.11$, and $\langle{\rm [K/Fe]}\rangle^{\rm 1D~NLTE}=0.05\pm0.14$. We found no statistically significant relations between the abundances of the three elements studied here. Also, there were no abundance trends with the distance from the cluster center, nor any statistically significant relations between the abundance/abundance ratios and absolute radial velocities of individual stars. All these facts suggest the similarity of K abundance in stars that belong to different generations in 47 Tuc which, in turn, may hint that evolution of K in this particular cluster was unrelated to the nucleosynthesis of Na and/or Mg.

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arXiv2017

New ultra metal-poor stars from SDSS: follow-up GTC medium-resolution spectroscopy

Context. The first generation of stars formed in the Galaxy left behind the chemical signatures of their nucleosynthesis in the interstellar medium, visible today in the atmospheres of low-mass stars formed afterwards. Sampling the chemistry of low-mass provides insight into the first stars. Aims. We aim to increase the samples of stars with extremely low metal abundances, identifying ultra metal-poor stars from spectra with modest spectral resolution and signal-to-noise ratio (S/N). Achieving this goal involves deriving reliable metallicities and carbon abundances from such spectra. Methods. We carry-out follow-up observations of faint, V>19, metal-poor candidates selected from SDSS spectroscopy and observed with OSIRIS at GTC. The SDSS and follow-up OSIRIS spectra have been analyzed using the FERRE code to derive effective temperatures, surface gravities, metallicities and carbon abundances. In addition, a well-known extremely metal-poor star has been included in our sample to calibrate the analysis methodology. Results. We have observed and analyzed five metal-poor candidates from modest-quality SDSS spectra. All stars in our sample have been confirmed as extremely metal-poor stars, in the [Fe/H]<-3.3 regime. We report the recognition of J173403+644632, a carbon-enhanced ultra metal-poor dwarf star with [Fe/H]=-4.3 and [C/Fe]=+3.1.

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arXiv2017

The Pristine survey II: a sample of bright stars observed with FEROS

Extremely metal-poor (EMP) stars are old objects formed in the first Gyr of the Universe. They are rare and, to select them, the most successful strategy has been to build on large and low-resolution spectroscopic surveys. The combination of narrow- and broad band photometry provides a powerful and cheaper alternative to select metal-poor stars. The on-going Pristine Survey is adopting this strategy, conducting photometry with the CFHT MegaCam wide field imager and a narrow-band filter centred at 395.2 nm on the CaII-H and -K lines. In this paper we present the results of the spectroscopic follow-up conducted on a sample of 26 stars at the bright end of the magnitude range of the Survey (g<=15), using FEROS at the MPG/ESO 2.2 m telescope. From our chemical investigation on the sample, we conclude that this magnitude range is too bright to use the SDSS gri bands, which are typically saturated. Instead the Pristine photometry can be usefully combined with the APASS gri photometry to provide reliable metallicity estimates.

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arXiv2017

Lithium abundance and 6Li/7Li ratio in the active giant HD123351 I. A comparative analysis of 3D and 1D NLTE line-profile fits

Current three-dimensional (3D) hydrodynamical model atmospheres together with NLTE spectrum synthesis, permit to derive reliable atomic and isotopic chemical abundances from high-resolution stellar spectra. Not much is known about the presence of the fragile 6Li isotope in evolved solar-metallicity RGB stars, not to mention its production in magnetically active targets like HD123351. From fits of the observed CFHT spectrum with synthetic line profiles based on 1D and 3D model atmospheres, we seek to estimate the abundance of the 6Li isotope and to place constraints on its origin. We derive A(Li) and the 6Li/7Li isotopic ratio by fitting different synthetic spectra to the Li-line region of a high-resolution CFHT spectrum (R=120 000, S/R=400). The synthetic spectra are computed with four different line lists, using in parallel 3D hydrodynamical CO5BOLD and 1D LHD model atmospheres and treating the line formation of the lithium components in non-LTE (NLTE). We find A(Li)=1.69+/-0.11 dex and 6Li/7Li=8.0+/-4.4 % in 3D-NLTE, using the line list of Meléndez et al. (2012), updated with new atomic data for V I, which results in the best fit of the lithium line profile of HD123351. Two other line lists lead to similar results but with inferior fit qualities. Our 2-sigma detection of the 6Li isotope is the result of a careful statistical analysis and the visual inspection of each achieved fit. Since the presence of a significant amount of 6Li in the atmosphere of a cool evolved star is not expected in the framework of standard stellar evolution theory, non-standard, external lithium production mechanisms, possibly related to stellar activity or a recent accretion of rocky material, need to be invoked to explain the detection of 6Li in HD123351.

