Geometric properties of galactic discs with clumpy episodes
  A scenario for the formation of the bi-modality in the chemical space
[$\alpha$/Fe] vs [Fe/H] of the Milky Way was recently proposed in which
$\alpha$-enhanced stars are produced early and quickly in clumps. Besides
accelerating the enrichment of the medium with $\alpha$-elements, these clumps
scatter the old stars, converting in-plane to vertical motion, forming a
geometric thick disc. In this paper, by means of a detailed analysis of the
data from smooth particle hydrodynamical simulations, we investigate the
geometric properties (in particular of the chemical thick disc) produced in
this scenario. For mono-age populations we show that the surface radial density
profiles of high-[$\alpha$/Fe] stars are well described by single exponentials,
while that of low-[$\alpha$/Fe] stars require broken exponentials. This break
is sharp for young populations and broadens for older ones. The position of the
break does not depend significantly on age. The vertical density profiles of
mono-age populations are characterized by single exponentials, which flare
significantly for low-[$\alpha$/Fe] stars but only weakly (or not at all) for
high-[$\alpha$/Fe] stars. For low-[$\alpha$/Fe] stars, the flaring level
decreases with age, while for high-[$\alpha$/Fe] stars it weakly increases with
age (although with large uncertainties). All these properties are in agreement
with observational results recently reported for the Milky Way, making this a
plausible scenario for the formation of the Galactic thick disc.
