# Stellar models and evolution

With sufficiently good data, the asteroseismic determination of mass and radius can be essentially independent of stellar models. For other quantities, particularly the age, the inferences involve fitting models to observables, and their accuracy and precision depends on our ability to model stellar evolution. Thus, the asteroseismic investigation of stellar structure and evolution is an essential part of the characterisation of planet hosts, and is necessary if we are to put planetary systems into an evolutionary context. Asteroseismic investigation of a large number of stars, with a range of masses and ages, is anecessary tool for constraining models of stellar interiors, identifying missing physics, and improving our understanding of stellar evolution.

### Surface convective transport and granulation

A longstanding problem in stellar modelling is the description of convective transport and energy flux. Prescriptions for such transport in 1D stellar model computation require knowledge of quantities that cannot be derived easily from first principles. One therefore needs information from numerical simulations and, most importantly, from observations of stars in various evolutionary stages and with differing chemical compositions. One example is the filling factor of downdraft plumes carrying energy from the top of a convection zone to the bottom. These types of observational constraints can only be obtained with highly precise seismic measurements (frequencies, amplitudes, mode lifetimes), and only for bright enough stars. Furthermore, the necessary seismic precision demands these stars to be observed over long runs.

Stellar parameters are derived by comparing observedoscillation frequencies with the theoretical frequencies. As well known in the solar case, the differences between observedand theoretical frequencies increase with the frequency. The higher the frequency of a mode, the closer its eigenfunctions peaks to the surface. The increasing difference between observed and theoretical frequencies therefore mainly reflects the imperfection with which the standard models treat the turbulent convective superficial layers but also how non-adiabatic pulsations are modelled and how the acoustic wave propagates across an heterogeneous, turbulent medium. These phenomena, referred to as near-surface effect, affect the accuracy of seismic measurements, which ultimately limits the power of seismic diagnosis. In order to correct for this effect, several authors have proposed empirical or semi-empirical corrections based on the assumption of adiabatic pulsations (e.g. Kjeldsen et al. 2008; Ball & Gizon 2014; Sonoi et al. 2015). Most of these corrections involved one or several free parameters. For instance, in Sonoi et al (2015), surface effects were modelled assuming adiabatic pulsation and using a coarse grid of 3D surface models. The predicted surface effects were next fitted with a Lorentzian function involving two free parameters. The authors have shown that these parameters vary in the HR diagram as a function of the surface parameters of the star, such as the effective temperature and the surface gravity. Another approach by Ball & Gizon (2014) based on a formula given by Gough (1990) consists in correcting the theoretical frequencies by a term proportional to ν3/I where ν and I are the frequency and inertia of the mode. The departure from adiabatic pulsations is also expected to contribute to the surface effects (Houdek 2010; Grighacene et al. 2012). However, the current non-adiabatictreatments of pulsations still rely on a crude description of the complex coupling between pulsation and convection (for reviews see Belkacem & Samadi, 2013; Houdek & Dupret 2015). All these models, however, involve free parameters. They must be confronted with the observations. This can be done with bright objects with seismic data of very high quality and detailed spectroscopic characterisation (Teff, metallicity). Available CoRoT and Keplerdata sets allow us to test the current theoretical modellingbut only for a small set of bright objects. This is insufficient to cover properly the free parameter space. On the other hand, PLATO will observe a high number of bright main-sequence and subgiant stars with various surface abundance and with different evolutionary status.

The granulationis a manifestation of convectionat the surface of the stars. Measuring the properties of surface granulation is an important step for a better understanding of the physics of surface convection and to provide constraints on its modelling. Using Keplerdata, scaling relations linking the granulation characteristics to the stellar parameters and seismic properties have been established (Kallinger et al. 2014; Mathur et al. 2011) and a theoretical basis to these scaling relations exists (Mathur et al. 2011; Samadi et al. 2013). It is theoretically expected that the turbulent Mach number controls the granulation properties to second order. The Keplerobservations unfortunately cannot confirm this trend. A confirmation requires observing a statistically larger number of dwarfs cooler than the Kepler stars. For hotter (F) stars, the comparison between theoretical models and observations indicates significant departures (Ludwig et al. 2009; Samadi et al. 2013), which are not yet explained. A possible explanation could be the partial inhibition of the convection by a magnetic field (cf. Ludwig et al 2009; Samadi et al 2013; Cranmer et al 2014). Another issue is the theoretically predicted dependence of granulation on the surface metal abundances (Ludwig et al. 2009; Tremblay et al 2013). Testing all these theoretical developments requires high quality observations – high signal-to-noise ratio and high frequency resolution ‒ for a sample of different types of stars larger than available with Kepler. The PLATO mission, on the other hand, is perfectly suited for such a purpose.