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arXiv2017

The Gaia-ESO Survey: Galactic evolution of sulphur and zinc

Due to their volatile nature, when sulfur and zinc are observed in external galaxies, their determined abundances represent the gas-phase abundances in the interstellar medium. This implies that they can be used as tracers of the chemical enrichment of matter in the Universe at high redshift. Comparable observations in stars are more difficult and, until recently, plagued by small number statistics. We wish to exploit the Gaia ESO Survey (GES) data to study the behaviour of sulfur and zinc abundances of a large number of Galactic stars, in a homogeneous way. By using the UVES spectra of the GES sample, we are able to assemble a sample of 1301 Galactic stars, including stars in open and globular clusters in which both sulfur and zinc were measured. We confirm the results from the literature that sulfur behaves as an alpha-element. We find a large scatter in [Zn/Fe] ratios among giant stars around solar metallicity. The lower ratios are observed in giant stars at Galactocentric distances less than 7.5 kpc. No such effect is observed among dwarf stars, since they do not extend to that radius. Given the sample selection, giants and dwarfs are observed at different Galactic locations, and it is plausible, and compatible with simple calculations, that Zn-poor giants trace a younger population more polluted by SN Ia yields. It is necessary to extend observations in order to observe both giants and dwarfs at the same Galactic location. Further theoretical work on the evolution of zinc is also necessary.

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arXiv2017

An in-depth spectroscopic examination of molecular bands from 3D hydrodynamical model atmospheres. II. Carbon-enhanced metal-poor 3D model atmospheres

Context. Tighter constraints on metal-poor stars we observe are needed to better understand the chemical processes of the early Universe. Computing a stellar spectrum in 3D allows one to model complex stellar behaviours, which cannot be replicated in 1D. Aims. We examine the effect that the intrinsic CNO abundances have on a 3D model structure and the resulting 3D spectrum synthesis. Methods. Model atmospheres were computed in 3D for three distinct CNO chemical compositions using the CO 5 BOLD model atmosphere code, and their internal structures were examined. Synthetic spectra were computed from these models using Linfor3D and they were compared. New 3D abundance corrections for the G-band and a selection of UV OH lines were also computed. Results. The varying CNO abundances change the metal content of the 3D models. This had an effect on the model structure and the resulting synthesis. However, it was found that the C/O ratio had a larger effect than the overall metal content of a model. Conclusions. Our results suggest that varying the C/O ratio has a substantial impact on the internal structure of the 3D model, even in the hot turn-off star models explored here. This suggests that bespoke 3D models, for specific CNO abundances should be sought. Such effects are not seen in 1D at these temperature regimes.

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arXiv2017

On the computation of eigenfrequencies for equilibrium models including turbulent pressure

The space-borne missions have provided a wealth of highly accurate data. However, our inability to properly model the upper-most region of solar-like stars prevents us from making the best of these observations. This problem is called "surface effect" and a key ingredient to solve it is turbulent pressure for the computation of both the equilibrium models and the oscillations. While 3D hydrodynamic simulations help to include properly the turbulent pressure in the equilibrium models, the way this surface effect is included in the computation of stellar oscillations is still subject to uncertainties. We aim at determining how to properly include the effect of turbulent pressure and its Lagrangian perturbation in the adiabatic computation of the oscillations. We also discuss the validity of the gas-gamma model (GGM) and reduced gamma model (RGM) approximations, which have been used to compute adiabatic oscillations of equilibrium models including turbulent pressure. We use a patched model of the Sun with an inner part constructed by a 1D stellar evolution code (CESTAM) and an outer part by the 3D hydrodynamical code (CO$^5$BOLD). Then, the adiabatic oscillations are computed using the ADIPLS code for the GGM and RGM and with the MAD code imposing the adiabatic condition on an existing time-dependent convection (TDC) formalism. We show that the computation of the oscillations using the TDC formalism in the adiabatic limit improves significantly the agreement with the observed frequencies compared to the GGM and RGM. Of the components of the turbulent pressure perturbation, the perturbation of the density and advection term is found to contribute most to the frequency shift. We propose a formalism to evaluate the frequency shift due to the inclusion of the term with the turbulent pressure perturbation in the variational principle in order to extrapolate our result to other stars.

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