Most 1.6 Earth-Radius Planets are not Rocky

The Kepler Mission, combined with ground based radial velocity (RV) follow-up and dynamical analyses of transit timing variations, has revolutionized the observational constraints on sub-Neptune-size planet compositions. The results of an extensive Kepler follow-up program including multiple Doppler measurements for 22 planet-hosting stars more than doubles the population of sub-Neptune-sized transiting planets that have RV mass constraints. This unprecedentedly large and homogeneous sample of planets with both mass and radius constraints opens the possibility of a statistical study of the underlying population of planet compositions. We focus on the intriguing transition between rocky exoplanets (comprised of iron and silicates) and planets with voluminous layers of volatiles (H/He and astrophysical ices). Applying a hierarchical Bayesian statistical approach to the sample of Kepler transiting sub-Neptune planets with Keck RV follow-up, we constrain the fraction of close-in planets (with orbital periods less than ~50 days) that are sufficiently dense to be rocky, as a function of planet radius. We show that the majority of 1.6 Earth-radius planets are too low density to be comprised of Fe and silicates alone. At larger radii, the constraints on the fraction of rocky planets are even more stringent. These insights into the size demographics of rocky and volatile-rich planets offer empirical constraints to planet formation theories, and guide the range of planet radii to be considered in studies of the occurrence rate of "Earth-like" planets, $\eta_{Earth}$.Comment: Accepted to ApJ, In pres

Evolutionary Analysis of Gaseous Sub-Neptune-Mass Planets with MESA

Sub-Neptune-sized exoplanets represent one of the most common types of planets in the Milky Way, yet many of their properties are unknown. Here, we present a prescription to adapt the capabilities of the stellar evolution toolkit Modules for Experiments in Stellar Astrophysics (MESA) to model sub-Neptune mass planets with H/He envelopes. With the addition of routines treating the planet core luminosity, heavy element enrichment, atmospheric boundary condition, and mass loss due to hydrodynamic winds, the evolutionary pathways of planets with diverse starting conditions are more accurately constrained. Using these dynamical models, we construct mass-composition relationships of planets from 1 to 400 $M_{\oplus}$ and investigate how mass-loss impacts their composition and evolution history. We demonstrate that planet radii are typically insensitive to the evolution pathway that brought the planet to its instantaneous mass, composition and age, with variations from hysteresis. We find that planet envelope mass loss timescales, $\tau_{\rm env}$, vary non-monotonically with H/He envelope mass fractions (at fixed planet mass). In our simulations of young (100~Myr) low-mass ($M_p\lesssim10~M_\oplus$) planets with rocky cores, $\tau_{\rm env}$ is maximized at $M_{\rm env}/M_p=1\%$ to $3\%$. The resulting convergent mass loss evolution could potentially imprint itself on the close-in planet population as a preferred H/He mass fraction of ${\sim}1\%$. Looking ahead, we anticipate that this numerical code will see widespread applications complementing both 3-D models and observational exoplanet surveys.Comment: 18 pages, 9 figures, and 4 tables. Accepted to the Astrophysical Journal on August 29th, 201

Transit Light Curves with Finite Integration Time: Fisher Information Analysis

Kepler has revolutionized the study of transiting planets with its unprecedented photometric precision on more than 150,000 target stars. Most of the transiting planet candidates detected by Kepler have been observed as long-cadence targets with 30 minute integration times, and the upcoming Transiting Exoplanet Survey Satellite (TESS) will record full frame images with a similar integration time. Integrations of 30 minutes affect the transit shape, particularly for small planets and in cases of low signal-to-noise. Using the Fisher information matrix technique, we derive analytic approximations for the variances and covariances on the transit parameters obtained from fitting light curve photometry collected with a finite integration time. We find that binning the light curve can significantly increase the uncertainties and covariances on the inferred parameters when comparing scenarios with constant total signal-to-noise (constant total integration time in the absence of read noise). Uncertainties on the transit ingress/egress time increase by a factor of 34 for Earth-size planets and 3.4 for Jupiter-size planets around Sun-like stars for integration times of 30 minutes compared to instantaneously-sampled light curves. Similarly, uncertainties on the mid-transit time for Earth and Jupiter-size planets increase by factors of 3.9 and 1.4. Uncertainties on the transit depth are largely unaffected by finite integration times. While correlations among the transit depth, ingress duration, and transit duration all increase in magnitude with longer integration times, the mid-transit time remains uncorrelated with the other parameters. We provide code in Python and Mathematica for predicting the variances and covariances at www.its.caltech.edu/~eprice

