Intrinsic properties of the bars formed by the bar instability in flat
  stellar discs
  The properties of the bars formed by the bar instability are examined for
flat stellar discs. The initial mass models chosen are Kuzmin--Toomre discs,
for which two types of exact equilibrium distribution function (DF) are
employed in order to realize different distributions of Toomre's $Q$ values
along the radius. First, the most linearly unstable, global two-armed modes
(MLUGTAMs) of these disc models are determined by numerically solving the
linearized collisionless Boltzmann equation. Next, we carry out $N$-body
simulations whose models are constructed from the DFs adopted above. The latter
simulations unravel that the MLUGTAMs corresponding to those obtained from the
former modal calculations are excited in the early phases of evolution, finally
being deformed into bars in the nonlinear regime by the bar instability. We
show that for simulated bars, the length increases and the axis ratio, in
essence, decreases as the amplitude increases. These correlations are almost
similar to those of the observed bars. In addition, we find that these bar
properties are tightly correlated with the initial typical $Q$ value,
irrespective of the DF. In conclusion, a disc with a smaller typical $Q$ value
produces a bar which is smaller in amplitude, shorter in length and rounder in
shape. This finding might suggest that the Hubble sequence for barred galaxies
is the sequence of decreasing $Q$ from SBa to SBc or SBd. The implied
correlations between the initial typical $Q$ value and each of the bar
properties are discussed on the basis of the characteristics of the MLUGTAMs.
