Aims. This series of papers aims at building a new formalism specifically tailored to study the impact of turbulence on the global modes of oscillation in solar-like stars. This first paper aims at deriving a linear wave equation that directly and co
nsistently contains the turbulence as an input to the model, and therefore naturally contains the information on the coupling between the turbulence and the modes, through the stochasticity of the equations. Methods. We use a Lagrangian stochastic model of turbulence based on Probability Density Function methods to describe the evolution of the properties of individual fluid particles through stochastic differential equations. We then transcribe these stochastic differential equations from a Lagrangian frame to an Eulerian frame, more adapted to the analysis of stellar oscillations. We combine this method with Smoothed Particle Hydrodynamics, where all the mean fields appearing in the Lagrangian stochastic model are estimated directly from the set of fluid particles themselves, through the use of a weighting kernel function allowing to filter the particles present in any given vicinity. The resulting stochastic differential equations on Eulerian variables are then linearised. Results. We obtain a stochastic, linear wave equation governing the time evolution of the relevant wave variables, while at the same time containing the effect of turbulence. The wave equation generalises the classical, unperturbed propagation of acoustic waves in a stratified medium to a form that, by construction, accounts for the impact of turbulence on the mode in a consistent way. The effect of turbulence consists in a non-homogeneous forcing term, responsible for the stochastic driving of the mode, and a stochastic perturbation to the homogeneous part of the wave equation, responsible for both the damping of the mode and the modal surface effects.
The use of the full potential of stellar seismology is made difficult by the improper modeling of the upper-most layers of solar-like stars and their influence on the modeled frequencies. Our knowledge on these emph{surface effects} has improved than
ks to the use of 3D hydrodynamical simulations but the calculation of eigenfrequencies relies on empirical models for the description of the Lagrangian perturbation of turbulent pressure: the reduced-$Gamma_1$ model (RGM) and the gas-$Gamma_1$ model (GGM). Starting from the fully compressible turbulence equations, we derive both the GGM and RGM models using a closure to model the flux of turbulent kinetic energy. It is found that both models originate from two terms: the source of turbulent pressure due to compression produced by the oscillations and the divergence of the flux of turbulent pressure. It is also demonstrated that they are both compatible with the adiabatic approximation but also imply a number of questionable assumptions mainly regarding mode physics. Among others hypothesis, one has to neglect the Lagrangian perturbation of the dissipation of turbulent kinetic energy into heat and the Lagrangian perturbation of buoyancy work.
The development of space-borne missions has significantly improved the quality of the measured spectra of solar-like oscillators. Their $p$-mode line profiles can now be resolved, and the asymmetries inferred for a variety of stars other than the Sun
. However, it has been known for a long time that the asymmetries of solar $p$-modes are reversed between the velocity and the intensity spectra. Understanding the origin of this reversal is necessary in order to use asymmetries as a tool for seismic diagnosis. For stars other than the Sun, only the intensity power spectrum is sufficiently resolved to allow for an estimation of mode asymmetries. We recently developed an approach designed to model and predict these asymmetries in the velocity power spectrum of the Sun and to successfully compare them to their observationally derived counterparts. In this paper we expand our model and predict the asymmetries featured in the intensity power spectrum. We find that the shape of the mode line profiles in intensity is largely dependent on how the oscillation-induced variations of the radiative flux are treated, and that modelling it realistically is crucial to understanding asymmetry reversal. Perturbing a solar-calibrated grey atmosphere model, and adopting the quasi-adiabatic framework as a first step, we reproduce the asymmetries observed in the solar intensity spectrum for low-frequency modes. We conclude that, unlike previously thought, it is not necessary to invoke an additional mechanism (e.g. non-adiabatic effects, coherent non-resonant background signal) to explain asymmetry reversal. This additional mechanism is necessary, however, to explain asymmetry reversal for higher-order modes.
In this paper, we aim to develop a predictive model for solar radial $p$-mode line profiles in the velocity spectrum. Unlike the approach favoured by prior studies, this model is not described by free parameters and we do not use fitting procedures t
o match the observations. Instead, we use an analytical turbulence model coupled with constraints extracted from a 3D hydrodynamic simulation of the solar atmosphere. We then compare the resulting asymmetries with their observationally derived counterpart. We find that stochastic excitation localised beneath the mode upper turning point generates negative asymmetry for $ u < u_text{max}$ and positive asymmetry for $ u > u_text{max}$. On the other hand, stochastic excitation localised above this limit generates negative asymmetry throughout the $p$-mode spectrum. As a result of the spatial extent of the source of excitation, both cases play a role in the total observed asymmetries. By taking this spatial extent into account and using a realistic description of the spectrum of turbulent kinetic energy, both a qualitative and quantitative agreement can be found with solar observations perfoemed by the GONG network. We also find that the impact of the correlation between acoustic noise and oscillation is negligible for mode asymmetry in the velocity spectrum.
