Exoplanet Phase Curves Research Articles

Is binning always sinning? The impact of time-averaging for exoplanet phase curves

ABSTRACT We explore how finite integration time or temporal binning can affect the analysis of exoplanet phase curves. We provide analytical formulae to account for this effect or, if neglected, to estimate the potential biases in the retrieved parameters. As expected, due to their smoother variations over longer time-scales, phase curves can be binned more heavily than transits without causing severe biases. In the simplest case of a sinusoidal phase curve with period P, the integration time Δt reduces its amplitude by the scaling factor sinc(πΔt/P), without altering its phase or shape. We also provide formulae to predict reasonable parameter error bars from phase-curve observations. Our findings are tested with both synthetic and real data sets, including unmodelled astrophysical signals and/or instrumental systematic effects. Tests with the Spitzer data show that binning can affect the best-fitting parameters beyond predictions, due to the correction of high-frequency correlated noise. Finally, we summarize key guidelines for speeding up the analysis of exoplanet phase curves without introducing significant biases in the retrieved parameters.

Analytic Light Curves in Reflected Light: Phase Curves, Occultations, and Non-Lambertian Scattering for Spherical Planets and Moons

We derive efficient, closed-form, differentiable, and numerically stable solutions for the flux measured from a spherical planet or moon seen in reflected light, either in or out of occultation. Our expressions apply to the computation of scattered light phase curves of exoplanets, secondary eclipse light) curves in the optical, or future measurements of planet–moon and planet–planet occultations, as well as to photometry of solar system bodies. We derive our solutions for Lambertian bodies illuminated by a point source, but extend them to model illumination sources of finite angular size and rough surfaces with phase-dependent scattering. Our algorithm is implemented in Python within the open-source starry mapping framework and is designed with efficient gradient-based inference in mind. The algorithm is ∼4–5 orders of magnitude faster than direct numerical evaluation methods and ∼10 orders of magnitude more precise. We show how the techniques developed here may one day lead to the construction of two-dimensional maps of terrestrial planet surfaces, potentially enabling the detection of continents and oceans on exoplanets in the habitable zone. 6 6 https://github.com/rodluger/starrynight

Closed-form ab initio solutions of geometric albedos and reflected light phase curves of exoplanets

Studying the albedos of the planets and moons of the Solar System dates back at least a century. Of particular interest is the relationship between the albedo measured at superior conjunction, known as the ``geometric albedo", and the albedo considered over all orbital phase angles, known as the ``spherical albedo". Determining the relationship between the geometric and spherical albedos usually involves complex numerical calculations and closed-form solutions are restricted to simple reflection laws. Here we report the discovery of closed-form solutions for the geometric albedo and integral phase function, which apply to any law of reflection that only depends on the scattering angle. The shape of a reflected light phase curve, quantified by the integral phase function, and the secondary eclipse depth, quantified by the geometric albedo, may now be self-consistently inverted to retrieve globally averaged physical parameters. Fully Bayesian phase curve inversions for reflectance maps and simultaneous light curve detrending may now be performed due to the efficiency of computation. Demonstrating these innovations for the hot Jupiter Kepler-7b, we infer a geometric albedo of $0.25^{+0.01}_{-0.02}$, a phase integral of $1.77 \pm0.07$, a spherical albedo of $0.44^{+0.02}_{-0.03}$ and a scattering asymmetry factor of $0.07^{+0.12}_{-0.11}$.

A survey of exoplanet phase curves with Ariel

The ESA-Ariel mission will include a tier dedicated to exoplanet phase curves corresponding to ~10% of the science time. We present here the current observing strategy for studying exoplanet phase curves with Ariel. We define science questions, requirements and a list of potential targets. We also estimate the precision of phase curve reconstruction and atmospheric retrieval using simulated phase curves. Based on this work, we found that full-orbit phase variations for 35-40 exoplanets could be observed during the 3.5-yr mission. This statistical sample would provide key constraints on atmospheric dynamics, composition, thermal structure and clouds of warm exoplanets, complementary to the scientific yield from spectroscopic transits/eclipses measurements.

