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.
