Michel Rieutord

New multi-dimensional modelling of stellar evolution requires a new formulation of the mathematical problem; instead of the classical Lagrangian approach of 1D stellar models, multi-dimensional ones, such as Evolution STEllaire en Rotation (ESTER) models, use an Eulerian description of fluid flows. In this approach a spectral discretisation of the equations is used. However, ESTER models are presently restricted to early-type stars since numerical difficulties arise when dealing with low-mass or solar-type stars. This study is aimed at analysing the origin of these numerical difficulties and offering new strategies to circumvent them. After showing that the difficulties of ESTER in the low-mass star range are associated with the strong variations of opacities with temperature, we designed a toy model based on a simple plane-parallel fluid layer in hydrostatic equilibrium, equipped with an opacity law that reproduces the temperature dependence of solar opacity well. We took the Sun as the typical example of a low-mass star. The toy model clearly reproduces the difficulties faced by ESTER models when the surface effective temperature drops below 10 000 K. In this case the non-linear differential equation that determines the temperature profile becomes stiff and the number of the Newton solver iterations increases dramatically. As this model allows for the introduction of an adiabatic layer when destabilising temperature gradients are met, we show that one additional difficulty is the determination of the interfaces between this layer and the neighbouring radiative layers. It is shown that convergence towards the solution is efficient if interface positions are updated only when Newton corrections are small enough. The toy model also indicates that for low-mass stars, Newton iterations should be initialised with a fully isentropic layer. Convergence is then faster and more robust than with an initial fully radiative model, as is actually the case for ESTER models. This toy model opens a pathway to 2D or 3D models describing the stellar evolution of low-mass stars. This path has now to be assessed.

Context. Surface convection in cool stars sets the entropy jump between the atmosphere and the deep convective envelope and therefore affects the radius and structure of one-dimensional stellar models. In standard local treatments, this entropy jump is controlled through an effective convective efficiency. Aims. We have developed a phase-space closure that relates the mean stratification to the local distribution of entropy and velocity among convective elements. Our aim is to describe how the surface entropy transition emerges from the statistics of these elements rather than prescribing it directly. Methods. We first constructed a maximum-entropy model for the convective bulk, assuming that the realised states are bounded by a deep entropy level S . This prediction was then confronted with three-dimensional radiation-hydrodynamics simulations from the M3DIS grid. The comparison motivated an extension to a two-population closure in which an optimal bulk component coexists with a radiatively cooled plume component. Results. The maximum-entropy construction predicts a one-sided exponential entropy distribution, and its low-entropy tail is recovered in the deep convective part of the simulations. Closer to the surface, the distribution is reorganised by radiative cooling and separates into two thermodynamic components. Using a radiative entropy branch, an adiabatically descending cooled branch, and a population fraction controlled by optical depth, the two-population closure reproduces the mean entropy stratification of the four reference simulations over most of the displayed range without adjusting the entropy profile itself. Conclusions. The surface transition can therefore be interpreted as a continuous population decomposition, rather than as a sharp boundary between radiative and convective layers. This provides a natural route towards one-dimensional stellar models in which the surface entropy jump is determined from the statistical organisation of the flow instead of being set by a calibrated mixing-length efficiency.

