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In this paper, we propose a new method to obtain the depletion factor$\gamma(z)$, the ratio by which the measured baryon fraction in galaxy clustersis depleted with respect to the universal mean. We use exclusively galaxycluster data, namely, X-ray gas mass fraction ($f_{gas}$) and angular diameterdistance measurements from Sunyaev-Zel'dovich effect plus X-ray observations.The galaxy clusters are the same in both data set and the non-isothermalspherical double $\beta$-model was used to describe their electron density andtemperature profiles. In order to compare our results with those from recentcosmological hydrodynamical simulations, we suppose a possible time evolutionfor $\gamma(z)$, such as, $\gamma(z) = \gamma_0(1 + \gamma_1z)$. As mainconclusions we found that: the $\gamma_0$ value is in full agreement with thesimulations. { On the other hand, although the $\gamma_1$ value found in ouranalysis is compatible with $\gamma_1=0$ within 2$\sigma$ c.l., our resultsshow a non-negligible time evolution for the depletion factor, unlike theresults of the simulations. However, we also put constraints on $\gamma (z)$ byusing the $f_{gas}$ measurements and angular diameter distances obtained fromthe flat $\Lambda$CDM model (Planck results) and from a sample of galaxyclusters described by a elliptical profile. For these cases no significant timeevolution for $\gamma(z)$ was found. Then, if a constant depletion factor is aninherent characteristic of these structures, our results show that thespherical double $\beta$-model used to describe the galaxy clusters considereddoes not affect the quality of their $f_{gas}$ measurements.

Is the non-isothermal double $\beta$-model incompatible with no time evolution of galaxy cluster gas mass fraction ?