### Constraints on internal stratifications

Glitch observational constraints are powerful tools for placing constraints on the stratification inside solar-type stars (Monteiro et al. 2002; Mazumdar 2005). With the availability of precise frequency sets for a large number of stars, the variation of the locations of acoustic glitches in a large ensemble of solar-type stars populating the main-sequence and subgiant branches can be followed (Mazumdar et al. 2014). This will provide very strong constraints on the various transport processes (diffusion, turbulent chemical mixing) taking place at the base of the convection zone (Christensen-Dalsgaard et al. 2011; Lebreton & Goupil, 2012; Zhang C. G. et al 2012, 2013; Zhang Q. S. & Li 2012a, b). This will lead to necessary improvements in the physics in order to ensure that all observables including the location of the base of the convection zone are well fitted. We must stress that the initial helium content is an essential input to computations of stellar models but cannot be directly observed for F to K dwarfs. Systematic studies about the helium content in a given environment will then require its seismic measurement for a large sample of bright stars as proposed by the PLATO mission. However,one must keep in mind that the seismic measurements determine the helium abundance of the envelope and that its relation with the initial helium content (input to stellar models) depends on the processes taking place at the bottom of the convective envelope, which therefore must be constrained as well.

### Constraints on the core structure

Targets for the PLATO main program will have a spectral type from F5 to K7, which corresponds, for solar metallicity and MS-phase, to the mass interval 0.6‒1.4M. One of the most important issues in dating main-sequence (MS) stars with masses larger than the solar one is the inability for estimating from first principles the extension of chemically mixed regions. The main contributors to the uncertainty ofthe extension of chemically mixed regions are the transport processes of chemicals such as those related to convection and rotation.  Before the development of asteroseismology of solar like oscillators from space, the current way to estimate the extension of central mixed regions was the comparison of theoretical isochrones with the HR diagram of open clusters of different turn-off mass (or age), and the study of eclipsing binaries. These studies have made clear that:

• the extension of the central mixed region is underestimated by the models using the classical local MLT theory to describe convection and the Schwarzschild (or Ledoux) criterion for defining the boundaries of stable regions against convection. To reach a good fit of observations, an amount of extra-mixing above the classic boundaries is usually introduced in the models by using an opposite parameter, called overshooting parameter (for a review see Chiosi 2007).
• the efficiency of this extra-mixing increases with the stellar mass significantly in the mass interval corresponding to the onset and development of a convective core (~ 1.0‒1.5M). The variation of the efficiency with mass is expected to depend also on stellar metallicity (Bressan et al. 2012). Such a dependence on mass of the extra-mixing efficiency has been found for some targets in the Kepler field (Deheuvels et al. 2016).

