The determination of planetary structure in tidally relaxed inclined systems

Abstract

[Abridged] The recent discovery of a transiting planet on a non-circular orbit with a massive highly eccentric companion orbiting HAT-P-13 offers the possibility of probing the structure of the short-period planet. The ability to do this relies on the system being in a quasi-equilibrium state in the sense that the eccentricities are constant on the usual secular timescale, and decay on a timescale which is much longer than the age of the system. Since the equilibrium eccentricity is effectively a function only of observable system parameters and the unknown Love number of the short-period planet, the latter can be determined with accurate measurements of the planet's eccentricity and radius. However, this analysis relies on the unlikely assumption that the system is coplanar. Here we generalize our recent analysis of this fixed-point phenomenon to mutually inclined systems and show that the fixed point of coplanar systems is replaced by a limit cycle, with the average value of the eccentricity decreasing and its amplitude of variation increasing with increasing mutual inclination. This behaviour significantly reduces the ability to unambiguously determine the Love number of the short-period planet if the mutual inclination is higher than around 10^o. We show that for Q-values less than 10^6, the HAT-P-13 system cannot have a mutual inclination between 54 and 126^o because Kozai oscillations coupled with tidal dissipation would act to quickly move the inclination outside this range, and that the behaviour of retrograde systems is the mirror image of that for prograde systems. We derive a relationship between the equilibrium radius of the short-period planet, its Q-value and its core mass, and show that given current estimates of e_b and the planet radius, the HAT-P-13 system is likely to be close to coplanar [...]