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.
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.
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.
Planets on eccentric orbits have a higher geometric probability of transiting their host star. By application of Bayes’ theorem, we reverse this logic to show that the eccentricity distribution of ...transiting planets is positively biased. Adopting the flexible Beta distribution as the underlying prior for eccentricity, we derive the marginalized transit probability as well as the a priori joint probability distribution of eccentricity and argument of periastron, given that a planet is known to transit. These results allow us to demonstrate that most planet occurrence rate calculations using Kepler
data have overestimated the prevalence of planets by ∼10 per cent. Indeed, the true occurrence of planets from transit surveys is fundamentally intractable without a prior assumption for the eccentricity distribution. Further more, we show that previously extracted eccentricity distributions using Kepler
data are positively biased. In cases where one wishes to impose an informative eccentricity prior, we provide a recursive algorithm to apply inverse transform sampling of our joint prior probability distribution. Computer code of this algorithm, ECCSAMPLES, is provided to enable the community to sample directly from the prior (availablehere).
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.
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.
In this work, we investigate the accuracy of various approximate expressions for the transit duration of a detached binary against the exact solution, found through solving a quartic equation. ...Additionally, a new concise approximation is derived, which offers more accurate results than those currently in the literature. Numerical simulations are performed to test the accuracy of the various expressions. We find that our proposed expression yields a >200 per cent improvement in accuracy relative to the most previously employed expression. We derive a new set of equations for retrieving the light-curve parameters and consider the effect of falsely using circular expressions for eccentric orbits, with particularly important consequences for transit surveys. The new expression also allows us to propose a new light-curve fitting parameter set, which minimizes the mutual correlations and thus improves computational efficiency. The equation is also readily differentiated to provide analytic expressions for the transit duration variation due to secular variations in the system parameters, for example due to apsidal precession induced by perturbing planets.
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
Observational biases for transiting planets Kipping, David M; Sandford, Emily
Monthly notices of the Royal Astronomical Society,
12/2016, Volume:
463, Issue:
2
Journal Article
Peer reviewed
Open access
Observational biases distort our view of nature, such that the patterns we see within a surveyed population of interest are often unrepresentative of the truth we seek. Transiting planets currently ...represent the most informative data set on the ensemble properties of exoplanets within 1 au of their star. However, the transit method is inherently biased due to both geometric and detection-driven effects. In this work, we derive the overall observational biases affecting the most basic transit parameters from first principles. By assuming a trapezoidal transit and using conditional probability, we infer the expected distribution of these terms both as a joint distribution and in a marginalized form. These general analytic results provide a baseline against which to compare trends predicted by mission-tailored injection/recovery simulations and offer a simple way to correct for observational bias. Our results explain why the observed population of transiting planets displays a non-uniform impact parameter distribution, with a bias towards near-equatorial geometries. We also find that the geometric bias towards observed planets transiting near periastron is attenuated by the longer durations which occur near apoastron. Finally, we predict that the observational bias with respect to ratio-of-radii is super-quadratic, scaling as (R
P
/R
⋆)5/2, driven by an enhanced geometric transit probability and modestly longer durations.
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