The space-borne missions CoRoT and Kepler have already brought stringent constraints on the internal structure of low-mass evolved stars, a large part of which results from the detection of mixed modes. However, all the potential of these oscillation
modes as a diagnosis of the stellar interior has not been fully exploited yet. In particular, the coupling factor or the gravity-offset of mixed modes, $q$ and $varepsilon_{rm g}$, are expected to provide additional constraints on the mid-layers of red giants, which are located between the hydrogen-burning shell and the neighborhood of the base of the convective zone. In the present paper, we investigate the potential of the coupling factor in probing the mid-layer structure of evolved stars. Guided by typical stellar models and general physical considerations, we modeled the coupling region along with evolution. We subsequently obtained an analytical expression of $q$ based on the asymptotic theory of mixed modes and compared it to observations. We show that the value of $q$ is degenerate with respect to the thickness of the coupling evanescent region and the local density scale height. A structural interpretation of the global variations in $q$ observed on the subgiant and the red giant branches, as well as on the red clump, was obtained in the light of this model. We demonstrate that $q$ has the promising potential to probe the migration of the base of the convective region as well as convective extra-mixing in evolved red giant stars with typically $ u_{rm max} lesssim 100~mu$Hz. We also show that the frequency-dependence of $q$ cannot be neglected in the oscillation spectra of such stars, which is in contrast with what is assumed in the current measurement methods. This analytical study paves the way for a more quantitative exploration of the link of $q$ with the internal properties of evolved stars using stellar models.
Space-borne missions CoRoT and Kepler have provided a rich harvest of high-quality photometric data for solar-like pulsators. It is now possible to measure damping rates for hundreds of main-sequence and thousands of red-giant. However, among the sei
smic parameters, mode damping rates remain poorly understood and thus barely used for inferring the physical properties of stars. Previous approaches to model mode damping rates were based on mixing-length theory or a Reynolds-stress approach to model turbulent convection. While able to grasp the main physics of the problem, those approaches are of little help to provide quantitative estimates as well as a definitive answer on the relative contribution of each physical mechanism. Our aim is thus to assess the ability of 3D hydrodynamical simulations to infer the physical mechanisms responsible for damping of solar-like oscillations. To this end, a solar high-spatial resolution and long-duration hydrodynamical 3D simulation computed with the ANTARES code allows probing the coupling between turbulent convection and the normal modes of the simulated box. Indeed, normal modes of the simulation experience realistic driving and damping in the super-adiabatic layers of the simulation. Therefore, investigating the properties of the normal modes in the simulation provides a unique insight into the mode physics. We demonstrate that such an approach provides constraints on the solar damping rates and is able to disentangle the relative contribution related to the perturbation of the turbulent pressure, the gas pressure, the radiative flux, and the convective flux contributions. Finally, we conclude that using the normal modes of a 3D numerical simulation is possible and is potentially able to unveil the respective role of the different physical mechanisms responsible for mode damping provided the time-duration of the simulation is long enough.
The CoRoT and Kepler missions have provided high-quality measurements of the frequency spectra of solar-like pulsators, enabling us to probe stellar interiors with a very high degree of accuracy by comparing the observed and modeled frequencies. Howe
ver, the frequencies computed with 1D models suffer from systematic errors related to the poor modeling of the uppermost layers of stars. These biases are what is commonly named the near surface effect. The dominant effect is related to the turbulent pressure that modifies the hydrostatic equilibrium and thus the frequencies. This has already been investigated using grids of 3D RMHD simulations, which also were used to constrain the parameters of the empirical correction models. However, the effect of metallicity has not been considered so far. We study the impact of metallicity on the surface effect across the HR diagram, and provide a method for accounting for it when using the empirical correction models. We computed a grid of patched 1D stellar models with the stellar evolution code CESTAM in which poorly modeled surface layers have been replaced by averaged stratification computed with the 3D RMHD code CO5BOLD. We found that metallicity has a strong impact on the surface effect: keeping T_eff and log g constant, the frequency residuals can vary by up to a factor two. Therefore, the influence of metallicity cannot be neglected. We found that a correct way of accounting for it is to consider the surface Rosseland mean opacity. It allowed us to give a physically-grounded justification as well as a scaling relation for the frequency differences at nu_max as a function of T_eff, log g and kappa. Finally, we provide prescriptions for the fitting parameters of the correction functions. We show that the impact of metallicity through the Rosseland mean opacity must be taken into account when studying and correcting the surface effect.