Jupiter as an Exoplanet: Insights from Cassini Phase Curves

Abstract Due to its proximity to Earth, Jupiter of the solar system serves as a unique case study for gas-giant exoplanets. In the current Letter, we perform fits of ab initio, reflective, semi-infinite, homogeneous model atmospheres to 61 phase curves from 0.40 to 1.00 μm, obtained from the Cassini spacecraft, within a Bayesian framework. We reproduce the previous finding that atmospheric models using classic reflection laws (Lambertian, Rayleigh, single Henyey–Greenstein) provide poor fits to the data. Using the double Henyey–Greenstein reflection law, we extract posterior distributions of the single-scattering albedo and scattering asymmetry factors and tabulate their median values and uncertainties. We infer that the aerosols in the Jovian atmosphere are large, irregular, polydisperse particles that produce strong forward scattering together with a narrow backscattering lobe. The near-unity values of the single-scattering albedos imply that multiple scattering of radiation is an important effect. We speculate that the observed narrow backscattering lobe is caused by coherent backscattering of radiation, which is usually associated with solar system bodies with solid surfaces and regolith. Our findings demonstrate that precise, multiwavelength phase curves encode valuable information on the fundamental properties of cloud/haze particles. The method described in this Letter enables single-scattering albedos and scattering asymmetry factors to be retrieved from James Webb Space Telescope phase curves of exoplanets.

Exoplanet phase curves from TESS: Results from the Primary Mission and future prospects

Exoplanet phase curves from TESS: Results from the Primary Mission and future prospects

2.5D retrieval of atmospheric properties from exoplanet phase curves: application to WASP-43b observations

ABSTRACT We present a novel retrieval technique that attempts to model phase curve observations of exoplanets more realistically and reliably, which we call the 2.5-dimensional (2.5D) approach. In our 2.5D approach we retrieve the vertical temperature profile and mean gaseous abundance of a planet at all longitudes and latitudes simultaneously, assuming that the temperature or composition, x, at a particular longitude and latitude (Λ, Φ) is given by $x(\Lambda ,\Phi) = \bar{x} + (x(\Lambda ,0) - \bar{x})\cos ^n\Phi$, where $\bar{x}$ is the mean of the morning and evening terminator values of x(Λ, 0), and n is an assumed coefficient. We compare our new 2.5D scheme with the more traditional 1D approach, which assumes the same temperature profile and gaseous abundances at all points on the visible disc of a planet for each individual phase observation, using a set of synthetic phase curves generated from a GCM-based simulation. We find that our 2.5D model fits these data more realistically than the 1D approach, confining the hotter regions of the planet more closely to the dayside. We then apply both models to WASP-43b phase curve observations of HST/WFC3 and Spitzer/IRAC. We find that the dayside of WASP-43b is apparently much hotter than the nightside and show that this could be explained by the presence of a thick cloud on the nightside with a cloud top at pressure <0.2 bar. We further show that while the mole fraction of water vapour is reasonably well constrained to (1–10) × 10−4, the abundance of CO is very difficult to constrain with these data since it is degenerate with temperature and prone to possible systematic radiometric differences between the HST/WFC3 and Spitzer/IRAC observations. Hence, it is difficult to reliably constrain C/O.

Kepler’s dark worlds: a low albedo for an ensemble of Neptunian and Terran exoplanets

Photometric phase curves provide an important window onto exoplanetary atmospheres and potentially even their surfaces. With similar amplitudes to occultations but far longer baselines, they have a higher sensitivity to planetary photons at the expense of a more challenging data reduction in terms of long-term stability. In this work, we introduce a novel non-parametric algorithm dubbed "phasma" to produce clean, robust exoplanet phase curves and apply it to 115 Neptunian and 50 Terran exoplanets observed by Kepler. We stack the signals to further improve signal-to-noise, and measure an average Neptunian albedo of Ag < 0.23 to 95% confidence, indicating a lack of bright clouds consistent with theoretical models. Our Terran sample provides the first constraint on the ensemble albedo of exoplanets which are most likely solid, constraining Ag < 0.42 to 95% confidence. In agreement with our constraint on the greenhouse effect, our work implies that Kepler's solid planets are unlikely to resemble cloudy Venusian analogs, but rather dark Mercurian rocks.

