Recent observations have shown that at least some close-in exoplanets maintain eccentric orbits despite tidal circularization timescales that are typically much shorter than stellar ages. We explore gravitational interactions with a more distant planetary companion as a possible cause of these unexpected 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 we confirm that each mode decays at its own rate. Tidal damping of the eccentricities drives orbits to transition relatively quickly between periods of pericenter circulation and libration, and the planetary system settles into a locked state in which the pericenters are nearly aligned or nearly anti-aligned. Once in the locked state, the eccentricities of the two orbits decrease very slowly because of 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 that the observed two-planet system HAT-P-13 might be in just such an apsidally locked state, with parameters that obey our constraint reasonably well. We also survey close-in single planets, some with and some without an indication of an outer companion. None of the dozen systems that we investigate provides compelling evidence for unseen companions. Instead, we suspect that (1) orbits are in fact circular, (2) tidal damping rates are much slower than we have assumed, or (3) a recent event has excited these eccentricities. Our method should prove useful for interpreting the results of both current and future planet searches.
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