Goodness-Of-Fit Test For Isochrone Fitting In The Gaia Era: Statistical Assessment Of The Error Distribution

Context. The increasing precision in observational data made available by recent surveys means that the reliability of stellar models can be tested. For this purpose, a firm theoretical basis is crucial for evaluating the agreement of the data and theoretical predictions.Aims This paper presents a rigorous derivation of a goodness-of-fit statistics for colour-magnitude diagrams (CMD). We discuss the reliability of the underlying assumptions and their validity in real-world testing.Methods. We derived the distribution of the sum of squared Mahalanobis distances of stellar data and theoretical isochrone for a generic set of data and models. We applied this to the case of synthetic CMDs that were constructed to mimic real data of open clusters in the Gaia sample. Then, we analysed the capability of distinguishing among different sets of input physics and parameters that were used to compute the stellar models. To do this, we generated synthetic clusters from isochrones computed with these perturbed quantities, and we evaluated the goodness-of-fit with respect to the reference unperturbed isochrone.Results. We show that when r magnitudes are available for each of the N observational objects and p hyperparameters are estimated in the fit, the error distribution follows a chi (2) distribution with (r-1)N-p degrees of freedom. We show that the linearisation of the isochrone between support points introduces negligible deviation from this result. We investigated the possibility of detecting the effects on stellar models that are induced when the following physical quantities were varied: convective core overshooting efficiency, N-14(p, gamma)O-15 reaction rate, microscopic diffusion velocities, outer boundary conditions, and colour transformation (bolometric corrections). We conducted the analysis at three different ages, 150 Myr, 1 Gyr, and 7 Gyr, and accounted for errors in photometry from 0.003 mag to 0.03 mag. The results suggest that it is possible to detect the effect induced by only some of the perturbed quantities. The effects induced by a change in the diffusion velocities or in the N-14(p, gamma)O-15 reaction rate are too small to be detected even when the smallest photometric uncertainty is adopted. A variation in the convective core overshooting efficiency was detectable only for photometric errors of 0.003 mag and only for the 1 Gyr case. The effects induced by the outer boundary conditions and the bolometric corrections are the largest; the change in outer boundary conditions is detected for photometric errors below about 0.01 mag, while the variation in bolometric corrections is detectable in the whole photometric error range. As a last exercise, we addressed the validity of the goodness-of-fit statistics for real-world open cluster CMDs, contaminated by field stars or unresolved binaries. We assessed the performance of a data-driven cleaning of observations, aiming to select only single stars in the main sequence from Gaia photometry. This showed that this selection is possible only for a very precise photometry with errors of few millimagnitudes.