SPIDERMAN: an open-source code to model phase curves and secondary eclipses

We present SPIDERMAN, a fast code for calculating exoplanet phase curves and secondary eclipses with arbitrary surface brightness distributions in two dimensions. Using a geometrical algorithm, the code solves exactly the area of sections of the disc of the planet that are occulted by the star. The code is written in C with a user-friendly Python interface, and is optimised to run quickly, with no loss in numerical precision. Approximately 1000 models can be generated per second in typical use, making Markov Chain Monte Carlo analyses practicable. The modular nature of the code allows easy comparison of the effect of multiple different brightness distributions for the dataset. As a test case we apply the code to archival data on the phase curve of WASP-43b using a physically motivated analytical model for the two dimensional brightness map. The model provides a good fit to the data; however, it overpredicts the temperature of the nightside. We speculate that this could be due to the presence of clouds on the nightside of the planet, or additional reflected light from the dayside. When testing a simple cloud model we find that the best fitting model has a geometric albedo of $0.32 \pm0.02$ and does not require a hot nightside. We also test for variation of the map parameters as a function of wavelength and find no statistically significant correlations.

Exoplanet phase curves at large phase angles. Diagnostics for extended hazy atmospheres

At optical wavelengths, Titan's brightness for large Sun-Titan-observer phase angles significantly exceeds its dayside brightness. The brightening that occurs near back-illumination is due to moderately large haze particles in the moon's extended atmosphere that forward-scatter the incident sunlight. Motivated by this phenomenon, here we investigate the forward scattering from currently known exoplanets, its diagnostics possibilities, the observational requirements to resolve it, and potential implications. An analytical expression is derived for the amount of starlight forward-scattered by an exponential atmosphere that takes into account the finite angular size of the star. We use this expression to tentatively estimate how prevalent this phenomenon may be. Based on numerical calculations that consider exoplanet visibility, we identify numerous planets with predicted out-of-transit forward scattering signals of up to tens of parts-per-million provided that aerosols of $>$1 micron size form over an extended vertical region near the optical radius level. We propose that the interpretation of available optical phase curves should be revised to constrain the strength of this phenomenon that might provide insight into aerosol scale heights and particle sizes. For the relatively general atmospheres considered here, forward scattering reduces the transmission-only transit depth by typically less than the equivalent to a scale height. For short-period exoplanets the finite angular size of the star severely affects the amount of radiation scattered towards the observer at mid-transit.

Stellar-coronagraph observations of the phase curves of exoplanets

Over the two decades that have passed since the discovery of the first extrasolar planet 51 Peg b, almost all of the newly discovered exoplanets have been found either by radial velocity observations or by the transit method. Here we discuss the possibilities for observation of exoplanets and their phase curves by stellar coronagraphy, which has proven effective in experiments.

EFFECT OF LONGITUDE-DEPENDENT CLOUD COVERAGE ON EXOPLANET VISIBLE WAVELENGTH REFLECTED-LIGHT PHASE CURVES

We use a planetary albedo model to investigate variations in visible wavelength phase curves of exoplanets. Thermal and cloud properties for these exoplanets are derived using one-dimensional radiative-convective and cloud simulations. The presence of clouds on these exoplanets significantly alters their planetary albedo spectra. We confirm that non-uniform cloud coverage on the dayside of tidally locked exoplanets will manifest as changes to the magnitude and shift of the phase curve. In this work, we first investigate a test case of our model using a Jupiter-like planet, at temperatures consistent to 2.0 AU insolation from a solar type star, to consider the effect of H2O clouds. We then extend our application of the model to the exoplanet Kepler-7b and consider the effect of varying cloud species, sedimentation efficiency, particle size, and cloud altitude. We show that, depending on the observational filter, the largest possible shift of the phase curve maximum will be ~2°–10° for a Jupiter-like planet, and up to ~30° (~0.08 in fractional orbital phase) for hot-Jupiter exoplanets at visible wavelengths as a function of dayside cloud distribution with a uniformly averaged thermal profile. The models presented in this work can be adapted for a variety of planetary cases at visible wavelengths to include variations in planet–star separation, gravity, metallicity, and source-observer geometry. Finally, we tailor our model for comparison with, and confirmation of, the recent optical phase-curve observations of Kepler-7b with the Kepler space telescope. The average planetary albedo can vary between 0.1 and 0.6 for the 1300 cloud scenarios that were compared to the observations. Many of these cases cannot produce a high enough albedo to match the observations. We observe that smaller particle size and increasing cloud altitude have a strong effect on increasing albedo. In particular, we show that a set of models where Kepler-7b has roughly half of its dayside covered in small-particle clouds high in the atmosphere, made of bright minerals like MgSiO3 and Mg2SiO4, provide the best fits to the observed offset and magnitude of the phase-curve, whereas Fe clouds are found to be too dark to fit the observations.

