Transiting planets - lightcurve analysis for eccentric orbits

Transiting planet lightcurves have historically been used predominantly for measuring the depth and hence ratio of the planet-star radii, p. Equations have been previously presented by Seager & Mallen-Ornelas (2003) for the analysis of the total and trough transit lightcurve times to derive the ratio of semi-major axis to stellar radius, a/R*, in the case of circular orbits. Here, a new analytic model is proposed which operates for the more general case of an eccentric orbit. We aim to investigate three major effects our model predicts: i) the degeneracy in transit lightcurve solutions for eccentricity, e>0 ii) the asymmetry of the lightcurve and the resulting shift in the mid-transit time, Tmid iii) the effect of eccentricity on the ingress and egress slopes. It is shown that a system with changing eccentricity and inclination may produce a long period transit time variation (LTTV). Furthermore, we use our model in a reanalysis of HD 209458 b archived data by Richardson et al. (2006), where we include the confirmed non-zero eccentricity and derive a 24 micron planetary radius of R_P = 1.275 +- 0.082 R_J (where R_J = 1 Jovian radius), which is 1% larger than is we assume a circular orbit.Comment: 9 pages, 7 figures, 1 table Equation A37 correcte

Transit timing effects due to an exomoon II

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 degrees. Here, we will consider the effect for the more general case of i <= 90 degrees and an exomoon inclined from the planet-star plane by Euler rotation angles $\alpha$, $\beta$ and $\gamma$. 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 TTV effects, $\eta$, 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.Comment: Accepted for publication in MNRA

An Analytic Model for Rotational Modulations in the Photometry of Spotted Stars

Photometric rotational modulations due to starspots remain the most common and accessible way to study stellar activity. In the Kepler-era, there now exists precise, continuous photometry of ~150,000 stars presenting an unprecedented opportunity for statistical analyses of these modulations. Modelling rotational modulations allows one to invert the observations into several basic parameters, such as the rotation period, spot coverage, stellar inclination and differential rotation rate. The most widely used analytic model for this inversion comes from Budding (1977) and Dorren (1987), who considered circular, grey starspots for a linearly limb darkened star. In this work, we extend the model to be more suitable in the analysis of high precision photometry, such as that by Kepler. Our new freely available Fortran code, macula, provides several improvements, such as non-linear limb darkening of the star and spot, a single-domain analytic function, partial derivatives for all input parameters, temporal partial derivatives, diluted light compensation, instrumental offset normalisations, differential rotation, starspot evolution and predictions of transit depth variations due to unocculted spots. Through numerical testing, we find that the inclusion of non-linear limb darkening means macula has a maximum photometric error an order-of-magnitude less than that of Dorren (1987), for Sun-like stars observed in the Kepler-bandpass. The code executes three orders-of-magnitude faster than comparable numerical codes making it well-suited for inference problems.Comment: 30 pages, 6 figures, 2 tables, accepted in MNRAS. Error corrected in transit depth variations section. Code available at http://www.cfa.harvard.edu/~dkipping/macula.htm

Bayesian priors for the eccentricity of transiting planets

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 to demonstrate that most planet occurrence rate calculations using Kepler data have overestimated the prevalence of planets by ~10%. 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.Comment: 9 pages, 5 figures. Accepted to MNRAS. Code available at http://www.cfa.harvard.edu/~dkipping/ECCSAMPLES.htm

Relativistic Light Sails

One proposed method for spacecraft to reach nearby stars is by accelerating sails using either solar radiation pressure or directed energy. This idea constitutes the thesis behind the Breakthrough Starshot project, which aims to accelerate a gram-mass spacecraft up to one-fifth the speed of light towards Proxima Centauri. For such a case, the combination of the sail's low mass and relativistic velocity render previous treatments formally incorrect, including that of Einstein himself in his seminal 1905 paper introducing special relativity. To address this, we present formulae for a sail's acceleration, first in response to a single photon and then extended to an ensemble. We show how the sail's motion in response to an ensemble of incident photons is equivalent to that of a single photon of energy equal to that of the ensemble. We use this 'principle of ensemble equivalence' for both perfect and imperfect mirrors, enabling a simple analytic prediction of the sail's velocity curve. Using our results and adopting putative parameters for Starshot, we estimate that previous relativistic treatments underestimate the spacecraft's terminal velocity by ~50m/s for the same incident energy, sufficient to miss a target by several Earth radii. Additionally, we use a simple model to predict the sail's temperature and diffraction beam losses during the laser firing period, allowing us to estimate that for firing times of a few minutes and operating temperatures below 300C (573K), Starshot will require a sail of which absorbs less than 1 in 260,000 photons.Comment: Accepted to AJ. This version corrects the comparison in Figure 4 between our prediction and that of previous works, by accounting for finite light travel time effects. Other results are unchange

