We introduce a generalized scaling law , M _ { tot } = 10 ^ { K } A ^ { a } B ^ { b } , to look for the minimum scatter in reconstructing the total mass of hydrodynamically simulated X-ray galaxy clusters , given gas mass M _ { gas } , luminosity L and temperature T .
We find a locus in the plane of the logarithmic slopes a and b of the scaling relations where the scatter in mass is minimized .
This locus corresponds to b _ { M } = -3 / 2 a _ { M } +3 / 2 and b _ { L } = -2 a _ { L } +3 / 2 for A = M _ { gas } and L , respectively , and B = T .
Along these axes , all the known scaling relations can be identified ( at different levels of scatter ) , plus a new one defined as M _ { tot } \propto ( LT ) ^ { 1 / 2 } .
Simple formula to evaluate the expected evolution with redshift in the self-similar scenario are provided .
In this scenario , no evolution of the scaling relations is predicted for the cases ( b _ { M } = 0 ,a _ { M } = 1 ) and ( b _ { L } = 7 / 2 ,a _ { L } = -1 ) , respectively .
Once the single quantities are normalized to the average values of the sample under considerations , the normalizations K corresponding to the region with minimum scatter are very close to zero .
The combination of these relations allows to reduce the number of free parameters of the fitting function that relates X-ray observables to the total mass and includes the self-similar redshift evolution .