  Observations of the S-stars, the cluster of young stars in the inner 0.1 pc
of the Galactic Center, have been crucial in providing conclusive evidence for
a supermassive black hole at the center of our galaxy. Since some of the stars
have orbits less than that of a typical human lifetime, it is possible to
observe multiple orbits and test the weak-field regime of general relativity.
Current calculations of S-star orbits require slow and expensive computations
in order to numerically solve geodesic equations for many small time steps. In
this paper, we present a computationally efficient, first-order post-Newtonian
model for the astrometric and spectroscopic data gathered for the S-stars. We
find that future, 30-m class telescopes -- and potentially even current large
telescopes with very high spectroscopic resolution -- may be able to detect the
Shapiro effect for an S-star in the next decade or so.
