Pointing to the minimum scatter: the generalized scaling relations for
  galaxy clusters
  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 ~ (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.
