  Tidal dissipation within a short-period transiting extrasolar planet
perturbed by a companion object can drive orbital evolution of the system to a
so-called tidal fixed point, in which the apsidal lines of the transiting
planet and its perturber are aligned, and for which variations in the orbital
eccentricities of both planet and perturber are damped out. Significant
contributions to the apsidal precession rate are made by the secular
planet-planet interaction, by general relativity, and by the gravitational
quadropole fields created by the transiting planet's tidal and rotational
distortions. The fixed-point orbital eccentricity of the inner planet is
therefore a strong function of the planet's interior structure. We illustrate
these ideas in the specific context of the recently discovered HAT-P-13
exo-planetary system, and show that one can already glean important insights
into the physical properties of the inner transiting planet. We present
structural models of the planet, which indicate that its observed radius can be
maintained for a one-parameter sequence of models that properly vary core mass
and tidal energy dissipation in the interior. We use an octopole-order secular
theory of the orbital dynamics to derive the dependence of the inner planet's
eccentricity, on its tidal Love number. We find that the currently measured
eccentricity, implies 0.116 < k2_{b} < 0.425, 0 M_{Earth}<M_{core}<120
M_{Earth}$, and Q_{b} < 300,000. Improved measurement of the eccentricity will
soon allow for far tighter limits to be placed on all three of these
quantities, and will provide an unprecedented probe into the interior structure
of an extrasolar planet.