First-ascent red giants with masses below about $2,M_odot$ ignite helium in their degenerate core as a flash. Stellar evolution codes predict that the He flash consists of a series of consecutive subflashes. The detection of mixed modes in red giants
from space missions CoRoT and Kepler has opened new opportunities to search for stars in this evolution stage. During a subflash, the He burning shell is convective, which splits the cavity of gravity modes in two. We here investigate how this additional cavity modifies the oscillation spectrum of the star. We calculate the asymptotic mode frequencies of stellar models going through a He subflash using the JWKB approximation. To predict the detectability of the modes, we estimate their expected heights, taking into account the effects of radiative damping in the core. Our results are then compared to the oscillation spectra obtained by calculating numerically the mode frequencies during a He subflash. We show that during a He subflash, the detectable oscillation spectrum mainly consists of modes trapped in the acoustic cavity and in the outer g-mode cavity. The spectrum should thus resemble that of a core-helium-burning giant. However, we find a list of clear, detectable features that could enable us to identify red giants passing through a He subflash. In particular, during a He subflash, several modes that are trapped in the innermost g-mode cavity are expected to be detectable. We show that these modes could be identified by their frequencies or by their rotational splittings. Other features, such as the measured period spacing of gravity modes or the location of the H-burning shell within the g-mode cavity could also be used to identify stars going through a He subflash. The features derived in this study can now be searched for in the large datasets provided by the CoRoT and Kepler missions.
Oscillation modes with a mixed character, as observed in evolved low-mass stars, are highly sensitive to the physical properties of the innermost regions. Measuring their properties is therefore extremely important to probe the core, but requires som
e care, due to the complexity of the mixed-mode pattern. This work aims at providing a consistent description of the mixed-mode pattern of low-mass stars, based on the asymptotic expansion. We also aim at studying the variation of the gravity offset $varepsilon_{g}$ with stellar evolution. We revisit previous work about mixed modes in red giants and empirically test how period spacings, rotational splittings, mixed-mode widths and heights can be estimated in a consistent view, based on the properties of the mode inertia ratios. From the asymptotic fit of the mixed-mode pattern of a large set of red giants at various evolutionary stages, we derive unbiased and precise asymptotic parameters. As the asymptotic expansion of gravity modes is verified with a precision close to the frequency resolution for stars on the red giant branch (10$^{-4}$ in relative values), we can derive accurate values of the asymptotic parameters. We decipher the complex pattern in a rapidly rotating star, and explain how asymmetrical splittings can be inferred, as well as the stellar inclinations. This allows us to revisit the stellar inclinations in two open clusters, NGC 6819 and NGC 6791: our results show that the stellar inclinations in these clusters do not have privileged orientation in the sky. The variation of the asymptotic gravity offset along with stellar evolution is investigated in detail. We also derive generic properties that explain under which conditions mixed modes can be observed.
Oscillation properties are usually measured by fitting symmetric Lorentzian profiles to the power spectra of Sun-like stars. However the line profiles of solar oscillations have been observed to be asymmetrical for the Sun. The physical origin of thi
s line asymmetry is not fully understood, although it should depend on the depth dependence of the source of wave excitation (convective turbulence) and details of the observable (velocity or intensity). For oscillations of the Sun, it has been shown that neglecting the asymmetry leads to systematic errors in the frequency determination. This could subsequently affects the results of seismic inferences of the solar internal structure. Using light curves from the {it Kepler} spacecraft we have measured mode asymmetries in 43 stars. We confirm that neglecting the asymmetry leads to systematic errors that can exceed the $1sigma$ confidence intervals for seismic observations longer than one year. Therefore, the application of an asymmetric Lorentzian profile is to be favoured to improve the accuracy of the internal stellar structure and stellar fundamental parameters. We also show that the asymmetry changes sign between cool Sun-like stars and hotter stars. This provides the best constraints to date on the location of the excitation sources across the Hertzsprung-Russel diagram.
