In this paper we describe an algorithm and deduce the related mathematical formulae that allows the computation of observed fluxes in stellar and planetary systems with arbitrary number of bodies being part of a transit or occultation event. The presented method does not have any limits or constraints for the geometry and can be applied for almost all of the available limb darkening models as well. As a demonstration, we apply this scheme to gather information for the orbital inclinations from multiple transiting planetary systems in cases when mutual transits occur. We also show here that these mutual events constrain inclinations unambiguously, yielding a complete picture for the whole system.
Early analyses of exoplanet statistics from the Kepler Mission revealed that a model population of multiple-planet systems with low mutual inclinations (${\sim1^{\circ}-2^{\circ}}$) adequately describes the multiple-transiting systems but underpredicts the number of single-transiting systems. This so-called "Kepler dichotomy" signals the existence of a sub-population of multi-planet systems possessing larger mutual inclinations. However, the details of these inclinations remain uncertain. In this work, we derive constraints on the intrinsic mutual inclination distribution by statistically expl…
The {\it Kepler} mission has detected thousands of planetary systems with 1-7 transiting planets packed within 0.7~au from their host stars. There is an apparent excess of single-transit planet systems that cannot be explained by transit geometries alone, when a single planetary mutual inclination dispersion is assumed. This suggests that the observed compact planetary systems have at least two different architectures. We present a scenario where the "Kepler dichotomy" may be explained by the action of an external giant planet or stellar companion misaligned with the inner multi-planet system.…
We present a fast and user friendly exoplanet transit light curve modelling package PyTransit, implementing optimised versions of the Gimenéz and the Mandel & Agol transit models. The package offers an object-oriented Python interface to access the two models implemented natively in Fortran with OpenMP parallelisation. A partial OpenCL version of the quadratic Mandel-Agol model is also included for GPU-accelerated computations. The aim of PyTransit is to facilitate the analysis of photometric time series of exoplanet transits consisting of hundreds of thousands of datapoints, and of multi-pass…
We develop and apply methods to extract planet masses and eccentricities from observed transit time variations (TTVs). First, we derive simple analytic expressions for the TTV that include the effects of both first- and second-order resonances. Second, we use N-body Markov chain Monte Carlo (MCMC) simulations, as well as the analytic formulae, to measure the masses and eccentricities of ten planets discovered by Kepler that have not previously been analyzed. Most of the ten planets have low densities. Using the analytic expressions to partially circumvent degeneracies, we measure small eccentr…
Here we present juliet, a versatile tool for the analysis of transits, radial-velocities, or both. juliet is built over many available tools for the modelling of transits, radial-velocities and stochastic processes (here modelled as Gaussian Processes; GPs) in order to deliver a tool/wrapper which can be used for the analysis of transit photometry and radial-velocity measurements from multiple instruments at the same time, using nested sampling algorithms which allows it to not only perform a thorough sampling of the parameter space, but also to perform model comparison via bayesian evidences.…
WASP-148 is a recently announced extra-solar system harbouring at least two giant planets. The inner planet transits its host star. The planets travel on eccentric orbits and are near the 4:1 mean-motion resonance, which implies significant mutual gravitational interactions. In particular, this causes transit-timing variations of a few minutes, which were detected based on ground-based photometry. This made WASP-148 one of the few cases where such a phenomenon was detected without space-based photometry. Here, we present a self-consistent model of WASP-148 that takes into account the gravitati…
This paper considers whether the population of known transiting exoplanets provides evidence for additional outer planets on inclined orbits, due to the perturbing effect of such planets on the orbits of inner planets. As such, we develop a semi-analytical method for calculating the probability that two mutually inclined planets are observed to transit. We subsequently derive a simplified analytical form to describe how the mutual inclination between two planets evolves due to secular interactions with a wide orbit inclined planet and use this to determine the mean probability that the two inn…
Eleven transit light curves for the exoplanet WASP-140b were studied with the primary objective to investigate the possibility of transit timing variations (TTVs). Previously unstudied MicroObservatory and Las Cumbres Global Telescope Network photometry were analysed using Markov Chain Monte Carlo techniques, including new observations collected by this study of a transit in December 2021. No evidence was found for TTVs. We used two transit models coupled with Bayesian optimization to explore the physical parameters of the system. The radius for WASP-140b was estimated to be $1.38^{+0.18}_{-0.…
