  Static spherically symmetric solutions to the Einstein-Euler equations with
prescribed central densities are known to exist, be unique and smooth for
reasonable equations of state. Some criteria are also available to decide
whether solutions have finite extent (stars with a vacuum exterior) or infinite
extent. In the latter case, the matter extends globally with the density
approaching zero at infinity. The asymptotic behavior largely depends on the
equation of state of the fluid and is still poorly understood. While a few such
unbounded solutions are known to be asymptotically flat with finite ADM mass,
the vast majority are not. We provide a full geometric description of the
asymptotic behavior of static spherically symmetric perfect fluid solutions
with linear and polytropic-type equations of state with index n>5. In order to
capture the asymptotic behavior we introduce a notion of scaled
quasi-asymptotic flatness, which encodes a form of asymptotic conicality. In
particular, these spacetimes are asymptotically simple.
