Limitation of symmetry breaking by gravitational collapse: the revisit
  of Lin-Mestel-Shu instability
  We revisit the topic of shape evolution during the spherical collapse of an
$N$-body system. Our main objective is to investigate the critical particle
number below which, during a gravitational collapse, the amplification of
triaxiality from initial fluctuations is effective, and above which it is
ineffective. To this aim, we develop the Lin-Mestel-Shu theory for a system of
particles initially with isotropic velocity dispersion and with a simple
power-law density profile. We first determine, for an unstable cloud, two radii
corresponding to the balance of two opposing forces and their fluctuations:
such radii fix the sizes of the non-collapsing region and the triaxial seed
from density fluctuations. We hypothesize that the triaxial degree of the final
state depends on which radius is dominant prior to the collapse phase leading
to a different scheme of the self-consistent shape evolution of the core and
the rest of the system. The condition where the two radii are equal therefore
identifies the critical particle number, which can be expressed as the function
of the parameters of initial state. In numerical work, we can pinpoint such a
critical number by comparing the virialized flattening with the initial
flattening. The difference between these two quantities agrees with the
theoretical predictions only for the power-law density profiles with an
exponent in the range $[0,0.25]$. For higher exponents, results suggest that
the critical number is above the range of simulated $N$. We speculate that
there is an additional mechanism, related to strong density gradients that
increases further the flattening, requiring higher $N$ to further weaken the
initial fluctuations.
