  Recent observations have shown that at least some close-in exoplanets
maintain eccentric orbits despite tidal circularization timescales that are
typically shorter than stellar ages. We explore gravitational interactions with
a distant planetary companion as a possible cause of these non-zero
eccentricities. For simplicity, we focus on the evolution of a planar
two-planet system subject to slow eccentricity damping and provide an intuitive
interpretation of the resulting long-term orbital evolution. We show that
dissipation shifts the two normal eigenmode frequencies and eccentricity ratios
of the standard secular theory slightly, and that each mode decays at its own
rate. Tidal damping of the eccentricities drives orbits to transition between
periods of pericenter circulation and libration, and the planetary system
settles into a locked state where the pericenters are nearly aligned or
anti-aligned. Once in the locked state, the eccentricities of the two orbits
decrease very slowly due to tides rather than at the much more rapid
single-planet rate, and thus eccentric orbits, even for close-in planets, can
often survive much longer than the age of the system. Assuming that an observed
close-in planet on an elliptical orbit is apsidally-locked to a more distant,
and perhaps unseen companion, we provide a constraint on the mass, semi-major
axis, and eccentricity of the companion. We find the observed two-planet system
HAT-P-13 might be in just such an apsidally-locked state, with parameters that
obey our constraint well. We also survey close-in single planets, and found
that none provide compelling evidence for unseen companions. Instead, we
suspect that (1) orbits are circular, (2) tidal damping rates are slower than
our assumption, or (3) a recent event has excited these eccentricities. Our
method should prove useful for interpreting the results of current and future
planet searches.
