  We present model-independent measurements of the excited-state population of
atoms in a sodium (Na) magneto-optical trap (MOT) using a hybrid ion-neutral
trap composed of a MOT and a linear Paul trap (LPT). We photoionize excited Na
atoms trapped in the MOT and use two independent methods to measure the
resulting ions: directly by trapping them in our LPT, and indirectly by
monitoring changes in MOT fluorescence. By measuring the ionization rate via
these two independent methods, we have enough information to directly determine
the population of MOT atoms in the excited-state. The resulting measurement
reveals that there is a range of trapping-laser intensities where the
excited-state population of atoms in our MOT follows the standard two-level
model intensity-dependence. However, an experimentally determined effective
saturation intensity must be used instead of the theoretically predicted value
from the two-level model. We measured the effective saturation intensity to be
$I_\mathrm{se}=22.9(3)\:\textrm{mW}/\textrm{cm}^2$ for the type-I Na MOT and
$I_\mathrm{se}=48.9(7)\;\textrm{mW}/\textrm{cm}^2$ for the type-II Na MOT,
approximately 1.7 and 3.6 times the theoretical estimate, respectively. Lastly,
at large trapping-laser intensities, our experiment reveals a clear departure
from the two-level model at a critical intensity that we believe is due to a
state-mixing effect, whose critical intensity can be determined by a simple
power broadening model.