Context. Convection remains one of the main uncertain links between multidimensional hydrodynamics and 1D stellar evolution. In particular, transition regions such as near-surface layers or convective boundaries require mean-field descriptions that remain connected to the underlying dynamics rather than to a prescribed mixing length. Aims. We aim to build a dynamical mean-field framework for stellar convection in which the averaging procedure, the hierarchy of moments, and the local stability diagnostic all follow from the motion of convective fluid elements. This provides the theoretical basis for the closure model developed in the companion paper. Methods. We describe the flow as a distribution of mesoscopic fluid particles in position–velocity space. A conservation law for this distribution defines the average and yields the Reynolds-Favre mean-field equations as velocity-space moments. Under standard interior conditions, the same dynamics can be written in Liouvillian form, which extends the Hamiltonian structure to stratified and dissipative media. Results. The Liouvillian formulation identifies the phase-space divergence, { s , T }, as a local measure of contraction or expansion of nearby trajectories. In the quasi-adiabatic limit, its sign reduces to the classical Schwarzschild stability criterion. Beyond this limit, the diagnostic remains velocity-resolved and can distinguish between different parts of the convective population within the same layer, for example in surface and penetration regions. Appendices show how rotation, magnetic fields, and composition changes can be incorporated through modifications of the phase-space structure. Conclusions. Phase-space averaging provides a dynamically grounded route from hydrodynamics to mean-field stellar convection equations. It also supplies a local trajectory-based stability diagnostic and a natural starting point for the maximum-entropy closures constructed in the companion paper.

Context. Be stars are currently viewed as B-type stars surrounded by a disc fuelled by the star itself during episodic excretion events. The origin of these events is poorly understood. Aims. This study aims to determine whether surface equatorial Kelvin waves can be unstable and therefore can play a role in triggering the Be phenomenon. Methods. We first derived an analytical expression for gravito-inertial modes in the shallow water framework. We then numerically investigated the evolution of equatorial Kelvin modes as the system parameters varied. We extended the study to thick-layer configurations with a constant-density fluid. We then analysed the stability of these modes under differential rotation and viscous effects. Results. We show that equatorial Kelvin waves still exist in a spherical shell of finite thickness, but their equatorial confinement is weaker. At low azimuthal wave numbers, Kelvin waves lie in the inertial-wave frequency band and therefore exhibit specificities of inertial waves, such as shear layers associated with singularities of the Poincaré equation. These shear layers constitute new dissipative structures for Kelvin waves. When a radial (shellular) differential rotation is imposed, we show that equatorial Kelvin waves can be destabilised, provided that differential rotation and viscosity are in an appropriate range. We trace back the non-monotonic behaviour of the instability growth rate to the rise of a critical layer where the fluid azimuthal velocity equals the phase speed of the surface waves. Conclusions. This study provides new insights into the behaviour of equatorial Kelvin waves in astrophysics, particularly in rapidly rotating stars. The results reinforce the idea that gravito-inertial waves, and more specifically the equatorial Kelvin waves, can be unstable and thus constitute key components in the mechanisms leading to the Be phenomenon.

Context. The fundamental parameters of rapidly rotating stars are key quantities required to understand the impact of rotation on stellar evolution. A few nearby early-type stars offer the possibility to take precise measurements of these parameters, which helped us to constrain newly available two-dimensional models. Aims. We propose a method to retrieve the fundamental parameters of a fast-rotating star (mass, rotation rate, and age) and the inclination of its rotation axis on the line of sight using five spectrophotometric observables along with a set of steady 2D models. Methods. From two photometric indices, the temperature T _IRFM derived by the infrared flux method, the projected equatorial velocity ( V sin i ), and the apparent luminosity, along with a grid of 2D steady-state models, we selected models that are compatible with all observational constraints and derived the most probable mass, rotation rate, core hydrogen mass fraction relative to that of the envelope, and inclination of the rotation axis on the line of sight of the targeted star. Results. We applied this method to two stars: Rasalhague ( α Oph, A5IV) and Alkaid ( η UMa, B3V). We confirm and improve on the fundamental parameters of Rasalhague previously determined by infrared interferometry and provide a new determination of its rotation axis inclination on the line of sight, which we find to be 68.9 ± 5.6°. Concerning Alkaid, about which little is known, we infer a mass of 5.071 ± 0.023 M _⊙ and a rotation rate of 0.413 ± 0.057 times the critical angular velocity, corresponding to an equatorial rotation period of 14.6 hours. We also find an inclination of the rotation axis of 41.9 ± 8.2° on the line of sight. We show that Alkaid is a very young star, presumably between 2 and 8 Myr off the zero-age main sequence. As a side result, using high resolution spectra and the Least Square Deconvolution method, we determined a precise value of the V sin i of Rasalhague, namely 224.3 ± 2.6 km/s. Similarly, we find V sin i = 154.3 ± 9.1 km/s for Alkaid. Conclusions. Our results show that fundamental parameters of rapidly rotating stars can be determined (or at least bracketed) using five observable quantities that can be measured from the spectrum of the star. This is of great interest for stars that are beyond reach of interferometric observations.

