  We explore test particle orbits in the orbital plane of eccentric stellar
binary systems, searching for ``invariant loops'': closed curves that change
shape periodically as a function of binary orbital phase as the test particles
in them move under the stars' gravity. Stable invariant loops play the same
role in this periodically-varying potential as stable periodic orbits do in
stationary potentials; in particular, when dissipation is weak, gas will most
likely follow the non-intersecting loops, while nearby particle orbits librate
around them. We use this method to set bounds on the sizes of disks around the
stars, and on the gap between those and the inner edge of a possible
circumbinary disk. Gas dynamics may impose further restrictions, but our study
sets upper bounds for the size of circumstellar disks, and a lower bound for
the inner radius of a circumbinary disk. We find that circumstellar disks are
sharply reduced as the binary's eccentricity grows. Disks change in size and
shape only marginally with the binary phase, with no strong preference to
increase or decrease at any particular phase. The circumstellar disks in
particular can be quite asymmetric. We compare our results with other numerical
and theoretical results and with observations of the $\alpha$ Centauri and
L1551 systems, finding very good agreement. The calculated changes in the
shapes and crowding of the circumstellar orbits can be used to predict how the
disk luminosity and mass inflow should vary with binary phase.
