  The problem of a test body in the Schwarzschild geometry is investigated in a
Keplerian limit. Beginning with the Schwarzschild metric, a solution to the
limited case of approximately elliptical (Keplerian) motion is derived in terms
of trigonometric functions. This solution is similar in form to that derived
from Newtonian mechanics, and includes first-order corrections describing three
effects due to general relativity: precession; reduced radial coordinate; and
increased eccentricity. The quantitative prediction of increased eccentricity
may provide an additional observational test of general relativity. By analogy
with Keplerian orbits, approximate orbital energy parameters are defined in
terms of a relativistic eccentricity, providing first-order corrections to
Newtonian energies for elliptical orbits. The first-order relativistic equation
of orbit is demonstrated to be a limiting case of a very accurate
self-consistent solution. This self-consistent solution is supported by exact
numerical solutions to the Schwarzschild geometry, displaying remarkable
agreement. A more detailed energy parameterization is investigated using the
relativistic eccentricity together with the apsides derived from the
relativistic effective potential in support of the approximate energy
parameters defined using only first-order corrections. The methods and
approximations describing this Keplerian limit are applied to more general
static spherically-symmetric geometries. Specifically, equations of orbit and
energy parameters are also derived in this Keplerian limit for the
Reissner-Nordstr\"{o}m and Schwarzschild-de Sitter metrics.
