  Massive eccentric disks (gaseous or particulate) orbiting a dominant central
mass appear in many astrophysical systems, including planetary rings,
protoplanetary and accretion disks in binaries, and nuclear stellar disks
around supermassive black holes in galactic centers. We present an analytical
framework for treating the nearly Keplerian secular dynamics of test particles
driven by the gravity of an eccentric, apsidally aligned, zero-thickness disk
with arbitrary surface density and eccentricity profiles. We derive a
disturbing function describing the secular evolution of coplanar objects, which
is explicitly related (via one-dimensional, convergent integrals) to the disk
surface density and eccentricity profiles without using any ad hoc softening of
the potential. Our analytical framework is verified via direct orbit
integrations, which show it to be accurate in the low-eccentricity limit for a
variety of disk models (for disk eccentricity < 0.1-0.2). We find that free
precession in the potential of a disk with a smooth surface density
distribution can naturally change from prograde to retrograde within the disk.
Sharp disk features - edges and gaps - are the locations where this tendency is
naturally enhanced, while the precession becomes very fast. Radii where free
precession changes sign are the locations where substantial (formally singular)
growth of the forced eccentricity of the orbiting objects occurs. Based on our
results, we formulate a self-consistent analytical framework for computing an
eccentricity profile for an aligned, eccentric disk (with a prescribed surface
density profile) capable of precessing as a solid body under its own
self-gravity.
