Stellar populations of galaxies in the ALHAMBRA survey up to $z \sim 1$.
  IV. Properties of quiescent galaxies on the stellar mass$-$size plane
  We perform a comprehensive study of the stellar population properties of
quiescent galaxies as a function of size and stellar mass to constrain the
physical mechanism governing the stellar mass assembly and the likely evolutive
scenarios that explain their growth in size. After selecting all the quiescent
galaxies from the ALHAMBRA survey by the dust-corrected stellar mass$-$colour
diagram, we built a shared sample of $\sim850$ quiescent galaxies with reliable
sizes from the HST. The stellar population properties were retrieved using the
SED-fitting code MUFFIT with various sets of composite stellar population
models. Age, formation epoch, metallicity, and extinction were studied on the
stellar mass$-$size plane as function of size through a Monte Carlo approach.
This accounted for uncertainties and degeneracy effects amongst stellar
population properties. The stellar population properties of quiescent galaxies
and their stellar mass and size since $z\sim1$ are correlated. At fixed stellar
mass, the more compact the quiescent galaxy, the older and richer in metals it
is ($1$Gyr and $0.1$dex, respectively). In addition, more compact galaxies may
present slight lower extinctions than their more extended counterparts at the
same stellar mass ($<0.1$ mag). By means of studying constant regions of
stellar population properties across the stellar mass$-$size plane, we obtained
empirical relations to constrain the physical mechanism that governs the
stellar mass assembly of the form $M_\star \propto r_\mathrm{c}^\alpha$, where
$\alpha$ amounts to $0.50-0.55 \pm 0.09$. There are indications that support
the idea that the velocity dispersion is tightly correlated with the stellar
content of galaxies. The mechanisms driving the evolution of stellar
populations can therefore be partly linked to the dynamical properties of
galaxies, along with their gravitational potential.