A SEMI-ANALYTICAL MODEL OF VISIBLE-WAVELENGTH PHASE CURVES OF EXOPLANETS AND APPLICATIONS TO KEPLER- 7 B AND KEPLER- 10 B

Kepler has detected numerous exoplanet transits by precise measurements of stellar light in a single visible-wavelength band. In addition to detection, the precise photometry provides phase curves of exoplanets, which can be used to study the dynamic processes on these planets. However, the interpretation of these observations can be complicated by the fact that visible-wavelength phase curves can represent both thermal emission and scattering from the planets. Here we present a semi-analytical model framework that can be applied to study Kepler and future visible-wavelength phase curve observations of exoplanets. The model efficiently computes reflection and thermal emission components for both rocky and gaseous planets, considering both homogeneous and inhomogeneous surfaces or atmospheres. We analyze the phase curves of the gaseous planet Kepler-7 b and the rocky planet Kepler-10 b using the model. In general, we find that a hot exoplanet's visible-wavelength phase curve having a significant phase offset can usually be explained by two classes of solutions: one class requires a thermal hot spot shifted to one side of the substellar point, and the other class requires reflective clouds concentrated on the same side of the substellar point. The two solutions would require very different Bond albedos to fit the same phase curve; atmospheric circulation models or eclipse observations at longer wavelengths can effectively rule out one class of solutions, and thus pinpoint the albedo of the planet, allowing decomposition of the reflection and the thermal emission components in the phase curve. Particularly for Kepler-7 b, reflective clouds located on the west side of the substellar point can best explain its phase curve. We further derive that the reflectivity of the clear part of the atmosphere should be less than 7% and that of the cloudy part should be greater than 80% (abridged)

Nightside pollution of exoplanet transit depths

Out of the known transiting extrasolar planets, the majority are gas giants orbiting their host star at close proximity. Both theoretical and observational studies support the hypothesis that such bodies emit significant amounts of flux relative to the host star, increasing towards infrared wavelengths. For the dayside of the exoplanet, this phenomenon typically permits detectable secondary eclipses at such wavelengths, which may be used to infer atmospheric composition. In this paper, we explore the effects of emission from the nightside of the exoplanet on the primary transit lightcurve, which is essentially a self-blend. Allowing for nightside emission, an exoplanet's transit depth is no longer exclusively a function of the ratio-of-radii. The nightside of an exoplanet is emitting flux and the contrast to the star's emission is of the order of ~10^(-3) for hot-Jupiters. Consequently, we show that the transit depth in the mid-infrared will be attenuated due to flux contribution from the nightside emission by ~10^(-4). We show how this effect can be compensated for in the case where exoplanet phase curves have been measured, in particular for HD 189733b. For other systems, it may be possible to make a first-order correction by using temperature estimates of the planet. Unless the effect is accounted for, transmission spectra will also be polluted by nightside emission and we estimate that a Spitzer broadband spectrum on a bright target is altered at the 1-sigma level. Using archived Spitzer measurements, we show that the effect respectively increases the 8.0um and 24.0um transit depths by 1-sigma and 0.5-sigma per transit for HD 189733b. Consequently, we estimate that this would be 5-10 sigma effect for near-future JWST observations.