  We study the process of runaway, unstable Roche lobe overflow in coalescing
binary systems and its dependence on the properties of the binary involved. We
create three-dimensional hydrodynamic models of binary coalescences, and follow
them through a phase of increasing Roche lobe overflow until the accretor is
engulfed by the donor at the onset of a common envelope phase. In these models,
we vary binary properties of mass ratio, donor structure and spin, and equation
of state through the gas adiabatic index. We compare the numerical results to
semi-analytic models of binary orbit evolution based on mass and angular
momentum exchange between two point masses. Using our hydrodynamic simulations,
we measure the key parameters: the donor mass loss rate and the angular
momentum exchanged per unit mass loss from the donor. Using these calibrations,
the semi-analytic model closely reproduces the escalating mass loss and runaway
orbital decay observed in the hydrodynamic models. The semi-analytic model
accurately reproduces the major differences in orbit evolution that arise with
varying mass ratio and donor structure. We encapsulate the semi-analytic model
in a publicly-released python package RLOF. We apply this model to the observed
period decay and subsequent merger of the binary V1309 Sco, and find that it
can simultaneously reproduce the observed orbital decay and time of outburst.
We further demonstrate that there is a relationship between period derivative
and second derivative that can be a useful metric for evaluating candidate
merging binaries.