Main-sequence stars more massive than about 1.1‒1.2M⊙ (depending on the chemical composition) develop a convective core on the main-sequence. A more or less efficient extra mixing then takes place at the boundary of convective cores. This directly influences the age of the star: in presence of extra mixing, the star is older at a given stage (central hydrogen content) along the main-sequence. The process of extra mixing is a 3D non-local process that cannot be fully modelled in 1D stellar evolutionary codes. The extension of the associated mixed (i.e. chemically homogeneous) region therefore is a free parameter in a 1D stellar model computation. A range of values for this parameter can be obtained by fitting clusters or eclipsing binaries, but when applied to individual stars, this generates large uncertainties on their ages. Besides, for small convective cores as for stars of interest in the PLATO P1 sample, the parameter seems to vary with mass of the star. One therefore needs to implement a modelling from 3D simulations, to provide calibrations that will have to be tested with a set of seismically well-characterised stars. It is indeed well known that the edge of this core produces an acoustic glitch (rapid local variation in the sound speed profile), to which the acoustic modes are sensitive (e.g. Cunha & Metcalfe 2007). This glitch induces an oscillation of the mode frequencies as a function of the radial order. The period of this oscillation is directly related to the location of the glitch, and thus to the size of the mixed core. It has thus been possible to measure seismically the extent of the mixed core and to obtain estimates of the efficiency of the extra mixing that takes place at the boundary of convective cores owing to various physical processes, such as core overshooting or rotational mixing. This has been achieved for several individual targets (Deheuvels et al. 2010; Goupil et al. 2011; Silva Aguirre et al. 2013) and more recently for a sample of eight Kepler targets (Deheuvels et al. 2016). These studies all showed the need for an extension of the mixed core beyond the Schwarzschild limit. Deheuvels et al. (2016) hinted that the extension of the mixed core increases with stellar mass for low-mass targets, and they suggested that this quantity could therefore be calibrated. This would help obtain more precise stellar ages, especially toward the end of the main-sequence, because the size of the mixed core determines the duration of the main-sequence.

So far, a seismic measurement of the size of the mixed core has been possible for only about ten stars. However, the PLATO mission will provide much more numerous targets for which such a method can be applied, thus making it possible to perform a calibration of the extension of convective cores. This will help produce more reliable ages for stellar models, and potentially reach a better understanding of the processes that are responsible for extending convective cores.

For the hottest main-sequence stars of the P1 sample such as F stars, the translation of seismic observations into constraints upon the central properties of the star may be difficult.However, we can have access to the information on the sizes of mixed regions during the MS phase (and hence main-sequence lifetime), from studying populations of stars during the post-MS evolution, that-is during H-shell burning phase (subgiant and low-luminosity red giant (RGB) stars) and during the central-He burning phase (Red Clump stars).

Subgiants, whose progenitors are massive enough to have had a convective core during the main-sequence, could provide precious constraints on the extent of convective cores. After the main-sequence turnoff, the rapid increase in the central density induces an increase in the Brunt-Vaisala frequency near the core. As a result, non-radial modes become mixed: they behave both as p-modes in the envelope and as g-modes in the core. This phenomenon occurs due to the coupling between the p-mode and g-mode cavities, which arises through the evanescent region that separates these cavities. It has been shown that the strength of this coupling depends crucially on the gradient of the mean molecular weight at the edge of the evanescent region (Deheuvels & Michel 2011). This gradient is in fact the result of the withdrawal of the convective core at the end of the main-sequence. Its location is therefore directly related to the size of the convective core during the main-sequence. For this reason, measuring the strength of the coupling using the observed frequencies of mixed modes can bring information about the extent of the mixed core during the main-sequence. The feasibility of such an approach has been demonstrated by Deheuvels & Michel (2011) using CoRoT data for the subgiant HD49385. This diagnostic can only be applied to early subgiants (shortly after the turnoff) because after this, the H-burning shell washes out the gradient of chemical composition that was left by the convective core during the main-sequence. The Kepler satellite has also observed such early subgiants. However, only the stars observed in the “short cadence” mode could provide detections of mixed modes for these stars. The PLATO mission will greatly increase the number of such targets, making it possible to measure the extent of the end-of-main-sequence convective core for stars of various masses and chemical composition. This will be a valuable addition to the measurement of the size of convective cores during the main-sequence by exploiting acoustic glitches.

Red giants: Transport processes leave in the stellar structure features that not always have a distinct signature on the classical or asteroseismic properties when the star evolves on the main-sequence. However, the effect of these features can be clearly seen in post-main sequence evolution phases. Moreover, some stars will be barely detected as solar-like oscillators during their MS (intermediate stellar masses) or during the sub-giant phase because of the rapid crossingof the HR diagrambefore starting to burn He. Therefore, the goal of improving stellar evolution models to reach the highest accuracy in the characterisation of planetary systems requires seismic information about stellar structure where it is available, hence in main-sequence as well as post-main sequence stars (sub-giants, RGB and He-burning phases).

