An Inverse Problem on Determining Second Order Symmetric Tensor for Perturbed Biharmonic Operator

This article offers a study of the Calder\'on type inverse problem of determining up to second order coefficients of the higher order elliptic operator. Here we show that it is possible to determine an anisotropic second order perturbation given by a symmetric matrix, along with a first order perturbation given by a vector field and a zero-th order potential function inside a bounded domain by measuring the Dirichlet to Neumann map of the perturbed biharmonic operator on the boundary of that domain

Orbital Architectures of Kepler Multis From Dynamical Instabilities

The high-multiplicity exoplanet systems are generally more tightly packed when compared to the solar system. Such compact multi-planet systems are often susceptible to dynamical instability. We investigate the impact of dynamical instability on the final orbital architectures of multi-planet systems using N-body simulations. Our models initially consist of six to ten planets placed randomly according to a power-law distribution of mutual Hill separations. We find that almost all of our model planetary systems go through at least one phase of dynamical instability, losing at least one planet. The orbital architecture, including the distributions of mutual Hill separations, planetary masses, orbital periods, and period ratios, of the transit-detectable model planetary systems closely resemble those for the multi-planet systems detected by Kepler. We find that without any formation-dependent input, a dynamically active past can naturally reproduce important observed trends including multiplicity-dependent eccentricity distribution, smaller eccentricities for larger planets, and intra-system uniformity. On the other hand, our transit-detectable planet populations lack the observed sub-population of eccentric single-transiting planets, pointing towards the Kepler dichotomy. These findings indicate that dynamical instabilities may have played a vital role in the final assembly of sub-Jovian planets.Comment: 15 pages, 15 figures; published in MNRA