We propose a method to be able to decide whether the planets of CoRoT-7 are moving on mutually inclined orbits in the order of i > 10 degrees. The extrasolar system CoRoT-7 is very special with respect to the closeness of the planets to the host star, which results in a fast dynamical development. It would therefore be possible to determine the inclination of the innermost planet CoRoT-7b with respect to the observer after an observation of at least three years from space with the satellite CoRoT with sufficient precision. Different inclinations would cause different duration of the transit ti…
We consider the effects of mutual transits by extrasolar planet-companion systems (in a true binary or a planet-satellite system) on light curves. We show that induced changes in light curves depend strongly on a ratio between a planet-companion's orbital velocity around their host star and a planet-companion's spin speed around their common center of mass. In both the slow and fast spin cases (corresponding to long and short distances between them, respectively), a certain asymmetry appears in light curves. We show that, especially in the case of short distances, occultation of one faint obje…
We present a solution for the light curve of two bodies mutually transiting a star with polynomial limb darkening. The term "mutual transit" in this work refers to a transit of the star during which overlap occurs between the two transiting bodies. These could be an exoplanet with an exomoon companion, two exoplanets, an eclipsing binary and a planet, or two stars eclipsing a third in a triple star system. We include analytic derivatives of the light curve with respect to the positions and radii of both bodies. We provide code that implements a photodynamical model for a mutual transit. We inc…
The transit timing variation technique (TTV) has been widely used to detect and characterize multiple planetary systems. Due to the observational biases imposed mainly by the photometric conditions and instrumentation and the high signal-to-noise required to produce primary transit observations, ground-based data acquired using small telescopes limit the technique to the follow-up of hot Jupiters. However, space-based missions such as Kepler and CoRoT have already revealed that hot Jupiters are mainly found in single systems. Thus, it is natural to question ourselves if we are properly using t…
As our ability to undertake more powerful Searches for Extraterrestrial Intelligence (SETI) grows, so does interest in the more controversial endeavour of Messaging Extraterrestrial Intelligence (METI). METI proponents point to the SETI Paradox - if all civilisations refrain from METI then SETI is futile. I introduce Mutual Detectability as a game-theoretic strategy aimed at increasing the success potential of targeted SETI. Mutual detectability is embodied by four laws: mutuality, symmetry, opportunity and superiority. These laws establish how SETI participants can engage each other using gam…
Previous analyses of Doppler and Kepler data have found that Sun-like stars hosting "cold Jupiters" (giant planets with $a\gtrsim 1\,\mathrm{AU}$) almost always host "inner super-Earths" (1-$4\,R_\oplus$, $a\lesssim1\,\mathrm{AU}$). Here, we attempt to determine the degree of alignment between the orbital planes of the cold Jupiters and the inner super-Earths. The key observational input is the fraction of Kepler stars with transiting super-Earths that also have transiting cold Jupiters. This fraction depends on both the probability for cold Jupiters to occur in such systems, and on the mutual…
The Kepler Object of Interest Network (KOINet) is a multi-site network of telescopes around the globe organised to follow up transiting planet candidate KOIs with large transit timing variations (TTVs). Its main goal is to complete their TTV curves, as the Kepler telescope no longer observes the original Kepler field. Combining Kepler and new ground-based transit data we improve the modelling of these systems. To this end, we have developed a photodynamical model, and we demonstrate its performance using the Kepler-9 system as an example. Our comprehensive analysis combines the numerical integ…
$π$ Men hosts a transiting super Earth ($P\approx6.27$ d, $m\approx4.82$ $M_{\oplus}$, $R\approx2.04$ $R_{\oplus}$) discovered by TESS and a cold Jupiter ($P\approx2093$ d, $m \sin I\approx10.02$ $M_{\rm{Jup}}$, $e\approx0.64$) discovered from radial velocity. We use Gaia DR2 and Hipparcos astrometry to derive the star's velocity caused by the orbiting planets and constrain the cold Jupiter's sky-projected inclination ($I_b=41-65^{\circ}$). From this we derive the mutual inclination ($ΔI$) between the two planets, and find that $49^{\circ}< ΔI < 131^{\circ}$ (1$σ$), and $28^{\circ} < ΔI < 152^…
