We analysed the likelihood maps to investigate the degeneracy in age, metallicity, reddening, and distance modulus. The likelihood values were obtained from a comparison of a given synthetic CMD with the observed one (see Sect. 3 for details).
To establish the standard deviation of the likelihood statistics, we proceeded as follows. A synthetic best model CMD was created with the same numbers of stars as the observed one. We call this a false observation. We then compared this CMD with another one with the same parameters but with many more stars (e.g. 105). From this model vs. model comparison, a likelihood value is obtained, as previously done for data vs. model comparions. Then simulating 300 false observations, and comparing them with the star rich synthetic CMD, we derived the standard deviation σ. This tells how much the likelihood can vary due only to stochastic effects, i.e., due to the statistical fluctuations on the position of stars in a CMD with a number of points compatible with that of the observed CMDs.
The likelihood maps of these comparisons are displayed in Figs. A.1 to A.5, in the space of parameters of [Fe/H] vs. log (age), log (age) vs. (m − M), and E(B − V) vs. (m − M), centred in the parameters of the best-fit. The colour scale is red, green, cyan or yellow, corresponding to a difference from the best fit solution up to 1-sigma, between 1- and 2-sigma, between 2- and 3-sigma, and larger than 3-sigma of the best fit, respectively. The central panel in each figure gives the value obtained for the two fixed parameters, since in each map two parameters are varied and the two others are fixed.
These maps clearly reveal the expected anti-correlation between age and metallicity, distance modulus and age, and between distance modulus and reddening. These degeneracies are directly reflected in the physical parameter uncertainties since they correspond to the standard deviation in each parameter for the set of models with likelihood values up to 1-sigma (red points) from the best-fit solution.
Likelihood from the fits for different combinations of age, metallicity, reddening, and distance modulus for AM 3. Upper panel: metallicity vs. age; middle panel: age vs. distance modulus; bottom panel: reddening vs. distance modulus. The colour scale is red, green, cyan or yellow, corresponding to a difference from the best-fit solution up to 1-sigma, between 1- and 2-sigma, between 2- and 3-sigma, and larger than 3-sigma of the best fit, respectively.
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© ESO, 2014