• Age
• Asteroseismology
• Old stars
• Metal poor
• Scepter
• Vertical method
• M4.

Understanding the formation and evolution of the Milky Way is a challenging problem investigated with many efforts by ambitious large scale observational surveys.
However, whereas observations provide a detailed picture of the photometric, chemical and dynamical structure of our Galaxy, the crucial information about the stellar ages, needed to give a chronological order to the evolution, can only be indirectly inferred.
One of the most adopted methods to estimate stellar ages is to fit observations by theoretical isochrones, which are the positions in the colour-magnitude diagram of stars with the same age and chemical composition but different masses.
This method is sufficiently precise when applied to clusters of stars, that are composed of one or more populations of stars of approximately the same age, chemical composition but different masses.
However, it does not succeed in constraining ages of field stars, especially in the evolutionary stages where isochrones are less sensitive in age.
Other age indicators obtained from classical observables (temperature, luminosity, metal content etc.) provide good estimates only for stars in given evolutionary stages and are not yet reliable for very old stars, fundamental to trace the early evolution of the Galaxy.
However, the study of oscillations of stellar structures, called asteroseismology, is promising for determining stellar ages also for the older population of our Galaxy
Asteroseismology has been greatly developed in recent years, mainly due to the improvement of observational capabilities.
In particular, photometric observations from satellites such as CoRoT, Kepler and TESS, allowed astronomers to detect periodic variations of stellar luminosity due to radial oscillations in a wide variety of stars.

Stars that show oscillatory behaviour of external regions stemming from the stochastic driving of turbulent convection are called solar-like oscillators because of their resemblance to the Sun in the oscillations driving mechanism.
The oscillation power spectrum of these stars presents a modulation in the height of the oscillation modes which is approximately Gaussian and the frequency of maximum oscillation is dependent on mass temperature and radius of the star.
Furthermore, oscillation frequencies of consecutive orders of the same degree are equally spaced in the power spectrum density, so it is possible to define the large frequency separation, which is connected to the sound speed inside the star.
These two seismic quantities (called global seismic parameters) can be employed with classical observables (effective temperature, metallicity etc.) to constrain stars' properties such as mass, radius and age through the comparison between observations and theoretical predictions.

In the first part of my research work I quantitatively analyse through statistical methods the precision and biases of age estimation for old, metal poor field stars (which inhabit the Milky Way halo) when both classical and asteroseismic observations are adopted.
I quantified the best precision obtainable with a set of typical uncertainties on observable quantities, by adopting the same input physics in both theoretical models and synthetic dataset.
To do this I made use of a synthetic sample of 10000 artificial stars extracted from 280 stellar evolutionary tracks (40 different masses and seven metallicities), that I computed adopting the Pisa Evolutionary Code.
For this synthetic sample I performed the recovery of stellar parameters using the R library SCEPtER developed in Pisa, that I implemented into a personalised routine for this thesis.

This work extends to lower metallicities and to the later evolutionary stage of shell hydrogen burning (Red Giant Branch phase, RGB) the research done by Valle et al. [2015].

This analysis highlighted a great disparity in precision of asteroseismical age estimates between stars in main sequence (evolutionary stage of a central H burning star) and in the RGB phase.
In particular, the former are recovered in age with a typical precision (15/20\%) that is on average four times better than what is achieved for the latter, mainly because RGB stars are found much more compacted in the observable parameters space, especially in temperature, meaning that is more difficult to discriminate between stellar models with a given set of observables.
I evaluated the effects of improving the precision of observational data, finding that the best precision possible, that is obtained with the present smallest observational errors for MS stars is ~10\%.
I also evaluated the effects of the most relevant uncertainties in the input physics adopted in theoretical stellar models, finding that they introduce small systematic biases in all cases, except for microscopic diffusion in low mass stars, for which the effect is more relevant.
Various works have previously investigated biases due to systematic differences between theoretical models and real world stars, however not for this low metallicity or not adopting synthetic stars, which allow to control the separate effect of each uncertainty source.

The comparison of these results with the ones by Valle et al. [2015] show that the age of less metallic stars are estimated with a better precision.
This is mainly due to the fact that lower initial metallicity stars undergo bigger variations of superficial metallicity due to microscopic diffusion, compared to metal-richer stars, thus models are more spread apart in the parameter space and it is easier to discriminate between stellar models with a given set of observables.
We can thus conclude that current precision for observable parameters allows reliable age estimates for old halo MS field stars, while for RGB field stars the recovery precision is at the level of 70/80\%.

In the second part of this work I applied the asteroseismic method to the age estimation of stars in a globular cluster to compare the results with the ones of the already mentioned classical isochrone method.
Until now, the only globular cluster for which asteroseismical parameters are available is the old and mildly metal-poor globular cluster M4, one of the closest to the Sun.
In fact globular clusters are so far from us that determination of asteroseismic properties are very difficult.
Global seismic parameters are available only for six stars in the RGB phase Miglio et al. [2016].
Literature values present discrepancies for the superficial temperature of these six stars, thus I provided a new estimate for this quantity by applying a method widely used in literature to the newest highly precise photometric data by the Gaia satellite.
The asteroseismic age of M4 has been compared to the one that I obtained adopting the isochrone fitting method, which allowed me to estimate also extinction along the line of sight of M4.
To perform the isochrone fitting, I carefully selected photometric data for the cluster comparing different dataset available in different photometric bands.
The results obtained with the classical method are in agreement with previous results available in the literature.

The estimated asteroseismic age is very uncertain even if the central value is in very good agreement with the results of the classical method and with previous determination of the age of the cluster.
This is because asteroseismic data are available only for RGB stars which gave less precise results, but above all due to the great uncertainty on asteroseismic parameters as well as surface temperature.
Thus, at present, due to the situation of observational data in globular clusters, asteroseismic age determination, which is the method of choice for field stars, is not competitive with classical methods for old clusters.
However, the method seems promising for the nearest young clusters.