  Spitzer Space Telescope infrared observations indicate that several
transiting extrasolar giant planets have thermal inversions in their upper
atmospheres. Above a relative minimum, the temperature appears to increase with
altitude. Such an inversion probably requires a species at high altitude that
absorbs a significant amount of incident optical/UV radiation. Some authors
have suggested that the strong optical absorbers titanium oxide (TiO) and
vanadium oxide (VO) could provide the needed additional opacity. However, if
regions of the atmosphere are cold enough for Ti and V to be sequestered into
solids they might rain out and be severely depleted. With a model of the
vertical distribution of a refractory species in gaseous and condensed form, we
address the question of whether enough TiO (or VO) could survive aloft in an
irradiated planet's atmosphere to produce a thermal inversion. We find that,
without significant macroscopic mixing, a heavy species such as TiO --
especially one that can condense in a cold-trap region -- cannot persist in a
planet's upper atmosphere. In our model, the amount of macroscopic mixing that
would be required to loft TiO to millibar pressures ranges from ~10^7 to
~10^{11} cm^2/s on HD 209458b, HD 149026b, TrES-4, OGLE-TR-56b, and WASP-12b,
depending on the planet and on assumed condensate sizes. Such large values may
be problematic for the TiO hypothesis. [Abridged]
