Ages of globular clusters: breaking the age-distance degeneracy with the luminosity function

We extend our previous method to determine globular cluster ages using the luminosity function (Jimenez \& Padoan 1996). We show that the luminosity function depends on both age and distance modulus and that it is possible to distinguish between the two. This method provides at the same time independent determinations of distance and age of a GC by simply counting the number of stars found inside specified luminosity bins. The main uncertainties in other traditional methods for determining GCs ages are absent (e.g. mixing length, color-$T_{\rm eff}$ calibration, morphology of the color-magnitude diagram ). The distance modulus is the biggest uncertainty in determining the age of GCs. Here we show that the age can be determined with small uncertainty for any value of distance modulus using the LF and that the LF allows a determination of the distance modulus itself. This is explained by the fact that the luminosity function is affected by a change in distance-modulus in a way that is different from its time evolution. If GC stellar counts with statistical errors not larger than $3\%$ are available, the age can be determined with an uncertainty of about 0.4 Gyr (independent of distance modulus, mixing length and color calibration) and the distance modulus with an uncertainty of about 0.04 mag.