We propose a methodological framework to perform forward asteroseismic modeling of stars with a convective core, based on gravity-mode oscillations. These probe the near-core region in the deep ...stellar interior. The modeling relies on a set of observed high-precision oscillation frequencies of low-degree coherent gravity modes with long lifetimes and their observational uncertainties. Identification of the mode degree and azimuthal order is assumed to be achieved from rotational splitting and/or from period spacing patterns. This paper has two major outcomes. The first is a comprehensive list and discussion of the major uncertainties of theoretically predicted gravity-mode oscillation frequencies based on linear pulsation theory, caused by fixing choices of the input physics for evolutionary models. Guided by a hierarchy among these uncertainties of theoretical frequencies, we subsequently provide a global methodological scheme to achieve forward asteroseismic modeling. We properly take into account correlations among the free parameters included in stellar models. Aside from the stellar mass, metallicity, and age, the major parameters to be estimated are the near-core rotation rate, the amount of convective core overshooting, and the level of chemical mixing in the radiative zones. This modeling scheme allows for maximum likelihood estimation of the stellar parameters for fixed input physics of the equilibrium models, followed by stellar model selection considering various choices of the input physics. Our approach uses the Mahalanobis distance instead of the often-used χ2 statistic and includes heteroscedasticity. It provides estimation of the unknown variance of the theoretically predicted oscillation frequencies.
Context. The theory of rotational and chemical evolution is incomplete, thereby limiting the accuracy of model-dependent stellar mass and age determinations. The γ Doradus ( γ Dor) pulsators are ...excellent points of calibration for the current state-of-the-art stellar evolution models, as their gravity modes probe the physical conditions in the deep stellar interior. Yet, individual asteroseismic modelling of these stars is not always possible because of insufficient observed oscillation modes. Aims. This paper presents a novel method to derive distributions of the stellar mass, age, core-boundary mixing efficiency, and initial rotation rates for γ Dor stars. Methods. We computed a grid of rotating stellar evolution models covering the entire γ Dor instability strip. We then used the observed distributions of the luminosity, effective temperature, buoyancy travel time, and near-core rotation frequency of a sample of 539 stars to assign a statistical weight to each of our models. This weight is a measure of how likely the combination of a specific model is. We then computed weighted histograms to derive the most likely distributions of the fundamental stellar properties. Results. We find that the rotation frequency at zero-age main sequence follows a normal distribution, peaking at around 25% of the critical Keplerian rotation frequency. The probability-density function for extent of the core-boundary mixing zone, given by a factor of f CBM times the local pressure scale height (assuming an exponentially decaying parameterisation), decreases linearly with increasing f CBM . Conclusions. Converting the distribution of fractions of critical rotation at the zero-age main sequence to units of d −1 , we find most F-type stars start the main sequence with a rotation frequency between 0.5 d −1 and 2 d −1 . Regarding the core-boundary mixing efficiency, we find that it is generally weak in this mass regime.
Abstract
The asteroseismic modelling of period spacing patterns from gravito-inertial modes in stars with a convective core is a high-dimensional problem. We utilize the measured period spacing ...pattern of prograde dipole gravity modes (acquiring Π0), in combination with the effective temperature (Teff) and surface gravity (log g) derived from spectroscopy, to estimate the fundamental stellar parameters and core properties of 37 γ Doradus (γ Dor) stars whose rotation frequency has been derived from Kepler photometry. We use two 6D grids of stellar models, one with step core overshooting and another with exponential core overshooting, to evaluate correlations between the three observables Π0, Teff, and log g and the mass, age, core overshooting, metallicity, initial hydrogen mass fraction, and envelope mixing. We provide multivariate linear model recipes relating the stellar parameters to be estimated to the three observables (Π0, Teff, log g). We estimate the (core) mass, age, core overshooting, and metallicity of γ Dor stars from an ensemble analysis and achieve relative uncertainties of ∼10 per cent for the parameters. The asteroseismic age determination allows us to conclude that efficient angular momentum transport occurs already early on during the main sequence. We find that the nine stars with observed Rossby modes occur across almost the entire main-sequence phase, except close to core-hydrogen exhaustion. Future improvements of our work will come from the inclusion of more types of detected modes per star, larger samples, and modelling of individual mode frequencies.