Parametrizing the exoplanet eccentricity distribution with the Beta distribution

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 and b = 3.03+/-0.17. 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 sigma and display a signature consistent with the effects of tidal circularization.Comment: 5 pages, 4 figures, 2 tables, accepted in MNRAS Letters. Code available at https://www.cfa.harvard.edu/~dkipping/betaprior.htm

In Search of Exomoons

Two decades ago, astronomers began detecting planets orbiting stars other than our Sun, so-called exoplanets. Since that time, the rate of detections and the sensitivity to ever-smaller planets has improved dramatically with several Earth-sized planets now known. As our sensitivity dives into the terrestrial regime, increasingly the community has wondered if the moons of exoplanets may also be detectable, so-called "exomoons". Their detection represents an outstanding challenge in modern astronomy and would provide deep insights into the uniqueness of our Solar System and perhaps even expand the definition of habitability. Here, I will briefly review theoretical studies exploring the formation and evolution of exomoons, which serve to guide observational searches and provide testable hypotheses. Next, I will outline the different methods which have been proposed to accomplish this challenging feat and their respective merits. Finally, initial results from observational efforts will be summarized with a view to future prospects as well.Comment: 18 pages, 5 figures; to appear in the proceedings for the Frank N. Bash Symposium 2013: New Horizons in Astronomy, held October 6-8, 2013 in Austin, T

Efficient, uninformative sampling of limb-darkening coefficients for a three-parameter law

Stellar limb-darkening impacts a wide range of astronomical measurements. The accuracy to which it is modelled limits the accuracy in any covariant parameters of interest, such as the radius of a transiting planet. With the ever growing availability of precise observations and the importance of robust estimates of astrophysical parameters, an emerging trend has been to freely fit the limb-darkening coefficients (LDCs) describing a limb-darkening law of choice, in order to propagate our ignorance of the true intensity profile. In practice, this approach has been limited to two-parameter limb-darkening laws, such as the quadratic law, due to the relative ease of sampling the physically allowed range of LDCs. Here, we provide a highly efficient method for sampling LDCs describing a more accurate three-parameter non-linear law. We first derive analytic criteria which can quickly test if a set of LDCs are physical, although naive sampling with these criteria leads to an acceptance rate less than 1%. We then show that the loci of allowed LDCs can be transformed into a cone-like volume, from which we are able to draw uniform samples. We show that samples drawn uniformly from the conal region are physically valid in 97.3% of realizations and encompass 94.4% of the volume of allowed parameter space. We provide Python and Fortran code (LDC3) to sample from this region (and perform the reverse calculation) at https://github.com/davidkipping/LDC3, which also includes a subroutine to efficiently test whether a sample is physically valid or not.Comment: 12 pages, 8 figures. Accepted to MNRAS. Code and LaTeX source at https://github.com/davidkipping/LDC

On the detectability of transiting planets orbiting white dwarfs using LSST

White dwarfs are one of the few types of stellar objects for which we know almost nothing about the possible existence of companion planets. Recent evidence for metal contaminated atmospheres, circumstellar debris disks and transiting planetary debris all indicate that planets may be likely. However, white dwarf transit surveys are challenging due to the intrinsic faintness of such objects, the short timescale of the transits and the low transit probabilities due to their compact radii. The Large Synoptic Survey Telescope (LSST) offers a remedy to these problems as a deep, half-sky survey with fast exposures encompassing approximately 10 million white dwarfs with $r<24.5$ apparent magnitude. We simulate LSST photometric observations of 3.5 million white dwarfs over a ten-year period and calculate the detectability of companion planets with $P<10$ d via transits. We find typical detection rates in the range of $5 \times 10^{-6}$ to $4 \times 10^{-4}$ for Ceres-sized bodies to Earth-sized worlds, yielding $\sim 50$ to $4\,000$ detections for a 100% occurrence rate of each. For terrestrial planets in the continuously habitable zone, we find detection rates of $\sim 10^{-3}$ indicating that LSST would reveal hundreds of such worlds for occurrence rates in the range of 1% to 10%.Comment: 9 pages, 6 figures, Submitted for publication in MNRAS on September 6, 201

Probabilistic Forecasting of the Masses and Radii of Other Worlds

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 $2.0_{-0.6}^{+0.7} M_\oplus$, 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.Comment: Accepted in ApJ. New robustness test added, results unchanged. Forecaster code is available at https://github.com/chenjj2/forecaste