  To understand the formations of singularities of the Euler-Poisson system
with vacuum, we revisit Makino's star model in this article. We first remedy,
to some extent, the inconveniences of Makino's star model and remove its
imposed nonphysically exterior free-falling velocity field by only specifying
the velocity field on a compact support of the density. Makino has coined the
presence of such an exterior velocity field as a ``Cheshire cat phenomenon''
and he [33] and Rendall [45] have both emphasized the difficulty of removing
this phenomenon. Moreover, we obtain an improved local existence and uniqueness
theorem for initial data for the density and velocity that both have compact
support. Finally, we are able to prove a localized strong continuation
criterion in which the breakdown of solutions is only controlled by quantities
defined on the compact support of the solution. In addition, the localized
strong continuation criterion also generalizes the continuation criterion from
the incompressible Euler equation in a bounded domain [16,47] to the
compressible Euler-Poisson system.
