Goodness-of-fit test for isochrone fitting in the Gaia era. Statistical
  assessment of the error distribution
  [...] 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. 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 constructed to mimic real data of open clusters in the GAIA
sample. We analysed the capability of distinguishing among different sets of
input physics and parameters that were used to compute the stellar models. We
generated synthetic clusters from isochrones computed with these perturbed
quantities, and we evaluated the goodness-of-fit with respect to the
unperturbed isochrone. 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 causes
negligible deviation from this result. We investigated the possibility of
detecting the effects on stellar models that are induced when varying
convective core overshooting efficiency, $^{14}$N$(p,\gamma)^{15}$O reaction
rate, microscopic diffusion velocities, outer BCs, and colour transformation.
The results suggest that it is possible to detect the effect induced by only
some of the perturbed quantities. [...] 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. [...] 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. [...]
