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 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.
Life emerged on Earth within the first quintile of its habitable window, but a technological civilization did not blossom until its last. Efforts to infer the rate of abiogenesis, based on its early ...emergence, are frustrated by the selection effect that if the evolution of intelligence is a slow process, then life’s early start may simply be a prerequisite to our existence, rather than useful evidence for optimism. In this work, we interpret the chronology of these two events in a Bayesian framework, extending upon previous work by considering that the evolutionary timescale is itself an unknown that needs to be jointly inferred, rather than fiducially set. We further adopt an objective Bayesian approach, such that our results would be agreed upon even by those using wildly different priors for the rates of abiogenesis and evolution—common points of contention for this problem. It is then shown that the earliest microfossil evidence for life indicates that the rate of abiogenesis is at least 2.8 times more likely to be a typically rapid process, rather than a slow one. This modest limiting Bayes factor rises to 8.7 if we accept the more disputed evidence of 13C-depleted zircon deposits E. A. Bell, P. Boehnke, T. M. Harrison, W. L. Mao, Proc. Natl. Acad. Sci. U.S.A. 112, 14518–14521 (2015). For intelligence evolution, it is found that a rare-intelligence scenario is slightly favored at 3:2 betting odds. Thus, if we reran Earth’s clock, one should statistically favor life to frequently reemerge, but intelligence may not be as inevitable.
It has been recently claimed that KOIs-268.01, 303.01, 1888.01, 1925.01, 2728.01, and 3320.01 are exomoon candidates, based on an analysis of their transit timing. Here, we perform an independent ...investigation, which is framed in terms of three questions: (1) Are there significant transit timing variations (TTVs)? (2) Is there a significant periodic TTV? (3) Is there evidence for a nonzero moon mass? We applied rigorous statistical methods to these questions alongside a reanalysis of the Kepler photometry and find that none of the Kepler objects of interest (KOIs) satisfy our three tests. Specifically, KOIs-268.01 and 3220.01 pass none of the tests and KOIs-303.01, 1888.01, and 1925.01 pass a single test each. Only KOI-2728.01 satisfies two, but fails the cross-validation test for predictions. Further, detailed photodynamical modeling reveals that KOI-2728.01 favors a negative-radius moon (as does KOI-268.01). We also note that we find a significant photoeccentric effect for KOI-1925.01 indicating an eccentric orbit of at least e > (0.62 0.06). For comparison, we applied the same tests to Kepler-1625b, which reveals that (1) and (3) are passed, but (2) cannot be checked with the cross-validation method used here, due to the limited number of available epochs. In conclusion, we find no compelling evidence for exomoons among the six KOIs. Despite this, we were able to derive exomoon mass upper limits versus semimajor axis, with KOI-3220.01 leading to particularly impressive constraints of MS/MP < 0.4% 2 at a similar semimajor to that of the Earth-Moon.
ABSTRACT 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
and b = 3.03
. 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 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.
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. 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. The TTV amplitude is ∝MSaS where MS is the exomoon mass and aS is the semimajor axis of the moon's orbit. Therefore, MS and aS 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 also provide two robust advantages. The TDV amplitude is ∝MSa−1/2S and therefore the ratio of TDV to TTV allows for MS and aS to be separately determined. TDV has a π/2 phase difference to the TTV signal, making it an excellent complementary technique.
Eclipsing systems, such as transiting exoplanets, allow one to measure the mean stellar density of the host star under various idealized assumptions. Asterodensity profiling (AP) compares this ...density to an independently determined value in order to check the validity of the assumptions and ultimately derive useful parameters. Several physical effects can cause said assumptions to become invalid, with the most well-known example being the so-called photoeccentric effect. In this work, we provide analytic expressions for five other effects which induce AP deviations: the photoblend, -spot, -timing, -duration and -mass effects. We find that these effects can easily reproduce large AP deviations and so we caution that extracting the eccentricity distribution is only viable with careful consideration of the prior distributions for these other effects. We also re-investigate the photoeccentric effect and derive a single-domain minimum eccentricity expression and the parameter range for which analytic formulae are valid. The latter result shows that the assumptions underlying the analytic model for the photoeccentric effect break down for close-in, highly eccentric planets, meaning that extreme care must be taken in this regime. Finally, we demonstrate that contaminated light fraction can be solved for, indicating that AP could be a potent tool for planet validation. PUBLICATION ABSTRACT
Exomoons are the natural satellites of planets orbiting stars outside our solar system, of which there are currently no confirmed examples. We present new observations of a candidate exomoon ...associated with Kepler-1625b using the Hubble Space Telescope to validate or refute the moon's presence. We find evidence in favor of the moon hypothesis, based on timing deviations and a flux decrement from the star consistent with a large transiting exomoon. Self-consistent photodynamical modeling suggests that the planet is likely several Jupiter masses, while the exomoon has a mass and radius similar to Neptune. Since our inference is dominated by a single but highly precise Hubble epoch, we advocate for future monitoring of the system to check model predictions and confirm repetition of the moon-like signal.
In our previous paper, we evaluated the transit duration variation (TDV) effect for a co–aligned planet–moon system at an orbital inclination of i= 90°. Here, we will consider the effect for the more ...general case of i≤ 90° and an exomoon inclined from the planet–star plane by Euler rotation angles α, β and γ. We find that the TDV signal has two major components, one due to the velocity variation effect described in our first paper and one new component due to transit impact parameter variation. By evaluating the dominant terms, we find the two effects are additive for prograde exomoon orbits, and deductive for retrograde orbits. This asymmetry could allow for future determination of the orbital sense of motion. We re–evaluate the ratio of TDV and transit timing variation effects, η, in the more general case of an inclined planetary orbit with a circular orbiting moon and find that it is still possible to directly determine the moon's orbital separation from just the ratio of the two amplitudes, as first proposed in our previous paper.
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