Context. Asymmetries in the observed rotational splittings of a multiplet contain information about the star’s rotation profile and internal magnetic field. Moreover, the frequency regularities of multiplets can be used for mode identification. However, to exploit this information, highly accurate theoretical predictions are needed. Aims. We aim to quantify the difference in the predicted mode asymmetries between a 1D perturbative method and a 2D method that includes a 2D stellar structure model, which takes rotation into account. We then place these differences between 1D and 2D methods in the context of asteroseismic measurements of internal magnetic fields. We only focus on the asymmetries and not on possible additional frequency peaks that can arise when the magnetic and rotation axis are misaligned. Methods. We coupled the 1D pulsation codes GYRE and StORM to the 2D stellar structure code ESTER and compared the oscillation predictions with the results from the 2D TOP pulsation code. We focused on zero-age main-sequence models representative of rotating β Cephei pulsators spinning at up to 20 per cent of the critical Keplerian rotation rate. Specifically, we investigated low-radial-order gravity and pressure modes. Results. We find a generally good agreement between the oscillation frequencies resulting from the 1D and 2D pulsation codes. We report differences in predicted mode multiplet asymmetries of mostly below 0.06 d ⁻¹ . Since the magnetic asymmetries are small compared to the differences in the rotational asymmetries resulting from the 1D and 2D predictions, accurate measurements of the magnetic field are in most cases challenging. Conclusions. Differences in the predicted mode asymmetries of a rotating star between 1D perturbative methods and 2D non-perturbative methods can greatly hinder accurate measurements of internal magnetic fields in main-sequence pulsators with low-order modes. Nevertheless, reasonably accurate measurements could be possible with n _pg ≥ 2 modes if the internal rotation is roughly below 10 per cent of the Keplerian critical rotation frequency for (aligned) magnetic fields of the order of a few hundred kilogauss. While the differences between the 1D and 2D frequency predictions are mostly too large for internal magnetic field detections, the rotational asymmetries predicted by StORM are in general accurate enough for asteroseismic modelling of the stellar rotation in main-sequence stars with identified low-order modes.

The Evolution STEllaire en Rotation (ESTER) code is the first 2D stellar structure code to be made open-source and freely available to the astronomy and astrophysics community. An important and novel advancement of this code is that it can reproduce the distorted shape and observable signatures (e.g., gravity darkening) of rapidly rotating stars. ESTER also calculates the steady-state large-scale flows within the star, namely their differential rotation and associated meridional circulation. In this report, we explore and document the physics implemented within version 1.1.0rc2 of the ESTER code, in a way that complements published descriptions. We illustrate this physics by plotting how stellar structure parameters vary through stellar interiors at a range of latitudes and at different angular velocities. We investigate how the thin convective envelopes of intermediate mass stars vary with latitude when rapidly rotating, becoming deeper and thicker near the equator. Simple comparisons of ESTER model predictions (e.g., central temperature and density, luminosity) with the output from the Modules for Experiments in Stellar Astrophysics (MESA) code [Paxton et al., 2010] shows generally good agreement. Additional comparisons provide important benchmarking and verification for ESTER as a comparatively young code. Finally, we provide a guide for installing and running the code on our local university cluster, aimed at helping students to begin work.

