Asteroseismic age estimates of RGB stars in open clusters: A statistical investigation of different estimation methods

Context. Open clusters (OCs) provide a classical target to calibrate the age scale and other stellar parameters. Despite their wide use, some issues remain to be explored in detail. Aims. We performed a theoretical investigation focused on the age estimate of red giant branch (RGB) stars in OCs based on mixed classical surface (Teff and [Fe/H]) and asteroseismic (Δν and νmax) parameters. We aimed to evaluate the performances of three widely adopted fitting procedures, that is, a pure geometrical fit, a maximum likelihood approach, and a single stars fit, in recovering stellar parameters. Methods. A dense grid of stellar models was computed, covering different chemical compositions and different values of the mixing-length parameter. Artificial OCs were generated from these data by means of a Monte Carlo procedure for two different ages (7.5 and 9.0 Gyr) and two different choices of the number of stars in the RGB evolutionary phase (35 and 80). The cluster age and other fundamental parameters were then recovered by means of the three methods previously mentioned. A Monte Carlo Markov chain approach was adopted for estimating the posterior densities of probability of the estimated parameters. Results. The geometrical approach overestimated the age by about 0.3 and 0.2 Gyr for true ages of 7.5 and 9.0 Gyr, respectively. The value of the initial helium content was recovered unbiased within the large random errors on the estimates. The maximum likelihood approach provided similar biases (0.1 and 0.2 Gyr) but with a variance reduced by a factor of between two and four with respect to geometrical fit. The independent fit of single stars showed a very large variance owing to its neglect of the fact that the stars came from the same cluster. The age of the cluster was recovered with no biases for 7.5 Gyr true age and with a bias of -0.4 Gyr for 9.0 Gyr. The most important difference between geometrical and maximum likelihood approaches was the robustness against observational errors. For the first fitting technique, we found that estimations starting from the same sample but with different Gaussian perturbations on the observables suffer from a variability in the recovered mean of about 0.3 Gyr from one Monte Carlo run to another. This value was as high as 45% of the intrinsic variability due to observational errors. On the other hand, for the maximum likelihood fitting method, this value was about 65%. This larger variability led most simulations - up to 90% - to fail to include the true parameter values in their estimated 1σ credible interval. Finally, we compared the performance of the three fitting methods for single RGB-star age estimation. The variability owing to the choice of the fitting method was minor, being about 15% of the variability caused by observational uncertainties. Conclusions. Each method has its own merits and drawbacks. The single star fit showed the lowest performances. The higher precision of the maximum likelihood estimates is partially negated by the lower protection that this technique shows against random fluctuations compared to the pure geometrical fit. Ultimately, the choice of the fitting method has to be evaluated in light of the specific sample and evolutionary phases under investigation.