  Infrared transmission and emission spectroscopy of exoplanets, recorded from
primary transit and secondary eclipse measurements, indicate the presence of
the most abundant carbon and oxygen molecular species (H2O, CH4, CO, and CO2)
in a few exoplanets. However, efforts to constrain the molecular abundances to
within several orders of magnitude are thwarted by the broad range of
degenerate solutions that fit the data. Here we explore, with radiative
transfer models and analytical approximations, the nature of the degenerate
solution sets resulting from the sparse measurements of "hot Jupiter"
exoplanets. As demonstrated with simple analytical expressions, primary transit
measurements probe roughly 4 atmospheric scale heights at each wavelength band.
Derived mixing ratios from these data are highly sensitive to errors in the
radius in planet (at a reference pressure), which are approximately a few
percent. For example, an uncertainty of 1% in the radius of a 1000 K and
H2-based exoplanet with Jupiter's radius and mass, causes an uncertainty of a
factor of ~100-10000 in the derived gas mixing ratios. The degree of
sensitivity depends on how the line strength increases with the optical depth
(i.e. the curve of growth) and the atmospheric scale height. Temperature
degeneracies in the solutions of the primary transit data, which manifest their
effects through scale height and absorption coefficients, are smaller. We argue
that these challenges can be partially surmounted by a combination of selected
wavelength sampling of optical and IR measurements and, when possible, the
joint analysis of transit and secondary eclipse data of exoplanets.
