Stellar limb darkening affects a wide range of astronomical measurements and is frequently modelled with a parametric model using polynomials in the cosine of the angle between the line of sight and the emergent intensity. Two-parameter laws are particularly popular for cases where one wishes to fit freely for the limb darkening coefficients (i.e. an uninformative prior) due to the compact prior volume and the fact that more complex models rarely obtain unique solutions with the present data. In such cases, we show that the two limb darkening coefficients are constrained by three physical boundary conditions, describing a triangular region in the two-dimensional parameter space. We show that uniformly distributed samples may be drawn from this region with optimal efficiency by a technique developed by computer graphical programming: triangular sampling. Alternatively, one can make draws using a uniform, bivariate Dirichlet distribution. We provide simple expressions for these parametrizations for both techniques applied to the case of quadratic, square-root and logarithmic limb darkening laws. For example, in the case of the popular quadratic law, we advocate fitting for q₁ = (u₁ + u₂)² and q₂ = 0.5u₁(u₁ + u₂)⁻¹ with uniform priors in the interval [0, 1] to implement triangular sampling easily. Employing these parametrizations allows one to derive model parameters which fully account for our ignorance about the intensity profile, yet never explore unphysical solutions, yielding robust and realistic uncertainty estimates. Furthermore, in the case of triangular sampling with the quadratic law, our parametrization leads to significantly reduced mutual correlations and provides an alternative geometric explanation as to why naively fitting the quadratic limb darkening coefficients precipitates strong correlations in the first place.

Mass and radius are two of the most fundamental properties of an astronomical object. Increasingly, new planet discoveries are being announced with a measurement of one of these quantities, but not both. This has led to a growing need to forecast the missing quantity using the other, especially when predicting the detectability of certain follow-up observations. We present an unbiased forecasting model built upon a probabilistic mass–radius relation conditioned on a sample of 316 well-constrained objects. Our publicly available code, Forecaster, accounts for observational errors, hyper-parameter uncertainties, and the intrinsic dispersions observed in the calibration sample. By conditioning our model on a sample spanning dwarf planets to late-type stars, Forecaster can predict the mass (or radius) from the radius (or mass) for objects covering nine orders of magnitude in mass. Classification is naturally performed by our model, which uses four classes we label as Terran worlds, Neptunian worlds, Jovian worlds, and stars. Our classification identifies dwarf planets as merely low-mass Terrans (like the Earth) and brown dwarfs as merely high-mass Jovians (like Jupiter). We detect a transition in the mass–radius relation at M _⊕, which we associate with the divide between solid, Terran worlds and Neptunian worlds. This independent analysis adds further weight to the emerging consensus that rocky super-Earths represent a narrower region of parameter space than originally thought. Effectively, then, the Earth is the super-Earth we have been looking for.

It is suggested that the distribution of orbital eccentricities for extrasolar planets is well described by the Beta distribution. Several properties of the Beta distribution make it a powerful tool for this purpose. For example, the Beta distribution can reproduce a diverse range of probability density functions (PDFs) using just two shape parameters (a and b). We argue that this makes it ideal for serving as a parametric model in Bayesian comparative population analysis. The Beta distribution is also uniquely defined over the interval zero to unity, meaning that it can serve as a proper prior for eccentricity when analysing the observations of bound extrasolar planets. Using nested sampling, we find that the distribution of eccentricities for 396 exoplanets detected through radial velocity with high signal-to-noise is well described by a Beta distribution with parameters a = 0.867$${^{+ 0.044}_{- 0.044}}$$ and b = 3.03$${^{+ 0.17}_{- 0.16}}$$. The Beta distribution is shown to be 3.7 times more likely to represent the underlying distribution of exoplanet eccentricities than the next best model: a Rayleigh + exponential distribution. The same data are also used in an example population comparison utilizing the Beta distribution, where we find that the short- and long-period planets are described by distinct Beta distributions at a confidence of 11.6σ and display a signature consistent with the effects of tidal circularization.

We explore how finite integration times or equivalently temporal binning induces morphological distortions to the transit light curve. These distorti

AU Microscopii (AU Mic) is the second closest pre-main-sequence star, at a distance of 9.79 parsecs and with an age of 22 million years¹. AU Mic possesses a relatively rare² and spatially resolved³ edge-on debris disk extending from about 35 to 210 astronomical units from the star⁴, and with clumps exhibiting non-Keplerian motion^5,6,7. Detection of newly formed planets around such a star is challenged by the presence of spots, plage, flares and other manifestations of magnetic ‘activity’ on the star^8,9. Here we report observations of a planet transiting AU Mic. The transiting planet, AU Mic b, has an orbital period of 8.46 days, an orbital distance of 0.07 astronomical units, a radius of 0.4 Jupiter radii, and a mass of less than 0.18 Jupiter masses at 3σ confidence. Our observations of a planet co-existing with a debris disk offer the opportunity to test the predictions of current models of planet formation and evolution.