Non self-similar Luminosity-temperature relation and dynamical friction

Extending the results of a previous paper \citep{DelPopolo2005}, by taking into account the role of dynamical friction, we recovered the luminosity-temperature relation (LTR). While by assuming self-similarity, a scaling law in which $L\propto T^2$ is obtained, observations show that the relation between luminosity and temperature is steeper, ${L \propto T^ {\simeq 3}}$. This difference can be explained in terms of energy input by non-gravitational processes, like pre-heating, supernovae feedback, and heating from AGN. In this paper, we studied the LTR by means of a modified version of the punctuated equilibria model \citep{Cavaliere1999}, taking into account in addition dynamical friction, thus extending the approach found in \citep{DelPopolo2005}. The result is a non-self-similar LTR with a bend at $\simeq 2$ keV, with a slope $2.76 \pm 0.18$ at larger energies and $3.4 \pm 0.18$ at energies smaller than 2 keV. This result is in agreement with the XXL survey \citep{Giles2016}. Moreover the steeper slopes at smaller energies is in agreement with some studies claiming a further steepening of the LTR at the low mass end. We also compared the results of our model with the 400d groups sample, finding that in groups the slope is slightly steeper than in clusters, %and namely $3.35 \pm 0.3$, in agreement with the \citep{Zou2016} study for the 400d groups sample, that gives a slope $3.29 \pm 0.33$.