  Triple body systems are prevalent in nature, from planetary to stellar to
supermassive black hole scales. In a hierarchical triple system, oscillations
of the inner orbit's eccentricity and inclination can be induced on secular
timescales. Over many cycles, the octupole-level terms in the secular equations
of motion can drive the system to extremely high eccentricities via the
Eccentric Kozai-Lidov (EKL) mechanism. The overall decrease in the inner
orbit's pericenter distance has potentially dramatic effects for realistic
systems, such as tidal disruption. We present an analytical approximation in
the test particle limit to describe the step-wise eccentricity evolution of the
inner orbit. We also integrate the equations of motion and present
approximations in both the test particle limit and in the general case to
describe the overall octupole-level time evolution of the eccentricity. The
analytical approximations are compared to numerical simulations to show that
the models accurately describe the form and timescale of the secular descent
from large distances to a close-encounter distance (e.g., the Roche limit). By
circumventing the need for numerical simulations to obtain the long-term
behavior, these approximations can be used to readily estimate descent
timescales for populations of systems. We demonstrate by calculating rates of
EKL-driven migration for Hot Jupiters in stellar binaries.
