  We present a study of the distribution of asymptotic burst oscillation
periods for a sample of 26 bursts from 4U 1636-53 observed with the Rossi X-ray
Timing Explorer (RXTE). The burst sample includes both archival and proprietary
data and spans more than 4.5 years. We also present new optical light curves of
V801 Arae, the optical counterpart of 4U 1636-53, obtained during 1998-2001. We
use these optical data to refine the binary period measured by Augusteijn et
al. (1998) to 3.7931206(152) hours. We show that a subset of ~70% of the bursts
form a tightly clustered distribution of asymptotic periods consistent with a
period stability of ~ 1/10,000. The tightness of this distribution, made up of
bursts spanning more than 4 years in time, suggests that the underlying period
is highly stable, with a time to change the period of ~30,000 years, comparable
to similar numbers derived for X-ray pulsars. We investigate the period and
orbital phase data for our burst sample and show that it is consistent with
binary motion of the neutron star with v sin i < 55 km/sec at 90% confidence.
We use this limit as well as previous radial velocity data to constrain the
binary geometry and component masses in 4U 1636-53.
