The Three Hundred Project: dissecting the fundamental plane of galaxy clusters up to z=1

We perform a systematic study of the recently discovered fundamental plane of galaxy clusters (CFP) using ~250 simulated clusters from The Three Hundred project, focusing on the stability of the plane against different temperature definitions and its dependence on the dynamical relaxation state of clusters. The CFP is characterised by $T\propto M_\mathrm{s}^\alpha\,r_\mathrm{s}^\beta$, defined with the gas temperature ($T$) and the characteristic halo scale radius and mass ($r_\mathrm{s}$ and $M_\mathrm{s}$) assuming a Navarro-Frenk-White halo description. We explore two definitions of weighted temperatures, namely mass-weighted and spectroscopic-like temperatures, in three radial ranges. The Three Hundred project clusters at $z=0$ lie on a thin plane whose parameters ($\alpha,\beta$) and dispersion ($0.015-0.030$ dex) depend on the gas temperature definition. The CFP for mass-weighted temperatures is closer to the virial equilibrium expectation ($\alpha=1,\beta=-1$) with a smaller dispersion. For gas temperatures measured within $500h^{-1}$ kpc, the resulting CFP deviates the most from the virial expectation and shifts towards the similarity solution for a secondary infall model ($\alpha=1.5,\beta=-2$). Independently of the temperature definition, we find that clusters at $z=1$ and relaxed clusters form a CFP similar to the virial expectation, unlike disturbed clusters exhibiting stronger evolution. Only systems formed over the last 4 Gyr present a CFP that is closer to the similarity solution. All these findings are compatible with the CFP obtained for a CLASH subsample excluding the hottest clusters with $T_\mathrm{X}>12$ keV.