Context. While rotation has a major impact on stellar structure and evolution, its effects are not well understood. Thanks to high-quality and long-time base photometric observations obtained with ...recent space missions, we are now able to study stellar rotation more precisely. Aims. We aim to constrain radial differential rotation profiles in γ Doradus (γ Dor) stars, and to develop new theoretical seismic diagnosis for such stars with rapid and potentially non-uniform rotation. Methods. We have derived a new asymptotic description which accounts for the impact of weak differential near-core rotation on gravity-mode period spacings. The theoretical predictions are illustrated from pulsation computations with the code GYRE and compared with observations of γ Dor stars. When possible, we also derived the surface rotation rates in these stars by detecting and analysing signatures of rotational modulation, and computed the core-to-surface rotation ratios. Results. Stellar rotation must be strongly differential before its effects on period spacing patterns can be detected, unless multiple period spacing patterns can be compared. Six stars in our sample exhibit a single unexplained period spacing pattern of retrograde modes. We hypothesise that these are Yanai modes. Finally, we find signatures of rotational spot modulation in the photometric data of eight targets. Conclusions. If only one period spacing pattern is detected and analysed for a star, it is difficult to detect differential rotation. A rigidly rotating model will often provide the best solution. Differential rotation can only be detected when multiple period spacing patterns have been found for a single star or its surface rotation rate is known as well. This is the case for eight of the stars in our sample, revealing surface-to-core rotation ratios between 0.95 and 1.05.
Context.
The physical mechanisms driving the transport of angular momentum in stars are not fully understood, as current models cannot explain the observed stellar rotation profiles across all stages ...of evolution.
Aims.
By making use of pulsating F-type dwarfs, we aim to (i) observationally calibrate the efficiency of angular momentum transport, assuming a constant uniform viscosity, and (ii) test how well state-of-the-art rotating stellar models with angular momentum (AM) transport by rotationally induced processes can explain observed rotation profiles. In both cases, the aim is to simultaneously reproduce the measured near-core rotation and core-to-surface rotation ratio.
Methods.
Asteroseismic modelling is applied to a sample of seven slowly rotating pulsators in order to derive (core) masses and ages from their gravity-mode oscillations. This work focuses on the main sequence (MS), using models that start with an initial uniform rotation frequency at the start of core-hydrogen burning, which is a free parameter. Two treatments of AM transport are considered: (i) a constant uniform viscosity, and (ii) rotationally induced processes (including the Spruit-Tayler dynamo). Next, the initial rotation frequency of each star is derived from the observed present-day near-core rotation frequency for both treatments.
Results.
Asteroseismic modelling of gravity mode periods reveals that all seven slowly rotating stars (one of which is not further modelled) in the sample are near the end of core-hydrogen burning. To explain the near-core rotation rate at the inferred age, initial rotation frequencies at the zero-age main sequence need to be below 10% of the initial critical break-up frequency. The derived initial rotation frequencies are consistent with previous works.
Conclusions.
A diffusive approximation of angular momentum transport can in general explain the observed rotation profiles of the six slowly rotating F-type dwarfs for average values of the viscosity of between 2 × 10
5
and 5 × 10
7
cm
2
s
−1
or when the viscosity is computed from rotationally induced mechanisms. Yet, for three stars in the sample, the core-to-surface rotation fraction from rotationally induced mechanisms is predicted to be higher than observed.
Context.
The
Kepler
and Transiting Exoplanet Survey Satellite (TESS) space telescopes delivered high-precision, long-duration photometric time series for hundreds of main-sequence stars, revealing ...their numerous gravito-inertial (
g
) pulsation modes. This high precision allows us to evaluate increasingly detailed theoretical stellar models. Recent theoretical work extended the traditional approximation of rotation, a framework to evaluate the effect of the Coriolis acceleration on
g
modes, to include the effects of the centrifugal acceleration in the approximation of slightly deformed stars, which so far have mostly been neglected in asteroseismology. This extension of the traditional approximation was conceived by re-deriving the traditional approximation in a centrifugally-deformed, spheroidal coordinate system.
