In galaxy clusters, the relations between observables in X-ray and millimeter wave bands and the total mass have normalizations, slopes and redshift evolutions that are simple to estimate in a self-similar scenario. We study these scaling relations and show that they can be efficiently expressed, in a more coherent picture, by fixing the normalizations and slopes to the self-similar predictions, and advocating, as responsible of the observed deviations, only three physical mass-dependent quantities: the gas clumpiness $C$, the gas mass fraction $f_g$ and the logarithmic slope of the thermal pressure profile $beta_P$. We use samples of the observed gas masses, temperature, luminosities, and Compton parameters in local clusters to constrain normalization and mass dependence of these 3 physical quantities, and measure: $C^{0.5} f_g = 0.110 (pm 0.002 pm 0.002) left( E_z M / 5 times 10^{14} M_{odot} right)^{0.198 (pm 0.025 pm 0.04)}$ and $beta_P = -d ln P/d ln r = 3.14 (pm 0.04 pm 0.02) left( E_z M / 5 times 10^{14} M_{odot} right)^{0.071 (pm 0.012 pm 0.004)}$, where both a statistical and systematic error (the latter mainly due to the cross-calibration uncertainties affecting the cxo and xmm results used in the present analysis) are quoted. The degeneracy between $C$ and $f_g$ is broken by using the estimates of the Compton parameters. Together with the self-similar predictions, these estimates on $C$, $f_g$ and $beta_P$ define an inter-correlated internally-consistent set of scaling relations that reproduces the mass estimates with the lowest residuals.
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