  With recent and archival Rossi X-Ray Timing Explorer (RXTE) X-ray
measurements of the heavily obscured X-ray pulsar EXO 1722-363 (IGR
J17252-3616), we carried out a pulse timing analysis to determine the orbital
solution for the first time. The binary system is characterized by a_x sin(i) =
101 +/- 3 lt-s and P_orb = 9.7403 +/- 0.0004 days (90% confidence), with the
precision of the orbital period being obtained by connecting datasets separated
by more than 7 years (272 orbital cycles). The orbit is consistent with
circular, and e < 0.19 at the 90% confidence level. The mass function is 11.7
+/- 1.2 M_sun and confirms that this source is a High Mass X-ray Binary (HMXB)
system. The orbital period, along with the previously known ~414 s pulse
period, places this system in the part of the Corbet diagram populated by
supergiant wind accretors. Using previous eclipse time measurements by Corbet
et al. and our orbital solution, combined with the assumption that the primary
underfills its Roche lobe, we find i > 61 degrees at the 99% confidence level,
the radius of the primary is between 21 R_sun and 37 R_sun, and its mass is
less than about 22 M_sun. The acceptable range of radius and mass shows that
the primary is probably a supergiant of spectral type B0I-B5I. Photometric
measurements of its likely counterpart are consistent with the spectral type
and luminosity if the distance to the system is between 5.3 kpc and 8.7 kpc.
Spectral analysis of the pulsar as a function of orbital phase reveals an
evolution of the hydrogen column density suggestive of dense filaments of gas
in the downstream wake of the pulsar, with higher levels of absorption seen at
orbital phases 0.5-1.0, as well as a variable Fe K_alpha line.
