Recovering the fundamental plane of galaxies by $f(R)$ gravity
  The fundamental plane (FP) of galaxies can be recovered in the framework of
$f(R)$ gravity avoiding the issues related to dark matter to fit the
observations. In particular, the power-law version $f(R)\propto R^n$, resulting
from the existence of Noether symmetries for $f(R)$, is sufficient to implement
the approach. In fact, relations between the FP parameters and the corrected
Newtonian potential, coming from $R^n$, can be found and justified from a
physical point of view. Specifically, we analyze the velocity distribution of
elliptical galaxies and obtain that $r_c$, the scale-length depending on the
gravitational system properties, is proportional to $r_e$, the galaxy effective
radius. This fact points out that the gravitational corrections induced by
$f(R)$ can lead photometry and dynamics of the system. Furthermore, the main
byproduct of such an approach is that gravity could work in different ways
depending on the scales of self-gravitating systems.
