  The heaviest particle-bound carbon isotope, $^{22}$C, is thought to have a
Borromean three-body structure. We discuss and compare four-body, i.e.
three-body projectile plus target, reaction model calculations of reaction
cross sections for such systems that use the fast adiabatic approximation.
These methods are efficient and well-suited for quantitative analyses of
reactions of neutron-rich nuclei with light target nuclei at secondary beam
energies of $\approx$300 MeV/nucleon, as are now becoming available. We compare
the predictions of the adiabatic model of the reaction both without and when
including the additional eikonal approximation that has been used extensively.
We show that the reaction cross section calculations have only limited
sensitivity to the size and structure of $^{22}$C and that the differences
arising from use of the eikonal approximation of the reaction mechanism are of
a similar magnitude.
