  A striking characteristic of non-Schwarzschild vacuum exteriors is that they
contain not only the total gravitational mass of the source, but also an {\it
arbitrary} constant. In this work, we show that the constants appearing in the
"temporal Schwarzschild", "spatial Schwarzschild" and
"Reissner-Nordstr{\"o}m-like" exteriors are not arbitrary but are completely
determined by star's parameters, like the equation of state and the
gravitational potential. Consequently, in the braneworld scenario the
gravitational field outside of a star is no longer determined by the total mass
alone, but also depends on the details of the internal structure of the source.
We show that the general relativistic upper bound on the gravitational
potential $M/R < 4/9$, for perfect fluid stars, is significantly increased in
these exteriors. Namely, $M/R < 1/2$, $M/R < 2/3$ and $M/R < 1$ for the
temporal Schwarzschild, spatial Schwarzschild and Reissner-Nordstr{\"o}m-like
exteriors, respectively. Regarding the surface gravitational redshift, we find
that the general relativistic Schwarzschild exterior as well as the braneworld
spatial Schwarzschild exterior lead to the same upper bound, viz., $Z < 2$.
However, when the external spacetime is the temporal Schwarzschild metric or
the Reissner-Nordstr{\"o}m-like exterior there is no such constraint: $Z <
\infty$. This infinite difference in the limiting value of $Z$ is because for
these exteriors the effective pressure at the surface is negative. The results
of our work are potentially observable and can be used to test the theory.