Probabilistic Mass-Radius Relationship for Sub-Neptune-Sized Planets

The Kepler Mission has discovered thousands of planets with radii $<4\ R_\oplus$, paving the way for the first statistical studies of the dynamics, formation, and evolution of these sub-Neptunes and super-Earths. Planetary masses are an important physical property for these studies, and yet the vast majority of Kepler planet candidates do not have theirs measured. A key concern is therefore how to map the measured radii to mass estimates in this Earth-to-Neptune size range where there are no Solar System analogs. Previous works have derived deterministic, one-to-one relationships between radius and mass. However, if these planets span a range of compositions as expected, then an intrinsic scatter about this relationship must exist in the population. Here we present the first probabilistic mass-radius relationship (M-R relation) evaluated within a Bayesian framework, which both quantifies this intrinsic dispersion and the uncertainties on the M-R relation parameters. We analyze how the results depend on the radius range of the sample, and on how the masses were measured. Assuming that the M-R relation can be described as a power law with a dispersion that is constant and normally distributed, we find that$M/M_\oplus=2.7(R/R_\oplus)^{1.3}$, a scatter in mass of$1.9\ M_\oplus$, and a mass constraint to physically plausible densities, is the "best-fit" probabilistic M-R relation for the sample of RV-measured transiting sub-Neptunes ($R_{pl}<4\ R_\oplus$). More broadly, this work provides a framework for further analyses of the M-R relation and its probable dependencies on period and stellar properties.Comment: 14 pages, 5 figures, 2 tables. Accepted to the Astrophysical Journal on April 28, 2016. Select posterior samples and code to use them to compute the posterior predictive mass distribution are available at https://github.com/dawolfgang/MRrelatio

Regrouping for Instruction in Reading: Effective Practices

It is the intent of this project to create a handbook providing information about a plan for regrouping for instruction in reading at Long Beach Elementary School. It will be presented to parents, teachers and administrators at workshops concerning regrouping for instruction in reading. The handbook will provide practical applications of curricular adaptation and instructional techniques that may be used to facilitate improvement in reading for students. The policies and procedures described in the manual will provide the following information: • Assessment • Placement • Transitions • Pace and Skill Levels • Curricular Adaptations • Teacher Collaboration and Consultation • Ongoing Student Evaluation • Flexibility of Grouping • Instructional Teaching Technique

Comparison of multivariate methods for measuring change from pretest to posttest

Three multivariate methods for measuring change from pretest to posttest are compared with respect to statistical power over various levels and combinations of effect size, alpha level, sample size, number of dependent variables, number of significantly different dependent variables, correlation between corresponding pretest and posttest scores, and correlation between unrelated pretest and posttest scores. The method utilizing posttests as the dependent variables and pretests as covariates was found to have superior statistical power in the majority of the scenarios examined. However, there were scenarios where the method utilizing change scores as dependent variables and the method utilizing only posttests as the dependent variables displayed greater power. Using results from the Monte Carlo simulations, comparisons are presented that reveal the conditions under which each of the three multivariate methods displayed greater statistical power than the other two. In addition to the immediate implications of the current study, suggested future avenues of research that could expand upon the current findings are discussed

Festschrift for Dean Simon

We write together about our dear friend Michelle Simon because of her enormous contribution to our professional and personal lives. Working with someone who becomes more than a colleague, but a friend as well, is a special gift. Michelle not only paved our path, she became a trail blazer in legal education. The professional trajectory of Michelle Simon speaks volumes to her talent, tenacity, and consensus building skills