  We use information about DIS and $J/ \psi$ production on hydrogen to model
the $t$-dependence of the $\gamma^{\ast} N$ scattering amplitude. We
investigate the profile function for elastic scattering of hadronic components
of the virtual photon off both a nucleon and heavy nuclear target, and we
estimate the value of the impact parameter where the black body limit is
reached. We also estimate the fraction of the cross section that is due to
hadronic configurations in the virtual photon wave function that approach the
unitarity limit. We extract, from these considerations, approximate lower
limits on the values of $x$ where the leading twist approximation in DIS is
violated. We observe that the black body limit may be approached within HERA
kinematics with $Q^{2}$ equal to a few GeV$^2$ and $x \sim 10^{-4}$.
Comparisons are made with earlier predictions by Munier {\it et al.}, and the
longitudinal structure function is compared with preliminary HERA data. The
principle advantage of our method is that we do not rely solely on the
$t$-dependence of $\rho$-meson production data. This allows us to extend our
analysis down to very small impact parameters and dipole sizes. Finally, we
perform a similar calculation with a $^{208}$Pb target, and we demonstrate that
the black body limit is already approached at $Q^{2} \sim 20$ GeV$^{2}$ and $x
\sim 10^{-4}$.
