The Detectability of Transit Depth Variations due to Exoplanetary Oblateness and Spin Precession

Knowledge of an exoplanet's oblateness and obliquity would give clues about its formation and internal structure. In principle, a light curve of a transiting planet bears information about the planet's shape, but previous work has shown that the oblateness-induced signal will be extremely difficult to detect. Here we investigate the potentially larger signals due to planetary spin precession. The most readily detectable effects are transit depth variations (T$\delta$V) in a sequence of light curves. For a planet as oblate as Jupiter or Saturn, the transit depth will undergo fractional variations of order 1%. The most promising systems are those with orbital periods of approximately 15--30 days, which is short enough for the precession period to be less than about 40 years, and long enough to avoid spin-down due to tidal friction. The detectability of the T$\delta$V signal would be enhanced by moons (which would decrease the precession period) or planetary rings (which would increase the amplitude). The Kepler mission should find several planets for which precession-induced T$\delta$V signals will be detectable. Due to modeling degeneracies, Kepler photometry would yield only a lower bound on oblateness. The degeneracy could be lifted by observing the oblateness-induced asymmetry in at least one transit light curve, or by making assumptions about the planetary interior.Comment: Accepted for publication in The Astrophysical Journa

Monotone Projection Lower Bounds from Extended Formulation Lower Bounds

In this short note, we reduce lower bounds on monotone projections of polynomials to lower bounds on extended formulations of polytopes. Applying our reduction to the seminal extended formulation lower bounds of Fiorini, Massar, Pokutta, Tiwari, & de Wolf (STOC 2012; J. ACM, 2015) and Rothvoss (STOC 2014; J. ACM, 2017), we obtain the following interesting consequences. 1. The Hamiltonian Cycle polynomial is not a monotone subexponential-size projection of the permanent; this both rules out a natural attempt at a monotone lower bound on the Boolean permanent, and shows that the permanent is not complete for non-negative polynomials in VNP$_{{\mathbb R}}$ under monotone p-projections. 2. The cut polynomials and the perfect matching polynomial (or "unsigned Pfaffian") are not monotone p-projections of the permanent. The latter, over the Boolean and-or semi-ring, rules out monotone reductions in one of the natural approaches to reducing perfect matchings in general graphs to perfect matchings in bipartite graphs. As the permanent is universal for monotone formulas, these results also imply exponential lower bounds on the monotone formula size and monotone circuit size of these polynomials.Comment: Published in Theory of Computing, Volume 13 (2017), Article 18; Received: November 10, 2015, Revised: July 27, 2016, Published: December 22, 201

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Includes bibliographical references