The Fundamental Plane in the hierarchical context

Context. The Fundamental Plane (FP) relation and the distribution of early-type galaxies (ETGs) in the FP projections, cannot be easily explained in the hierarchical framework, where galaxies grow up by merging and star formation episodes. Aims. We want to show here that both the FP and its projections arise naturally from the combination of the Virial Theorem (VT) and a new time-dependent relation, describing how luminosity and stellar velocity dispersion change during galaxy evolution. This relation has the form of the Faber-Jackson (FJ) relation but a different physical meaning: the new relation is $L = L'_0(t) \sigma^{\beta(t)}$, where its coefficients $L'_0$ and $\beta$ are time-dependent and can vary considerably from object to object, at variance with those obtained from the fit of the $L - \sigma$ plane. Methods. We derive the equations of the FP and its projections as a function of $\beta$ and $L'_0$, by combining the VT and the $L = L'_0(t) \sigma^{\beta(t)}$ law. Then, from their combination we derive the expression of the FP as a function of $\beta$ and the solutions for $\beta$ and $L'_0$. Results. We demonstrate that the observed properties of ETGs in the FP and its projections can be understood in terms of variations of $\beta$ and $L'_0$. These two parameters encrypt the history of galaxy evolution across the cosmic epochs and determine the future aspect of the FP and its projections. Using the solutions found for $\beta$ and $L'_0$ for each galaxy at the present epoch, we derive the coefficients of the FP (and FJ relation) and show that, the values of the coefficients coming form the fit, obtained in the literature, originate from the average of the single FP coefficients derived for each galaxy. In addition, we show that the variations of beta naturally explain the curvature observed in the FP projections and the correct position of the Zone of Exclusion.