Low-mass red giant stars all have a similar He-core mass (~0.48M) and populate the so-called Red Clump. Red giants with higher masses populate the secondary Red Clump. In young stellar clusters populated enough, such as clusters in the Magellanic Clouds, these two groups are clearly identified from their luminosity. For red giants in the field, seismic indexes for red giants such as the dipole period spacing (DP1) can be used to classify red giant stars as RGB, He-burning phases (RC1 and RC2). DP1is closely linked to the mass of the He-core, with two different linear laws for RGB and He-burning phases. The chemical composition and/or MS extra-mixing process determine the stellar mass distribution in the RC2 population (Montalban et al. 2013). From comparison with stellar models, the seismic indexes (Dν, νmaxand DP1) will determine the efficiency of chemical mixing processes acting during the MS phase, essential for stellar dating. Therefore, seismic and spectroscopic studies of RC2 populations are a key tool to characterise the efficiency of extra-mixing processes for MS stars in the galactic field.

### Constraints on angular momentum transport

Another main source of uncertainty affecting age determination is the presence and efficiency of transport mechanisms in radiative zones (Zahn 1992; Maeder 2009). While these mechanisms can have a significant impact on the main-sequence lifetime, they are still poorly understood and crudely modelled. The rotationally induced chemical mixing is believed to be directly linked to the evolution of angular momentum inside the stars (Zahn 1992; Maeder & Zahn 1998).  Indeed, angular momentum and chemical elements can be both transported in the radiative zones of rotating stars through meridional circulation and hydrodynamical instabilities. This results in a change of the global and asteroseismic properties of stars when rotational effects are taken into account, and in particular to an increase of the main-sequence lifetime due to the transport of fresh hydrogen fuel in the stellar core (e.g. Eggenberger et al. 2010). These changes depend on the poorly known efficiency of rotational mixing, which can be constrained by obtaining information about the internal rotation profiles in stellar radiative zones. Radial differential rotation can be inferred by asteroseismology for stars that have mixed modes (e.g. Suárez et al. 2006). These modes have a g-mode character in the core and a p-mode character in the envelope. They are therefore sensitive to the core, while having amplitudes large enough to be detected at the surface. Mixed modes are present in subgiant and red giant stars (e.g. Beck et al. 2011), and depth variation of internal rotation has already been measured using Kepler data (e.g. Beck et al. 2012; Deheuvels et al. 2012, 2014). This brings valuable constraints on the transport processes during and, perhaps more importantly, prior to the post-main sequence stage.

Core rotation rate as a function of the stellar radius. The open symbols correspond to the stars studied by Mosser et al. (2012b, circles: RGB stars, squares: clump stars). The filled symbols indicate the stars that were studied in Deheuvels et al. (2014), and the cross corresponds to the young giant KIC 7341231 studied by Deheuvels et al. (2012). Figure from Deheuvels et al. (2014).

The interpretation of Kepler data has shown that the core of red giants is spinning down, in spite of its contraction (Mosser et al. 2012b). This has brought clear evidence that an efficient transport of angular momentum takes place in these stars while its origin remains unknown. On the contrary, the contracting core spins up during the subgiant phase (Deheuvels et al. 2012, 2014), which thus corresponds to a period of relative decoupling between the core and the envelope. Late subgiants and early red giants correspond to the intermediate phase between these two regimes, and probing the internal rotation profile of these stars can provide estimates of the timescale over which angular momentum is redistributed. Moreover, inversions of internal rotation profiles using Keplerdata have hinted at the existence of a sharp rotation gradient near the H-burning shell in early red giants (Deheuvels et al. 2014). If it is confirmed, this will bring strong constraints on the mechanism that transports angular momentum in these stars. Late subgiants and early red giants are ideal targets to test this further. However, only the longest observing runs provided sufficient precision to derive the information on the core rotation of these evolved stars, and on the evolution of core rotation with time. The sample of available Kepler sub-giant stars is too small to allow for an unbiased statistical study. To proceed further, we will need a larger sample of bright post-main sequence stars. The necessary high quality needed to study these stars and this for a large number of such stars will only be available with the PLATO mission.

#### References

PLATO – Revealing habitable worlds around solar-like stars
Definition Study Report, ESA-SCI(2017)1, April 2017