This paper employs a simple model, considering just geometry and linear or quadratic limb darkening, to fit Kepler transit data via a Markov Chain Monte Carlo (MCMC) methodology for Kepler-1b, 5b, 8b, 12b, 77b, 428b, 491b, 699b, 706b, and 730b. Additional fits were made of the systems using the more sophisticated modeller Winfitter, which gives results in general agreement with the simpler model. Analysis of data with longer integration times showed biasing of the derived parameters, as expected from the literature,} leading to larger estimates for radii and reducing estimates of the system in…
When exoplanets pass in front of their stars, they imprint a transit signature on the stellar light curve which to date has been assumed to be symmetric in time, owing to the planet being modelled as a circular area occulting the stellar surface. However, this signature might be asymmetric due to different temperature/pressure and/or chemical compositions in the different terminator regions of the transiting planet. catwoman is a Python package that allows to model these asymmetric transit lightcurves, calculating lightcurves for any radially symmetric stellar limb darkening law, and where pla…
Near ultraviolet observations of WASP-12b have revealed an early ingress compared to the optical transit lightcurve. This has been interpreted as due to the presence of a magnetospheric bow shock which forms when the relative velocity of the planetary and stellar material is supersonic. We aim to reproduce this observed early ingress by modelling the stellar wind (or coronal plasma) in order to derive the speed and density of the material at the planetary orbital radius. From this we determine the orientation of the shock and the density of compressed plasma behind it. With this model for the …
A large proportion of transiting planetary systems appear to possess only a single planet as opposed to multiple transiting planets. This excess of singles is indicative of significant mutual inclinations existing within a large number of planetary systems, but the origin of these misalignments is unclear. Moreover, recent observational characterization reveals that mutual inclinations tend to increase with proximity to the host star. These trends are both consistent with the dynamical influence of a strong quadrupolar potential arising from the host star during its early phase of rapid rotati…
We report twelve new transit observations of the exoplanet WASP-4b from the Transit Monitoring in the South Project (TraMoS) project. These transits are combined with all previously published transit data for this planet to provide an improved radius measurement of Rp = 1.395 +- 0.022 Rjup and improved transit ephemerides. In a new homogeneous analysis in search for Transit Timing Variations (TTVs) we find no evidence of those with RMS amplitudes larger than 20 seconds over a 4-year time span. This lack of TTVs rules out the presence of additional planets in the system with masses larger than …
We have added to the Chroma+ suite of stellar atmosphere and spectrum modelling codes the ability to synthesize the exo-planet transit lightcurve for planets of arbitrary size up to 10% of the host stellar radius, and arbitrary planetary and stellar mass and orbital radius (thus determining orbital velocity) and arbitrary orbital inclination. The lightcurves are computed in situ, integrated with the radiative transfer solution for the radiation field emerging from the stellar surface, and there is no limb-darkening parameterization. The lightcurves are computed for the Johnson-Bessel photometr…
We present a new open source python package, based on PyLightcurve and PyTorch, tailored for efficient computation and automatic differentiation of exoplanetary transits. The classes and functions implemented are fully vectorised, natively GPU-compatible and differentiable with respect to the stellar and planetary parameters. This makes PyLightcurve-torch suitable for traditional forward computation of transits, but also extends the range of possible applications with inference and optimisation algorithms requiring access to the gradients of the physical model. This endeavour is aimed at foste…
A significant fraction of an exoplanet transit model evaluation time is spent calculating projected distances between the planet and its host star. This is a relatively fast operation for a circular orbit, but slower for an eccentric one. However, because the planet's position and its time derivatives are constant for any specific point in orbital phase, the projected distance can be calculated rapidly and accurately in the vicinity of the transit by expanding the planet's $x$ and $y$ positions in the sky plane into a Taylor series at mid-transit. Calculating the projected distance for an elli…
The physical characteristics and atmospheric chemical composition of newly discovered exoplanets are often inferred from their transit spectra which are obtained from complex numerical models of radiative transfer. Alternatively, simple analytical expressions provide insightful physical intuition into the relevant atmospheric processes. The deep learning revolution has opened the door for deriving such analytical results directly with a computer algorithm fitting to the data. As a proof of concept, we successfully demonstrate the use of symbolic regression on synthetic data for the transit rad…
