  It is standard lore that gravitational radiation reaction circularizes the
orbits of inspiralling binary systems. But in recent papers, Loutrel et al.
have argued that at late times in such inspirals, one measure of eccentricity
actually increases, and that this could have observable consequences. We show
that this variable, the magnitude of the Runge-Lenz vector ($e_{\rm RL}$), is
not an appropriate measure of orbital eccentricity, when the eccentricity is
smaller than the leading non-Keplerian perturbation of the orbit. Following
Loutrel et al., we use Newtonian equations of motion plus the leading
gravitational radiation-reaction terms, the osculating-orbits approach for
characterizing binary orbits, and a two-timescale analysis for separating
secular from periodic variations of the orbit elements. We find that $e_{\rm
RL}$ does grow at late times, but that the actual orbital variables $r$ and
$dr/dt$ show no such growth in oscillations. This is in complete agreement with
Loutrel et al. We reconcile this apparent contradiction by pointing out that it
is essential to take into account the direction of the Runge-Lenz vector, not
just its magnitude. At late times in an inspiral, that direction, which defines
the pericenter angle, advances at the same rate as the orbital phase. The
correct picture is then of a physically circular orbit whose osculating
counterpart is indeed eccentric but that resides permanently at the orbit's
latus rectum at $-90^{\rm o}$, therefore exhibiting no oscillations. Including
first post-Newtonian effects in the equations of motion, we show that $e_{\rm
RL}$ grows even more dramatically. But the phase of the Runge-Lenz vector again
rotates with the orbit at late times, but now the osculating orbit resides at
"perpetual apocenter", so again the physical orbit circularizes.
