Journal Article

The determination of planetary structure in tidally relaxed inclined systems

Rosemary A. Mardling

in Monthly Notices of the Royal Astronomical Society

Published on behalf of The Royal Astronomical Society

Volume 407, issue 2, pages 1048-1069
Published in print September 2010 | ISSN: 0035-8711
Published online September 2010 | e-ISSN: 1365-2966 | DOI: https://dx.doi.org/10.1111/j.1365-2966.2010.16814.x
The determination of planetary structure in tidally relaxed inclined systems

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The recent discovery of a transiting short-period planet on a slightly non-circular orbit with a massive highly eccentric companion orbiting the star 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 time-scale (typically, a few thousand years), and decay on a time-scale 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 assumption that the system is coplanar, a situation which seems unlikely given the high eccentricity of the outer planet. Here we generalize our recent analysis of this fixed-point phenomenon to mutually inclined systems in which the outer body dominates the total angular momentum, and show that (1) the fixed point of coplanar systems is replaced by a limit cycle in eb–η space, where eb is the eccentricity of the inner planet and η is the angle between the periapse lines, with the average value of eb, e(av)b, 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°. (2) We show that for Q-values less than 106, the HAT-P-13 system cannot have a mutual inclination between 54° and 126° because Kozai oscillations coupled with tidal dissipation would act to quickly move the inclination outside this range, and (3) that the behaviour of retrograde systems is the mirror image of that for prograde systems in the sense that (almost) identical limit cycles exist for a given mutual inclination and π minus this value. (4) We derive a relationship between e(av)b, the equilibrium radius of the short-period planet, its Q-value and its core mass, and show that given current estimates of eb and the planet radius, as well as the lower bound placed on the Q-value by the decay rate of e(av)b, the HAT-P-13 system is likely to be close to prograde coplanar, or have a mutual inclination between 130° and 135°. Lower rather than higher core masses are favoured. (5) An expression for the time-scale for decay of the mutual inclination is derived, revealing that it evolves towards a non-zero value as long as eb > 0 on a time-scale which is much longer than the age of the system. (6) We conclude with a scattering scenario for the origin of the HAT-P-13 system and show that almost identical initial conditions can result in significantly different outer planet eccentricities, stellar obliquities and planet radii. The implications for systems with high stellar obliquities such as HAT-P-7 and WASP-17 are briefly discussed.

Keywords: methods: analytical; celestial mechanics; planets and satellites: dynamical evolution and stability; planets and satellites: formation; planets and satellites: interiors; planetary systems

Journal Article.  13273 words.  Illustrated.

Subjects: Astronomy and Astrophysics

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