Aims.
We explore the effect of the centrifugal acceleration on
g
modes and assess its detectability in space-based photometric observations.
Methods.
We implemented the new theoretical framework to calculate the centrifugal deformation of pre-computed 1D spherical stellar structure models and computed the corresponding
g
-mode frequencies, assuming uniform rotation. The framework was evaluated for a grid of stellar structure models covering a relevant parameter space for observed
g
-mode pulsators.
Results.
The centrifugal acceleration modifies the effect of the Coriolis acceleration on
g
modes, narrowing the equatorial band in which they are trapped. Furthermore, the centrifugal acceleration causes the pulsation periods and period spacings of the most common
g
modes (prograde dipole modes and
r
modes) to increase with values similar to the observational uncertainties of the measured period spacing values in
Kepler
and TESS data.
Conclusions.
The effect of the centrifugal acceleration on
g
modes is formally detectable in modern space photometry. The implementation of the used theoretical framework in stellar structure and pulsation codes will allow for more precise asteroseismic modelling of centrifugally deformed stars in order to assess its effect on mode excitation, trapping, and damping.
Context.
Asteroseismic studies show that cores of post-main-sequence stars rotate more slowly than theoretically predicted by stellar models with purely hydrodynamical transport processes. Recent ...studies of main-sequence stars, particularly Gamma Doradus (
γ
Dor) stars, have revealed the internal rotation rates for hundreds of stars, offering a counterpart on the main sequence for studies of angular momentum transport.
Aims.
We investigate whether such a disagreement between observed and predicted internal rotation rates is present in main-sequence stars by studying angular momentum transport in
γ
Dor stars. Furthermore, we test whether models of rotating stars with internal magnetic fields can reproduce their rotational properties.
Methods.
We computed rotating models with the Geneva stellar evolution code taking into account meridional circulation and shear instability. We also computed models with internal magnetic fields using a general formalism for transport by the Tayler-Spruit dynamo. We then compared these models to observational constraints for
γ
Dor stars that we compiled from the literature, thus combining the core rotation rates, projected rotational velocities from spectroscopy, and constraints on their fundamental parameters.
Results.
We show that combining the different observational constraints available for
γ
Dor stars enable us to clearly distinguish the different scenarios for internal angular momentum transport. Stellar models with purely hydrodynamical processes are in disagreement with the data, whereas models with internal magnetic fields can reproduce both core and surface constraints simultaneously.
Conclusions.
Similarly to results obtained for subgiant and red giant stars, angular momentum transport in radiative regions of
γ
Dor stars is highly efficient, in good agreement with predictions of models with internal magnetic fields.
Context. Galactic archaeology largely relies on precise ages of distant evolved stars in the Milky Way. Nowadays, asteroseismology can deliver ages for many red giants observed with high-cadence, ...high-precision photometric space missions such as CoRoT, Kepler , K2, TESS, and soon PLATO. Aims. Our aim is to quantify the age uncertainties of currently slowly rotating red giants due to the cumulative effect of their fast rotation during core-hydrogen burning: their rotation in earlier evolutionary phases caused mixing of elements, resulting in heavier helium cores and the prolongation of their main-sequence lifetime. These rotational effects are usually ignored when age-dating red giants, despite our knowledge of fast rotation for stars with M ≥ 1.3 M ⊙ . Methods. We used a sample of 490 F-type gravito-inertial pulsators ( γ Doradus stars) with precise asteroseismic estimates of their internal rotation rate from Kepler asteroseismology and with luminosity estimates from Gaia . For this sample, which includes stars rotating from nearly zero to about 60% of the critical rate, we computed the cumulative effect on the age in their post-main-sequence evolution caused by rotational mixing on the main sequence. We used stellar model grids with different physical prescriptions that mimic rotational mixing to assess systematic uncertainties on the age. Results. With respect to non-rotating models, the sample of 490 γ Doradus stars, as red giant progenitors, reveals age differences up to 5% by the time they start hydrogen-shell burning when relying on the theory of rotationally induced diffusive mixing as included in the MIST isochrones. Using rotational mixing based on an advective-diffusive approach that includes meridional circulation leads to an age shift of 20% by the time of the tip of the red giant branch. Conclusions. The age-dating of red giants is affected by the cumulative effect of rotational mixing during the main sequence. Such rotationally induced age shifts should be taken into account in addition to other effects if the aim is to perform Galactic archaeological studies at the highest precision.
