New Perspectives on the Exoplanet Radius Gap from a Mathematica Tool and Visualized Water Equation of State

Recent astronomical observations obtained with the Kepler and TESS missions and their related ground-based follow-ups revealed an abundance of exoplanets with a size intermediate between Earth and Neptune (1 R ⊕ ≤ R ≤ 4 R ⊕). A low occurrence rate of planets has been identified at around twice the size of Earth (2 × R ⊕), known as the exoplanet radius gap or radius valley. We explore the geometry of this gap in the mass–radius diagram, with the help of a Mathematica plotting tool developed with the capability of manipulating exoplanet data in multidimensional parameter space, and with the help of visualized water equations of state in the temperature–density (T–ρ) graph and the entropy–pressure (s–P) graph. We show that the radius valley can be explained by a compositional difference between smaller, predominantly rocky planets (<2 × R ⊕) and larger planets (>2 × R ⊕) that exhibit greater compositional diversity including cosmic ices (water, ammonia, methane, etc.) and gaseous envelopes. In particular, among the larger planets (>2 × R ⊕), when viewed from the perspective of planet equilibrium temperature (T eq), the hot ones (T eq ≳ 900 K) are consistent with ice-dominated composition without significant gaseous envelopes, while the cold ones (T eq ≲ 900 K) have more diverse compositions, including various amounts of gaseous envelopes.

Star-shaped order for distributions characterized by several parameters and some applications

The star-shaped ordering between probability distributions is a common way to express aging properties. A well-known criterion was proposed by Saunders and Moran [(1978). On the quantiles of the gamma and F distributions. Journal of Applied Probability 15(2): 426–432], to order families of distributions depending on one real parameter. However, the lifetime of complex systems usually depends on several parameters, especially when considering heterogeneous components. We extend the Saunders and Moran criterion characterizing the star-shaped order when the multidimensional parameter moves along a given direction. A few applications to the lifetime of complex models, namely parallel and series models assuming different individual components behavior, are discussed.

Parametric Nonlinear Model Reduction Using K-Means Clustering for Miscible Flow Simulation

This work considers the model order reduction approach for parametrized viscous fingering in a horizontal flow through a 2D porous media domain. A technique for constructing an optimal low-dimensional basis for a multidimensional parameter domain is introduced by combining K-means clustering with proper orthogonal decomposition (POD). In particular, we first randomly generate parameter vectors in multidimensional parameter domain of interest. Next, we perform the K-means clustering algorithm on these parameter vectors to find the centroids. POD basis is then generated from the solutions of the parametrized systems corresponding to these parameter centroids. The resulting POD basis is then used with Galerkin projection to construct reduced-order systems for various parameter vectors in the given domain together with applying the discrete empirical interpolation method (DEIM) to further reduce the computational complexity in nonlinear terms of the miscible flow model. The numerical results with varying different parameters are demonstrated to be efficient in decreasing simulation time while maintaining accuracy compared to the full-order model for various parameter values.