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 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 q 1 ≡ (u 1 + u 2 ) 2 and q 2 ≡ 0.5u 1 (u 1 + u 2 ) −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.

Most stars become white dwarfs after they have exhausted their nuclear fuel (the Sun will be one such). Between one-quarter and one-half of white dwarfs have elements heavier than helium in their atmospheres, even though these elements ought to sink rapidly into the stellar interiors (unless they are occasionally replenished). The abundance ratios of heavy elements in the atmospheres of white dwarfs are similar to the ratios in rocky bodies in the Solar System. This fact, together with the existence of warm, dusty debris disks surrounding about four per cent of white dwarfs, suggests that rocky debris from the planetary systems of white-dwarf progenitors occasionally pollutes the atmospheres of the stars. The total accreted mass of this debris is sometimes comparable to the mass of large asteroids in the Solar System. However, rocky, disintegrating bodies around a white dwarf have not yet been observed. Here we report observations of a white dwarf--WD 1145+017--being transited by at least one, and probably several, disintegrating planetesimals, with periods ranging from 4.5 hours to 4.9 hours. The strongest transit signals occur every 4.5 hours and exhibit varying depths (blocking up to 40 per cent of the star's brightness) and asymmetric profiles, indicative of a small object with a cometary tail of dusty effluent material. The star has a dusty debris disk, and the star's spectrum shows prominent lines from heavy elements such as magnesium, aluminium, silicon, calcium, iron, and nickel. This system provides further evidence that the pollution of white dwarfs by heavy elements might originate from disrupted rocky bodies such as asteroids and minor planets.

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 terms, 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 am 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 upon 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 2.0 +0.7 −0.6 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 11.6 σ and display a signature consistent with the effects of tidal circularization.

As the number of known exoplanets continues to grow, the question as to whether such bodies harbour satellite systems has become one of increasing interest. In this paper, we explore the transit timing effects that should be detectable due to an exomoon and predict a new observable. We first consider transit time variation (TTV), where we update the model to include the effects of orbital eccentricity. We draw two key conclusions: 1) In order to maintain Hill stability, the orbital frequency of the exomoon will always be higher than the sampling frequency. Therefore, the period of the exomoon cannot be reliably determined from TTV, only a set of harmonic frequencies. 2) The TTV amplitude is proportional to M_S a_S where M_S is the exomoon mass and a_S is the semi-major axis of the moon's orbit. Therefore, M_S and a_S cannot be separately determined. We go on to predict a new observable due to exomoons - transit duration variation (TDV). We derive the TDV amplitude and conclude that its amplitude is not only detectable, but the TDV signal will provide two robust advantages: 1) The TDV amplitude is proportional to M_S a_S^{-1/2} and therefore the ratio of TTV and TDV allows for a separate determination of M_S and a_S. 2) TDV has a 90 degrees phase difference to the TTV signal, making it an excellent complementary technique.Comment: 10 pages, 3 figures, 1 table, notation update

We explore how finite integration times or equivalently temporal binning induces morphological distortions to the transit light curve. These distortions, if uncorrected for, lead to the retrieval of erroneous system parameters and may even lead to some planetary candidates being rejected as ostensibly unphysical. We provide analytic expressions for estimating the disturbance to the various light-curve parameters as a function of the integration time. These effects are particularly crucial in light of the long-cadence photometry often used for discovering new exoplanets by, for example, Convection Rotation and Planetary Transits (CoRoT) and the Kepler Missions (8.5 and 30 min). One of the dominant effects of long integration times is a systematic underestimation of the light-curve-derived stellar density, which has significant ramifications for transit surveys. We present a discussion of numerical integration techniques to compensate for the effects and produce expressions to quickly estimate the errors of such methods, as a function of integration time and numerical resolution. This allows for an economic choice of resolution before attempting fits of long-cadence light-curves. We provide a comparison of the short-and long-cadence light curves of TrES-2b and show that the retrieved transit parameters are consistent using the techniques discussed here.

Two decades ago, empirical evidence concerning the existence and frequency of planets around stars, other than our own, was absent. Since this time, the detection of extrasolar planets from Jupiter-sized to most recently Earth-sized worlds has blossomed and we are finally able to shed light on the plurality of Earth-like, habitable planets in the cosmos. Extrasolar moons may also be frequent habitable worlds but their detection or even systematic pursuit remains lacking in the current literature. Here, we present a description of the first systematic search for extrasolar moons as part of a new observational project called "The Hunt for Exomoons with Kepler " (HEK). The HEK project distills the entire list of known transiting planet candidates found by Kepler (2326 at the time of writing) down to the most promising candidates for hosting a moon. Selected targets are fitted using a multimodal nested sampling algorithm coupled with a planet-with-moon light curve modelling routine. By comparing the Bayesian evidence of a planet-only model to that of a planet-with-moon, the detection process is handled in a Bayesian framework. In the case of null detections, upper limits derived from posteriors marginalised over the entire prior volume will be provided to inform the frequency of large moons around viable planetary hosts, η . After discussing our methodologies for target selection, modelling, fitting and vetting, we provide two example analyses.

The Transit Timing Variations (TTVs) can be used as a diagnostic of gravitational interactions between planets in a multi-planet system. Many Kepler Objects of Interest (KOIs) exhibit significant TTVs, but KOI-142.01 stands out among them with an unrivaled, ≃12-hour TTV amplitude. Here we report a thorough analysis of KOI-142.01's transits. We discover periodic Transit Duration Variations (TDVs) of KOI-142.01 that are nearly in phase with the observed TTVs. We show that KOI-142.01's TTVs and TDVs uniquely detect a non-transiting companion with a mass ≃0.7 that of Jupiter (KOI-142c). KOI-142.01's mass inferred from the transit variations is consistent with the measured transit depth, suggesting a Neptune class planet (KOI-142b). The orbital period ratio P c /P b = 2.03 indicates that the two planets are just wide of the 2:1 resonance. The present dynamics of this system, characterized here in detail, can be used to test various formation theories that have been proposed to explain the near-resonant pairs of exoplanets.