  We study analytically the development of gravitational instability in an
expanding shell having finite thickness. We consider three models for the
radial density profile of the shell: (i) an analytic uniform-density model,
(ii) a semi-analytic model obtained by numerical solution of the hydrostatic
equilibrium equation, and (iii) a 3D hydrodynamic simulation. We show that all
three profiles are in close agreement, and this allows us to use the first
model to describe fragments in the radial direction of the shell. We then use
non-linear equations describing the time-evolution of a uniform oblate spheroid
to derive the growth rates of shell fragments having different sizes. This
yields a dispersion relation which depends on the shell thickness, and hence on
the pressure confining the shell. We compare this dispersion relation with the
dispersion relation obtained using the standard thin-shell analysis, and show
that, if the confining pressure is low, only large fragments are unstable. On
the other hand, if the confining pressure is high, fragments smaller than
predicted by the thin-shell analysis become unstable. Finally, we compare the
new dispersion relation with the results of 3D hydrodynamic simulations, and
show that the two are in good agreement.
