Gravitational Fragmentation of Expanding Shells. I. Linear Analysis
  We perform a linear perturbation analysis of expanding shells driven by
expansions of HII regions. The ambient gas is assumed to be uniform. As an
unperturbed state, we develop a semi-analytic method for deriving the time
evolution of the density profile across the thickness. It is found that the
time evolution of the density profile can be divided into three evolutionary
phases, deceleration-dominated, intermediate, and self-gravity-dominated
phases. The density peak moves relatively from the shock front to the contact
discontinuity as the shell expands. We perform a linear analysis taking into
account the asymmetric density profile obtained by the semi-analytic method,
and imposing the boundary conditions for the shock front and the contact
discontinuity while the evolutionary effect of the shell is neglected. It is
found that the growth rate is enhanced compared with the previous studies based
on the thin-shell approximation. This is due to the boundary effect of the
contact discontinuity and asymmetric density profile that were not taken into
account in previous works.
