  About 30-50% of the baryons in the local Universe are unaccounted for and are
likely in a hot phase, 10^5.5-10^8 K. A hot halo (10^6.3 K) is detected around
the Milky Way through the O VII and O VIII resonance absorption and emission
lines in the soft X-ray band. Current instruments are not sensitive enough to
detect this gas in absorption around other galaxies and galaxy groups, the two
most likely sites. We show that resonant line absorption by this hot gas can be
detected with current technology, with a collecting area exceeding about 300
cm^2 and a resolution R > 2000. For a few notional X-ray telescope
configurations that could be constructed as Explorer or Probe missions, we
calculate the differential number of O VII and O VIII absorbers as a function
of equivalent width through redshift space, dN/dz. The hot halos of individual
external galaxies produce absorption that should be detectable out to about
their virial radii. For the Milky Way, one can determine the radial
distribution of density, temperature, and metallicity, after making optical
depth corrections. Spectroscopic observations can determine the rotation of a
hot gaseous halo.
