  It is well known that a spherically symmetric constant density static star,
modeled as a perfect fluid, possesses a bound on its mass m by its radial size
R given by 2m/R \le 8/9 and that this bound continues to hold when the energy
density decreases monotonically. The existence of such a bound is intriguing
because it occurs well before the appearance of an apparent horizon at m = R/2.
However, the assumptions made are extremely restrictive. They do not hold in a
humble soap bubble and they certainly do not approximate any known
topologically stable field configuration. We show that the 8/9 bound is robust
by relaxing these assumptions. If the density is monotonically decreasing and
the tangential stress is less than or equal to the radial stress we show that
the 8/9 bound continues to hold through the entire bulk if m is replaced by the
quasi-local mass. If the tangential stress exceeds the radial stress and/or the
density is not monotonic we cannot recover the 8/9 bound. However, we can show
that 2m/R remains strictly bounded away from unity by constructing an explicit
upper bound which depends only on the ratio of the stresses and the variation
of the density.