Context.
The efficiency of the transport of angular momentum and chemical elements inside intermediate-mass stars lacks proper calibration, thereby introducing uncertainties on a star’s evolutionary ...pathway. Improvements require better estimation of stellar masses, evolutionary stages, and internal mixing properties.
Aims.
Our aim was to develop a neural network approach for asteroseismic modelling, and test its capacity to provide stellar masses, ages, and overshooting parameter for a sample of 37
γ
Doradus stars; these parameters were previously determined from their effective temperature, surface gravity, near-core rotation frequency, and buoyancy travel time Π
0
. Here our goal is to perform the parameter estimation from modelling of individual periods measured for dipole modes with consecutive radial order rather than from Π
0
. We assess whether fitting these individual mode periods increases the capacity of the parameter estimation.
Methods.
We trained neural networks to predict theoretical pulsation periods of high-order gravity modes (
n
∈ 15, 91), and to predict the luminosity, effective temperature, and surface gravity for a given mass, age, overshooting parameter, diffusive envelope mixing, metallicity, and near-core rotation frequency. We applied our neural networks for Computing Pulsation Periods and Photospheric Observables (
C-3PO
) to our sample and compute grids of stellar pulsation models for the estimated parameters.
Results.
We present the near-core rotation rates (from the literature) as a function of the inferred stellar age and critical rotation rate. We assessed the rotation rates of the sample near the start of the main sequence assuming rigid rotation. Furthermore, we measured the extent of the core overshoot region and find no correlation with mass, age, or rotation. Finally, for one star in our sample, KIC 12066947, we find indications of mode coupling in the period spacing pattern which we cannot reproduce with mode trapping.
Conclusions.
The neural network approach developed in this study allows the derivation of stellar properties dominant for stellar evolution, such as mass, age, and extent of core-boundary mixing. It also opens a path for future estimation of mixing profiles throughout the radiative envelope, with the aim of inferring these profiles for large samples of
γ
Doradus stars.
Contact.
Rotation is a key ingredient in the theory of stellar structure and evolution. Until now, stellar evolution codes operate in a one-dimensional framework for which the validity domain in ...regards to the rotation rate is not well understood.
Aims.
In this Letter, we present the first results of self-consistent stellar models in two spatial dimensions, which compute the time evolution of a star and its rotation rate along the main sequence (MS). We also present a comparison to observations.
Methods.
We make use of an extended version of the
ESTER
code, which solves the stellar structure of a rotating star in two dimensions with time evolution, including chemical evolution, and an implementation of rotational mixing. We computed evolution tracks for a 12
M
⊙
model, once for an initial rotation rate equal to 15% of the critical frequency, and once for 50%.
Results.
We first show that our model initially rotating at 15% of the critical frequency is able to reproduce all the observations of the
β
Cephei star HD 192575, which was recently studied with asteroseismology. Beyond the classical surface parameters, such as effective temperature or luminosity, our model also reproduces the core mass along with the rotation rate of the core and envelope at the estimated age of the star. This particular model also shows that the meridional circulation has a negligible influence on the transport of chemical elements such as nitrogen, for which the abundance may be increased at the stellar surface. Furthermore, it shows that in the late MS, nuclear evolution is faster than the relaxation time needed to reach a steady state of stellar angular momentum distribution.
Conclusions.
We demonstrate that we have successfully taken a new step towards two-dimensional evolutionary modelling of rotating stars. This opens new perspectives on the understanding of the dynamics of fast rotating stars and on the way rotation impacts stellar evolution.
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