Context. The rotational evolution of stars remains an open question in stellar physics because numerous phenomena contribute to the distribution of angular momentum. Aims. This paper aims to determine the timescale over which a rotating early-type star relaxes to a steady baroclinic state or, equivalently, the conditions under which its nuclear evolution is slow enough to allow the star's evolution to be modelled as a series of quasi-steady states. Methods. We investigate the damping timescale of baroclinic and viscous eigenmodes that are potentially excited by the continuous forcing of nuclear evolution. We first examine this with a spherical Boussinesq model. Since much of the dynamics is concentrated in the radiative envelope of the star, we then improve the realism of the modelling by using a polytropic model of the envelope that incorporates a realistic density profile. Results. The polytropic model of the envelope highlights the key role of the region at the core-envelope interface. The results of evolutionary models recently obtained with two-dimensional axisymmetric ESTER models appear to arise from the slow damping of viscous modes. Using a vanishing Prandtl number appears to be too strong an approximation to explain the models’ dynamics. Baroclinic modes, previously thought to be good candidates for this relaxation process, are found to be too rapidly damped. Conclusions. The dynamical response of rotating stars to the slow forcing of their nuclear evolution appears as a complex combination of non-oscillating eigenmodes. Simple Boussinesq approaches are not sufficiently realistic to explain this reality. This study underlines the key role of layers near the core-envelope interface in early-type stars as well as the importance of angular momentum transport mechanisms-here represented by viscosity-for early-type stars to reach critical rotation, which is presumably associated with the Be phenomenon.

The rotational evolution of stars is still an open question of stellar physics because of the numerous phenomena that can contribute to the distribution of angular momentum. This paper aims at determining the time scale over which a rotating early-type star relaxes to a steady baroclinic state or, equivalently, in which case its nuclear evolution is slow enough to let the evolution of the star be modelled by a series of quasi-steady states. We investigate the damping time scale of baroclinic and viscous eigenmodes that are potentially excited by the continuous forcing of nuclear evolution. We first investigate this problem with a spherical Boussinesq model. Since much of the dynamics is concentrated in the radiative envelope of the star, we then improve the realism of the modelling by using a polytropic model of the envelope that takes into account a realistic density profile. The polytropic model of the envelope underlines the key role of the region at the core-envelope interface. The results of evolutionary models recently obtained with two-dimensional axisymmetric ESTER models turn out to be a consequence of the slow damping of viscous modes. Using a vanishing Prandtl number appears to be too strong an approximation to explain the dynamics of the models. Baroclinic modes previously thought as good candidates of this relaxation process turn out to be too quickly damped. The dynamical response of rotating stars to the slow forcing of its nuclear evolution appears as a complex combination of non-oscillating eigenmodes. Simple Boussinesq approach are not realistic enough to explain this reality. The present work underlines the key role of layers near the core-envelope interface in an early-type star and also the key role of any angular momentum transport mechanism, here played by viscosity, for early-type stars to reach critical rotation, presumably associated with the Be phenomenon.

Context. Understanding the effects of rotation in stellar evolution is key to modelling early-type stars, half of which have equatorial velocities over 100 km s⁻¹. The nearby star Altair is an example of such fast-rotating stars, and furthermore, it has the privilege of being modelled by a detailed 2D concordance model that reproduces most of its observables. Aims. The aim of this paper is to include new asteroseismic frequencies to improve our knowledge of Altair, especially its age. Methods. We processed images of Altair obtained during July 2022 by the Transiting Exoplanet Survey Satellite using the halo photometry technique to obtain its light curve over this observation period. Results. By analysing the light curve, we derived a set of 22 new frequencies in the oscillation spectrum of Altair and confirmed 12 previously known frequencies. Compared with model predictions, we could associate ten frequencies with ten axisymmetric modes. This identification is based on the modelled visibility of the modes. Moreover, nine of the modelled frequencies can be adjusted to simultaneously match their corresponding observed frequencies, once the core hydrogen mass fraction of the concordance model is set to X_core/X_ini ≃ 0.972, with X_ini = 0.739. Using the combined results of a 1D MESA model computing the pre-main sequence and a 2D time-dependent ESTER model computing the main sequence, we find that this core hydrogen abundance sets the age of Altair to 88 ± 10 Myr, which is slightly younger than previous estimates.